Liquids, Solutions, and Interfaces: From Classical Macroscopic

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Liquids, Solutions, and Interfaces

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Liquids, Solutions, and Interfaces From Classical Macroscopic Descriptions to Modern Microscopic Details

W. RONALD FAWCETT University of California, Davis

1 2004

3

Oxford New York Auckland Bangkok Buenos Aires Cape Town Chennai Dar es Salaam Delhi Hong Kong Istanbul Karachi Kolkata Kuala Lumpur Madrid Melbourne Mexico City Mumbai Nairobi Sa˜o Paulo Shanghai Taipei Tokyo Toronto

Copyright # 2004 by Oxford University Press, Inc. Published by Oxford University Press, Inc. 198 Madison Avenue, New York, New York, 10016 www.oup.com Oxford is a registered trademark of Oxford University Press All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of Oxford University Press. Library of Congress Cataloging-in-Publication Data Fawcett, W. Ronald. Liquids, solutions, and interfaces / W. Ronald Fawcett. p. cm. Includes bibliographical references and index. ISBN 0-19-509432-8 1. Solution (Chemistry) 2. Interfaces (Physical sciences). I. Title. QD541.F39 2004 541.3 0 4—dc21 2003013539

Cover illustration: A Li+cation solvated by four tetrahedrally disposed acetonitrile molecules. The Li+cation is the principal component of the non-aqueous electrolyte solution used in rechargeable lithium ion batteries found in laptop computers and cell phones. Acetonitrile is often used as one of the solvents in these batteries.

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Printed in the United States of America on acid-free paper

To my three muses, Zuzana, Natalka, and Tetiana

Time is a never ending Fugue An Interplay of Slow and Fast Of Silence . . . And of Light and Dark That Flickers with Hypnotic Rhythm

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Preface

This book developed from a series of lectures given to freshmen graduate students specializing in analytical and physical chemistry and chemical engineering at the University of California, Davis. The purpose of these lectures is to introduce the students to modern topics in solution chemistry. Solutions are involved in every practical chemistry laboratory, in chemical analysis, in biochemistry, in clinical chemistry, and in chemical synthesis. When I was a student, solution chemistry occupied a major fraction of physical chemistry textbooks. At that time it dealt mainly with classical thermodynamics, phase equilibria, and non-equilibrium phenomena, especially those related to electrochemistry. Much has happened in the intervening period with the development of important new experimental techniques. At the present time, solutions are examined experimentally at the molecular level. In X-ray and nuclear diﬀraction experiments, the structure of liquids and solutions is described in atomic detail to give the time average of the distribution and orientation of the component molecules and ions. Laser spectroscopy provides a route to the time resolution of molecular events occurring in the femtosecond time range. Non-linear spectroscopic techniques are being used to study the molecular composition and structure of interfaces. Both theory and experiment have seen tremendous advances since the 1950s. The purpose of this book is to bring the student through these developments from the classical macroscopic descriptions to the modern microscopic details. The subject matter in this monograph falls into three general areas. The ﬁrst of these involves liquids and solutions at equilibrium. These subjects are discussed in chapters 1–5, and include the thermodynamics of solutions, the structure of liquids, electrolyte solutions, polar solvents, and the spectroscopy of solvation.

viii

PREFACE

An attempt is made to familiarize the student with the fundamental background material together with important aspects of current research in each of these areas. Chapters 6 and 7 deal with non-equilibrium properties of solutions, and the kinetics of reactions in solutions. In the latter chapter, emphasis is placed on fast reactions in solution and femtochemistry. Chapters 8–10 involve important aspects of solutions at interfaces. These include liquids and solutions at interfaces, electrochemical equilibria, and the electrical double layer. This subject matter is of interest to analytical chemists, physical chemists, biochemists, and chemical and environmental engineers. The material in this book should be suitable for senior undergraduates and graduate students who have completed two semesters or three quarters of chemical thermodynamics and statistical mechanics. A solid background in mathematics and physics is also necessary to understand the subject matter. The instructor will note that detailed derivations of the thermodynamic equations have been given. In the case of quantum mechanics and statistical mechanics, some derivations are not given but instead, the background for the theory is carefully described. In addition, the student is referred to the original literature. The material is arranged in such a way that it may be presented at diﬀerent levels. Thus, if one wishes to present only the fundamentals of describing the structure of liquids from chapter 2, the material dealing with the statistical mechanical description of liquid systems can be largely avoided. The same comment applies to most of the material in this monograph. Introductions to thermodynamics, statistical thermodynamics, and chemical kinetics are included in chapters 1, 2, and 7, respectively. This material is given to help students review the subject matter from earlier courses in physical chemistry and to introduce the symbols used in this book. General references giving other sources for the material covered in a given chapter, and to compilations of experimental data are listed at the end of each chapter. As far as possible only SI units have been used in writing equations and presenting experimental data. Angstroms and calories, which still appear in the scientiﬁc literature, are avoided. Instead, nanometers and picometers are used for atomic and molecular dimensions, and joules for units of energy. Pressure is discussed in terms of pascals and bars rather than torrs and atmospheres. Equations involving the molecular dipolar properties, namely the dipole moment and polarizability, assume units of coulomb meters and farad square meters, respectively, for these quantities. However, tabulated data are given in the more familiar cgs system with debyes for the dipole moment and cubic nanometers for the polarizability. This follows the usage in most data tabulations at the present time. The connection between the SI and cgs units is explained in chapter 2. The symbols recommended by the International Union of Pure and Applied Chemistry [1] are used as much as possible. This preface would not be complete without an expression of gratitude to those who made this project possible. First, I am grateful to those who introduced me to the topics discussed in this book during my student years at the University of Toronto. There are many teachers in this group but special mention goes to Professors Frank Wetmore and Mike Dignam in the Department of Chemistry.

PREFACE

ix

Second, I would like to thank my students, especially those at UC Davis who listened to and discussed the lectures on which this book is based. Third, I thank my typists Elizabeth Bogren and Zuzana Kova´cˇova´, who patiently and faithfully prepared the manuscript including all the complicated equations. Special thanks are also due to Alex Tikanen and Dmytro Verbovy for their work with the diagrams. Finally, I would like to thank those who reviewed individual chapters including Rafael Andreu, Imre Bako, Bob de Levie, Dennis Evans, Manuel Galan, Douglas Henderson, Andrzej Lasia, Shiraz Markarian, Roger Parsons, Oleg Petrii, Dino Tinti, Nancy True, and Galina Tsirlina. I am especially indebted to Oldrˇ ich and Eva Fischer, who have gone through the text with painstaking care and helped to eliminate numerous small errors and to establish a consistent system of symbols. The helpful comments of all those involved in the review process resulted in a signiﬁcant improvement in this monograph during the long gestation period. Reference 1. Mills, I.; Cvitasˇ , T.; Homann, K.; Kallay, N.; Kuchitsu, K. Quantities, Units, and Symbols in Physical Chemistry; Blackwell Scientiﬁc Publications: Oxford, U.K., 1986.

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Contents

Fundamental Constants xvi 1. The Thermodynamics of Liquid Solutions 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14

Most Liquid Solutions Are Not Ideal, 3 Concentration Units, 4 Thermodynamic Quantities, 6 Partial Molar Quantities, 9 Ideal Solutions—Raoult’s Law, 15 Thermodynamics of Ideal Solutions, 16 Non-Ideal Solutions, 18 Thermodynamics of Non-Ideal Solutions, 21 Regular Solutions, 24 An Empirical Approach to Non-Ideal Solutions, 30 Ideally Dilute Solutions, 33 Thermodynamics of Ideally Dilute Solutions, 34 Experimental Determination of Solution Activities, 38 Concluding Remarks, 40 References, 41 Problems, 41

2. The Structure of Liquids 2.1 2.2 2.3 2.4 2.5

3

45

What Is a Liquid?, 45 The Statistical Thermodynamics of Liquids, 47 Intermolecular Forces, 52 Distribution and Correlation Functions, 61 The Experimental Study of Liquid Structure, 65 xi

xii

CONTENTS

2.6 The Direct Correlation Function and the Mean Spherical Approximation, 70 2.7 Computer Simulations of Simple Liquids, 73 2.8 Estimation of Thermodynamic Properties from the Pair Correlation Function, 75 2.9 The Properties of a Hard-Sphere Fluid, 79 2.10 The Structure of Water, 84 2.11 Distribution Functions for Liquid Solutions, 88 2.12 Concluding Remarks, 90 References, 91 Problems, 92 3. Electrolyte Solutions 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12

95

Electrolyte Solutions Are Always Non-Ideal, 95 Ionic Size in Solutions, 97 The Thermodynamics of Ion–Solvent Interactions, 100 Ion–Solvent Interactions According to the Born Model, 102 Ion–Solvent Interactions According to the Mean Spherical Approximation, 106 The Thermodynamics of Electrolyte Solutions, 111 The Experimental Determination of Activity Coeﬃcients for Electrolytes, 116 Ion–Ion Interactions According to the Debye–Hu¨ckel Model, 121 Ion–Ion Interactions According to the MSA, 130 The Thermodynamics of Ion Association, 135 Ion Association According to the MSA, 140 Concluding Remarks, 143 References, 144 Problems, 145

4. Polar Solvents

148

4.1 What Constitutes a Polar Liquid?, 148 4.2 Some Important Properties of Polar Solvents, 149 4.3 The Static Solvent Permittivity on the Basis of Continuum Models, 153 4.4 The Static Solvent Permittivity According to the MSA, 162 4.5 Dielectric Relaxation Phenomena, 169 4.6 The Permittivity of Electrolyte Solutions, 176 4.7 The Dielectric Relaxation Parameters, 180 4.8 Ion Solvation in Polar Solvents, 184

CONTENTS

xiii

4.9 Polar Solvents as Lewis Acids and Bases, 191 4.10 Concluding Remarks, 199 References, 200 Problems, 201 5. Spectroscopic Studies of Liquid Structure and Solvation

204

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10

What Spectroscopic Techniques Are Available?, 205 X-Ray and Neutron Diﬀraction Experiments, 206 Nuclear Magnetic Resonance Spectroscopy in Solutions, 213 NMR Studies of Ion Solvation in Water, 219 NMR Studies of Ion Solvation in Non-Aqueous Solvents, 223 Vibrational Spectroscopy in Solutions, 226 Infrared Spectroscopy of Polar Solvents, 232 Infrared Spectroscopy of Non-Electrolyte Solutions, 239 Infrared Spectroscopy of Electrolyte Solutions, 242 Ultraviolet–Visible Spectroscopy and Solvatochromic Eﬀects, 245 5.11 Concluding Remarks, 250 References, 251 Problems, 252 6. Non-Equilibrium Phenomena in Liquids and Solutions 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12

254

Non-Equilibrium Processes Are Usually Complex, 254 The Thermodynamics of Irreversible Processes, 255 The Viscosity of Liquids, 259 Isothermal Diﬀusion in Solutions, 264 Linear Diﬀusion from a Wall, 266 The Electrochemical Potential, 271 The Conductivity of Electrolyte Solutions, 274 Experimental Studies of Conductivity, 283 The Debye–Onsager Model for Conductivity, 288 Transport Phenomena in Non-Aqueous Solutions, 294 Proton Transport Phenomena, 298 Concluding Remarks, 300 References, 301 Problems, 301

7. Chemical Reaction Kinetics in Solution

304

7.1 What Time Scales Are Involved for Chemical Reactions in Solution?, 304 7.2 Fundamental Concepts, 305

xiv

CONTENTS

7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12

General Types of Solution Reactions, 312 Temperature Eﬀects and Transition State Theory, 323 Diﬀusion-Controlled Rapid Reactions, 329 Relaxation Techniques for Rapid Reactions, 332 Laser Spectroscopy and Femtochemistry in Solutions, 338 The Theory of Homogeneous Electron Transfer, 346 NMR Spectroscopy and Chemical Exchange Reactions, 358 Medium Eﬀects in Solution Reactions, 366 Linear Gibbs Energy Relationships, 375 Concluding Remarks, 377 References, 378 Problems, 380

8. Liquids and Solutions at Interfaces

383

8.1 The Molecular Environment at the Interface Is Diﬀerent than in the Bulk, 383 8.2 The Interfacial Tension of Liquids, 385 8.3 The Thermodynamics of Fluid Interfaces, 390 8.4 The Electrical Aspects of Interfaces, 395 8.5 The Work Function for Electrons in Metals, 398 8.6 The Liquid|Gas Interface and the Adsorption Isotherm, 401 8.7 Experimental Measurement of the Volta Potential Diﬀerence at Interfaces, 408 8.8 The Metal|Solution Interface, 422 8.9 The Liquid|Liquid Interface, 426 8.10 Surface Films on Liquids, 433 8.11 Spectroscopy at Liquid Interfaces, 437 8.12 Concluding Remarks, 442 References, 443 Problems, 444 9. Charge Transfer Equilibria at Interfaces

447

9.1 Electrochemical Equilibria Occur at a Wide Variety of Interfaces, 447 9.2 Electrochemical Cells, 448 9.3 The Thermodynamic Basis of the Nernst Equation, 456 9.4 The Absolute Electrode Potential, 461 9.5 Experimental Studies of Electrochemical Cells, 464 9.6 Electrochemical Cells for Electroanalysis, 474 9.7 The Liquid Junction Potential, 477 9.8 Membrane Potentials and the Donnan Eﬀect, 484

CONTENTS

xv

9.9 Ion-Selective Electrodes, 494 9.10 p-Functions and the Deﬁnition of pH, 502 9.11 Concluding Remarks, 504 References, 505 Problems, 505 10.

The Electrical Double Layer

508

10.1 The Electrical Double Layer Is an Example of Electrostatic Equilibrium, 508 10.2 The Thermodynamics of the Ideally Polarizable Interface, 510 10.3 The Experimental Study of the Double Layer, 516 10.4 The Structure of the Double Layer, 530 10.5 The Potential of Zero Charge and the Role of the Metal, 535 10.6 The Gouy–Chapman Model of the Diﬀuse Double Layer, 542 10.7 The Structure of the Inner Layer in the Absence of Adsorption, 552 10.8 The Speciﬁc Adsorption of Ions, 558 10.9 The Adsorption of Molecules at Electrodes, 569 10.10 Concluding Remarks, 576 References, 577 Problems, 579 Appendix A. Mathematical Background A.1 A.2 A.3 A.4

582

Laplace Transforms, 582 Fourier Transforms, 584 Complex Numbers and Functions, 585 Power Series, 586

Appendix B. The Laws of Electricity and Magnetism Appendix C. Numerical Methods of Data Analysis C.1 C.2 C.3 C.4 C.5 C.6 C.7

The Principle of Least Squares, 595 Linear Regression, 599 Multiple Linear Regression, 605 Numerical Methods, 608 Numerical Interpolation, 610 Numerical Integration, 612 Numerical Diﬀerentiation, 614

Index 617

589 595

Fundamental Constants

Fundamental Constants Velocity of light Fundamental charge Electron rest mass Proton rest mass Neutron rest mass Avogadro constant Planck constant Boltzmann’s constant Gas constant Faraday constant Permittivity of vacuum

c e0 me mp mn NL h kB R F "0

2:997925 108 m s1 1:602177 1019 C 9:109390 1031 kg 1:672623 1027 kg 1:67493 1027 kg 6:022137 1023 mol1 6:626076 1034 Js 1:380658 1023 J K1 8:314510 J mol1 K1 96485.3 C mol1 8:854188 1012 C2 N1 m2

T g

273.15 K 9.80665 m s2

atm cal A˚

101,325 Pa 4.184 J 100 pm 107 J 101.325 J 1:6022 1019 J 2:9979 109 esu 3:3356 103 esu 8:9875 1011 esu 3:3356 1030 C m 104 Tesla

Deﬁned Constants Temperature of H2O freezing Standard acceleration of free fall Conversion Factors Standard atmosphere Thermochemical calorie Angstrom 1 erg 1 litre atm 1 eV 1C 1V 1F 1 Debye 1 Gauss

From Cohen and Taylor, Rev. Mod. Phys., 59 (1987) 1121 xvi

Liquids, Solutions, and Interfaces

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1

The Thermodynamics of Liquid Solutions

Joel Hildebrand was born in Camden, New Jersey, in 1881. His interest in science developed in high school and he went on to study physics and chemistry at the University of Pennsylvania, obtaining a B.S. degree in 1903. He stayed at Pennsylvania for a doctoral degree in chemistry, which he obtained in 1906. Like several young American scientists of that period, he went to Germany to study the emerging ﬁeld of physical chemistry. Hildebrand spent a year at the University of Joel Henry Hildebrand Berlin in the laboratory of Walther Nernst and attended lectures given by Nernst and van’t Hoﬀ. He then returned to the University of Pennsylvania, where he held the position of instructor in chemistry until 1913. Then, at the invitation of G. N. Lewis, he joined the faculty of the University of California at Berkeley, where he remained for the rest of his scientiﬁc career. Hildebrand’s main scientiﬁc research was in the area of the physical chemistry of liquids and non-electrolyte solutions. He was a major contributor to the theory of regular solutions. Much of his work in this ﬁeld is summarized in his monograph with Robert Scott, The Solubility of Non-electrolytes [1]. He was famous at Berkeley for his lectures, especially to freshman classes. His scientiﬁc work was recognized by many awards during his long career, including the Priestley Medal of the American Chemical Society in 1962. Hildebrand was also an avid sportsman and was particularly fond of skiing in the Sierra Nevada mountains. One of his last papers was published in the Annual Review of Physical Chemistry in 1981 at the age of 100. He died in 1983.

1.1 Most Liquid Solutions Are Not Ideal Chemistry in the laboratory very often involves the use of liquid solutions. This is especially true in chemical analysis, where the amount of analyte is easily manipulated when it is dissolved in a solution. Solutions are often the medium for chemical reactions which form the basis of titrations. Other simple analytical procedures are based on absorption spectroscopy, which is used to determine the concentrations of an analyte in solution. 3

4

LIQUIDS, SOLUTIONS, AND INTERFACES

Most liquid solutions, also called liquid mixtures, are non-ideal. This follows from the fact that the components are in intimate contact with one another, and that the forces between the various species are usually not the same. As a result, the physical properties of the solution, for example, the vapor pressure of a given component, are usually not simply related to its concentration. This non-ideality leads to the concept of the activity of a solution component. As far as the analytical chemist is concerned, only concentration is ultimately of interest. Thus, if an analysis is based on the measurement of a physical property which in turn depends on the activity of a component, it is very important that the relationship between activity and concentration be understood for the system in question. Activity and its relationship to concentration is deﬁned within the context of chemical thermodynamics. Using the laws which govern phase equilibria and the laboratory observations relating to these processes one can develop a detailed understanding of this relationship. In this chapter the macroscopic concepts of chemical thermodynamics which are relevant to solutions are reviewed. In addition, some simple models based on molecular concepts are discussed. The examples chosen are mainly limited to non-electrolyte solutions, especially those involving polar molecules.

1.2 Concentration Units Concentration of one component in a two-component system can be expressed in several ways: as a weight/weight ratio, as a volume/volume ratio, or as a weight/ volume ratio. Physical chemists clearly prefer to express concentration as a weight/weight ratio because then one has the possibility of estimating the number of moles of both components in the solution. In this case, solution composition is independent of temperature and pressure. On the other hand, the analytical chemist prefers to use a weight/volume ratio. This is usually because one component, namely, the analyte, is present at low concentration. Then, one refers to this component as the solute, and the majority component as the solvent. However, in this case the concentration changes when temperature or pressure is changed. Consider the simple example of a solution of acetonitrile in water formed by mixing 10 g of acetonitrile with 90 g of water. The concentration of acetonitrile can be simply stated as 10% by weight. Another way of expressing the concentration is in terms of the relative number of moles of these molecules. Given that the molecular mass of acetonitrile is 41.04 g, the number of moles of acetonitrile nB used to form the solution is 10.000/41.04 ¼ 0.2437. The corresponding number of moles of water nA is 90.000/18.02 ¼ 4.9945 where 18.02 is its molecular mass. Thus, one may express the concentration as the mole fraction, xB , of acetonitrile where xB ¼

nB 0:2437 ¼ 0:0465 ¼ nA þ nB 4:9945 þ 0:2437

ð1:2:1Þ

Another commonly used concentration unit in physical chemistry is molality. It is deﬁned as the number of moles of component B per 1000 g of pure component A, which is regarded here as the solvent. In the present case, the molality is

THE THERMODYNAMICS OF LIQUID SOLUTIONS

5

0:2437 1000=90 ¼ 2:708 m. This is still a weight/weight ratio but has units of mol kg1 . The relationship between mole fraction and molality can be written xB ¼

mB ð1000=MA Þ þ mB

ð1:2:2Þ

where MA is the molecular weight of the solvent, that is, component A. In dilute solutions for which mB 1000/MA this relationship becomes xB ﬃ

m B MA 1000

(dilute solutions)

ð1:2:3Þ

For example, a 0.02 m solution of acetonitrile in water corresponds to a mole fraction xB equal to 3:6 104 . The concentration unit used for analysis is molarity, that is, the number of moles of solute per liter of solution. It should be noted that the molarity involves a weight/volume ratio, and that the volume involved is that of the total solution. In order to determine the molarity of the system being considered here, one must know the density of the solution. In general, this property cannot be determined from the densities of the individual components but must be found in an independent experiment. The density of acetonitrile–water solutions as a function of the weight fraction of acetonitrile is shown in ﬁg. 1.1. From these data one ﬁnds that the density of the solution made of 10 g acetonitrile and 90 g water is 0.979 g mL1 at 25 C. Thus, the volume of the same solution is 102.15 mL and the corresponding molarity, 0.2437/0.10215 ¼ 2.386 M. The relationship between the molarity cB and mole fraction xB is xB ¼

cB ð1000 r cB MB Þ=MA þ cB

ð1:2:4Þ

Fig. 1.1 Plots of the density of aqueous solutions of acetonitrile (AcN,*) and sodium hydroxide (^) against their weight fraction in the solution.

6

LIQUIDS, SOLUTIONS, AND INTERFACES

Table 1.1 Concentration of Acetonitrile in Aqueous Solutions Using Different Expressions for the Relative Amounts of Acetonitrile and Water at 25 C Weight Percent

Mole Fraction/xB

Molality, mB /mol kg1

Molarity, cB /mol L1

Solution Density, r=g L1

0.01 0.1 1 10

4:39 105 4:39 104 4:42 103 0.0465

2:44 103 2:44 102 0.246 2.708

2:44 103 2:44 102 0.245 2.386

0.997 0.997 0.995 0.979

where r is the density of the solution in g mL1 . When the solution is dilute, one may neglect the terms in the denominator involving cB so that the expression for xB becomes xB ﬃ

cB MA 1000 rA

(dilute solutions)

ð1:2:5Þ

where rA is the density of the pure solvent A. On comparing equations (1.2.3) and (1.2.5), one sees that for aqueous solutions where rA ﬃ 1:0, the molarity is equal to the molality when the solution is dilute. These calculations are illustrated in table 1.1 for a change in acetonitrile concentration by a factor of 1000 in the range of dilute solutions. It is clear that molality and molarity are equal for dilute aqueous solutions. However, if water is not the solvent, the density of the solution is probably suﬃciently diﬀerent from unity that these quantities are no longer equal. Notice also that the deﬁnition of molarity is temperature dependent because the volume of the system depends on temperature. Thus, the analyst should always cite the temperature at which solutions were prepared. A plot of the density of sodium hydroxide solutions of varying composition is also shown in ﬁg. 1.1. Electrolyte solutions are considerably more dense than water when they are concentrated. The important point to remember about these systems is that their compositions cannot be varied over the whole range because the solute is normally a solid at room temperature. Thus, the range of the weight fraction scale is determined by the solubility of the solid solute. In order to convert from molality to molarity, the solution density must be determined. Density data for common solutions can be found in data compilations such as the Landolt–Bo¨rnstein tables.

1.3 Thermodynamic Quantities The composition of a solution is obviously one of its important properties. In the preceding section various ways of describing the composition of a two-component system were described. Other properties include its volume, V, internal energy, U, and entropy, S. In order to specify any one of these, one must specify not only the amounts of each of the components but also the temperature, T, and pressure, P. These quantities are known as the independent variables of the system. They are

THE THERMODYNAMICS OF LIQUID SOLUTIONS

7

to some extent arbitrary, but nevertheless convenient. If the system possesses more than two components, the number of variables required to be speciﬁed to determine any quantity such as V is N þ 2, where N is the number of components. Consider ﬁrst of all the volume of the solution. The volume of any solution may be estimated from the mass of each component and its density. Volume is an extensive property, since its value depends on the total amount of solution. A quantity of more fundamental interest is the speciﬁc volume, Vs , that is, the volume per gram. It is simply the reciprocal of the density. This is an intensive quantity, since its value does not depend on the size of the solution, only on its composition, temperature, and pressure. From the point of view of chemists, an even better way to describe this property is in terms of the molar volume, that is, the volume per mole of solution. For a two-component solution, the molar volume Vm is related to the density as follows: Vm ¼

M A xA þ M B xB r

ð1:3:1Þ

Notice that the units of this quantity are L mol1 if the density is expressed in g L1 . To calculate the volume from the molar volume one must know the number of moles of each component, nA and nB . Thus, V ¼ ðnA þ nB ÞVm

ð1:3:2Þ

where the relative amounts of nA and nB are those required to give the mole fractions xA and xB . It was pointed out above that the volume is a function of the number of moles of each component, temperature, and pressure. Thus, one may write for a twocomponent system V ¼ VðnA ; nB ; T; PÞ

ð1:3:3Þ

It follows that the total derivative of the volume dV is given by @V @V @V @V dnA þ dnB þ dT þ dP dV ¼ @nA nB ;T;P @nB nA ;T;P @T nA ;nB ;P @P nA ;nB ;T ð1:3:4Þ where each partial derivative speciﬁes the change in the volume with a given independent variable, holding the other independent variables constant. It is very important that this speciﬁcation be made; otherwise one does not know exactly what change is being measured. The internal energy of the system, U, is deﬁned on the basis of the ﬁrst law of thermodynamics, which, in simple terms, states that energy cannot be created or destroyed. For a closed system, the gain in internal energy during a process involving a change in the values of its independent variables is equal to the heat gained by the system, q, plus the work done on the system, w. Mathematically, this is expressed as U ¼ q þ w

ð1:3:5Þ

If the changes in q and w are inﬁnitesimally small, this relationship may be written dU ¼ dq þ dw

ð1:3:6Þ

8

LIQUIDS, SOLUTIONS, AND INTERFACES

Furthermore, if the work is limited to pressure–volume changes, then dU ¼ dq PdV

ð1:3:7Þ

The relationship dw ¼ PdV reﬂects the fact that when mechanical work is done on the system, its volume decreases. Otherwise stated, if the volume of the system increases during a change in state, the system must do work against the surrounding pressure, which leads to a net loss in its internal energy. The second law of thermodynamics states that all spontaneous processes lead to an increase in disorder, which is quantitatively measured by means of the system’s entropy, S. For an inﬁnitesimally small process involving a ﬂow of heat into the system, dq, the entropy change, dS, is given by dS ¼

dq T

ð1:3:8Þ

Combining equations (1.3.7) and (1.3.8), one obtains an important result summarizing the ﬁrst and second laws in diﬀerential form: dU ¼ TdS PdV

ð1:3:9Þ

One must remember that this expression applies to a closed system, that is, one in which no matter enters or leaves (dni ¼ 0). If one relaxes this condition for a twocomponent system, then the general expression for the change in internal energy becomes @U @U dnA þ dn ð1:3:10Þ dU ¼ TdS PdV þ @nA nB ;S;V @nB nA ;S;V B where the last two derivatives describe the change in internal energy with the number of moles of each component. Since equation (1.3.10) gives the total diﬀerential of U, one arrives at the following deﬁnitions of temperature and pressure: @U T¼ ð1:3:11Þ @S nA ;nB ;V @U ð1:3:12Þ P¼ @V nA ;nB ;S Furthermore, equation (1.3.10) suggests that the internal energy is best described as a function of entropy, volume, and the number of moles of each component, that is U ¼ UðS; V; nA ; nB Þ

ð1:3:13Þ

As a result, chemists have introduced other thermodynamic functions so that the properties of a system may be considered with respect to more convenient independent variables, especially temperature and pressure. The three remaining thermodynamic variables and their deﬁnitions are as follows: the enthalpy, H ¼ U þ PV the Helmholtz energy,

ð1:3:14Þ

THE THERMODYNAMICS OF LIQUID SOLUTIONS

9

A ¼ U TS

ð1:3:15Þ

G ¼ U TS þ PV

ð1:3:16Þ

and the Gibbs energy, The diﬀerential form of the enthalpy for a closed system may be found by writing the total derivative: dH ¼ dU þ PdV þ VdP

ð1:3:17Þ

Combining this result with equation (1.3.9), one obtains dH ¼ TdS þ VdP

ð1:3:18Þ

For the Helmholtz energy, the corresponding result is dA ¼ SdT PdV

ð1:3:19Þ

dG ¼ SdT þ VdP

ð1:3:20Þ

and for the Gibbs energy It follows that the Gibbs energy for a closed system is conveniently described as a function of temperature and pressure. Relaxing the condition that the system be closed, the total derivative of G in an open two-component system becomes @G @G dn þ dn ð1:3:21Þ dG ¼ SdT þ VdP þ @nA nB ;T;P A @nB nA ;T;P B This is a very important equation in discussing chemical equilibria, and is the starting point for deriving other important results. One sees that the entropy can be deﬁned from the temperature dependence of the Gibbs energy, @G S¼ ð1:3:22Þ @T P;nA ;nB and the volume from the pressure dependence @G V¼þ @P T;nA ;nB

ð1:3:23Þ

The thermodynamic variables, U; S; H; A; and G introduced above are extensive quantities like the volume V. Thus, the amount of internal energy in a sulfuric acid solution depends on whether one has 250 mL beaker, a 4 L bottle, or a full railway tank car. Just as for volume, it is necessary to deﬁne intensive variables giving the internal energy per gram, Us , or the internal energy per mole, Um . Since the present discussion is concerned with chemistry, we will use only the molar quantities Um , Sm , Hm , Am , and Gm which are deﬁned from the corresponding extensive quantity by equations like equation (1.3.2).

1.4 Partial Molar Quantities Partial molar quantities are used to describe the change in properties of a multicomponent system when one component is added at constant temperature, pres-

10

LIQUIDS, SOLUTIONS, AND INTERFACES

sure, and amounts of all other components. In the present section, partial molar quantities will be considered with respect to the volume of a two-component system. For example, the partial molar volume of component A is deﬁned as @V A ¼ ð1:4:1Þ @nA nB ;T;P and that of B as

B ¼

@V @nB

ð1:4:2Þ nA ;T;P

It follows from equation (1.3.4) that, at constant temperature and pressure, a change in the volume of a two-component solution is given by dV ¼ A dnA þ B dnB

ð1:4:3Þ

This expression may be integrated under conditions that the relative amounts of nA and nB , that is, the composition of the solution, do not change: V ¼ A n A þ B n B

ð1:4:4Þ

The resulting equation states that the volume of the solution may be calculated given the number of moles of each component and their partial molar volumes. In terms of the molar volume, this equation becomes Vm ¼ A xA þ B xB

ð1:4:5Þ

It can be shown that there is a relationship between the partial molar volumes for a given solution composition. Taking the total derivative of the volume on the basis of equation (1.4.4), one obtains at constant temperature and pressure dV ¼ A dnA þ nA dA þ B dnB þ nB dB

ð1:4:6Þ

Comparing equations (1.4.3) and (1.4.6), one ﬁnds that nA dA þ nB dB ¼ 0

ð1:4:7Þ

or dividing by the total number of moles, nA þ nB , xA dA þ xB dB ¼ 0

ð1:4:8Þ

This equation, which is one example of the Gibbs–Duhem equation, shows that changes in the partial molar volume of one component may be related to changes in the same quantity for the other component. Experimentally, it means that one only has to measure one partial molar volume as a function of composition provided one has a value of the second partial molar volume at a reference point. In order to illustrate this point, equation (1.4.8) is written in a form suitable for calculating A from B : ð ð xB d ð1:4:9Þ dA ¼ xA B If the reference point for the integration is a solution consisting of pure component A (xB ¼ 0), then the integration constant required is the molar volume of pure component A, and one may write

THE THERMODYNAMICS OF LIQUID SOLUTIONS

11

Fig. 1.2 Plots of the molar volume of aqueous solutions of acetonitrile (AcN) and methanol (MeOH) against their mole fraction in solution. xðB

A VmA ¼ xB ¼0

xB d xA B

ð1:4:10Þ

Since the data tabulated in the literature for thermodynamic quantities are intensive, one needs a method of determining partial molar quantities from these data. However, the partial molar quantity involves the ﬁrst derivative of an extensive quantity such as the volume with respect to the number of moles of a particular component. Although the resulting derivative is intensive in nature, it itself implies that it comes from extensive quantities. The raw data that are used to estimate @V=@ni are normally the molar volume Vm as a function of the mole fraction of a given component xi . Thus, it is reasonable to examine the relationship between @V=@ni and @Vm =@xi . A plot of Vm against xB , the mole fraction of acetonitrile, is shown in ﬁg. 1.2 using the data given in table 1.2 for the acetonitrile–water system. Considering the relationship between V and Vm (equation (1.3.2)), @V m 1 @V V ¼ ð1:4:11Þ @nA nB nA þ nB @nA nB ðnA þ nB Þ2 Furthermore,

@xB @nA

¼ nB

nB ðnA þ nB Þ2

ð1:4:12Þ

Now dividing equation (1.4.11) by (1.4.12), one obtains the result that @V m V ¼ Aþ @xB x B nB This equation is rearranged to give

ð1:4:13Þ

12

LIQUIDS, SOLUTIONS, AND INTERFACES

Table 1.2 Molar Volume Data for the Acetonitrile–Water System at 25 C Partial Molar Volumes Mole Fraction Molar Volume Excess of Acetonitrile of the Solution Molar Volume Water Acetonitrile 1 xAcN Vm /mL mol1 V ex AcN / mL mol1 w /mL mol1 m /mL mol 0 0.2 0.4 0.6 0.8 1.0

18.07 24.56 31.48 38.60 45.85 53.01

0 –0.50 –0.56 –0.44 –0.17 0

Vm ¼ A þ xB

— 51.83 52.51 53.03 53.08 53.01

@V m @xB

18.07 17.75 17.46 16.95 16.91 —

ð1:4:14Þ

One may interpret this equation by relating it to the equation for a simple straight line. Referring to ﬁg. 1.2, it follows that a line drawn with a slope equal to @Vm =@xB at the point (Vm , xB ) will intersect the Vm axis at A , which is the y intercept. Such a line is shown on the molar volume plot in ﬁg. 1.2 at the point ^ (xB ¼ 0.5) for the methanol–water system. In a similar way, it is easily shown that Vm ¼ B þ xA

@V m @xA

ð1:4:15Þ

Since a plot of the molar volume against the mole fraction of B is easily converted to one against the mole fraction of A (xA ¼ 1 xB ), equation (1.4.15) shows that the intercept on the right-hand ordinate of ﬁg. 1.2 (xB ¼ 1, xA ¼ 0) gives the partial molar volume of component B. On the basis of the above analysis it has been shown the partial molar quantities are easily obtained from intensive quantities like the molar volume Vm when this quantity is plotted as a function of an intensive composition variable like the mole fraction. The plots in ﬁg. 1.2 show that the molar volume is almost a linear function of the mole fraction of solute. If the curves in ﬁg. 1.2 were actually perfect straight lines, the partial molar volumes would be constant independent of solution composition. Such a situation would arise if the solution were perfectly ideal. In reality, very few solutions are ideal, as will be seen from the discussion in the following section. In order to see more clearly the departure from ideality, one deﬁnes and calculates a quantity called the excess molar volume. This quantity is equal to the actual molar volume less the molar volume for the solution if it were ideal. The latter can be considered as the volume of the solution that would be found if the molecules of the two components form a solution without expansion or contraction. Thus, the ideal molar volume can be deﬁned as V id m ¼ xA VmA þ xB VmB

ð1:4:16Þ

and is calculated directly from the molar volumes of the pure components A and B. It follows that the excess molar volume is given by

THE THERMODYNAMICS OF LIQUID SOLUTIONS id V ex m ¼ Vm V m ¼ Vm xA VmA xB VmB

13

ð1:4:17Þ

The excess molar volume is also called the molar volume of mixing. Plots of V ex m against the mole fraction of solute are shown for the acetonitrile– water and methanol–water systems in ﬁg. 1.3. The V ex m function focuses attention on the non-ideality of the solution. The plots demonstrate clearly that the behavior of these solutions is not simple. Both systems have negative excess volumes. This indicates that the molecules occupy a smaller volume in solution than they do as pure liquids. Such an observation is undoubtedly due to attractive forces between the two components, which are stronger than the forces between molecules in the pure liquids. The partial molar volumes may be calculated from the data for the excess molar volume in a manner similar to that used with the molar volume data. On the basis of equation (1.4.17), one may write @V ex @V m m ¼ þ VmA VmB @xB @xB

ð1:4:18Þ

Then substituting equation (1.4.13), one obtains after rearrangement A ¼ Vm xB

@V ex m þ xB ðVmA VmB Þ @xB

ð1:4:19Þ

Using the deﬁnition of V ex m given by equation (1.4.17), this simpliﬁes to A ¼ V ex m xB

@V ex m þ VmA @xB

ð1:4:20Þ

In a similar way, one may show that

Fig. 1.3 Plots of the excess molar volume for the acetonitrile (AcN)–water and methanol (MeOH)–water systems against the mole fraction of these solutes.

14

LIQUIDS, SOLUTIONS, AND INTERFACES

B ¼ V ex m xA

@V ex m þ VmB @xA

ð1:4:21Þ

EXAMPLE

The application of equations (1.4.20) and (1.4.21) to determine partial molar volumes in the methanol–water system is now illustrated. The excess molar volume for this system is fairly symmetrical with respect to the mole fraction of methanol, and can be ﬁtted with reasonable accuracy by a cubic equation using least squares. The result is V ex m ¼ 0:0286 4:163xB þ 4:341xB2 0:1977xB3

ð1:4:22Þ

The resulting curve is shown in ﬁg. 1.3, drawn through the experimental points. On the basis of this analytical expression, one may now write an analytical expression for the ﬁrst derivative at any point on the curve. This equation is @V ex m ¼ 4:163 þ 8:682xB 0:5931xB2 @xB

ð1:4:23Þ

Now, suppose one wants the values of the partial molar volumes at xB ¼ 0.4. 1 Substituting into equation (1.4.22), one ﬁnds that V ex m ¼ 0:955 mL mol ; ex 1 similarly, from equation (1.4.23), @V m =@xB ¼ 0:785 mL mol . The molar volume of pure water, VmA , is 18.07 mL mol1 , and that of pure methanol, VmB , 40.72 mL mol1 . Calculating the partial molar volume of water from equation (1.4.20), one obtains A ¼ 0:955 þ 0:4 0:785 þ 18:07 ¼ 17:43 mL mol1

ð1:4:24Þ

Similarly, from equation (1.4.21) for the partial molar volume of methanol, B ¼ 0:955 0:6 0:785 þ 40:72 ¼ 39:29 mL mol1

ð1:4:25Þ

These calculations can be carried out for any value of xB . Partial molar quantities can be deﬁned for any of the remaining thermodynamic functions including the internal energy U, the enthalpy H, the Helmholtz energy A, and the Gibbs energy G. Those most used in chemistry are the chemical potentials, which are deﬁned from the Gibbs energy for the system. Thus, the chemical potential of component A is deﬁned as @G ð1:4:26Þ mA ¼ @nA nB ;T;P and that for component B as

@G mB ¼ @nB

ð1:4:27Þ nA ;T;P

THE THERMODYNAMICS OF LIQUID SOLUTIONS

15

These quantities are connected to the molar Gibbs energy for the solution by the equation Gm ¼ xA mA þ xB mB

ð1:4:28Þ

The application of these quantities to understanding physical and chemical equilibria in solutions is investigated in the sections which follow.

1.5 Ideal Solutions—Raoult’s Law The concept of an ideal solution is important in the development of an understanding of the properties of real solutions. In a liquid solution, molecules are in intimate contact with one another so that the question of ideality is determined by the nature of the intermolecular forces. Suppose a solution is formed by mixing two liquids, A and B. Then, the solution is ideal if the intermolecular forces between A and B molecules are no diﬀerent from those between A and A, or B and B molecules. An indication of whether or not the above condition for ideality is met is obtained from the vapor pressure of the solution. At a given temperature, the vapor pressure of a pure liquid is a measure of the ability of molecules to escape from the liquid to the gas phase. By studying the vapor pressure of a solution as a function of its composition at constant temperature one may assess the solution’s ideality or its degree of departure from ideality. For an ideal solution, the tendency of molecule A to escape is proportional to its mole fraction, that is, to its concentration expressed in terms of the fraction of molecules which are of type A. The proportionality constant must be the vapor pressure of pure component A because this vapor pressure is reached when the mole fraction is unity. This result is Raoult’s law, which is expressed mathematically as PA ¼ PA xA

ð1:5:1Þ

Similarly, for the other component B, PB ¼ PB xB

ð1:5:2Þ

A system which is close to ideal in its behavior is a solution of benzene and toluene. These molecules are very similar in structure so that the intermolecular forces between benzene and toluene do not diﬀer greatly from those between benzene molecules or toluene molecules in each pure liquid. The vapor pressure diagram for this system at 25 C is shown in ﬁg. 1.4. Since toluene has a higher molecular weight, its vapor pressure as a pure liquid is lower (3.25 kPa) than that of benzene (12.69 kPa). The total vapor pressure PT is obtained by adding those of components A and B, so that PT ¼ PA þ PB ¼ PA þ ðPB PA ÞxB

ð1:5:3Þ

The total vapor pressure also a linear function of the mole fractions xA and xB . It is clear from the above that Raoult’s law deﬁnes ideality by relating the properties of the liquid solution to the vapor with which it is in equilibrium. Since one knows how to deal with the thermodynamic properties of ideal gaseous

16

LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 1.4 Vapor pressure of benzene, PBZ , and toluene, PTL , and total vapor pressure, PT , plotted against the mole fraction of toluene xTL for benzene–toluene solutions at 25 C.

solutions, one now has a route to develop the thermodynamics of ideal liquid solutions.

1.6 Thermodynamics of Ideal Solutions The thermodynamic equations for ideal solutions are derived by considering the equilibrium between a given component in the vapor phase and liquid solution. Thus, for component A in a solution containing two components, Aliquid Ð Avapor

ð1:6:1Þ

The thermodynamic condition for equilibrium is that the chemical potential of A in the liquid phase be equal to that of A in the vapor, that is, msA ¼ mvA

ð1:6:2Þ

where msA is the chemical potential of A in the liquid solution, and mvA , that for A in the vapor phase. If one now assumes that the vapor phase behaves ideally, one may write mvA ¼ mv; A þ RT ln PA

ð1:63Þ

where mv; A is the standard chemical potential of A in the vapor phase measured when the partial pressure of A is 1 bar, and PA , the actual partial pressure of A. Since Raoult’s law applies when the liquid solution is ideal, one may also write v; msA ¼ mvA ¼ mA þ RT ln xA PA

ð1:6:4Þ

THE THERMODYNAMICS OF LIQUID SOLUTIONS

17

where xA is the mole fraction of A and PA the partial pressure of the pure liquid v; and PA are constants for any given temperature and pressure, A. Since mA equation (1.6.4) may be rewritten as msA ¼ ms; A þ RT ln xA

ð1:6:5Þ

s; ¼ mv; mA A þ RT ln PA

ð1:6:6Þ

where s; mA

Thus, is the standard chemical potential of component A in the liquid solution, which can be measured when the mole fraction of A is one (pure A). A similar analysis for the other component B leads to the equation msB ¼ ms; B þ RT ln xB

ð1:6:7Þ

v; ms; B ¼ mB þ RT ln PB

ð1:6:8Þ

where

The thermodynamic quantities associated with the mixing of pure liquids to form a solution are important in assessing solution properties. Suppose nA moles of component A are combined with nB moles of component B to form a solution. The Gibbs energy change associated with this process is given by s; nB ms; mix G ¼ nA msA þ nB msB nA mA B

ð1:6:9Þ

On the basis of equations (1.6.5) and (1.6.7), this may be rewritten as mix G ¼ nA RT ln xA þ nB RT ln xB

ð1:6:10Þ

Dividing both sides by the total number of moles, nA þ nB , one obtains mix Gm ¼ xA RT ln xA þ xB RT ln xB

ð1:6:11Þ

where the subscript ‘‘m’’ indicates that the Gibbs energy change is given on a molar (intensive) basis. It is easy to see that this result can be generalized to a multicomponent system with n components by writing mix Gm ¼

n X

xi RT ln xi

ð1:6:12Þ

i¼1

Since the mole fraction xi is less than one, its logarithm is negative. Thus, mix Gm is a negative quantity, indicating that the mixing process is spontaneous, as one would expect. The entropy change associated with mixing can be obtained by taking the temperature derivative of mix Gm . Accordingly, mix Sm ¼

n X @mix Gm ¼ xi R ln xi @T i¼1

ð1:6:13Þ

This quantity is clearly positive, since the mixing process results in an increase in entropy. On the basis of the deﬁnitions of Gibbs energy and enthalpy (equations (1.3.14) and (1.3.16), the enthalpy of mixing is given by

18

LIQUIDS, SOLUTIONS, AND INTERFACES

mix Hm ¼ mix Gm þ Tmix Sm

ð1:6:14Þ

Substituting in equations (1.6.12) and (1.6.13), one obtains mix Hm ¼ 0

ð1:6:15Þ

This result gives one of the important properties of ideal solutions, namely, that the mixing process does not involve any heat. Since the components of the solution interact with each other in exactly the same manner that they interact with themselves in the pure liquid, mixing is neither exothermic nor endothermic. By taking the pressure derivative of mix Gm , one may determine the volume change associated with mixing: mix Vm ¼

@mix Gm ¼0 @P

ð1:6:16Þ

The fact that the volume change associated with mixing the components is zero gives another important property of an ideal solution. On the other hand, a volume change does accompany the formation of most solutions. One example was analyzed above in section 1.4. This change is another reﬂection of the fact that the energy due to the intermolecular forces between the components changes with solution composition. In summary, there are three important characteristics of ideal solutions that one should remember in assessing the properties of any non-ideal system: (i) the vapor pressure of each component is proportional to its mole fraction in solution over the whole composition range (Raoult’s law); (ii) the enthalpy of mixing is zero; (iii) the volume change associated with mixing is zero. The sections which follow deal with non-ideal solutions.

1.7 Non-Ideal Solutions Most solutions are non-ideal. This is simply a result of diﬀerences in the chemical nature of the molecular components in the solution, and in the way in which they interact with each other. A convenient way of examining intermolecular forces in a pure liquid is in terms of its internal pressure, Pi , which is deﬁned as @U ð1:7:1Þ Pi ¼ @V T This quantity was investigated extensively by Hildebrand [1], who showed that the internal pressure is approximately equal to the enthalpy of vaporization divided by the molar volume. Thus, Pi ﬃ

vap H m Vm

ð1:7:2Þ

Values of the internal pressure for some commonly used solvents are given in table 1.3. It is apparent that the internal pressure varies considerably from one solvent to another, water having the highest value among those considered.

THE THERMODYNAMICS OF LIQUID SOLUTIONS

19

Table 1.3 Values of Internal Pressure for Some Common Solvents at 25 C

Solvent Acetonitrile Acetone Carbon tetrachloride Chloroform Dimethylsulfoxide Hexane Nitrobenzene Methanol Propylene carbonate Water

Molar Volume Vm =cm3 mol1

Enthalpy of Vaporization vap Hm / kJ mol1

Internal Pressure Pi /kJ dm3

52.9 74.0 97.1 80.7 71.3 131.6 102.7 40.7 85.2 18.07

33.2 30.8 32.4 32.2 52.9 31.55 52.5 37.43 42.8 43.99

628 416 334 399 742 240 511 917 502 2434

Thus, one has a clear indication that intermolecular forces are signiﬁcantly diﬀerent in these liquids. Under these circumstances, a solution formed from two of them would not be ideal and would generally exhibit positive deviations from Raoult’s law. It is also obvious that not every pair of liquids formed from those shown in the table are miscible. Thus, a very non-polar solvent such as hexane is immiscible with a very polar one like water. For those which are miscible, the diﬀerence in internal pressure gives a good indication of the extent of departure from ideality. A very few systems show a negative deviation from Raoult’s law behavior. This occurs when there is a strong attraction between the two molecules forming the solution, a well-known example being the chloroform–acetone system. An example of a system exhibiting a small positive deviation from Raoult’s law is a methanol–water solution (see ﬁg. 1.5). It should be noted that when the concentration of methanol is small (xMeOH < 0.1), the vapor pressure of water is close to the value expected on the basis of Raoult’s law. Similarly, for a dilute solution of water in methanol (xMeOH > 0.9, xw < 0.1), the vapor pressure of methanol is approximately equal to that in an ideal solution. These conditions are often observed in dilute solutions and have important consequences with respect to their thermodynamic properties, as will be seen in the following section. A much more complex behavior is demonstrated by the acetonitrile–water system [2] (ﬁg. 1.6). The vapor pressure curves show an interesting change in slope at a mole fraction of acetonitrile close to 0.8. However, when the mole fraction of acetonitrile is less than 0.06, the vapor pressure of water is close to the value predicted by Raoult’s law. As mentioned above, a few systems show negative deviations from Raoult’s law, a well-known example being the acetone–chloroform system (ﬁg. 1.7). In this case there is attractive interaction between the two components, speciﬁcally, between the electron-rich oxygen in acetone, and the hydrogen atom in chloroform. As a result, the escaping tendency of either molecule from the solution is

20

LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 1.5 Vapor pressure of methanol and water for methanol–water solutions plotted against the mole fraction of methanol at 40 C. The left-hand ordinate scale gives the vapor pressure of water and the right-hand scale that of methanol.

less than it would be if the chemical interaction did not occur. These systems are similar to those showing positive deviations from Raoult’s law in that the vapor pressure of the predominant component approaches ideal behavior when the mole fraction of the minority component is very small, that is, with a mole fraction less than 0.1. Now we proceed to examine the thermodynamic properties of non-ideal solutions.

Fig. 1.6 Vapor pressure of acetonitrile and water for acetonitrile–water solutions at 25 C plotted against the mole fraction of acetonitrile. The left-hand ordinate scale gives the vapor pressure of water and the right-hand scale, that of acetonitrile.

THE THERMODYNAMICS OF LIQUID SOLUTIONS

21

Fig. 1.7 Vapor pressure of acetone and chloroform for acetone–chloroform solutions at 35 C plotted against the mole fraction of acetone.

1.8 Thermodynamics of Non-Ideal Solutions The starting point for developing the thermodynamics of non-ideal solutions is the same as that for ideal solutions. Thus, one considers the equilibrium between each component in the liquid solution and its vapor. It follows from section 1.6 that one may write in general for component A, msA ¼ mvA ¼ mv; A þ RT ln PA

ð1:8:1Þ

This equation states that chemical potentials of component A in the liquid solution and vapor are equal and that each relates to the vapor pressure of A. However, one would like to have a way of relating the chemical potential of A to its mole fraction in solution. This is achieved by relating the vapor pressure of A to its mole fraction in the liquid solution using a correction factor to make the value of PA predicted by Raoult’s law equal to the true value. Thus, one writes PA ¼ gA xA PA

ð1:8:2Þ

where gA is the correction factor, known as the activity coeﬃcient of A. It is clear that gA is greater than unity when the system exhibits positive deviations from Raoult’s law, and less than unity when the deviations are negative. Furthermore, the value of gA depends on xA , since the extent of departure varies with solution composition (see ﬁgs. 1.5–1.7). Combining equations (1.8.1) and (1.8.2), one obtains msA ¼ ms; A þ RT ln gA xA

ð1:8:3Þ

s; is given by equation (1.6.6). It is emphasized that the standard state for where mA this deﬁnition of the activity coeﬃcient is based on the properties of pure component A. As the mole fraction of A approaches unity, the activity coeﬃcient gA also approaches unity, as can be seen from the vapor pressure plots presented earlier. It is also emphasized that the standard chemical potential has the same

22

LIQUIDS, SOLUTIONS, AND INTERFACES

value as it would have in an ideal solution because it only depends on the properties of pure component A. The product gA xA is known as the activity of component A in solution. Thus, one may also write s; þ RT ln aA msA ¼ mA

ð1:8:4Þ

a A ¼ g A xA

ð1:8:5Þ

where

In a similar way, one may write for component B, s; msB ¼ ms; B þ RT ln aB ¼ mB þ RT ln gB xB

ð1:8:6Þ

where mBs; is given by equation (1.6.8) and gB ¼ PB =ðxB PB Þ

ð1:8:7Þ

It is interesting to evaluate the thermodynamic functions of mixing for the nonideal solution. On the basis of equations (1.8.3) and (1.8.6), mix G ¼ nA RT ln gA xA þ nB RT ln gB xB

ð1:8:8Þ

mix Gm ¼ xA RT ln gA xA þ xB RT ln gB xB

ð1:8:9Þ

or on a molar basis

In the case of non-ideal solutions the mixing functions are often referred to the value they would have in an ideal solution, mix Gid m , thereby deﬁning the excess Gibbs energy of solution formation: id mix Gex m ¼ mix Gm mix Gm ¼ xA RT ln gA þ xB RT ln gB

ð1:8:10Þ

In order to calculate the other excess functions one must know the temperature and pressure derivatives of the activity coeﬃcients gA and gB . The excess entropy of mixing is given by mix Sex m ¼ xA R ln gA xA RT

@ ln gA @ ln gB xB R ln gB xB RT @T @T

ð1:8:11Þ

and the excess enthalpy by 2 mix H ex m ¼ xA RT

@ ln gA @ ln gB xB RT 2 @T @T

ð1:8:12Þ

Finally, the excess volume of mixing is obtained from the pressure derivatives of the activity coeﬃcients: mix Vmex ¼ xA RT

@ ln gA @ ln gB þ xB RT @P @P

ð1:8:13Þ

EXAMPLE

The vapor pressure of acetonitrile above an acetonitrile–water solution with a mole fraction of 0.395 in acetonitrile is 9.727 kPa at 25 C. The corresponding vapor pressure of water is 2.874 kPa. At the same temperature the vapor

THE THERMODYNAMICS OF LIQUID SOLUTIONS

23

pressures of the pure liquids are 11.983 kPa and 3.166 kPa for acetonitrile and water, respectively. Estimate the Raoult law activity coeﬃcients for each component, the molar Gibbs energy of mixing, and the excess value of this function. The enthalpy of mixing for this solution is 876.1 J mol1 . Estimate the entropy of mixing and its excess value. The Raoult law activity coeﬃcient for acetonitrile is gAcN ¼

9:727 ¼ 2:055 0:395 11:98

ð1:8:14Þ

The corresponding quantity for water is gw ¼

2:874 ¼ 1:500 0:605 3:166

ð1:8:15Þ

The molar Gibbs energy of mixing is mix Gm ¼ 0:395 2479:4 lnð2:055 0:395Þ þ 0:605 2479:4 lnð1:500 0:605Þ

ð1:8:16Þ

¼ 349:9 J mol1 The excess molar Gibbs energy of mixing is mix Gex m ¼ 0:395 2479:4 lnð2:055Þ þ 0:605 2479:4 lnð1:500Þ ¼ 1313:6 J mol1

ð1:8:17Þ

The enthalpy of mixing is also the excess enthalpy of mixing because an ideal solution has zero enthalpy of mixing. Now, the entropy of mixing can be calculated: mix S m ¼

mix H m mix Gm 876:1 þ 349:9 ¼ 4:111 J K1 mol1 ¼ 298:2 T

ð1:8:18Þ

and mix Sex m ¼

ex mix H ex 876:1 1313:6 m mix Gm ¼ 1:467 J K1 mol1 ¼ 298:2 T ð1:8:19Þ

Values of the excess Gibbs energy, enthalpy, and entropy for the acetonitrile– water system which show signiﬁcant departures from ideality are shown as a function of solution composition in ﬁg. 1.8. The excess enthalpy is positive over the whole composition range, reaching a maximum value of 1067 J mol1 in the vicinity of an acetonitrile mole fraction equal to 0.7. These data give a direct measure of the endothermic nature of the mixing process. The excess entropy displays a rather complex behavior, being negative at lower concentrations of acetonitrile and positive for values of xAcN greater than 0.7. Excess thermodynamic data such as those shown in ﬁg. 1.8 provide a convenient way of recording the properties of non-ideal solutions and are often found in tables for liquidvapour equilibria. However, they provide information about the solution as a

24

LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 1.8 Excess Gibbs energy, enthalpy, and entropy for acetonitrile–water solutions at 25 C plotted against the mole fraction of acetonitrile.

whole, not about the individual components. If one wants the activity coeﬃcients for each component, one must also have vapor pressure data for each component as a function of solution composition. In assessing the above treatment of non-ideal solutions, it must be kept in mind that it is applicable to a limited number of systems. This follows from the fact that we are often dealing with solutions of solids in liquids, and also because not all liquids are miscible over the whole composition range. Under these circumstances it is not convenient to deﬁne the standard state for one component in terms of the pure substance. Thus, for the majority of solutions, the majority component is treated as the solvent and its thermodynamics discussed with respect to its pure state within the context of Raoult’s law. The other minority component, which is the solute, is discussed using a standard state based on the properties of an ideally dilute solution. These systems are considered in more detail later in this chapter.

1.9 Regular Solutions In examining the properties of non-ideal solutions, it became clear that some systems diﬀer from ideality in a manner which could be treated by quite simple statistical mechanical models. The solution non-ideality is often reﬂected in the experimental observation that the enthalpy of mixing is not zero, so that A–B interactions are diﬀerent from A–A and B–B interactions. On the basis of his study of the properties of a large number of liquid solutions, Hildebrand [1, 3] introduced the concept of a regular solution. This is a system for which the enthalpy of mixing is non-zero and the entropy of mixing has its ideal value so that Sex is zero. This is equivalent to assuming that the molecules are randomly distributed in the mixture so that the diﬀerences in the intermolecular forces which lead to the non-zero value of H ex cannot be large. Guggenheim [4, 5]

THE THERMODYNAMICS OF LIQUID SOLUTIONS

25

discussed strictly regular solutions which have the additional restriction that mix V ex m be zero. These types of systems are considered in this section. Consider a mixture of two molecules A and B which are both approximately spherical in shape. As perfect spheres they can pack together in a face-centered cubic lattice to form a liquid with a coordination number c of 12 in each pure liquid. The mixture packs in the same way provided the molecular sizes are not too diﬀerent, more speciﬁcally, provided the molecular volumes do not diﬀer by more than a factor of two [4]. If the free volume between molecules in the mixture does not diﬀer from the sum of those in the two pure liquids used to form the mixture, then the volume of mixing is eﬀectively zero, and the interaction energy experienced by a given molecule in the mixture may be calculated by summing the contributions from nearest neighbors. Mixtures with these properties are strictly regular. Let us now consider how one can estimate the enthalpy of mixing given the enthalpies associated with A–A, A–B, and B–B interactions at the molecular level. If each molecule has c nearest neighbors, then the number of interactions experienced by type A molecules is cnA /2, and the number of B molecules, cnB /2, where the factor of two appears in order to avoid counting the interactions twice. If one deﬁnes the number of A–A interactions as nAA , the number of B–B interactions as nBB , and the number of A–B interactions as nAB , it follows that cnA ¼ 2nAA þ nAB

ð1:9:1Þ

cnB ¼ 2nBB þ nAB

ð1:9:2Þ

and

By adding these equations, one obtains an expression for twice the total number of interactions in the solution. Suppose that the enthalpy associated with an A–A interaction is hAA , that with B–B, hBB , and that with an A–B interaction, hAB . Then the enthalpy of nA molecules in pure A associated with intermolecular interactions is cn ð1:9:3Þ HA ¼ A hAA 2 Similarly, for pure B, HB ¼

cnB h 2 BB

ð1:9:4Þ

The enthalpy of the solution formed from these pure liquids, associated with intermolecular interactions is Hsl ¼ nAA hAA þ nBB hBB þ nAB hAB

ð1:9:5Þ

As a result, the enthalpy of mixing becomes mix H ¼ Hsl HA HB hAA þ hBB ¼ nAB hAB 2

ð1:9:6Þ

In order to develop this model further one has to obtain an expression for nAB in terms of nA and nB . The assumption used is that the molecular composition

26

LIQUIDS, SOLUTIONS, AND INTERFACES

around a given molecule is completely random and therefore reﬂects the overall solution composition. This is precisely the assumption used by Hildebrand to deﬁne a regular solution. Thus, the number nAB can be calculated from the mole fractions deﬁning solution composition, and is given by nAB ¼ cnxA xB

ð1:9:7Þ

where n is the total number of molecules (n ¼ nA þ nB ). The enthalpy of mixing can now be expressed as mix H ¼ cnxA xB h

ð1:9:8Þ

where h ¼ hAB

hAA þ hBB 2

ð1:9:9Þ

Since the mixing process is completely random, one may use the value of Smix for an ideal solution, which for the present system is given by mix S ¼ nA R ln xA nB R ln xB

ð1:9:10Þ

Thus, the expression for the Gibbs energy of mixing becomes mix G ¼ nA RT ln xA þ nB RT ln xB þ cðnA þ nB ÞxA xB h

ð1:9:11Þ

One can now derive expressions for the chemical potentials of the individual components. Since the Gibbs energy of the solution is s; þ nB mBs; þ mix G Gsl ¼ nA mA

ð1:9:12Þ

the chemical potential of A is obtained by diﬀerentiating Gsl with respect to nA (equation (1.4.26)): 2 mA ¼ ms; A þ RT ln xA þ cxB h

ð1:9:13Þ

Similarly, one ﬁnds for B, mB ¼ mBs; þ RT ln xB þ cx2A h

ð1:9:14Þ

Thus, according to the model for a regular solution the activity coeﬃcients gA and gB are given by ln gA ¼

cx2B h RT

ð1:9:15Þ

ln gB ¼

cx2A h RT

ð1:9:16Þ

and

These quantities, in turn, may be related to the vapor pressure of each component over the solution using equations. (1.8.2) and (1.8.7): ! cx2B h PA ¼ xA PA exp ð1:9:17Þ RT and

THE THERMODYNAMICS OF LIQUID SOLUTIONS

PB ¼ xB PB exp

cx2A h RT

!

27

ð1:9:18Þ

A plot of vapor pressure data for a hypothetical system, assuming ch=RT is unity, is shown in ﬁg. 1.9. The positive deviations from ideality indicate that the mixing process is endothermic. As the parameter h is increased, the deviations increase in the positive direction. Obviously, the theory also predicts negative deviations from Raoult’s law when h is negative, that is, when the mixing process is exothermic. Under these conditions the intermolecular forces between the two species A and B are attractive, and the escaping tendency of each is less than it would be if the solution were ideal (h ¼ 0). EXAMPLE

Estimate the vapor pressure of the two components in a regular solution for which ch=ðRTÞ ¼ 1 and xA ¼ 0.4 given that the vapor pressure of pure component A is 15.0 kPa and that of pure B, 20.0 kPa. Also calculate the Raoult law activity coeﬃcients. Repeat the calculation for the case that ch=ðRT Þ ¼ 1. From equation (1.9.17) at xA ¼ 0.4, the vapor pressure of A is PA ¼ 0:4 15:0 exp ð1 0:36Þ ¼ 8:60 kPa

ð1:9:19Þ

The vapor pressure of B is PB ¼ 0:6 20:0 exp ð1 0:16Þ ¼ 14:08 kPa

ð1:9:20Þ

The activity coeﬃcients are gA ¼

PA 8:60 ¼ 1:433 ¼ xA pA 0:4 15:0

ð1:9:21Þ

Fig. 1.9 Vapor pressure for a hypothetical regular solution for which chðRTÞ ¼ 1 plotted against the mole fraction of component B. The vapor pressure of pure component B is 26.7 kPa, and that of component A, 20.0 kPa. The broken lines show Raoult law behavior.

28

LIQUIDS, SOLUTIONS, AND INTERFACES

and gB ¼

PB 14:08 ¼ 1:173 ¼ xB PB 0:6 20:0

ð1:9:22Þ

In the case that the mixing process is exothermic and ch=ðRTÞ ¼ 1, the vapor pressures are PA ¼ 0:4 15:0 expð1 0:36Þ ¼ 4:19 kPa

ð1:9:23Þ

PB ¼ 0:6 20:0 expð1 0:16Þ ¼ 10:23 kPa

ð1:9:24Þ

and

The activity coeﬃcients are gA ¼

PA 4:19 ¼ 0:698 ¼ xA PA 0:4 15:0

ð1:9:25Þ

gA ¼

PB 10:23 ¼ 0:853 ¼ xB PB 0:6 20:0

ð1:9:26Þ

and

It is clear that there must be a limit to the endothermicity associated with the mixing process, if a stable solution is to be formed. As the enthalpy of mixing increases at a given composition, eventually a value is reached where the Gibbs energy of mixing is zero. This follows from the fact that the entropy of mixing for a regular solution is constant for ﬁxed solution composition and equal to the ideal value (equation (1.9.10)). Values of mix Gm for a regular solution as a function of composition for increasing values of ch=ðRTÞ are shown in ﬁg. 1.10. When this parameter reaches a value of 3, mix Gm increases over part of the composition

Fig. 1.10 The molar Gibbs energy of mixing in units of RT plotted against the mole fraction of component B for regular solutions with increasing values of the ratio w ¼ chðRTÞ.

THE THERMODYNAMICS OF LIQUID SOLUTIONS

29

range; this result indicates that a solution does not form. One may regard the diagrams in this ﬁgure as referring to a unique system at diﬀerent temperatures. Thus, as temperature decreases, and ch=ðRTÞ increases, one eventually reaches a temperature at which the solution separates into its component liquids. The temperature at which phase separation begins to take place is called the critical temperature, and for regular solutions corresponds to ch=RT equal to 2. It is easily apparent that the value of mix Gm is approximately constant over most of the composition range for this value of ch=ðRTÞ. The critical temperature at which phase separation begins is deﬁned by the conditions [4] @2 ½mix Gm ðRTÞ ¼0 @x2B

ð1:9:27Þ

@3 ½mix Gm =ðRT Þ ¼0 @x3B

ð1:9:28Þ

and

In other words, the slope of a plot mix Gm against xB must be constant and equal to zero over the composition region where the critical phenomenon is observed. From equation (1.9.11), it follows that @2 ½mix Gm =ðRTÞ 1 1 2ch ¼0 ¼ þ xA xB RT @x2B

ð1:9:29Þ

@3 ½mix Gm =ðRTÞ 1 1 ¼ 2 þ 2 ¼0 3 @xB xA xB

ð1:9:30Þ

Thus, at the critical temperature, when xA ¼ xB , ch ¼2 RT

ð1:9:31Þ

At lower temperatures, this ratio is larger, and phase separation occurs. The important feature of the above treatment, which is also known in statistical mechanics as the Bragg–Williams approximation [6], is that the molecular composition around a given molecule reﬂects the bulk composition. This cannot be the case in general because of the diﬀerences in intermolecular forces between the solution components. Thus, if molecule A interacts more strongly with molecule B than with itself, the local composition of B around A is higher than the average value. Recognition of this fact leads to more complex descriptions of mixing phenomena such as that based on the quasi-chemical or Bethe approximation [5]. However, as soon as one accepts that the local molecular composition is not the same as the average bulk composition, it follows that the entropy of mixing is not given by the ideal value (equation (1.9.10)) and that the solution is no longer regular. The quasi-chemical model and other models for non-ideal molecular solutions have been considered in some detail in the development of theories for molecular liquid solutions but are not considered further here.

30

LIQUIDS, SOLUTIONS, AND INTERFACES

1.10 An Empirical Approach to Non-Ideal Solutions Most real solutions are neither ideal nor regular. As a result a realistic description of their thermodynamic properties must consider the fact that both the excess enthalpy of mixing, mix H ex , and excess entropy, mix S ex , are non- zero. Wilson [7] has proposed an empirical description of the excess thermodynamic properties of non-ideal systems which provides an excellent description on the basis of two adjustable parameters. His approach includes systems in which the component molecules have diﬀerent sizes, and estimates the Gibbs energy of mixing on the basis of the local volume fractions of each component. It is presented here for the case of binary mixtures but can easily be extended to systems with more components. Consider the solution composed of two molecules A and B. The number of A–A interactions, nAA , with respect to the number of A–B interactions, nAB , is given by the overall ratio of A to B in the solution weighted by factors which account for the enthalpy associated with these interactions, namely, hAA and hAB . Thus, one writes nAA nA exp½hAA =ðRTÞ ¼ nAB nB exp½hAB =ðRTÞ

ð1:10:1Þ

Similarly, estimating the ratio of the number of B–B interactions to A–B interactions, one obtains nBB nB exp½hBB =ðRTÞ ¼ nAB nA exp½hAB =ðRTÞ

ð1:10:2Þ

The volume fractions of each molecule, A and B , are then deﬁned using these ratios and the molar volumes of the two pure components, VmA and VmB : A ¼

nA V mA exp½ hAA =ðRTÞ nA V mA exp½ hAA =ðRTÞ þ nB V mB exp½ hAB =ðRTÞ

ð1:10:3Þ

B ¼

nB V mB exp½ hBB =ðRTÞ nA V mA exp½ hAB =ðRTÞ þ nB V mB exp½ hBB =ðRTÞ

ð1:10:4Þ

and

The Gibbs energy of mixing is then assumed to be mix Gm ¼ RTxA ln A þ RTxB ln B

ð1:10:5Þ

Subtracting oﬀ the Gibbs energy of mixing for the ideal solution (equation (1.6.11)), one obtains for mix Gex m : mix Gex m ¼ RTxA lnðA =xA Þ þ RTxB lnðB =xB Þ

ð1:10:6Þ

This may be rewritten as mix Gex m ¼ RTxA lnð1 rBA xB Þ RTxB lnð1 rAB xA Þ

ð1:10:7Þ

where rBA ¼ 1

V mB exp½ hBB =ðRTÞ V mA exp½ hAB =ðRTÞ

ð1:10:8Þ

THE THERMODYNAMICS OF LIQUID SOLUTIONS

31

and rAB ¼ 1

V mA exp½ hAA =ðRTÞ V mB exp½ hAB =ðRTÞ

ð1:10:9Þ

The parameters rAB and rBA are treated as adjustable and are chosen to obtain a good ﬁt with the experimental data. One may now derive expressions for the activity coeﬃcients of the two components. On the basis of equation (1.8.10) mix Gex ¼ nA RT ln gA þ nB RT ln gB

ð1:10:10Þ

so that @ mix Gex =ðRTÞ @ ln gA @ ln gB ¼ ln gA þ nA þ nB @nA @nA @nA

ð1:10:11Þ

@ mix Gex =ðRTÞ @ ln gA @ ln gB ¼ ln gB þ nA þ nB @nB @nB @nB

ð1:10:12Þ

and

From the Gibbs–Duhem relationship (see equation (1.4.8) and associated discussion), the last two terms in equations (1.10.11) and (1.10.12) are equal to zero. It follows that the derivatives of mix Gex =ðRTÞ with respect to nA and nB give directly the activity coeﬃcients of components A and B. Thus, diﬀerentiating equation (1.10.7) with respect to nA and simplifying, one obtains ln gA ¼ lnð1 rBA xB Þ

xA xB rBA x2B rAB þ 1 rBA xB 1 rAB xA

ð1:10:13Þ

In a similar way, the expression for ln gB is ln gB ¼ lnð1 rAB xA Þ

xA xB rAB x2A rBA þ 1 rAB xA 1 rBA xB

ð1:10:14Þ

Wilson [7] demonstrated the eﬀectiveness of this model using data for both binary and ternary systems, one example being the carbon tetrachloride–acetonitrile system. This system shows positive deviations from Raoult’s law, as shown from the data for mix Gex m presented in ﬁg. 1.11. Wilson [7] found that the best values of the parameters rBA and rAB are 0.6118 and 0.8287 at 25 C, respectively, where B refers to carbon tetrachloride and A to acetonitrile. The ﬁtted curve is also shown in the ﬁgure, from which it is clear that there is excellent agreement between the model and experiment. EXAMPLE

Using Wilson’s parameters for the carbon tetrachloride–acetonitrile system, estimate the Raoult law activity coeﬃcients for each component in a equimolar solution. Then estimate the molar Gibbs energy of mixing. If acetonitrile is component A, then its activity coeﬃcient is given by equation (1.10.13) so that

32

LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 1.11 Plot of the excess Gibbs energy of mixing for the carbon tetrachloride–acetonitrile system against the mole fraction of acetonitrile at 45 C. The points show the experimental results and the solid curve was calculated using equation (1.10.7) with the parameters given by Wilson [7] (see text).

ln gA ¼ lnð1 0:6118 0:5Þ

0:52 0:6118 0:52 0:8287 þ 1 0:6118 0:5 1 0:8287 0:5

¼ 0:3651 0:2204 þ 0:3538 ¼ 0:4985

ð1:10:15Þ

The value of gA is 1.646. Similarly, for carbon tetrachloride which is component B ln B ¼ lnð1 0:8287 0:5Þ

0:52 0:8287 0:52 0:6118 þ 1 0:8287 0:5 1 0:6118 0:5

¼ 0:5350 0:3538 þ 0:2204 ¼ 0:4016

ð1:10:16Þ

The value of gB is 1.494. The molar Gibbs energy of mixing is given by equation (1.8.9) so that mix Gm ¼ 0:5 8:3145 298:2 lnð1:646 0:5Þ þ 0:5 8:3145 298:2 lnð1:494 0:5Þ ¼ 603:1 J mol1

ð1:10:17Þ

Equation (1.10.7) can also be used to derive expressions for the excess entropy and enthalpy functions. Diﬀerentiating this equation with respect to temperature, the expression for mix Sex m is mix Sex m ¼ RxA lnð1 rBA xB Þ þ RxB lnð1 rAB xA Þ

RTxA xB @rBA RT xA xB @rAB 1 rBA xB @T 1 rAB xA @T

ð1:10:18Þ

THE THERMODYNAMICS OF LIQUID SOLUTIONS

33

Combining equations (1.10.7) and (1.10.18), the expression for mix H ex m is mix H ex m ¼

RT 2 xA xB @rBA RT 2 xA xB @rAB 1 rBA xB @T 1 rAB xA @T

ð1:10:19Þ

The two derivatives @rBA =@T and @rAB =@T constitute two additional parameters which are obtained by ﬁtting mix H ex m data to the model. Obviously, description of the entropy requires four parameters. The eﬀectiveness of Wilson’s model lies in the fact that only two parameters are required to describe the Gibbs energy at a given temperature. Its weakness lies in the fact that there is no clear molecular interpretation of these parameters. Wilson’s approach works for a great variety of systems but when the departures from ideality are complex, more detailed models are required. Some extensions of Wilson’s work have been discussed by Renon and Prausnitz [8] but they require introduction of more adjustable parameters.

1.11 Ideally Dilute Solutions For many solutions, it is not possible to vary the composition of the components over the whole range of mole fractions. This is obviously true of solutions made up of a solid and a liquid. For these systems it is better to choose a standard state which is based on the properties of a dilute solution. This leads to the deﬁnition of an ideally dilute solution. Such a system is easily deﬁned on a molecular basis as one in which the solute molecule only comes in contact with solvent molecules, and never with another solute molecule. In the previous discussion of regular solutions it was concluded that, when the two components are of equal size, the coordination number for the other molecules around a central one is twelve. This suggests that an ideally dilute solution must have a solute mole fraction which is less than 1/13, that is, 0.08. The above approximate guideline for an ideally dilute solution can only be made more exact by examining vapor pressure data for a speciﬁc system. The case of the methanol–water system discussed earlier is used as an illustration. For very dilute solutions, that is, when xMeOH is less than 0.04, the vapor pressure of methanol is linear in its mole fraction. This is the region where Henry’s law is obeyed. As the mole fraction increases, the actual vapor pressure falls below that predicted by Henry’s law, quite signiﬁcant deviations being found when xMeOH reaches 0.1. Henry’s law for component B in a two-component system of A and B may be expressed as PB ¼ kH xB

ð1:11:1Þ

where kH is the slope of the Henry law line. Raoult law behavior is also shown in ﬁg. 1.12. For this non-ideal system, which exhibits positive deviations from Raoult’s law, the slope of the Raoult law line, which is equal to the vapor pressure of pure methanol, is much less than that for the Henry law line. It should be remembered that in the concentration range over which Henry’s law holds for the solute, Raoult’s law is valid for the solvent (see ﬁg. 1.5). This fact gives one

34

LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 1.12 Vapor pressure of methanol for dilute solutions of methanol in water plotted against the mole fraction of methanol. The straight line shows the vapor pressure according to Henry’s law, and the broken line, that according to Raoult’s law.

another convenient guideline for judging the concentration range for ideally dilute behavior. The value of the Henry law constant and concentration range over which this law is valid depends very much on the system. This is easily seen by comparing the behavior of the methanol–water and acetonitrile–water systems (ﬁgs. 1.5 and 1.6). In the latter case ideally dilute solution behavior is observed for a lower range of mole fractions of the solute, that is, when xAcN is less than 0.03. The most important application of low concentration behavior of solutes is for solid solutes, especially electrolytes. Electrolyte solutions are examined in detail in chapter 3. Some general thermodynamic methods for describing the properties of very dilute solutions are considered in the following section.

1.12 Thermodynamics of Ideally Dilute Solutions Just as was done previously, one develops the thermodynamic description of an ideally dilute solution by considering the equilibrium between the dilute solute component in the liquid solution and in the vapor phase. If the minority component is designated B, then one may write its chemical potential as msB ¼ mvB ¼ mBv; þ RT ln PB

ð1:12:1Þ

Since Henry’s law holds when the solution is ideally dilute, this can be rewritten as msB ¼ ms; B þ RT ln xB

ð1:12:2Þ

mBs; ¼ mBv; þ RT ln kH

ð1:12:3Þ

where

THE THERMODYNAMICS OF LIQUID SOLUTIONS

35

It should be noted that ms; B gives the standard chemical potential for the ideally dilute solute in a hypothetical system in which the mole fraction of B is unity. This is obviously a ﬁctitious state which is impossible in reality but whose properties are obtained by extrapolating the Henry’s law line to xB ¼ 1 (see ﬁg. 1.12). When Henry’s law is not obeyed, an activity coeﬃcient gH B is introduced so that the product gH k x is equal to the vapor pressure P B . The activity of the dilute B H B H is deﬁned to be g x . Thus, the general expression for the concomponent aH B B B centration dependence of msB becomes H s; H msB ¼ ms; B þ RT ln aB ¼ mB þ RT ln gB xB

ð1:12:4Þ

where gH B is the Henry’s law activity coeﬃcient on the mole fraction scale. Because of the inconvenient nature of the standard state deﬁned above, the concentration units used to describe the concentration dependence of the chemical potential are usually diﬀerent. More convenient choices for concentration are molality and molarity. When the solution is dilute the relationship between mole fraction and molality is quite simple (see equation (1.2.3)). In terms of molality, the expression for the concentration dependence of the chemical potential of component B becomes msB ¼ mBs; þ RT ln mB where

v; ms; B ¼ mB þ RT ln

kH MA 1000

ð1:12:5Þ ð1:12:6Þ

and MA is the molecular weight of the solvent. Now, the standard chemical potential is that for a hypothetical system which obeys Henry’s law for a solute concentration of 1 m. For the methanol–water system, xB is equal to 0.018 at this concentration. Not only is this a dilute solution by the criteria that have been discussed here, but it is also a system which approximately obeys Henry’s law on the basis of the data shown in ﬁg. 1.12. It should be emphasized that this will not always be the case, signiﬁcant departures from Henry’s law being observed for very low concentrations for some systems, for example, acetonitrile–water mixtures. When the dilute system does not obey Henry’s law, one introduces an activity coeﬃcient as above to correct to the experimentally observed value. Thus, in general, one may write PB ¼

gH B kH MA mB 1000

ð1:12:7Þ

where gH B is the Henry’s law activity coeﬃcient for component B on the molality scale. Then, on the basis of equation (1.12.1), the chemical potential of component B may be written s; H msB ¼ mBs; þ RT ln aH B ¼ mB þ RT ln gB mB

ð1:12:8Þ

where aH B is the activity of B on the molality scale with the standard state deﬁned according to Henry’s law (equation (1.12.6)).

36

LIQUIDS, SOLUTIONS, AND INTERFACES

Alternatively, one may use molarity as the concentration unit. In this case, one must know the density of the solution to order to relate mole fraction to molarity. For very dilute solutions, using equation (1.2.5), the concentration dependence of the chemical potential is given by msB ¼ mBs; þ RT ln cB

ð1:12:9Þ

where mBs; ¼ mv; B þ RT ln

k H MA 1000rA

ð1:12:10Þ

rA being the density of the solvent. When the system does not obey Henry’s law, one introduces an activity coeﬃcient, gH B which is the multiplicative correction factor required to make the vapor pressure of B predicted by Henry’s law equal to that observed experimentally. Thus, gH B is deﬁned by the equation PB ¼

gH k H M A c 1000rA B

ð1:12:11Þ

and the expression for the concentration dependence of msB becomes H s; H msB ¼ ms; B þ RT ln aB ¼ mB þ RT ln gB cB

ð1:12:12Þ

EXAMPLE

Carbon–tetrachloride–benzene solutions can be regarded as regular with an enthalpy parameter ch equal to 324 J mol1 at 25 C. Given that the vapor pressure of pure carbon tetrachloride is 14.13 kPa, determine the Henry’s law constant for this component by examining its vapor pressure at mole fractions in the range 0.001 to 0.1. Then estimate the Henry’s law activity coeﬃcient at a mole fraction of 0.1 on the mole fraction and molality scales. From the theory for regular solutions, the Raoult law activity coeﬃcient for component B in a solution of A and B is ln gR B ¼

chx2A RT

ð1:12:13Þ

Values of gR B were estimated for values of xB in the range 0.001 to 0.005 and are recorded in the following table. Then, the partial pressure of B (carbon tetrachloride) was estimated using the relationship PB ¼ gR B PB xB

ð1:12:14Þ

Finally, the ratio PB =xB was also calculated. xB

gR B

PB / Pa

PB =100xB

mB

0.001 0.002 0.003 0.004 0.005

1.139 1.139 1.139 1.138 1.138

16.1 32.1 48.3 64.4 80.4

161 161 161 161 161

0.013 0.026 0.039 0.051 0.064

It is clear from these results that the Henry’s law constant is 16,100 Pa.

THE THERMODYNAMICS OF LIQUID SOLUTIONS

37

At xB ¼ 0.1, PB is equal to 1571 Pa. Henry’s law predicts that PA is 1610 Pa. Therefore, the Henry’s law activity coeﬃcient is 0.976. The molality is easily calculated from the mole fraction using the relationship mB ¼

1000xB MA xA

ð1:12:15Þ

where MA is the molecular mass of benzene (78.1). Values of the molality for the dilute solutions are also recorded in the table. The Henry’s law constant on this scale is 1253 Pa kg mol1 . A mole fraction of 0.1 in carbon tetrachloride corresponds to a molality of 1.42 m. Thus, the predicted vapor pressure by Henry’s law is 1780 Pa. As a result the Henry’s law activity coeﬃcient is 0.883. On the basis of the above, the deﬁnitions of the standard state (equations (1.12.3), (1.12.6), and (1.12.10)) and of the activity coeﬃcient gH B (equations (1.12.4), (1.12.7), and (1.12.11)) depend on the choice of concentration units used for the dilute solution component. It is emphasized that one must always state the concentration units in deﬁning these two quantities. This point is further illustrated in table 1.4, where activity coeﬃcients for the components in dilute aqueous solutions of methanol are tabulated. Notice ﬁrst of all that the molality of methanol has reached 6.2 m when the mole fraction is 0.1. This is due to the diﬀerence between the molecular mass of water (18 g) and that of methanol (32 g). On the basis of the Raoult law activity coeﬃcients the behavior of water is ideal over most of this concentration range, the activity coeﬃcient being 1.000 to four Table 1.4 Raoult Law and Henry Law Activity Coefﬁcients for Dilute Solutions of Methanol in Water at 40 C Activity Coeﬃcients Methanol Concentration Raoult’s Law

xMeOH

mMeOH /mol kg1

gR MeOH

gR w

0.0001 0.0002 0.0005 0.001 0.002 0.005 0.01 0.02 0.05 0.1

0.006 0.011 0.028 0.056 0.111 0.279 0.561 1.133 2.921 6.167

1.7054 1.704 1.704 1.704 1.702 1.696 1.687 1.669 1.618 1.539

1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.001 1.005

Henry’s Law* gH gH MeOH MeOH ðx ¼ s:sÞ (m ¼ s:sÞ 1.000 1.000 1.000 1.000 0.999 0.996 0.990 0.979 0.949 0.903

1.000 1.000 1.000 1.000 1.000 0.997 0.987 0.967 0.908 0.819

*The ﬁrst column gives the Henry law activity coefﬁcient for methanol on the basis of a standard state of xMeOH ¼ 1 and the second on the basis of a standard state of mMeOH ¼ 1.

38

LIQUIDS, SOLUTIONS, AND INTERFACES

signiﬁcant ﬁgures up to a mole fraction of 0.02. At the same time, the behavior of methanol with respect to Raoult’s law is non-ideal over the range considered. Since deviations from ideal behavior are positive (see ﬁg. 1.5), the Raoult law activity coeﬃcient for methanol is greater than one; in addition, it changes signiﬁcantly in this concentration range, decreasing steadily in magnitude as the mole fraction increases. On the other hand, the values of gH MeOH are unity at the lowest concentrations in the region where Henry’s law holds. As concentration increases, these activity coeﬃcients fall below unity and diﬀer according to the choice of standard state. By multiplying the concentration in particular units by the appropriate activity coeﬃcient, one obtains the Henry law activity. For instance when xMeOH ¼ 0.1, the activity of methanol on the mole fraction scale is 9:90 103 . On the molality scale, the activity is 0.554 mol kg1 of water. By multiplying the latter quantity by MA =1000, that is, 0.018 kg mol1 for water, one obtains approximately the activity on the mole fraction scale (see equation (1.2.3)). The results recorded in table 1.4 emphasize the necessity of clearly stating the standard conditions for deﬁning activity coeﬃcients in liquid solutions. As stated above, the molality scale is preferred to the mole fraction scale for most systems, especially those which are not miscible over the whole composition range. The molality scale is preferred over the molarity scale because the deﬁnition of solution concentration is independent of temperature and pressure. On the other hand, the molarity concentration scale is so popular in chemistry that one often ﬁnds activity coeﬃcients also recorded in the literature using a Henry’s law scale on the basis of molarity. This is especially true of electrolyte solutions, which are always nonideal. When one recognizes that the majority of solutions involve solid solutes, the importance of the Henry’s law deﬁnition of the standard state becomes clear. These solutes usually have a negligible vapor pressure for the conditions that the solutions are used. Thus, a question arises regarding determination of the activity of non-volatile solutes. This is dealt with in the following section.

1.13 Experimental Determination of Solution Activities In order to determine the activity of a component in solution, one must measure its vapor pressure. In the case of volatile liquids such as those discussed in most of this chapter, vapor pressure measurement is not a problem so that very accurate determination of activity is possible over the whole composition range for which a solution is formed. However, many solutes, for example, most solids, have negligible vapor pressures. Under these circumstances, one makes use of the Gibbs– Duhem relationship between the activities of the two-components in solution. Since the vapor pressure of the solvent can be measured, its activity can be determined, and then used to estimate the activity of the solute. On the basis of equation (1.4.28), for any inﬁnitesimal change in a twocomponent system, one may write dGm ¼ xA dmA þ mA dxA þ xB dmB þ mB dxB

ð1:13:1Þ

From the ﬁrst and second laws of thermodynamics for this system (equation (1.3.21)), one has

THE THERMODYNAMICS OF LIQUID SOLUTIONS

dGm ¼ Sm dT þ Vm dP þ mA dxA þ mB dxB

39

ð1:13:2Þ

Therefore, in general, one may write Sm dT Vm dP þ xA dmA þ xB dmB ¼ 0

ð1:13:3Þ

and at constant temperature and pressure xA dmA þ xB dmB ¼ 0

ð1:13:4Þ

This is the form of the Gibbs–Duhem equation needed to relate the activity of component B in solution to that of component A. Choosing the Raoult law activity for the solvent A, and the Henry law activity for the solute B, equation (1.13.4) may be rewritten as xA d ln aH dðln aR B ¼ AÞ xB

ð1:13:5Þ

In the limit of very dilute solutions, both activity coeﬃcients approach unity so that the activity of A can be replaced by xA , and the activity of B by its molality (assuming that one has chosen molality as the concentration unit). However, as xB becomes very small the ratio xA /xB becomes very large. Thus, in practice, one may not choose the inﬁnitely dilute solution as a reference point but instead a very dilute solution for which Henry’s law is valid. Then, integrating equation (1.13.5) between this very dilute concentration designated m1 , to any other concentration m2 , one obtains ð2 H ln aH B ¼ ln gB m2 ¼ ln m1 1

xA R d ln aA xB

ð1:13:6Þ

An example of application of the Gibbs–Duhem relationship to determination of the activity coeﬃcients of sucrose in aqueous solutions is shown in table 1.5. Sucrose (C12 H22 O11 ) has a very high molecular mass (342 g) compared to water so that solutions with very low solute mole fractions have relatively high molalities. By measuring the water vapor pressure as a function of sucrose concentration down to very low concentrations, one is able to determine its activity coeﬃcient and thus its activity on the Raoult law scale. It should be noted that the vapor pressure of water begins to depart from ideal behavior at a quite low sucrose mole fraction (0.005). By carrying out the integration deﬁned by equation (1.13.6) one is able to calculate the Henry’s law activity for sucrose and, thus, the activity coeﬃcients given in table 1.5. It should be noted that the values of gH quickly become non-unity and are greater than one. This is indicative of strong attractive solute–solvent interactions and negative deviations from Raoult law behavior. In the case of the methanol– water system for which positive deviations from Raoult’s law is observed (table 1.4), the Henry law activity coeﬃcients are less than one. The above procedure for determining activity coeﬃcients for solid solutes is often applied to electrolytes. This important class of solutes always behaves nonideally. The properties of electrolyte solutions are considered in detail in chapter 3.

40

LIQUIDS, SOLUTIONS, AND INTERFACES

Table 1.5 Activity Coefﬁcient Data for Sucrose and Water as Determined from Water Vapor Pressure Measurements for Their Solutions at 25 C Concentration of Sucrose

Activity Coeﬃcients

Water Sucrose Molality (Raoult’s Law) (Henry’s Law) gR gH Mole Fraction /mol kg1 w s 1 104 1 103 5 103 0:02 0:04 0:07 0:1

5:6 103 0:056 0:279 1:134 2:315 4:182 6:173

1:000 1:000 0:9999 0:998 0:990 0:968 0:939

1:000 1:001 1:042 1:206 1:52 2:15 2:91

1.14 Concluding Remarks The material in this chapter explains the relationship between the concentration of a solution component and its activity. The activity is monitored through the vapor pressure of the components, which are volatile for most of the examples considered. Thus, it is very easy to understand why the activity of a given component can also be deﬁned as its escaping tendency. It is obvious from the fact that most solutions are non-ideal that the relationship between activity and concentration is not simple. When the solution is very dilute, Henry’s law holds for the solute and Raoult’s law for the solvent. Then the activity is proportional to the concentration over a ﬁnite concentration range which must be determined for each system. A review of chemical thermodynamics, especially as it relates to the properties of liquid solutions, has also been presented. Partial molar quantities such as the chemical potential are an important feature of the treatment of this subject. It is often the case that the activity and chemical potential of one quantity is relatively easy to determine directly by experiment, whereas that of another component is not. Under these circumstances, the change in chemical potential of one component can be related to that of another through the Gibbs–Duhem equation. This relationship and its use in estimating thermodynamic properties are extremely important in solution chemistry. Two simple models for non-ideal solutions have been discussed. These are the Hildebrand–Guggenheim model for regular solutions, and Wilson’s empirical approach to non-ideal solutions. They give the ﬂavor of the subject but represent only a very small fraction of the theoretical work done in this area. The extension of the model for regular solutions using the quasi-chemical approximation was mentioned earlier. In this way, the approximation that the distribution of molecules is random in solution is relaxed. Another subject which has been examined in detail in the literature is solutions with strong association. This association is often due to hydrogen bonding and therefore is important in understanding the properties of solutions formed with water and the alcohols. Polymer solutions

THE THERMODYNAMICS OF LIQUID SOLUTIONS

41

have also been considered and their thermodynamic properties derived. More details regarding this subject and an introduction to the original literature in this ﬁeld can be found in the monographs by Prigogine [9] and Marcus [10]. General References G1. Hirata, M.; Ohe, S.; Nagahama, K. Computer Aided Data Book of Liquid-Vapor Equilibria; Elsevier: Amsterdam, 1975. G2. Levine, I. N. Physical Chemistry, 4th ed.; McGraw-Hill: New York, 1995. G3. Landolt-Bo¨rnstein. Densities of Binary Aqueous Systems and Heat Capacities of Liquid Systems; New Series, Group IV; Springer-Verlag: Berlin, 1977; Vol. 1b. References 1. Hildebrand, J. H.; Scott, R. L. The Solubility of Non-Electrolytes, 3rd ed.; Dover: New York, 1964. 2. Treiner, C.; Tzias, P.; Chemla, M.; Poltoratsky, G. M. J. Chem. Soc., Faraday Trans. I 1976, 72, 2007. 3. Hildebrand, J. H. J. Am. Chem. Soc. 1929, 51, 66. 4. Fowler, R. H.; Guggenheim, E. A. Statistical Thermodynamics; Cambridge Press, 1939; Chapter 8. 5. Guggenheim, E. A. Mixtures; Oxford University Press 1952; Chapter 4. 6. Rushbrooke, G. A. Introduction to Statistical Mechanics; Oxford University Press, 1949. 7. Wilson, G. M. J. Am. Chem. Soc., 1964, 86, 127. 8. Renon, H.; Prausnitz, J. M. AIChE J. 1968, 14, 135. 9. Prigogine, I.; Bellemans, A.; Mathot, V. The Molecular Theory of Solutions; North Holland: Amsterdam, 1957. 10. Marcus, Y. Introduction to Liquid State Chemistry; Wiley-Interscience: New York, 1977.

Problems 1. The density of LiClO4 solutions in dimethylsulfoxide at 25 C is given by r ¼ 1.0965 þ 6.204 102 c where r is the density in g mL1 and c, the concentration of LiC1O4 in M. Calculate the molality and mole fraction of the solute for solutions with concentrations of 0.1, 0.2, and 0.5 M. 2. The following density data are reported for the carbon tetrachloride–acetonitrile system at 25 C. Mole Fraction Acetonitrile

Density /g mL1

0 0.1657 0.3145 0.4406 0.5498 0.6473 0.7348 0.8113

1.5844 1.5066 1.4248 1.3441 1.2640 1.1827 1.1003 1.0194

42

LIQUIDS, SOLUTIONS, AND INTERFACES

0.8797 0.9430 1.0000

0.9390 0.8575 0.7766

Calculate the excess molar volume for the solution and partial molar volumes of the two components at mole fractions equal to 0, 0.2, 0.4, 0.6, 0.8, and 1.0. Use the appropriate numerical interpolation and diﬀerentiation techniques (see appendix C). 3 The following results are reported for the carbon tetrachloride–acetonitrile system at 45 C. Mole Fraction CCl4

Total Vapor Presure

Liquid

Vapor

/kPa

0 0.0347 0.1914 0.3752 0.4790 0.6049 0.8069 0.9609 1.000

0 0.1801 0.4603 0.5429 0.5684 0.5936 0.6470 0.8001 1.000

27.783 33.062 44.796 48.604 49.274 49.466 48.362 41.908 34.501

Use an interpolation method to obtain the values of the mole fraction of CCl4 in the vapor and the total vapor pressure for values of the mole fraction in the liquid phase equal to 0.2, 0.4, 0.6, and 0.8. Then calculate the vapor pressure and the activity coeﬃcient of each component for the same values. Finally estimate the molar Gibbs energy of mixing on the Raoult law scale at these four points. 4. The following data are reported for the molar enthalpy of mixing for the system discussed in question 3. Mole Fraction of CCl4 Liquid

mix Hm /J mol1

0 0.128 0.317 0.407 0.419 0.631 0.821 1.0

0 414 745 862 858 930 736 0

Determine the value of mix Hm at xCCl4 equal to 0.2, 0.4, 0.6, and 0.8. Combine these data with those obtained in question 3 to prepare a plot of mix Gm , mix Hm , and Tmix Sm against xCCl4 .

THE THERMODYNAMICS OF LIQUID SOLUTIONS

43

5. Calculate the excess Gibbs energy, entropy, and enthalpy of mixing for the carbon tetrachloride–acetonitrile system discussed in questions 3 and 4. Prepare a plot of these data and compare the results with those obtained in the previous question. 6. The following data are available for solutions of acetone and chloroform at 50 C. Mole Fraction of Acetone

Total Pressure

Liquid

Vapor

/kPa

0 0.10 0.20 0.30 0.38 0.40 0.50 0.60 0.70 0.80 0.90 1.00

0 0.071 1.165 0.279 0.380 0.408 0.550 0.684 0.789 0.890 0.955 1.000

69.5 66.0 63.2 61.7 61.1 61.3 62.5 65.2 68.1 72.0 76.8 81.6

Calculate the Raoult law activity coeﬃcients of both components and plot them as a function of the mole fraction of acetone. Determine the range of composition with respect to acetone that the solution can be regarded as regular. Calculate the enthalpy parameter for acetone–chloroform interactions on the basis of a one-parameter least-squares ﬁt of the data in this range using an appropriate plot. 7. J. J. Van Laar gave a useful semiempirical equation for the molar excess Gibbs energy of solutions, mix Gex m ¼

b12 x1 x2 b1 x1 þ b2 x2

where b12 , b1 , and b2 are characteristic constants. Show that the Van Laar relation implies that pﬃﬃﬃﬃﬃﬃﬃﬃ A12 x1 1 1=2 1 ¼ þ pﬃﬃﬃﬃﬃﬃﬃﬃ ln g1 B12 x2 A12 pﬃﬃﬃﬃﬃﬃﬃ 1 1=2 1 B12 x2 ¼ þ pﬃﬃﬃﬃﬃﬃﬃ ln g2 A12 x1 B12

where A12 ¼ b12 =ðb2 RTÞ and B12 ¼ b12 =ðb; RT Þ

44

LIQUIDS, SOLUTIONS, AND INTERFACES

8. The following vapor pressure data are found for methanol–water solutions at 40 C. xMeOH

PMeOH =kPa

0.04 0.08 0.12 0.2 0.4 0.6 0.8 1.0

2.1 4.3 6.4 9.9 17.2 23.3 29.3 36.0

Calculate the Raoult law and Henry law activity coeﬃcients for methanol on the mole fraction scale.

2

The Structure of Liquids

Douglas (Doug) Henderson was born in Calgary, Alberta, Canada in 1934. He grew up in Vancouver, British Columbia and attended the University of British Columbia where he obtained a bachelor’s degree in mathematics in 1956. Henderson then went to the University of Utah in Salt Lake City to work with Henry Eyring on the theory of liquids. He obtained his Ph.D. in physics under Eyring’s direction in 1961. After leaving graduate school he held faculty positions at universities in both the Douglas James Henderson United States and Canada. During 1966–67, Henderson was an invited scientist at the Commonwealth Scientiﬁc and Industrial Research Organization (CSIRO) in Melbourne, Australia. There he began a long and very successful collaboration with John Barker. One of the major results of their work was the perturbation theory of liquids, some of which was outlined in Reviews of Modern Physics in 1976 in their well-known paper ‘‘What is liquid? Understanding the states of matter’’ [1]. Barker and Henderson moved together to the IBM Almaden Research Center in San Jose, California as research scientists in 1969. Henderson’s research interests in the statistical mechanics of condensed phases involve many aspects, an important one in recent years being the electrical double layer. He was instrumental in bringing our understanding of interfacial phenomena at polarizable interfaces beyond the primitive level. He is currently professor of chemistry at Brigham Young University in Provo, Utah. His career is noteworthy for the international collaborations which he has established in many countries where research in statistical mechanics is carried out. He has published over 400 papers in this area. In addition, he has won numerous awards for his scientiﬁc work, the most recent being the Hildebrand Award of the American Chemical Society in 1999 for his work on the structure of liquids.

2.1 What Is a Liquid? It is well known from studies of the properties of matter that the liquid state is much more complex than either the gaseous or solid states. Studies of the properties of gases quickly lead to the ideal gas law, which describes the properties of real gases at low pressures and high temperatures. This success is clearly due to 45

46

LIQUIDS, SOLUTIONS, AND INTERFACES

the fact that the molecules in a dilute gas are far from one another so that the eﬀects of intermolecular forces and of the ﬁnite volume occupied by the gas molecules are negligible. As the pressure of a gas is increased and its temperature lowered, the eﬀects of non-ideality become apparent, and the equation of state becomes more complex. These changes are those required to convert the gas to a liquid. As the molecules come closer together, the inﬂuence of intermolecular forces becomes greater and the free volume available for the gas molecules is signiﬁcantly reduced because of the space occupied by the molecules themselves. The statistical mechanical description of a gas relies upon the concept that the molecules are in constant movement with trajectories determined by collisions with the walls of the container and with other molecules. The probability of ﬁnding another molecule in the immediate vicinity of a given molecule is extremely low and does not vary signiﬁcantly with distance from the reference molecule. On the other hand, solids are characterized by a very ordered structure in which each ion or molecule is surrounded by a ﬁxed number of neighbors whose nature and orientation are determined by the interparticle forces in the crystal. These may be chieﬂy ion–ion interactions, as in an ionic crystal, or intermolecular forces, as in a molecular crystal. Because of the high state of order in crystals it is a reasonably straightforward problem to calculate their thermodynamic properties on the basis of quite simple statistical mechanical models. One way of conceptualizing a liquid is as a very disordered solid. If one disrupts the structure of the nearest neighbors around a reference molecule in a molecular crystal, the eﬀect of the disruption extends quite far. As a result, there is some local order around the reference point but the extent of order falls oﬀ rapidly with distance so that at distances equivalent to four or ﬁve molecular diameters the system does not possess order with respect to the reference point. Theories of the liquid state based on an approach involving disordered solids were pursued from the 1930s to the 1960s but did not meet with much success. On the other hand, the liquid may be regarded as an extremely imperfect gas. In this approach, which has been quite successful, the statistical mechanical techniques used to describe the properties of non-ideal gases are extended to liquids. Considerable advances have been made since the 1950s in developing the theory of liquids [G1, G2]. The purpose of this chapter is to give an introduction to this subject, outlining the main theoretical and experimental topics. Rigorous development of the theory is not possible without appealing to more sophisticated mathematics. However, an understanding of the basic concepts involved in this subject is helpful, not only in reading further in the area of liquid structure, but also in developing the other topics considered in this monograph. Liquids may be classiﬁed according to the intermolecular (or interatomic) forces existing between the components. In this way, seven diﬀerent kinds of liquids may be identiﬁed: 1. The simplest liquids are those formed by the inert gases, He, Ne, Ar, etc. These atoms interact via van der Waals forces and strongly repel each other at short interatomic distances.

THE STRUCTURE OF LIQUIDS

47

2. Homonuclear diatomic molecules such as H2, N2, and Cl2 form the second group. They are similar to the ﬁrst in that they do not possess a dipole moment but have electrical quadrupole moments and are not spherical. 3. Liquid metals such as Hg make up the third group. Because of the mobility of electrons in metals, these systems have long-range coulombic forces, and are ‘‘softer’’ with respect to short-range repulsive forces. In addition, electrical screening eﬀects are important in liquid metals. 4. Molten salts are systems in which the components are ions but which are electrically neutral on a local scale. Coulombic forces are long range in molten salts and electrical screening is important. The complexity of these systems depends on the nature of the ions, that is, whether they are monoatomic or polyatomic. For polyatomic ions, other electrostatic forces may be involved. 5. Aprotic polar liquids such as dimethyl sulfoxide and acetonitrile make up another group. These molecules have high dipole moments, so that dipole– dipole interactions are an important part of the description of intermolecular forces. 6. Another group is composed of protic polar liquids such as water and the alcohols. In these systems, hydrogen bonding adds a further complication to the description of intermolecular forces. 7. Finally, non-polar molecular liquids, such as carbon tetrachloride and the hydrocarbons, form a group. Many of these systems possess no permanent dipole moment so that the intermolecular forces are similar to those existing in simple atomic liquids such as liquid argon. However, internal modes of motion are important in describing the properties of the molecular liquid.

Chemists are obviously concerned mainly with liquids in the last three groups. However, they are the most diﬃcult to model from the point of view of theory. Much of the theoretical eﬀort has been directed to understanding the properties of the simplest liquids, namely, the inert gases. In the following sections, the statistical mechanical approach developed to understand liquid properties is outlined. The purpose of this subject is to establish a connection between the properties of the individual atoms or molecules in the liquid and the bulk properties of the system. An important part of this development is the experimental study of liquid structure which is also outlined in the following discussion.

2.2 The Statistical Thermodynamics of Liquids Statistical thermodynamics uses statistical arguments to develop a connection between the properties of individual molecules in a system and its bulk thermodynamic properties. For instance, consider a mole of water molecules at 25 C and standard pressure (1 bar). The thermodynamic state of the system has been deﬁned on the basis of the number of molecules, the temperature, and the pressure. In order to relate the macroscopic thermodynamic properties such as U, G, H and A to the properties of the individual molecules, one would have to solve the Schro¨dinger wave equation (SWE) for a system composed of 6 1023 interacting water molecules. This is an impossible task at present but if it were possible, one would obtain a wave function, j , and an energy, “j , for the system. Moreover,

48

LIQUIDS, SOLUTIONS, AND INTERFACES

the values of j and “j would ﬂuctuate with time because of the changing positions of the molecules and their interactions. The task of statistical thermodynamics is to relate the time average properties of the many microstates which arise in the ﬂuctuating system to its macrostate. In the following section the basic principles of statistical thermodynamics are outlined, ﬁrst for non-interacting systems, such as ideal gases, and then for interacting systems, that is, non-ideal gases and liquids. The present discussion concerns the canonical ensemble. An ensemble is a hypothetical collection of an large number of non-interacting systems, each of which is in the same thermodynamic state (macrostate) but in a diﬀerent microstate. A fundamental postulate of statistical thermodynamics is that the measured time average of a macroscopic property of the system is equal to the average value of that property in the ensemble. Thus, time averaging can be replaced by averaging over the systems in the ensemble. The fact that the ensemble is canonical means that each system in the ensemble has the same volume, temperature, and composition (number of molecules). This means that each system is rigid (constant volume), and has impermeable walls (constant composition), but walls which are thermally conductive. All of the systems can be imagined as being immersed in constant temperature bath. The internal energy of the macrosystem, U, is found by averaging the energy “j for each microsystem in the ensemble. The averaging is carried out over all possible quantum states of the microsystem, taking into account the probability that the total energy is equal to “j . A second postulate used in this process is that for a thermodynamic system of ﬁxed volume, composition, and temperature, all quantum states that have equal energy have equal probability of occurring. Estimation of the probability for a given quantum state leads to the deﬁnition of the canonical partition function of the macrosystem: X eb“j ð2:2:1Þ Q¼ j

where b ¼ 1=ðkB TÞ and kB is the Boltzmann constant. The partition function is of central importance in determining the thermodynamic properties of the macrosystem given its microscopic properties. Sometimes the summation is deﬁned in terms of available energy levels rather than quantum states. In this case, each term in the summation must be multiplied by the degeneracy of the corresponding energy level. On the basis of the deﬁnition of the partition function, the probability that the microsystem is in quantum state “j is given by fj ¼

eb“j Q

Since the internal energy, U, is given by X U¼ fj “ j it is easily shown that U¼

@ ln Q @ ln Q ¼ kB T2 @b V;N @T V;N

ð2:2:2Þ

ð2:2:3Þ

ð2:2:4Þ

49

THE STRUCTURE OF LIQUIDS

In the same way, one may estimate the system’s pressure P by noting that X P¼ fj Pj ð2:2:5Þ where Pj is the pressure for the microsystem in microstate j. It follows that 1 @ ln Q P¼ ð2:2:6Þ b @V T;N An expression for the entropy S of the ensemble can be found by relating the temperature derivative of the entropy to the heat capacity at constant volume using equation (1.3.9). Accordingly, @U @S CV ¼ ¼T ð2:2:7Þ @T V;N @T V;N Since CV ¼ it follows that

@ @ ln Q kB T2 @T @T V;N

ð2:2:8Þ

! @S @ ln Q @2 ln Q ¼ 2kB þk T @T V @T V;N B @T2

ð2:2:9Þ V;N

The corresponding diﬀerential equation to obtain the entropy is ! @ ln Q @2 ln Q dS ¼ 2kB dT þ kB T dT @T V;N @T2 V;N @ ln Q @ ln Q ¼ kB dT þ kB d T @T V;N @T V;N

ð2:2:10Þ

Integrating with respect to temperature from 0 K, one obtains the expression S S0 ¼ kB ln Q kB ln Q0 þ kB T

@ ln Q @T

ð2:2:11Þ

where S0 and Q0 are the entropy and partition function at 0 K. The partition function Q0 is simply the degeneracy of the lowest available energy state of the system and is a temperature-independent quantity. Equating S0 and kB ln Q0 , the resulting expression for the entropy is S¼

U þ kB ln Q T

ð2:2:12Þ

On the basis of these deﬁnitions, one easily obtains expressions for the Helmholtz energy A ¼ U TS ¼ kB T ln Q and for the Gibbs energy

ð2:2:13Þ

50

LIQUIDS, SOLUTIONS, AND INTERFACES

@ ln Q G ¼ U TS þ PV ¼ kB T ln Q þ kB TV @V T;N

ð2:2:14Þ

The thermodynamic functions are introduced here for a one-component system, but are easily extended to multicomponent systems [G3, G4]. Two other ensembles are commonly deﬁned in statistical thermodynamics, namely, the microcanonical ensemble and the grand canonical ensemble. In the microcanonical ensemble, the number of moles, volume, and internal energy, U, are ﬁxed in each system. This means that each system is totally isolated, being unable to exchange either energy or matter with the surroundings. In the case of the grand canonical ensemble, the volume, temperature, and chemical potential of the single component in each system are held constant. Thus, for such an ensemble, both energy and matter can be exchanged with the surroundings. Further details regarding these ensembles and the relationships between the three types introduced can be found elsewhere [G3, G4]. The treatment of the canonical ensemble for the special case of an ideal gas is now outlined. First of all the partition function for the macrosystem is related to the properties of the individual molecules. At the same time, the energy of a given state in the ensemble “j can be related to the energy of an individual molecule i . In the case of a gas or a liquid the particles in the system are indistinguishable because of random motion. Furthermore, if the particles do not interact with one another as in an ideal gas, it is easily shown that qN ð2:2:15Þ N! where q is the partition function for a given molecule, which is given by X b q¼ e i ð2:2:16Þ Q¼

i

The summation for q is made over the available quantum states for the molecule as determined by solution of the SWE. In order to proceed further one must write down the Hamiltonian for the molecule, that is, the expression for its kinetic and potential energy. The former contains up to ﬁve contributions, namely, for its translational, rotational, vibrational, electronic, and nuclear energies. Since the molecules do not interact with one another the potential energy is zero. Thus, one may write H ¼ Htrans þ Hrot þ Hvib þ Helec þ Hnuc

ð2:2:17Þ

For present purpose this is more conveniently written as H ¼ Htrans þ Hint

ð2:2:18Þ

where Hint represents all of the internal modes of possessing energy. With the Hamiltonian operator H one obtains the SWE whose solution gives the energy of an individual molecule, : ¼ trans þ int

ð2:2:19Þ

where trans and int are the contributions of the translational and internal modes to the energy, respectively. These may then be used to estimate the molecular partition function, q:

THE STRUCTURE OF LIQUIDS

q ¼ qtrans qint

51

ð2:2:20Þ

Expressions for the individual partition functions have also been developed. From the solution of the SWE for a particle in a box it is well known that 2pmkB T 3=2 V qtrans ¼ V¼ 3 ð2:2:21Þ h2 where m is the mass of the particle, V, the volume of the box and h, the Planck constant and ¼

h ð2pmkB TÞ1=2

ð2:2:22Þ

The expression for the canonical partition function of an ideal gas thus becomes Q¼

N qN qN V N trans Qint ¼ int 3N N! N!

ð2:2:23Þ

Another way of writing the molecular partition function is to use the classical expression for Htrans in an integral form. By classical mechanics, Htrans for each molecule is given by 1 ðP2 þ P2y þ P2z Þ ð2:2:24Þ Htrans ¼ 2m x where Pi is the linear momentum in the ith direction. Then, ððð 1 qtrans ¼ 3 expðbHtransÞ dPdr ð2:2:25Þ h where dP ¼ dPx dPy dPz and dr ¼ dxdydz. The factor 1/h3 appears in this expression in order to keep the classical result the same as that from the quantummechanical summation. Integration with respect to the volume element dr yields the volume V of the container. Substituting equation (2.2.24) into equation (2.2.25), one may write 21 33 ð V4 ð2pmkB TÞ3=2 qtrans ¼ 3 expðbP2 =2mÞdp5 ¼ V ð2:2:26Þ h h3 1

Thus, the classical expression for qtrans yields the same result as that obtained from the SWE (equation (2.2.21)). It is useful to keep the classical way of expressing the partition function and extend its application to more complex situations. First of all, one may write an expression for the canonical partition function of an ideal gas in terms of the partition function for each molecule. On the basis of the total Hamiltonian for each molecule (equation (2.2.18)), Hj , the canonical partition function is ð ð ð N 1 expðbHj ÞdPj drj Q¼ ð2:2:27Þ N!h3N Each integral within the brackets is sixfold with three directional and three momentum coordinates involved in the integration process. Alternatively, the partition function may be expressed as

52

LIQUIDS, SOLUTIONS, AND INTERFACES

Q¼

ð ð 1 . . . expðbHÞdP1 . . . dPN dr1 . . . drN N!h3N

where H¼

X

Hj

ð2:2:28Þ

ð2:2:29Þ

j

is the Hamiltonian of the entire system. In the notation used, particle 1 at position r1 (a vector) is located in volume element dr1, and so on. Now the partition function is expressed in terms of one 6N-fold integral. The above treatment of an ideal gas is easily extended to a non-ideal system, at least, in a conceptual fashion. In the non-ideal system the molecules interact with each other so that the system has a potential energy, U, which depends on the coordinates of each molecule in the system. In the real system in which molecular positions change with time, the potential energy also ﬂuctuates. This is reﬂected in the canonical ensemble by variation in the potential energy with the molecular conﬁguration in a given microstate. The Hamiltonian for such a system containing N molecules is H¼

N N X 1 X ðP2xj þ P2yj þ P2zj Þ þ Hintj þ Uðr1 ; r2 ; . . . ; rN Þ 2m j¼1 j¼1

ð2:2:30Þ

Performing the integration in equation (2.2.27), one obtains the following expression for the canonical partition function Q: Q¼

qN int Z N!3N

where Z is the conﬁgurational partition function, that is, ð ð Z ¼ . . . expðbUÞdr1 dr2 . . . drN

ð2:2:31Þ

ð2:2:32Þ

It should be noted that it is assumed that the intermolecular forces do not aﬀect the internal degrees of freedom so that qint is independent of whether these forces are present or not. When they are absent (U ¼ 0), the integral Z collapses to VN and equation (2.2.31) becomes the same as equation (2.2.23). The important task of the statistical thermodynamics of imperfect gases and liquids is to evaluate Z. This subject is discussed in detail later in this chapter. However, the nature of the intermolecular forces which give rise to the potential energy U is considered next.

2.3 Intermolecular Forces Molecules exhibit relatively long-range attractive forces between themselves which give rise to the cohesive forces in liquids. These forces arise because the electronic distribution in the molecule or atom making up the liquid is not uniform either on a time-averaged basis or with respect to its instantaneous value. Non-uniformity in the time-averaged electronic distribution in a molecule is a well-known phenomenon, and is discussed in terms of the experimentally measured dipole

THE STRUCTURE OF LIQUIDS

53

moment, and higher-order moments such as the quadrupole and octapole moments. The major attractive component of intermolecular forces is due to dipole–dipole interactions, of which there are three principle contributions, namely, that due to permanent dipole interactions, that due to induced dipole– permanent dipole interactions, and the dispersion contribution arising from interactions between the instantaneous dipole moments. These are collectively called van der Waals forces and give rise to an attractive potential between two molecules which is inversely proportional to the distance between them to the sixth power. The nature of van der Waals forces is now examined in more detail. van der Waals forces are usually treated using the physics of ideal dipoles. A real dipole consists of equal and opposite charges separated by a ﬁnite distance d. This concept is easily applied to molecules where diﬀerences in atomic electronegativities leads to polar molecules in which the equal and opposite charges are separated by distances the order of a typical bond length. A simple example is carbon monoxide, which has a dipole moment of 0.12 debye because of the polarity associated with the C–O bond. An ideal dipole is one in which the ﬁnite distance separating the charges shrinks to zero but the charge separation is maintained. Thus, the ideal dipole is a point with a vector character associated with the charge separation. The dipole moment is measured in coulomb meters in SI units and in debyes in traditional units.* The dipole vector points along the axis of the dipole from the negative to the positive charge. Because of the vector character of the dipole moment, the electrostatic equations describing its properties are much more complex than those for a point charge. The potential due to an ideal dipole p at a distance r from its center is p ¼

pr p cos y ¼ 3 4pe0 r 4pe0 r2

ð2:3:1Þ

where y is the angle between the dipole vector and the line from the dipole to the point where the potential is measured, and "0, the permittivity of free space. The maximum potential is obtained in the direction of the dipole vector where cos y is equal to one. On the other hand the potential perpendicular to the dipole is zero (cos 90 ¼ 0). The corresponding equation for a point charge q is q ð2:3:2Þ q ¼ 4pe0 r In this case the potential is independent of the direction from the charge at which it is measured. The dipole potential in a given direction falls oﬀ as 1=r2 , whereas that due to a point charge, as 1=r. The ﬁeld due to the ideal dipole is found by taking the gradient of the potential. Thus,

*All electrical equations in this monograph are written in SI units. As a result, the dipole moment must be given in C m, not in its usual units of debyes (1 debye ¼ 3.336 1030 C m). In addition, the molecular polarizability, which is usually tabulated in m3, has units of C2 m N1 in the SI system ða 0 ðcgsÞ ¼ aðSIÞ=ð4pe0 ÞÞ. Thus, a polarizability of 1030 m3 in traditional units is equivalent to 1.113 1040 C m2 V1 in SI units. Although the equations are written for SI units, dipole moments and polarizabilities are recorded in the tables in this monograph in the traditional units of debyes and m3.

54

LIQUIDS, SOLUTIONS, AND INTERFACES

Ep ¼ gradq ¼

3p r p r 4pe0 r5 4pe0 r3

ð2:3:3Þ

The ﬁrst term on the right-hand side gives the component of the ﬁeld in the direction r deﬁned by the vector from the ideal dipole to the point of measurement. The second term gives the component in the direction deﬁned by the dipole. The net ﬁeld, Ep , which is also a vector, points in a direction between p and r deﬁned by their vector sum. When the ﬁeld is measured along the direction of the ideal dipole, the expression simpliﬁes to Ep ðy ¼ 0Þ ¼

p 2pe0 r3

ð2:3:4Þ

Now one may estimate the energy of two dipoles on the basis of the work done to bring dipole p2 from inﬁnity to a point located at distance r from dipole p1. This is u12 ¼ p2 E ¼

3p1 r p p ðp rÞ þ 1 2 3 4pe0 r5 2 4pe0 r

ð2:3:5Þ

In polar coordinates, equation (2.3.5) becomes u12 ðrÞ ¼

p1 p2 ½2 cos y1 cos y2 sin y1 sin y2 cosðf1 f2 Þ 4pe0 r3

ð2:3:6Þ

where yi is the angle that dipole i makes with the z-axis, chosen as the direction deﬁned by the line between the dipoles, and fi is the angle that dipole i makes with the x-axis (see ﬁg. 2.1). There are several important conﬁgurations that are among all possible conﬁgurations of two dipoles. The ﬁrst corresponds to the case that the two dipoles point along the z-axis with the head of one dipole pointing to the tail of the other (ﬁg. 2.1). In this case y1 ¼ y 2 ¼ 0

ð2:3:7Þ

and u12 ðrÞ ¼

2p1 p2 4pe0 r3

ð2:3:8Þ

Fig. 2.1 Schematic diagram showing two dipoles pa and pb separated by a distance r along the z-axis with orientations deﬁned by the angles ya, fa and yb, fb. The four simple conﬁgurations for two dipoles and their interaction energies are also illustrated with ur ¼ pa pb =ð4pe0 r3 Þ:

THE STRUCTURE OF LIQUIDS

55

On the basis of equation (2.3.6), this is the conﬁguration of minimum energy. The conﬁguration of maximum energy occurs when the two dipoles are oriented with their heads or tails pointing to one another. This occurs when y1 ¼ 0

y2 ¼ 1808

and

ð2:3:9Þ

and gives an energy u12 ðrÞ ¼

2p1 p2 4pe0 r3

ð2:3:10Þ

Two other conﬁgurations are important in assessing dipole–dipole interactions. The ﬁrst occurs when the two dipoles are parallel to each other but point in opposite directions. This is obtained when y1 ¼ 908;

y2 ¼ 2708;

f1 ¼ f2

and

ð2:3:11Þ

with the result that u12 ðrÞ ¼

p1 p2 4pe0 r3

ð2:3:12Þ

Finally, when the two dipoles are parallel and point in the same direction y1 ¼ 908;

y2 ¼ 908;

and

f1 ¼ f2

ð2:3:13Þ

so that u12 ðrÞ ¼

p1 p2 4pe0 r3

ð2:3:14Þ

The important result from this analysis is that the dipole–dipole interaction energy depends on the product of the two dipole moments and decreases with distance as 1=r3 . In order to estimate the contribution of interactions between permanent dipoles to intermolecular forces, one must consider all possible relative orientations at a given separation distance r. The interaction energy is then averaged, taking into account the probability of a given orientation on the basis of a Boltzmann factor. In this way, orientations with attractive interaction are weighted in favor of those resulting in repulsive interaction. The expression giving the net average interaction energy between two dipoles is ÐÐ u ðrÞ exp½u12 ðrÞ=kB Tdo1 do2 Ð 12 Ð ð2:3:15Þ udd ðrÞ ¼ exp½u12 ðrÞ=kB Tdo1 do2 where doi ¼ sin yi dydf

ð2:3:16Þ

Integration over all possible values of y and f for the case that u12 ðrÞ is small with respect to kB T gives the following result for the average dipole–dipole interaction energy: udd ðrÞ ¼

2p21 p22 3ð4pe0 Þ2 kB Tr6

ð2:3:17Þ

56

LIQUIDS, SOLUTIONS, AND INTERFACES

In the averaging process, the factor p1 p2 =ð4pe0 Þr3 ; which appears in the expression for the energy of a speciﬁc orientation, is squared, so that the average attractive interaction depends on 1/r6. Thus it is important to distinguish between the interaction energy of two isolated dipoles, ﬁxed with respect to their relative orientations, which depends on the distance separating them as 1=r3 , and the interaction energy between the same two dipoles, when it is averaged over all possible mutual orientations, which depends on 1=r6 . As an example, consider the case of two water molecules in contact, each with a dipole moment equal to 1.85 debyes (6.17 1030 C m) and represented as spheres with diameters of 274 pm. The attractive energy between these molecules when they are oriented in the conﬁguration of maximum attraction is 3:25 1020 J, that is, 7:9kB T at room temperature. On the other hand, when the attractive energy is estimated using Boltzmann averaging, it is 4:30 1020 J or 10:4kB T. The Boltzmann-averaged value is more than 50% of the maximum attractive energy. It follows that dipole–dipole interactions represent a major component of the cohesive energy which leads to formation of the liquid state in water. A second component of van der Waals forces is that resulting from the induced dipole moment formed in one molecule in the presence of another molecule possessing a permanent dipole moment. If a1 and a2 are the polarizabilities of the two molecules, the potential energy due to dipole–induced dipole interactions is uid ðrÞ ¼

p21 a2 þ p22 a1 ð4pe0 Þ2 r6

ð2:3:18Þ

This is derived by considering the interaction of two polarizable dipoles in a general orientation and then averaging over all possible orientations. Assuming a molecular polarizability equal to 1.46 1030 m3 (1.62 1040 C m2 V1) for water, the value of uid ðrÞ for water molecules in contact is 2:3 1021 J, or 0:6kB T at room temperature. As expected, this contribution is much smaller than that due to the interaction of the permanent dipole moments. The third contribution to the attractive forces comes from London dispersion forces. These arise from the instantaneous dipoles on the molecules formed by the moving electrons. Thus, inert gas atoms such as helium and argon possess instantaneous dipole moments formed by the electron cloud, which is constantly in motion. Dispersion forces were analyzed using quantum mechanics by London [2] who derived the following expression for the attractive potential: udis ¼

3I1 I2 a1 a2 2ðI1 þ I2 Þð4pe0 Þ2 r6

ð2:3:19Þ

where I1 is the ionization potential of molecule 1, and I2 , that of molecule 2. For two water molecules in contact, the dispersion energy is 1:85kB T at room temperature given that the ionization potential of a water molecule is 12.6 eV or 2.02 1018 J molec1 .

THE STRUCTURE OF LIQUIDS

57

EXAMPLE

Estimate the contributions to the van der Waals energy in dimethylsulfoxide at 25 C by considering two molecules in contact as hard spheres with diameters of 491 pm. The dipole moment of dimethyl sulfoxide is 3.96 debyes, its polarizability, 7.99 103 nm3, and its ionization potential, 9.01 eV. The dipole–dipole attractive potential energy for two identical spherical molecules in contact is udd ¼

2p4 3ð4pe0 Þ2 kB Ts6

ð2:3:20Þ

where s is the molecular diameter. For dimethyl sulfoxide, udd ¼

2ð3:96 3:336 1930 Þ4 3ð1:113 1010 Þ2 1:38 1023 298:2 ð4:91 1010 Þ6

ð2:3:21Þ

¼ 2:84 1020 J molec1 ; or 6:91kB T Similarly, for the dipole–induced dipole interactions, uid ¼ ¼

2p2 a ð4pe0 Þ2 s6 2ð3:96 3:336 1030 Þ2 ð7:99 1030 1:113 1010 Þ ð1:113 1010 Þ2 ð4:91 1010 Þ6

ð2:3:22Þ

¼ 1:79 1021 J molec1 ; or 0:43kB T For the London dispersion energy, udis ¼ ¼

3Ia2 4ð4pe0 Þ2 s6 3ð9:01 1:602 1019 Þð7:99 1030 Þ2 4 ð4:91 1010 Þ6

ð2:3:23Þ

¼ 4:93 1021 J molec1 ; or 1:20kB T Note that the calculations of uid and udis in which the polarizability is involved are simpler when the SI polarizability a is replaced by a 0 ¼ =4pe0 , which has units of m3. Then equations (2.3.22) and (2.3.23) may be written as uid ¼

2p2 a 0 4pe0 s6

ð2:3:24Þ

udis ¼

3Ia 02 4s6

ð2:3:25Þ

and

Values of the dipole–dipole, dipole–induced dipole, and London dispersion contributions for two molecules in contact are summarized for a range of liquids

58

LIQUIDS, SOLUTIONS, AND INTERFACES

Table 2.1 Estimation of van der Waals Interaction Energies for Two Molecules at Contact Using a Spherical Representation of the Molecules Atom or Molecule Argon Nitrogen Hexane Benzene Water Acetone Acetonitrile Dimethyl sulfoxide

Molecular Dipole Molecular Ionization van der Waals Energy* Diameter Moment Polarizability Potential s=pm p=debye a=nm3 I=eV udd =kB T uid kB T udis =kB T 370 400 592 563 274 476 427 491

0 0 0.09 0 1.82 2.88 3.92 3.96

1:63 103 1:76 103 0:0118 0:0103 1:46 103 6:41 103 4:41 103 7:99 103

15.8 15.6 10.1 9.25 12.6 9.71 12.2 9.01

0.0 0.0 0.0 0.0 10.38 2.33 15.34 6.91

0.0 0.0 0.0 0.0 0.55 0.22 0.54 0.43

0.48 0.34 0.95 0.90 1.85 1.00 1.14 1.20

Estimated at 25 C.

*

in table 2.1. It should be kept in mind that equations (2.3.17)–(2.3.19) are generally not applicable at such short distances, since they were derived on the basis of point dipole models. However, they give a good idea of the relative magnitude of the three components and how they change when one goes from an inert gas like argon to very polar molecules like water and dimethyl sulfoxide. It is also easily seen from these equations that the intermolecular potential energy falls oﬀ rapidly with increase in distance between the molecules. Thus, if the separation is two molecular diameters, the value of the potential energy drops by a factor of 1/26, that is, 1/64; for three molecular diameters this decreases to 1/36, or 1/729, and so on. The above simple picture of the attractive component of intermolecular forces is certainly not complete. Since a complete electrostatic description of a molecule usually requires the introduction of higher-order moments, one must also consider the potential energy due to dipole–quadrupole interactions, quadrupole– quadrupole interactions, and interactions of moments of higher order. These lead to terms proportional to r8 , r10 , and so on. Obviously, the potential energy due to these components falls oﬀ much more rapidly than those due to dipole– dipole interactions, and therefore, is less important. Another factor not considered in the above discussion is the role of hydrogen bonding, which is very important for protic molecules such as water and the alcohols. It is well recognized that hydrogen atoms bonded to more electronegative atoms such as oxygen and nitrogen can form a bond with another electron-rich atom in an adjacent molecule. The best-known example of hydrogen bonding is that in water, where the hydrogen atom on one molecule bonds with the oxygen atom on an adjacent molecule. Other well-known examples are the alcohols, certain amides like formamide and acetamide, and carboxylic acids. In order to estimate the strength of hydrogen bonding, one must develop a quantum-mechanical model of the system. This is a complex topic and is not considered further here. It is suﬃcient to point out that hydrogen bonding accounts for an important fraction of the cohesive energy in liquids when it is present.

THE STRUCTURE OF LIQUIDS

59

Hydrogen bonding is an example of a relatively short-range attractive force between speciﬁc types of molecules. In general, one must consider another shortrange force which is experienced by all atoms and molecules and which is repulsive. This repulsive interaction arises when the electronic clouds in two molecules are so close that they repel one another. It may be treated quantum mechanically in the adiabatic approximation in which the nuclei in the molecules are assumed to be motionless, and the wave functions for the electrons calculated. An approximate form for the repulsive energy between two molecules is urep ¼ AR ear

ð2:3:26Þ

where AR and a are constants depending on the properties of the individual molecules. This is more conveniently approximated by a function of the twelfth power of the intermolecular distance, r, so that equation (2.3.26) becomes urep ¼

AR r12

ð2:3:27Þ

When one compares this term with the attractive van der Waals forces considered above it is clear that the repulsive energy falls oﬀ much more rapidly with distance than the attractive one. When one adds the attractive van der Waals potential terms to the repulsive term, one obtains the Lennard–Jones expression for the intermolecular potential energy for a simple ﬂuid such as an inert gas like argon. On the basis of the above, the Lennard–Jones potential function may be written uðrÞ ¼

ALJ BLJ 6 r12 r

ð2:3:28Þ

where the second term represents the attractive van der Waals component. Obviously, there is a minimum potential energy, LJ , corresponding to the optimum distance between any two molecules. In addition, one may identify the molecular diameter s as corresponding to the distance at which the intermolecular potential energy is zero. As a result, the constants ALJ and BLJ are connected by the equation ALJ =BLJ ¼ s6

ð2:3:29Þ

duðrÞ ALJ B ¼ 12 13 þ 6 LJ ¼0 dr r7e re

ð2:3:30Þ

At the optimum separation re

so that r6e ¼

2ALJ BLJ

ð2:3:31Þ

It follows that at the minimum LJ ¼

ALJ BLJ B2 B2 B2 6 ¼ LJ LJ ¼ LJ 12 4ALJ 2ALJ 4ALJ re re

ð2:3:32Þ

Solving equations (2.3.29) and (2.3.32) for the constants ALJ and BLJ , one obtains

60

LIQUIDS, SOLUTIONS, AND INTERFACES

ALJ ¼ 4s12 LJ

ð2:3:33Þ

BLJ ¼ 4s6 LJ

ð2:3:34Þ

and

It follows that the Lennard–Jones equation is

s 12 s6 uðrÞ ¼ 4LJ r r

ð2:3:35Þ

It is easily seen that the function r12 was chosen to replace ear for the repulsive component in order that a simple mathematical result could be obtained. EXAMPLE

Estimate the Lennard–Jones energy of molecular nitrogen at the optimum separation re and at twice this value. Use the parameters given in table 2.2. From the table, LJ ¼ 95:2 1:381 1023 ¼ 1:245 1021 J molec1

ð2:3:36Þ

When r is equal to re , uðrÞ is equal to LJ . The relationship between re and s is r6e ¼ 2s6

ð2:3:37Þ

so that re is 420.4 pm, given that s is 374.5 pm. When r is equal to 2re " #

s12 s6 1 1 uðrÞ ¼ 4LJ ¼ LJ 12 5 2 2 ½2ð21=6 sÞ12 ½2ð21=6 sÞ6

ð2:3:38Þ

¼ 0:031 1:245 1021 ¼ 3:86 1023 J molec1 Thus, the Lennard–Jones energy at 2re is 3:86 1023 J molec1 : A plot of the Lennard–Jones potential against distance between two molecules is shown in ﬁg. 2.2. It is clear that the repulsive component dominates for values of r less than the molecular diameter, where it rises to values over 5LJ for r ¼ 0:9s. The stable minimum occurs at r ¼ 1:12s. By the time the separation between the two molecules is equal to three times their diameter, the intermolecular

Table 2.2 Lennard–Jones Parameters for Simple Molecules Atom or Molecule Ar Xe N2 CO2

s=ðpmÞ

LJ k1 B =K

350 410 374.5 433

117.7 222.3 95.2 198.2

*kB is the Boltzmann constant, so that LJ k1 B has units of temperature.

THE STRUCTURE OF LIQUIDS

61

Fig. 2.2 Plot of the Lennard–Jones potential uðrÞ in units of the attractive potential energy LJ against intermolecular distance, r, in units of the molecular diameter s. The vertical and horizontal straight lines at r ¼ s show the potential energy for a hard-sphere representation of the system.

potential is negligible. This potential energy function has often been used in the development of the theory of simple liquids such as the noble gases at low temperature. For many liquids that chemists use, a much more complex expression for the intermolecular potential is required. However, the properties of a Lennard– Jones liquid are reasonably easily derived and form a convenient point of reference with respect to which the properties of more complex systems can be considered.

2.4 Distribution and Correlation Functions The experimental and theoretical descriptions of liquid structure are most conveniently achieved in terms of distribution functions. This is because there is short-range structure in the liquid, but at large distances from the point of reference, the distribution of molecules is random. In this section, the fundamental aspects of distribution and correlation functions, especially, the pair correlation function, are outlined. Using the conﬁgurational partition function derived earlier (equation (2.2.32)), one may write an expression for the probability of any given conﬁguration in which the particle located at r1 is in volume element dr1, that at r2 in dr2, and so on (ﬁg. 2.3). This probability is given by f ðNÞ ðr1 ; . . . rN Þdr1 . . . drN ¼

expðbUÞ dr1 . . . dN Z

ð2:4:1Þ

The ratio exp(ðbUÞ=Z is the important part of this expression, giving the fraction of the conﬁguration integral associated with a given potential energy U. Otherwise, the expression is of little practical use other than as a starting point for deriving simpler distribution functions. The most elementary of these is the single-particle distribution function f ð1Þ ðr1 Þ, which gives the probability of ﬁnding a particle or molecule at position r1 . It is found by integrating the N-particle density function f ðNÞ over the remaining coordinates r2 ; . . . rN and multiplying by the number of ways of choosing particle 1, namely N. The result is

62

LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 2.3 Coordinate system in which the location of each particle is designated by a vector ri with its associated volume element dri. Three particles are shown at r1, r2, and r3. The scalar distances between them are r12 ¼ jr1 r2 j and r13 ¼ jr1 r3 j.

f ð1Þ ðr1 Þ ¼

ð ð N . . . expðbUÞdr2 . . . drN Z

ð2:4:2Þ

Since the system is uniform, the potential energy is a function of relative coordinates only so that Z can be written as the product of the integral in the numerator times the volume of the system V. It follows that the singlet distribution function is simply the number of particles or molecules in the ﬂuid N per unit volume V, that is, f ð1Þ ðr1 Þ ¼

N V

ð2:4:3Þ

One may also deﬁne the pair distribution function f ð2Þ ðr1 ; r2 Þ as the probability of ﬁnding a particle in the volume element dr2 at r2 given that there already is a particle in the volume element dr1 at position r1 . By analogy with equation (2.4.2), the pair distribution function is given by ð ð NðN 1Þ ð2Þ . . . expðbUÞdr3 . . . drN f ðr1 ; r2 Þ ¼ ð2:4:4Þ Z By integrating f ð2Þ ðr1 ; r2 Þ over dr2 , that is, over the volume of the system, one obtains the singlet distribution function: ð NðN 1Þ ð2:4:5Þ f ð2Þ ðr1 ; r2 Þdr2 ¼ ðN 1Þf ð1Þ ðr1 Þ ¼ V In the same way, one may deﬁne higher-order distribution functions. For instance, f ð3Þ ðr1 ; r2 ; r3 Þ is the probability of ﬁnding a particle at r3 given that there are already particles at r1 and r2 . A deﬁning relationship for this function is ð f ð3Þ ðr1 ; r2 ; r3 Þdr3 ¼ ðN 2Þf ð2Þ ðr1 ; r2 Þ ð2:4:6Þ The pair distribution function leads to the pair correlation function, which illustrates how the local order found near a given molecule is lost as distance from the molecule increases. This quantity is of fundamental theoretical interest and may be determined in X-ray and neutron scattering experiments, as discussed below. The deﬁnition of the pair correlation function gðr12 Þ is

THE STRUCTURE OF LIQUIDS

gðr12 Þ ¼

N2 f ð2Þ ðr1 ; r2 Þ V2

63

ð2:4:7Þ

This function gives the normalized probability for ﬁnding a molecule at r2 when another molecule is at r1 . Usually, the reference molecule is assumed to be at the origin of the coordinate system and the subscript ‘‘12’’ is dropped so that the pair correlation function becomes simply gðrÞ. In the limit that r goes to inﬁnity, gðrÞ goes to unity, demonstrating that there is no correlation between the molecule at the origin and one inﬁnitely far away. Another way of thinking about this function is to consider gðrÞ as a factor which multiplies the average bulk density, N=V, to give the local density, NgðrÞ=V, about a central molecule. The correlation function for a ﬂuid obeying the Lennard–Jones potential as calculated by Verlet [3] is shown in ﬁg. 2.4. At the optimum distance from the central molecule or particle, which is slightly larger than the molecular diameter s, gðrÞ is close to three because of attractive intermolecular forces. As distance increases, the correlation function oscillates, reﬂecting the ﬁnite size of the particles in the system. A second maximum is found at twice the optimum distance but its height is signiﬁcantly reduced with respect to that of the ﬁrst. Eventually, the oscillations die away so that, at a distance equal to four or ﬁve particle diameters, the eﬀect of the central molecule is scarcely felt. Several important quantities may be obtained from gðrÞ. One is the potential of mean force WðrÞ. This is deﬁned by the equation WðrÞ ¼ kB T ln gðrÞ

ð2:4:8Þ

The problem of obtaining a relationship between WðrÞ and the intermolecular potential uðrÞ is central to the statistical mechanics of non-ideal gases and liquids. Various methods have been described for making this connection as discussed in the statistical mechanical literature [G1–G4]. As with any potential energy, one may obtain the force acting on the central particle in the direction of any other particle by diﬀerentiation with respect to the distance vector r1 . Thus, the mean force acting on the central particle due to a molecule at r1 is

Fig. 2.4 The correlation function gðrÞ for a ﬂuid obeying the Lennard–Jones potential with LJ ¼ 1:69kB T and a molecular density N=V ¼ 0:88s3 [3].

64

LIQUIDS, SOLUTIONS, AND INTERFACES

Fðr1 Þ ¼

@WðrÞ kB T @gðrÞ ¼ @r1 gðrÞ @ðr1 Þ

ð2:4:9Þ

Another function commonly used in discussing the structure of liquids is the total correlation function hðrÞ. It is simply related to gðrÞ by the equation hðrÞ ¼ gðrÞ 1

ð2:4:10Þ

Clearly, when the separation between two molecules is suﬃciently large, hðrÞ is zero (see ﬁg. 2.4). From the total correlation function, one obtains the structure factor SðkD Þ by Fourier transformation. Thus, ð SðkD Þ ¼ 1 þ ðN=VÞ hðrÞeikD r dr ð2:4:11Þ where kD is the reciprocal distance. The structure factor may be determined experimentally from X-ray and neutron diﬀraction experiments, and thus provides a direct route to information about the pair correlation function in liquids. A plot of SðkD Þ against reciprocal distance kD based on the data obtained by Yarnell et al. [4] for liquid argon near its triple point is shown in ﬁg. 2.5. Damped oscillations of the same kind are found for the structure factor as are seen for gðrÞ. More information about Fourier transforms is found in appendix A. One still has the problem of relating the potential energy of the system U, which appears in the conﬁgurational integral (equation (2.2.32)) and in the pair correlation function (equations (2.4.4) and (2.4.7)) to the intermolecular potential energy between any two molecules, uij . The potential energy can be expanded in terms of

Fig. 2.5 An example of an experimentally determined structure factor, namely, that for liquid argon near its triple point [4]. (Reproduced from Physical Review A, by permission.)

THE STRUCTURE OF LIQUIDS

65

a sum of interactions between pairs of molecules, triplets of molecules, and so on. It follows that Uðr1 ; . . . ; rN Þ ¼

N X i<j¼1

uðrij Þ þ

N X

vðrij ; rik ; rjk Þ þ . . .

ð2:4:12Þ

i<j
where uðrij Þ is the pair interaction energy already discussed, and vðrij ; rik ; rjk Þ the triplet interaction energy. The contributions of the triplet and higher-order terms given by the second and higher sums in equation (2.4.12) constitute about 5% of the total potential energy. Estimation of eﬀects beyond the pair potential contribution represents an important problem in the statistical thermodynamics of liquids and several strategies have been developed to solve this problem. More detailed discussion of this topic, including the use of cluster diagrams, can be found elsewhere [G1, G2, G4]. In this chapter, only the integral equation approach is considered and is presented later.

2.5 The Experimental Study of Liquid Structure The structure of pure liquids and liquid solutions is conveniently studied using diﬀraction techniques [5]. The most common of these is X-ray diﬀraction. Two other useful techniques are neutron and electron diﬀraction. In these experiments, radiation, which is usually monoenergetic, penetrates the liquid sample and is scattered through an angle y (see ﬁg. 2.6). The analytical information is obtained by studying the intensity of the scattered radiation as a function of this angle. In

Fig. 2.6 Schematic representation of a scattering experiment in which the intensity of scattered radiation is measured as a function of the scattering angle y (From P. A. Egelstaﬀ, An Introduction to the Liquid State, 2nd edition, Oxford University Press, New York (1992), by permission.)

66

LIQUIDS, SOLUTIONS, AND INTERFACES

X-ray diﬀraction, the X-rays are scattered by the electron clouds around individual atoms. Since the atoms and molecules of the liquid sample are not ﬁxed in space, the information resulting from the diﬀraction experiment must be interpreted in terms of statistical averages. The neutrons used in a neutron diﬀraction experiment are scattered by the nuclei of the atoms in the liquid sample so that the scattering pattern is quite diﬀerent from that for X-rays. In electron diﬀraction, the electrical potential, which depends on the spatial conﬁguration of the nuclei and electronic density distribution, determines the diﬀraction pattern. Early experiments involved simple monoatomic liquids such as the inert gases and liquid metals. However, many molecular liquids have also been studied, including polar liquids such as water, the alcohols, and amides [5]. In this section, attention is focused on two of these techniques, namely, X-ray and neutron diﬀraction. The incoming radiation in the experiment is characterized by its wavelength, l, or its frequency n, and by its intensity I0 . Special eﬀorts must be made to achieve monochromatic radiation. The wavelength is chosen so that it is commensurate with the average distance between atoms, but much smaller than the sample dimensions. A typical wavelength for neutrons is the order of 100 pm, which corresponds to a kinetic energy of 0.08 eV; the wavelength and the linear momentum are connected through the de Broglie equation: l¼

h P

ð2:5:1Þ

where P is the momentum of the neutrons. The scattering results are presented as a function of the variable kD , which has the dimensions of reciprocal distance: kD ¼

4p sinðy=2Þ l

ð2:5:2Þ

Consider ﬁrst of all the scattering of neutrons by the nuclei in a monoatomic liquid. This process is characterized by a scattering cross-section, sD , which, in turn, is related to bD , the bound atom scattering length. For slow neutrons, sD ¼ 4pbD 2

ð2:5:3Þ

Two types of scattering are observed in these experiments, namely, coherent and incoherent scattering. The former is directly related to the structure of the sample and can be used to determine the structure factor SðkD Þ of a simple monoatomic liquid. Incoherent scattering arises due to isotope or spin eﬀects. The presence of diﬀerent isotopes in a sample leads to diﬀerent values of scattering length bD , which depends in a complex way on the number of nucleons in the nucleus from which it is scattered. In addition, a given nucleus may possess a spin which is randomly oriented in the sample. This characteristic of the system contributes to incoherent scattering. The incoherence only gives information about an individual atom and no information about the collective motion of atoms. The relationship between the scattered intensity and the structure factor for a monoatomic liquid is IðyÞ ¼ aD ðyÞhbD i2 ðSðkD Þ þ D Þ

ð2:5:4Þ

THE STRUCTURE OF LIQUIDS

67

where aD ðyÞ is a factor which depends on the properties of the instrument used for the experiments and inelastic scattering from the sample, and D gives the contribution from incoherent scattering. hbD i denotes the value of the atomic scattering length averaged with respect to isotopic composition and spin state. The two unknown quantities in equation (2.5.4), namely, aD ðyÞ and D are determined by measuring the intensity at y ¼ 0 and y ¼ 1. The structure factor of undeﬂected neutrons is related to the sample atomic density NA and isothermal compressibility, T by the following equation: Sð0Þ ¼ NA T kB T

ð2:5:5Þ

At very large angles, the structure factor goes to unity (see ﬁg. 2.6) Sð1Þ ¼ 1

ð2:5:6Þ

As a result, one may show for monoatomic liquids that SðkD Þ ﬃ

IðkD Þ Ið0Þ Ið1Þ Ið0Þ

ð2:5:7Þ

In the case of X-rays, scattering occurs from electrons in the atoms of the liquid sample, and depends on the electron density of the isolated atom, Ne ðrÞ=V, where Ne ðrÞ is the number of electrons at distance r. The quantity corresponding to hbD i2 in the neutron diﬀraction experiment is the atomic scattering intensity for isolated atoms, fD 2 ðkD Þ, where ðra Ne ðrÞ sin kD r 4pr2 dr ð2:5:8Þ fD ðkD Þ ¼ V kD r 0

In contrast to hbD i depends on the angle of scattering y as expressed through the parameter kD . The structure factor SðkD Þ is simply the ratio of the scattered intensity IðkD Þ to the incident intensity I0 divided by the intensity ratio that would be observed for the same number of atoms of the same atomic mass as individuals. The resulting expression for SðkD Þ is 2

; f 2D ðkD Þ

SðkD Þ ¼

IðkD Þ Z2 I0 NZ f 2D ðkD Þ

ð2:5:9Þ

where NZ is the number of atoms of atomic number Z. Both the X-ray and neutron diﬀraction techniques are subject to experimental diﬃculties. It is important to be able to carry out the experiments to high values of kD so that Fourier transformation can be performed accurately. A practical limitation for the value of kD in X-ray and classical neutron diﬀraction experiments is 180 nm1 . It is also diﬃcult to measure SðkD Þ for very small values of kD . Problems due to incoherent scattering with neutrons and with absorption of radiation also have to be dealt with. However, it is apparent from ﬁg. 2.5 that data of high precision can be obtained for simple systems such as the rare gases in the liquid state. Initially, diﬀraction experiments were performed on monoatomic liquids such as argon and neon, or on liquid metals such as mercury. The analysis of scattering data from molecular liquids is more diﬃcult. Nuclei with diﬀerent atomic masses are present, and the incident radiation is scattered by diﬀerent amounts by each

68

LIQUIDS, SOLUTIONS, AND INTERFACES

type of nucleus. In the case of symmetrical molecules such as methane or carbon tetrachloride, the scattering data can be interpreted in a reasonably straightforward fashion. However, in the case of molecules like water, the resulting spectra are much more complex. For liquids with molecules containing diﬀerent atoms, the change in intensity with angle y is given by " # X dI 2 ¼ nA ci f i ðkD Þ þ FT ðkD Þ ð2:5:10Þ dy i where nA is the number of diﬀerent atoms (nuclei) in the sample, ci , the molar concentration of atom i; fi ðkD Þ, the inherent scattering amplitude of the same atom, and FT ðkD Þ, the structure function for the liquid being studied. For example, in the case of water there are two atoms, the H atom and the O atom. In neutron diﬀraction, the scattering amplitude fi ðkD Þ is an inherent property of the atom and is tabulated for each atom on the basis of its nuclear composition. That means that it is diﬀerent for normal hydrogen, 1 H, and heavy hydrogen (deuterium), 2 H. Since ordinary water contains some heavy water, an ordinary water sample actually contains three diﬀerent atoms. In the above discussion of scattering from monoatomic liquids, fi ðkD Þ was referred to as bD ; the simpler notation was used because the scattering fraction is independent of kD in neutron experiments. The structural information from the scattering experiment is contained in the structure function FT ðkD Þ which is deﬁned as follows for both X-ray and neutron experiments: !2 XX X FT ðkD Þ ¼ ð2 dij Þci cj fi ðkD Þfj ðkD Þ½S ij ðkD Þ 1= ci fi ðkD Þ ð2:5:11Þ ij

j

i

where S ij ðkD Þ is the partial structure factor for atoms i and j, and dij is the Kronecker delta. In the case of a monoatomic ﬂuid such as liquid argon, there is only one kind of atom ði ¼ jÞ and the structure factor S Ar is easily extracted from the structure function. In the case of liquid water, ignoring minor isotopic components, there are three partial structure factors, namely, S OO ; S OH , and S HH . The individual structure factors must be separated in the structure function in order to obtain the corresponding pair correlation functions. This separation is usually achieved using isotopic substitution techniques as described below. It should be noted that the partial structure factors S OH and S HO are the same because of the nature of the experiment. It follows that there are six independent partial structure factors for a triatomic molecule such as HOD, namely, S OO ; S OD ; S OH ; S DD ; S DH , and S HH . If the individual partial structure factors S ij ðkD Þ can be determined, one can calculate the individual partial pair correlation function gij ðrÞ on the basis of a Fourier transformation: 1 ð V gij ðrÞ ¼ 1 þ ½S ij ðkD Þ 1kD sinðkD rÞ dkD ð2:5:12Þ 2p2 Nr 0

THE STRUCTURE OF LIQUIDS

69

However, if this is not possible, one is forced to Fourier transform the whole structure function FT ðkD Þ to obtain the overall correlation function GðrÞ, which is deﬁned as 1 ð V GðrÞ ¼ FT ðkD Þ sinðkD rÞ dkD ð2:5:13Þ 2p2 Nr 0

Because the fractions fi ðkD Þ and fj ðkD Þ depend on the nature of the atom, GðrÞ is not a linear combination of the individual gij ðrÞ functions, so that their separate determination is quite diﬃcult in general. As pointed out above, X-radiation is scattered by the electrons around each atom, and the scattering amplitude increases monotonically with the atomic number Z of the atom [G2]. The value of fi ðkD Þ for the hydrogen atom is so small that the contribution of partial structure factors involving this atom to FT ðkD Þ can be neglected. As a result, in an X-ray scattering experiment in pure water, the only signiﬁcant contribution to FT ðkD Þ is from S OO ðkD Þ. The overall correlation function GðrÞ obtained from these experiments shows a single sharp maximum at 285 pm [6, 7]. More detailed analysis of the data indicates that each water molecule is surrounded on the average by four nearest neighbors. In the case of neutron diﬀraction, the radiation is scattered by the atomic nuclei, not by the electrons. It turns out that nucleons such as 1 H and 2 H have very diﬀerent scattering amplitudes. This means that isotope eﬀects are very important in developing experimental strategies. Soper and Phillips [8] used data for the structure function obtained in mixtures of normal and heavy water to extract values of the partial structure factors for water. In this way they were able to determine all of the pair distribution functions for water from their diffraction data. These are gHH ðrÞ; gOH ðrÞ, and gOO ðrÞ. More details of their experimental results are given in section 2.10. X-ray and neutron diﬀraction studies have also been carried out in other liquids with hydrogen bonding. Results with three or more diﬀerent atoms become quite complex. A good example is methanol, which has three diﬀerent atoms, namely, C, O, and H. Moreover, there are two diﬀerent types of H atom, that on oxygen, which is involved in hydrogen bonding, and those in the methyl group, which are not. On this basis there are ten distinct contributions to the structure function, that is, S CC ; S CO ; S CH ; S CM ; S OO ; S OH ; S OM ; S HH ; S HM ; and S MM , where M designates a hydrogen atom in the methyl group, and H, the hydrogen atom of the hydroxyl group. In an X-ray experiment, only three of these are signiﬁcant, namely, S CC ; S CO , and S OO . Fourier transformation of the structure function FT ðkD Þ for methanol gives a correlation function GðrÞ with one sharp peak at 280 pm [9]. This corresponds to the distance between two oxygen atoms in adjacent methanol molecules. The peak is sharp because hydrogen bonding results in a strong correlation between oxygen atoms in liquid methanol. Further analysis of the data demonstrates that there are, on the average, two other molecules associated with the central one. This result is interesting because it demonstrates the importance of hydrogen bonding in the structure of protic solvents. Water, which has two sites for hydrogen bonding per molecule, has approximately four nearest neighbors associated by hydrogen bonding, whereas

70

LIQUIDS, SOLUTIONS, AND INTERFACES

methanol, which can form one hydrogen bond per molecule, has two. More recently, a complete description of the ten pair correlation functions in methanol has appeared [10]. The experimental data were obtained using neutron diﬀraction and involved the use of isotope techniques to separate the component structure factors. These were then Fourier transformed to give the individual pair correlation functions. Other polar liquids which have been studied using X-ray and neutron diﬀraction are acetonitrile [11, 12], formamide [13, 14], and dimethyl sulfoxide [15]. The analysis of X-ray data for molecules containing the methyl group is simpler because this group can be treated as a simple atom. On this basis, acetonitrile contains only three distinguishable atoms, namely, CH3, C, and N. As a result there are only six partial structure factors to be considered in analyzing the X-ray diﬀraction data. Studies of protic solvents such as formamide show that there are hydrogen-bonded chains in this liquid. On the other hand, aprotic solvents like acetonitrile and dimethyl sulfoxide show evidence of dimerization due to strong dipole–dipole interactions. Diﬀraction studies of liquid structure constitute an active area of contemporary research. When suitable isotopes are available the data may be resolved to give the individual structure factors and pair correlation functions. These studies help elucidate the role of intermolecular forces in determining liquid structure, and hence bulk properties.

2.6 The Direct Correlation Function and the Mean Spherical Approximation One of the important methods of developing a model for the potential energy of the liquid system is that based on integral equations. These include the Kirkwood integral equation, the Born–Green–Yvon equation, and the Ornstein–Zernike equation. The last approach leads to the deﬁnition of the direct correlation function cðr12 Þ. It is the approach which is most frequently used and is the one which is considered here. In earlier discussion, the total correlation function hðr12 Þ was deﬁned to be gðr12 Þ 1. This quantity is a measure of the total inﬂuence molecule 1 has on molecule 2 at a distance r12 . Ornstein and Zernike [16] proposed that this inﬂuence could be considered as composed of two parts, a direct part and an indirect part. The direct part, which measures the direct inﬂuence of molecule 1 on molecule 2, is given by cðr12 Þ. The indirect part is the inﬂuence propagated by molecule 1 on molecule 3, which then aﬀects molecule 2 either directly or indirectly through other molecules. The indirect eﬀect is weighted by the density and averaged over all positions of molecule 3. As a result one may write ð N hðr12 Þ ¼ cðr12 Þ þ cðr13 Þhðr23 Þdr3 ð2:6:1Þ V v

This equation is the Ornstein–Zernike (OZ) equation and gives the mathematical deﬁnition of cðr12 Þ with the indirect eﬀect being expressed as a convolution integral of h and c. By Fourier transformation, one obtains

THE STRUCTURE OF LIQUIDS

Nc~ðkD Þh~ðkD Þ h~ðkD Þ ¼ c~ðkD Þ þ V

71

ð2:6:2Þ

where c~ and h~ are the Fourier transforms of cðrÞ and hðrÞ, respectively. Using the deﬁnition of the structure factor, one may also write Nc~ðkD Þ SðkD Þ 1 ¼ V SðkD Þ

ð2:6:3Þ

It follows that the direct correlation function may be calculated from the experimental structure factor after Fourier transformation of the expression on the right-hand side of this equation. Three other important equations in the statistical thermodynamics of liquids involve the direct correlation function and provide a connection between the intermolecular potential uðrÞ for two molecules and the potential of mean force WðrÞ. One way of deriving these equations is by writing an approximate expression for the indirect contribution to the pair correlation function gðrÞ. Keeping in mind that WðrÞ is deﬁned as ln gðrÞ=b, and deﬁning gind ðrÞ as the contribution to gðrÞ from indirect interactions, an approximate expression for cðrÞ is cðrÞ ¼ gðrÞ gind ðrÞ ¼ gðrÞ expfb½WðrÞ uðrÞg

ð2:6:4Þ

The approximation consists of assuming that the potential of mean force WðrÞ less the direct interaction energy uðrÞ gives a correct estimate of the indirect eﬀect of the other molecules in the system. Equation (2.6.4) may also be written as cðrÞ ¼ gðrÞf1 exp½buðrÞg

ð2:6:5Þ

This result is known as the Percus–Yevick (PY) approximation [17]. It has been applied with considerable success to the evaluation of the properties of a liquid composed of hard spheres. In this case, the interaction energy uðrÞ is zero so that cðrÞ ¼ 0;

r>s

ð2:6:6Þ

Another approximation for cðrÞ is obtained by expanding the exponential in equation (2.6.4). Assuming that the ﬁrst two terms in the expansion suﬃce, one obtains cðrÞ ¼ gðrÞ 1 þ bWðrÞ buðrÞ ¼ hðrÞ ln gðrÞ buðrÞ

ð2:6:7Þ

This expression for cðrÞ is called the hypernetted chain (HNC) approximation [18]. On the basis of cluster diagrams [G4] it has been shown that the HNC equation provides a better description of most systems than the PY equation. It has been applied extensively in the development of the theory of liquids and electrolyte solutions. Its drawback is that the OZ equation can only be solved numerically using this approximation. Values of the direct correlation function estimated for the Lennard–Jones ﬂuid described earlier (ﬁg. 2.4) using the PY equation are shown in ﬁg. 2.7. It is apparent that cðrÞ rises to a maximum close to the ﬁrst maximum in the total correlation function hðrÞ and then rapidly dies away, reaching a value of zero close to r ¼ 2:5s. When cðrÞ is estimated by the HNC equation, qualitatively similar results are obtained but the exact values of cðrÞ diﬀer by a few percent.

72

LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 2.7 Total correlation function hðrÞ () and the direct correlation function cðrÞ () for a liquid obeying the Lennard–Jones potential with the same parameters as in ﬁg. 2.4.

Since cðrÞ is short ranged, an even simpler approximation can often be applied to estimate it in the region outside of a hard sphere. The logarithmic term in equation (2.6.7) can be written as ln[1 þ hðrÞ]. Expanding this using only the ﬁrst two terms, the expression for cðrÞ becomes cðrÞ ¼ buðrÞ;

r>s

ð2:6:8Þ

This approximation is known as the mean spherical approximation (MSA). For the case of a hard-sphere ﬂuid for which uðrÞ ¼ 0, the MSA is equivalent to the PY approximation. For the case that the hard spheres have embedded point charges, the function uðrÞ is simply Coulomb’s law. Although the MSA provides the least detailed expression for cðrÞ, it is popular because the OZ equation can often be solved using this approximation to yield an analytical expression for gðrÞ. The equation for gðrÞ within a hard sphere is gðrÞ ¼ 0;

r<s

ð2:6:9Þ

Equations (2.6.8) and (2.6.9) give what is known in statistical mechanics as the closure conditions. To solve a particular problem given uðrÞ, one must ﬁrst determine cðrÞ for r < s. After Fourier transformation, one can calculate the total correlation function in Fourier space, namely, h~ðkÞ, using equation (2.6.2). Inverse Fourier transformation then gives hðrÞ and gðrÞ. Wertheim [19] showed that cðrÞ within a non-interacting hard sphere on the basis of the MSA is given by cðrÞ ¼ l1 where

62 zr zl1 r3 ; s 2s3

r<s

ð2:6:10Þ

THE STRUCTURE OF LIQUIDS

73

Fig. 2.8 The pair correlation function gðrÞ for a ﬂuid composed of hard spheres at a packing fraction of z ¼ 0:49 calculated as a function of distance, r, using the Ornstein– Zernike equation with the direct correlation function given by equations (2.6.6) and (2.6.10). The data shown as () are from a Monte Carlo calculation.

l1 ¼

ð1 þ 2zÞ2 ð1 zÞ4

ð2:6:11Þ

ð1 þ z=2Þ2 ð1 zÞ4

ð2:6:12Þ

and l2 ¼

The parameter z is the packing fraction given by z¼

Nps3 6V

ð2:6:13Þ

The radial distribution function is now obtained using the OZ equation. The resulting expressions for gðrÞ are rather lengthy and are not written out here. However, it is clear that the calculated gðrÞ is very close to the value that one obtains on the basis of a computer simulation (ﬁg. 2.8). A brief description of computer calculations is given in the following section. As will be seen in later chapters, the integral equation approach has been applied to other important problems relating to liquids and solutions. The MSA used to deﬁne cðrÞ in the region outside of a given sphere has proven to be especially useful because of its simplicity.

2.7 Computer Simulations of Simple Liquids Two types of reference ﬂuids have been studied extensively by computer experiments. One system is composed of the non-interacting hard spheres discussed

74

LIQUIDS, SOLUTIONS, AND INTERFACES

above and is often referred to as Percus–Yevick (PY) ﬂuid. The other system is made up of spheres which interact by van der Waals forces according to a Lennard–Jones potential. Such a ﬂuid corresponds to a good approximation to a noble gas in the liquid state, for example, liquid argon. In a computer experiment, the ﬂuid is imagined to consist of a cell with a known number of particles N and volume V. The value of N is usually chosen to be between 100 and 1000, its exact magnitude depending on the speed and memory capabilities of the computer. The real liquid is imagined to be made up of many such cells, all being identical (see ﬁg. 2.9). Surface eﬀects in the experiment are minimized by assuming that, if a particle leaves the cell on one side, it re-enters from an image cell on the opposite side. In this way, surface eﬀects are minimized and calculations based on a relatively small number of particles are capable of estimating accurately the properties of a macroscopic system containing the order of 1023 molecules. The two simulation techniques commonly used are the Monte Carlo method [20, 21] and the method of molecular dynamics [22, 23]. In Monte Carlo experiments, an initial conﬁguration for the N particles is chosen and the corresponding potential energy Uðr1 ; r2 ; . . . ; rN Þ is calculated. This conﬁguration, chosen by the programmer, is usually something simple, for instance, a face-centered cubic arrangement of the particles. Then, the position of one particle is changed at random, generating a new conﬁguration. The potential energy U is calculated and compared with that of the previous conﬁguration. If it is smaller, the new conﬁguration is accepted as a contribution to the estimation of the conﬁgurational partition function for the ensemble, Z. However, if U is greater than that for the previous conﬁguration, a choice is made on the basis of the increment in the potential energy, U. The fraction expðbUÞ is calculated and compared with a random number r in the interval 0 to 1 generated by the computer. If the fraction is greater than r, the new conﬁguration is accepted and added to the chain of conﬁgurations describing the system. If the fraction is less than the random number r, the new conﬁguration is rejected and the previous conﬁguration is added again into the chain of conﬁgurations describing the system. This process is repeated many times, thereby generating a chain of conﬁgurations. The overall properties of the system then converge to those of a canonical ensemble for which

Fig. 2.9 Schematic diagram illustrating the periodic boundary conditions used in computer simulations. The lines represent cell boundaries. The positions of the particles in each cell, assumed to be ﬁve in this example, are identical so that surrounding cells may be considered images of a central one. Each particle is given a diﬀerent symbol to clarify the imaging eﬀects.

THE STRUCTURE OF LIQUIDS

75

V, N, and T are held constant. In this way, the conﬁgurational integral Z is calculated. This immediately allows one to calculate the partition function Q for the ensemble (equation (2.2.31)) and thus the thermodynamic functions for the system. In addition, the pair correlation function for the ﬂuid is obtained in this calculation. The acceptance or rejection of conﬁgurations with high potential energy is a special feature of modern Monte Carlo techniques [20, 21]. In the case of hardsphere systems, this choice is particularly simple because U is equal to zero when spheres do not overlap, and inﬁnity when they do. Details about application of this technique to more complex systems such as molecular ﬂuids involve some variations which are described in the original literature. In a molecular dynamics calculation, both the position and velocity of the particles in the cell are speciﬁed. The movement of the particles is governed by Newton’s laws of motion, and the initial velocities are chosen from a Maxwell distribution. This computer experiment is particularly easy for non-interacting hard spheres because the molecules move in straight lines and undergo elastic collisions. The velocity after the collision is found through the laws of conservation of momentum and energy. Since the total energy of the system is constant, it corresponds to a microcanonical ensemble with constant U, V, and N. The temperature is found from the time average of the molecular kinetic energy. Since velocity is included in the description of the system both equilibrium and transport properties can be obtained from this type of computer calculation. When these calculations are performed for systems with continuous functions for the potential energy U, ﬁnite diﬀerence techniques are used. This type of calculation is more complex and takes more computer time. In the end one obtains the same information as in the Monte Carlo experiment, namely, the pair correlation function and the thermodynamic properties of the ﬂuid. The additional information from molecular dynamics allows one to calculate velocity correlation functions, and ﬂow properties such as viscosity and diﬀusion coeﬃcients. The results of computer experiments are often used to test approximate theories of liquids such as those based on the OZ equation discussed above. For this reason they are an important part of understanding the structure of liquids and how structure aﬀects liquid properties.

2.8 Estimation of Thermodynamic Properties from the Pair Correlation Function In section 2.2 it was shown that the thermodynamic properties of a system may be estimated from its partition function, Q. In a liquid, estimation of Q requires knowledge of the conﬁgurational partition function, Z. In the following section the relationship between Z and the pair correlation function gðrÞ is examined. In principle, gðrÞ may be determined from experiment. Therefore, it is important to understand the connection between gðrÞ and the thermodynamic properties of the system. The starting point in the derivation is the partition function Q (equation (2.2.31)), which is written in terms of logarithms:

76

LIQUIDS, SOLUTIONS, AND INTERFACES

ln Q ¼ N ln qint þ ln Z ln N! 3N ln l

ð2:8:1Þ

Now the internal energy U can be found by taking the derivative with respect to b, @ ln Q @ ln qint @ ln Z @ ln l U¼ ¼ N þ3N ð2:8:2Þ @b V;N @b V;N @b V;N @b V;N On the basis of the deﬁnition of l (equation (2.2.22)) and qint , this may be rewritten as 3NkB T @ ln Z þ Neint ð2:8:3Þ U¼ 2 @b V;N where the ﬁrst term on the right-hand side gives the average kinetic energy of the system, the second, the contribution from the internal energy of the molecules, and the last, the average potential energy of the system due to intermolecular interactions. Diﬀerentiating equation (2.2.32) with respect to b, the following expression is obtained for the average potential energy: ð ð @ ln Z 1 ¼ . . . U expðbUÞdr1 dr2 . . . drN hUi ¼ ð2:8:4Þ @b Z The multiple integral in this equation expresses the fact that the averaging process is carried out by multiplying the potential energy of each conﬁguration U by its probability given by the factor expðbUÞ=Z. In order to proceed further, the important assumption is made that U is made up of the sum of potentials between pairs of molecules: X NðN 1Þ uðr12 Þ U¼ uðrij Þ ¼ ð2:8:5Þ 2 1 i j N where uðr12 Þ is the pair potential energy. After substitution of equation (2.8.5) into equation (2.8.4), one obtains ð ð NðN 1Þ hUi ¼ . . . uðr12 Þ expðbUÞdr1 dr2 . . . drN 2Z ð

ðð ð ð2:8:6Þ NðN 1Þ ¼ uðr12 Þ . . . expðbUÞ dr3 . . . drN dr1 dr2 2Z Comparing the inner integral with the deﬁnition of the pair distribution function given by equations (2.4.4) and (2.4.7), one may now write ðð 1 hUi ¼ uðr12 Þf ð2Þ ðr1 r2 Þdr1 dr2 2 ð2:8:7Þ ðð N2 ¼ uðr12 Þgðr12 Þdr1 dr2 2V 2 Integrating over the coordinates of particle 1, the following result is obtained for the average potential energy: N2 hUi ¼ 2V

1 ð

uðrÞgðrÞ4pr2 dr 0

ð2:8:8Þ

THE STRUCTURE OF LIQUIDS

77

The integral in this equation gives the total intermolecular potential energy between a central molecule and other molecules in the ﬂuid located between r and r þ dr. Thus, evaluation of hUi involves adding up contributions for all values of r and multiplying by the factor N=2, since any one of the N molecules can be considered as central. The factor of two arises so that each pair interaction is not counted twice. The ﬁnal expression for the internal energy is 3NkB T N2 þ Neint þ U¼ 2 2V

1 ð

uðrÞgðrÞ4pr2 dr

ð2:8:9Þ

0

It should be remembered that the approximation involved in equation (2.8.5) may not always be valid. When three-body interactions are important they must be included in the estimation of U. Under most circumstances higher-order interactions account for approximately 5% of the potential energy so that equation (2.8.5) represents a reasonable approximation. The next quantity calculated from the radial distribution function is the pressure. On the basis of equations (2.2.6) and (2.2.31), one may write 1 @ ln Z ð2:8:10Þ P¼ b @V T;N where it is assumed that the partition function due to internal degrees of freedom is independent of volume. Derivation of the relationship between the volume derivative of ln Z and the radial distribution function gðrÞ is presented elsewhere [G3, G4] and is not reproduced here. The result expressed as the equation of state is PV N ¼1 NkB T 6VkB T

1 ð

gðrÞ

duðrÞ 4pr3 dr dr

ð2:8:11Þ

0

Estimation of the pressure requires that the derivative of the pairwise interaction energy with respect to distance be known. This is certainly not a problem for simple systems. Equation (2.8.11) was also derived on the basis of the assumption that the potential energy U can be written as a sum of two-body interactions. To this extent the result is approximate. In order to complete the thermodynamic description, the Helmholtz energy A is now calculated. On the basis of equation (1.3.15), one may write

so that

A U ¼ S T T

ð2:8:12Þ

A dU 1 þ Ud d ¼ dS T T T

ð2:8:13Þ

dU P dV ¼ dS T T

ð2:8:14Þ

Since

78

LIQUIDS, SOLUTIONS, AND INTERFACES

it follows that

@ðA=TÞ ¼U @ð1=TÞ V

ð2:8:15Þ

Thus, integration of equation (2.8.9) with respect to 1=T yields the Helmholtz energy A. This in turn requires that the radial distribution function gðrÞ be known as a function of temperature. Since gðrÞ is generally not available as a function of temperature, equation (2.8.15) is not a convenient route to obtain A. On the basis of equation (1.3.19), one obtains the relationship @A ¼ P ð2:8:16Þ @V T Thus, another route to estimating A is by integration of the pressure equation (equation (2.8.11)) with respect to volume. This calculation requires that the dependence of gðrÞ on volume or density be known. Since this is usually not known, this method of estimating A is also not convenient in most cases. Another method used to estimate the Helmholtz energy in a ﬂuid with intermolecular forces is to introduce a coupling parameter, x. This quantity, which varies between 0 and 1, measures the extent to which a central reference molecule is coupled to all other molecules in the system. Thus, when x ¼ 0, one imagines that the reference molecule experiences no intermolecular interactions, whereas all other molecules interact with each other in the normal fashion. In the case of van der Waals interactions, this is equivalent to setting the central molecule’s dipole moment equal to zero at all moments of time. As the parameter x increases from 0 the dipolar characteristics of the central molecule are gradually ‘‘turned on’’ until when x ¼ 1, the system returns to its normal state. In order to evaluate the eﬀects of this imaginary process, one needs to evaluate the pair correlation function for diﬀerent values of x. This quantity is designated gðr; xÞ where gðr; 1Þ is equal to gðrÞ. Of course, the value of the pair correlation function during this imaginary process cannot be determined experimentally, but it can be estimated using various theoretical models, for instance, the MSA model discussed earlier. When the number of molecules is very large, the chemical potential m can be calculated from the increase in Helmholtz energy obtained by adding molecules, one by one, to the system. Thus, @A ¼ AðN; V; TÞ AðN 1; V; TÞ ð2:8:17Þ m¼ @N V;T From equations (2.2.13) and (2.2.31), one may write AðN; V; TÞ ¼ NkB T ln qint kB T ln ZN þ kB T ln N! þ 3NkB T ln

ð2:8:18Þ

where ZN is the conﬁguration integral for the system with N molecules. It follows that ZN m ¼ kB T ln qint kB T ln ð2:8:19Þ þ kB T ln N þ 3kB T ln l ZN1 On the basis of the deﬁnition of the conﬁguration integral (equation (2.2.32)) it can be shown that

THE STRUCTURE OF LIQUIDS

ð1 ZN @ ln ZN dx ln ¼ ln V þ ZN1 @x

79

ð2:8:20Þ

0

Now making use of the relationship between the average potential energy hUi and the pair correlation function (equation 2.8.8), one obtains

ð ð1 1 ZN N ln uðrÞgðr; xÞ4pr2 drdx ¼ ln V ZN1 VkB T

ð2:8:21Þ

0 0

The resulting expression for the chemical potential is ! ð ð1 1 Nl3 N uðrÞgðr; xÞ4pr2 drdx m ¼ kB T ln þ Vqint VkB T

ð2:8:22Þ

0 0

If one has a model for gðr; x) one may estimate the chemical potential m, and hence, all of the other thermodynamic functions. The above analysis demonstrates the importance of the pair correlation function in estimation of the thermodynamic properties of simple liquids. In the following section, the properties of the simplest ﬂuid, namely, one based on non-interacting hard spheres, are developed on the basis of the relationships presented in this section.

2.9 The Properties of a Hard-Sphere Fluid It is useful to examine the properties of a ﬂuid made up of non-interacting hard spheres. Such a system may be regarded as an important reference liquid, albeit ﬁctitious, with respect to which the properties of real systems can be compared. Its properties are most easily obtained on the basis of the Percus–Yevick (PY) approximation. Since the spheres do not interact, the interaction energy uðrÞ is zero outside any sphere: uðrÞ ¼ 0;

r>s

ð2:9:1Þ

Since the spheres are hard, that is, they do not penetrate one another, uðrÞ ¼ 1;

r<s

ð2:9:2Þ

It is apparent that the interaction energy is discontinuous at the boundary of a sphere. From the discussion in section 2.6, discontinuities are also possessed by the pair correlation function gðrÞ and the direct correlation function cðrÞ at the boundary of a sphere. In order to remove problems associated with these discontinuities, one introduces a new function, yðrÞ deﬁned as follows: yðrÞ ¼ gðrÞ expðbuðrÞÞ

ð2:9:3Þ

Outside of a sphere where uðrÞ is zero, one ﬁnds that yðrÞ ¼ gðrÞ;

rs

ð2:9:4Þ

80

LIQUIDS, SOLUTIONS, AND INTERFACES

Referring to the PY approximation (equation (2.6.5)), inside a sphere where gðrÞ is zero, yðrÞ ¼ cðrÞ;

r s

ð2:9:5Þ

Although cðrÞ and gðrÞ are discontinuous at r ¼ s; yðrÞ is continuous. The ﬁrst step in obtaining the properties of the hard-sphere system is to calculate its equation of state on the basis of equation (2.8.11). From equations (2.9.1) and (2.9.2), the derivative duðrÞ=dr is zero for all values of r except r ¼ s where it is a negative-going delta function. This follows from the fact that uðrÞ drops from 1 to 0 at the boundary of a hard sphere. It can be shown that the integral in the equation of state (equation (2.8.11)) simpliﬁes to the value of the integral at r ¼ s multiplied by kB T so that one obtains PV 2pNs3 ¼1þ gðsÞ ¼ 1 þ 4zyðsÞ NkB T 3V

ð2:9:6Þ

where the packing fraction z is introduced to simplify the result (see equation (2.6.13)). On the basis of equations (2.9.4) and (2.9.5), one may write gðsþÞ ¼ yðsÞ ¼ cðsÞ

ð2:9:7Þ

where sþ and s indicate that the function is evaluated at the positive and negative side of the discontinuity at s, respectively. Thus, gðsÞ may be calculated on the basis of equations (2.6.10)–(2.6.12), giving the result that gðsÞ ¼

1 þ z=2 ð1 zÞ2

ð2:9:8Þ

Substituting equation (2.9.8) into equation (2.9.6) one obtains the following equation of state: PV 1 þ 2z þ 3z2 ¼ NkB T ð1 zÞ3

ð2:9:9Þ

In the limit of very low densities, the parameter z goes to zero and the ratio PV=ðNkB TÞ approaches one. This is the expected ideal gas limit. At higher densities, the ratio PV=ðNkB TÞ is greater than unity and goes to inﬁnity when z ¼ 1. In fact, for a real system, the value pﬃﬃﬃ of 3z is limited by the density of hard spheres in a close-packed system, which is 2=s . The corresponding value of the parameter pﬃﬃﬃ z is 2p=6, or 0.74, and that of PV=ðNkB TÞ, 61. The equation of state may also be calculated from an equation relating the compressibility of the system to the pair correlation function gðrÞ. The result is PV 1 þ z þ z2 ¼ NkB T ð1 zÞ3

ð2:9:10Þ

For low densities, this equation gives the same value of PV=ðNkB TÞ as that estimated from the pressure equation (equation (2.9.9)). However, at high densities, the estimate from the compressibility equation is much higher (see ﬁg. 2.10). The disagreement between the equations of state obtained by the two diﬀerent methods clearly is a result of the approximation made in deriving the PY equation.

THE STRUCTURE OF LIQUIDS

81

Values of PV=ðNkB TÞ have been calculated for the hard-sphere system by computer simulation. At high densities, the results obtained in this way fall between those estimated by the pressure and compressibility equations. This led Carnahan and Starling [24] to propose an equation of state obtained by combining one-third of the result from the pressure equation (equation (2.9.9)) together with two-thirds of the compressibility result (equation (2.9.10)). The result is PV 1 þ z þ z2 z3 ¼ NkB T ð1 zÞ3

ð2:9:11Þ

The Carnahan–Starling equation of state agrees well with the result of computer simulations over the range shown in ﬁg. 2.10 and is used in all further calculations presented here. EXAMPLE

Estimate the value of PV=ðNkB TÞ for a hard-sphere ﬂuid using the Carnahan and Starling equation of state assuming a concentration of 10 M and a hardsphere diameter of 300 pm. The concentration 10 M corresponds to 10 6:023 1023 ¼ 6:023 1027 molec m3 103

ð2:9:12Þ

The packing fraction z is equal to z¼

3:1416 6:023 1027 ð3 1010 Þ3 ¼ 8:514 102 6

ð2:9:13Þ

Thus, PV 1 þ 8:514102 þ ð8:514102 Þ2 þ ð8:514 102 Þ3 ¼ ¼ 1:427 NkB T ð1 8:514102 Þ3

ð2:9:14Þ

At this concentration the hard-sphere ﬂuid has properties which are signiﬁcantly diﬀerent from those of an ideal ﬂuid (PV=ðNkB TÞ ¼ 1).

Fig. 2.10 Plot of PV=ðNkB TÞ for a hard sphere ﬂuid against the ratio of its density N=V to the pﬃﬃﬃ closepacked density ð 2=s3 Þ. The data designated () were obtained using the pressure equation (2.9.9), those designated (^) using the compressibility equation (2.9.10), and the smooth curve, using the Carnahan– Starling equation (2.9.11).

82

LIQUIDS, SOLUTIONS, AND INTERFACES

The Helmholtz energy of the system is now calculated using equations (2.8.16) and (2.9.11). Accordingly, ð ð A P 1 þ z þ z2 ¼ z3 dV ð2:9:15Þ ¼ dV ¼ NkB T NkB T Vð1 zÞ3 Recalling that z¼

pNs3 6V

ð2:9:16Þ

it follows that dV ¼

pNs3 dz 6z2

ð2:9:17Þ

so that the integral in equation (2.9.15) can be written in terms of z. The result is ð A 1 þ z þ z2 z3 ¼ dz ð2:9:18Þ NkB T zð1 zÞ3 This integral can be broken up into four integrals, all of which can be found in standard tables. The integration is carried out from a ﬁnite but very low density where the parameter z is z0 , and the properties of the system are those of an ideal gas, to any higher value of z. The result is A A0 z 3 þ 2z 3z2 3 ð2:9:19Þ ¼ þ ln þ NkB T NkB T z0 2 2ð1 zÞ2 where A0 is the Helmholtz energy of an ideal gas. On the basis of equations (2.2.13) and (2.2.23) this contribution is A0 ¼ ln V0 þ ln N 1 þ 3 ln l NkB T

ð2:9:20Þ

where V0 is the volume of the ideal gas. Noting that z V ¼ 0 V z0 the expression for the Helmholtz energy becomes A N 3 þ 2z 3z2 3 þ ¼ 3 ln l þ ln NkB T V 2 2ð1 zÞ2

ð2:9:21Þ

ð2:9:22Þ

One may now derive the remaining thermodynamic functions. First of all, the internal energy is obtained by diﬀerentiation of A=T with respect to the reciprocal temperature:

@ðA=TÞ @ ln l 3NkB T U¼ ¼ ð2:9:23Þ ¼ 3NkB @ð1=TÞ V @ð1=TÞ 2

THE STRUCTURE OF LIQUIDS

83

The internal energy is simply the kinetic energy of the system and therefore does not diﬀer from the result for an ideal gas containing the same number of molecules. One may now write an expression for the entropy of the system: S UA N 3 þ 2z 3z2 ¼ 3 ln l ln ¼ þ3 ð2:9:24Þ NkB NkB T V 2ð1 zÞ2 Finally, the Gibbs energy is obtained by adding PV to the Helmholtz energy: G A þ PV N 3 þ 2z 3z2 1 þ z þ z2 z3 3 ¼ ¼ 3 ln l þ ln þ NkB T NkB T V 2 2ð1 zÞ2 ð1 zÞ3 ð2:9:25Þ For an ideal monoatomic gas, the Gibbs energy at constant temperature and pressure varies with the logarithm of the molecular density. The additional terms in z in equation (2.9.25) give the contribution to G due to the non-ideality of the hard-sphere system with respect to an ideal gas. EXAMPLE

Estimate the internal energy, Helmholtz energy, and entropy of a hard-sphere ﬂuid in units of NkB T assuming a concentration of 10 M, a diameter of 300 pm, a molecular mass of 30 g, and a temperature of 258C. The concentration of hard spheres is 6:023 1027 m3: The mass of each sphere is 30 103 ¼ 4:98 1026 kg 6:022 1023 The parameter l (equation (2.2.22)) is l¼

6:626 1034 ð2 3:1416 4:98 1026 1:38 1023 298:2Þ1=2

¼ 1:847 1011 m

ð2:9:26Þ

3 ln l ¼ 74:14 N ln ¼ 63:97 V

ð2:9:27Þ ð2:9:28Þ

From the previous example, z is equal to 8:514 102 : Thus, 3 þ 2z 3z2 ¼ 7:52 2ð1 zÞ2

ð2:9:29Þ

A ¼ 74:14 þ 63:97 þ 7:52 1:5 ¼ 4:15 NkB T

ð2:9:30Þ

and

The internal energy in units of NkB T is 1.5. The entropy is given by

84

LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 2.11 Plots of the Helmholtz energy for a hard-sphere ﬂuid (s ¼ 0:37 nm) against the logarithm of the pressure at constant temperature as indicated.

S UA ¼ 1:5 4:15 ¼ 2:65 ¼ NkB NkB T

ð2:9:31Þ

Values of the Helmholtz energy estimated as a function of pressure at constant temperature on the basis of equation (2.9.22) are shown in ﬁg. 2.11. These plots are reasonably linear in the logarithm of the pressure at low pressures. This is to be expected, since the density is proportional to the pressure under these conditions, and the eﬀects of non-ideality are relatively unimportant. However, at higher pressures the value of A, starts to rise sharply due to non-ideality. Eventually, one reaches positive values of A, indicating that the ﬂuid is not stable. It has been shown that the hard-sphere system undergoes a phase transition from ﬂuid to solid when Ns3 =V ¼ 0:943. For the system considered in ﬁg. 2.11, this corresponds to a pressure of 760 bars at 300 K. Many interesting calculations relating to phase transitions and critical phenomena in hard-sphere and Lennard–Jones ﬂuids have been carried out. Just as the ideal gas forms a convenient point of reference in discussing the properties of real gases, so does the hard-sphere ﬂuid in discussing the properties of liquids. This is especially true at low densities, where the role of intermolecular forces in real systems is not so important. In this limit, the hard-sphere model is useful in developing the theory of solutions, as will be seen in chapter 3.

2.10 The Structure of Water The discussion in this chapter has largely concerned very simple liquids such as a hypothetical ﬂuid composed of non-interacting hard spheres, or spheres interacting via the Lennard–Jones potential function. The most common liquid, namely, water, is much more complex. First, it is a molecule with three atoms, and has a

THE STRUCTURE OF LIQUIDS

85

large permanent dipole moment. Second, it does not possess a center of symmetry, unlike methane, or carbon tetrachloride. Finally, it is more complex than other polar solvents because of the strong hydrogen bonding. In spite of its complexity, the structure of liquid water has been determined with precision using neutron diﬀraction techniques [8]. The structure factor may be analyzed to obtain correlation functions for each type of atom–atom interaction in the system. For water, the three atom–atom distribution functions are gOO ðrÞ; measuring the distribution of oxygen–oxygen separations, gOH ðrÞ; measuring the oxygen–hydrogen distribution, and gHH ðrÞ, measuring the hydrogen– hydrogen distribution. Since the scattering amplitude for oxygen is much smaller than that for hydrogen, the gOO ðrÞ distribution is the most diﬃcult to determine. The experimental results are illustrated in ﬁg. 2.12. This function rises from a value of zero at 200 pm to a sharp maximum of 3.09 at a distance of 288 pm. This means that the local density of oxygen atoms is 3.09 times greater than the average value. However, in order to estimate of the oxygen–oxygen coordination number one must consider the distribution density over the whole region of the maximum in gOO ðrÞ. One way of achieving this is to integrate the distribution function up to the ﬁrst minimum, considering shells of volume 4pr2 dr. The coordination number is then deﬁned as rmin ð

ro gOO ðrÞ 4pr 2 dr

nc ¼

ð2:10:1Þ

0

where ro is the density of oxygen atoms in the liquid. When this integration is carried out to a distance of 330 pm from the central oxygen atom, the result is that nc ¼ 4:3. This suggests that the central water molecule is approximately in a tetrahedral environment surrounded on the average by four nearest neighbors.

Fig. 2.12 Oxygen–oxygen distribution function in liquid water determined at 258C by neutron diﬀraction [8].

86

LIQUIDS, SOLUTIONS, AND INTERFACES

Beyond the ﬁrst minimum, gOO ðrÞ oscillates with a second maximum at 450 pm. In order to understand these results in more detail, one needs to carry out a molecular dynamics calculation. However, it is clear that the behavior of water is roughly similar to that of much more simple liquids if one considers the oxygen atom alone. A much diﬀerent pattern is seen for the oxygen–hydrogen distribution function (ﬁg. 2.13). These results have been corrected for the intramolecular contribution to gOH ðrÞ, and show only the eﬀects of the intermolecular correlation. A fairly sharp maximum occurs at 185 pm, and a broader but higher one at 330 pm. The ﬁrst maximum is clearly associated with the hydrogen atoms in surrounding molecules which are hydrogen bonded to the oxygen atom in a central molecule. Integration out to the ﬁrst minimum gives a coordination number of 1.7. The second maximum occurs at the ﬁrst minimum on the gOO ðrÞ distribution curve. At this point, the total number of hydrogen atoms around the central oxygen is close to eight, that is, two times the number of oxygen atoms in the same volume of solution. These observations all follow what is expected on the basis of the molecular composition of water. The distribution function for hydrogen atoms with respect to a central hydrogen is shown in ﬁg. 2.14. The function gHH is the easiest to determine experimentally because of the large scattering amplitude for hydrogen with respect to that for oxygen. These results have also been corrected for the intramolecular contribution to gHH ðrÞ. The ﬁrst maximum at 240 pm is followed by one which is less high at 390 pm. The coordination number estimated by integrating out to the ﬁrst minimum at 310 pm is 5.8. These hydrogen atoms are associated with a central molecule via hydrogen bonding. Comparing gOH ðrÞ and gHH ðrÞ, one sees that the maxima on the H–H distribution function are located further from the central atom by 60 pm with respect to those on the O–H distribution. This is perfectly reasonable because of the O–H bond length in water. Another method of studying water structure is Raman spectroscopy [25]. Using this technique, one is able to distinguish spectral features which arise from intra-

Fig. 2.13 Oxygen–hydrogen distribution function in liquid water determined at 258C by neutron diﬀraction [8].

THE STRUCTURE OF LIQUIDS

87

Fig. 2.14 Hydrogen–hydrogen distribution function in liquid water determined at 258C by neutron diﬀraction [8].

molecular and intermolecular vibrational modes of the water molecules. Interpretation of the data is assisted by comparing spectra obtained in H2O and D2O, and by changing the temperature. A picture which is consistent with both the Raman spectra and the neutron diﬀraction data shows considerable order around a given water molecule. Because of hydrogen bonding, one molecule is surrounded on the average by four hydrogen-bonded nearest neighbors (see ﬁg. 2.15). The orientation of these neighbors can be such that the system has a high degree of symmetry. However, keeping in mind the dynamic character of liquid structure, rotation of the molecules hydrogen bonded to the central molecule can reduce the symmetry. The dynamic character of the system also results in a very

Fig. 2.15 Diagram of a water molecule surrounded by four nearest neighbors whose orientation is determined by hydrogen bonding. Hydrogen bonds are denoted by broken lines and O–H bonds by solid lines. (Reproduced from reference 25, with permission.)

88

LIQUIDS, SOLUTIONS, AND INTERFACES

weak correlation between the central molecule and those located further away than the four nearest neighbors. The vibrational spectra of water are discussed in more detail in section 5.7A. It is clear that the physical properties of water are very much inﬂuenced by the important role played by hydrogen bonding in determining its structure. These properties include the high dielectric permittivity, which cannot be explained on the basis of dipole–dipole interactions alone. It is also clear that electrolytes have a very disruptive eﬀect on water structure. Cations are solvated by the lone electron pairs on the oxygen atom of the water molecule and thus cause considerable disruption in the local water structure. This leads to changes in the bulk physical properties of water, such as its permittivity. Many aspects of the structure and properties of aqueous solutions can be understood in terms of the qualitative structural changes that occur because of solute–solvent interactions. This subject is discussed in more detail in the following chapters.

2.11 Distribution Functions for Liquid Solutions The techniques used to describe the properties of pure liquids can be extended in a fairly straightforward fashion to liquid solutions [26, 27]. This treatment is normally restricted to liquids in which the molecules behave as non-interacting hard spheres or as dipolar species interacting via a Lennard–Jones potential. The discussion here is limited to two-component mixtures but it is easily extended to more complex systems. The energy equation for a liquid mixture or solution is 3NkB T N2 X þ 2p xx U¼ 2 V ij i j

1 ð

uij ðrÞgij ðrÞr2 dr

ð2:11:1Þ

0

where x1 is the mole fraction of component 1, x2 , that of component 2, and N ¼ N1 þ N2 . The summation involves three terms and requires speciﬁcation of the interaction energies u11 ; u12 and u22 as well as the pair correlation functions g11 ; g12 and g22 . In equation (2.11.1), the contribution to U from the internal degrees of freedom of the two molecules in the solution has not been included. Several approaches to estimation of the internal energy have been described. These involve assuming speciﬁc relationships between the pair correlation functions gij and the form of the interaction energy uij . The simplest of these is based on the van der Waals treatment of ﬂuids and its application of the law of corresponding states. Examination of typical radial distribution functions for mixtures such as those shown in ﬁg. 2.16 reveals that the maximum in each distribution function gðrÞ occurs close to the diameter s describing the distance of closest approach for the two molecules involved. Thus, it is better to describe the radial distribution function in terms of the reduced distance r=s instead of the distance r. This conclusion leads to the assumption that

THE STRUCTURE OF LIQUIDS

89

Fig. 2.16 Typical radial distribution functions of a two-component mixture plotted against the reduced distance for the 2–2 distribution. (From reference 26, with permission.)

g11

r r r ¼ g22 ¼ g12 s11 s22 s12

ð2:11:2Þ

When the interaction energy is given by the Lennard–Jones potential, one may write uij ðrÞ ¼ 4eij

h s n s m i ij ij r r

ð2:11:3Þ

where n is a coeﬃcient describing repulsive interactions and m, one describing attractive interactions. As discussed earlier, n is often assumed to be 12, and m is 6 for dipole–dipole interactions (see equation (2.3.35)). This expression includes the case of a hard-sphere ﬂuid when n ¼ 1 and the term in m is neglected. As far as liquids are concerned the more important contribution comes from the repulsive component which predominates for molecules in close contact. The Lennard– Jones function is conformal because it has the same form independent of the molecule, that is, of the values of eij and sij . Changing the variable in equation (2.11.1) from r to y ¼ r=sij and substituting the expression for uij ðrÞ one obtains 3NkB T N2 X þ 8p U¼ xxe s 3 2 V ij i j ij ij If one deﬁnes

1 ð

1 1 g ðyÞy2 dy yn ym ij

0

ð2:11:4Þ

90

LIQUIDS, SOLUTIONS, AND INTERFACES

s3mix ¼

X

xi xj sij 3

ð2:11:5Þ

ij

and emix ¼

X

xi xj eij sij 3 =

X

ij

xi xj sij 3

ð2:11:6Þ

ij

the energy equation simpliﬁes to 3NkB T N2 þ 8p e s3 U¼ 2 V mix mix

1 ð

1 1 gðyÞy2 dy yn y m

ð2:11:7Þ

0

where gðyÞ is any one of the distribution functions deﬁned in equation (2.11.2). This equation shows that the composition dependence of the internal energy U is speciﬁed only through the quantities emix and smix For this reason, the properties of the system at a given composition are those of a hypothetical pure ﬂuid with properties deﬁned by emix and smix . Thus, the model is called a one-ﬂuid model. In the case of mixtures of simple molecules, the Gibbs energy for a one-ﬂuid system may be written as G ¼ emix ðkB T=emix Þ 3NkB T ln smix

ð2:11:8Þ

where the function ðkB T=emix Þ is found by curve-ﬁtting the experimentally observed data for G for the pure liquids as a function of temperature. The other thermodynamic functions are determined by diﬀerentiation. The van der Waals one-ﬂuid theory is quite successful in predicting the properties of mixtures of simple molecules. Unfortunately, the systems usually considered by chemists are considerably more complex, and often involve hydrogen bonding and other chemical interactions. Nevertheless, the material presented here outlines how one could proceed to develop models for more complex systems on the basis of the integral equation approach.

2.12 Concluding Remarks The fundamental quantity used to describe liquid structure is the pair correlation function gðrÞ. It has been estimated both experimentally and on the basis of computer calculations for a great variety of liquid systems. Simple monoatomic liquids such as the inert gases were of great interest to physicists because they allowed them to test theories based on fairly simple models. Intermolecular forces can be described using the Lennard–Jones potential and the thermodynamic properties of these liquids can be estimated reasonably well. On the other hand, chemists are interested in much more complex polyatomic systems. When the constituent molecule is polar, description of the intermolecular forces must at least include dipole–dipole interactions and often higher-order multipole interactions. In the case of protic liquids such as water, hydrogen bonding is a very important factor in determining its structure as a liquid. This was seen directly in the case of water from the correlation functions derived from neutron diﬀrac-

THE STRUCTURE OF LIQUIDS

91

tion data and from the structural information obtained from Raman spectroscopy. In discussing the statistical thermodynamics of liquids, emphasis has been placed on the Ornstein–Zernike equation and the integral equation approach to obtaining thermodynamic properties from molecular ones. This is by no means the only approach, an alternative one being based on the perturbation theory of liquids. Perturbation theory has been developed in some detail and is quite successful in describing the properties of simple liquids [28, 29]. In addition, in discussing the integral equation approach, simple approximations such as the mean spherical approximation (MSA) have been emphasized, since they often give analytical expressions for the thermodynamic functions. A good example of such an outcome is the Percus–Yevick results for a hard-sphere ﬂuid discussed in section 2.9. The nature of the approximations used in these methods can be understood in detail using cluster diagrams. More about this method of describing the integrals which arise in a detailed description of intermolecular interactions can be found in monographs dealing with liquid structure [G1, G2]. In conclusion, the material in this chapter is meant to give only an introduction to the subject of liquid structure. Much of what has been presented has dealt with systems which can be represented as point dipoles embedded in hard spheres. Very few liquid systems of chemical interest can be described in such simple terms. However, the simple models can often be modiﬁed to make them more realistic. For example, the eﬀects of chemical interactions can be introduced by assuming that the hard sphere experiences ‘‘sticky’’ interactions in a given direction with respect to the central dipole. Other methods are available for dealing with the eﬀects of non-sphericity. Thus, the simple models can often be made relevant to chemical systems after suitable modiﬁcation. In the following chapters, some of the theories introduced here are used to discuss other systems, including polar solvents and electrolyte solutions. The statistical mechanical tools introduced here are important because they help one to develop an understanding of the way that molecular properties of a given system inﬂuence its macroscopic properties.

General References G1. Egelstaﬀ, P. A. An Introduction to the Liquid State, 2nd ed.; Oxford University Press: New York, 1992. G2. Watts, R. O. and McGee, I. J. Liquid State Chemical Physics; John Wiley: New York, 1976. G3. Eyring, H.; Henderson, D.; Stover, B. J.; Eyring, E. M. Statistical Mechanics and Dynamics, 2nd ed.; John Wiley: New York, 1982. G4. McQuarrie, D. A. Statistical Mechanics; Harper and Row: New York, 1976.

References 1. Barker, J. A.; Henderson, D. Rev. Mod. Phys. 1976, 48, 587. 2. London, F. Trans. Faraday Soc. 1937, 33, 8. 3. Verlet, L. Phys. Rev. 1968, 165, 201. 4. Yarnell, J. L.; Katz, M. J.; Wenzel, R. G.; Koenig, S. H. Phys. Rev. 1973, A7, 2130.

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LIQUIDS, SOLUTIONS, AND INTERFACES

5. Kalman, E.; Palinkas, G. In The Chemical Physics of Solvation, Part B, Spectroscopy of Solvation; Dogonadze, R. R., Kalman, E., Kornyshev A. A., Ulstrup, J., eds.; Elsevier: Amsterdam, 1986, Chapter10. 6. Narten, A. H.; Levy, H. A. J. Chem. Phys. 1971, 55, 2263. 7. Hajdu, F.; Lengyel, S.; Palinkas, G. J. Appl. Crystallogr. 1976, 9, 194 . 8. Soper, A. K.; Phillips, M. G. Chem. Phys. 1986, 107, 47; Soper, A. K. Chem. Phys. 2000, 258, 121. 9. Narten, A. H.; Habenschuss, A. J. Chem. Phys. 1984, 80, 3387. 10. Yamaguchi, T.; Hidaka, K.; Soper, A. K. Mol. Phys. 1999, 97, 603. 11. Bertagnolli, H.; Zeidler, M. D. Mol. Phys. 1978 35, 177. 12. Radnai, T.; Itoh, S.; Ohtaki, H. Bull. Chem. Soc. Jpn. 1988, 61, 3845. 13. Kalman, E.; Serke, I.; Palinkas, G.; Zeidler, M. D.; Weisman, F. J.; Bertagnolli, H.; Chieux, P. Z. Naturforsch. 1983, 38a, 231. 14. Bellisent Funel, M.C.; Nasr, S.; Bosio, L. J. Chem. Phys. 1997, 106, 7913. 15. Luzar, A.; Soper, A. K.; Chandler, D. J. Chem. Phys. 1992, 96, 8460. 16. Ornstein, L. S.; Zernike, F. Proc. Acad. Sci. Amsterdam 1914, 17, 793. 17. Percus, J. K.; Yevick, G. J. Phys. Rev. 1958, 110, 1. 18. van Leeuwen, J. M. J.; de Boer, J. Physica (Utrecht) 1959, 25, 792; Green, M. S. J. Chem. Phys. 1960, 33, 1403. 19. Wertheim, M. S. J. Math. Phys. 1964, 5, 643. 20. Metropolis, N.; Rosenbluth, A. W.; Rosenbluth, M. N.; Teller, E.; Teller, A. H. J. Chem. Phys. 1953, 21, 1087. 21. McDonald, I. R.; Singer, K. Quart. Rev. 1970, 24, 238. 22. Rahman, A. Phys. Rev. 1964, 136, A405. 23. Alder, B. J.; Wainwright, T. E. J. Chem. Phys. 1957, 27, 1208; 1959, 31, 459; 1960, 33, 1439. 24. Carnahan, N. F.; Starling, K. E. J. Chem. Phys. 1970, 53, 600. 25. Walrafen, G. E. J. Chem. Phys. 1964, 40, 3249. 26. Henderson, D.; Leonard, P. J. In Physical Chemistry, an Advanced Treatise; Eyring, H., Henderson, D., Jost, W., eds.; Academic Press: New York, 1971; Vol. VIII B, Chapter 7. 27. McDonald, I. R. In Statistical Mechanics, Specialist Periodical Reports; Chemical Society: London, 1973; Vol. 1, p 134. 28. Henderson, D.; Barker, J. A. In Physical Chemistry, an Advanced Treatise Eyring, H., Henderson, D., Jost, W., eds.; Academic Press: New York, 1971; Vol. VIII B. 29. Smith, W. R. In Statistical Mechanics, Specialist Periodical Reports; Chemical Society: London, 1973; Vol. 1, p 71.

Problems 1. The following molecules form liquids at room temperature. Calculate the energy involved between two similar molecules at contact (r ¼ s) due to dipole–dipole, induced dipole–dipole and London dispersion forces. The molecules are assumed to be spherical with a diameter s. Express the results in units of kB T at 258C. Molecule

Diameter s=pm

Dipole Moment p/debye

Polarizability 1030 a=m3

Ionization Potential I=MJ mol1

CCl4 CH2Cl2 CH3OH

538 450 371

0 1.60 2.87

10.49 6.48 3.26

1.107 1.090 1.047

THE STRUCTURE OF LIQUIDS

93

2. Calculate the Lennard–Jones interaction potential for the following molecules and plot them on the same graph. The calculations should be carried out for increments of r by 0.02s in the range 0.9–1.2s, and by 0.1s in the range 1.2–2.4s. Molecule

LJ k1 B =K

s=pm

Tb =K

423 333 298

592 491 431

342 462 374

Hexane Dimethylsulfoxide Nitromethane

Does the interaction potential give a good measure of cohesive forces in the corresponding liquids? Discuss in terms of the boiling point. 3. The following table gives the distribution function gðrÞ for liquid argon at 85 K. The Lennard–Jones parameters for argon are s ¼ 350 pm and LJ k1 B ¼ 118 K. Estimate the Lennard–Jones potential at each value of r, and then use the Percus–Yevick equation to calculate the direct correlation function cðrÞ (equation (2.6.5)). Plot gðrÞ and cðrÞ against r. r=pm

gðrÞ

r=pm

gðrÞ

r=pm

gðrÞ

r=pm

gðrÞ

0 100 200 300 320 327 334 341 347 354 361 368 375 381 388 395 402 409 415 422 429 436 443 449 456

0 0 0 0 0.072 0.321 0.736 1.283 1.886 2.441 2.851 3.054 3.040 2.850 2.555 2.234 1.947 1.722 1.555 1.424 1.306 1.186 1.062 0.946 0.847

463 477 504 518 552 579 592 613 627 647 667 688 715 729 756 769 797 804 810 817 824 831 844 872 878

0.773 0.686 0.580 0.560 0.609 0.729 0.804 0.914 1.004 1.140 1.229 1.269 1.254 1.230 1.125 1.056 0.947 0.921 0.894 0.870 0.850 0.836 0.825 0.836 0.845

855 892 899 906 913 919 926 933 940 947 953 960 967 981 994 1008 1022 1042 1062 1075 1090 1103 1124 1144 1165

0.859 0.876 0.894 0.911 0.928 0.943 0.959 0.976 0.994 1.014 1.033 1.049 1.063 1.083 1.099 1.108 1.104 1.084 1.065 1.045 1.020 0.999 0.973 0.944 0.934

1219 1253 1273 1293 1345 1376 1410 1444 1485 1566 1655 1805 1900 1975 2057 2118 2227 2295 2377 2479 2547 2620 2683 1

0.954 0.984 1.014 1.034 1.046 1.026 1.007 0.989 0.971 0.993 1.020 0.988 1.000 1.009 1.000 0.994 1.000 1.004 1.000 0.998 1.000 1.002 1.000 1.000

94

LIQUIDS, SOLUTIONS, AND INTERFACES

4. Use the above data to calculate the internal energy per mole of argon at 85 K given that its density is 21.25 atoms nm3 . The contribution to the internal energy from internal degrees of freedom (equation (2.8.9)) should be neglected. Develop a condition in your computer program so that numerical integration is terminated when the integral becomes constant. 5. Estimate the direct correlation function for liquid argon at 85 K using the hypernetted chain approximation with the data given in problem 3. Compare the result with that found using the Percus–Yevick approximation. 6. Estimate the packing fraction for a hard-sphere liquid with a density of 21.25 atoms nm3 and a hard-sphere diameter of 350 pm. Use this result to calculate the Percus–Yevick product for the system at 85 K using the Carnahan– Starling equation of state (equation (2.9.11)). 7. Calculate the internal energy and entropy of the hard-sphere system described in problem 6. Assume that the atomic weight of the hard sphere is 39.95 g mol1 . 8. The following table gives the distribution function gOO ðrÞ in liquid water at 258C. Use these data together with the density of water (0.997 g cm3 ) to calculate the coordination number for a central oxygen atom assuming that the pair correlation function should be integrated up to the ﬁrst minimum. r=pm

gOO ðrÞ

r=pm

gOO ðrÞ

r=pm

gOO ðrÞ

200 213 223 233 243 248 253 258 263 268 273

0 0.008 0.040 0.116 0.233 0.306 0.399 0.541 0.782 1.179 1.746

278 283 288 293 298 303 308 313 318 323 333

2.388 2.907 3.092 2.869 2.351 1.758 1.273 0.966 0.813 0.752 0.734

343 353 363 373 383 393 403 413 423 433 443

0.749 0.773 0.807 0.850 0.898 0.950 1.003 1.050 1.089 1.116 1.333

3

Electrolyte Solutions

Wilhelm Ostwald was born to a German family in 1853 in Riga, Latvia, where he grew up and attended school. In 1872 he entered Dorpat University (now Tartu University in Estonia) where he studied chemistry, obtaining the diploma in 1875. At that time the Baltic States were part of the Russian Empire with some institutions of higher learning operating in German for the German minority who lived in that region of Europe. Ostwald stayed at Friedrich Wilhelm Ostwald Dorpat to study physics under Arthur von Oettingen and chemistry under Carl Schmidt. In 1877 he was appointed lecturer at Dorpat, and he obtained his doctoral degree in 1878. He moved back to Riga in 1881 as professor of chemistry at the Polytechnic Institute. In 1887 he accepted the Chair of Physical Chemistry at Leipzig University and remained there until 1906. Ostwald is generally considered to be the founder of modern physical chemistry. He began the ﬁrst journal dealing with this subject, namely, Zeitschrift fu¨r physikalische Chemie in 1887, and edited it until 1922. The main area of research interest for Ostwald was electrolyte solutions, but he also contributed to other areas of physical chemistry, especially reaction kinetics and catalysis. He had several famous associates including S. A. Arrhenius, J. H. van’t Hoﬀ, and W. Nernst, all of whom won the Nobel Prize in Chemistry. Many young North American chemists studied in his laboratory, and then went on to develop the new area of physical chemistry at home. These included G. N. Lewis, A. A. Noyes, and W. Lash Miller. Ostwald spent much of his time during his long retirement studying the scientiﬁc basis for color. He won the Nobel Prize in Chemistry in 1909 for his work on catalysis, chemical equilibria, and reaction kinetics. He died in 1932.

3.1 Electrolyte Solutions Are Always Non-Ideal Electrolyte solutions are important in all branches of chemistry, but especially in analytical chemistry, and biochemistry. These systems by their nature are always non-ideal, and represented an early challenge to theoreticians interested in describing their thermodynamic properties. The solute components are ions, cations, and anions, which carry opposite charges and thus interact very diﬀer95

96

LIQUIDS, SOLUTIONS, AND INTERFACES

ently with one another. The existence of electrolyte solutions depends on the polar properties of the solvent through which the individual ions are stabilized. When one recognizes the molecular nature of the solvent, one must also consider the interactions between solvent dipoles and the ion. This results in changes in solvent structure in the immediate vicinity of the ions. It follows that a complete description of an electrolyte solution at the molecular level requires the consideration of ion–dipole, ion–ion, and dipole–dipole interactions. In addition to these simple electrostatic interactions, one must also consider the role of hydrogen bonding in protic solvents like water. In very dilute electrolyte solutions, the most important consideration is ion– dipole interactions. One expects these interactions to be diﬀerent for cations and anions. This follows from the fact that the solvent molecule is not a simple dipole in the electrostatic sense but instead it has a chemical structure which is diﬀerent at each end of the molecular dipole. Each ion interacts locally with four to six solvent molecules in its immediate surroundings. In the case of water, the concentration of water molecules in the pure liquid is 55.5 M; it follows that the number of water molecules experiencing direct interaction with ions in dilute solutions represents a small fraction of the total number. As the electrolyte concentration increases, ion–ion interactions become more important in determining the thermodynamic properties of the solution. The electrostatic ﬁeld of an ion is long ranged, decreasing with the reciprocal of the distance from the charge center of the ion. As a result a given ion has an ionic atmosphere in which the concentration of oppositely charged ions in its vicinity is slightly greater on the average than that of ions of the same charge. The properties of the ionic atmosphere depend on temperature, that is, on the randomizing eﬀects of thermal motion. Thus, the composition of the ionic atmosphere ﬂuctuates with time, and the relative diﬀerence between the local concentration of cations and anions in the vicinity of a given ion varies with both temperature and distance. As the electrolyte concentration increases, the average distance between ions and the thickness of the ionic atmosphere both decrease. Under some circumstances ion–ion interactions can be more important than ion–dipole interactions. This is especially true when the valence of the ion is greater than one, and the electrolyte concentration is high. Then, the formation of ion pairs and higher aggregates is possible. Two types of ion pairs have been recognized, namely, contact ion pairs in which the cation and anion are in physical contact, and solvent-separated ion pairs in which one or two solvent molecules are situated between the cation and anion. Ion pairing must be considered in developing a complete picture of an electrolyte solution. As was seen in chapter 2, both dipole–dipole interactions and hydrogen bonding are important in determining the structure and thermodynamic properties of pure water. In the immediate vicinity of an ion, the solvent structure is disrupted so that local dipole–dipole interactions and hydrogen bonding are diﬀerent than they are in pure water. These changes are also important because they aﬀect the local permittivity and the strength of ion–ion interactions. In the present chapter, the properties of electrolyte solutions in water are discussed in detail. Initially the solvation of ions in inﬁnitely dilute solutions is considered on the basis of the Born theory. Then, the Debye–Hu¨ckel model for

ELECTROLYTE SOLUTIONS

97

ion–ion interactions in dilute solutions is described. Finally, the role of ion pairing is outlined. Methods of improving the description of electrolyte solutions on the basis of the MSA are also presented.

3.2 Ionic Size in Solutions In order to apply the models which are introduced in this chapter, one needs an estimate of the size of the ions which make up the electrolyte. This is most easily done for monoatomic ions, since they are spherical, so that ionic size is known by determining the ionic radius. Furthermore, accurate interparticle distances are available from X-ray diﬀraction studies of ionic crystals, and more importantly, from X-ray and neutron diﬀraction studies of aqueous electrolyte solutions [1]. One of the most widely known and used set of ionic radii are those estimated by Pauling [2] on the basis of interionic distances in ionic crystals. He noted that repulsive eﬀects between ions of the same charge depend on the relative size of the cation and anion in the crystal, and also took into consideration the coordination number of the ion with oppositely charged neighbors in the crystal lattice. The results obtained for the alkali metal and halide ions for the case that the coordination number is six (rock salt structure) are summarized in table 3.1. As the collection of X-ray diﬀraction data became more extensive, it was possible to describe the electron density distribution in ionic crystals in more detail. Using these data one can divide up the internuclear distance in the crystal on the basis of the minimum in the electron density between the two oppositely charged ions [3, 4]. For example, in the case of NaCl for which the internuclear distance is 281 pm, the minimum in the electron density leads to radii of 117 pm for Naþ and 164 pm for Cl. Radii derived on this basis are larger for cations and smaller for anions than those of Pauling.

Table 3.1 Radii for the Alkali Metal and Halide Ions for a Coordination Number of Six Estimated from Crystallographic Data Together with Ion–Oxygen Atom Distances from Diffraction Studies of Aqueous Solutions Ionic Radius / pm Ion

Ion-Oxygen Distance in Pauling [2] Shannon and Prewitt [5] Aqueous Solution [1] / pm

Liþ Naþ Kþ Rbþ Csþ

61 96 133 148 166

88 116 152 163 184

210 242 280 — 315

F Cl Br I

134 181 195 217

119 167 182 206

262 310 334 363

98

LIQUIDS, SOLUTIONS, AND INTERFACES

Shannon and Prewitt [5] considered a very large collection of crystallographic data for metal oxides and metal ﬂuorides. Included in these data were transition metal compounds in which the bonding between cation and anion is both electrostatic and covalent in character. They noted that the eﬀective radius for a given ion depends on its coordination number in the crystal. In addition, for transition metal ions the radius depends on whether the d electrons in the ion are in a high or low spin state. By assigning a radius of 126 pm to the O2 ion and 119 pm to the F ion when they are surrounded by six counter ions, they found that the eﬀective radii of the alkali metal ions and halide ions are very close to those obtained from electron density maps for the corresponding crystals. Their results for these ions are also summarized in table 3.1. Another source of information directly relevant to ionic size in solution is X-ray and neutron diﬀraction studies of interatomic distances in aqueous electrolyte solutions [1]. Values of the ion–oxygen distance found in these experiments are given in table 3.1. In the case of the alkali metal ions, the diﬀerence between the ion–oxygen distance and the Shannon–Prewitt radius increases from 122 pm in the case of Liþ to 131 pm in the case of Csþ, the average diﬀerence being 127 pm. The average value is very close to the eﬀective radius of oxygen on the Shannon–Prewitt scale. In the case of Liþ, the average number of water molecules around this cation is probably closer to four in concentrated solutions [6]. Thus, a more appropriate value for the Liþ ion radius in water is 82 pm. As the atomic number of the alkali metal cation increases, the average number of associated water molecules increases and the eﬀective radius in solution changes with respect to the Shannon and Prewitt estimate accordingly. In the case of the halide ions, the diﬀerence between the ion–oxygen distance and the Shannon–Prewitt radius increases from 143 pm for F, to 157 pm for I, the average diﬀerence being 149 pm. Assessment of this result is more diﬃcult because water molecules are oriented around anions via hydrogen bonding (see ﬁg. 3.1). The fact that the average distance between an ion and the oxygen atom in a water molecule is larger for an anion than a cation of the same size is easily explained on the basis of the expected diﬀerence in water orientation. Some increase in the average number of water molecules associated with a given anion is expected with increase in the anion’s atomic number. Studies of pure water have resulted in the conclusion that water has a diameter of 284 pm in a spherical representation [G3]. Thus, if the electrolyte solution is represented as a collection of hard spheres, the distance between the center of a

Fig. 3.1 The orientation of a deuterated water molecule at a Liþ ion and Cl ion as determined by neutron diﬀraction. (From reference G3, with permission.)

ELECTROLYTE SOLUTIONS

99

Kþ ion and that of an adjacent water molecule is 294 pm. The corresponding distance for a Cl ion is 309 pm. Examination of these estimates with respect to the experimental diﬀraction data reveals that they are reasonable. However, if the hard-sphere model relies on a point dipole description of the water molecule, the chemical nature of the ion–solvent interactions is clearly neglected. Shannon–Prewitt radii for other monoatomic ions at various coordination numbers are recorded in table 3.2. The choice of radius is important when estimating the solvation parameters of highly charged ions such as Al3þ. It is also important to note that the estimated radii for transition metal ions such as Mn2þ and Fe3þ depend on the spin state of the d electrons in the ion. This also leads to important diﬀerences in solvation energies. Radii have also been tabulated for polyatomic ions [7]. Since these species are not truly spherical, such radii must be regarded as eﬀective. However, it is useful to have estimates of eﬀective radii when comparing ionic solvation parameters. In the following section, methods of determining the solvation parameters of single ions in inﬁnitely dilute electrolyte solutions are considered. This is followed by a discussion of simple models of ionic solvation in which ionic size is an important factor.

Table 3.2 Shannon and Prewitt Radii for Monoatomic Ions Determined from Crystallographic Data for Various Coordination Numbers* [5] Ion Agþ Al3þ Ba2þ Ca2þ Cd2þ Co2þ Co3þ Cr2þ Cr3þ Cu2þ FFe2þ Fe3þ Ga3þ

Coordination Number 6 4 6 6 6 4 6 6 6 6 6 6 6 6 6 4 6 6 6 6 4 6

(ls) (hs) (ls) (hs) (ls) (hs)

(ls) (hs) (ls) (hs)

Radius /pm 129 53 67 150 114 98 109 79 88 67 75 87 96 76 87 117 75 91 69 79 61 76

Ion Hg2þ In3þ Liþ Mg2þ Mn2þ Mn3þ Naþ Ni2þ O2Pb2þ S2 Se2 Sr2þ Tlþ Tl3þ V2þ V3þ Zn2þ

Coordination Number 6 6 4 4 6 6 6 6 6 4 6 6 6 6 6 6 6 6 6 6 4 6

(ls) (hs) (ls) (hs)

Radius /pm 116 93 73 63 86 81 96 72 79 113 84 126 132 170 184 130 164 102 93 78 74 89

*hs refers to the high-spin conﬁguration of the d electrons in a transition metal cation, and ls, to the low-spin conﬁguration.

100

LIQUIDS, SOLUTIONS, AND INTERFACES

3.3 The Thermodynamics of Ion–Solvent Interactions A very simple experiment that has been carried out for many electrolytes in water is the measurement of the enthalpy associated with the dissolution of the electrolyte, which is often a solid, in water. This process can be either exothermic or endothermic, and has an enthalpy change which depends on the relative amounts of electrolyte and water. By studying the enthalpy of solution for one mole of electrolyte as a function of the number of moles of water, which increase from one experiment to the next, one can determine the enthalpy of solution associated with the formation of an inﬁnitely dilute solution. In the case of NaCl, the relevant process is NaClcry þ 1H2 Ol ! NaClsl ; 1H2 O

ð3:3:1Þ

with an enthalpy change equal to 3.9 kJ mol1, where the subscript ‘‘cry’’ represents the crystalline state, ‘‘l,’’ the liquid state, and ‘‘sl,’’ the solution. Under these experimental conditions, and assuming complete dissociation of the electrolyte, the value of the enthalpy change reﬂects ion–solvent interactions in solution and the lattice energy of the NaCl crystal. Since the solution is inﬁnitely dilute, ion–ion interactions do not play a role in determining the magnitude of the enthalpy change. A much more suitable point of reference for assessing ion–solvent interactions is the unsolvated ion in the gas phase. Thus, one would prefer to know the enthalpy change associated with the reaction Naþ g þ Clg þ 1H2 Ol ! NaClsl ; 1H2 O

ð3:3:2Þ

where Naþ g and Clg are gas phase ions. In order to determine this quantity, the enthalpy change associated with the reaction NaClcry ! Naþ g þ Clg

ð3:3:3Þ

must be found. On the basis of tabulated thermodynamic data [G4], the enthalpy change for reaction (3.3.3) is 787.4 kJ mol1. It follows that the enthalpy of reaction (3.3.2) is –783.4 kJ mol1. This quantity is called the enthalpy of solvation of NaCl, sH. On the basis of other thermodynamic experiments, one can determine the entropy change, sS, and the Gibbs energy change, sG, associated with the same process. Thermodynamic data for electrolyte solvation have been found experimentally for many diﬀerent electrolytes. Ultimately, one would like to be able to analyze these results further to obtain separate contributions from the cation and anion. However, that is not possible without making an extrathermodynamic assumption. As a result, a scale of single ion solvation parameters has been deﬁned relative to those for the Hþ ion. For example, the enthalpy associated with the process Hþ g þ Clg þ / H2 O ! HCl; 1H2 O

ð3:3:4Þ

is –1469.9 kJ mol1. If the enthalpy of solvation of Hþ is deﬁned to be zero, then the enthalpy of solvation of Cl is –1469.9 kJ mol1. Using this result and the

ELECTROLYTE SOLUTIONS

101

value of sH for NaCl, the enthalpy of solvation of the Naþ ion is 686.5 kJ mol1. The resulting scale of solvation enthalpies is called the conventional scale. Another way of describing solvation parameters on the conventional scale is by means of an exchange reaction between the ion in question and the proton. Thus, the solvation parameters for the Naþ ion are deﬁned by the reaction þ þ þ Naþ g þ H ; / H2 O ! Na ; 1H2 O þ Hg

ð3:3:5Þ þ

From thermodynamic tables, the enthalpy of formation of H in an inﬁnitely dilute aqueous solution is zero by deﬁnition; the same quantity for Naþ from tabulated data is –240.1 kJ mol1. On the basis of these results, the enthalpy associated with reaction (3.3.5) is 686.7 kJ mol1. This result agrees within experimental error with that obtained by comparing the heats of formation of inﬁnitely dilute aqueous solutions of NaCl and HCl. Estimation of the entropy of solvation requires calculation of the entropy of the ion in the gas phase. For a monoatomic ion, the main contribution to the entropy comes from its translational energy. Simple ions formed from the main group elements have the electronic structure of an inert gas and therefore do not have an electronic contribution to the entropy. On the other hand, ions formed from transition metals may have an electronic contribution to the gas phase entropy, which depends on the electronic conﬁguration of the ion’s ground state and of any other electronic states which are close in energy to the ground state. The translational entropy is given by the Sackur–Tetrode equation, which is obtained from the solution of the SWE for a particle in a box (see section 2.2) " # 5R ð2pmkB T Þ3=2 kB T þ R ln Strans ¼ ð3:3:6Þ 2 P h3 Here m is the mass of the ion and the other symbols have their usual meaning. For standard conditions of 25 C and 1 bar pressure, this equation becomes Strans ¼ 108:856 þ 12:472 ln M

ð3:3:7Þ

where M is the atomic mass for the ion expressed in grams. The entropies of the ions in solution have been estimated from a variety of thermodynamic data and are given in standard tables [G4]. EXAMPLE

Estimate the enthalpy, entropy, and Gibbs energy of solvation of the S2 ion at 25 C and 1 bar using tabulated thermodynamic data together with the electron aﬃnities of sulfur in the gas phase, which are Sg þ e ! S g

H ¼ 200:4 kJ mol1

ð3:3:8Þ

H ¼ 456 kJ mol1

ð3:3:9Þ

and 2 S g þ e ! Sg

The enthalpy of vaporization of solid sulfur is 278.8 kJ mol1. The enthalpy of formation of Hþ in the gas phase is 1536.2 kJ mol1. The enthalpy of formation of S2 in an inﬁnitely dilute aqueous solution is 33.1 kJ mol1. The enthalpy of solvation of S2 is deﬁned by the process

102

LIQUIDS, SOLUTIONS, AND INTERFACES 2 2Hþ g þ Sg ! H2 S; / H2 O

ð3:3:10Þ

On the basis of the above data, sH for S2 is –3573.7 kJ mol1. Using the Sackur–Tetrode equation the entropies for Hþ and S2 in the gas phase are 108.95 and 152.10 J K1 mol1, respectively. The entropy of S2 in an inﬁnitely dilute aqueous solution is –14.60 J K1 mol1 and that for Hþ, zero by deﬁnition. It follows that sS for S2 is –384.61 J K1 mol1. The Gibbs energy of solvation is obtained using the relationship s G ¼ s H Ts S

ð3:3:11Þ

On the basis of the above results, s G for S2 is –3459.0 kJ mol1 Values of sG and sS on the conventional scale are summarized in table 3.3. Large variations are found in these quantities with ionic size and charge but they are diﬃcult to interpret without conversion to an absolute scale. Various methods have been described to determine the absolute values of the solvation parameters but they all require an extrathermodynamic assumption. On the other hand, the Gibbs energy and enthalpy of formation of small water clusters containing a single ion may be determined using mass spectrometry [8]. The number of water molecules in a cluster is small, varying from one to six. By extrapolating these results to obtain the values of G and H for an inﬁnite number of water molecules in the cluster, one obtains the absolute solvation parameters, sG(abs) and sH(abs). A method of carrying out this extrapolation has been developed [9] which leads to the conclusion that sG(abs) is –1104.5 kJ mol1 and sH(abs), –1150 kJ mol1 for Hþ ions at 25 C. The corresponding value of sS(abs) is –153 J K1 mol1. The value of sG(abs) obtained by the mass spectrometric method is quite close to that estimated on the basis of measurements of the work function for the Hþ ion in an inﬁnitely dilute solution. The latter estimate, which is –1096 kJ mol1 at 25 C is discussed in more detail in chapter 8. On the basis of the values of sG(abs) and sS(abs) obtained in the mass spectrometric experiments one may convert the results obtained on the conventional scale to the absolute scale. These results are also recorded in table 3.3. Of course, it is an easy matter to obtain the value of sH on either scale using relationship (3.3.11). Having obtained absolute values of the thermodynamic properties describing ion solvation it is interesting to examine various theories for ion solvation. The best known of these is the Born model, which is described in the following section.

3.4 Ion–Solvent Interactions According to the Born Model The Born model [11] provides a means of estimating the Gibbs energy of solvation for an ion in an inﬁnitely dilute solution. It is based on a continuum description of the solvent as a uniform dielectric with a relative permittivity of es. The work of transferring the ion from vacuum to the dielectric medium is estimated on the basis of the following three-step process: (a) the ion is reversibly discharged in vacuum; (b) the discharged ion, which is assumed to be a sphere of radius, ri, is

Table 3.3 Solvation Parameters for Monoatomic Ions in Water at 25 C and 1 bar [10] Conventional Values Ion

Absolute Values

sG / kJ mol1 sS / J K1 mol1 sG / kJ mol1 sS / J K1 mol1

Cations from the Main Group Elements Hþ Liþ Naþ Kþ Rbþ Csþ Be2þ Mg2þ Ca2þ Sr2þ Ba2þ Al3þ Ga3þ In3þ Tlþ Tl3þ Sn2þ Pb2þ

0.0 575.1 680.8 752.6 775.2 798.4 289.3 277.6 600.7 729.9 857.3 1362.4 1370.2 820.7 746.1 822.0 619.1 681.6

0.0 10.7 20.0 56.9 66.1 72.2 48.1 68.8 9.9 20.7 57.3 144.8 165.9 7.9 59.3 40.3 32.5 53.0

1104 529 424 352 329 306 2498 1931 1608 1479 1352 4676 4684 4134 358 4135 1590 1527

153 164 133 96 87 81 354 375 296 285 249 604 625 451 94 499 273 253

84.5 9.5 174.4 29.4 64.0 125.5 74.6 89.0 11.1 57.2 14.4 55.2 23.1 10.7

3946 1933 4544 1868 1957 4429 2017 2095 583 2109 489 2057 1829 1859

543 315 633 335 370 584 380 395 164 363 139 361 329 295

429 304 278 243 1238

115 53 37 14 79

Transition Metal Cations Sc3þ Cr2þ Cr3þ Mn2þ Fe2þ Fe3þ Co2þ Ni2þ Cuþ Cu2þ Agþ Zn2þ Cd2þ Hg2þ

632.7 276.0 1230.7 340.7 251.5 1115.8 192.2 113.5 521.1 100.2 615.8 152.2 379.8 349.8

Anions from the Main Group Elements F Cl Br I S2

1533.6 1408.5 1382.2 1347.1 3446.6

268.3 205.8 190.1 166.9 384.6

104

LIQUIDS, SOLUTIONS, AND INTERFACES

transferred from vacuum to the dielectric liquid; and (c) the sphere is reversibly charged up to become the original ion in the dielectric medium. In order to estimate the associated work, one must be able to calculate the potential on the surface of the ionic sphere due to its charge. According to Coulomb’s law, the potential due to a charge zie0 at a distance ri is given by zi e0 ¼ ð3:4:1Þ 4pemed e0 ri where zi is the ion charge number, e0, the electronic charge, emed, the relative permittivity of the medium, and e0, the permittivity of free space. The work done in charging up the sphere from zero charge to a ﬁnal charge of zie0 is then zð i e0

w¼ 0

q dq 4pemed e0 ri

ð3:4:2Þ

where q is any intermediate value of the charge. Carrying out the integration one obtains w¼

ðzi e0 Þ2 8pemed e0 ri

ð3:4:3Þ

The total work associated with the above three-step process for solvation can be written ws ¼ wa þ wb þ wc

ð3:4:4Þ

Assuming that wb, the work associated with introducing the uncharged sphere into the dielectric liquid, is zero, then ws ¼

ðzi e0 Þ2 ðz e Þ2 þ i 0 8pe0 ri 8pes e0 ri

ð3:4:5Þ

where the ﬁrst term corresponds to the work associated with discharging the sphere in vacuum (emed ¼ 1) and the second, to charging up the sphere in the liquid (emed ¼ es). The work done per mole of ions is equated to the Gibbs solvation energy of the ion, which may be expressed as N ðz e Þ2 1 s Gi ¼ L i 0 1 ð3:4:6Þ es 8pe0 ri where NL is Avogadro’s constant. This is the basic result of the Born model. Having obtained the Gibbs energy of solvation, one may now determine the other thermodynamic parameters. The entropy is obtained from the temperature derivative of s Gi : s S i ¼

ds Gi NL ðzi e0 Þ2 1 des ¼ dT 8pe0 ri e2s dT

ð3:4:7Þ

where des/dT is the temperature derivative of the relative solvent permittivity. It should be noted that the radius of the ion has been assumed independent of temperature, a reasonable assumption for monatomic ions. Since the solvent permittivity decreases with increase in temperature, the Born model predicts

ELECTROLYTE SOLUTIONS

105

that the entropy of solvation is negative. This is an expected result, since the solvation process is imagined to involve reorganization of local solvent molecules around the ion to stabilize its presence in the medium. The Gibbs energy and entropy may now be combined to obtain an expression for the enthalpy of solvation: N ðz e Þ2 1 T de ð3:4:8Þ s Hi ¼ s Gi þ Ts Si ¼ L i 0 1 2 s 8pe0 ri es es dT The enthalpy of solvation is clearly larger in magnitude (more negative) than the Gibbs energy because the sign of the entropy and Gibbs energy contributions are the same. Estimates of sG and sS according to the Born model using the Shannon and Prewitt radii are given in table 3.4. From the results, it is clear that the Born estimates of sGi for the alkali metal cations are all too large in magnitude, the diﬀerence for the smallest ion, Liþ, being the largest. In the case of the halide anions, the diﬀerence between the Born estimate and experiment is not as great, the Born estimate always being larger in magnitude. On the other hand, the Born estimates of the entropy change accompanying solvation are usually too small. The generally accepted explanation for the failure of the Born model is that the structure of the solvent is signiﬁcantly altered near the ion by the eﬀect its ﬁeld has on the local water molecules. This eﬀect is especially important for cations, which are stabilized by the electron density on the oxygen atoms of water. One can describe the local disruption of water structure as resulting in an eﬀective decrease in the dielectric constant of the solvent near the ion. As a result, the work associated with the charging the ionic system in solution is less negative, and the Born estimates are too high in magnitude. In the case of anions, hydrogen bonds from surrounding water molecules are important in stabilizing the ion. Disruption of the local solvent structure also occurs but it is not as severe. As a result the Born Table 3.4 Experimental Value and Estimates According to the Born Model and Mean Spherical Approximation for the Gibbs Energy and Entropy of Solvation of the Alkali Metal Cations and Halide Anions at 25 C Gibbs Energy s Gi / kJ mol1

Entropy s Si / J K1 mol1

Ion

Expt*

Born

MSA

Expt*

Born

MSA

Liþ Naþ Kþ Rbþ Csþ F Cl Br I

529 424 352 329 306 429 304 278 243

779 591 451 421 373 576 411 377 333

483 403 333 316 288 396 310 291 264

164 133 96 87 81 115 53 37 14

46 35 27 25 22 34 24 22 20

199 143 101 92 78 138 89 79 66

*Absolute value of the Gibbs energy of solvation from table 3.3.

106

LIQUIDS, SOLUTIONS, AND INTERFACES

estimates for anions are closer to the experimental quantities than for cations. It is also apparent from table 3.4 that the Born model underestimates the magnitude of the entropy change resulting from ion solvation especially for the smaller ions. This is speciﬁcally because it does not take into account the local restructuring of water, which increases the orderliness of the system near the ion. Many attempts have been made to improve the Born description of ion solvation. Most of these rely upon continuum descriptions of the solvent in which the permittivity varies from a low value near the ion to the bulk value farther away. This variation is described mathematically either as two or three regions with a constant permittivity in each, or as a solvent with a continuously varying permittivity over a region of a few molecular diameters thick. Macroscopic concepts such as the permittivity are not really valid at molecular dimensions. For this reason these models are not considered further here. Instead, in the next section, a model based on a discrete description of the electrolyte solution as a collection of hard spheres is discussed.

3.5 Ion-Solvent Interactions According to the Mean Spherical Approximation In the Born model, the solvent was considered to be a continuum with uniform dielectric properties, right up to an ion, which carries a discrete charge. As soon as one recognizes the molecular nature of the solvent, one can imagine how the local solvent structure near an ion is disrupted by the high electrical ﬁeld near the ion. Another way of representing a polar solvent is as a collection of close-packed hard spheres with centrally located point dipoles. Such a description ignores the chemical composition of the solvent molecule but takes into account its most important electrostatic feature, namely, that it has a dipole moment. No consideration is given to the presence of higher-order moments, or to the possibility that the solvent may also be involved in hydrogen bonding. The simplest non-primitive model for representing a dilute electrolyte solution is the mean spherical approximation (MSA). In its present application, the system consists of hard spheres of two sizes, one size for the solvent molecules, and another for the ions, both cations and anions. Since this model is only applied here in the limit of inﬁnite dilution the restriction that cations and anions have equal radii is of no consequence. On the basis of the discussion in chapter 2, the MSA requires that the pair correlation function for any two spheres goes to zero within the reference sphere because they are hard. Outside of any sphere, the direct correlation function is set equal to the interaction energy between the spheres divided by kBT. Solution of the statistical mechanical problem involves applying the Ornstein–Zernike (OZ) equation to obtain the pair correlation function outside of a given sphere. Since the system consists of a mixture of ions and dipoles, the OZ equations appropriate for a mixture must be used with consideration of the ion–ion, ion–dipole, and dipole–dipole interactions that are present. Once the pair correlation functions have been found, the thermodynamic properties of the system may be calculated. In the following presentation, the equations deﬁning the problem are given as well as the solution, no attempt being made to

ELECTROLYTE SOLUTIONS

107

present all of the mathematics required to obtain the solution. The interested reader is referred to the original literature [12–14] for the mathematical details. In the following, the radius of a solvent molecule represented as a sphere is denoted by rs, and that of either a cation or anion by ri. The conditions which deﬁne the direct correlation function are the following. For ion–ion interactions, gij ðrÞ ¼ 0;

r < 2ri

ð3:5:1Þ

and cij ðrÞ ¼ buij ðrÞ ¼

zi zj be20 ; 4pe0 r

r > 2ri

ð3:5:2Þ

where zi and zj are the charge numbers on ions i and j, and b ¼ 1=ðkB TÞ. For ion–dipole interactions, gid ðr; oÞ ¼ 0;

r < ri þ rs

ð3:5:3Þ

and cid ðr; oÞ ¼

zi e0 pbso sr ; 4pe0 r2

r > ri þ rs

ð3:5:4Þ

where p is the dipole moment of the solvent molecule, so , a unit vector in the direction of the dipole, sr , one in the direction r, and o, an angular variable deﬁning the dipole orientation. Finally, for dipole–dipole interactions, gdd ðo1 ; r; o2 Þ ¼ 0;

r < 2rs

ð3:5:5Þ

and cdd ðo1 ; r; o2 Þ ¼

bp2 ½ðs1 sr Þ ðs2 sr Þ s1 s2 ; 4pe0 r3

r > rs

ð3:5:6Þ

where the dipole directions are deﬁned by unit vectors s1 and s2 . Using the Ornstein–Zernike equation (2.6.1), one can now determine the total correlation function for each of these interactions. Only one result from this model is presented here, namely, the Gibbs energy of ion–dipole interactions in the limit of inﬁnite dilution, Gid . Further, if one identiﬁes this with the Gibbs solvation energy, the result is N L ðzi e0 Þ2 1 1 ð3:5:7Þ s G i ¼ 1 8pe0 ri es 1 þ rs =ls ri where ls is the polarization parameter for the solvent. This parameter is obtained from the dielectric properties of the solvent, and on the basis of the work of Wertheim [15] is given by l2s ð1 þ ls Þ4 ¼ 16es

ð3:5:8Þ

By using the bulk dielectric permittivity, the value of ls includes consideration not only of the dipole moment of the solvent but also its polarizability and other features which are necessary to describe ion–dipole and dipole–dipole interac-

108

LIQUIDS, SOLUTIONS, AND INTERFACES

tions. In the case of water at 298 K, the value of ls estimated from es (78.46) is 2.65. It should be noted that the role of dipole–dipole interactions in determining s Gi is ignored in equation (3.5.7). A more convenient way of writing the Gibbs solvation energy is N ðz e Þ2 1 1 s G i ¼ L i 0 ð3:5:9Þ 1 8pe0 es r i þ ds where ds is equal to rs/ls and depends only on the nature of the solvent. If ds is zero, the expression for s Gi becomes equal to that given by the Born model. It is clear that ds is a correction to the ionic radius and results in estimates of s Gi which are considerably smaller than the Born estimates. A reasonable value of rs for water is 142 pm, so that the value of ds is 53.6 pm. This is not insigniﬁcant with respect to the radii of simple monoatomic ions (see tables 3.1 and 3.2). EXAMPLE

Acetonitrile is a polar solvent with a relative permittivity of 35.9. It may be represented as a hard sphere with a diameter of 427 pm. Estimate the Gibbs energy of solvation of Naþ in acetonitrile according to the Born and MSA models. Compare the theoretical estimates with the experimental estimate given that the Gibbs energy of transfer for Naþ from water to acetonitrile is 15.1 kJ mol1. The Gibbs energy of solvation according to the Born model is given by equation (3.4.6). The constant NLe02/8pe0 is equal to 6:945 105 J m mol1. The radius of Naþ according to Shannon and Prewitt is 116 pm (table 3.1). The factor (1 1/es) is equal to 0.972. The resulting value of s Gi is –581.9 kJ mol1. In order to estimate s Gi according to the MSA, one must ﬁrst ﬁnd the value of ls using equation (3.5.8). Solving this cubic equation, one obtains the result that ls is equal to 2.26. Using a radius of 213.5 pm for acetonitrile, the corresponding value of ds is 94.5 pm. Thus, the MSA estimate of s Gi for Naþ is –320.7 kJ mol1 (equation (3.5.9)). The absolute value of the Gibbs energy of solvation of Naþ in water according to experiment is –424 kJ mol1 (table 3.4). After transfer to acetonitrile, s Gi is –409 kJ mol1. The MSA estimate of s Gi is closer to the experimental value than the Born estimate. The entropy of solvation is obtained from the temperature derivative of s Gi [16]. Recognizing that ls depends on temperature, the resulting expression is " # ds Gi N L ðzi e0 Þ2 1 1 des 1 1 dds s S i ¼ ¼ 1 ð3:5:10Þ es dT dT 8pe0 e2s ðri þ ds Þ dT ðri þ ds Þ2 where the temperature coeﬃcient of ds is given by dds r dl ¼ s2 s dT ls dT

ð3:5:11Þ

ELECTROLYTE SOLUTIONS

109

dls/dT can be determined by diﬀerentiation of the Wertheim equation with respect to temperature, which gives

dls ls ð1 þ ls Þ 1 des ¼ ð3:5:12Þ 3ls þ 1 2es dT dT Since the dielectric permittivity decreases with temperature, the coeﬃcient dls/dT is negative, and dds/dT is positive. As a result, both terms in the square brackets in equation (3.5.10) give negative contributions to s S i . Given that des/dT for water is –0.360 K1, the value of dls/dT is 2:48 103 K1 and dds/dT is 0.050 pm K1. Values of s Gi and s S i calculated according to the MSA for the alkali metal cations and halide anions are recorded in table 3.4. The estimates of s Gi for the cations are much better than those by the Born model and give acceptable results. On the other hand, the results for the anions are signiﬁcantly lower than the experimental values. The MSA estimates of s S i for most ions are too high in magnitude. In general, the MSA provides a simple extension of the Born continuum model and gives better values for the fundamental thermodynamic quantities relating to ionic solvation. The major reason the simple MSA model fails is that it has ignored dipole–dipole interactions in the estimation of s Gi and s S i . Dipole–dipole interactions in the immediate vicinity of an ion are expected to be repulsive in character and reduce the estimates of s Gi ands S i from those given by equations (3.5.9) and (3.5.10). An expression for the dipole–dipole contribution to the solvation energy has been derived on the basis of the MSA [12, 13] but it is generally felt to overestimate the repulsive eﬀect. When one analyzes the functional form of the dipole– dipole term, it can be shown that it is similar to that for the ion–dipole contribution. As a result, a simple way of writing an improved expression for the Gibbs solvation energy is N L ðzi e0 Þ2 1 1 1 f dd s Gi ¼ ð3:5:13Þ 8pe0 ðri þ ds Þ es where fdd is a term depending on dipole–dipole interactions which is much less than one. Another problem with the MSA is that it does not distinguish between the solvation of cations and anions of the same size. Thus, although the Kþ and F ions have approximately the same radius, the F anion is more strongly solvated than the Kþ cation (table 3.4). This can be understood in terms of the eﬀect that each ion has on local water structure. The Kþ ion disrupts this structure more so that the stabilizing eﬀect of the local ion–dipole interactions is oﬀset by the work done to break up the water structure, that is, to disrupt attractive dipole–dipole interactions and hydrogen bonding between local water molecules. This means that the parameter ds should be diﬀerent for cations and anions in the same solvent (table 3.5). It is interesting to examine the data for ions from the main group elements on the basis of equation (3.5.13) using the absolute values of s Gi given in table 3.3. In order to do this, the equation is rearranged as follows:

110

LIQUIDS, SOLUTIONS, AND INTERFACES

Table 3.5 MSA Parameters Relating to the Solvation of Monoatomic Ions of the Main Group Elements in Water on the Basis of Equation (3.5.14) Polarization Parameter, ds / pm

Dipole–Dipole Interaction Parameter, fdd

49 0

0 0.26

Cations Anions

z2i ¼ ks ðri þ ds Þ s Gi

ð3:5:14Þ

where ks ¼

N L e20 ð1

8pe0 es f dd Þðes 1Þ

ð3:5:15Þ

It follows that a plot of z2i ðs Gi Þ1 should be linear against the ionic radius ri, and that fdd and ds can be calculated from the slope and radius, respectively. The quality of the ﬁt of equation (3.5.14) to the experimental data is illustrated in ﬁg. 3.2. It is clear from this plot that the behavior of cations and anions is signiﬁcantly diﬀerent. On the other hand, the linear plots obtained for each species are very good. The radius correction parameter ds for cations is slightly smaller than the value predicted by the Wertheim equation but that for anions is considerably smaller. This is clearly a result of the fact that cations disrupt the structure of water much more than anions. On the other hand, dipole–dipole

Fig. 3.2 Plots of z2i ðs Gi Þ1 against ionic radius ri for the alkali metal cations (*), alkaline earth metal cations (&), halide anions (~) and the sulﬁde anion (!). The parameters ds and f dd are determined from the intercept and slope of these plots, respectively, according to equation (3.5.14). The plot for cations has been shifted vertically by 1 mol MJ1 for the sake of clarity.

ELECTROLYTE SOLUTIONS

111

interactions are negligible for water molecules arranged around a cation, whereas they are signiﬁcant in the solvation of anions. The latter result reﬂects the role of hydrogen bonding in anion solvation. In assessing the results one must keep in mind the fact that it has been assumed that the appropriate radii for all ions are those estimated for a coordination number of six. This is probably not correct for smaller ions such as Liþ and Mg2þ. For these ions, the radii estimated for a coordination number of four may be more appropriate. If smaller radii are chosen for the smaller ions, the characteristics of the plots in ﬁg. 3.2 change. This would aﬀect both the polarization parameter ds and the dipole–dipole interaction parameter fdd. The above empirical model can be extended to estimation of the entropy of solvation. When equation (3.5.13) is diﬀerentiated with respect to temperature, one must consider the temperature variation of ds and fdd as well as the temperature dependence of es. However, the number of data points is too few to determine all the temperature coeﬃcients which arise. In summary, the empirical approach to ionic solvation based on the MSA is quite successful for monoatomic ions of the main group elements. It helps one to understand the important diﬀerences between the way cations and anions are solvated in water. It can also be applied to other ions, including polyatomic ions, provided the solvation is essentially electrostatic in character. Thus, one may estimate eﬀective radii for anions such as nitrate and perchlorate from the Gibbs solvation energy using the value of ds calculated for the halide ions. Considering the simplicity of the model, it provides an useful means of understanding the thermodynamics of solvation.

3.6 The Thermodynamics of Electrolyte Solutions Electrolyte solutions are non-ideal by their nature. This follows from the fact that the interactions among the three species in solution, namely, cations, anions, and solvent molecules, are all diﬀerent. Ions with the same charge strongly repel one another, whereas oppositely charged ions attract one another. Interactions of ions with the solvent molecules depend on ionic charge. From a fundamental point of view one would like to understand each interaction in detail at the molecular level. However, thermodynamically one may study the properties of the electrolyte as a whole, or of the solvent. In this section, the thermodynamic conventions for dealing with the properties of electrolytes are presented. As will be seen, these conventions depend largely on the fact that cations and anions are always present at concentrations such that the net charge in the solution is zero. In general one distinguishes two types of electrolyte solutions. The ﬁrst is formed from strong electrolytes, also called ionophores. Strong electrolytes exist as ions in the solid state, a well-known example being NaCl. When they dissolve in a polar solvent such as water, the individual ions are stabilized by ion–dipole interactions with solvent molecules. Otherwise the properties of the system reﬂect long-range electrostatic ion–ion interactions. On the other hand, a weak electrolyte or ionogene is a compound which is molecular in nature as a pure substance. It forms ions by interaction with water molecules. A familiar weak electrolyte is

112

LIQUIDS, SOLUTIONS, AND INTERFACES

acetic acid, which interacts with water to form solvated protons and acetate ions. Usually, dissociation of the molecule into ions is not complete, so that a dissociation constant is required to determine the ionic concentrations in solution. It is important to note that ionophores are not always completely dissociated. For example, when NaCl is dissolved in a solvent of lower relative permittivity, such as methanol, it is ion paired to some extent. The thermodynamics of systems with ion pairing is considered separately in section 3.10. Under these circumstances the ionophore behaves in the same way as a weak electrolyte. On the other hand, all ionogenes are not weak electrolytes. For example, HCl, which is a molecule in the gas phase, is completely dissociated in water and therefore is a strong electrolyte. Acetic acid is completely dissociated in liquid ammonia, which is a much stronger base than water. Thus, the solvent plays an important role in determining the extent of electrolyte dissociation in solution. In the following discussion the traditional terms, strong and weak electrolytes, are used. A. Strong Electrolytes Consider a simple 1–1 electrolyte MX which is completely dissociated in dilute solutions. The chemical potential of MX can be written as mMX ¼ mMX þ RT ln aMX

ð3:6:1Þ

where aMX is the activity of the electrolyte. Activity is related to the concentration of the electrolyte on the appropriate scale. Although one cannot directly measure separately the cationic activity aM and anionic activity aX, one can consider these quantities conceptually. They are deﬁned by the relationship aM aX ¼ aMX

ð3:6:2Þ

and are not necessarily equal, in spite of the fact that the concentrations of M and X must be equal. The electrolyte concentration is commonly expressed either in terms of molality or molarity. Using the former, the activity of ion i is given by ai ¼ gi m e

ð3:6:3Þ

where gi is the activity coeﬃcient on the molal scale and me, the electrolyte molality. The mean ionic activity coeﬃcient is deﬁned by the equation g2 ¼ gM gX

ð3:6:4Þ

Since individual ionic activity coeﬃcients cannot be measured experimentally, only the mean quantity is tabulated. It follows that the activity of a 1–1 electrolyte is given by ð3:6:5Þ aMX ¼ g2 m2e and its chemical potential by þ 2RT ln g me mMX ¼ mMX

ð3:6:6Þ

The concentration dependence of g is an important feature involved in the experimental and theoretical evaluation of electrolyte behavior. The chemical potential of the standard state, mMX , is that for a hypothetical one-molal solution in which all real interactions are imagined to be absent (g ¼ 1.00). Thus, the

ELECTROLYTE SOLUTIONS

113

standard state for the electrolyte solution is chosen on the basis of the Henry law convention (see section 1.11). One may also write expressions for the chemical potential of the electrolyte on the molarity and mole fraction scales. In the former case, the expression is mMX ¼ mMX þ 2RT ln y ce

ð3:6:7Þ

mMX

is the standard chemical potential and y, the corresponding activity where coeﬃcient. On the mole fraction scale, one writes mMX ¼ mMX þ 2RT ln f xe D

ð3:6:8Þ

D

where mMX is the standard chemical potential, f, the corresponding activity coeﬃcient, and xe, the mole fraction of the electrolyte. As with the molal scale, the standard states are deﬁned with respect to hypothetical ideal solutions of unit concentration. The mole fraction scale is not often used for electrolytes for the obvious reason that the electrolyte concentration cannot be varied over the whole range of mole fractions because of its limited solubility. On the other hand, the molar scale is popular. In order to convert values of activity coeﬃcients which are usually tabulated on the molal scale [G1] to the molar scale, one needs to know the concentrations on the two scales, ce and me, and the density of the pure solvent, rs . The relationship between the activity coeﬃcients is c g ¼ e y ð3:6:9Þ me rs The above equations for a 1–1 electrolyte are easily extended to a more complex electrolyte MaXb. Dissociation of this electrolyte give cations Mbþ and anions Xa-: Ma Xb ! nþ Mbþ þ n Xa

ð3:6:10Þ

The stoichiometric numbers nþ and n give the number of moles of cations and anions, respectively, resulting from the dissolution of one formula of electrolyte. Their sum is designated as n: n ¼ nþ þ n

3:6:11Þ

On the basis of equation (3.6.2) the electrolyte activity is ae ¼ ðaþ Þnþ ða Þn

ð3:6:12Þ

Each of these activities is the product of an activity coeﬃcient and a concentration. On the molality scale, one may write ae ¼ ðgþ mþ Þnþ ðg m Þn

ð3:6:13Þ

m þ ¼ nþ m e

ð3:6:14Þ

m ¼ n m e

ð3:6:15Þ

Now,

and

The mean molality m is deﬁned on the basis of a geometric average so that

114

LIQUIDS, SOLUTIONS, AND INTERFACES n

ðm Þn ¼ ðmþ Þnþ ðm Þn ¼ nþþ nn mne

ð3:6:16Þ

Now one deﬁnes a mean activity coeﬃcient using the equation ðg Þn ¼ ðgþ Þnþ ðg Þn

ð3:6:17Þ

It follows that the electrolyte activity is ae ¼ ðnþ Þnþ ðn Þn ðg Þn mne

ð3:6:18Þ

One may also deﬁne a mean electrolyte activity, a , which is given by n

a ¼ ðnþþ nn Þ1=n g me

ð3:6:19Þ

It is easily seen that this expression yields the correct result for a 1–1 electrolyte when nþ and n are set equal to unity. EXAMPLE

A 0.1 m solution of MgCl2 in water has a density of 1.0047 g mL1 at 258C. The mean ionic activity coeﬃcient on the molal scale is 0.528. Calculate the mean activity and electrolyte activity on this scale. Repeat the calculations for the molarity scale. For this electrolyte, the concentration of Mg2þ ions is 0.1 m ðnþ ¼ 1Þ and that of Cl ions 0.2 m ðn ¼ 2Þ. Thus the mean ionic molality on the basis of equation (3.6.16) is m ¼ ð1 22 Þ1=3 me ¼ 41=3 ð0:1Þ ¼ 0:1587

ð3:6:20Þ

The mean activity is a ¼ g m ¼ 0:528 0:1587 ¼ 0:0838

ð3:6:21Þ

The electrolyte activity on the molality scale is ae ¼ a3 ¼ ð0:0838Þ3 ¼ 5:89 104

ð3:6:22Þ

The molecular mass of MgCl2 is 95.22 g. Thus, 9.522 g of MgCl2 are dissolved in 1000 g of water to form a 0.1 m solution. Given that the density of the solution is 1.0047, its volume is V¼

9:522 þ 1000 ¼ 1004:8 mL 1:0047

ð3:6:23Þ

0:1 ¼ 0:0995 M 1:0048

ð3:6:24Þ

Its molar concentration is ce ¼

The mean ionic concentration is c ¼ 41=3 ð0:0995Þ ¼ 0:1579 M

ð3:6:25Þ 1

Given that the density of pure water at 258C is 0.997 g mL , the activity coeﬃcient on the molar scale (equation (3.6.9)) is y ¼

me rso g 0:1 0:997 0:528 ¼ 0:529 ¼ 0:0995 ce

Thus, the mean electrolyte activity is

ð3:6:26Þ

ELECTROLYTE SOLUTIONS

a ¼ y c ¼ 0:529 0:1579 ¼ 0:0835

115

ð3:6:27Þ

The electrolyte activity on the molarity scale is ae ¼ a3 ¼ 5:83 104

ð3:6:28Þ

It is important to remember that the mean ionic concentration and activity are based on the geometrical mean of the concentrations of the two ions in non-symmetrical electrolytes, not on their arithmetic mean. B. Weak Electrolytes In order to develop the thermodynamics of a weak electrolyte, one extra parameter is required, namely the dissociation constant Kdiss . Using the same terminology as above, the dissociation process for a weak 1–1 electrolyte is MX ( + M þ þ X The equilibrium constant for this process is a a Kdiss ¼ M X aMX

ð3:6:29Þ

ð3:6:30Þ

where aM and aX are the activities of the ions, and aMX , that of the undissociated molecule. On the molality scale, this may be written as Kdiss ¼

g2 m2 gMX mMX

ð3:6:31Þ

where g is the mean ionic activity coeﬃcient on the molality scale, m , the mean ionic molality, gMX , the activity coeﬃcient of the molecular species, and mMX , its molality. This relationship is often written in terms of the fraction of weak electrolyte which is dissociated, namely, ai . If me is the stoichiometric molality of the weak electrolyte, that is, the molality of the undissociated molecule plus that of the dissociated form, then equation (3.6.31) bcomes Kdiss ¼

g2 a2i me gMX ð1 ai Þ

ð3:6:32Þ

An expression can now be written for the chemical potential of the weak electrolyte system. On the basis of the composition of the solution mMX ¼ ai mþ M þ ai mX þ ð1 ai ÞmMX

mþ M

ð3:6:33Þ

m X

and are the chemical potentials of the free ions and mMX , that of the where undissociated molecular. Writing each chemical potential in terms of its standard value and the corresponding activity, one obtains mMX ¼ mMX þ ai RT ln aM aX þ ð1 ai ÞRT ln aMX

ð3:6:34Þ

mMX ¼ ai mMþ þ ai mX þ ð1 ai ÞmMX

ð3:6:35Þ

where In these equations, mi is the standard potential of species i and ai , its activity. Finally, from equilibrium (3.6.29), one may write that

116

LIQUIDS, SOLUTIONS, AND INTERFACES mMX ¼ mþ M þ mX

ð3:6:36Þ

RT ln Kdiss ¼ mMX mMþ mX

ð3:6:37Þ

so that By substituting equations (3.6.30) and (3.6.37) into equation (3.6.34), a much simpler expression for mMX is obtained: mMX ¼ mMþ þ mX þ RT ln aM aX

ð3:6:38Þ

This result shows that the chemical potential of the weak electrolyte system may be expressed in terms of the activities of the ions only, without explicitly including the activity of the undissociated molecule. Equation (3.6.38) is no diﬀerent in form from those for a strong electrolyte (equations (3.6.1) and (3.6.2)). Of course, the activities of the ions are much less for the weak electrolyte than those for the strong electrolyte for a given molality. Thus, on the basis of the present analysis for a weak electrolyte aM aX ¼ a2i g2 m2e

ð3:6:39Þ

When these equations are written in terms of the concentration (molarity) the results are mMX ¼ mMþ þX þRT ln aM aX

ð3:6:40Þ

aM aX ¼ a2i y2 c2e

ð3:6:41Þ

where mMþ

mX

and are the standard potentials of the two ions on the molarity scale, y , the mean ionic activity coeﬃcient, and ce , the molarity of the weak electrolyte system.

3.7 The Experimental Determination of Activity Coefﬁcients for Electrolytes In general, one may distinguish two methods of determining the activity of an electrolyte in solution. One type is based on the direct measurement of electrolyte activity. The most popular technique in this category involves electromotive force measurements with electrochemical cells. This subject is not discussed here, but later in the chapter dealing with electrochemical equilibria (chapter 9). The second type of measurement involves determination of the activity of the solvent, for instance, by measuring its vapor pressure. These activity data are used to calculate the activity of the electrolyte on the basis of the Gibbs–Duhem relationship. Methods related to determination of the solvent’s vapor pressure include the determination of the lowering of the melting point, and the elevation of the boiling point. The latter two techniques have the inconvenience that the activity coeﬃcients are determined at the freezing point or boiling point of water, and not at room temperature or the usual experimental temperature of 258C. Thus, the most often used technique is the direct determination of the vapor pressure of the solvent as a function of electrolyte concentration at 258C. The experimental techniques have been described in detail by Robinson and Stokes [G1].

ELECTROLYTE SOLUTIONS

117

Having obtained the solvent vapor pressure, one may immediately relate it to the activity of the electrolyte using the Gibbs–Duhem relationship. Thus, P ð3:7:1Þ ne d ln ae ¼ ns d ln as ¼ ns d ln s Ps where the subscript ‘‘e’’ denotes electrolyte, and the subscript ‘‘s,’’ solvent. Furthermore, if one works with molalities, such that ne is equal to me and ns to 1000 M 1 s where Ms is the molecular mass of the solvent, then 1000 nme d ln g me ¼ d ln as ð3:7:2Þ Ms At this point, it is convenient to introduce a quantity called the osmotic coeﬃcient, j. It is deﬁned according to the equation 1000 ln as ð3:7:3Þ nme j ¼ Ms One often ﬁnds values of j tabulated for electrolyte solutions as well as the mean activity coeﬃcient g [G1, G5]. Taking the total derivative of equation (3.7.3), one obtains 1000 d ln as ð3:7:4Þ nme dj þ nj dme ¼ Ms Combining this with equation (3.7.2), one may write d ln g ¼ dj þ ðj 1Þ d ln me

ð3:7:5Þ

Remembering that both g and j go to unity as me goes to zero, one obtains m ðe ð3:7:6Þ ln g ¼ ðj 1Þ þ ðj 1Þ d ln me 0

This relationship is used to calculate mean ionic activity coeﬃcients using values of the osmotic coeﬃcient, j. Alternatively, one may calculate the osmotic coeﬃcient given the mean ionic activity coeﬃcient. From equations (3.7.2) and (3.7.3) dðnme jÞ ¼

1000 d ln as ¼ nme d ln g me Ms

ð3:7:7Þ

Integrating from zero molality, one obtains m ðe

nme j ¼

nme d ln g me

ð3:7:8Þ

0

or 1 j¼1þ me EXAMPLE

m ðe

me d ln g

ð3:7:9Þ

0

The activity of water (Pw/Pw ) in a 6:4 103 m NaCl solution is 0.9997754. Estimate the osmotic coeﬃcient and mean ionic activity coeﬃcients for the

118

LIQUIDS, SOLUTIONS, AND INTERFACES

electrolyte. In more dilute solutions, it may be assumed [G1] that the relationship between the osmotic coeﬃcient and molality is j ¼ 1 0:330 m1=2 e

ð3:7:10Þ

From equation (3.7.3), the osmotic coeﬃcient is j¼

1000 1 ln ð0:9997754Þ ¼ 0:9738 18:02 2 6:4 103

ð3:7:11Þ

Note that the value of the osmotic coeﬃcient agrees with that estimated from equation (3.7.10). Combining equations (3.7.6) and (3.7.10), one obtains m ðe

ln g ¼ 0:0262

0:330 m1=2 e d ln me 0 m ðe

¼ 0:0262

0:330 me1=2 dme

ð3:7:12Þ

0

¼ 0:0262 2 0:330 me1=2 ¼ 0:0790 The value of g is 0.924. Values of the water activity and osmotic coeﬃcient for NaCl solutions in water for concentrations up to 6 m are shown in ﬁg. 3.3. First, it is apparent that the vapor pressure of water decreases continuously with increase in salt concentration as one would expect. The value of aw at the most dilute concentration considered (0.1 m) is 0.99665. Measurements of vapor pressure lowering for lower concentrations are extremely diﬃcult. The concentration dependence of the osmotic coeﬃcient j reported in the same ﬁgure is quite diﬀerent from that of the water activity. This quantity at ﬁrst decreases, and then rises to values which are signiﬁcantly

Fig. 3.3 Plots of the water activity, aw, and corresponding osmotic coeﬃcient, j, for aqueous solutions of sodium chloride at 25 C.

ELECTROLYTE SOLUTIONS

119

greater than unity. Tables of the osmotic coeﬃcients for a large number of electrolytes in water at 25 C are available in the monograph by Robinson and Stokes [G1], and in the more recent compilation by Lobo [G5]. In order to calculate the mean ionic activity coeﬃcient, g, from the osmotic coeﬃcient, j, one must perform a numerical integration to evaluate the integral in equation (3.7.6). The value of the integrand, chosen to be ð1 jÞ=me , is shown as a function of electrolyte molality using the NaCl data in ﬁg. 3.4. This function is clearly ﬁnite in the limit that me goes to zero, and is equal to one. As a result, there is no problem in evaluating the integral by numerical techniques, and thereby estimating ln g. In many cases, it is more convenient to use activity coeﬃcients on the molarity scale. Not only is molarity more commonly used as a concentration unit in chemistry but values of y are more directly related to the results of statistical mechanical theories of electrolyte solutions discussed later in this chapter. For a given molality, me, one must calculate the corresponding molarity, ce, using the relationship ce ¼

1000me rs me Me þ 1000

ð3:7:13Þ

where rs is the density of the solution and M e, the molecular mass of the electrolyte. The solution density is not related in a simple way to either the molality or molarity. It has been tabulated [G5] for many electrolyte solutions but often not at the speciﬁc concentrations of interest. In this regard, there is a useful empirical relationship which can be used to describe the variation in solution density with molality [G6]: ln rs ¼ ln rs þ

Ae me Me me Me þ 1000

ð3:7:14Þ

Fig. 3.4 Plots of the function (1j)/me, where j is the osmotic coeﬃcient against solute molality, me, for solutions of sodium chloride in water at 25 C.

120

LIQUIDS, SOLUTIONS, AND INTERFACES

Table 3.6 Values of the Constant Ae in Equation (3.7.14) which Relates the Solution Density to Its Molality for Aqueous Solutions of the Alkali Metal Halides Anion F

Cl

Br

I

— 1.0260 0.8471 0.9710 1.2595

0.5667 0.6938 0.6323 0.7748 0.8308

0.8248 0.8388 0.7624 0.9077 0.8536

0.8697 0.9146 0.8310 0.848 0.6420

Cation Liþ Naþ Kþ Rbþ Csþ

Values of the constant Ae determined via least-squares ﬁts of density data for aqueous solutions of the alkali metal halides are given in table 3.6. Similar ﬁts may be made for other electrolyte solutions so that the conversion of molality to molarity is easily carried out. Once the concentration units are known, values of y are easily calculated from g using equation (3.6.9). These quantities are given in table 3.7 for the NaCl system for concentrations up to 2.4 m. The above results illustrate the importance of non-ideality for electrolyte solutions and also of the use of the Gibbs–Duhem relationship in obtaining electrolyte

Table 3.7 Mean Activity Coefﬁcients for the Aqueous NaCl System on the Molality and Molarity Scales Molality/me

g

Molarity/ce

y

0.010 0.020 0.050 0.100 0.200 0.300 0.500 0.600 0.700 0.800 0.900 1.000 1.200 1.400 1.600 1.800 2.000 2.200 2.400

0.903 0.872 0.822 0.779 0.735 0.710 0.681 0.673 0.667 0.662 0.659 0.657 0.654 0.654 0.657 0.661 0.668 0.675 0.683

0.010 0.020 0.050 0.100 0.199 0.298 0.494 0.592 0.689 0.786 0.883 0.979 1.170 1.360 1.548 1.734 1.919 2.101 2.282

0.900 0.869 0.820 0.780 0.738 0.714 0.687 0.680 0.675 0.672 0.670 0.669 0.669 0.671 0.677 0.684 0.694 0.705 0.716

ELECTROLYTE SOLUTIONS

121

activity coeﬃcients from vapor pressure measurements for the solvent. Although activity coeﬃcients can be measured directly for some electrolytes, they can always be obtained indirectly from the solvent activity. In the next section a model developed to account for non-ideality is considered and its ability to predict experimental results assessed.

3.8 Ion–Ion Interactions According to the ¨ ckel Theory Debye–Hu The ﬁrst successful theory of electrolyte solutions was proposed by Debye and Hu¨ckel. It is based on a model in which the ions are represented as point charges embedded in a dielectric continuum. The model uses fundamental electrostatics to calculate the average local potential at any given ion due to its ionic atmosphere [17]. The main problem is to determine a distribution function for the ions in the atmosphere. This is achieved by assuming that the probability of ﬁnding an ion at a distance, r, from the central ion is given by the Boltzmann distribution law which accounts for the randomizing eﬀects of thermal motion in the presence of the electrical ﬁeld. The potential at the reference ion may then be used to calculate the excess Gibbs energy due to the ion–ion interactions present in the electrolyte solution. In the Debye–Hu¨ckel model, solution non-ideality is attributed to the ion–ion interactions and related to the excess Gibbs energy through the activity coeﬃcient, yi for ion i. It is important to recognize that the average potential in a uniform phase, such as an electrolyte solution, is constant in the absence of an external ﬁeld, such as an electrical ﬁeld. However, the problem solved by Debye and Hu¨ckel is the estimation of the average local potential which is diﬀerent from the average potential. The local potential is sometimes called the micropotential, whereas the average potential is called the macropotential. It will be seen that the micropotential ﬂuctuates in the solution, being more positive at a cation and more negative at an anion. The theory presented below gives an estimate of how the ionic atmosphere aﬀects these ﬂuctuations. Two equations are required to solve the problem in question. The ﬁrst is the Poisson equation, which describes how the micropotential, , varies in space. As a partial diﬀerential equation it is written as r2 ¼ div grad ¼

rz es e0

ð3:8:1Þ

where rz is the average charge density at a distance, r, from the central ion. It is derived by combining two fundamental laws of electrostatics, namely, Coulomb’s law and Gauss’ law. In vector notation, these are E ¼ grad

ð3:8:2Þ

div D ¼ rz

ð3:8:3Þ

and

122

LIQUIDS, SOLUTIONS, AND INTERFACES

where E is the electrical ﬁeld and D, the electric displacement (D ¼ ee0 E) (see Appendix B). The Laplacian of , r2 , describes the spatial variation of the potential and in Cartesian coordinates is given by r2 ¼

@2 @2 @2 þ 2 þ 2 @x2 @y @z

ð3:8:4Þ

However, the present problem is more conveniently solved in spherical coordinates because an estimate of the average charge density at a distance r is sought. In this case, depends only on r and not on the Eulerian angles, which deﬁne an exact location in spherical coordinates. As a result, equation (3.8.1) can be written as 1 d 2 d r 2 ð3:8:5Þ r ¼ 2 r ¼ z dr dr e r s e0 The second equation is obtained by applying Boltzmann statistics to estimate the charge density, rz. The local concentration of ion i at a distance r from the central reference ion, cri , diﬀers from its average concentration, c i , because the potential, , is diﬀerent from the average potential. Using Boltzmann statistics these quantities are connected by the relationship cri ¼ c i expðzi f Þ

ð3:8:6Þ

where c i the concentration in moles per cubic meter and f ¼ e0/(kBT) ¼ F/(RT). If the central ion is a cation, and is positive, then the local concentration of other cations is lower and that of anions higher. The extent to which the local concentration diﬀers from the average concentration depends on the relative values of the electrostatic energy, zie0, and the thermal kinetic energy, kBT. For a simple electrolyte solution containing one cation with charge number zþ and one anion with charge number z, the charge density at distance r is rz ¼ zþ Fc þ expðzþ f Þ þ z Fc expðz f Þ

ð3:8:7Þ

where F is the Faraday constant (F ¼ NLe0). In a more general situation where the solution contains more than two types of ions, one may write rz ¼

n X

zi Fc i expðzi f Þ

ð3:8:8Þ

i¼1

Combination of equations (3.8.5) and (3.8.8) gives the Poisson–Boltzmann equation for the system: n X 1 d 2 d zi Fc i r ¼ expðzi f Þ ð3:8:9Þ dr ee r2 dr i¼1 s 0 There are two important problems with this equation. First, it is a non-linear diﬀerential equation because of the exponential terms in and therefore cannot be solved in a simple way. The second problem is that the solution of the statistical problem is not consistent with the fundamental laws of electrostatics. These problems were overcome by Debye and Hu¨ckel [17] in a fairly simple way. In the case that the electrostatic energy is small in comparison to the thermal kinetic

ELECTROLYTE SOLUTIONS

123

energy, the exponential term in equation (3.8.9) can be expressed by the ﬁrst two terms in its Taylor series expansion. Thus, the expression for the charge density becomes rz ¼

n X

zi Fc i

i¼1

n X

z2i Ffc i

ð3:8:10Þ

i¼1

The ﬁrst term on the right-hand side of equation (3.8.10) gives the sum of the ionic charges in the solution, which must add to zero. From the second term, one deﬁnes an important quantity used in assessing the properties of electrolyte solutions, namely, the ionic strength, I. The deﬁnition is I¼

n 1X z2 c 2 i¼1 i i

ð3:8:11Þ

For a 1–1 electrolyte, the concentrations of cations and anions are equal to the overall bulk concentration c e and the charge numbers zi are þ1 and 1. Then, the ionic strength is also equal to the bulk concentration c e . For a 2–1 electrolyte, such as ZnCl2, the cationic concentration is equal to c e , whereas the anionic concentration is 2c e . In this case, the ionic strength is 3c e and recognizes the higher charge on the cations. For a 2–2 electrolyte, such as ZnSO4, the ionic strength is four times the bulk concentration. Of course, the ionic strength may be expressed in other concentration units, including the molality me. It is an important property of an electrolyte solution, especially in evaluating the extent of departure of its properties from ideality. When the Debye–Hu¨ckel approximation for the charge density rz is used, the Poisson–Boltzmann equation can be written as 1 d 2 d 2FfI r ¼ ¼ k2 ð3:8:12Þ 2 dr dr es e0 r where the constant k is given by k¼

2FfI 1=2 es e0

ð3:8:13Þ

Dimensional analysis shows that k has units of reciprocal length, and it is called the Debye–Hu¨ckel reciprocal distance. It depends on the ionic strength of the solution, the dielectric properties of the solvent, and temperature. For an aqueous solution containing a 1–1 electrolyte at a concentration of 1 M (1000 mol m3) at 25 C, k is equal to 3.288 nm1. As will be seen below, 1/k corresponds to the eﬀective thickness of the ionic atmosphere, which would be 304 pm for a 1 M solution. The solution to the Poisson–Boltzmann equation is now rather easily obtained. First of all, one makes the substitution y ¼ r Then,

ð3:8:14Þ

124

LIQUIDS, SOLUTIONS, AND INTERFACES

dy d ¼r þ dr dr

ð3:8:15Þ

d2 y d d2 þr 2 ¼2 2 dr dr dr

ð3:8:16Þ

and

The left-hand side of equation (3.8.12) gives 1 d 2 d 2 d d2 r þ 2 ¼ dr r dr r2 dr dr

ð3:8:17Þ

so that the Poisson–Boltzmann equation becomes d2 y ¼ k2 y dr2

ð3:8:18Þ

This is a well-known diﬀerential equation in mathematical physics, whose general solution is y ¼ r ¼ k1 ekr þ k2 ekr

ð3:8:19Þ

The constants k1 and k2 are determined by the boundary conditions for the system. One of these states that the potential, , must go to zero as the distance, r, goes to inﬁnity. This leads to the conclusion that term in ekr cannot contribute to the potential ; as a result the constant k2 is zero. The solution to the Poisson– Boltzmann equation with one undetermined constant k1 is then ¼

k1 kr e r

ð3:8:20Þ

In order to determine the constant k1, one makes use of the relationship between the charge density rz and local potential . From equations (3.8.10) and (3.8.11), n X rz ¼ z2i Ffc i ¼ 2FfI ¼ es e0 k2 ð3:8:21Þ i¼1

The total charge in the solution outside a given reference ion must be equal and opposite to the charge on the ion in order to maintain electroneutrality. Each ion has an excluded volume around it into which the center of another ion may not enter. For spherical ions this volume may be considered also to be spherical and to have a radius a related to the sum of the radii of the cation and anion in a simple electrolyte but not necessarily equal to that sum (see ﬁg. 3.5). The cutoﬀ distance a deﬁnes the lower limit for integrating the charge density outside of a reference ion in a calculation which allows one to determine the constant k1. If the charge on the reference ion is zie0, one may write 1 ð

4pr2 rz dr ¼ zi e0

ð3:8:22Þ

a

where the integration involves adding the charge density in spherical shells of volume 4pr2dr. Combining this with equations (3.8.20) and (3.8.21), one obtains

ELECTROLYTE SOLUTIONS

125

Fig. 3.5 Models for the excluded volume around (a) a contact ion pair and (b) a solvent-separated ion pair with consideration of interaction of their solvation atmospheres. The radius of the excluded volume is designated as ‘a’. 1 ð

4prk1 es e0 k2 ekr dr ¼ zi e0

ð3:8:23Þ

a

Integrating by parts, the result is 4pk1 es e0 ðkaeka þ eka Þ ¼ zi e0

ð3:8:24Þ

so that k1 ¼

zi e0 eka 1 4pe0 es 1 þ ka

The solution of the Poisson–Boltzmann equation is thus ka zi e0 ekr e ¼ 4pe0 es r 1 þ ka

ð3:8:25Þ

ð3:8:26Þ

To proceed further, one must separate the contribution to due to the ionic atmosphere from the contribution that the ion makes itself in the absence of other ions, that is, the so-called self-atmosphere potential. The latter quantity is given by ze self ¼ i 0 ð3:8:27Þ 4pe0 es r It follows that atm estimated at r ¼ a, the distance to which other ions are absent, is zi e0 1 zi e0 zi e0 k ¼ atm ¼ ð3:8:28Þ 4pe0 es a 1 þ ka 4pe0 es a 4pe0 es 1 þ ka The remaining part of the problem involves relating the work done in forming the ionic atmosphere to the activity coeﬃcient yi. If all departure from ideality is attributed to the eﬀects of ion–ion interactions, and the work done is estimated in an imaginary charging process in which the reference ion acquires its charge in the presence of all of the other ions, then zð i e0

0

z2i e20 k atm dðzi e0 Þ ¼ 8pe0 es 1 þ ka

ð3:8:29Þ

For a mole of these ions, the relationship between the contribution to the Gibbs energy due to the non-ideality and the work done is

126

LIQUIDS, SOLUTIONS, AND INTERFACES

RT ln yi ¼

NL z2i e20 k 8pe0 es 1 þ ka

ð3:8:30Þ

Note that the activity coeﬃcient estimated is designated yi because the concentration of the ions is expressed in molarity units. Since k ¼ 3.288 I1/2 nm1 at 25 C in water, the expression for ln yi for these conditions is ln yi ¼

1:174 z2i I 1=2 1 þ 3:288 a I 1=2

ð3:8:31Þ

ADH z2i I 1=2 1 þ BDH aI 1=2

ð3:8:32Þ

In general one may write ln yi ¼

where ADH and BDH are the well-known Debye–Hu¨ckel constants. If the ionic strength is expressed in moles per liter, then these constants are given by the expressions

1=2 NL e20 F 2000 ð3:8:33Þ ADH ¼ 8p ðe0 es RTÞ3 and 2000 1=2 ð3:8:34Þ BDH ¼ F e0 es RT where the factor of 1000 is introduced to convert moles per liter to moles per cubic meter. Values of ADH and BDH for water in the temperature range 0 to 50 C are given in table 3.8. The Debye–Hu¨ckel theory is also used to estimate activity coeﬃcients for dilute solutions on the molality scale. In this case, equation (3.8.32) becomes ln gi ¼

0 ADH z2i I 1=2 0 1 þ BDH aI 1=2

ð3:8:35Þ

where I 1/2, A 0 , and B 0 are now the ionic strength and the Debye–Hu¨ckel constants on the molality scale. Thus, Table 3.8 Values of the Debye–Hu¨ckel Constants on the Molarity Scale Together with the Density and Relative Permittivity of Water in the Temperature Range 0–50 C T/ C

rs/kg L1

es

ADH / L1/2 mol1/2

BDH/nm1 L1/2 mol1/2

0 10 20 25 30 40 50

0.99984 0.99970 0.99820 0.99705 0.99565 0.99222 0.98804

87.81 83.99 80.27 78.46 76.67 73.22 69.90

1.1309 1.1455 1.1638 1.1741 1.1856 1.2100 1.2374

3.2473 3.2611 3.2784 3.2881 3.2987 3.3212 3.3462

ELECTROLYTE SOLUTIONS

127

1X 2 z i mi ð3:8:36Þ 2 0 0 where mi is the molality of ion i. The constants ADH , and BDH for dilute solutions (see equation (3.7.13)) are given by 1=2 0 ADH ¼ ADH rs ð3:8:37Þ I¼

and 0 BDH ¼ BDH ðrs Þ1=2

0 ADH

1=2

0 BDH

ð3:8:38Þ

¼ 1:1724 kg mol and ¼ 3:2832 nm kg mol1=2 Thus, at 25 C, on the molality scale. The Debye–Hu¨ckel equation as presented above is often called the extended Debye–Hu¨ckel law (EDHL) because a simpler expression is used for very dilute solutions. When the ionic strength is less than 0.001 M, the term in the denominator of equation (3.8.32) goes to one, and one may write 1=2

ln yi ¼ ADH z2i I 1=2

1

1=2

ð3:8:39Þ

This result is known as the limiting Debye–Hu¨ckel law (LDHL) and gives a remarkably simple way of estimating the eﬀects of non-ideality in very dilute electrolyte solutions. Although the model does not recognize the ﬁnite size of the ions in solution or the structure of the solvent in a molecular sense, it points out the importance of the ionic strength in determining the degree of departure from ideality. In order to examine the predictions of the model with respect to experimental data, one must estimate the mean molar activity coeﬃcient. On the basis of equation (3.6.17), this can be obtained from the relationship n ln y ¼ nþ ln yþ þ n ln y ¼ nþ z2þ ADH I 1=2 n z2 ADH I 1=2

ð3:8:40Þ

Electroneutrality requires that jnþ zþ j ¼ jn z j

ð3:8:41Þ

jnþ z2þ j ¼ jn zþ z j

ð3:8:42Þ

jn z2 j ¼ jnþ zþ z j

ð3:8:43Þ

so that

and

Using equation (3.6.11), the expression for the LDHL becomes ln y ¼ ADH jzþ z jI 1=2

ð3:8:44Þ

In the same way, the corresponding expression for the EDHL is ln y ¼

ADH jzþ z jI 1=2 1 þ BDH aI 1=2

ð3:8:45Þ

128

LIQUIDS, SOLUTIONS, AND INTERFACES

EXAMPLE

Estimate the mean ionic activity coeﬃcient for 0.04 M CaCl2 using the LDHL and the EDHL at 258C assuming that the ion size parameter a is 0.47 nm. Compare with the experimental value, which is 0.597. The ionic strength of the solution is I ¼ ð1=2Þ½0:04 4 þ 0:08 1 ¼ 0:12 M

ð3:8:46Þ

According to the LDHL, ln y ¼ Ajzþ z jI 1=2 ¼ 1:174 2 ð0:12Þ1=2 ¼ 0:8134 or y ¼ 0:443

ð3:8:47Þ

According to the EDHL, ln y ¼

0:8134 ¼ 0:5298 1 þ 3:288 0:47 ð0:12Þ1=2

or y ¼ 0:589

ð3:8:48Þ

The result from the EDHL is much closer to the experimental value, as one would expect. Experimental values of ln y for NaCl are plotted against the square root of the ionic strength in ﬁg. 3.6. Also shown are the values of ln y predicted by the LDHL. It is clear that the limiting law agrees with the experimental data only for the most dilute solutions (I < 0:001). At higher concentrations, the experimental results fall above the limiting law predictions, mainly because one must

Fig. 3.6 Plot of y on a logarithmic scale against the square root of the ionic strength, I 1=2 for aqueous NaCl at 258C. The straight line shows the prediction of the limiting Debye– Hu¨ckel law (equation (3.8.39)).

ELECTROLYTE SOLUTIONS

129

consider the eﬀect of ionic size as expressed through the extended Debye–Hu¨ckel equation. For electrolytes containing ions of higher charge the ﬁt between theory and experiment is limited to even more dilute solutions. In order to examine the eﬀect of ionic size on the value of ln y , the experimental data are plotted in another way. Equation (3.8.45) may be rearranged to give

Ajzþ z jI 1=2 ¼ 1 þ BaI 1=2 ln y

ð3:8:49Þ

Accordingly, a plot of the function on the left-hand side should give a straight line when plotted against I 1=2 with a slope equal to Ba and an intercept of unity on the y-axis. Such a plot with data for NaCl is shown in ﬁg. 3.7. It is apparent that these conditions are met for ionic strengths less than 0.1 M ðI 1=2 < 0:3Þ. At higher concentrations there are positive deviations from the value of the ordinate predicted by the EDHL. The slope of the initial linear portion is 1.39, from which the estimate of the ion size parameter is 420 pm. Several reasons can be cited for the failure of the Debye–Hu¨ckel theory at higher concentrations. One important reason is the fact that this model only considers the fact that the central ion has size, and ignores the ﬁnite size of the other ions in the atmosphere. As a result the thickness of the ionic atmosphere is underestimated in concentrated solutions. In addition, the extra work involved in introducing additional electrolyte ions into a concentrated solution while maintaining constant volume is neglected. This fact is mainly responsible for the experimentally observed increase in ln y with increase in ionic strength (see ﬁg. 3.6). Finally, the structure of the solvent, water, is strongly aﬀected by ion–solvent interactions. As the concentration of the electrolyte increases, the fraction of

Fig. 3.7 Plot of the activity coeﬃcient data for aqueous NaCl at 258C according to equation (3.8.49). The value of the ion size parameter a from the slope in the limit of low concentrations is 420 pm.

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LIQUIDS, SOLUTIONS, AND INTERFACES

water molecules associated with ions increases, and the dielectric permittivity of the solution decreases. This results in an increase in the strength of ion–ion interactions and eventually can lead to ion pairing. In the following section a method for estimating the activity coeﬃcients of electrolytes in more concentrated solutions is considered, using a version of the MSA which ignores the molecular nature of the solvent.

3.9 Ion-Ion Interactions According to the MSA Although the Debye–Hu¨ckel theory of electrolyte solutions gives a good description of their physical properties when they are very dilute, it fails in a serious way at higher concentrations, especially for electrolytes involving ions of higher charge. In these systems the neglect of the ﬁnite size of all constituent ions is an important defect and leads to overestimation of the eﬀects of the ion–ion interactions. One way of overcoming this defect is to represent the electrolyte solution as a collection of hard spheres of varying sizes, corresponding to the constituent cations, anions, and solvent molecules. This approach was described earlier in the discussion of ion–solvent interactions (section 3.5), and has also been applied to the assessment of ion–ion interactions, especially in very dilute solutions [12]. Such an approach is called non-primitive because it considers the discrete nature of the solvent molecules and ions which make up the electrolyte solution. A method of treating these systems which falls between the Debye–Hu¨ckel model and a non-primitive model represents the ions as hard spheres and the solvent as a dielectric continuum. Such an approach is deﬁnitely primitive because it neglects the molecular nature of the solvent, but it allows one to consider the eﬀects of ionic size more carefully than in the Debye–Hu¨ckel model, in which only the size of the central ion was taken into account. In the MSA, all of the ions have a ﬁnite size. As a result the distribution of ions around the central one is determined not only by the local electrostatic potential but also by how well they can be packed in space. It follows that the ionic distribution functions are not always smooth but can have an oscillatory character, especially near the point of reference at the central ion. When the cation and anion are assumed to have equal sizes the present MSA treatment is called a restricted primitive model. Although presentations of the unrestricted version of the MSA are available [18–20] the discussion here is limited to the restricted version. In the following model, the electrolyte is assumed to consist of ions having radius ri and charge number zi . Since the solution is electrically neutral, the sum of the concentration of each ion, c i times its valence zi must equal zero, that is X zi ci ¼ 0 ð3:9:1Þ i

The ions are embedded in a dielectric continuum of relative permittivity es . The MSA boundary conditions for this system are gij ðrÞ ¼ 0; and

r < 2ri

ð3:9:2Þ

ELECTROLYTE SOLUTIONS

cij ðrÞ ¼ buij ðrÞ ¼

zi zi be20 ; 4pe0 es r

r > 2ri

131

ð3:9:3Þ

On the basis of these conditions and the Ornstein–Zernike equation, one obtains the distribution functions gij ðrÞ and cij ðrÞ for all values of r. An important parameter obtained in the solution of this problem is , which is related to the thickness of the ionic atmosphere: !1=2 NL e20 b X z2i c i ð3:9:4Þ ¼ 4e0 es i ð1 þ 2ri Þ2 The concentration c i is expressed in moles per cubic meter. For the case that the ions all have the same radius ri , 2ri is equal to the distance of closest approach, a, deﬁned in ﬁg. 3.5. Then, the summation in equation (3.9.4) can be related to the ionic strength and the expression for can be written as k ð3:9:5Þ ¼ 2ð1 þ aÞ where k is the Debye–Hu¨ckel reciprocal distance deﬁned earlier (equation (3.8.13)). It is obvious that in the limit that the ion size parameter a goes to zero, 2 is equal to the reciprocal distance k. Solving equation (3.9.5) for , one obtains ¼

ð1 þ 2akÞ1=2 1 2a

ð3:9:6Þ

In dilute solutions, the pair distribution function is gij ðrÞ ¼

bzi zj e20 2r e 4pe0 es r

ð3:9:7Þ

This is exactly the form expected for gij ðrÞ on the basis of the Debye–Hu¨ckel model if 2 replaces k as the screening parameter. The thermodynamic properties of the electrolyte solution are derived by ﬁrst calculating the excess internal energy due to electrostatic interactions, Ues . This quantity is given by N e2 X z2i c i ð3:9:8Þ Ues ¼ L 0 4pe0 es 1 þ 2ri i where Ues has units of J m3 molec1 . For the case that all ions have the same size, this reduces to Ues ¼

NL e20 I 2pe0 es 1 þ a

ð3:9:9Þ

The corresponding excess Helmholtz energy is estimated using the thermodynamic relationship @ðbAes Þ ¼ Ues @b

ð3:9:10Þ

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LIQUIDS, SOLUTIONS, AND INTERFACES

After some algebra, one obtains the result Aes ¼ Ues þ

3 3pb

ð3:9:11Þ

Since b ¼ 1=ðkB TÞ, the term kB 3 =ð3pÞ can be identiﬁed as the excess entropy per unit volume associated with electrostatic interactions. One may now extract the electrostatic contribution to the single ion activity coeﬃcient, which is @ðbAes Þ be20 z2i bz2 Ues ðln yi Þes ¼ ¼ i ð3:9:12Þ ¼ NL @ci 4pe0 es ð1 þ aÞ 2NL I After application of equation (3.8.42), the expression obtained for the mean ionic activity coeﬃcient is ðln y Þes ¼

bjzþ z jUes 2NL I

ð3:9:13Þ

In order to complete the MSA estimate of ln yi one must add the hard-sphere contribution, which accounts for the fact that work must be done to introduce the ions as hard spheres into the solution. It is obtained from the Percus–Yevick model for non-interacting hard spheres. For the case that all ions (spheres) have the same radius, the result is (see equation (3.9.22)) ðln yi Þhs ¼

6z 3z2 2z þ þ 2 3

ð3:9:14Þ

where ¼1z

ð3:9:15Þ

and NL pa3 z¼

P

c i

i

6

ð3:9:16Þ

Since ðln yi Þ is equal to ðln yi Þhs when all the ions have the same size, the resulting expression for the mean ionic activity coeﬃcient is ln y ¼ ðln y Þes þ ðln y Þhs

ð3:9:17Þ

Estimation of this quantity requires that only one adjustable parameter be speciﬁed, namely the mean ionic diameter a. EXAMPLE

Estimate the mean activity coeﬃcient for 0.25 M NaCl according to the MSA assuming an ion size parameter equal to 360 pm and a temperature of 258C. For an ionic strength of 0.25 M, the Debye–Hu¨ckel parameter k is equal to 1.644 nm1 . The MSA is now estimated using equation (3.9.6): ¼

1 þ 2 0:36 1:644Þ1=2 1 ¼ 0:6635 nm1 2 0:36

ð3:9:18Þ

ELECTROLYTE SOLUTIONS

133

Now, Ues is calculated using equation (3.9.9). The constant NL e20 =ð2pe0 es Þ is equal to 3:541 106 J m mol1 so that Ues ¼

3:541 106 6:635 108 0:25 1000 1 þ ð0:6635 0:36Þ

ð3:9:19Þ

¼ 4:741 105 J m3 mol1 From equation (3.9.13) ðln y Þes ¼

4:741 105 ¼ 0:382 2 8:3144 298:2 250

ð3:9:20Þ

In order to estimate the hard-sphere contribution the packing fraction z is calculated ﬁrst: z¼

6:022 1023 p ð3:6 1010 Þ3 250 2 ¼ 7:36 103 6

ð3:9:21Þ

Using equation (3.9.14), ðln y Þhs ¼

6 7:36 103 3 ð7:36 103 Þ2 2 7:36 103 þ þ ¼ 0:060 0:9926 ð0:9926Þ2 ð0:9926Þ3 ð3:9:22Þ

Thus, ln y ¼ 0:382 þ 0:060 ¼ 0:322

ð3:9:23Þ

The resulting estimate of y (0.725) agrees very well with the experimental value (0.726). The MSA model was ﬁtted to the activity coeﬃcient data for aqueous NaCl solutions on the molarity scale following the procedure used in the above example. The results are shown in ﬁg. 3.8 for the case that the mean ionic diameter is assumed to be 360 pm. An excellent ﬁt was found for concentrations up to 0.3 M. At higher concentrations, the estimated value of ln y is higher than that found experimentally. This is clearly a result of the hard-sphere contribution, which is positive. Results obtained using the extended Debye–Hu¨ckel model are shown in the same ﬁgure. This model requires that a higher average ionic diameter be assumed and produces results which agree with experiment up to 0.1 M. The Debye–Hu¨ckel model overestimates the eﬀects of non-ideality at higher concentrations because it ignores the eﬀects due to the size of all the ions in the system, considering only the eﬀect of the size of the central ion. The ﬁt of the MSA to activity coeﬃcient data for aqueous electrolyte solutions can be considerably improved if one takes into consideration the decrease in solvent permittivity which accompanies the increase in electrolyte concentration. This phenomenon is clearly related to the eﬀect that ions have on solvent structure and was studied originally in aqueous solutions by Hasted et al. [21, 22]. More recently, data have been collected for a large number of electrolytes by Barthel and coworkers [23]. In the case of NaCl solutions, the change in dielectric permittivity with electrolyte concentrations up to 2 M is given by

134

LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 3.8 Plot of the mean ionic activity coeﬃcient for NaCl at 258C on a logarithmic scale against the square root of the molar ionic strength. The points give the experimental data; the curves give the theoretical ﬁts for the MSA and extended Debye–Hu¨ckel models with ion size parameters as indicated.

es ¼ 78:46 19ce þ 5c3=2 e

ð3:9:24Þ

At a concentration of 0.1 M, the value of es is 76.7 but when the concentration is 1 M, es falls to 64.5. The extent to which the permittivity changes with concentration also depends on the nature of the electrolyte. This aspect of solvent properties is discussed in more detail in section 4.6. When the variation in es with electrolyte concentration is considered the expression for ðln yi Þes becomes ðln yi Þes ¼

bz2i Ues be20 I @es þ 2NL I 2e0 e2s ð1 þ aÞ @c i

bz2 Ues bUes @es ¼ i 2NL I NL es @c i

ð3:9:25Þ

Making use of equation (3.9.13), the mean ionic activity coeﬃcient is ðln y Þes ¼

jzþ z jbUes bUes @es 2NL I nNL es @c e

ð3:9:26Þ

where @es @e @e ¼ nþ s þ n s @c e @cþ @c

ð3:9:27Þ

@es =@c e can be measured experimentally and in the present example is calculated from the concentration derivative of equation (3.9.24). The ﬁt of the MSA model with varying solvent permittivity is shown in ﬁg. 3.9. These results demonstrate the importance of considering the true solvent permit-

ELECTROLYTE SOLUTIONS

135

Fig. 3.9 As in ﬁg. 3.8, but with a varying dielectric permittivity for the solvent in the MSA model (see equation (3.9.26)).

tivity in estimating ion–ion interactions. The value of the size parameter a which yields the best ﬁt (436 pm) is somewhat larger than that found for the simpler MSA ﬁt with a constant permittivity equal to that of the pure solvent. At concentrations higher than 1 M, the MSA curve lies above the experimental results. This undoubtedly reﬂects a small extent of ion pairing in concentrated solutions of 1–1 electrolytes so that the true ionic strength is somewhat less than that estimated on the basis of the stoichiometric concentration. It should be noted that redeﬁning the Debye–Hu¨ckel constants so that they change with electrolyte concentration to reﬂect the corresponding change in es does not extend the concentration range over which this model ﬁts experimental data. This result emphasizes that it is important to include the ﬁnite size of all ions in a model which is applied in a concentration range greater than 0.1 M. Inclusion of the change in solvent permittivity in the MSA description is an eﬀective way of dealing with the change of solvent properties which accompany the addition of an electrolyte to a polar solvent. Since permittivity data are now available for a large number of electrolyte solutions in water [23], the MSA model can be applied to a wide variety of systems. However, there is one feature of electrolyte solutions which has been neglected in the treatments presented up to this point, namely, the existence of ion aggregates. This feature of electrolyte solutions is discussed in the following sections of this chapter.

3.10 The Thermodynamics of Ion Association Two important factors which determine whether ionic association is present in an electrolyte solution are the charge on the ions and the dielectric permittivity of the solvent. Their roles are clearly seen on the basis of Coulomb’s law, which shows

136

LIQUIDS, SOLUTIONS, AND INTERFACES

that the force between two ions increases with increase in the product of their charges and with decrease in the solvent’s permittivity, that is Fc ¼

zi zj e20 4pe0 es r2ij

ð3:10:1Þ

where zi and zj are the valences of the two ions, and rij , the distance between them. Another factor to be considered is chemical bonding between two ions to form a molecule. This phenomenon falls under the general classiﬁcation of weak electrolytes, which were discussed in section 3.6. Derivation of the thermodynamic equations for an electrolyte system with ion pairing follows the same procedure given for a weak electrolyte. However, in the following the ion pairing equilibrium is deﬁned in terms of an association process. For a 1–1 electrolyte, ion pairing is described as Mþ þ X ( + MXip

ð3:10:2Þ

where MXip represents the neutral ion pair. The association equilibrium constant Kas is given by aip ð3:10:3Þ Kas ¼ aM aX where aM and aX are the activities of the cation and anion, respectively, and aip , that of the ion pair. The association constant may be written in terms of the standard potentials of each of these species as follows: RT ln Kas ¼ mMþ þ mX mip

ð3:10:4Þ

Here mMþ and mX are the standard chemical potentials for the two ions on the molarity scale, and mip , that for the ion pair. As a result of ion pairing, a fraction ai of the dissolved electrolyte remains in solution as ions and a fraction ð1 ai Þ exists as ion pairs. Thus, the chemical potential of the electrolyte can be expressed as mMX ¼ ai mMþ þ ai mX þ ð1 ai Þmip

ð3:10:5Þ

where mMþ and mX are the chemical potentials of the ions and mip , that of the ion pair. Substituting equations (3.10.3) and (3.10.4) into equation (3.10.5), one obtains mMX ¼ mMþ þ mX þ RT ln aM aX

ð3:10:6Þ

This result shows that the chemical potential of the electrolyte can be expressed in terms of the activities of the two ions without considering that of the ion pair. It was obtained earlier for weak electrolytes as equations (3.6.38) and (3.6.40). Because of ion pairing the ionic activities aM and aX are less than they would be on the basis of the stoichiometric concentration of the electrolyte, ce . Expanding the activities in equation (3.10.3) in terms of activity coeﬃcients and concentrations, one can write Kas ¼

yip cip ð1 ai Þyip ¼ yM c M yX c X y2 a2i ce

ð3:10:7Þ

ELECTROLYTE SOLUTIONS

137

where yip is the activity coeﬃcient of the ion pair, cip , its concentration, yM and yX , the activity coeﬃcients for the individual ions, y , the mean ionic activity coeﬃcient, and cM and cX , the concentration of the two free ions, which are equal to each other. One may also write the activity product aM aX as aM aX ¼ a2i y2 c2e

ð3:10:8Þ

It is clear that the value of Kas must be known in order to assess the thermodynamic properties of an electrolyte system in which ion pairing takes place. The above equations may also be written on the molality scale (see section 3.6). It should be noted that experimental activity coeﬃcient data for systems known to undergo ion pairing are tabulated ignoring ion pairing. In this case the tabulated ionic activity coeﬃcient on the molality scale must be regarded as giving ai g , not . For example, moderate ion pairing occurs in NaF solutions. Given that the tabulated mean ionic activity coeﬃcient for 0.1 m NaF is 0.765 and assuming Kas ¼ 1 and gip ¼ 1, the value of ai in this solution is 0.941 on the basis of equation (3.10.7). It follows that the true mean ionic activity coeﬃcient is 0.765/0.941, that is, 0.813. This calculation requires a value of Kas which must be obtained on the basis of additional experimental information. Small changes in the assumed value of Kas have a large eﬀect on the result of the calculation, especially at high electrolyte concentrations. EXAMPLE

A reasonable estimate of the thermodynamic association constant for MgSO4 is 164. The experimental value of g in 0.4 m MgSO4 is 0.0756. Estimate the true value of g assuming that the activity coeﬃcient of the ion pair is unity. On the basis of equation (3.10.7), one may write on the molal scale with gip ¼ 1, 1a ¼ Kas ¼ 164 g2 a2 me

ð3:10:9Þ

Substituting for ai g , 0.0756, and for me , 0.4, one obtains 1 ai ¼ 0:375

ð3:10:10Þ

Thus, the fraction dissociated ai is 0.625 and the true activity coeﬃcient g is 0.121. The above example shows that the same analysis can be applied to a symmetrical electrolyte containing ions of higher charge, for example MgSO4. In the case of non-symmetrical electrolytes such as MgCl2, the ion association process can be more complex. For the 2–1 electrolyte MX2, two association steps are possible: M2þ þ X ( + MXþ

Kas1

ð3:10:11Þ

+ MX2 MXþ þ X (

Kas2

ð3:10:12Þ

and +

Formation of the ion pair MX is expected to take place under most circumstances. However, formation of the ion triplet MX2 may not be signiﬁcant, especially in aqueous solutions. Important factors determining the magnitude of the

138

LIQUIDS, SOLUTIONS, AND INTERFACES

association constants are the charges on the ions and the solvent permittivity, as can be seen from equation (3.10.1). An analysis of the properties of a system containing more than one association step to obtain the concentrations and activities of the component ions is obviously a complicated problem. However, if the association constants are known, methods exist for estimating the activity coeﬃcients of the system over a wide concentration range [24]. Considerable eﬀort has been made to develop a model for the parameter Kas on the basis of statistical theories using simple electrostatic concepts. The ﬁrst of these was proposed by Bjerrum [25]. It contains important ideas which are worth reviewing. He assumed that all oppositely charge ions within a certain distance of a central ion are paired. The major concept in this model is that there is a critical distance from the central ion over which ion association occurs. Obviously, it must be suﬃciently small that the attractive Coulombic forces are stronger than thermal randomizing eﬀects. Bjerrum assumed that at such short distances there is no ionic atmosphere between the central ion and a counter ion so that the electrostatic potential due to the central ion may be calculated directly from Coulomb’s law. The value of this potential at a distance r is ¼

zi e0 4pe0 es r

ð3:10:13Þ

where zi e0 is the charge on the central ion. The concentration of counter ions with charge zj e0 at the same distance is estimated using Boltzmann statistics so that crj ¼ c j expðzj f Þ

ð3:10:14Þ

where c j is the bulk concentration in ions per cubic meter and f ¼ e0 b. Combining these equations one obtains an expression for the probability PðrÞ of ﬁnding two oppositely charged ions at a distance r from each other: ! zi zj e20 b 2 PðrÞ ¼ 4pr cj exp ð3:10:15Þ 4pe0 es r This function has two opposing trends: ﬁrstly, as r increases the exponential term decreases due to the decrease in Coulombic attraction; secondly, the volume associated with the sphere containing possible counter ions increases as 4pr2 . Thus, the probability has a minimum whose location can be found by diﬀerentiating PðrÞ with respect to r. The condition for a minimum is

q

q dPðrÞ q B B ¼ 8prc j exp þ 4pr2 c j B2 exp ¼0 ð3:10:16Þ dr r r r where zi zj e20 b ð3:10:17Þ 4pe0 es Deﬁning rB as the distance corresponding to the minimum, it follows that qB ¼

rB ¼

zi zj e20 b qB ¼ 8pe0 es 2

ð3:10:18Þ

ELECTROLYTE SOLUTIONS

139

The value of the Bjerrum distance rB depends on the ionic charges, the nature of the solvent and the temperature but not on electrolyte concentration. For aqueous solutions of 1–1 electrolytes at 258C, it is equal to 357 pm. Bjerrum proposed that all ions contained in a sphere with this radius are paired. This is a reasonable proposal for aqueous systems, since ions of typical size would be close to contact at such small separations. However, if the solvent has a lower dielectric permittivity, the distance over which ions are considered to be paired increases and the assumption that they are in contact is more diﬃcult to accept. For example, when the solvent’s relative permittivity is 10, the Bjerrum distance rB increases to 2800 pm. A plot of the Bjerrum probability function against distance r is shown for an aqueous solution in ﬁg. 3.10. The probability of ﬁnding an oppositely charged ion rises rapidly from the minimum at 357 pm as the distance between ions gets smaller. An increase in probability is also seen as distance increases from the minimum. The function PðrÞ is limited for small values of r by the fact that oppositely charged ions can approach the central ion up to the distance of closest approach a. However, there is no limit in PðrÞ for large values of r. Bjerrum obtained an estimate of the number of paired ions by integrating PðrÞ from the distance of closest approach a to the minimum on the probability curve rc . In this way, he was able to obtain an estimate of the association constant Kas . Further details may be found in the monograph by Robinson and Stokes [G1]. As suggested by the above discussion, there are serious problems with the Bjerrum model. One of these relates to the fact that unreasonably large critical distances are involved in deﬁning an ion pair in solutions of low permittivity. The second relates to the fact that the probability distribution is not normalized and continues to increase with increase in distance r. The latter problem is eﬀectively avoided by considering only those values of PðrÞ up to the minimum in the curve.

Fig. 3.10 Bjerrum probability function PðrÞ for oppositely charged ions plotted against r, the distance from a central ion for a 1–1 electrolyte at 1 M in water at 258C with es ¼ 78:46:

140

LIQUIDS, SOLUTIONS, AND INTERFACES

Fuoss [26] revised the distribution function taking into account the fact that the probability should go to zero for large values of r. However, the modiﬁed function describing PðrÞ is essentially the same as that given by Bjerrum for small values of r. Therefore, the simpler theory is that usually used. Fuoss developed a new theory of ion association in 1958 [27] which overcame some of the diﬃculties associated with the Bjerrum approach. The cations in the solution were assumed to be conducting spheres of radius a and the anions to be point charges. The ions are assumed to be immersed in a dielectric continuum of permittivity es . Only oppositely charged ions separated by the distance a are assumed to form ion pairs. The resulting expression for the association constant is 4000pa3 NL jqB j Kas ¼ exp 3 a

ð3:10:19Þ

The most important quantity determining the magnitude of Kas is the ion size parameter a, which appears to the third power in the numerator and also in the exponential term. The role of the solvent and the charges on the two ions is expressed through the parameter qB (equation (3.10.17)). This quantity increases with the charge product jzi zj j and with decrease in the solvent permittivity. In the case of MgSO4, assuming a is equal to 450 pm, the value of jqB j=a is 6.35. The estimate of Kas according to equation (3.10.19) is 132. This result is the correct order of magnitude on the basis of other experimental evidence. The Fuoss estimate of Kas is based on a more reasonable model than that of Bjerrum and therefore is preferred. However, there are also problems with the Fuoss treatment in so far as it considers the solvent to be a dielectric continuum. Dielectric saturation eﬀects are expected to be important, especially near the ions involved in ion pair formation. The second problem relates to the choice of the eﬀective size for the ions. In the calculation made here the value of a for MgSO4 was chosen to be much bigger than the crystallographic radius of Mg2þ . This presumably is because the cation is strongly hydrated in aqueous solution. One is then faced with the question whether the ion pair involves contact of the two ions or whether it is better considered to be a species in which the two ions are separated by at least one water molecule. These questions can only be properly resolved using other experimental methods. In summary, both models are useful from a qualitative point of view in assessing the strength of ion pairing in a given system. They demonstrate the importance of the ionic charge and the dielectric permittivity of the medium in determining Kas . In order to obtain a good quantitative estimate of Kas , as, the size of the ions involved must be carefully determined.

3.11 Ion Association According to the MSA In the treatment of ionic activity coeﬃcients according to the MSA it was emphasized that the ﬁnite size of the ions is an important factor in estimating their Gibbs energy. Accordingly, the work done to introduce an additional ion to the solution

ELECTROLYTE SOLUTIONS

141

increases at constant solution volume with increase in electrolyte concentration. This is the hard-sphere contribution to the activity coeﬃcient which depends on the size of the ions and on their concentration. When ion pairing is present there are three solute species, the additional one being the ion pair. This species is expected to be larger than either the cation or the anion, and to be a factor in determining the hard-sphere contribution to the Gibbs energy. In more traditional treatments of ion pairing, the activity coeﬃcient of the ion pair was assumed to be unity. On the basis of the MSA it is greater than one, and depends on electrolyte concentration. For the case of symmetrical electrolytes with ion pairing, the electrolyte activity is given by aM aX ¼ a2i y2 c2e ð3:11:1Þ where ai is the fraction of the ions remaining free, y , the mean activity coeﬃcient on the molarity scale, and ce , the electrolyte molarity. On the basis of the ion pairing equilibrium Kas ¼

ð1 ai Þyip y2 a2i ce

ð3:11:2Þ

where Kas is the ion pairing equilibrium constant and yip , the activity coeﬃcient of the ion pair. Estimation of the activity coeﬃcients and the fraction ai requires knowledge of two parameters, namely Kas , and a, the mean diameter of the three solute species. As outlined earlier (section 3.9), y consists of an electrostatic contribution and a hard-sphere contribution. The ﬁrst is ðln y Þes ¼

jzþ z jbUes bUes @es 2NL I nNL es @c e

ð3:11:3Þ

where Ues is deﬁned by equation (3.9.9) and c e is the concentration in moles per cubic meter. The hard sphere contribution is given by equations (3.9.14)–(3.9.16) in the restricted approximation. It is then equal for all solute species so that ln yip ¼ ðln y Þhs ¼ ðln yi Þhs

ð3:11:4Þ

Finally, the experimentally available activity coeﬃcient is (y Þexp where ðy Þexp ¼ ai y ¼ ai ðy Þes ðy Þhs

ð3:11:5Þ

Given values of a and Kas , the activity coeﬃcient y is calculated in the following way. Initially assuming ai ¼ 1, the value of (y Þes is estimated using equations (3.11.3) and (3.11.4) with the values of c e and a. Now equation (3.11.2) may be solved for ai given Kas . The process is repeated using the new estimate of ai until successive estimates of y agree to the required precision. Results of such an analysis for the MgSO4 system are shown in ﬁg. 3.11. The best values of a and Kas for this system are 610 pm and 185 L mol1 , respectively. On the basis of dielectric relaxation experiments, the permittivity of MgSO4 solutions as a function of electrolyte concentration is given by es ¼ 78:5 31:8ce þ 12:8c3=2 e

ð3:11:6Þ

An excellent ﬁt between the MSA model and experimental data is possible for concentrations up to 1.5 M (see ﬁg. 3.11). At higher concentrations, the theory

142

LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 3.11 Plot of the eﬀective mean ionic activity coeﬃcient for MgSO4 at 258C on a logarithmic scale against the square root of the molar ionic strength. The curve gives the ﬁt with the modiﬁed MSA assuming a ¼ 610 pm and Kas ¼ 185 L mol1 ; the straight line shows the prediction of the limiting Debye–Hu¨ckel law.

predicts values of y which are higher than those found experimentally. This is probably because the eﬀective size of the ions, especially the strongly hydrated Mg2þ ion, decreases at high concentrations, where there are fewer water molecules available per ion. It is very interesting to examine how the mean activity coeﬃcient for the ions, y , the ion pair, yip , and the fraction of electrolyte dissociated, ai , vary with electrolyte concentration. These quantities are plotted against ce for the range over which the theory is successful in ﬁg. 3.12. The mean ionic activity coeﬃcient

Fig. 3.12 Plots of yip , y , and ai on a logarithmic scale for the MgSO4 system according to the modiﬁed MSA model at 258C against electrolyte concentration ce in water.

ELECTROLYTE SOLUTIONS

143

y falls to very low values as one would expect for a 2–2 electrolyte at high ionic strength. In this regard it should be remembered that a concentration of 1.5 M corresponds to an ionic strength of 6 M, so that the theory has been very successful in describing the electrostatic interactions. At the same time the activity coeﬃcient of the ion pair increases reaching values greater than 2. This has an important inﬂuence on the estimate of ai and the ﬁnal estimate of the overall mean electrolyte activity coeﬃcient ai y . The fraction of electrolyte dissociated is initially close to 0.6 and then slowly increases. This reﬂects changes in the apparent association constant with ionic strength and is an important factor in determining the ionic strength. On the basis of equation (3.11.2) the apparent association constant may be deﬁned as Kap ¼

Kas y2 yip

ð3:11:7Þ

It decreases with increasing ionic strength due to the corresponding decrease in y and increase in yip . As a result, there is a slow increase in ai , the fraction dissociated, at higher concentrations. In conclusion, the MSA provides an excellent description of the properties of electrolyte solutions up to quite high concentrations. In dilute solutions, the most important feature of these systems is the inﬂuence of ion–ion interactions, which account for almost all of the departure from ideality. In this concentration region, the MSA theory does not diﬀer signiﬁcantly from the Debye–Hu¨ckel model. As the ionic strength increases beyond 0.1 M, the ﬁnite size of all of the ions must be considered. This is done in the MSA on the basis of the hard-sphere contribution. Further improvement in the model comes from considering the presence of ion pairing and by using the actual dielectric permittivity of the solution rather than that of the pure solvent.

3.12. Concluding Remarks The theory of electrolyte solutions developed in this chapter relies heavily on the classical laws of electrostatics within the context of modern statistical mechanical methods. On the basis of Debye–Hu¨ckel theory one understands how ion–ion interactions lead to the non-ideality of electrolyte solutions. Moreover, one is able to account quantitatively for the non-ideality when the solution is suﬃciently dilute. This is precisely because ion–ion interactions are long range, and the ions can be treated as classical point charges when they are far apart. As the concentration of ions increases, their ﬁnite size becomes important and they are then described as point charges within hard spheres. It is only when ions come into contact that the problems with this picture become apparent. At this point one needs to add quantum-mechanical details to the description of the solution so that phenomena such as ion pairing can be understood in detail. The above comments also apply to the description of ion–solvent interactions. The traditional treatment of a solvent molecule is to represent it as a hard sphere with a central point dipole. Such a model ignores the details of the

144

LIQUIDS, SOLUTIONS, AND INTERFACES

chemical nature of the interactions between the ion and the surrounding solvent molecules. In summary, the models discussed in this chapter focus on the physical aspects of electrolyte solutions but they ignore the chemical aspects. This is especially apparent in the treatment of ion solvation where an empirical correction to the MSA model was applied to treat the diﬀerences in behavior seen for cations and anions in water. The same problem arises in using classical electrostatics to describe ion pairing. In spite of the fact that the Bjerrum and Fuoss models give a good qualitative description of an ion association, this phenomenon can only be understood in detail by using quantum-mechanical methods. Needless to say, such calculations in condensed media are much more diﬃcult to carry out. The problems outlined here can be greatly elucidated using spectroscopic methods. With the appropriate technique one can probe the strength of ion–solvent interactions, and measure the extent of contact ion pairing. Spectroscopic studies of electrolyte solutions have certainly greatly improved the understanding of these important systems. Major spectroscopic methods and results of their application to these systems are considered in detail in chapter 5.

General References G1. Robinson R. A.; Stokes, R. H. Electrolyte Solutions, 2nd ed.; Butterworths: London, 1968. G2. Petrucci, S. Ionic Interactions; Academic Press: New York, 1971; Vols. I and II. G3. Marcus, Y. Ion Solvation; Wiley-Interscience: New York, 1985. G4. Wagman, D. D.; Evans, W. H.; Parker, V. B.; Schumm, R. H.; Halow, I.; Bailey, S. M.; Churney, K.L.; Natall, R. L. J. Phys. Chem. Ref. Data 1982, 11, Supp. 2. G5. Lobo, V. M. M. Handbook of Electrolyte Solutions; Elsevier: New York, 1989. G6. Horvath, A. L. Handbook of Aqueous Electrolyte Solutions; Ellis Horwood: Chichester, 1985.

References 1. Ohtaki, H.; Radnai, T. Chem. Rev. 1993, 93, 1157. 2. Pauling, L. The Nature of the Chemical Bond, 3rd ed.; Cornell University Press: Ithaca, NY, 1960; Chap. 13. 3. Gourary, B. S.; Adrian, F. J., Solid State Phys. 1960, 10, 125. 4. Morris, D. F. C. Struct. Bonding 1968, 4, 63. 5. Shannon, R. D.; Prewitt, C. T. Acta Crystallogr. 1969, B25, 925. 6. Szasz, G. I.; Heinzinger, K.; Palinkas, G. Chem. Phys. Lett. 1981, 78, 194. 7. Jenkins, H. D. B.; Thakur, K. P. J. Chem. Educ. 1979, 56, 576. 8. Bowen, K. H.; Castleman, A. W. J. Phys. Chem. 1996, 100, 12911. 9. Tissandier, M. D.; Cowen, K. A.; Feng, W. Y.; Gundlach, E.; Cohen, M. H.; Earhart, A. D.; Coe, J. V.; Tuttle, Jr., T. R. J. Phys. Chem. A 1988, 102, 7787. 10. Fawcett, W. R. J. Phys. Chem. B 1999, 103, 11181. 11. Born, M. Z. Phys. 1920, 1, 45. 12. Chan, D. Y. C.; Mitchell, D. J.; Ninham, B. W. J. Chem. Phys. 1979, 70, 2946. 13. Garisto, F.; Kusalik, P. G.; Patey, G. N. J. Chem. Phys. 1983, 79, 6294. 14. Blum, L.; Fawcett, W. R. J. Phys. Chem. 1992, 96, 408. 15. Wertheim, M. S. J. Chem. Phys. 1971, 55, 4291. 16. Fawcett, W. R.; Blum, L. J. Chem. Soc. Faraday Trans. 1992, 88, 3339.

ELECTROLYTE SOLUTIONS

17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.

145

Debye, P.; Hu¨ckel, E. Phys. Z. 1923, 24, 185. Blum, L. Mol. Phys. 1975, 30, 1529. Blum, L.; Hoye, J. S. J. Phys. Chem. 1977, 81, 1311. Humﬀray, A. A. J. Phys. Chem. 1983, 87, 5521. Hasted, J. B.; Ritson, D. M.; Collie, C. H. J. Chem. Phys. 1948, 16, 1. Haggis, G. H.; Hasted, J. B.; Buchanan, T. J. J. Chem. Phys. 1952, 20, 1452. Barthel, J.; Buchner, R.; Mu¨nsterer, M. Electrolyte Data Collection, Part 2: Dielectric Properties of Water and Aqueous Electrolyte Solutions; Dechema: Frankfurt/Main, 1995. Pitzer, K. S. Activity Coeﬃcients in Electrolyte Solutions, 2nd ed.; CRC Press: Boca Raton, FL, 1991. Bjerrum, N. Kg. Danske Videnskab. Selskab. 1926, 7, 9. Fuoss, R. M. Trans. Faraday Soc. 1934, 30, 967. Fuoss, R. M. J. Am. Chem. Soc. 1958, 80, 5059.

Problems 1. Nitromethane (NM) is a polar solvent with a relative permittivity of 35.8 at 258C. It has a diameter of 431 pm when represented as a sphere. Estimate the Gibbs energy of solvation of K+, whose diameter is 276 pm, in NM according to the Born model and MSA at 258C. Compare these estimates with the experimental estimate, which is 300 kJ mol1 . 2. Estimate the entropy of solvation of K+ in NM given that the temperature coeﬃcient of the permittivity is 0:161 K1 . Use both the Born model and the MSA. Compare these with the experimental estimate of 181 J K1 mol1 at 258C. What are the contributions to long-range interactions which depend on des =dT, and local interactions which depend on dds=dT, in this estimate? 3. Estimates of the Gibbs energy of solvation of the alkali metal ions in acetonitrile are 464 (Li+), 374 (Na+), 307 (K+), 288 (Rb+), and 257 kJ mol1 (Cs+) at 258C. Estimate the MSA parameter ds for acetonitrile using a plot of ðs Gm Þ1 against cation radius. Compare this result with the value estimated from the permittivitry which is equal to 35.9. 4. The activity coeﬃcient on the molality scale for NaI at 258C is 0.938 at 0.1 m and 0.991 at 1 m. Estimate the corresponding concentrations in mol L1 using the density data summarized in table 3.6, and hence, the activity coeﬃcients on the molarity scale. 5. Use the extended Debye–Hu¨ckel theory to estimate the mean ionic activity coeﬃcient for Na2SO4 at concentrations of 0.01 and 0.1 M and 258C assuming an ion size parameter of 400 pm. Also calculate the mean electrolyte activity and the electrolyte activity.

146

LIQUIDS, SOLUTIONS, AND INTERFACES

6. Given the following activity coeﬃcient data determine the average ionic radius for each electrolyte according to the extended Debye–Hu¨ckel equation. All data were obtained in water at 258C. y Conc/M

NaCl

CaCl2

0.001 0.002 0.005 0.01 0.02 0.05 0.1 0.2 0.5

0.965 0.953 0.927 0.900 0.869 0.820 0.780 0.738 0.687

0.888 0.851 0.787 0.729 0.664 0.577 0.517 0.469 0.444

Rearrange the extended Debye–Hu¨ckel equation so that the experimental data may be plotted as a straight line depending on the square root of the ionic strength whose slope is proportional to the average ionic radius. 7. On the basis of the equation ln y ¼

ADH zA zB I 1=2 þ CI 1 þ BDH aI 1=2

derive an expression for the ionic strength at which ln y is a minimum. Given that the minimum value of y for CaCl2 is 0.444, and using the results obtained in question 6, determine the appropriate value of C to ﬁt the equation to these data. Calculate y for ce ¼ 1 and 3 M. Then plot ln y versus I 1=2 showing the prediction of the LDHL on the graph. 8. The true solubility product (I ¼ 0) for calcium hydroxide is 6:5 106 at 258C. Estimate its solubility in (a) water, and (b) 0.1 M NaNO3 using the EDHL ðBDH a ¼ 1Þ to estimate activity coeﬃcients. Calculate the pH of the solutions in each case. 9. Estimate the activity coeﬃcient for KBr in water at 258C using the MSA with an ion size parameter of 470 pm at concentrations of 0.1 M and 0.5 M. Estimate the solution permittivity using equation (3.9.18). Compare with the experimental values, which are 0.818 (0.1 M) and 0.666 (0.5 M). 10(a). Use the Fuoss equation (3.10.19) to estimate an association constant for ZnSO4 assuming an ion size parameter of 650 pm. (b). Estimate the fraction a of free ions using the formula

ELECTROLYTE SOLUTIONS

Kas ¼

147

1a g2 a2 ce

which assumes that the activity coeﬃcient of the ion pair is unity. It is necessary to use an iterative technique to solve this problem. 11(a). Estimate the Debye–Hu¨ckel constants ADH and BDH in dimethylsulfoxide at 258C given that the density of DMSO is 1.096 g cm3 and its permittivity 46.7. (b). Use the following activity coeﬃcient data for LiCl in DMSO to estimate an ion size parameter for the electrolyte on the basis of the EDHL. m

g

0.005 0.01 0.02 0.03 0.05 0.08 0.10 0.12

0.859 0.803 0.736 0.698 0.644 0.600 0.578 0.559

4

Polar Solvents

Peter Debye was born in 1884 at Maastricht, the Netherlands, where he also grew up. After secondary school he studied electrical technology at the Institute of Technology in Aachen and graduated in 1905. After two years as an assistant in the same institute, Debye moved to Munich University to study physics and obtained a Ph.D. in physics in 1908. In 1911, he was appointed Professor of Theoretical Physics at Zurich University. He Peter Joseph Wilhelm Debye held chairs at Utrecht, and Go¨ttingen, before moving to ETH Zurich in 1920 as Professor of Physics and Principal. In 1927, he held the same post at Leipzig University and in 1934 he was appointed Director of the Max Planck Institute at the Kaiser Wilhelm Institute of Physics in Berlin. With the outbreak of war in Europe, Debye moved to Cornell University in the United States as Professor of Chemistry and Chair of the Chemistry Department. He held that position until his retirement in 1952. Debye’s work is remembered in several, quite diﬀerent, areas of physical chemistry. One of his ﬁrst important contributions was made in 1912 to the theory of speciﬁc heat of solids, in which he extended the earlier work of Einstein. In 1923, while at Zurich, he published the famous theory of strong electrolytes, together with Erich Hu¨ckel. He worked extensively on the dielectric properties of matter and published his famous monograph Polar Molecules [1] in 1929. Another area in which he contributed signiﬁcantly was X-ray crystallography. Debye was awarded the Nobel Prize for Chemistry in 1936 for ‘‘his contribution to explaining the structure of molecules, his research into dipole moments, and his work on the electronic diﬀraction of gases.’’ His devotion to science gained him many distinctions and international awards. He died in Ithaca, New York, in 1966.

4.1 What Constitutes a Polar Liquid? Polar solvents are those liquids whose relative permittivity is suﬃciently high that electrolytes can be dissolved in them. The best-known example of such a liquid is water. The oxygen end of this simple molecule is electron-rich and can stabilize cations. The hydrogen atoms are electron-poor and thus are involved in the solvation of anions. The structure of pure water is very much inﬂuenced by the 148

POLAR SOLVENTS

149

hydrogen bonding between the negative end of the molecular dipole at oxygen and a hydrogen atom on an adjacent molecule. The special properties of water as a solvent for electrolytes are the central reason for its importance in living systems. There are many other solvents which can be classiﬁed as polar. Some of them, such as the alcohols, have the same polar group as the water molecule, namely, the hydroxyl group –OH. These solvents are also involved in hydrogen bonding, and are generally classiﬁed as protic. Other examples of protic solvents are simple amides such as formamide and acetamide. In these systems, the protic group is –NH2, the hydrogen atom being involved in hydrogen bonding with the oxygen atom in the carbonyl group on an adjacent molecule. There are other polar solvents which are not protic. These involve liquids with large dipole moments. Some examples are acetonitrile, propylene carbonate, and dimethylsulfoxide. In each case, the solvent molecule possesses an electronegative group which is rich in electrons. The opposite end of the molecule is electron deﬁcient but does not have acidic hydrogen atoms which can participate in hydrogen bonding. This class of solvents is called aprotic. In this chapter, the properties of polar solvents are discussed, especially as they relate to the formation of electrolyte solutions. Polar solvents are arbitrarily deﬁned here as those liquids with a relative permittivity greater than 15. Solvents with zero dipole moment and a relative permittivity close to unity are non-polar. These include benzene, carbon tetrachloride, and cyclohexane. Solvents with relative permittivities between 3 and 5 are weakly polar, and those with values between 5 and 15 are moderately polar. The latter systems are not considered in the discussion in this chapter. Initially, some relevant thermodynamic and molecular properties of polar solvents are considered. Then, their dielectric properties are considered in detail. Ion solvation in these solvents is also discussed with emphasis on some non-thermodynamic methods of dividing experimentally measured data for electrolytes into contributions for the cation and anion. Finally, the important characteristics of the solvent in its direct interaction with the solute, namely, its acidity and basicity, are also described.

4.2 Some Important Properties of Polar Solvents A number of properties, including dielectric properties, should be considered in assessing a given polar solvent. Important bulk properties are density, vapor pressure, thermodynamic properties related to vaporization, heat capacity, viscosity, compressibility, and surface tension. Some of these are summarized in table 4.1 and are discussed brieﬂy in this section. Water is unique among polar solvents in that it is a small molecule with a low molar volume. As a result, the concentration of water in pure water is 55.5 M. This means that the mole fraction of water in dilute aqueous solutions is close to one, and the partial molar quantities in these solutions are close to the corresponding quantities for the pure solvent. Other solvents have considerably higher molar volumes, and therefore, lower concentrations in the pure solvent. The

Table 4.1 Thermodynamic Properties of Selected Polar Solvents at 25 Ca

Solvent

Molecular Mass /g mol1

Molar Volume /cm3 mol1

Enthalpy of Vaporization /kJ mol1

Entropy of Vaporizationb /K1 mol1

Molar Heat Capacity /J K1 mol1

18.02 32.04 46.07 60.10 74.12 45.04 59.07

18.07 40.73 58.69 75.14 92.0 39.89 59.14

44.0 37.4 42.3 47.5 52.4 65.0 59.6

108.9 104.8 110.1 111.3 110.4 98.1 99.8

75.3 81.5 112.3 143.8 177.1 107.6 123.8

58.08 41.05 103.12 69.11 87.12 73.09 78.13 179.20 99.13 123.11 61.04 55.08 102.08 120.17 116.16

74.04 52.86 103.06 87.87 93.05 77.4 71.3 175.7 96.4 102.7 54.0 70.9 85.2 95.3 120.3

31.3 32.9 55.5 39.3 49.2 47.5 52.9 61.1 54.0 55.0 38.3 36.1 65.3 79.5 45.5

88.4 84.0 98.9 88.1 98.7 90.0 93.3 112.2 94.0 84.2 91.9 83.6 93.1 107.3 101.4

124.9 91.5 190.3 155.8 176 148.4 153.2 321.3 166.4 177.1 105.8 119.7 160.2 180.0 229.6

Protic 1. 2. 3. 4. 5. 6. 7.

Water (W) Methanol (MeOH) Ethanol (EtOH) 1-Propanol (PrOH) 1-Butanol (BuOH) Formamide (F) N-methylformamide (NMF)

Aprotic 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. a

Acetone (AC) Acetonitrile (AcN) Benzonitrile (BzN) Butyronitrile (BuN) Dimethylacetamide (DMA) Dimethylformamide (DMF) Dimethylsulfoxide (DMSO) Hexamethylphosphoramide (HMPA) N-methylpyrrolidinone (NMP) Nitrobenzene (NB) Nitromethane (NM) Propionitrile (PrN) Propylene carbonate (PC) Tetramethylenesulphone (TMS) Tetramethylurea (TMU)

Obtained from compilations by Marcus [2] and Riddick et al. [G4]. Estimated at the normal boiling point.

b

POLAR SOLVENTS

151

concentration of methanol in pure methanol is 24.6 M; the corresponding concentration of 1-butanol is 10.9 M. The largest solvent molecule considered here is hexamethylphosphoramide, which has a concentration of only 5.7 M in the pure liquid. The molar volume of the solvent is obviously an important parameter in assessing the degree of departure from ideality of its solutions. The enthalpy of vaporization is an important parameter which can be used to assess intermolecular forces in a given liquid. As discussed earlier, the ratio vap Hm =Vm gives the internal pressure of the liquid (section 1.7). The entropy of vaporization is also useful. When it is measured at the normal boiling point, this quantity is close to 89 kJ mol1 in an unstructured solvent according to Trouton’s rule. The data in table 4.1 show that vap S is signiﬁcantly greater than 89 J K1 mol1 for the protic solvents, which are hydrogen-bonded liquids. However, this quantity is also larger in some aprotic solvents for which dipole– dipole interactions are strong. The dielectric properties of the solvents considered here are summarized in table 4.2. These properties are important in evaluating the solvation of ions in polar solvents under both static and dynamic conditions. The relative permittivity of a solvent at high frequencies, eop, can be calculated from the refractive index, nop, the relationship being eop ¼ n2op

ð4:2:1Þ

Relaxation parameters obtained by studying the frequency dependence of the relative permittivity are also summarized in this table. In the case of the aprotic solvents the summarized parameters were usually obtained by analyzing the experimental data on the basis of the Debye model, which is discussed in detail later in this chapter. Protic solvents follow a more complex behavior because of hydrogen bonding and are generally considered to exhibit multiple relaxations. For these systems the relaxation time listed in table 4.2 is that for the slowest solvent relaxation, whereas the value of the high-frequency permittivity, e1, is that for the fastest relaxation process. More details about the behavior of these systems are given later. It should be noted that e1, which is usually measured in the far infrared region, is slightly higher then eop, which is measured at visible optical frequencies. The distinction between these quantities results from the fact that the Debye treatment is based on a continuum model which does not recognize the molecular nature of the solvent and the features of its vibrational spectrum in the infrared region. Some molecular properties of polar solvents are summarized in table 4.3. The dipole moment and molecular polarizability are the molecular parameters which lead to the solvent permittivity. The other parameters listed are the molecular diameter and the Lennard–Jones interaction energy, eLJ. These are of interest in assessing the role of van der Waals forces in determining the properties of a polar liquid. In assembling the properties of polar solvents, attention has been focused on their dipolar properties, that is, their ability to stabilize electrolytes in solution. As will be seen in the following chapter, it is not suﬃcient to consider only the electrical properties of these molecules as deﬁned in classical electrostatics. The

Table 4.2 Dielectric Properties of Selected Polar Solvents at 25 Ca

Solvent

Relative Static Permittivity es

Temperature Coeﬃcient des =dT

Refractive Index nop

Temperature Coeﬃcient dnop =dT 104

Debye Relaxation Timec tD =ps

Temperature Coeﬃcientb HD =kJ mol1

High-Frequency Permittivity e1

78.46 32.70 24.55 20.33 17.51 111.0 182.4

0.360 0.197 0.147 0.142 0.132 0.72 1.6

1.3325 1.3265 1.3594 1.3837 1.3973 1.4468 1.4300

1.05 3.83 4.0 3.72 3.9 1.44 3.8

8.32 51.5 163 329 517 37.3 128

18.2 15.8 18.0 20.0 23.8 15.2 19.3

4.49 2.79 2.69 2.44 2.22 4.48 3.20

20.7 35.9 25.2 24.5 37.78 36.71 46.68 30.0 32.0 34.82 35.8 28.2 66.1 43.7 23.1

0.0977 0.160 0.091 0.108 0.213 0.178 0.106 0.175 0.140 0.180 0.161 0.119 0.240 0.115 —

1.3560 1.3416 1.5259 1.3820 1.4356 1.4282 1.4773 1.45 1.4680 1.5500 1.3796 1.3636 1.4209 1.4820 1.4493

5.0 4.5 4.8 4.3 5.4 4.6 3.58 3.6 5.0 4.6 4.5 4.5 3.75 3.4 —

3.2 3.2 37.9 6.6 16.0 10.4 18.9 80.0 21.0 45.0 4.0 5.0 43.1 — 31

6.5 8.3 (10.6) 8.8 11.6 10.0 11.7 (14.6) (12.1) 13.1 6.6 8.65 15.8 — —

1.89 2.26 3.80 (2.5) 3.04 2.94 5.3 3.3 4.06 4.10 2.0 (2.4) 4.14 — 4.5

Proticc 1. 2. 3. 4. 5. 6. 7.

W [5, 6] MeOH [5, 6] EtOH [5, 6] PrOH [5, 7] BuOH [7, 8] F [6, 9] NMF [6, 9]

Aprotic 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. a

AC [10] AcN [9, 11] BzN [12] BuN [11] DMA [9, 13] DMF [6, 9] DMSO [14] HMPA [15] NMP [16] NB [12] NM [6] PrN [11] PC [6, 9] TMS TMU [17]

The majority of the data were obtained from compilations by Marcus [2], Riddick et al. [G4], and Karapetyan and Eychis [3]. Speciﬁc references to relaxation data are given with each solvent. Numbers in parentheses are estimates. b Temperature coefﬁcient of D deﬁned as R @ln D/@(1/T). c In the case of the protic solvents which have several relaxation processes, the value of D is for the ﬁrst relaxation.

POLAR SOLVENTS

153

Table 4.3 Molecular Properties of Selected Polar Solvents

Solvent

Dipole Momenta p/debye

Polarizabilityb 103 ap /nm3

Diameterc s/nm

Lennard–Jones Energyd eLJ k1 B =K

1.85 1.70 1.69 1.58 1.66 3.73 3.83

1.47 3.26 5.13 6.96 8.79 4.23 6.05

0.274 0.371 0.436 0.498 0.540 0.382 0.452

79 234 324 — — — 320

2.88 3.92 4.18 4.07 3.80 3.82 3.96 5.54 4.09 4.22 3.46 4.05 4.98 4.81 3.47

6.41 4.42 12.54 8.11 9.63 7.90 7.99 18.97 10.62 12.97 4.95 6.26 8.56 10.77 12.80

0.476 0.427 0.574 0.532 0.548 0.517 0.491 0.698 0.569 0.574 0.431 0.477 0.536 0.581 0.544

362 275 520 — 450 380 333 670 — 609 298 — 400 — —

Protic 1. 2. 3. 4. 5. 6. 7.

W MeOH EtOH PrOH BuOH F NMF

Aprotic 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

AC AcN BzN BuN DMA DMF DMSO HMPA NMP NB NM PrN PC TMS TMU

From gas phase data when possible [4] or from the compilations of Marcus [2] (1 debye is equal to 3:3356 1030 C m). b Estimated using the Lorentz–Lorenz equation (4.3.21). c Estimated from gas phase solubilities [18]; when these data were not available, the solvent’s diameter was estimated on the basis of a linear correlation between existing experimental data and the cube root of the molar volume. d From the compilations of Marcus [2]. a

speciﬁc chemical interactions between an ion and polar solvent molecule are also important. Methods of including consideration of both electrical and chemical interactions are outlined later in this chapter.

4.3 The Static Solvent Permittivity on the Basis of Continuum Models The task of statistical thermodynamics in describing the dielectric properties of a liquid is to develop a model for the liquid permittivity in terms of the dipole moment and polarizability of the component molecules. The usual approach has been to assume that the liquid molecule can be represented as a polarizable hard

154

LIQUIDS, SOLUTIONS, AND INTERFACES

sphere with a point dipole which is embedded in a dielectric continuum. In the presence of an applied external ﬁeld E, the molecule experiences a local ﬁeld Ee . An important feature of the model is the method used to estimate the local ﬁeld. A. The Debye Model The ﬁrst solution to this problem as given by Debye [1] for the case of dipolar molecules in the gas phase. The potential energy of a molecule with dipole moment p in a ﬁeld E is equal to u ¼ pE cos y

ð4:3:1Þ

where y is the angle between the ﬁeld vector E and the dipole. The degree of orientation of the molecules in the ﬁeld varies with the temperature upon which randomizing thermal motion depends. The average value of the angle y is now determined on the basis of the Boltzmann distribution law. The number of molecules lying within a solid angle, d, in the direction y, is given by dNV ¼ AN ebu d

ð4:3:2Þ

where AN is a constant related to the total number of molecules per unit volume, NV and b ¼ 1/(kB T). It follows that 2p ð

NV ¼

AN expðbp E cos yÞd

ð4:3:3Þ

0

where the integration is carried out over all possible values of the solid angle, . The increment d can be estimated by considering cones of semi-angle y and y þ dy deﬁned with respect to the ﬁeld vector, E. On this basis it follows that d ¼ 2p sin y dy

ð4:3:4Þ

The component of the dipole moment in the direction of the ﬁeld is p cos y. To obtain the average value hpi, one adds up values of p cos y over all possible angles. Thus, the average is Ðp hpi ¼

0

p cos y expðbp E cos yÞ sin y dy Ðp

ð4:3:5Þ expðbp E cos yÞ sin y dy

0

After integration by parts one obtains hpi ¼ p cothðbpEÞ

1 bE

ð4:3:6Þ

The function coth x 1=x is called the Langevin function so that the average dipole moment is equal to its magnitude times the Langevin of bpE. The value of bpE is usually much smaller than unity. For instance, the value of bp for a molecule with a dipole moment of 3 debyes is 2.4 109 m V1 at 25 C. The applied ﬁeld would have to be 4.1 108 V m1 in order to make bpE unity. The

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155

applied ﬁeld is normally much smaller, so that the average value of the dipole moment can be estimated using the value of coth(bpE) in the limit that bpE is small. Expanding the exponential terms in coth(bpE) retaining the ﬁrst four terms in the expansion, one obtains b2 p2 E2 1 bpE 2! þ cothðbpEÞ

3 b3 p3 E3 bpE 2bpE þ 2 3! 2þ2

ð4:3:7Þ

Thus, the approximate expression for hpi is hpi ¼

bp2 E 3

ð4:3:8Þ

The total polarization of the medium per unit volume considering also the molecular polarizability becomes ! bp2 P ¼ NE p þ ð4:3:9Þ 3 The units of the polarization P are coulombs per square meter; this quantity gives the eﬀective dipole of the medium per unit volume containing NV molecules. EXAMPLE

A parallel-plate capacitor is set up at 25 C with a voltage of 105 V between the plates, which are separated by 1 mm. Dimethylsulfoxide (DMSO) vapor at low pressure is introduced into the gap between the plates. Estimate the average dipole moment of each DMSO molecule in the direction of the ﬁeld. The ﬁeld in the capacitor gap is 108 V m1 . Since a dilute gas is present in the gap the local ﬁeld can be assumed to be equal to the applied ﬁeld. The dipole moment of DMSO is 3.96 debyes (table 4.3): p ¼ 3:96 3:336 1030 ¼ 1:32 1029 C m

ð4:3:10Þ

The average dipole moment in the direction of the ﬁeld is hpi ¼

ð1:32 1029 Þ2 108 ¼ 1:41 1030 C m 3 1:38 1023 298:2

ð4:3:11Þ

This is equivalent to 0.423 debyes. Even though the ﬁeld is quite high, the eﬀects of thermal motion result in the average dipole moment of the molecule in the direction of the ﬁeld being much less than the permanent dipole moment. In applying this relationship to dipolar molecules in dense gases and liquids the ﬁeld E is understood to be the local or eﬀective ﬁeld Ee. In order to estimate Ee one must solve the statistical mechanical problem which relates the local quantity to the macroscopic applied ﬁeld E. The problem is solved by estimating the local ﬁeld in a spherical cavity within the dielectric (ﬁg. 4.1). The cavity is assumed to have molecular dimensions with diameter a. The material within the sphere is considered in terms of individual molecules, whereas that outside the sphere is

156

LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 4.1 Schematic diagram of a spherical cavity within a homogenous dielectric between the plates of a parallel-plate capacitor.

assumed to have homogeneous properties with an isotropic relative permittivity es. One also imagines the dielectric to be between the plates of a parallel-plate capacitor so that the ﬁeld and polarization vectors have a deﬁned direction. In order to estimate Ee one must consider three contributions, namely, the ﬁeld E1 imposed on the medium via the charge on the plates of the capacitor, the ﬁeld E2 due to the polarization charge on the surface of the spherical cavity, and ﬁnally, the ﬁeld E3 due to the molecules within the sphere. E1 is obviously identiﬁed with the experimentally controlled ﬁeld E. The ﬁeld E2 is calculated by integrating the polarization in the direction of the ﬁeld over the surface of the spherical cavity. This gives ðp E2 ¼ 0

ðP cos yÞð2pr2 sin y dyÞ 4pe0 r2

ð4:3:12Þ

where 2pr2 sin y dy is the area of an element on the sphere deﬁned by the angles y and y þ dy. The result is E2 ¼

P 3e0

ð4:3:13Þ

In order to estimate E3, one must know how the individual molecules are arranged within the cavity. For example, if they are located on a cubic lattice or when the molecules form an ideal gas, E3 is zero. In the solution to the problem given here, E3 is assumed to be zero. Thus, Ee ¼ E þ

P 3e0

ð4:3:14Þ

This relationship was originally derived by Lorentz [19]. In order to proceed further one must make use of the fundamental electrostatic relationship between the electric displacement D, the electric ﬁeld E, and the polarization P, which is D ¼ e0 E þ P ¼ e0 es E

ð4:3:15Þ

P ¼ e0 Eðes 1Þ

ð4:3:16Þ

It follows that

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157

It is clear from these relationships that the electric displacement D has the same units as the polarization P, namely, C m2 . Now, from equation (4.3.14) Ee ¼

es þ 2 E 3

ð4:3:17Þ

Identifying the ﬁeld in equation (4.3.9) as the local ﬁeld Ee when this expression is applied to liquids, and substituting in equation (4.3.16) for P, one obtains the result ! es 1 N bp2 ¼ þ ð4:3:18Þ es þ 2 3e0 p 3 for N molecules. For one mole of molecules, one may write Nv ¼

NL N ¼ Lr Vm M

ð4:3:19Þ

where NL is the Avogadro constant, Vm , the molar volume of the liquid, r, its mass density, and M, its molecular mass. Equation (4.3.18) is then rewritten in the form ! es 1 M N L bp2 ¼ Pm ¼ þ ð4:3:20Þ es þ 2 r 3e0 p 3 where the quantity on the left-hand side is called the molar polarization. It is familiar from the analysis of dielectric permittivity data for liquids and solids, and is usually expressed in units of cubic centimeters per mole. Equation (4.3.20) is the well-known Debye equation. For pure liquids, the Debye equation suggests that the molar polarization should be a linear function of the reciprocal temperature. Furthermore, one should be able to analyze relative permittivity data for a polar liquid like water as a function of temperature to obtain the dipole moment and polarizability from the slope and intercept, respectively. In fact, if one constructs such a plot using data for a polar solvent, one obtains results which are unreasonable on the basis of known values of p and ap from gas phase measurements. The reason for the failure of the Debye model in liquids is the fact that it neglects the ﬁeld E3 due to dipoles in the immediate vicinity of a given molecule. However, it provides a reasonable description of the dielectric properties of dilute polar gases. In liquids, relatively strong forces, both electrostatic and chemical, determine the relative orientation of the molecules in the system, and lead to an error in the estimation of the orientational component of the molar polarization. There are two circumstances under which equation (4.3.20) proves useful. The ﬁrst of these is at frequencies suﬃciently high that the orientational component of Pm is eﬀectively zero. Then, one can write eop 1 M N L ap Rm ¼ ¼ ð4:3:21Þ eop þ 2 r 3e0 This is called the Lorentz–Lorenz equation, and is used to estimate the molecular refraction Rm from the refractive index or eop. Since the polarizability ap is often

158

LIQUIDS, SOLUTIONS, AND INTERFACES

not known from gas phase data, it has been customary to estimate it for many molecules using equation (4.3.21). EXAMPLE

Given that the refractive index of water at 25 C is 1.3325 and its molar volume 18.07 cm3, estimate the molecular polarizability of the water molecule. Then estimate the same quantity at 50 C where the value of n is 1.3291, and the molar volume 18.234 cm3. At 25 C eop ¼ n2 ¼ 1:7756

ð4:3:22Þ

eop 1 0:7756 18:07 ¼ 3:712 cm3 mol1 Vm ¼ 3:7756 eop þ 2

ð4:3:23Þ

Thus,

The corresponding value of ap is ap ¼

3:712 106 3e0 ¼ 1:637 1040 C m2 V1 NL

ð4:3:24Þ

This result may be expressed in the more customary units of cubic meters or cubic nanometers by dividing by 4pe0 . Thus, ap is equal to 1:472 103 nm3 at 258C. When the calculation is repeated at 508C the result is ap ¼ 1:471 103 nm3 . One expects the polarizability to be independent of temperature in a range where the electrons in the molecule remain in the same molecular orbitals. The small change in the polarizability reﬂects the weakness of the Lorentz–Lorenz model, which is based on continuum concepts. However, the estimated change is small, so that one may assume that the model is reasonably good. The second situation where equation (4.3.20) is often applied is to the analysis of data for the dielectric permittivity of dilute solutions of a polar compound in a non-polar solvent such as benzene or carbon tetrachloride. Under these circumstances local structure due to dipole–dipole interactions can be neglected, and the dipole moment of the polar solute can be calculated from the change in molar polarization with concentration of the polar solute. Then, the molar polarization can be written es 1 ðM1 x1 þ M2 x2 Þ ð4:3:25Þ Pm ¼ x1 Pm1 þ x2 Pm2 ¼ es þ 2 r where Pm1 is the molar polarization of the non-polar solvent, Pm2 , that due to the polar solute, and r, the density of the solution. Pm1 is assumed to be constant, independent of solution composition, and Pm2 is determined as a function of the mole fraction x2 . From the experimental data, one extrapolates to x2 ¼ 0 to obtain the limiting value, P0m2 . The dipole moment is then calculated using the Debye equation with the polarization component being estimated from the corresponding data for the refractive index. Although this method has been often

POLAR SOLVENTS

159

applied, it can be criticized on fundamental grounds [G1]. Therefore, molecular data obtained using this approach should be regarded as approximate. EXAMPLE

Using the data given below for dilute solutions of nitromethane in benzene, calculate the molar polarization and refraction of the solute in the limit of inﬁnite dilution. Then, estimate the dipole moment of nitromethane on the basis of the Debye model. The following data were reported by Balakier [20] at 293.2 K. Mole Fraction of Nitromethane x2

Solution Density r/g cm3

Relative Permittivity es

Refractive Index nop

0.0000 0.0197 0.0392 0.0593 0.0798

0.8789 0.8830 0.8860 0.8893 0.8926

2.284 2.587 2.863 3.188 3.457

1.5011 1.4992 1.4976 1.4958 1.4943

Given that the molecular weight of nitromethane is 61.04 and that of benzene, 78.12, one ﬁrst calculates the molar volume of the solution. For example, when x2 ¼ 0:0197, Vm ¼

61:04 0:0197 þ 78:12 0:9803 ¼ 88:09 cm3 mol1 0:8830

ð4:3:26Þ

The molar polarization is estimated using the dielectric constant following equation (4.3.20): Pm ¼ 88:09

1:587 ¼ 30:48 cm3 mol1 4:587

ð4:3:27Þ

The molar refraction is calculated in the same way using the optical dielectric constant, that is, the square of the refractive index: Rm ¼ 88:09

ð1:5011Þ2 1 ¼ 25:87 cm3 mol1 ð1:5011Þ2 þ 2

ð4:3:28Þ

The results obtained as a function of mole fraction are tabulated below. Mole Fraction of Nitromethane x2

Molar Volume Vm =cm3 mol1

Molar Polarization Pm =cm3 mol1

Molar Refraction Rm /cm3 mol1

0.0000 0.0197 0.0392 0.0593 0.0798

88.88 88.09 87.42 86.71 85.99

26.64 30.48 33.49 36.57 38.72

26.19 25.87 25.61 25.32 25.05

160

LIQUIDS, SOLUTIONS, AND INTERFACES

Now one calculates the molar polarization due to the solute. For example, when x2 ¼ 0:0197, Pm2 ¼

30:48 26:64 0:9803 ¼ 221:4 cm3 mol1 0:0197

ð4:3:29Þ

Similarly, for the molar refraction due to the solute: Rm2 ¼

25:87 26:19 0:9803 ¼ 10:1 cm3 mol1 0:0197

ð4:3:30Þ

Values of Pm2 and Rm2 are plotted against the mole fraction x2 in ﬁg. 4.2. Extrapolation to x2 ¼ 0 gives the values of these parameters in the limit of inﬁnite dilution. They are Pm20 ¼ 232:7 cm3 mol1 and Rm20 ¼ 9:8 cm3 mol1 . From equation (4.3.20), one may write Pm20 Rm20 ¼

NL bp2 9e0

ð4:3:31Þ

Thus, 9 8:854 1012 1:381 1023 293:2 ð232:7 9:8Þ 106 p¼ 6:022 1023 ¼ 1:09 1029 C m ¼ 3:27 debyes

!1=2

ð4:3:32Þ

B. Other Models for the Static Permittivity Another approach to a continuum model for the electric permittivity which considers the ﬁeld E3 was given by Onsager [21]. This model also makes use of a

Fig. 4.2 Plot of the molar polarization Pm2 and molar refraction Rm2 for nitromethane against its mole fraction in benzene.

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161

spherical cavity but limits it to one molecule. As a result it is applicable only to liquids without strong directional forces. The equation obtained by Onsager is ðes eop Þð2es þ eop Þ2 es ðeop þ 2Þ2

Vm ¼

NL bp2 9e0

ð4:3:33Þ

where Vm is the molar volume of the liquid. It is clear that the distortional contribution to the polarization is not explicitly available in this expression but is included in a complex way in the quantity on the left-hand side. The model corrects some of the problems of the Debye model but in the end it still fails to account for the correlation of a given dipole to its neighbors. Further developments in the theory of the structure of polar liquids included estimates of the correlation of a given dipole to its neighbors. Important contributions were made in this direction by Kirkwood [22] and Fro¨hlich [23]. In Kirkwood’s model, the ﬁeld E3 is calculated by considering all possible orientations of surrounding dipoles in a spherical cavity for a ﬁxed orientation of the central dipole. By averaging over these orientations, Kirkwood obtained an improved estimate of the polarization of the medium. For the case of nonpolarizable dipoles the result is ðes 1Þð2es þ 1Þ N g bp2 Vm ¼ L k 3e0 3es

ð4:3:34Þ

where gK is a correlation parameter describing the degree of local order. This parameter is unity when the average dipole moment of the molecules immediately surrounding a central molecule which is held ﬁxed is the same as the dipole moment of an isolated molecule. In other words, ﬁxing the position of one molecule does not aﬀect the positions of the surrounding dipoles except through longrange electrostatic forces. On the other hand, if the neighboring dipoles tend to line up in a parallel fashion as a result of the ﬁxing of a central dipole, the parameter gK rises to values greater than unity. Similarly, if ﬁxing one dipole tends to line up the surrounding dipoles in an antiparallel direction, the parameter gK will be less than one. Equation (4.3.34) diﬀers from the Onsager equation (equation (4.3.33)) in two ways. If the distortional contribution in equation (4.3.33) is neglected by setting eop equal to one, and the correlation between dipoles is neglected by setting gK equal to one in equation (4.3.34), these expressions are the same. The distortional contribution is not an important factor for polar molecules so that the version of the Kirkwood equation given here may be applied to the systems considered in this chapter. Inclusion of the correlation parameter gK in this equation gives an important improvement in the description of the interactions between molecules in these systems. This is especially important when the interactions are chemical in nature, for example, when hydrogen bonding is present. In principle, it is possible to calculate gK if the structure of the liquid is known, but in general, it is considered to be an adjustable parameter. Dielectric permittivity data for water and acetonitrile in the temperature range from 0 to 508C are plotted according to the Kirkwood equation in ﬁg. 4.3. The straight lines shown are based on one-parameter least-squares ﬁts, the slopes giving the value of NL gK p2 =3kB e0 . On the basis of this analysis for water, assum-

162

LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 4.3 Plot of the molar polarization according to the Kirkwood model against 1000/T, where T is the absolute temperature for water (*) and acetonitrile (^) in the temperature range 0–508C.

ing that its dipole moment is 1.85 D, the value of the correlation parameter gK is 4.5. This result is higher than one would estimate on the basis of the structure of liquid water determined by neutron diﬀraction and Raman spectroscopic studies (see section 2.10). The high value reﬂects the eﬀects of hydrogen bonding in this very structured liquid. In the case of acetonitrile, assuming p ¼ 3:92 D, the value of gK is 1.3. In this solvent the degree of correlation between adjacent molecules is much weaker. Since hydrogen bonding is absent in acetonitrile, the only source of correlation would be the formation of dimers resulting from the antiparallel association of the dipole vectors due to the –CN groups in this molecule. The above examples illustrate that continuum models such as the Kirkwood model are reasonably successful in describing the static permittivity, provided one has an independent means of estimating the correlation parameter gK . Unfortunately, these estimates are available for only a few polar solvents, so that gK must be considered an independent parameter. The version of Kirkwood’s theory presented here only considers orientational polarization. When distortional polarization, that is, the eﬀect of molecular polarizability, is included, interpretation of experimental results is less clear. Since the approach taken here involves continuum concepts, it is necessarily limited. In the following section, a simple model based on a molecular description of a polar liquid is presented.

4.4 The Static Solvent Permittivity According to the MSA The approach taken in non-primitive models for the dielectric properties of polar solvents is quite diﬀerent from that described above. The ﬁrst step in a molecular model is to calculate the pair correlation function for molecules in the liquid.

POLAR SOLVENTS

163

Then, when one knows the dipole moment and polarizability, one should be able to estimate the dielectric constant of the medium using the position and average orientation of each molecule in the system. Since dipole–dipole interactions are long range, the problem is not simple and approximations are made in order to ﬁnd a solution. The usual approximation is to represent each molecule as a hard sphere with a central point dipole p and polarizability ap . Such a representation of the system should give a good representation of aprotic solvents but does not have any feature which is able to describe hydrogen bonding. A treatment for polar solvents on the basis of the mean spherical approximation was ﬁrst given by Wertheim [24, 25]. The closure conditions are based simply on the dipole–dipole interaction energy between the polar molecules in the system. Neglecting molecular polarizability, these conditions are gðo1 ; r; o2 Þ ¼ 0

for

r > ss

ð4:4:1Þ

and cðo1 ; r; o2 Þ ¼

bp2 ½ðs1 sr Þðs2 sr Þ s1 s2 Þ 4pe0 r3

for

r > ss

ð4:4:2Þ

where ss is the diameter of a hard sphere, sr , a unit vector in the direction r, and s1 and s2 , unit vectors in the directions of the two dipoles which give rise to the interaction energy. By solving the OZ equation (see section 2.6), using this simple expression for the direct correlation function, cðrÞ, one can ﬁnd the form of the pair correlation function gðrÞ as a function of distance from any given hard sphere, averaging out the angular dependence. One may then estimate the thermodynamic properties of the system using the methods described earlier (see section 2.8). On the basis of Wertheim’s solution of the MSA for dipolar hard spheres it is convenient to deﬁne a polarization parameter ls which is obtained directly from the relative static permittivity, that is, by solving the equation l2s ð1 þ ls Þ4 ¼ 16es

ð4:4:3Þ

The value of ls for a typical polar solvent is approximately two. This equation was introduced earlier in the development of the MSA for ion–solvent interactions (section 3.5). It was seen that the MSA gives an improved description of ion solvation parameters with respect to the Born model. However, it fails to distinguish between the solvation of cations and anions of the same size. In other words, it fails to distinguish between the short-range chemical interactions which stabilize ions of diﬀering charge. It is obvious that a model for the dielectric properties of a polar solvent must consider the eﬀects of hydrogen bonding and other chemical interactions which are not included in a point dipole model for solvent molecules. A simple way of doing this is to introduce a directional sticky parameter [26]. Thus, each solvent molecule is represented by a hard sphere with a central point dipole which is polarizable. The surface of the hard sphere is modiﬁed so that it can interact attractively or repulsively with the surface of an adjacent sphere. ‘‘Stickiness’’ is an eﬀective method of introducing short-range chemical forces. It amounts to assuming that interactions along the line of the dipole vector may be stronger

164

LIQUIDS, SOLUTIONS, AND INTERFACES

or weaker than one estimates on the basis of the simple point dipole model. Equation (4.4.3) then becomes t0 2 2 4 ¼ 16es ð4:4:4Þ ls ð1 þ ls Þ 1 ls where t0 is the longitudinal stickiness parameter. Further development of the theory leads to an expression for the excess internal energy in terms of molecular solvent parameters and ls . The result is ex ¼ Um

dp2 b2 bVm 2ps3s

ð4:4:5Þ

where b2 ¼

6ðls 1Þ ls þ 2

ð4:4:6Þ

Vm is the molar volume of the solvent, and dp , a parameter related to the strength of dipole–dipole interactions. With consideration of dipole polarizability, dp is given by " # NL a2 bp2 a ap 2 dp ¼ þ ð4:47Þ Vm 3e0 e0 where a¼

3s2s 3s3s ½b2 ap ðpe0 Þ

ð4:4:8Þ

and NL is the Avogadro constant. The parameter dp may also be estimated on the basis of the polarization parameter ls and the stickiness parameter t0 deﬁned by equation (4.4.4). This relationship is " # l2s ðls þ 2Þ2 t0 2 16 2 1 dp ¼ 2 ð4:4:9Þ 9 ls ls ð1 þ ls Þ4 Equations (4.4.4)–(4.4.9) give a relationship between the molecular parameters p and ap and the macroscopic relative permittivity es via the two MSA parameters ls and t0 . Equations (4.4.4) and (4.4.9) are non-linear equations from which one obtains ls and t0 once the parameter dp has been estimated. The above equations can be solved using iterative techniques to obtain values of ls , t0 , and dp which are self-consistent. First of all, equation (4.4.4) is solved for ls , assuming that t0 is zero. This value of ls is used to obtain an initial estimate for dp using equations (4.4.6), (4.4.7), and (4.4.8). Equation (4.4.9) is now solved for t0 . The entire calculation is repeated using this estimate of t0 to obtain a second estimate of ls from equation (4.4.4). By going through an iterative scheme several times, the solution converges to give constant values of ls and t0 . In the case of water, the revised estimate of ls is 2.50 and the stickiness parameter is 0:40. The negative value of t0 indicates that dipole–dipole interactions are enhanced by the sticky chemical interactions. In the case of water, dp2 is 30% larger with these interactions than when estimated without them.

POLAR SOLVENTS

165

EXAMPLE

Estimate the MSA parameters for DMSO using the dielectric properties given in table 4.2 and the molecular properties in table 4.3. The ﬁrst step is to estimate ls on the basis of the Wertheim equation (4.4.3) and the value of es (46.68). The equation is ls ð1 þ ls Þ2 ¼ ð16 46:68Þ1=2 ¼ 27:33

ð4:4:10Þ

The solution of this cubic equation gives ls ¼ 2:385. The molecular properties of DMSO are ss ¼ 4:91 1010 m p ¼ 3:96 3:336 1030 ¼ 1:32 1029 C m ap ¼ ð4pe0 Þ 7:99 1030 C m2 V1 The second step is to estimate dp from equation (4.4.9). The estimate of b2 (equation (4.4.6)) is b2 ¼

6 1:385 ¼ 1:895 4:385

ð4:4:11Þ

The estimate of a (equation (4.4.8)) is a¼

3 ð4:91 1010 Þ3 ¼ 1:206 3 ð4:91 1010 Þ3 4 7:99 1030 1:895

ð4:4:12Þ

The number density for the solvent molecules is NL 6:022 1023 ¼ ¼ 8:45 1027 molec m3 Vm 7:13 105

ð4:4:13Þ

It follows that ð1:206Þ2 8:45 1027 ð1:32 1029 Þ2 þ ð4p 1:206 8:45 3 8:854 1012 4:115 1021 1027 7:99 1030 Þ ¼ 19:60 þ 1:02 ¼ 20:62 ð4:4:14Þ

dp2 ¼

From equation (4.4.9) 9dp2 t0 2 16 1 ¼ 2 þ 2 2 ls ls ðls þ 2Þ ls ð1 þ ls Þ4 9 20:61 16 ¼ þ ¼ 1:718 2 2 ð2:385Þ ð4:385Þ ð2:385Þ2 ð3:385Þ4 and t0 ¼ 0:74: Now a new value of s is estimated using equation (4.4.4): 16 46:68 1=2 ls ð1 þ ls Þ2 ¼ ¼ 20:85 1:718

ð4:4:16Þ

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LIQUIDS, SOLUTIONS, AND INTERFACES

The result is ls ¼ 2:129. dp2 is now estimated again, the result being 19.56. The iteration process is repeated until successive values are the same to a reasonable level of precision. The results for DMSO are ls ¼ 1:85 and t0 ¼ 1:53. Values of ls and t0 have been estimated for the polar solvents considered in this chapter and are summarized in table 4.4. Also included in this table are values of the Kirkwood correlation parameter gK . It is deﬁned in the MSA model as gK ¼

ðes 1Þð2es þ 1Þ 3es dp2

ð4:4:17Þ

As can be seen from the results the values of ls are usually close to two. In general, they are higher for the protic solvents than the aprotic ones. The stickiness parameter is negative for aprotic solvents and positive for most protic solvents. A negative value of t0 indicates that the degree of order in the solvent is higher than one predicts on the basis of the simple point dipole–dipole model. It also reﬂects the fact that the dipole moment is not centrally located in the solvent molecules considered. Table 4.4 MSA Parameters for Selected Polar Solvents

Solvent

Polarization Parameter ls

Stickiness Parameter t0

Molecular Polarizability 103 ap /nm3

Kirkwood Correlation Parameter gK

2.50 2.58 2.62 2.68 2.69 2.08 3.28

0.40 0.80 1.11 1.36 1.47 2.36 0.33

1.48 3.29 5.21 7.10 8.99 4.26 6.20

2.79 2.99 3.08 3.23 3.26 2.04 4.52

1.70 1.47 1.52 1.39 1.99 1.80 1.85 1.71 1.74 1.77 1.55 1.45 1.98 1.74 1.81

0.79 2.53 1.65 1.93 0.77 1.30 1.53 1.28 1.29 1.32 2.14 2.17 1.70 1.77 0.62

6.51 4.48 13.05 8.29 9.91 8.12 8.24 19.61 10.99 13.54 5.04 6.37 8.80 11.19 13.16

1.49 1.18 1.23 1.09 1.89 1.60 1.67 1.44 1.52 1.56 1.38 1.15 1.86 1.53 1.64

Protic 1. 2. 3. 4. 5. 6. 7.

W MeOH EtOH PrOH BuOH F NMF

Aprotic 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

AC AcN BzN BuN DMA DMF DMSO HMPA NMP NB NM PrN PC TMS TMU

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167

A trend in the value of t0 can be seen for a series of solvents with the same polar groups and alkyl groups of increasing size. For example, for water and the alcohols, t0 increases from 0:40 for water to 1.47 for butanol. A similar trend is seen for the nitriles where t0 increases from 2:54 for acetonitrile to 1:67 for benzonitrile. These values of t0 are plotted in ﬁg. 4.4 against b2p =d 3 e0 , the quantity which characterizes the strength of dipole–dipole interactions in the medium. There is a clear linear relationship between these parameters. As the size of the alkyl groups increases, the dipolar group moves further away from the center of the molecule. This results in a decrease in dipole moment, an increase in molecular size, and a corresponding decrease in the strength of dipole–dipole interactions. These changes require that the stickiness parameter change as well to account for the fact that the dipole–dipole interactions cannot be treated by the simple point dipole model. Examination of the data for aprotic solvents in table 4.4 also reveals a general trend according to which t0 becomes more negative as the strength of dipole–dipole interactions increases [26]. The estimates of the correlation parameter gK give an idea of the degree of local structure in the polar solvent. In the case of water and the alcohols, this parameter is relatively high and increases with the length of the hydrocarbon chain. The result for water, namely, 2.79, is remarkably close to the value estimated by Kirkwood [22] on the basis of the well-known tetrahedral structure for the surrounding hydrogen-bonded water dipoles (2.65) (see also ﬁg. 2.14) [G2]. The increase in gK in the alcohols suggests a tendency to form hydrogen-bonded chains in these systems. The value of gK is also large for N-methyl formamide, a strongly structured solvent [27]. In the case of the aprotic nitriles, the average value of gK is 1.2. This result indicates that local ordering is minimal in these systems. Dimethylsulfoxide, which has approximately the same dipole moment as acetonitrile, is known to be strongly dimerized in the liquid state [28]. As a result it

Fig. 4.4 Plot of the stickiness parameter t0 against the values of bp2 =ðe0 d3 ), which characterizes the strength of dipole–dipole interactions for two selected solvent groups.

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LIQUIDS, SOLUTIONS, AND INTERFACES

has a higher value of gK . An even higher value is found for propylene carbonate (1.86), which is also associated in the liquid state. In summary, the values of gK give a useful indication of local order in polar solvents and may be related to more detailed structural information when this is available. One may also use the MSA model to describe the permittivity of the system at optical frequencies. Under these circumstances the system responds to the electrical ﬁeld only through electronic polarization, the orientational component being ‘‘frozen.’’ The directional stickiness of the dipoles is then unimportant so that t0 is eﬀectively zero at very high frequencies. Under these circumstances, the polarization parameter is given by l2op ð1 þ lop Þ4 ¼ 16eop

ð4:4:18Þ

where lop is the value of the polarization parameter at optical frequencies. The expression for dp , the parameter which characterizes the excess energy of the system, becomes dp2 ¼

NL aap e0 V m

ð4:4:19Þ

where a is given by equation (4.4.8) and b2 ¼

6ðlop 1Þ lop þ 2

Alternatively, dp may be estimated from the following equation: l2op ðlop þ 2Þ2 1 1 dp2 ¼ eop 9

ð4:4:20Þ

ð4:4:21Þ

Since values of the molecular polarizability are not available from gas phase measurements for most of the polar solvents considered here, they may be estimated using the MSA in the following way. First, a value of lop is calculated by solving equation (4.4.18) using the experimental value of eop . Then, the parameter dp is estimated using equation (4.4.21). Finally, the polarizability is found from equation (4.4.19). The results obtained are summarized in table 4.4. They are very close to estimates obtained on the basis of the Lorentz–Lorenz equation (see equation (4.3.21) and table 4.3), usually being a few percent higher. In summary, the MSA provides a remarkably simple and eﬀective way of describing the dielectric properties of polar solvents. By including stickiness in the model one has a very easy method of dealing with the eﬀects of non-sphericity, the non-central location of the polar group, and hydrogen bonding. These eﬀects collectively make up the chemical part of the problem of estimating the permittivity of a polar liquid from its dipole moment and polarizability. One should note that the polarization parameter used here is quite diﬀerent from that introduced in section 3.4 in the discussion of ion–solvent interactions. Although it arises in the model in the same way, it is diﬀerent because the chemical interaction between a solvent dipole and an ion depends on the chemical nature of the polar group in the solvent and whether it is able to participate in hydrogen bonding. This leads to the necessity of introducing diﬀerent values of this parameter for cations and

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anions. On the other hand, the polarization parameter used here describes the interaction of one dipole with neighboring dipoles. These examples show that simple modiﬁcation of the MSA leads to an excellent description of the properties of real systems without recourse to complex mathematical treatments.

4.5 Dielectric Relaxation Phenomena When a polar solvent is placed in a changing electrical ﬁeld, the molecules must realign so that their dipole vectors maintain the orientation corresponding to minimum energy. Because of intermolecular forces, this process does not occur inﬁnitely fast but on a time scale which depends on the properties of the medium and which is usually on the order of 1–100 ps. Dielectric relaxation experiments provide very useful information about molecular motion in polar liquids and the ability of the solvent molecules to respond to changing electrical conditions. The theory of dielectric relaxation is based on a macroscopic model which considers how the polarization of the medium changes with time. As seen earlier, the polarization is made up of an orientational and a distortional component so that one may write P ¼ Po þ Pd

ð4:5:1Þ

When an electrical ﬁeld is applied to the system, the distortional component relaxes very quickly on a time scale corresponding to electronic motion. On the other hand, the orientational component relaxes on a time scale consistent with molecular motion. In deriving an expression for the kinetics of the relaxation process, it is assumed that the rate of relaxation is proportional to the departure of the orientational polarization from its equilibrium value, that is, dPo Poe Po ¼ dt t

ð4:5:2Þ

where Poe is the equilibrium value of the orientational polarization and t is the relaxation time. The reciprocal of t is the equivalent of the ﬁrst-order rate constant for the relaxation process. From the fundamental laws of electrostatics, the relation between the polarization and the electrical ﬁeld is P ¼ e0 Eðe 1Þ

ð4:5:3Þ

where e represents the value of the permittivity at time t. If E is changing, then the polarization depends on the time dependence of both E and e. At equilibrium the total polarization is given by Pe ¼ e0 E ðes 1Þ

ð4:5:4Þ

This equation shows that the static permittivity is the appropriate value when the system is equilibrated. At very high frequencies only the distortional component of the polarization remains so that Pd ¼ e0 E ðe1 1Þ

ð4:5:5Þ

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LIQUIDS, SOLUTIONS, AND INTERFACES

The permittivity e1 is the value reached at the highest frequency used in the relaxation experiment. Normally this corresponds to microwave frequencies of the order of 1 THz but it can extend into the far infrared. Now, one may estimate the orientational polarization as Po ¼ P Pd ¼ e0 E ðe e1 Þ

ð4:5:6Þ

At equilibrium, this contribution is Poe ¼ Pe Pd ¼ e0 E ðes e1 Þ

ð4:5:7Þ

It is also important to be able to relate the orientational polarization to the electric displacement D. On the basis of equation (4.3.15), one may write D ¼ e0 E þ Po þ Pd

ð4:5:8Þ

P o ¼ D e0 e1 E

ð4:5:9Þ

so that

Under the usual circumstances, dielectric relaxation is studied at an electrical ﬁeld which is controlled, for example, by controlling the voltage drop across a parallel-plate condenser. On the basis of equation (4.5.2), substituting for Po and Poe using equations (4.5.7) and (4.5.9), one obtains the following diﬀerential equation in D: t

dðD e0 e1 EÞ þ D e0 e1 E ¼ e0 Eðes e1 Þ dt

ð4:5:10Þ

Suppose that the dielectric system relaxes due to the application of a constant ﬁeld E at time t ¼ 0. Then equation (4.5.10) simpliﬁes to t

dD þ D ¼ e0 es E ¼ D e dt

ð4:5:11Þ

where D0 is the equilibrium value of the electric displacement. The diﬀerential equation may also be written as dðD De Þ dt ¼ D De t

ð4:5:12Þ

D ¼ De þ ðD0 De Þet=t

ð4:5:13Þ

for which the solution is

where Do is the initial value of D. This equation describes an exponential decay in D which is characterized by a relaxation time t, from the initial value D0 to the equilibrium value De . Most experimental studies involve measurements of the electrical permittivity as a function of frequency using an alternating voltage or ﬁeld E. Then, the ﬁeld varies with time according to the equation E ¼ Em cos ot

ð4:5:14Þ

where Em is the amplitude of the a.c. ﬁeld and o is its angular frequency. These experiments are commonly discussed in terms of the frequency f , where

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f ¼

o 2p

171

ð4:5:15Þ

The electrical displacement also varies in a sinusoidal fashion but is not necessarily in phase with the electrical ﬁeld. The relationship between these quantities is obtained by solving equation (4.5.10). When the alternating ﬁeld is initially applied to the system, there are some transients which describe the initial relaxation. The solution to the diﬀerential equation presented here ignores the initial eﬀects, and considers only the properties of the system at a steady state. Under such circumstances any ﬂuctuation in the system occurs at the same frequency as that of the perturbing ﬁeld E. In order to make the mathematics easier, the diﬀerential equation is solved using a complex number representation of the sinusoidally varying quantities (see appendix A). Thus, the ﬁeld E is written as E ¼ Em ejot

ð4:5:16Þ

where ejot ¼ cos ot þ j sin ot ð4:5:17Þ pﬃﬃﬃﬃﬃﬃﬃ and j is equal to 1. By comparing the magnitudes to the quantities associated with cos ot and sin ot, one is able to estimate the phase angle of the alternating quantity. The steady-state solution to the diﬀerential equation for D is given by D ¼ AD ejot

ð4:5:18Þ

Substituting this expression for D into equation (4.5.10) one obtains jotðAD ejot e0 e1 Em ejot Þ þ AD ejot e0 e1 Em ejot ¼ e0 Em ejot ðes e1 Þ

ð4:5:19Þ

Simplifying and solving for AD , the result is AD ¼

e0 Em ½es þ j!tðe1 es Þ þ o2 t2 e1 1 þ o2 t2

ð4:5:20Þ

Under non-equilibrium conditions D ¼ e e0 E

ð4:5:21Þ

A D ¼ e e0 E m

ð4:5:22Þ

It follows that

Now one can write the following equation for the varying permittivity: e e e e ð4:5:23Þ e ¼ e1 þ s 212 jot s 212 1þo t 1þo t This result obtained by Debye [1] describes the frequency dependence of the permittivity e. In the following discussion the relaxation time measured with the electrical ﬁeld E as the controlled variable is called the Debye relaxation time and is given the symbol tD . From equation (4.5.23) the in-phase component of the permittivity is e e ein ¼ e1 þ s 212 ð4:5:24Þ 1 þ o tD

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LIQUIDS, SOLUTIONS, AND INTERFACES

At very low or static frequencies (o ¼ 0Þ, the in-phase component is equal to es as one would expect. In the limit of very high frequencies, the in-phase component is equal to e1 . For angular frequencies which are close to 1=tD , ein drops rapidly from es to e1 , reaching the value (es þ e1 Þ=2 when ot is equal to one. The out-of-phase component is eout ¼

otD ðes e1 Þ 1 þ o2 t2D

ð4:5:25Þ

It is zero in the limits of very low and very high frequencies, and reaches a maximum value equal to (es e1 Þ when ot is one. The fact that this term is multiplied by j in the overall expression shows that the out-of-phase component lags the in-phase component by 908. Plots of the in- and out-of-phase components of the permittivity of a hypothetical solvent with typical parameters (es ¼ 50, e1 ¼ 2, and tD ¼ 20 ps) are shown as a function of the logarithm of the frequency in ﬁg. 4.5. The frequency range over which most of the change in these quantities occurs is from 100 MHz to 1 THz. The upper limit is beyond the range of most microwave experiments, which is about 300 GHz. The out-of-phase component reaches a maximum value when otD ¼ 1, which occurs at a frequency of 8 GHz in this example. This is also the frequency at which the rate of change in ein with frequency is a maximum. Obviously, the frequency range shown in this plot could not be covered in a normal microwave experiment. Thus, extrapolation techniques are often used to estimate e1 and tD from real experimental data. An alternative experiment is one in which permittivity changes are studied at constant electrical displacement D, for example, for constant charge on the plates of a capacitor. The resulting relaxation phenomena are then characterized by a

Fig. 4.5 Plots of the in-phase, ein , and out-of-phase, eout , components of the dielectric permittivity of a hypothetical Debye solvent against the logarithm of frequency. The parameters assumed are ein =50, eout =2, and tD ¼ 20 ps.

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173

diﬀerent relaxation time and it is useful to examine this situation further. On the basis of equation (4.5.10) when D is constant, tD e0 e1

dE þ D ¼ e0 es E dt

ð4:5:26Þ

Rearranging, one may write tD e1 dE D e0 es E ¼ e es es 0 dt

ð4:5:27Þ

dðe0 es E DÞ ¼ D e0 es E dt

ð4:5:28Þ

or tL where tL ¼

e1 tD es

ð4:5:29Þ

tL is known as the longitudinal relaxation time. Integration of equation (4.5.28) leads to the result that D e0 es E ¼ ðD e0 es E0 Þet=tL

ð4:5:30Þ

where E0 is the initial value of E. Since the electric displacement is constant and equal to the equilibrium value De , equation (4.5.30) can be rewritten as E Ee ¼ ðE0 Ee Þet=tL

ð4:5:31Þ

where De is related to Ee through equation (4.5.21). Equation (4.5.31) shows that the decay of E back to its equilibrium value Ee is described by a rate constant 1=L . It is analogous to equation (4.5.13), which was obtained for the case of constant electrical ﬁeld E. The relaxation properties of a polar solvent may be studied by carrying out experiments in which the electrical ﬁeld E or displacement D are changed suddenly in a step fashion. In the case that the ﬁeld is the independently controlled quantity, one follows the change in D with time. An analysis of the transient observed experimentally allows one to determine the Debye relaxation time tD on the basis of equation (4.5.13). Similarly, if one studies the relaxation transient under conditions that D is suddenly changed, for instance, by suddenly changing the charge on the plates of a capacitor, one observes a transient involving the electrical ﬁeld E. Analysis of these data allows one to determine the longitudinal relaxation time tL . If both experiments are conducted, comparison of tD and tL allows one to estimate the high-frequency permittivity e1 assuming that the static permittivity is already known. However, dielectric relaxation experiments are usually conducted in the presence of sinusoidally varying electrical ﬁelds as a function of frequency. Many such experiments have been carried out in various pure liquids and solutions [G5, 4–16, 29], and values of the in-phase and out-of-phase permittivity reported as a function of frequency over a wide range. Analysis of these data to obtain es , e1 , and tD can be carried out in a variety of ways. On the basis of equations (4.5.24)

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LIQUIDS, SOLUTIONS, AND INTERFACES

and (4.5.25), eliminating the common factor ðes e1 Þ=ð1 þ o2 t2D Þ, one obtains the equation ein ¼ e1 þ

eout otD

ð4:5:32Þ

This shows that a plot of ein against eout =o should be linear, with e1 being obtained from the intercept and 1=tD from the slope. This method of data analysis normally involves extrapolation because the experimental frequency range does not include values of ein close to e1 . An alternative way of analyzing the data results in estimates of es and tD from a linear plot. Multiplying equation (4.5.25) by otD and adding (es e1 Þ=ð1 þ o2 t2D Þ to both sides, one obtains otD eout þ

es e1 ¼ es e1 1 þ o2 t2D

ð4:5:33Þ

Combining this with equation (4.5.24), the result is ein ¼ es otD eout

ð4:5:34Þ

Accordingly, a plot of ein against oeout should be linear with a slope of tD . Such a plot is shown in ﬁg. 4.6 using data obtained for water at 258C [G5]. A linear relationship is obtained with the data for frequencies less than 40 GHz. On the basis of a straight line ﬁtted in this region, the value of es from the intercept is 78.20 and the value of tD from the slope, 8.38 ps ðr ¼ 0:9998Þ. A plot of the same data according to equation (4.5.32) is shown in ﬁg. 4.7. A straight line can be drawn through the points obtained at frequencies below 40 GHz. The slope of this line gives 1=tD from which the value of tD is 8.37 ps. This is in excellent agreement with the result from the previous plot. The intercept gives the value of e1 , which is 6.02 for this system.

Fig. 4.6 Plot of ein against oeout using dielectric relaxation data for water at 258C [G5]. The straight line is drawn considering only the data obtained at frequencies below 40 GHz.

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175

Fig. 4.7 Plot of ein against eout =o using dielectric relaxation data for water at 258C [G5]. The straight line was ﬁtted using the data points shown as ﬁlled circles.

An important question is why the data obtained at frequencies above 40 GHz do not follow the simple Debye model. A plot of eout against ein for data in the high-frequency region is shown in ﬁg. 4.8. It is clear that the experimental results lie above the values predicted by the Debye model using the best values of the parameters found from data at lower frequencies. This suggests that another relaxation process involving much higher frequencies is present in water. In the alcohols [4–7], two high-frequency relaxation processes are found. The complex nature of dielectric relaxation in this group of solvents is attributed to hydrogen bonding.

Fig. 4.8 Plot of eout against ein using dielectric relaxation data for water in the frequency range 60–410 GHz [G5]. The solid line shows the contribution from the low-frequency relaxation process.

176

LIQUIDS, SOLUTIONS, AND INTERFACES

It is often possible to describe the dielectric data for complex liquids in terms of multiple relaxation processes, each of which follows a Debye relaxation based on equation (4.5.2). The resulting equations for ein and eout are ein ¼ en1 þ

n X ei0 ei1 i¼1

1 þ o2 t2i

ð4:5:35Þ

and eout ¼

n X ðei0 ei1 Þoti i¼1

1 þ o2 t2i

ð4:5:36Þ

ei0 and ei1 are the low- and high-frequency values of the permittivity for the relaxation process with relaxation time ti . The static permittivity es is equal to the low- frequency permittivity for the ﬁrst relaxation process e10 . In addition, the high-frequency permittivity for the ith relaxation ei1 is equal to the low-frequency permittivity for the (i þ 1)th relaxation eðiþ1Þ0 . Thus, for a liquid having n relaxation processes, there are n values of ei0 which must be speciﬁed together with n diﬀerent relaxation times, ti , and ﬁnally the high-frequency permittivity for the last relaxation process, en1 . In the case of water, it has been shown that the dielectric data can be described in terms of two relaxation processes. Further analysis of the data shown in ﬁg. 4.8 leads to the following parameters: e20 ¼ 6:02, e21 ¼ 4:57, and t2 ¼ 0:97 ps [G5]. The second relaxation process is much faster than the ﬁrst and is characterized by frequencies in the terahertz range. The slow relaxation process which is characterized by a large amplitude (e10 e11 ¼ 72Þ is attributed to a cooperative process involving clusters of water molecules connected through hydrogen bonding. The fast process is attributed to hydrogen bond formation and breakup. More will be said about the nature of these processes in section 4.7. Other relationships which have been used to describe dielectric relaxation data include the Cole–Cole and Cole–Davidson equations [29]. These are preferred when a distribution of relaxation times rather than a single relaxation time is more appropriate to describe the data in a given frequency range. Nevertheless, the Debye model in its simple version or multiple relaxation versions works quite well for most of the solvents considered here. The principal problem with existing data in the literature is that they often are not available over a suﬃciently wide frequency range so that a reasonable analysis at the high-frequency end cannot be made. Only recently have experiments been carried out above 100 GHz. For solvents with rapid relaxation processes experiments above 1 THz in the far infrared region are required.

4.6 The Permittivity of Electrolyte Solutions When an electrolyte is added to a polar solvent, the resulting solution has very diﬀerent electrical properties than those of the pure solvent. The most obvious change is the fact that the solution has an easily measured conductivity due to the presence of the ions. The solvent is a dielectric which behaves as a capacitor but its

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dielectric properties change in the presence of the electrolyte. This is partially due to the fact that the structure of the solvent is disrupted in the immediate vicinity of the ions. Thus, in terms of an equivalent circuit, an electrolyte solution consists of a resistor and capacitor in parallel. By studying the frequency dependence of the impedance of an electrolyte solution one can learn something about both ion–solvent and ion–ion interactions and how these aﬀect solution structure. Dielectric relaxation data have been collected for a number of electrolyte solutions, both aqueous and non-aqueous [G5, 29]. These data are obtained by measuring the impedance of the solution and then correcting for solution conductivity. When simple electrolytes which do not ion pair are involved, the permittivity data often can be treated by the Debye model. A typical example is KCl in water. Values of ein plotted as a function of oeout for this system at two diﬀerent concentrations are shown in ﬁg. 4.9. On the basis of equation (4.5.34), these plots give the value of es from the intercept and tD from the slope. In this case, es is 76.58 at a concentration of 0.1 M, and 67.05 at 1 M. The static permittivity decreases noticeably with increase in electrolyte concentration. This result is attributed to two factors. When the electrolyte ions do not possess a dipole moment, as is the case for monoatomic ions, they do not respond to the electrical ﬁeld applied to the system in the relaxation experiment. However, they occupy volume in the system, so that the overall permittivity of the solution decreases with increase in the mole fraction of the electrolyte. The second eﬀect related to the addition of electrolyte is connected to the breakup of solvent structure due to solvation of the ions. This is especially important in water solutions, where hydrogen bonding plays an important role in determining solvent structure. The value of the relaxation time also decreases with increase in electrolyte concentration. In the present case, tD varies from 8.26 ps for 0.1 M KCl to 8.05 ps for 1 M KCl.

Fig. 4.9 Plot of ein against oeout using dielectric relaxation data for aqueous KCl solutions: (*) 0.1 and (^) 1 M.

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LIQUIDS, SOLUTIONS, AND INTERFACES

The extent to which the dielectric permittivity of the solution departs from that of the pure solvent depends not only on the electrolyte concentration but also on the nature of the electrolyte. Considering the alkali metal halides, the largest drop in es at a given concentration is found for Liþ electrolytes [G5, 29]. This reﬂects the disruptive eﬀect of the small cation which causes reorientation of the solvent molecules around it and a subsequent breaking of hydrogen bonds. The dielectric decrement decreases with increase in cation atomic number for a given anion. On the other hand, in the case of anions the dielectric decrement increases with increase in anion atomic number for a given cation. Anions are less disruptive of water structure, so that the main eﬀect of changing the anion is a change in the volume fraction of solution occupied by ions. Representative plots of es as a function of concentration are shown in ﬁg. 4.10. The principal feature depending on electrolyte concentration is the initial slope which is called the dielectric decrement. In addition all plots are curved, suggesting that a limiting permittivity is reached at high electrolyte concentration. The dependence of es on electrolyte concentration ce has been examined by Friedman [30], who derived the following relationship: es ¼ es0 de ce þ be c3=2 e

ð4:6:1Þ

where es0 is the dielectric permittivity of the pure solvent, de , the dielectric decrement, and be , a curvature parameter. This equation describes the concentration dependence of es quite well for most 1–1 electrolytes over a wide concentration range. The curvature parameter is small for 1–1 electrolytes, usually falling in the range 2–5 L3=2 mol3=2 . When be is set equal to 5 L3=2 mol3=2 , the values of de for the alkali metal halides fall in a range from 14 to 22 L mol1 [31].

Fig. 4.10 Plots of the static permittivity of three 1–1 electrolyte solutions against electrolyte concentration. The ordinate scale is correct for the NaCl data; for KF, the data have been shifted up by 10 units, and for KBr, down by 10 units for the sake of clarity.

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179

EXAMPLE

Use the following permittivity data for aqueous NaClO4 solutions at 258C to determine the best values of the parameters de and be for this electrolyte: ce =M es 0.5 1.0 1.5 2.0 3.0 4.0 5.0

70.41 63.56 58.17 53.44 47.28 40.29 35.21

The permittivity of pure water at 258C is 78.46. Thus, equation (4.6.1) can be rearranged as follows: 78:46 es ¼ de be c1=2 e ce

ð4:6:2Þ

The parameters de and be are determined from a plot of (78:46 es Þ=ce against ce1=2 . The required data are given in the following table: 1=2 c1=2 e =M

1 ð78:46 es Þc1 e =M

0.707 1.000 1.225 1.414 1.732 2.000 2.236

16.1 14.9 13.5 12.5 10.4 9.5 8.7

These data are ﬁtted to a straight line by least squares. From the intercept, de ¼ 19:4 M1 and be ¼ 5:1 L3=2 mol3=2 . When ion pairing is present an additional relaxation is observed at low frequencies. A typical example is the MgSO4 system in water. The ion pair has a dipole moment and therefore this species reorients in the alternating electrical ﬁeld. The relaxation time associated with the reorientation is much longer than that associated with the reorientation of water molecules. It depends not only on reorientation of the ion–pair dipole but also on the kinetics of its formation and decomposition. For this reason, the parameter for the low-frequency relaxation process is strongly concentration dependent. Dielectric relaxation data for a 0.08 M Mg2 SO4 solution are shown in ﬁg. 4.11. On the basis of an analysis of these data by Barthel and coworkers [29, 32], three relaxation processes may be discerned. The ﬁrst one, involving the ion pair, occurs between permittivity values of 82.9 and 75.2 and involves a relaxation time of 181 ps. The second process, which is attributed to the slow reorientation of water clusters, takes place between the permittivity values of 75.2 and 8.4 with a relaxation time of 8.4 ps. Finally, the high-frequency process, which occurs between 8.4

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LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 4.11 Plot of ein against eout for dielectric relaxation data obtained with 0.08 M MgSO4 in water. The semicircles show the resolved Debye relaxation processes for (a) the ion pair; (b) the slow reorientation of water clusters; (c) the fast process in water.

and 4.6 with a relaxation time of 1.5 ps, is attributed to the formation and breaking of hydrogen bonds. Obviously, the dielectric behavior of systems with ion pairing is much more complex than those without it. The presence of the ion pair gives the solution a higher permittivity at lower frequencies than it otherwise would have. This feature is important in understanding the equilibrium properties of these solutions. The permittivity data for the low-frequency process may be used to determine the ion pairing equilibrium constant and the rate constants for formation and breakup of this species. Thus, dielectric relaxation experiments in electrolytes provide valuable information about ion association equilibria. A variety of dielectric relaxation data are now available for both aqueous and non-aqueous solutions. These results help one understand the properties of these solutions in more detail. They are complementary to information obtained from thermodynamic, spectroscopic, and conductivity experiments, and provide an important basis for understanding solution structure.

4.7 The Dielectric Relaxation Parameters According to the Debye model there are three parameters associated with dielectric relaxation in a simple solvent, namely, the static permittivity es , the Debye relaxation time tD , and the high-frequency permittivity e1 . The static permittivity has already been discussed in detail in sections 4.3 and 4.4 . In this section attention is especially focused on the Debye relaxation time tD and the related quantity, the longitudinal relaxation time tL . The signiﬁcance of these parameters for solvents with multiple relaxation processes is considered. The high-frequency permittivity e1 and its relationship to the optical permittivity eop is also discussed.

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181

As deﬁned, the Debye relaxation time is the reciprocal of a ﬁrst-order rate constant. Thus, it is expected to depend on temperature according to the usual Arrhenius relationship HD tD ¼ AD exp ð4:7:1Þ RT where AD is a pre-exponential factor and HD , the activation enthalpy associated with the relaxation process. Values of HD are available for many polar solvents and fall in the range from 5 to 25 kJ mol1 (see table 4.2). For unassociated polar solvents the relaxation process is assumed to involve rotation of a molecular dipole in the alternating electrical ﬁeld and is called rotational diﬀusion. The major barrier to this process is dipole–dipole interactions. Thus, it was shown that HD is a linear function of the enthalpy associated with dipole–dipole interactions in aprotic solvents [6]. Although tD is not a molecular parameter, it is related in a complex way to the molecular events which take place in a polar liquid subjected to an alternating electrical ﬁeld [G2]. The temperature dependence of the longitudinal relaxation time L is also an important quantity. For a Debye solvent, tL is given by the relationship e ð4:7:2Þ tL ¼ 1 tD es The enthalpy associated with the temperature dependence is deﬁned as HL ¼ R

@ ln tL @ ln e1 @ ln es ¼ HD þ R R @ð1=TÞ @ð1=TÞ @ð1=TÞ

ð4:7:3Þ

The temperature derivatives of the high- and low-frequency contributions to the permittivity are easily estimated from experimental data (see table 4.2). In some cases the temperature derivative of e1 is not available but one may assume that it is approximately equal to the temperature derivative of eop . Unfortunately, the Debye model provides only an approximate description of aprotic solvents. It has been applied extensively to determine their relaxation properties quite successfully, mainly because permittivity data are available over a limited frequency range. As a result, the high-frequency parameter is usually obtained by a long extrapolation. As experimental methods have become available at frequencies above 50 GHz, it has been found that the behavior of aprotic solvents is more complex [9]. Acetonitrile (AcN) can be described by the Debye model using data up to 40 GHz [9]. When the frequency range is extended to 90 GHz, the data can only be ﬁtted if a distribution of relaxation times is assumed. Dimethylsulfoxide (DMSO) and propylene carbonate (PC) have been studied up to 89 GHz [9]. These data can be ﬁtted to the Debye model with two relaxation processes. DMSO is known to form dimers in the bulk [28]. This association is undoubtedly the reason for its complex relaxation behavior. The same is true for PC, a molecule with a very high dipole moment. The dielectric relaxation data for dimethylformamide (DMF) and dimethylacetamide (DMA) can be described by two Debye processes [9]. The lowfrequency, high-amplitude process is attributed to rotational diﬀusion. For

182

LIQUIDS, SOLUTIONS, AND INTERFACES

these systems, the high-frequency relaxation is ascribed to hindered intramolecular rotation about the C–N bond [9]. All protic solvents undergo multiple relaxation processes due to the presence of hydrogen bonding. In the case of water and formamide (F), the data can be described in terms of two Debye relaxations. For the alcohols and N-methylformamide (NMF), three Debye relaxations are required for the description. In all of these solvents, the low-frequency process involves the cooperative motion of hydrogen-bonded clusters. In the case of water and the alcohols the highfrequency process involves the formation and breaking of hydrogen bonds. The intermediate process in the alcohols is ascribed to rotational diﬀusion of monomers. Studies of dielectric relaxation in these systems have been carried out for the n-alkyl alcohols up to dodecanol [8]. Values of the relaxation parameters for water and the lower alcohols are summarized in table 4.5. F and NMF are also highly structured solvents as a result of hydrogen bonding. The low-frequency relaxation process in these systems can be attributed to the cooperative motion of hydrogen-bonded clusters. The process at the highest frequencies has a similar relaxation time to those observed in DMF and DMA. Thus, it is probably due to intramolecular rotation about the C–N bond in the monomer. The intermediate relaxation observed in NMF is attributed to rotational diﬀusion of a monomer. Relaxation parameters for F and NMF are also summarized in table 4.5. Estimation of the longitudinal relaxation time in solvents with multiple Debye relaxation processes is not straightforward. In fact, tL is a function of time in these systems [33, 34], and varies between two limiting values. For a solvent with two relaxation processes, the low-frequency limit for tL is e tL0 ¼ 1 ðf1 tD1 þ f2 tD2Þ ð4:7:4Þ es where fi ¼

ei0 ei1 e s e1

ð4:7:5Þ

At high frequencies, the limiting value of tL is e1 f 1 f2 1 tL1 ¼ þ es tD1 tD2

ð4:7:6Þ

Table 4.5 Dielectric Relaxation Parameters for Some Protic Solvents at 258C [5, 9] Solvent

es ¼ e10

t1 =ps

e20

t2 =ps

e30

t3 =ps

e1

W MeOH EtOH PrOH F NMF

78.45 32.63 24.35 20.44 109.5 186.0

8.32 51.5 163 329 37.3 128

6.18 5.91 4.49 3.74 7.08 6.13

1.02 7.09 8.97 15.1 1.16 7.93

— 4.90 3.82 3.20 — 4.60

— 1.12 1.81 2.40 — 0.78

4.49 2.79 2.69 2.44 4.48 3.20

POLAR SOLVENTS

183

The time characterizing transition between tL1 and tL0 is tm , which is given by tm ¼ f1 tD1 þ f2 tD2

ð4:7:7Þ

It is interesting to calculate these parameters for water on the basis of the data given in table 4.5. For this solvent, the fractions f1 and f2 are 0.977 and 0.023, respectively. As a result, the low-frequency value of tL , tL0 , is equal to 0.47 ps. Because f1 is close to one, the inﬂuence of the high-frequency process on tL0 is minimal. At high frequencies, the limiting value of tL , tL1 , is 0.41 ps. For this system the diﬀerence between the limiting values of tL is very small and beyond the limit of experimental detection under most circumstances. The time characterizing transition between tL1 and tL0 is 1.19 ps. Thus, for processes occurring at frequencies greater than 10 THz, the appropriate value of tL is tL1 . For frequencies lower than 100 GHz, the corresponding value is tL0 . When the solvent has three relaxation processes, the above relationships for tL become tL0 ¼

e1 ðf t þ f2 tD2 þ f3 tD3 Þ es 1 D1

ð4:7:8Þ

e1 f1 f f 1 þ 2 þ 3 es tD1 tD2 tD3

ð4:7:9Þ

and tL1 ¼

Variation of tL between the high- and low-frequency limits is characterized by two exponentially decaying functions with diﬀerent transition times [34]. In the case of the alcohols the diﬀerence between these limits is signiﬁcant and can lead to complex relaxation behavior in these solvents. For methanol, the highfrequency value of tL is 1.0 ps and the low-frequency value, 4.0 ps. For 1-propanol, the diﬀerence is greater, the corresponding values being 5.3 and 36.5 ps, respectively. The high-frequency permittivity e1 reﬂects the value of this property when reorientation of the solvent dipoles no longer contributes to polarization of the medium. As the frequency of the a.c. electrical ﬁeld increases, the ability of the molecules to respond rapidly decreases. Eventually, the solvent dipoles become ‘‘frozen’’ when the a.c. frequency is very high. This occurs typically for frequencies close to 1 THz, corresponding to relaxation times in the order of 1 ps. Experimentally, these processes occur in the far infrared at frequencies of 30 cm1 or greater. The molecules now respond to the electromagnetic ﬁeld through distortional and electronic polarization. Distortional polarization corresponds to movement of the molecular nuclei with respect to one another and is equivalent to molecular vibration. As the frequency of the electromagnetic ﬁeld increases into the infrared, discontinuities are found in the permittivity of the liquid corresponding to the frequency at which a given molecular vibration occurs. The permittivity after each of these vibrations is activated decreases until the distortional contribution to the permittivity is lost. Eventually in the visible region, the molecules only respond via electronic polarization, that is, response of the electronic cloud in the molecules to the alternating electrical ﬁeld. This response is normally

184

LIQUIDS, SOLUTIONS, AND INTERFACES

determined by measuring the refractive index of the liquid at the sodium D line (589 nm) and thereby estimating eop . Because of the distortional contribution to the permittivity, e1 is always larger than eop . For example, in the case of water, e1 is 4.49 and eop , 1.78. Relaxation parameters such as tD and tL are used in the analysis of the kinetics of very fast processes in solution. As one might expect, movement of solvent molecules can be inﬂuential in determining the rates of these processes. Thus, the study of dielectric relaxation not only provides valuable information about solvent structure but also relaxation parameters relevant to fast solution kinetics. This subject is discussed in more detail in chapter 7.

4.8 Ion Solvation in Polar Solvents The variation in ion solvation with solvent nature is an interesting subject because it is relevant to basic ideas about the structure of electrolyte solutions. According to the classical Born model, the Gibbs energy of ion solvation depends on solvent nature only through its static dielectric permittivity es . When this subject was examined for aqueous solutions in chapter 3, it was concluded that local ion– solvent interactions are very important in determining the magnitude of the Gibbs energy of solvation and that these interactions result in disruption of local solvent structure. Therefore, it is interesting to examine ion solvation in a wide variety of polar solvents in order to assess the relative importance of long-range electrostatic and short-range chemical forces. The most direct way of determining s G is by measuring the work associated with transferring the electrolyte from the gas phase, where the ions are inﬁnitely far apart from one another, to the liquid solution. However, from a practical point of view, it is much simpler to solve this problem starting with the solid electrolyte. Consider a simple 1–1 electrolyte MX which dissolves in solvent S: MXcry !MXsl ðSÞ

ð4:8:1Þ

The Gibbs energy change associated with formation of the solution may be written as sl G ¼ sl G þ RT ln aM aX

ð4:8:2Þ

where sl G is the value of sl G at unit activity on the molarity scale. When the dissolution process (4.8.1) is at equilibrium, the electrolyte solution is saturated and one may write sl G ¼ RT lnðaM aX Þsat ¼ RT ln Ksp

ð4:8:3Þ

where Ksp is the thermodynamic solubility product. Following the usual thermodynamic conventions for electrolytes, sl G is the standard Gibbs energy change associated with dissolution of solid electrolyte in a hypothetical 1 M solution in which all real interactions are absent (see section 3.6). In practice, solubility products are reported in the literature in terms of concentrations. Thus, determi-

POLAR SOLVENTS

185

nation of sl G usually requires that one estimate the mean activity coeﬃcient of the electrolyte in the saturated solution. One way of avoiding this problem is to use an electrochemical cell to determine Ksp with electrodes reversible to the cation and anion of the electrolyte (see chapter 9). In order to remove the contribution to sl G from the solid phase, values of this quantity are compared in two diﬀerent solvents. If one imagines that the same salt is dissolved in two diﬀerent solvents S1 and S2 , then one may estimate the Gibbs energy of transfer of MX according to the process MXsl ðS1 Þ!MXsl ðS2 Þ

ð4:8:4Þ

The corresponding standard Gibbs energy change is called the Gibbs energy of transfer: tr G ¼ sl G ðS2 Þ sl G ðS1 Þ

ð4:8:5Þ

tr G depends only on the diﬀerences in the interaction of the ions with the two solvents, the contribution from the solid phase having been lost in subtraction. Values of sl G have been determined for many electrolytes in most of the polar solvents considered in this chapter. Various reference solvents have been considered for reporting tr G [35–38] but that used in most recent work is water. Thus, tr G gives the Gibbs energy associated with transferring one mole of the electrolyte in the hypothetical standard state from water to the given non-aqueous solvent. This quantity is positive for most simple 1–1 electrolytes because the electrolyte is less soluble in the non-aqueous medium [36, 37]. Obviously, it is more interesting to have single ion quantities for the Gibbs energy of transfer. This requires that an extrathermodynamic assumption be made to achieve separate cation and anion contributions. Many assumptions have been considered and evaluated in the literature [38, 39]. Only one is described here and its use may be considered a convenient convention which leads to reasonable separation of the cationic and anionic contributions to tr G . By choosing an electrolyte with very large ions whose charge is buried in the center of the ion, it can be assumed that the cation and anion have equal contributions to tr G . The electrolyte used is tetraphenylarsonium tetraphenylborate (TATB) which is sparingly soluble in water, but more soluble in non-aqueous media. As a result, the values of tr G for this electrolyte are negative. The TATB assumption is expressed quantitatively through the equation tr G ðTAþ Þ ¼ tr G ðTB Þ ¼

tr G ðTATBÞ 2

ð4:8:6Þ

By choosing a series of electrolytes which includes salts containing the tetraphenylarsonium cation and the tetraphenylborate anion, the individual contributions to tr G from the more common ions may be extracted. EXAMPLE

The following data are for the Gibbs energy of transfer from water to propylene carbonate for 1–1 electrolytes at 258C [37]. Use these data to estimate the corresponding quantities for the individual ions on the basis of the TATB assumption.

186

LIQUIDS, SOLUTIONS, AND INTERFACES

Electrolyte LiI NaCl KCl KI TAI TATB

tr G =kJ mol1 43.7 53.1 44.4 23.4 19.2 72.0

From the value of tr G for TATB, tr G TAþ Þ ¼ tr G ðTB Þ ¼ 72:0=2 ¼ 36:0 kJ mol1

ð4:8:7Þ

From the value of tr G for TAI, tr G ðI Þ ¼ 19:2 þ 36:0 ¼ 16:8 kJ mol1

ð4:8:8Þ

From the data for the other iodides, tr G ðLiþ Þ ¼ 43:7 16:8 ¼ 26:9 kJ mol1

þ

tr G ðK Þ ¼ 23:4 16:8 ¼ 6:6 kJ mol

1

ð4:8:9Þ ð4:8:10Þ

From the data for the chlorides, tr G ðCl Þ ¼ 44:4 6:6 ¼ 37:8 kJ mol1

ð4:8:11Þ

tr G ðNaþ Þ ¼ 53:1 37:8 ¼ 15:3 kJ mol1

ð4:8:12Þ

and

These results are slightly diﬀerent than the values recorded in tables 4.6 and 4.7. The tabulated data are based on a larger collection of results for 1–1 electrolytes and therefore are better estimates of the single ion values. Values of tr G for the alkali metal cations together with the value for the tetraphenylarsonium ion are given in table 4.6 for the solvents considered in this chapter. For the alkali metal ions, tr G is both positive and negative, the latter values indicating that the ion is more stable in the non-aqueous environment. In the case of the Naþ ion, tr G varies from 17 kJ mol1 in hexamethylphosphoramide (HMPA) to 36 kJ mol1 in nitrobenzene. It is clear that there is no correlation between tr G and the static permittivity es or a related function of es , such as that found in the Born equation (3.4.6). The solvent with the highest permittivity among those considered is NMF. Its value of tr G for Naþ is negative but not as large as that seen for solvents like HMPA or DMSO. Similarly, BuOH has a very low permittivity but its value of tr G is not as large as that of NB. This point is further demonstrated by considering three aprotic solvents with approximately the same values of es , namely, AcN, DMF, and NM. The values of tr G for the Naþ ion in these solvents are 15.1, 9:6, and 32 kJ mol1 , respectively. This shows emphatically that chemical interactions are

POLAR SOLVENTS

187

Table 4.6 Gibbs Energy of Transfer for Monovalent Cations from Water to a Non-Aqueous Solvent at 258C [39] tr G /kJ mol1 Solvent

Liþ

Naþ

Kþ

Rbþ

Csþ

Ph4 Asþ

(529) 4.4 11 11 11 10 20

(424) 8.2 14 17 19 8 7

(352) 9.6 16.4 17 20 4.3 6

(329) 9.6 16 19 23 5 8

(306) 8.9 15 17 19 6 7

— 24.1 21.2 25 20 23.9 33

10 25 36 — 22 10 15 — 35 38 48 27 23.8 6 23

10 15.1 21 16 12.1 9.6 13.4 17 15 36 31.6 15 14.6 3 14

4 8.1 19 — 11.7 10 13.0 16 11 21 15.4 11 5.3 4 13

4 6.3 15 — 8 9.7 10.4 10 8 19 11.0 — 1 9 10

4 6.0 13 — 7 10.8 13.0 7 10 18 5.6 — 7 10 —

32 32.8 35 — 40 38.5 37.4 39 40 36 32.6 — 36 36 —

Protic 1. 2. 3. 4. 5. 6. 7.

Wa MeOH EtOH PrOH BuOH F NMF

Aprotic 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

AC AcN BzN BuNb DMA DMF DMSO HMPA NMP NB NM PrN PC TMS TMUb

a

Values in parentheses give the absolute Gibbs energy of solvation in water estimated from experiment. b Estimated from polarographic halfwave potential data [40].

much more important in determining solvation in these systems than long-range electrostatic eﬀects. Values of tr G for some halide ions, the perchlorate ion, and the tetraphenylborate ion are given in table 4.7. This quantity is always positive for the halide ions, and is much larger in aprotic solvents than in protic systems. Thus, small monoatomic anions are especially diﬃcult to solvate when hydrogen bonding is not available in the system. The same is not true for the perchlorate ion, which actually is more stable in some aprotic solvents than in water. Because the negative charge in the ion is distributed over ﬁve atoms, it dissolves more easily than the Cl anion in all non-aqueous media. For this reason perchlorate salts are often used to introduce cations into a non-aqueous solvent. Values of tr G for the tetraphenylborate anion are always large and negative, and by deﬁnition equal to the values for the tetraphenylarsonium cation.

188

LIQUIDS, SOLUTIONS, AND INTERFACES

Table 4.7 Gibbs Energy of Transfer for Monovalent Anions from Water to a Non-Aqueous Solvent at 258C [39] tr G /kJ mol1 Solvent

Cl

Br

I

ClO 4

Ph4 B

(304) 13.2 20.2 26 29 13.7 —

(278) 11.1 18.2 22 24 10.7 —

(243) 7.3 12.9 19 22 7.3 —

(165) 6.1 10 17 22 12 —

— 24.1 21.2 25 20 23.9 33

57 42.1 47 — 54.9 48.3 40.3 58 51 35 37.7 — 39.8 47 —

42 31.3 37 — 44.0 36.2 27.4 46 37 29 29 — 30 35 —

25 16.8 25 — 21 20.4 10.4 30 19 18 18.9 — 13.7 21 —

6 2 13 — — 4 1 7 12 10 4.7 — 3 — —

32 32.8 35 — 40 38.5 37.4 39 40 36 32.6 — 36.0 36 —

Protic 1. 2. 3. 4. 5. 6. 7.

Wa MeOH EtOH PrOH BuOH F NMF

Aprotic 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

AC AcN BzN BuN DMA DMF DMSO HMPA NMP NB NM PrN PC TMS TMU

a Values in parentheses give the absolute Gibbs energy of solvation in water estimated from experiment.

Extensive data also exist for the enthalpy of transfer of 1–1 electrolytes from water to non-aqueous solvents. These data are obtained in calorimetric experiments in which the enthalpy of solution is measured as a function of electrolyte concentration. The value of the enthalpy of solution per mole at inﬁnite dilution is obtained by extrapolation. The enthalpy of transfer is deﬁned in the same way as the Gibbs energy of transfer according to process (4.8.4). Thus, the standard enthalpy of transfer is given by tr H ¼ sl H ðS2 Þ sl H ðS1 Þ

ð4:8:13Þ

The reference solvent in common use is water [39]. Several extrathermodynamic assumptions have been considered to separate experimental values of tr H into cationic and anionic contributions [35]. The one commonly used is the TATB assumption, namely, that the contributions to

POLAR SOLVENTS

189

tr H for the large tetraphenylarsonium and tetraphenylborate ions are equal. This leads to the values shown in tables 4.8 and 4.9 for the alkali metal and halide ions. Values for tr H for the alkali metal cations are negative in almost all cases. Exceptions are those for Liþ and Na+ in NM, and Liþ in PC and TMS. It follows that the interaction of these monoatomic monovalent cations is more exothermic with most polar solvents than with water. On the other hand, the enthalpy of transfer of the smaller anions, Cl and Br , is almost always positive. This is a direct indication that these species are diﬃcult to dissolve, especially in aprotic solvents. In the case of I and ClO 4 , the enthalpy of transfer is usually negative. Comparison of the enthalpy data with the Gibbs energy data recorded in tables 4.6 and 4.7 shows that these quantities are very diﬀerent. When both Gibbs energies and enthalpies of transfer are available, one can calculate the entropy of transfer [39].

Table 4.8 Enthalpy of Transfer for Monovalent Cations from Water to a Non-Aqueous Solvent 258C [39] tr H =kJ mol1 Solvent

Liþ

Naþ

Kþ

Rbþ

Csþ

Ph4 Asþ

(578) 21.7 20.2 18.4 — 6.0 —

(464) 20.7 19.4 18.8 — 16.5 22.5

(381) 19.0 19.6 18.3 — 17.9 —

(355) 16.5 — — — 17.8 —

(330) 14.1 11.8 12.7 — 17.7 —

— 2.2 0 1.5 — 0.5 7.6

— 8.0 — — — 25.4 27.1 57.4 26 — 25.8 — 2.8 12 —

— 13.3 — — 41.2 32.4 29.2 50.6 41 — 11.5 — 10.5 16 —

— 22.9 — — — 35.7 35.4 46.6 47 — 18.5 — 22.5 26 —

— 24.6 — — — 36.1 38.1 43.8 55 — 17.7 — 24.9 28 —

— 26.1 — — — 34.6 33.0 45.0 50 — 16.6 — 27.5 26 —

— 11.1 — — 13.7 17.2 10.6 25.8 17 — 6.9 — 13.1 11 —

Protic 1. 2. 3. 4. 5. 6. 7.

Wa MeOH EtOH PrOH BuOH F NMF

Aprotic 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

AC AcN BzN BuN DMA DMF DMSO HMPA NMP NB NM PrN PC TMS TMU

a Values in parentheses give the absolute enthalpy of solvation in water estimated from experiment.

190

LIQUIDS, SOLUTIONS, AND INTERFACES

Table 4.9 Enthalpy of Transfer for Monovalent Anions from Water to a Non-Aqueous Solvent 258C [39] tr H /kJ mol1 Solvent

Cl1

Br

I

ClO 4

Ph4 B

(320) 8.4 10.4 8.4 — 3.5 14.2

(289) 4.5 5.5 2.5 — 1.5 —

(247) 1.0 0.7 1.5 — 6.8 —

(176) 3.1 2.7 0 — 20 11.8

— 0.9 0 1.5 — 0.5 7.6

— 19.3 — — 35.6 17.9 20.0 38.2 27 — 21.2 — 26.2 27 —

— 8.0 — — 17.2 0.6 4.6 17.7 13 — 7.1 — 15.2 13 —

— 7.6 — — 0.7 15.0 11.5 5.8 2 — 7.8 — 1.6 8 —

— 16.0 — — 15.3 23.4 18.2 20.7 11 — 19.2 — 16.3 20 —

— 11.1 — — 13.7 17.2 10.6 25.8 17 — 6.9 — 13.1 11 —

Protic 1. 2. 3. 4. 5. 6. 7.

Wa MeOH EtOH PrOH BuOH F NMF

Aprotic 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

AC AcN BzN BuN DMA DMF DMSO HMPA NMP NB NM PrN PC TMS TMU

a Values in parentheses give the absolute enthalpy of solvation in water estimated from experiment.

Using the absolute values of the Gibbs energy of solvation for the individual ions in water one can calculate the absolute value of this quantity in any solvent using the relationship s GðSÞ ¼ s GðWÞ þ tr G

ð4:8:14Þ

where s GðSÞ is the Gibbs energy of solvation in solvent S and s G(W) the corresponding quantity in water. According to the Born model for ion solvation presented in section 3.4, the Gibbs solvation energy for a given ion depends on solvent nature only through the dielectric permittivity es . Speciﬁcally, this model predicts that s G(S) should be linear in (1 1=es ) provided the ionic radius ri does not depend on solvent nature. However, plots based on the data in tables 4.6 and 4.7 for simple cations and anions show no correlation between s G(S) and (1 1=es ). These results emphasize that a simple electrostatic model cannot account for the variation in the Gibbs energy of solvation with solvent nature.

POLAR SOLVENTS

191

As will be seen in the following section, the chemical nature of the interaction between the ion and the solvent must be considered in order to understand this variation. The data in tables 4.6–4.9 are very helpful in predicting the solubility of a given electrolyte in a solvent. In addition, they give an indication of the extent to which ion pairing is important. If the solvent interacts strongly with the ion, then its tendency to form ion pairs is reduced. Data are available for more ions than those considered here, and these can be consulted to assess the properties of other electrolytes in a given solvent [39, 40].

4.9 Polar Solvents as Lewis Acids and Bases Examination of the variation in thermodynamic properties of electrolytes in polar solvents shows conclusively that continuum electrostatic models are not able to account for the experimental data. As a result, chemists have looked for other ways of explaining the variation in these quantities with solvent nature. Since ions are stabilized by the dipolar character of the solvent molecule, a method of measuring the ability of a given solvent to solvate cations and anions was sought. Cations are stabilized by the electronegative end of the solvent dipole. Thus, the important property of the solvent molecule is its ability to donate a pair of electrons, that is, to act as a Lewis base. On the other hand, anions are stabilized by the positive end of the solvent dipole. The relevant molecular property is solvent acidity or the ability to accept a pair of electrons. In the case of Lewis acidity, the ability to form hydrogen bonds is also a contributing factor. The fact that the chemical nature of the ends of the molecular dipole is quite diﬀerent means that a given solvent can have a very diﬀerent Lewis acidity and basicity. Early work by Dimroth and Reichardt [41, 42] led to the development of a polarity scale ET based on a solvatochromic dye. This dye, namely, 4-(2,4,6triphenyl-pyridinium)-2,6-diphenylphenoxide undergoes a pp transition in the visible or ultraviolet which is accompanied by a large decrease in the molecular dipole moment (see ﬁg. 4.12). In the ground state, the dye interacts strongly with the solvent through the electronegative oxygen atom. The wavelength associated with the pp transition depends strongly on the nature of the solvent and is clearly related to solvent acidity, that is, to the solvent’s ability to solvate the electronegative oxygen atom in the ground state. Thus, the pp transition is observed at 810 nm in a non-polar solvent such as diphenyl ether, whereas it occurs at 453 nm in water. ET is deﬁned on the basis of the energy associated with this transition in kJ mol1 . It provides an important empirical parameter for measuring solvent acidity (see table 4.10). Another scale for measuring solvent acidity was formulated by Mayer et al. [43]. It is called the solvent acceptor number (AN) and is based on the relative values of the 31P NMR chemical shifts produced by a given solvent with a strong Lewis base, triethylphosphine oxide (ﬁg. 4.13). The data were normalized so that the acceptor number of hexane is zero and that for the 1:1 adduct with the strong Lewis acid, SbCl5, 100 when dissolved in 1-2 dichloroethane. The attractive feature of this scale is that it varies over a wide range for the polar solvents con-

192

LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 4.12 Chemical structure of the solvatochromic dye used to deﬁne the ET polarity scale: left, ground state; right, excited p state.

sidered here, namely, from 10.6 for hexamethylphosphoramide to 54.8 for water (table 4.10). As one would expect, there is a strong linear correlation between the parameters ET and AN, the relationship being

Table 4.10 Empirical Parameters for Solvent Acidity and Basicity Acidity Solvent

Basicity

AN

ET

DN

Bsc

54.8 41.5 37.9 37.3 36.8 39.8 32.1

264.0 231.8 217.1 212.1 210.0 236.8 226.4

18.0 19.0 19.2 19.8 19.5 24 27

591 589 589 589b 589 598 604

12.5 18.9 15.5 15.2a 13.6 16.0 19.3 10.6 13.3 14.8 20.5 22.7a 18.3 19.2 10.7a

176.6 190.8 173.6 180.3 182.8 183.3 188.7 171.1 176.6 172.4 193.7 195.0 184.1 171.5 171.5

17.0 14.1 11.9 16.6 27.8 26.6 29.8 38.8 27.3 4.4 2.7 16.1 15.1 14.8 31

569 573 572 573b 608 602 613 633 604b 522 530 572b 554 562 596

Protic 1. 2. 3. 4. 5. 6. 7.

W MeOH EtOH PrOH BuOH F NMF

Aprotic 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. a

AC AcN BzN BuN DMA DMF DMSO HMPA NMP NB NM PrN PC TMS TMU

Estimated on the basis of equation (4.9.1). Estimated on the basis of equation (4.9.2).

b

POLAR SOLVENTS

193

Fig. 4.13 Lewis base, triethylphosphine oxide, used to deﬁne the AN scale.

ET ¼ 150:6 þ 1:954 AN

ð4:9:1Þ

with a correlation coeﬃcient of 0.964. The slope of this relationship demonstrates that the variation in ET is only 47% of that in AN, so that AN is the preferred parameter to assess solvent acidity. Equation (4.9.1) can be used to estimate values of AN which were missing from the original compilation [43]. These are also given in table 4.10. A popular method of estimating solvent basicity is based on the donor number scale, DN, introduced by Gutmann [44, 45]. The donor number is deﬁned as the value of the heat of reaction of the polar solvent with the strong Lewis acid SbCl5 when these reactants are dissolved in 1,2-dichloromethane (ﬁg. 4.14). The scale suﬀers from the problem that DN cannot be measured directly for protic solvents because of the instability of SbCl5 in these systems. However, values of DN for these solvents have been estimated by a variety of other techniques [46]. Another method of estimating solvent basicity makes use of the solvatochromism of a Cu2þ complex, namely Cu(II) N;N;N 0 ;N 0 -tetramethylethylenediamine acetoacetonate [47]. This parameter which is designated Bsc is especially convenient because it may be measured directly for all the solvents considered here, both protic and aprotic. The relationship between Bsc and DN is Bsc ¼ 525:2 þ 2:89DN

ð4:9:2Þ

with a correlation coeﬃcient of 0.942. Unfortunately, experimental values of Bsc are not available for all the solvents considered here but they may be estimated using equation (4.9.2). It was pointed out earlier that the Gibbs energy of transfer of simple monatomic ions such as Naþ and Cl1 is not related in any simple way to the dielectric permittivity of the solvent. On the other hand, tr G for the alkali metal cations is correlated with solvent basicity. A plot of this quantity for the Naþ ion against the solvent’s donor number DN is shown in ﬁg. 4.15. Although the correlation is not excellent, it shows a clear trend in which tr G becomes more negative as DN increases. This is exactly what one expects because the ability of the solvent to stabilize the cation in solution increases with increase in solvent basicity. On the basis of the MSA, the molar Gibbs solvation energy for a monovalent ion i in solvent S is

Fig. 4.14 Lewis acid, antimony pentachloride, used to deﬁne the DN scale.

194

LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 4.15 Plot of the Gibbs energy of transfer of Naþ from water to various non-aqueous solvents against the Gutmann donor number DN.

s Gi ðSÞ ¼

NL e20 1 1 1 e s r i þ ds 8pe0

ð4:9:3Þ

where NL is the Avogadro constant, e0 , the fundamental charge, e0 , the permittivity of free space, es , the solvent’s dielectric permittivity, ri , the ion’s radius, and ds , the MSA parameter correcting ri . The latter quantity depends not only on the nature of the solvent but also on whether the ion is a cation or an anion. Deﬁning 1/ds as Bs or solvent basicity, for the case of a monovalent cation, equation (4.9.3) may be rewritten as NL e20 1 Bs ð4:9:4Þ s GMþ ðSÞ ¼ 1 es 1 þ Bs ri 8pe0 This shows that the Gibbs solvation energy changes with solvent nature due to changes in es and Bs . Since the factor (1 1=es ) is close to unity for the polar solvents discussed here, and assuming that Bs ri does not change signiﬁcantly with respect to one, then the main reason for the correlation seen in ﬁg. 4.15 is the change in Bs . Thus, the reciprocal of the MSA parameter for cations is directly related to solvent basicity [48, 49]. Values of Bs estimated using the data for Naþ ion given in table 4.6 are plotted against the Gutmann DN in ﬁg. 4.16. A good linear correlation is found with a correlation coeﬃcient of 0.921. The equation relating Bs to DN is Bs ¼ 16:30 þ 0:314DN

ð4:9:5Þ

Thus, Bs provides an alternate empirical scale for solvent basicity. In the case of the halide anions, the Gibbs energy of transfer is correlated with solvent acidity. A plot of tr G for the chloride anion is shown as a function of the Gutmann AN in ﬁg. 4.17. A very good correlation is found based on the data

POLAR SOLVENTS

195

Fig. 4.16 Plot of the MSA parameter Bs estimated in various solvents for the Naþ ion (equation (4.9.4)) against the Gutmann donor number DN.

which are available for 18 solvents (r ¼ 0:954). According to the MSA, the Gibbs solvation energy is given by equation (4.9.3) with the provision that the parameter ds depends on the nature of the solvent but is not the same as that found for the alkali metal cations [48, 49]. Deﬁning 1/ds as As or solvent acidity for the case of a halide anion, equation (4.9.3) can be rewritten as s GX ðSÞ ¼

NL e20 1 As 1 es 1 þ As ri 8pe0

ð4:9:6Þ

Fig. 4.17 Plot of the Gibbs energy of transfer of Cl from water to various non-aqueous solvents against the Gutmann acceptor number AN.

196

LIQUIDS, SOLUTIONS, AND INTERFACES

As ﬁlls the role of an acidity parameter because the factors (1 1=es ) and As ri do not vary markedly for the polar solvents considered in the present analysis. As a result, s GX ðSÞ correlates well with the acidity parameter AN. Values of As estimated using the data for Cl1 ion given in table 4.7 are plotted against the Gutmann acceptor number in ﬁg. 4.18. A very good linear correlation with a correlation coeﬃcient of 0.978 is obtained. The relationship between As and AN is As ¼ 7:826 þ 0:172AN

ð4:9:7Þ

As provides another empirical scale for solvent acidity. The above treatment may be extended to the analysis of the Gibbs energy of transfer of simple electrolytes and related thermodynamic quantities. Because the electrolyte contains both cations and anions, the thermodynamic property is expected to depend on both the acidity and basicity of the solvent. A simple way of expressing this dependence is Q ¼ Q0 þ as AN þ bs DN

ð4:9:8Þ

where Q is the solvent-dependent thermodynamic quantity, Q0 , its value in a solvent with zero acceptor number and zero donor number, as , the response to solvent acidity and bs , the response to solvent basicity [50]. Values of Q0 , as , and bs may be determined using a three-parameter least-squares ﬁt to express Q as a linear function of both AN and DN. Graphical presentation of the results of this ﬁt requires using a three-dimensional plot. Results of analyzing Gibbs energy of transfer data on the basis of equation (4.9.8) for ﬁve alkali metal halides and TATB from water to various non-aqueous solvents are summarized in table 4.11. Acceptable ﬁts to equation (4.9.6) are obtained for these systems, the correlation coeﬃcient r decreasing with increase in the size of the alkali metal cation and halide anion. However, there is a problem in assessing the properties of the ﬁt on the basis of the response factors as and bs .

Fig. 4.18 Plot of the MSA parameter As estimated in various solvents for the Cl ion (equation (4.9.5)) against the Gutmann acceptor number AN.

POLAR SOLVENTS

197

Table 4.11 Analysis of the Dependence of the Gibbs Energy of Transfer of 1–1 Electrolytes [36, 37] on Solvent Acidity and Basicity (Equation (4.9.8))

Salt LiCl NaCl KCl KBr CsBr TATB*

No. of Solvents

Q0 /kJ mol1

as

bs

as

9 10 8 10 10 12

133.1 106.2 87.9 74.2 67.4 94.6

1:48 1:31 1.13 0.87 0.82 1.60

2:82 1:86 1.33 1.38 1.15 0.35

0.55 0.62 0.62 0.50 0.53 0.88

bs

Stand. Dev. /kJ mol1

Corr. Coeﬀ. r

0.45 0.38 0.38 0.50 0.47 0.12

2.6 2.8 4.2 6.1 9.7 6.9

0.996 0.993 0.987 0.963 0.894 0.959

*TATB refers to (C6 H5 )4 As (C6 H5 )4 B.

Since the range over which AN and DN vary depends on the solvents chosen in the study, the relative importance of solvent acidity and basicity in determining the Gibbs energy of transfer is not easily assessed. By calculating relative partial regression coeﬃcients, the role of solvent acidity and basicity in determining the thermodynamic quantity can be clearly seen [50]. In order to do this, one must estimate the variance for the independent and dependent variable involved in the multiparameter analysis. For the parameter Q, the variance is deﬁned as !2 X X 2 nQ ¼ Qi Qi =n ð4:9:9Þ i

i

where the Qi s are the individual values of Q in a data set containing n diﬀerent values. The variances of AN and DN, nAN and nDN , respectively, are calculated in a similar way. The partial regression coeﬃcients are deﬁned as a0s ¼ jas jðnAN =nQ Þ1=2

ð4:9:10Þ

b0s ¼ jbs jðnDN =nQ Þ1=2

ð4:9:11Þ

and

The parameters a0s and b0s are now normalized to the same scale so that their relative values indicate the relative importance of solvent acidity and basicity in the given analysis. A more direct way of assessing this importance is in terms of relative partial regression coeﬃcients as and bs , which are deﬁned as follows: a s ¼

a0s

a0s þ b0s

ð4:9:12Þ

a0s

r0s þ b0s

ð4:9:13Þ

and bs ¼

198

LIQUIDS, SOLUTIONS, AND INTERFACES

The parameter as gives the fraction of the explained variation in the parameter Q, which is due to solvent acidity. In the same way, b s gives the corresponding fraction due to solvent basicity. The values of as and bs recorded in table 4.11 show that solvent acidity is usually more important than solvent basicity in determining solvation of the electrolytes considered. The result for TATB for which as ¼ 0:88 and bs ¼ 0:12 is especially surprising. According to the TATB assumption used to estimate single ion Gibbs energies of transfer, these quantities should each be equal to 0.50. If results for the three protic solvents W, MeOH, and F are removed from the data set, and the analysis performed with the remaining nine aprotic solvents, the values of as and bs are then equal to 0.50. This result suggests that the TATB assumption is valid for aprotic solvents but not very good for protic solvents with strong hydrogen bonding [48]. A more general relationship for analyzing solvent eﬀects for thermodynamic quantities was given by Koppel and Palm [51]. They extended equation (4.9.9) to include non-speciﬁc solvent eﬀects related to solvent permittivity and polarizability. Their equation can be written as Q ¼ Q0 þ as AN þ bs DN þ gs X þ ds Y

ð4:9:14Þ

where X is the solvent’s polarity, Y, its polarizability, and gs and ds , the corresponding response factors. Solvent polarity is deﬁned to be X¼

es 1 es þ 2

ð4:9:15Þ

and is directly related to the molar polarization deﬁned in Debye’s theory of the dielectric permittivity of polar ﬂuids (see equation (4.3.20)). The polarizability Y is given by Y¼

eop 1 eop þ 2

ð4:9:16Þ

According to the Lorentz–Lorenz equation (4.3.21) for the molar refraction at optical frequencies, Y is directly proportional to the molecular polarizability ap . The Koppel–Palm equation has also been applied to the analysis of solvent eﬀects on thermodynamic quantities related to the solvation of electrolytes [48, 49]. In the case of the systems considered in table 4.11, addition of the parameter X to the linear equation describing the solvent eﬀect improves the quality of the ﬁt to the experimental data, especially in the case of alkali metal halide electrolytes involving larger ions. The parameter Y is not important for these systems but does assist in the interpretation of other thermodynamic quantities which are solvent dependent [48, 49]. Addition of these parameters to the analysis is only possible when the solvent-dependent phenomenon has been studied in a large number of solvents. In conclusion, solvent acidity and basicity are useful parameters for the interpretation of solvent eﬀects related to the solvation of electrolytes in polar media. This role demonstrates the importance of the chemical nature of the interaction between the ion and solvent molecule. Chemical interactions cannot be rationalized simply in terms of concepts from electrostatics based on point charges and

POLAR SOLVENTS

199

point dipoles. Although the parameters used to assess solvent acidity and basicity are empirical they are related in a direct way to the molecular description of ion solvation according to the MSA.

4.10 Concluding Remarks Attention in this chapter has been focused on the dielectric properties of polar solvents and how they relate to the solvation of electrolytes. The important problem discussed is how one uses statistical mechanics to derive the solution permittivity from the molecular properties of polar liquids, namely, the dipole moment and polarizability. In the classical or continuum approach developed by Debye, Onsager, Fro¨hlich, and Kirkwood, it became apparent that description of the molecular properties within a spherical cavity in the liquid required the introduction of a correlation parameter gK . This parameter takes into consideration the fact that the orientation of molecular dipoles with respect to one another is not just determined by dipole–dipole interactions but also by chemical considerations such as hydrogen bonding. This fact is especially apparent for liquid water in which the presence of strong hydrogen bonding results in a rather high solvent permittivity. In the molecular model based on the MSA, an additional parameter is required namely the molecular diameter in a spherical representation of the individual molecule. This model is only successful when the molecular sphere with its embedded point dipole also is ‘‘sticky’’ in a given direction. The stickiness factor plays the same role as the correlation parameter gK in the classical approach. It accounts for chemical interactions such as hydrogen bonding and the eﬀects of non-sphericity in the real system. The success of the MSA model is that it predicts values of gK which are consistent with structural information about the liquid obtained spectroscopically. Dielectric relaxation experiments provide valuable information about the properties of polar solvents in an alternating electrical ﬁeld. The important parameter characterizing the ability of the solvent dipole to respond to the changing ﬁeld is the Debye relaxation time. Solvents with hydrogen bonding have several relaxation times, one of which corresponds to the rotational diﬀusion of clusters of molecules. The information provided by these experiments helps one to understand the role of the solvent in fast reactions occurring in solution. More about this aspect of solvent eﬀects is discussed in chapter 7. When these experiments are conducted in electrolyte solutions one is able to assess ion pairing by examining relaxation data at lower frequencies. Ion solvation in polar solvents is also an important aspect of the subject matter considered here. This is traditionally studied by measuring the Gibbs energy and enthalpy of transfer of a variety of electrolytes from water to another polar solvent. Single ion quantities are then derived on the basis of the TATB assumption. Study of these quantities for simple monoatomic ions like the alkali metal cations and the halide anions leads to the conclusion that speciﬁc molecular properties, namely, Lewis acidity and basicity, are important in ion solvation. On the other hand, the dielectric permittivity, a non-speciﬁc bulk property,

200

LIQUIDS, SOLUTIONS, AND INTERFACES

does not play an important role. The analysis presented points out the importance of the chemical structure of the solvent in stabilizing ions in the electrolyte solution. Thus, the conclusions reached here are in accord with those from chapter 3. Physical laws such as Coulomb’s law and bulk parameters such as the permittivity help one to understand long-range ion–ion interactions. However, to understand close-range interactions, one must understand the chemical properties of the solvent. In order to assess Lewis acidity and basicity, empirical parameters such as the Gutmann acceptor number and donor numbers have proven to be quite useful. Ion–solvent interactions are conveniently studied by several spectroscopic techniques. These provide information about the structure of the solvent around the ion including the solvation number, and the strength of the ion–solvent bond. Information about these techniques is presented in the following chapter.

General References G1. Bo¨ttcher, C. J. F.; Bordwijk, P. Theory of Electric Polarization, 2nd ed.; Elsevier: Amsterdam, 1978. G2. Hill, N. E.; Vaughn, W. E.; Prince, A. H.; Davies, M. Dielectric Properties and Molecular Behaviour; Van Nostrand: London, 1969. G3. Politzer, P., Murray, J. S. Quantitative Treatments of Solute-Solvent Interactions; Elsevier: Amsterdam, 1994. G4. Riddick, J. A.; Bunger, W. B.; Sakano, T. K. Organic Solvents, 4th ed.; Wiley: New York, 1986. G5. Barthel, J.; Buchner, R.; Mu¨nsterer, M. Electrolyte Data Collection, Part 2: Dielectric Properties of Water and Aqueous Electrolyte Solutions; Dechema: Frankfurt-amMain, 1995.

References 1. Debye, P. Polar Molecules; Chemical Catalog Co.: New York, 1929. 2. Marcus, Y. Properties of Solvents; Wiley: New York, 1998. 3. Karapetyan, Yu, A.; Eychis, V. N. Physico-chemical Properties of Non-Aqueous Electrolyte Solutions (in Russian); Khimia: Moscow, 1989; Chapter 1. 4. Lide, D. R., Ed. Handbook of Physics and Chemistry, 84th ed.; CRC Press: Boca Raton, FL, 2003–2004. 5. Barthel, J.; Bachhuber, K; Buchner, R.; Hetzenauer, H. Chem. Phys. Lett. 1990, 165, 369. 6. Fawcett, W. R. Chem. Phys. Lett. 1990, 174, 167. 7. Castner, E. W., Jr.; Bagchi, B.; Maroncelli, M.; Webb, S. P.; Ruggiero, A. J.; Fleming, G. R. Ber. Bunsen-Ges. Phys. Chem. 1988, 92, 363. 8. Garg, S. K.; Smyth, C. P. J. Phys. Chem. 1965, 69, 1294. 9. Barthel, J.; Bachhuber, K.; Buchner, R.; Gill, J. B.; Kleebauer, M. Chem. Phys. Lett. 1990, 167, 62. 10. Calderwood, J. H.; Smyth, C. P. J. Am. Chem. Soc. 1956, 78, 1295. 11. Krishnaji A.; Mansingh, A. J. Chem. Phys. 1964, 41, 827. 12. Davies, G. J.; Evans, G. J.; Evans, M. W. J. Chem. Soc., Faraday Trans. 2, 1979, 75, 1428. 13. Brownsell, V. L.; Price, A. H. J. Phys. Chem. 1970, 74, 4004. 14. Elie, V. Bull. Chem. Soc. Belges 1984, 93, 839. 15. Behret, H.; Schmidthals, F.; Barthel, J. Z. Phys. Chem. NF 1975, 96, 73.

POLAR SOLVENTS

16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51.

201

Dachwitz, E.; Stockhausen, M. Ber. Bunsen-Ges. Phys. Chem. 1985, 89, 959. Ga¨uman, T. Helv. Chim Acta 1958, 41, 1933. Wilhelm, E.; Battino, R. J. Chem. Phys. 1989, 55, 4012. Lorentz, H. The Theory of Electrons; Dover: New York, 1952. Balakier, G. Pol. J. Chem. 1980, 54, 2297. Onsager, L. J. Am. Chem. Soc. 1936, 58, 1486. Kirkwood, J. G. J. Chem. Phys. 1939, 7, 911. Fro¨hlich, H. Theory of Dielectrics; Oxford University Press: London, 1949. Wertheim, M. S. J. Chem. Phys. 1971, 55, 4291. Wertheim, M. S. Mol. Phys. 1973, 25, 211; 1973, 26, 1425. Blum, L.; Fawcett, W. R. J. Phys. Chem. 1993, 97, 7185; 1996, 100, 10423. Reid, D. S.; Vincent, C. A. J. Electroanal. Chem. 1968, 18, 427. Fawcett, W. R.; Kloss, A. A. J. Chem. Soc., Faraday Trans. 1996, 92, 3333. Barthel, J.; Buchner, R. Pure Appl. Chem. 1991, 63, 1473. Friedman, H. J. Chem. Phys. 1982, 76, 1092. Fawcett, W. R.; Tikanen, A. C. J. Phys. Chem. 1996, 100, 4251. Barthel, J.; Hetzenauer, H.; Buckner, R. Ber. Bunsen-Ges. Phys. Chem. 1992, 96, 988. Hynes, J. T. J. Chem. Phys. 1986, 90, 3701. Fawcett, W. R. Chem. Phys. Lett. 1992, 199, 153. Krishnan, C. V.; Friedman, H. L. In Solute-Solvent Interactions; Coetzee, J. F., Ritchie, C. D., eds.; Marcel Dekker: New York, 1976; Vol. 2, Chapter 9. Cox, B. G.; Hedwig, G. R.; Parker, A. J.; Watts, D. W. Aust. J. Chem. 1974, 27, 477. Cox, B. G.; Waghorne, W. E. Chem. Soc. Rev. 1980, 9, 381. Alexander, R.; Parker, A. J.; Sharp, J. H.; Waghorne, W. E. J. Am. Chem. Soc. 1972, 94, 1148. Marcus, Y.; Kamlet, M. J.; Taft, R. W. J. Phys. Chem. 1988, 92, 3613. Gritzner, G. Pure Appl. Chem. 1990, 62, 1839. Dimroth, K.; Reichardt, C. Liebigs Ann. Chem. 1969, 727, 93. Reichardt, C. Solvents and Solvent Eﬀects in Organic Chemistry, 2nd ed.; VCH Publishers: New York, 1988. Mayer, U.; Gutmann, V.; Gerger, W. Monatsh. Chem. 1975, 106, 1235. Gutmann, V.; Wychera, E. Inorg. Nucl. Chem. Lett. 1966, 2, 257. Gutmann, V. Coord. Chem. Rev. 1976, 19, 225. Marcus, Y. J. Solution Chem. 1984, 13, 599. Sandstro¨m, M.; Persson, I.; Persson, P. Acta Chem. Scand. 1990, 44, 653. Fawcett, W. R. J. Phys. Chem. 1993, 97, 9540. Fawcett, W. R. In Quantitative Treatments of Solute/Solvent Interactions; Politzer, P. Murray, J. S., eds.; Elsevier: Amsterdam, 1994; Chapter 6. Krygowski, T. M.; Fawcett, W. R. J. Am. Chem. Soc. 1975, 97, 2143. Koppel, I. A.; Palm, V. A. In Advances in Linear Free Energy Relationships; Chapman, N. B., Shorter, J., eds.; Plenum: London, 1972; Chapter 5.

Problems 1. Show that there is a relationship between the molar volume (table 4.1) and molecular diameter (table 4.3) for the solvents considered in these compilations. Check to see whether the slope of the least-squares line has the expected value. 2. The following data were obtained for dilute solutions of nitromethane in carbon tetrachloride at 208C.

202

LIQUIDS, SOLUTIONS, AND INTERFACES

Mole fraction NM

Density r/kg m3

Relative Permittivity

Refractive Index

0 0.0343 0.0486 0.0693 0.1202

1594.0 1584.1 1580.6 1573.8 1557.1

2.238 2.710 2.958 3.257 4.038

1.4605 1.4582 1.4569 1.4559 1.4529

Using the method outlined in section 4.3A, estimate the molar polarization and refraction in the limit of inﬁnite dilution. Use these results to estimate the dipole moment and polarizability of NM and compare with the results in table 4.3. 3. Estimate the Kirkwood polarization parameter for dimethylsulfoxide using the MSA model described in section 4.4. Your result should agree with that given in table 4.4. 4. Given the following dielectric relaxation data for methanol, ﬁnd the values of the es , e1 , and tD , assuming there is only one relaxation process. Frequency /MHz

In-Phase e0

Out-of-Phase e00

5 1520.5 2016 4183 5216 7122

32.30 27.45 25.04 15.42 13.45 9.57

0 10.30 11.55 13.10 12.12 10.40

5. The following data were obtained for the permittivity of aqueous solutions of LiBr. Conc./M

es

0.5 1.0 1.5 2.0 3.0 4.0 5.0

69.29 62.75 56.53 51.85 42.56 35.66 30.44

Determine the best value of de and be in equation (4.6.1) using a least-squares procedure, and assuming that es for pure water is 78.46. Compare these values with the results for NaCl (equation (3.9.24)) and comment. 6. The following data are available for methanol assuming one relaxation process:

POLAR SOLVENTS

Temperature/8C

es

e1

tD /ps

43.4 37 25 0 30

50.6 47.9 44.4 37.7 31.9

6.9 6.7 6.9 6.4 6.0

333 263 186 95 47

203

Estimate the longitudinal relaxation time at 25, 0, and 258C. Estimate the activation enthalpy associated with both tD and tL . 7. The following data have been obtained for the free energy of transfer of Liþ and Br ions from water to the given non-aqueous solvent. Analyze the dependence of tr G on solvent using the equation Q ¼ Q0 þ as AN þ bs DN. tr G =kJ mol1 Solvent

Liþ

Br

AN

DN

H2 O MeOH PC DMF AcN NM NB DMSO

0 4.4 23.8 10 25 48 38 15

0 11.1 30.0 36.2 31.3 30 29 27.4

54.8 41.5 18.3 16.0 18.9 20.5 14.8 19.3

18.0 19.0 15.1 26.6 14.1 2.7 4.4 29.8

Determine the relative contributions of solvent acidity and basicity to the observed variation in tr G . Compare the ﬁt with the above equation to a simple linear ﬁt involving the more important parameter. Is the addition of the second parameter justiﬁed? 8. The following data are reported for the Gibbs energy of transfer of the salts LiCl and KBr. Perform the analysis used in question 7 with these data. What do you conclude about the suitability of equation (4.9.8)? tr G =kJ mol1 Solvent

LiCl

KBr

H2 O MeOH F NMF DMF AN DMSO PC NB NM

0 16.7 5.86 13.8 31.0 66.1 20.1 61.9 — —

0 20.5 2.93 — 22.6 38.5 13.4 35.6 57.2 44.1

5

Spectroscopic Studies of Liquid Structure and Solvation

Ga´bor Pa´linka´s

Erika Ka´lma´n

Ga´bor Pa´linka´s was born in Budapest, Hungary, in 1942. He entered Eo¨tvo¨s Lora´nd University in Budapest, graduating with a Ph.D. in physics in 1968. He then joined the Laboratory of Structural Chemistry of the Hungarian Academy of Sciences, where he undertook X-ray diﬀraction studies of water and aqueous electrolyte solutions. In 1973 he began a collaboration with Erika Ka´lma´n to develop electron diﬀraction methods for studying the structure of liquids. Their work resulted in the ﬁrst experimental derivation of the pair correlation functions in liquid water using existing neutron, X-ray, and electron diﬀraction data. He expanded his interests in 1975 to include molecular dynamics simulations of these systems in a collaboration with Professor Karl Heinzinger at the Max Planck Institute in Mainz, Germany. He has published extensively in the area of solution structure, both aqueous and non-aqueous. He is currently Director of the Chemical Research Center of the Hungarian Academy of Sciences, and holds the position of Professor at the Technical University in Budapest. Erika Ka´lma´n was born in Budapest in 1942, where she grew up and went to school. She attended university in Dresden, Germany and obtained a Ph.D. in chemistry in 1970 after studying in the laboratory of Professor Kurt Schwabe. She then returned to Budapest to begin work on the study of liquid structure at the Hungarian Academy of Sciences. In 1973 she began a very fruitful collaboration with Ga´bor Pa´linka´s. She spent a sabbatical year in the United States in 1975, where she worked in the laboratories of Harold Friedman at Stony Brook and Alan Narten at Oak Ridge National Laboratories. In the mid 1980s she helped edit an important series of books on 204

SPECTROSCOPIC STUDIES OF LIQUID STRUCTURE AND SOLVATION

205

the chemical physics of solvation. She has expanded her interests in recent years to include studies of corrosion and nanochemistry. She currently directs the Surface Chemistry and Corrosion Research group at the Chemical Research Center of the Hungarian Academy of Sciences and holds the position of Professor at the Technical University in Budapest. Erika Ka´lma´n and Ga´bor Pa´linka´s were married in 1974 and have two children.

5.1 What Spectroscopic Techniques Are Available? Spectroscopy involves the study of the interactions of electromagnetic radiation with matter. In the case of liquids, radiation of a wide range of frequencies, and thus energies, has been used, all the way from radio-frequency waves to X-rays. Experiments involving neutrons, which are associated with very short wavelengths, are also important. In the spectroscopic experiment the incident radiation may be either absorbed or scattered and the experimental information is obtained by examining the intensity and direction of the radiation after it has passed through the sample. Several spectroscopic techniques will be considered in this chapter. X-ray and neutron diﬀraction techniques are powerful tools for studying the structure of liquids and have been introduced in chapter 2. They may also be used to study the structure of solutions and determine distribution functions for both the solute and solvent. The feasibility of these experiments depends on the number of diﬀerent nuclei involved in the system. UV-visible spectroscopy is mainly used to study electronic transitions in polyatomic species. These species are often complex ions formed between the electrolyte and the solvent, or between the cation and one or more anions. Vibrational spectroscopy involves electromagnetic radiation of lower energy, usually in the infrared region. It is used to study intramolecular vibrational modes and how they are altered by the environment in solution. It can also be used to study the bonds formed between solute and solvent in the solvation process. Finally, nuclear magnetic resonance spectroscopy and its application to the study of solvation will be discussed. This is a particularly powerful technique because it provides information about the environment of a given nucleus, and experiments speciﬁc to a given nucleus can be carried out provided the nucleus has a non-zero magnetic moment. Several other spectroscopic techniques are commonly used [G1] but those considered here provide a representative picture of what can be learnt from those experiments. One should remember that the atoms and molecules in liquids are not motionless but in a state of ﬂux determined by the intermolecular interactions and temperature. From the study of microwave spectroscopy discussed in chapter 4, it was found that rotational diﬀusion processes in liquids are characterized by relaxation times the order of a few picoseconds. When a liquid is irradiated with UV or visible radiation which involves frequencies greater than 1015 Hz, the atoms and molecules appear to be motionless because the frequency of the electromagnetic radiation is much greater than that associated with molecular motion (1012 Hz and lower). The same is true for infrared spectroscopy except in the

206

LIQUIDS, SOLUTIONS, AND INTERFACES

far infrared where the frequency of the probing radiation is sometimes close to that of the dynamic processes in the liquid. In the case of NMR spectroscopy, radio-frequency radiation is used to study the relaxation processes of a given nucleus in a magnetic ﬁeld. Since most relaxation processes in solution take place at higher frequencies than the resonant frequencies in NMR, only the average environment of a given nucleus in solution is involved in the experimental observations. In this chapter much of the discussion involves electrolyte solutions in both aqueous and non-aqueous media. The role of spectroscopy in elucidating the chemical interactions which cannot be understood on the basis of classical electrostatics is emphasized. More details about each of the four spectroscopic techniques and their application are now given.

5.2 X-Ray and Neutron Diffraction Studies The important experimental features of X-ray and neutron diﬀraction were discussed earlier in chapter 2. The incident radiation is characterized by an intensity I0 and a wavelength l. In the case of neutrons the wavelength is calculated from the momentum of the particles using the de Broglie equation. The neutron diffraction experiments discussed earlier involved monochromatic particles, that is, neutrons with the same velocity and thus the same kinetic energy. Another type of neutron diﬀraction experiment involves neutrons with a range of energies. This is called a time-of-ﬂight experiment, and it requires detectors which respond to particles of diﬀerent energies at diﬀerent scattering angles. The quantity of interest in a diﬀraction experiment is the intensity I(y) observed at a scattering angle y or at wave number kD deﬁned as kD ¼

4p sinðy=2Þ l

ð5:2:1Þ

where kD has the dimension of reciprocal length. The scattered intensity at angle y arises from three sources, namely, coherent, incoherent, and multiple scattering. As was seen earlier in section 2.5, the structure function FT(kD) for the liquid sample is determined from the coherent scattering. The relationship between FT(kD) and the individual partial structure factors S ij(kD) for any two diﬀerent atoms i and j in the liquid is !2 XX X FT ðkD Þ ¼ ð2 dij Þci cj fi ðkD Þ fj ðkD Þ S ij ðkD Þ 1 = ci fi ðkD Þ ð5:2:2Þ ij

j

i

The ultimate goal of the diﬀraction experiment is to determine each contribution S ij (kD) to the structure function FT(kD). Then one can determine the individual pair correlation functions gij(r) by Fourier transformation. As pointed out in section 2.5, there are signiﬁcant diﬀerences between results from X-ray and neutron diﬀraction experiments because of the way these species are scattered by the atoms of the liquid. In the case of X-rays the scattering is due to the electrons around each nucleus, and the scattering amplitude increases with

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207

the atomic number of the atom. This means that when X-rays interact with liquid water, they are signiﬁcantly scattered only by the oxygen atoms. As a result one partial structure factor, namely, S OO(kD), makes a contribution to FT(kD). On the other hand, in a neutron diﬀraction experiment neutrons are scattered by both the proton 1 H, and the oxygen atom 18 O. X-ray and neutron diﬀraction experiments diﬀer signiﬁcantly in another important way. In an X-ray experiment the scattering amplitude for atom i, fi (kD), depends strongly on the reciprocal distance or wave number kD. As kD increases fi (kD) decreases monotonically [1]. This means that the disentanglement of the individual partial structure factors S ij(kD) requires knowledge of fi (kD). On the other hand, in a neutron diﬀraction experiment fi (kD) is independent of kD and depends only on the nature of the atom and its nuclear components. As a result, fi (kD) is given diﬀerent names in these experiments. In an X-ray experiment, fi (kD) is usually referred to as the form factor, whereas fi (kD) in a neutron diﬀraction experiment is called the coherent scattering length [1]. Because of the important diﬀerences between the two diﬀraction experiments, the strategies used in carrying out these studies are not the same. X-ray experiments are certainly more common since the equipment used is easily obtained. Neutron experiments are carried out at a nuclear reactor site or at an accelerator with the appropriate facilities. In the following sections, some results from diffraction experiments are presented with emphasis on the structural information which has been obtained regarding ion solvation in electrolyte solutions. A. X-Ray Diffraction Studies of Electrolyte Solutions X-ray diﬀraction has been used extensively to study the structure of aqueous electrolyte solutions [G1]. When an electrolyte with monoatomic ions is dissolved in water the number of diﬀerent atoms in the system is four, namely, M and X, from the electrolyte, and H and O from water. As a result there are ten independent partial structure factors. They are S OO, S OM, S OX, S OH, S HH, S HM, S HX, S MM, S MX, and S XX. In a more complex system with n diﬀerent atoms, the number of independent partial structure factors is ½nðn þ 1Þ=2. In an X-ray experiment, all partial structure factors involving the H atom are not important so that the number is reduced from ten to six. Of these, two more may be neglected, namely, those involving only cations or anions, that is, S MM and S XX. Obviously, ions of the same sign tend to be as far as possible away from each other. Finally, if ion pairing is negligible, S MX may also be neglected. Thus, three signiﬁcant partial structure factors, namely, S OO, S OM, and S OX usually contribute to the observed structure function FT(kD). Because the scattering amplitudes fi (kD) vary with reciprocal distance, a normalized structure function H(kD) is usually deﬁned in reporting X-ray diﬀraction data. Thus, HðkD Þ ¼ FT ðkD Þ=MðkD Þ

ð5:2:3Þ

where M(kD) is chosen to change the scale to that characteristic of one molecule and remove some of the breadth due to the electron distribution in the molecule [2]. A convenient choice in the case of electrolyte solutions is

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LIQUIDS, SOLUTIONS, AND INTERFACES

MðkD Þ ¼

X

!2 xi fi ðkD Þ

ð5:2:4Þ

i

where xi is the mole fraction of atom i [3]. The value of kDH(kD) oscillates in an irregular fashion and with decreasing amplitude. These data have been analyzed on the basis of the ﬁrst-neighbors model [3] according to which HðkD Þ ¼ xM HOM ðkD Þ þ xX HOX ðkD Þ þ xW HOO ðkD Þ

ð5:2:5Þ

where xM, xX, and xW are the mole fractions of cation–water aggregates, anion– water aggregates, and ‘‘free’’ water, respectively, and Hij is the normalized structure factor for atoms i and j. Accordingly, there are two states for water molecules, those bound in hydration shells and those in the normal (‘‘free’’) water structure. In addition, the hydration shells only involve nearest neighbors which are arranged in a very symmetrical fashion. Using this model, Pa´linka´s et al. [3] resolved the data for FT(kD) to obtain a model for alkali and alkaline earth metal halide solutions. In this way they developed a model for ion hydration in these systems. Their results are shown for 2 m solutions of the alkali metal chlorides in ﬁgs 5.1 and 5.2. It is clear from these data that gOM(r) has a well-deﬁned maximum at a value of r which is greater than 200 pm. In addition, the position of the maximum shifts to higher values as the atomic mass of the alkali metal cation increases. On the other hand, the position of the maximum for Cl on the gOX(r) function is approximately independent of cation nature and occurs at 314 pm.

Fig. 5.1 Values of the normalized structure factor kDH(kD) for 2 m solutions of the alkali metal chlorides in water as a function of reciprocal distance kD [3]. The data for KCl, NaCl, and LiCl have been shifted vertically by 20, 40, and 60 nm1, respectively, for the sake of clarity.

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209

Fig. 5.2 Pair correlation functions for the cation–water, gOM(r), and anion– water interactions, gOX(r), in 2 m aqueous solutions of the alkali metal halides [3]. The data for NaCl, KCl, and CsCl have been shifted vertically by 5, 10, and 15 units, respectively, for the sake of clarity.

Many X-ray diﬀraction studies of electrolyte solutions have been carried out in aqueous solutions [G1, 4, 5]. Values of the most probable distance, roi, between the oxygen atom in water and a number of monoatomic ions are summarized in table 5.1. In the case of the cations, this distance reﬂects the radius of the cation plus the eﬀective radius of the water molecule measured in the direction of the lone pairs on oxygen. In the case of alkali metals, the eﬀective radius of water increases from 122 pm for Liþ to 131 pm for Csþ when the Shannon and Prewitt radii are assumed for the cations (see section 3.2), the average value being 127 pm. This result can be attributed to the observation that the coordination number for water molecules around an alkali metal or alkaline metal earth cation changes with cation size and electrolyte concentration. In the case of the Liþ ion, this number decreases from six in very dilute solutions to four in concentrated solutions [5]. Because of the electrostatic character of the interaction between the cation and water molecules, these molecules exchange rapidly with other water molecules in their vicinity. For this reason, the solvation coordination number should be considered as an average. In the case of transition metal cations, the coordinated water molecules are covalently bonded to the cation, and the coordination number can be interpreted as ﬁxed in the absence of other ligands. If the cations Mn2þ, Fe2þ, Co2þ, and Ni2þ are assumed to be in the high spin state, then the eﬀective radius of coordinated water molecules in these systems is 123 pm on the basis of the X-ray data. Interpretation of the roi data for monoatomic anions is much more diﬃcult because of the fact that anions are solvated in water via hydrogen bonding (see

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Table 5.1 Most Probable Distance roi Between the Oxygen Atom in Water and Monoatomic Ion i in Aqueous Electrolyte Solutions [5] Ion

Distance roi / pm

Coordination Number

Solution

Liþ Naþ Kþ Csþ Agþ

210 242 280 315 242

6 6 6 6 4

2.2 M LiI 2 M NaCl 2 M KCl 2 M CsCl 3.3 m AgNO3

Mg2þ Ca2þ Sr2þ

212 244 264

6 6 8

1.1 M MgCl2 2 m CaBr2 1.6 m SrCl2

Al3þ In3þ

187 215

6 6

3.8 m Al(NO3)3 5.0 m In(ClO4)3

Mn2þ Fe2þ Co2þ Ni2þ Cu2þ

220 212 210 207 194

6 6 6 6 4

2 m MnSO4 2.2 m Fe(ClO4)2 3.7 m Co(ClO4)2 3 m NiCl2 2.8 m Cu(ClO4)2

F Cl Br I-

262 310 334 363

4.5 6 6 6.9

4.2 m KF 3 m CoCl2 2 m CaBr2 2.2 m LiI

ﬁg. 3.1). Thus, the ion–oxygen atom distance cannot be interpreted in a simple way as it is for cations [5]. X-ray diﬀraction studies have also been carried out in non-aqueous electrolyte solutions. In the case of methanol, there are two atoms which scatter X-rays, namely, carbon and oxygen. When a monoatomic electrolyte is added the number of scattering atoms increases to four. As a result, such a system has ten partial structure factors. If the ion–ion correlations are neglected this reduces to seven. A system which has been analyzed in some detail is MgCl2 in methanol [6, 7]. Analysis of the data gives a Mg–O distance of 207 pm and a Cl–O distance of 318 pm for coordination numbers of six [6]. As the number of atoms in the solvent molecule increases disentanglement of the partial structure factors becomes more diﬃcult. One way of making the analysis easier is to use diﬀerent isotopes of the atoms involved. This technique has been applied with success in neutron diﬀraction experiments which are described in the following section. B. Neutron Diffraction Studies of Electrolyte Solutions In the case of neutron diﬀraction, the radiation is scattered by the atomic nuclei, not by the electrons [1]. As a result, the scattering amplitude fi (kD) is determined by the nucleons and depends in a complex way on their total number. This means

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211

that isotope eﬀects are very important in developing experimental strategies, and that very light atoms such as hydrogen and deuterium have signiﬁcant values of fi (kD) [8, 9]. As discussed earlier there are ten partial structure factors which can contribute to the structure function Fi (kD) for a monoatomic electrolyte MX dissolved in water. By using isotope substitution techniques, six of these can be factored out, and only two of the remaining contributions are normally signiﬁcant. Consider as an example the studies of LiCl in D2O by neutron diﬀraction [8]. The isotopes 6Li and 7Li have very diﬀerent scattering amplitudes, so that the structure functions Fi (kD) are not the same for solutions of 6LiCl and 7LiCl. Moreover, the scattering amplitude fi (kD) is independent of kD, that is, of scattering angle y, and depends only on the nature of atom i. By subtracting the overall structure functions obtained in the presence of diﬀerent Li isotopes, the contributions from partial structure factors not depending on Li are removed. Thus, FT ðkD ; LiÞ ¼ FT ðkD ; 7 LiÞ FT ðkD ; 6 LiÞ

ð5:2:6Þ

depends only on the partial structure factors S LiO, S LiD, S LiLi, and S LiCl. Furthermore, it is assumed that solution structure is independent of isotopic nature, so that the partial structure factors for 7Li are exactly the same as those for 6Li. Two of the above partial structure factors are negligible, namely those for Li–Li interactions, and Li–Cl interactions assuming that ion pairing is not signiﬁcant. Fourier transformation of FT(kD, Li) gives the overall correlation function GLi(r) which depends on the pair correlation functions gLiO and gLiD. A plot of GLi(r) against distance r determined for LiCl in D2O is shown in ﬁg. 5.3. Two clearly resolved peaks are seen. The ﬁrst corresponds to the most probable distance between the Liþ ion and the oxygen on the nearest water molecules (195 pm) and the second to the corresponding distance between the Liþ ions and D atoms (255 pm). If the same diﬀerence is determined for 6LiCl and 7LiCl solutions in light water, the diﬀerence pair correlation function depends on gLiO and gLiH.

Fig. 5.3 The ﬁrst-order diﬀerence function GLi(r) for LiCl in D2O as a function of interatomic distance r. (From reference 8, with permission.)

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LIQUIDS, SOLUTIONS, AND INTERFACES

Using the data obtained in light and heavy water, the individual pair correlation functions can be separated. For the data in D2O, one has GLi ¼ 2cLi cO ðfLi ÞfO ½gLiO 1 þ 2cLi cD ðfLi ÞfD ½gLiD 1

ð5:2:7Þ

where the independence of fi from kD has been noted. Similarly in H2O, one has GLi ¼ 2cLi cO ðfLi ÞfO ½gLiO 1 þ 2cLi cH ðfLi ÞfH ½gLiD 1

ð5:2:8Þ

These are two equations in the unknowns gLiO and gLiD, assuming that gLiD and gLiH are exactly the same [8, 9]. Thus, the two pair correlation functions may be obtained separately. This method requires that careful and precise experiments be carried out. Another type of isotope experiment can be used to determine the eﬀect of the electrolyte on water structure. If one compares the overall pair correlation functions for a given LiCl concentration in H2O and D2O one obtains the diﬀerence function GLi(r). This function depends on the values of gij(r) which are due to the two H isotopes. They are gOH, gLiH, gClH, and gHH, where the subscript ‘‘H’’ designates both 1H and 2H. To a good approximation, one expects to ﬁnd changes only in gHH with change in electrolyte concentration. Experiments have shown [10] that the pair correlation function gHH in 1 m LiCl is almost the same as that in pure water, thereby demonstrating that the hydrogen bonding in the solution is very little perturbed by the added electrolyte. However, when the electrolyte concentration is increased to 10 m, there is a signiﬁcant change in gHH, indicating that the number of hydrogen bonds in the very concentrated solution is about 70% lower than in pure water. Using the second diﬀerence technique the partial pair correlation functions gClH and gClO have been determined for the Cl ion in water [11]. The experiments involve ﬁrst determining the diﬀerence function GCl(r) using chloride salts rich in 35Cl or 37Cl. Then each of these experiments is done in either H2O or D2O to achieve separation of the gClH and gClO partial pair correlation functions. The results conﬁrm that the orientation of the water molecule with respect to the Clion is that shown in ﬁg. 3.1. It is useful to consider the general characteristics of the partial pair correlation function for a metal ion coordinated to water molecules. Given suﬃciently precise data and metal isotopes with diﬀerent scattering fractions fi(kD), the pair correlation function gMO is the ultimate goal of experiments designed to study ion solvation. The shape of gMO for the cases of strongly and weakly coordinated water molecules is shown in ﬁg. 5.4. When the water is tightly bound to the metal ion, the ﬁrst peak in gMO(r) is high and sharp, reaching values greater than 2. The position of the peak gives the M–O distance. By integrating gMO(r) up to the ﬁrst minimum at rs, one obtains an estimate of the coordination number for the ion (see equation (2.10.1)). For Ni2þ in water, this number is 6 as one would expect [8, 9]. In addition, the average residence time for such a water molecule is quite long, speciﬁcally, the order of 10 ms. When the water molecules are weakly bound to the central metal ion the appearance of the partial pair correlation function gMO(r) changes signiﬁcantly (see ﬁg. 5.4). The ﬁrst peak on this curve is lower and broader and the subsequent minimum much less deep. Cations with this type of

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213

Fig. 5.4 Generic radial distribution functions gMO for the metal ion Mnþ in water for cases of (a) strong coordination and (b) weak coordination. rMO is the optimum metal ion– oxygen distance in the solvated species. The coordination number is estimated by integrating the distribution function out to the minimum at rs. (From reference 8, with permission.)

gMO are less strongly hydrated and have water residence times in the ﬁrst solvation layer which are of the order of 10 ps. Not only does neutron diﬀraction allow one to determine ionic size and hydration numbers in solution but it can also be used to assess changes in hydration with concentration. In the case of Liþ, the hydration number is 6 in dilute solutions but it drops to values below 4 in very concentrated solutions. Similar conclusions have been reached regarding divalent cations such as Ca2þ for which the ion–solvent interactions are mainly electrostatic in nature. For this system the hydration number decreases from 10 in 1 M CaCl2 to 6 in a 4.5 M solution of the same salt. The important limitation of neutron diﬀraction experiments is that the necessary isotopes are often not available so that the diﬀerence technique described here cannot be applied. Thus, neutron diﬀraction studies have been carried out to study solvation of Liþ and Kþ cations but not Naþ. In the latter case, X-ray techniques, which do not provide information about hydrogen bonding, are used. In summary, diﬀraction techniques provide a powerful means of investigating the structure of electrolyte solutions. They give information about the pair correlation functions which can be directly related to modern theoretical techniques such as molecular dynamics calculations. This information can also be used to improve the statistical thermodynamic models of electrolyte solutions discussed in chapter 3.

5.3 Nuclear Magnetic Resonance Spectroscopy in Solutions Nuclear magnetic resonance (NMR) spectroscopy is based on the magnetic properties of the nuclei which make up the molecules and ions in solution. Each nucleus has a spin quantum number I which depends on the number of protons

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LIQUIDS, SOLUTIONS, AND INTERFACES

and neutrons in the nucleus and the way in which the spins of these nucleons interact. The spin quantum number can have integral or half-integral values so that I may equal 0, 1/2, 1, 3/2, . . . NMR spectroscopy was ﬁrst applied to protons (1H) for which I is equal to 1/2. In the case of deuterium, which has one proton and one neutron, the nuclear spin quantum number is 1. Atoms with an even number of protons and an even number of neutrons have I equal to zero; examples are 12C, 16O, and 32S. The magnetic properties of some nuclei which have been studied experimentally are summarized in table 5.2. The magnetic energy level is degenerate with a total of 2I þ 1 quantum states. In the presence of an external magnetic ﬁeld these quantum states have diﬀerent energies, each with a quantum number mM. For example, the proton for which I ¼ 1/2 has two energy levels with mM ¼ 1/2 or 1/2, and the deuteron for which I ¼ 1 has three energy levels with mM ¼ 1, 0, or 1. Another important property of a magnetic nucleus is its magnetogyric ratio gM , which is also given in table 5.2. This gives a measure of how strongly or weakly the nucleus interacts with an external magnetic ﬁeld, and can be considered as the magnetic analog of the extinction coeﬃcient in optical spectroscopy. Using this quantity and the spin quantum number one can calculate the magnetic moment for the nucleus, which is given by lM ¼

gM Ih 2p

ð5:3:1Þ

In the presence of a magnetic ﬁeld B0 the energy levels split according to the individual values of lM. The diﬀerence in energy or energy gap between these levels is given by hn ¼

lM B0 gM hB0 ¼ I 2p

ð5:3:2Þ

and depends on the strength of the magnetic ﬁeld. This is illustrated for the proton in ﬁg. 5.5. Table 5.2 Magnetic Properties of Some Nuclei Nucleus 1

H H 7 Li 13 C 14 N 17 O 19 F 23 Na 35 Cl 33 S 2

Spin Quantum Number I

Magnetogyric Ratio gM 107 =rad T1 s1

1/2 1 3/2 1/2 1 5/2 1/2 3/2 3/2 3/2

26.75 4.11 10.40 6.73 1.93 3.63 25.18 7.08 2.64 2.05

SPECTROSCOPIC STUDIES OF LIQUID STRUCTURE AND SOLVATION

215

Fig. 5.5 Energy level splitting for a proton (I ¼ 1=2) in an applied magnetic ﬁeld B0.

EXAMPLE

Calculate the magnetic moment for a proton from its magnetogyric ratio. What is the frequency corresponding to the energy gap in a magnetic ﬁeld of 4.69 T? The magnetic moment lM is lM ¼

26:75 107 ð1=2Þ 6:626 1034 ¼ 1:41 1026 J T1 2p

ð5:3:3Þ

The energy gap is hn ¼

1:413 1026 4:69 ¼ 1:325 1025 J ð1=2Þ

ð5:3:4Þ

This corresponds to a frequency of 200 MHz. From a classical point of view a nucleus which possesses a magnetic moment (I 6¼ 0) can be considered as a spinning charged particle. If I ¼ 1=2, the nucleus behaves as a spinning charged sphere; if I is larger, then the nucleus is equivalent to a non-spherical spinning charge, and, in addition, possesses a quadrupole moment. When a magnetic nucleus is placed in a magnetic ﬁeld B0 it tends to adopt the direction of the ﬁeld. However, because of its spin, it is not aligned exactly with the external ﬁeld but instead at an angle to it, as shown in ﬁg. 5.6. The net result is that the magnetic moment vector lM precesses about the magnetic ﬁeld vector B0. The angular velocity for this precession is called the Larmor precessional frequency and is given by oL ¼ gM B0

ð5:3:5Þ

If oL is expressed as a normal frequency n, it is clear that n is given by equation (5.3.2). Now consider the eﬀect of a small additional magnetic ﬁeld B1 applied perpendicular to the main ﬁeld B0 . If the ﬁeld B1 rotates around B0 at the Larmor precessional frequency oL , a torque is produced on vector lM which acts to

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LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 5.6 Magnetic moment vector lM precessing about the applied ﬁeld vector B0 (a); the same system with addition of a rotating magnetic ﬁeld B1 perpendicular to B0 (b).

change the angle y (ﬁg. 5.6). On the other hand, for other frequencies, the interaction between lM and B1 varies depending on the relative phases of the two motions and there is no sustained eﬀect. When B1 rotates at the Larmor frequency the nuclear magnetic moment can absorb energy and assume a higher energy level in which lM points in the opposite direction. The absorption of energy by the nucleus under these circumstances is called nuclear magnetic resonance. In the NMR experiment the sample containing the magnetic nuclei is exposed to a ﬁxed magnetic ﬁeld B0 and a rotating ﬁeld B1 , as described above. The frequency of the rotating ﬁeld is swept through a range of values including the Larmor frequency for the nucleus in question. The absorption of energy at this frequency is observed when a plot of energy absorbed against frequency is made. This is the NMR spectrum. The rotating magnetic ﬁeld is generated using a linearly oscillating ﬁeld of radio frequency. The linearly oscillating ﬁeld can be resolved into two rotating magnetic ﬁelds of equal magnitude but opposite rotational directions. Only the component rotating in the same direction as lM interacts with the nuclei. The above experiment is called a frequency sweep experiment. Equivalent results are obtained if the radio frequency is held constant and the magnetic ﬁeld B0 is changed. This is called the ﬁeld sweep technique. The relative populations of the separate energy levels for a nucleus in a magnetic ﬁeld and the relaxation processes from a higher energy level to a lower one are also of fundamental interest. On the basis of Boltzmann’s law and equation (5.3.2) the ratio of the number of nuclei in the higher energy level to that in the lower level for a proton (I ¼ 1=2) is Nu gM hB0 ð5:3:6Þ ¼ exp N1 2pkB T Since the energy gap between these levels is small with respect to kB T for typical values of B0 , the exponential can be replaced by the ﬁrst two terms in its expansion as an inﬁnite series. Thus,

SPECTROSCOPIC STUDIES OF LIQUID STRUCTURE AND SOLVATION

Nu g hB ¼1 M 0 N1 2pkB T

217

ð5:3:7Þ

EXAMPLE

Estimate the ratio of Nu =N1 for protons in a magnetic ﬁeld of 4.69 T at 258C. The energy gap between the two levels is hn ¼

26:75 107 6:626 1034 4:69 ¼ 1:325 1025 J 2p

ð5:3:8Þ

The value of kB T at 258C is kB T ¼ 1:381 1023 298:2 ¼ 4:118 1021 J

ð5:3:9Þ

Therefore the ratio Nu =N1 is Nu 1:325 1025 ¼1 ¼ 1 3:22 105 N1 4:118 1021

ð5:3:10Þ

Thus the diﬀerence in population for these two levels corresponds to 32.2 ppm. There are two kinds of relaxation processes in NMR, namely, spin–lattice and spin–spin relaxation. The lattice is the general collection of atoms and molecules surrounding the nuclei which interact with the magnetic ﬁeld. The lattice system is involved in vibrational and rotational motion reﬂecting the thermal energy of the system. Interaction between a nucleus which has absorbed energy from the magnetic ﬁeld and the lattice causes this nucleus to return to the ground state, thereby transferring energy which increases the vibrational and rotational motion of the lattice. This is spin–lattice relaxation which is a ﬁrst-order process characterized by a relaxation time T1 . It is also called longitudinal relaxation because it is associated with change in the component of the magnetization vector of the nucleus Mz in the direction of the applied magnetic ﬁeld B0 . The relaxation time T1 depends on the magnetogyric ratio of the nucleus gM and thus on the Larmor frequency at which it absorbs energy. It also depends on the characteristics of the lattice. The fact that T1 is ﬁnite results in broadening of the lines in the NMR spectrum. In liquids, values of T1 fall in the range from 102 to 102 s. The relationship between the width of the NMR line and T1 is n ¼

1 2pT1

ð5:3:11Þ

The other mechanism by which the system returns to equilibrium after absorption of energy is spin–spin relaxation. This is attributed to interaction of the spin vectors on adjacent nuclei. The precessing moment on nucleus 1 sets up an oscillating ﬁeld at nucleus 2; this ﬁeld momentarily comes into phase with that of the other nucleus which absorbs energy from nucleus 1. The resulting transfer of energy between the nuclei is called spin–spin relaxation. The other mechanism of line broadening associated with spin–spin relaxation is due to the local inhomogeneity of the magnetic ﬁeld B0 . Because the environment of each nucleus contains other magnetic nuclei, the local value of B0 , Blocal , is slightly diﬀerent. As a result there are a range of Larmor frequencies for the responding nuclei

218

LIQUIDS, SOLUTIONS, AND INTERFACES

depending on the range of values of Blocal . Spin–spin relaxation can also be described in terms of the magnetization vectors in the ðx; yÞ-plane perpendicular to the main magnetic ﬁeld B0 . These vectors Mx and My achieve ﬁnite values after the absorption of energy by the magnetic nuclei. They decay to zero, their equilibrium values, with a rate constant 1=T2 . Thus, spin–spin relaxation is also called transverse relaxation. Both longitudinal and transverse relaxation of the magnetization vectors Mx , My , and Mz are illustrated in ﬁg. 5.7. An understanding of relaxation processes in the resonance process allows one to explain the line shapes observed experimentally. As will be seen in chapter 7, chemical reactions, for example, proton transfer reactions, can lead to band broadening in the NMR spectrum. Thus, this feature of the spectra has very important applications in the study of fast chemical reactions in solution. In the discussion of the Larmor frequency above, the eﬀect of the magnetic environment on the absorbing nucleus was ignored. The eﬀectiveness of NMR spectroscopy in chemistry is deﬁnitely a result of the sensitivity of the absorbing nucleus to the nature of the nuclei in its immediate environment. Consider for example the case of ethanol, C2H5OH. One may distinguish three diﬀerent protons with respect to environment, namely, the proton in the –OH group, the protons in the methyl group, and the protons in the –CH2 or methylene group. Each of these species experiences a diﬀerent magnetic environment, so that the NMR spectrum of ethanol has three absorption peaks at low resolution. The shielding eﬀect of the environment is expressed quantitatively in terms of the shielding constant s. Thus, if oL is the Larmor angular frequency in the absence of a magnetic environment, the angular velocity in the presence of this environment is o ¼ gM B0 ð1 sÞ

ð5:3:12Þ

Fig. 5.7 Illustration of the magnetization vectors Mx , My , and My for a system (a) in the presence of an external magnetic ﬁeld B0 in the z-direction but no transverse ﬁeld B1 ; (b) after application of B1 in the (x; y)-plane and absorption of energy; and (c) after partial relaxation back to the equilibrium conﬁguration in (a).

SPECTROSCOPIC STUDIES OF LIQUID STRUCTURE AND SOLVATION

219

The observed peaks in the NMR spectrum are usually cited in terms of chemical shifts measured with respect to the peak for a reference compound. In the case of 1 H spectra this compound is usually tetramethylsilane, (CH3)4Si, which is added to the sample placed in the spectrometer. The protons in this reference are all equivalent and therefore give one NMR peak. The chemical shifts for the other peaks observed are deﬁned as o or d ¼ 10 or 6

ð5:3:13Þ

where or is the angular frequency at which the reference peak is observed. Values of d are given in parts per million through the factor 106. Positive contributions to s are termed diamagnetic, whereas negative contributions are paramagnetic. The description of NMR spectroscopy given here is applicable to a continuous wave instrument in which the radio-frequency is swept continuously over the appropriate range. Most modern instruments are pulsed or Fourier transform spectrometers (FT-NMR). In the pulsed instrument, the sample is irradiated with periodic pulses of radio-frequency energy that are directed through the sample at right angles to the magnetic ﬁeld. The short pulse, which lasts the order of 10 ms, results in a change in the net magnetization of the sample system away from the z-direction of the permanent magnetic ﬁeld. When the pulse ends, the magnetization relaxes back to its equilibrium position in a process called free induction decay (FID). The FID signal is detected by the spectrometer and stored digitally. By repeating this process many times, the signal-to-noise ratio of the FID is improved. Finally, the average FID which gives the NMR spectrum in the time domain is Fourier transformed to give the frequency domain spectrum. In the early stages of NMR spectroscopy, most work involved studies of protons in organic compounds. However, instruments are now available for studying a wide variety of other nuclei. In this chapter, attention is focused on NMR studies of ion solvation. Work carried out in both aqueous and non-aqueous solutions is considered in the following sections.

5.4. NMR Studies of Ion Solvation in Water One of the most important applications of NMR spectroscopy is the determination of the solvation number of ions in aqueous solutions. This can be accomplished in some cases using 1H NMR or using 17O NMR in isotopically enriched water. An example involving the highly charged Al3+ ion is shown in ﬁg. 5.8. The proton spectrum of an aqueous solution of Al(ClO4)3 at a temperature of 46:38C shows two peaks, a large one corresponding to the protons in bulk water, and a lower one shifted downﬁeld corresponding to the protons associated with the Al3+ ion. By integrating the area under these peaks one can determine the solvation number of Al3+ which is 6. As the temperature is raised, both lines broaden and move toward coalescence due to proton exchange processes, so that at 26:48C the absorption due to coordinated water molecules is scarcely visible [G1, G3].

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Fig. 5.8 Proton NMR spectra of aqueous 2.1 m Al(ClO4)3 at (a) 46:38 and (b) 26:48C. (From reference G1, with permission.)

The above experiment only works when the rate of exchange of the solvent is slow with respect to the frequency separation between the coordinated and bulk water molecules. If this frequency is 1000 Hz, then the relaxation time characterizing solvent exchange should be signiﬁcantly greater than 1 ms. This is the case for highly charged main group cations or for most transition metal ions for which water is a ligand. Data for some diamagnetic cations are summarized in table 5.3. It can be seen that d increases roughly with the ratio zi =ri , that is, with the ratio of ionic charge to ionic radius. However, this is not a complete explanation, as can be seen from the Al3+, Ga3+, In3+ series. The downﬁeld shift also increases with the polarizability of the ion. Paramagnetic cations such as Mn2+, Fe2+, Co2+, and Ni2+ cause a much larger downﬁeld chemical shift of the protons in coordinated water molecules. In some cases 1H NMR data can be used to study hydration for these systems in 17Oenriched water. Since the gyromagnetic ratio for 17O is much less than that for 1H the downﬁeld chemical shift in hertz is not as large. 17O NMR spectroscopy has the advantage that 17O exchange processes between adjacent water molecules are much slower than the corresponding 1H exchange processes. As a result, one source of line width broadening is greatly reduced. When the rate of solvent exchange around the ion is fast, only one line is seen in the NMR spectrum. In this case, the 1H nucleus in the solvent may experience all possible environments in the solution within the time corresponding to the Table 5.3 1H NMR Data for Water in the Primary Solvation Shell of Diamagnetic Cations Cations

Cation Chemical Shifta / ppm

Solvation Number

Be2+ Mg2+ Al3+ Ga3+ In3+ (Water)

8.04 5.55 8.74 8.98 7.22 4.13

4 6 6 6 6

a

Measured at low temperatures with respect to ethane gas as reference.

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221

radio frequency of the NMR experiment. Such a situation holds in a solution of NaCl. To a good approximation, the proton in the water molecule experiences three environments, that near the cation, that near an anion, and that in the bulk solvent far away from either ionic component. The experimentally observed chemical shift is an average for the protons in each of these environments. As a result the observed chemical shift changes with electrolyte concentration [12, 13]. Proton chemical shifts observed on addition of simple electrolytes to water are given in table 5.4. They depend on the nature of the electrolyte and can be either positive or negative. In the limit of dilute solutions the shifts are linear in concentration, reﬂecting the fact that solvent–solvent interactions are being replaced by ion–solvent interactions. The non-linearity at higher concentrations is attributed to ion pair formation. Two phenomena contribute to the observed chemical shift. One involves the breaking of hydrogen bonds, which leads to a positive shift. The other involves polarization of the water molecule in the ﬁeld of the ion, which gives a negative shift. The former eﬀect predominates for most electrolytes. Obviously, the chemical shifts given in table 5.4 depend on both cation and anion, and it is interesting to try to separate their contributions. The eﬀect of the cation was studied by Davies et al. [13], who measured the limiting slope of the chemical shift against concentration plots for a series of alkali metal and alkaline earth metal chlorides (see ﬁg. 5.9). When the molal chemical shift is plotted against the polarizing strength of the cation estimated as the charge/radius ratio, z=r, one ﬁnds that the most positive shifts in these series are for K+ and Ba2+. As z=r increases the molal shift becomes more negative due to the increase in polarizing strength of the cation. In the alkali metal series, this trend changes direction at K+, the molal shifts for Cs+ and Rb+ being less positive than one would expect on the basis of a simple linear correlation. This probably reﬂects the polarizability of the largest cations which should also be considered in estimating cationic polarizing strength. The trends for the alkali metals and alkaline earth metals are separated by 0.033 ppm. As argued by Davies et al. [13], this diﬀerence gives the eﬀect of the extra Cl ion which is in the alkaline earth metal solution (MCl2). By shifting the dependence for the alkaline earth metal chloride solutions down by two times 0.033 ppm, and that for the alkali metal chlorides down by 0.033 pm, one trend is

Table 5.4 Molal Chemical Shifts for the Proton Resonance of Water in Electrolyte Solutions at 258C [13] Electrolyte NaF KF LiCl NaCl KCl RbCl CsCl a

Chemical Shifta d / ppm kg mol1 0.021 0.008 0.047 0.096 0.111 0.092 0.080

With respect to that for pure water.

Electrolyte NaBr NaI NaCN NaNO3 NaBF4 NaSCN NaClO4

Chemical Shifta d / ppm kg mol1 0.128 0.175 0.113 0.081 0.123 0.143 0.140

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LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 5.9 Molal chemical shifts for the alkali metal and alkaline earth metal chlorides in aqueous solution plotted against the polarizing strength of the cation estimated as z=r (Pauling scale). The data shown as () have been corrected for the contribution of Cl , and represent estimates of the cation contribution (see text).

found for both series. This analysis gives an approximate method of separating cation and anion eﬀects. In assessing it, one should remember that the anion interacts directly with the proton whose magnetic resonance is being measured, whereas the cation interacts indirectly through the oxygen atom in the water molecule. The eﬀect of the halide ions for a given alkali metal ion is such that the molal chemical shift becomes increasingly more negative with increase in polarizing strength. Anions are known to have very little disruptive eﬀect on water structure, so that the trend observed mainly reﬂects the polarization of the proton by the anion. Another way of studying solvation eﬀects for simple diamagnetic ions is to measure chemical shifts for the solute ion. In the case of aqueous solutions of Na+ electrolytes, experiments have shown that the 23Na+ chemical shift is proportional to the mole fraction of the electrolyte over a wide concentration range [14, 15], so that it may be expressed as d ¼ d e xe

ð5:4:1Þ

where de is the value of the chemical shift characterizing the given electrolyte at unit mole fraction and xe , the mole fraction of the electrolyte. Similar results are available for 39K [16] and other alkali metal cations [14]. These results shed considerable light on the nature of the interactions of these simple cations with their environment. Values of de for seven Na+ electrolytes are summarized in table 5.5. A positive shift occurs when the interaction of the Na+ cation with the anion is less than that with a water molecule. This situation occurs in NaClO4 solutions over a very wide range of concentrations. At a mole fraction of 0.25, every Na+ ion is accompa nied by one ClO 4 and three molecules of water in solution, so that ClO4 ions are

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223

Table 5.5 Value of de for various 1–1 electrolytes containing 23Na+ [15] Electrolyte NaClO4 NaNO3 NaBPh4 NaCl NaBr NaI NaOH

de /ppm 24 19 0 7 10 13 22

expected to be in the immediate environment of the cation. The NMR results show that there is no signiﬁcant interaction between the electron clouds of these two ions. As the basicity of the donor ion or molecule increases, the chemical shift becomes more negative according to the following series: ClO 4 < NO3 < H2 O < Cl < Br < I < OH . This series also follows an increasing tendency to form ion pairs. Thus, in the case of Na+ and I , signiﬁcant overlap can occur between the electron clouds of these ions, so that I shields the 23Na+ nucleus more eﬀectively than a water molecule. As a result there is a moderate tendency to form ion pairs in concentrated NaI solutions. Studies of the nuclear resonances of 35Cl, 81Br, and 127I have also been carried out in various alkali metal halide solutions [17]. The magnitude of the chemical shift increases with electrolyte concentration and also with atomic number of the anion. In the case of the alkali metal ions the chemical shift becomes more positive in the series Naþ < Kþ < Liþ < Rbþ < Csþ . The results were attributed to direct interaction between the cation and anion in solutions containing K+, Rb+, and Cs+. With the smaller cations, interactions between the halide ion and the water molecules solvating the cation are more important. The above discussion provides a brief introduction to the NMR spectroscopy of solvation in water. Much more has been learnt about the kinetic aspects of solvation by studying relaxation times associated with the magnetic resonance lines. Detailed information is available in several reviews [G1, 18, 19]. Clearly, NMR spectroscopy is a powerful tool in the study of ion–solvent and ion–ion interactions in aqueous solutions and has helped greatly to improve the understanding of electrolyte solutions at the microscopic level.

5.5 NMR Studies of Ion Solvation in Non-Aqueous Solvents It is not surprising that 1H NMR was also applied to the study of cation solvation in non-aqueous media. Most solvent molecules contain protons, but in the case of

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aprotic solvents these protons often do not interact directly with the ions of the electrolyte. Proton NMR has been used to determine solvation numbers of highly charged cations such as Al3+ in solvents such as methanol, dimethylformamide, and dimethylsulfoxide [G3]. Since the solvent molecules are tightly bound to the cation, their exchange with molecules in the bulk is slow and two peaks are seen in the NMR spectrum, one corresponding to the solvent molecules coordinated to the cation and the other to free solvent molecules. By comparing peak areas, the number of coordinating solvent molecules per cation can be determined. This is typically six for cations such as Mg2+, Zn2+, Al3+, Ga3+, Co2+, and Ni2+ [G3]. Studies have also been carried out with systems where the exchange of solvent molecules between the solvation shells of the ions and the bulk is rapid. In this case the chemical shift of the nucleus being probed, for example protons, depends on the nature of the electrolyte and its concentration. A typical example is the 1H chemical shifts observed for some 1–1 electrolytes in acetonitrile [20], which are summarized in table 5.6. In all cases the shift is to a lower frequency, indicating that the protons at the positive end of the molecular dipole are more strongly shielded in the presence of the electrolyte than in pure acetonitrile. Distinct trends are diﬃcult to discern due to complications resulting from diﬀerent degrees of ion pairing, but the perchlorate anion has the least eﬀect among the anions considered. Among the halide anions, the trend follows the order Cl Br > I . The eﬀect of the large tetraalkylammonium ions is small because of their size and the fact that they interact with the negative end of the molecular dipole. The inﬂuence of the alkali metal cations increases approximately in the order Kþ < Naþ < Liþ , that is, in the order of their polarizing power. Another method of examining ion–solvent interactions is to study the chemical shifts of the solute ions. This type of investigation has been carried out for the 7 + 23 Li , Na+, and 205Tl+ cations and for the 35Cl, 81Br, and 127I anions.

Table 5.6 Molar Chemical Shifts for the Proton Resonance of Acetonitrile in Electrolyte Solutions at 258C [20] Electrolytea LiI NaI KI LiClO4 NaClO4 Et4 NCl Et4 NBr Bu4 NCl Bu4 NBr Bu4 NI

Chemical Shiftb d/Hz M1 6.5 7.5 6.0 4.0 3.0 7.5 7.4 6.3 6.9 5.0

Et4 Nþ is tetraethylammonium and Bu4 Nþ , tetrabutylammonium. b With respect to that for pure acetonitrile. a

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225

Variation in the 23Na+ chemical shift with solvent nature has been very helpful in elucidating the nature of the ion–solvent interactions, especially for aprotic solvents. Popov [21] showed that there is a linear correlation between the chemical shift observed for 23Na+ in an aprotic solvent and the Gutmann donor number DN when an electrolyte with minimal ion pairing is used. Values of d obtained in NaPh4B are plotted against DN for 11 aprotic solvents in ﬁg. 5.10. A very good correlation is found with a correlation coeﬃcient of 0.976. Data points for formamide, water, and methanol are also shown in this plot. The result for formamide falls on the regression line but those for water and methanol are clearly below it. This led to the suggestion [21] that the eﬀective donor numbers of water and the alcohols are much higher than the values given by Gutmann [22] when these solvents have their bulk structure with the accompanying hydrogen bonding. However, the Gutmann values of DN for these solvents are conﬁrmed by independent estimates of solvent basicity (see section 4.9). Chemical shifts for the 7Li nucleus have also been studied as a function of solvent nature for a variety of electrolytes including lithium perchlorate and lithium tetraphenylborate [21]. Although they change with solvent nature, there is no correlation with solvent donicity. Popov attributed the failure of the 7Li system to monitor basic properties of the solvent to the fact that the paramagnetic and diamagnetic screening constants for this atom are approximately equal, so that they cancel one another. In the case of 23Na, the paramagnetic screening constant is much larger than the diamagnetic one, so that this system can be used to monitor the basicity of aprotic solvents. Solvation of the halide ions in various solvents has been studied by measuring the chemical shifts for 35Cl, 79Br, and 127I in water and several other polar solvents [23]. The chemical shifts are strongly solvent dependent and correlate to the UV absorption band energy corresponding to charge transfer to the solvent.

Fig. 5.10 Chemical shift for 23Na using sodium tetraphenylborate dissolved in various solvents [21]. The regression line was drawn considering only aprotic solvents (ﬁlled circles). Abbreviations for solvents given in chapter 4: PYR ¼ pyridine.

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LIQUIDS, SOLUTIONS, AND INTERFACES

Polar molecular solutes have been used to probe the donor–acceptor properties of polar solvents. 19F chemical shifts have been measured for interaction between triﬂuoroiodomethane and the solvent molecule as electron pair donor [24]. As interaction between the donor molecule and the iodine atom in this molecule increases, electron density at the ﬂuorine atoms increases with a resulting positive chemical shift in the 19F NMR signal. An excellent correlation between these shifts and the Gutmann donor number was reported [24]. The 31P chemical shifts in triethylphosphine oxide measured in polar solvents is used to deﬁne the acceptor number scale for solvent acidity [25]. In this case, the oxygen atom in the P¼O bond acts as an electron pair donor to the solvent as a Lewis acid. The resulting inductive eﬀect lowers the electron density at the phosphorus atom and results in a chemical shift which depends on solvent acidity. In summary, NMR techniques based on the chemical shifts of a solute atom provide a useful way of probing solute–solvent interactions. They demonstrate clearly that the chemical nature of these interactions is important, and must be considered in developing an overall understanding of the structure of both electrolyte solutions and solutions of polar molecules.

5.6 Vibrational Spectroscopy in Solutions Vibrational spectroscopy is a powerful tool for studying solvation in polar solvents. These solvent molecules have permanent dipole moments and therefore absorb radiation in the infrared region of the electromagnetic spectrum due to the vibrational modes of motion. In the present section, the principal experimental techniques used in vibrational spectroscopy are outlined with emphasis on methods for analyzing solvent structure and solvation. A non-linear polyatomic molecule with N atoms has 3N 6 degrees of vibrational freedom. Each vibrational mode is characterized by a frequency which is determined by the force constant characterizing the vibrational motion, and the reduced mass of the system in a harmonic oscillator representation. For example, acetonitrile (CH3CN) has six atoms and 12 degrees of vibrational freedom. A very important vibrational mode in this molecule is the CN stretch, which is associated with the polar cyano group. If this group interacts with an ionic or molecular solute in the solvation process, the electron density in the bond is aﬀected, and a change in the vibrational frequency is observed. Other vibrations in this molecule are associated with the methyl groups and the C–C bond. EXAMPLE

Estimate the force constant for the C–C bond in acetonitrile using the harmonic oscillator approximation, given that the frequency of the C–C stretching mode expressed in wave numbers is 918 cm1 . In the harmonic oscillator approximation the vibrational frequency is given by n¼

1 k 1=2 2p m

ð5:6:1Þ

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227

where k is the force constant and m, the reduced mass. The reduced mass is estimated assuming that the vibrating groups are methyl (15.04 g) and cyano (26.02 g): m¼

0:01504 0:02602 ¼ 1:583 1026 kg molec1 6:022 1023 ð0:01504 þ 0:02602Þ

ð5:6:2Þ

In SI units, the frequency is n ¼ 918 2:998 1010 ¼ 2:752 1013 s1

ð5:6:3Þ

On the basis of equation (5.6.1), the force constant is k ¼ 4p2 1:583 1026 ð2:752 1013 Þ2 ¼ 473 N m1 molec1

ð5:6:4Þ

There are two kinds of vibrational spectroscopy commonly used to study solvent structure and solvation, namely, infrared and Raman spectroscopy. These methods diﬀer signiﬁcantly with respect to the nature of the vibrational features which may be observed. Therefore, some fundamental concepts and experimental methods are introduced before speciﬁc results are presented.

A. Infrared Spectroscopy Infrared spectroscopy is normally carried out in the mid-infrared region, that is, from 4000 to 200 cm1 . The near infrared region is at higher frequencies (12,500 to 4000 cm1 ) and the far infrared, at lower frequencies (200 to 10 cm1 ). Experiments in these regions require special techniques and are not considered further here. Infrared radiation is only absorbed by the irradiated molecule at the appropriate frequency if the corresponding vibration results in a change in molecular dipole moment. This means that not all vibrational modes are infrared active. An analysis of which vibrational modes in a polyatomic molecule are active is based on group theory and the symmetry properties of the molecule. More details about this subject may be found in monographs devoted to spectroscopy [G4]. Modern infrared spectroscopy is normally carried out in the Fourier transform mode (FTIR). Thus, the experimental information is gathered as an interferogram in the time domain and then Fourier transformed to obtain absorbance as a function of wavelength. A number of interferograms may be recorded in a few seconds. These are averaged before Fourier transformation. In this way, the signal-to-noise ratio in the resulting spectrum is signiﬁcantly enhanced with respect to a spectrum obtained in a traditional dispersive spectrometer In a simple transmission experiment the liquid sample is examined in a cell made of a suitable infrared transparent medium. These include sodium chloride, potassium bromide, zinc selenide, cadmium telluride, and germanium. Materials like sodium chloride should not be used to study solutions in protic solvents like methanol and water. In the case of strongly absorbing media, infrared experiments are more conveniently carried out using internal reﬂectance techniques. An example of an

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Fig. 5.11 Schematic diagrams illustrating (A) an attenuated total reﬂection experiment at the interface between a hemispherical window and a test solution and (B) the evanescent wave, which decays rapidly with distance into the test medium.

experimental setup to carry out such an experiment is shown in ﬁg. 5.11. The infrared radiation enters a hemispherical window at an angle y to the normal drawn from the window j solution interface. When the refractive index of the solution is less than that of the window, the radiation is totally reﬂected at the interface if the angle y is greater than the critical angle yc . On the basis of the wave nature of the radiation the beam actually penetrates the optically less dense medium a very short distance. This penetrating wave is called the evanescent wave because it decays exponentially with distance. From the theory of physical optics the depth of penetration is the order of the wavelength. EXAMPLE

Estimate the depth of penetration of infrared light with a frequency of 2000 cm1 into an aqueous solution given that it is transmitted through a ZnSe hemisphere (nop ¼ 2:93) at an angle 28 greater than the critical angle. The critical angle is given by s nop yc ¼ sin nw op 1

ð5:6:5Þ

where nsop is the refractive index of the solution and nw op , that of the window. Assuming that nsop ¼ 1:333, 1:333 yc ¼ sin1 ¼ 278 ð5:6:6Þ 2:93 The depth of penetration is given by dp ¼

lw 2 1=2 2pðsin y ðnsop =nw op Þ Þ 2

ð5:6:7Þ

where lw is the wavelength of the radiation in the window. The required wavelength is

SPECTROSCOPIC STUDIES OF LIQUID STRUCTURE AND SOLVATION

lw ¼

2:93 101 ¼ 1:465 105 m 2000

229

ð5:6:8Þ

Thus, for an angle of incidence equal to 298 dp ¼

1:465 105 ¼ 1:39 105 m 2pð0:235 0:207Þ1=2

ð5:6:9Þ

For this system the depth of penetration is 14 mm. This distance is comparable to the wavelength of the infrared radiation which is 5 mm. The internal reﬂectance technique is usually called attenuated total reﬂection (ATR) spectroscopy. It is especially useful for studying strongly absorbing media, for example, aqueous solutions. When the infrared radiation is absorbed in the test medium, one obtains a spectrum similar to that from a transmission experiment. However, there are distortions in the ATR spectrum, especially in the region of intense bands. One reason for distortion is the fact that the depth of penetration varies with wavelength. The other eﬀect is due to the change of the refractive index of the solution in the region of the intense band. ATR spectra should be corrected for these eﬀects so that they may be compared to normal transmission spectra. The combination of ATR and FTIR spectroscopies gives a powerful tool for studying the eﬀects of solvation in solutions made from polar solvents. Some recent data obtained in experiments using these methods are discussed later in this chapter. B. Raman Spectroscopy In Raman spectroscopy, the vibrational modes of a molecule are activated but in an entirely diﬀerent way. In such an experiment the system is irradiated with a strong source of light, usually from a laser which emits in the visible region. A molecule whose major dimension is less than the wavelength of the light will scatter the light if it is not absorbed. The electrons in the irradiated molecule oscillate as a result of this interaction and thereby produce radiation of the same frequency but scattered in all directions with respect to the incident beam. This phenomenon is known as Rayleigh scattering. However, a small fraction of the incident radiation can interact with the molecule via its vibrational degrees of freedom. If the vibrational motion results in a change in molecular polarizability, then the quantum of scattered light is slightly diﬀerent in energy from that of the stimulating radiation. When this occurs, the phenomenon is known as Raman scattering. EXAMPLE

Given that carbon dioxide is a linear molecule, which of its vibrational modes are infrared active, and which, Raman active? As a linear triatomic molecule, carbon dioxide has four degrees of vibrational motion. These are the symmetrical stretch (n1 ), the asymmetrical stretch (n3 ), and the bending mode (n2 ). The later vibration is doubly degenerate and can be described in two directions perpendicular to the interatomic axis.

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Because of its symmetry with respect to the central carbon atom, carbon dioxide has no net dipole moment. In a symmetrical stretching vibration, the dipole moment of the molecule remains zero. Therefore the n1 mode is not infrared active. However, the electron density along the interatomic axis is alternately elongated and condensed. Thus, the molecular polarizability changes with symmetrical stretching and the n1 mode is Raman active. Exactly the opposite conclusions are reached for the asymmetric stretching mode (n3 ). Since one C–O bond contracts while the other stretches, the molecule develops a dipole moment during vibration. Thus, the n3 mode is infrared active. At the same time the increase in polarizability due to the stretching bond is oﬀset by the decrease due to the contracting bond. As a result there is no net change in polarizability and the n3 is Raman inactive. The same kind of arguments apply to the bending modes (n2 ). Thus, n2 is infrared active and Raman inactive. This example demonstrates that infrared and Raman spectroscopies are often complementary. By applying a normal mode analysis using group theory, one may determine which vibrational modes in more complex molecules are infrared or Raman active. The shift in frequency of the Raman scattered radiation from the stimulating radiation may result in quanta of lower or higher frequency. The former process is known as Stokes scattering and the latter as anti-Stokes scattering. The quantummechanical theory of the Raman eﬀect involves the formation of a ‘‘virtual’’ excited quantum state at an energy level higher than the ground state by an amount hn0 , where n0 is the frequency of the stimulating radiation. The vast majority of molecules which reach the ‘‘virtual’’ excited state decay back to the ground state with the emission of Rayleigh scattered light of exactly the same frequency. However, a very few molecules in the virtual excited state decay down to the ﬁrst excited vibrational level of the ground state, thereby giving rise to Stokes scattered light. The energy of the quantum of scattered light will be less than hn0 by an amount corresponding to the energy diﬀerence between the ground vibrational state and the ﬁrst excited vibrational state. This is hnvib . It follows that the frequency of the Stokes line is nst ¼ n0 nvib

ð5:6:10Þ

A very few molecules in the sample are in the ﬁrst excited vibrational level when they interact with the stimulating radiation. They reach a virtual excited state which is correspondingly higher in energy than most excited molecules. The majority of these fall back to the same vibrational level, thereby emitting scattered Rayleigh light of frequency n0 . However, some also reach the ground vibrational level and emit a quantum of anti-Stokes scattered light. Because of the total energy change involved, the frequency of the anti-Stokes line is naSt ¼ n0 þ nvib

ð5:6:11Þ

Because the population of the ﬁrst excited vibrational level is much less than that of the ground state, the intensity of the anti-Stokes lines is less than that of the

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231

Stokes lines. In most Raman experiments, only the Stokes part of the spectrum is examined. The Raman spectrum of carbon tetrachloride is shown in ﬁg. 5.12. The stimulating radiation, which has a wavelength of 488.0 nm, was obtained from an argon-ion laser. On the wave number scale this corresponds to 20,492 cm1 , which is blue light. Three lines are resolved on each side of the central peak, which is itself due to Rayleigh scattering. These lines correspond to ﬁve vibrational modes in the CCl4 molecule which are Raman active. The intensities of the Stokes lines which occur on the low-frequency side are higher than those of the anti-Stokes lines. In addition, the shift of each Stokes line from the central Rayleigh line is exactly equal to that for the corresponding anti-Stokes line. Thus, the features of the spectrum conform to the quantum-mechanical analysis discussed earlier. Raman spectroscopy is an important technique used in the study of solvent structure and solvation. Because of its complementary nature it is often used in connection with infrared spectroscopy. Although most of the examples discussed in the following sections involve infrared experiments, many examples of Raman experiments can be found in the literature.

Fig. 5.12 Raman spectrum of carbon tetrachloride obtained using an argon-ion laser operating at 488.0 nm. Each line is labeled in terms of its shift from the central Rayleigh line. (From reference G5, with permission.)

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5.7 Infrared Spectroscopy of Polar Solvents All polar solvents considered in chapter 4 are polyatomic species and therefore have vibrational spectra. Some vibrational modes are infrared active and others are Raman active. Especially interesting are those modes associated with the molecular dipole because they give information about intermolecular interactions. In this section, attention is focused on the infrared spectra of pure polar solvents. A. Protic Solvents The most common protic solvent is water. It is also one of the most complex from the point of view of vibrational spectroscopy because of its highly structured nature. Since water is a triatomic, non-linear molecule it has three vibrational modes, which are illustrated in ﬁg. 5.13. The n1 mode is the symmetrical stretch; n2 is the bending mode; and n3 is the asymmetrical stretch. All three vibrational modes for water are active in the infrared because they involve changes in the dipole moment. Activity in the Raman spectrum requires that the polarizability of the molecule changes during vibration. Analysis of this aspect of molecular properties is more diﬃcult but it shows that all three modes are also Raman active. A summary of the frequencies of these vibrations for H2O, and the isotopes D2O, and HOD determined from gas phase spectra are given in table 5.7. The infrared and Raman spectra of liquid water are broad and complicated by several types of vibrational interactions. Three regions can be distinguished in the infrared spectrum (see ﬁg. 5.14). The O–H stretching region due to the n1 and n3 modes is observed between 2800 and 3800 cm1 . The broad character of this band is a result of the large, ﬂuctuating distribution of hydrogen bonds. In fact, one can distinguish two classes of O–H stretching vibrations which depend on whether the O–H group is involved in a weak or strong hydrogen bond. The second region involves the n2 bending mode observed between 1500 and 1800 cm1 . A librational or restricted rotational region is observed between 250 and 800 cm1 . Finally, there is a translational or hydrogen bond stretching region in the far infrared from 20 to 250 cm1 which is not shown in ﬁg. 5.14. The diﬃculties in interpreting the vibrational spectrum of water are obvious. Because of the broad nature of the bands, the two stretching modes n1 and n3 overlap. In addition, overtones complicate the assignment of bands. Thus, the ﬁrst overtone of the n2 bending mode, 2n2 , lies close in frequency to the stretching modes n1 and n3 . For these reasons, there are advantages in studying HOD, a molecule for which the vibrational frequencies are quite diﬀerent. As a result,

Fig. 5.13 Normal modes of vibration of a water molecule. The arrows indicate the relative motion of each atom.

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Table 5.7 Vibrational Frequencies for H2O, D2O, and HOD from Gas Phase Measurements Frequency/cm1 Molecule Mode H2O HOD D2O

n1

n2

n3

3657 1595 3756 2727 1402 3707 2671 1178 2788

overtone and combination bands are more widely separated. Thus, the vibrational properties of water are conveniently studied using dilute solutions of HOD in H2O or D2O [18]. Protic solvents always have more complex infrared spectra because of the presence of hydrogen bonding in the liquid state. In methanol, this involves interaction of the acidic proton on the OH group in one molecule with the oxygen atom in an adjacent molecule (ﬁg. 5.15). The infrared spectrum shows a wide band centered at 3346 cm1 which is due to the –OH stretch. When methanol is dissolved as a dilute solute in carbon tetrachloride, this band is sharp and appears at 3644 cm1 . An –OH bending mode appears at 1449 cm1 . Another broad band due to –OH out-of-plane deformation is centered at 663 cm1 . The other features of the methanol spectrum are due to the vibrational modes of the CH3– group or to skeletal vibrations [27]. Another example of a protic solvent is formamide in which hydrogen bonding involves the proton in the –NH2 group and the carbonyl oxygen in an adjacent molecule (see ﬁg. 5.16). A broad band centered at 3320 cm1 is due to the asym-

Fig. 5.14 Infrared spectrum of puriﬁed water in the frequency range 4000–400 cm1 . (From reference 26, with permission.)

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LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 5.15 Ball-and-stick diagram of a methanol molecule together with its infrared spectrum as a liquid. (From reference 26, with permission.)

metric and symmetric stretching vibrations of the –NH2 group. Other bands due to this group occur at 1605, 1050, and 602 cm1 . An important band for this molecule is the CO stretching vibration at 1685 cm1 . However, this band is not sharp, other spectral features being very close in frequency. The band at 1392 cm1 is assigned to the symmetric OC N stretching mode and that at 1308 cm1 , to the C–H deformation [27, 28]. As one would expect, the frequencies of the bands due to vibrational modes associated with hydrogen bonding change when formamide is a dilute solute in an aprotic solvent. Then, the –NH2 band at 3320 cm1 and the CO band at 1683 cm1 shift to higher frequencies. B. Aprotic Solvents As one would expect the vibrational spectra of aprotic liquids are usually much simpler than those for protic liquids. Acetonitrile is an example of an aprotic solvent whose polar properties are due to the large dipole moment associated 1 with the C N stretch at 2254 cm is a prominent feature N bond. The C

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235

Fig. 5.16 Ball-and-stick diagram of a formamide molecule together with its infrared spectrum as a liquid. (From reference 26, with permission.)

of the IR spectrum of liquid acetonitrile (ﬁg. 5.17). Other important bands include 1 1 the C C N bending mode at 748 cm , the C C stretch at 918 cm , the CH3 1 rocking mode at 1040 cm and CH3 deformation modes at 1376 and 1445 cm1 . The CH3 stretching modes are seen at 2945 and 3003 cm1 . The spectrum contains two Fermi resonance bands in the region shown, an important one arising at 2293 cm1 . This band is due to a combination of the bands at 918 cm1 and 1 1375 cm1 , and interacts somewhat with the C N band at 2254 cm because it 1 is only separated by 39 cm . Because of the electron density on the nitrogen atom, this end of the molecule interacts strongly with cations and with molecular solutes which are good Lewis acids. For this reason, the C N stretching band is often used to assess interactions of acetonitrile as a solvent with ionic and molecular solutes. More information about this aspect of the IR spectroscopy of the acetonitrile molecule is given in the following section. Acetone is an aprotic liquid with a very diﬀerent structure. Its infrared spec stretch. trum possesses a strong band at 1716 cm1 due to the asymmetric CO Other prominent features of its spectrum include the CH3– asymmetric and sym-

236

LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 5.17 Ball-and-stick diagram of an acetonitrile molecule together with its infrared spectrum as a liquid. (From reference 26, with permission.)

metric deformation modes at 1420 and 1364 cm1 , respectively, the C C C asymmetric stretch at 1222 cm1 , and the CO deformation mode at 530 cm1 (ﬁg. 5.18). The CH3– stretching vibrations appear around 3000 cm1 . The CO stretching band can be used to monitor solute–solvent interactions in acetone solutions but it is not as sharp as the C N stretching band in acetonitrile. This shows that several overlapping bands occur in this region of the acetone spectrum. The two principal components are due to the CO stretch of the acetone monomer, and to the same stretching mode in an acetone dimer which occurs at slightly smaller wave numbers. Because of the high dipole moment associated with the double bond in this molecule, dimers are formed in two diﬀerent conﬁgurations. Dimerization in acetone has been studied by both Raman [29] and infrared spectroscopy [30]. Dimethylformamide (DMF) has a much more complex structure than either acetonitrile or acetone, but it is interesting to compare its infrared spectrum with that of formamide. By replacing the two acidic protons on the amino group by methyl groups, one obtains an aprotic liquid. As can be seen by comparing ﬁgs 5.16 and 5.19, the infrared spectrum of DMF is much simpler than that of for-

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237

Fig. 5.18 Ball-and-stick diagram of acetone together with its infrared spectrum as a liquid. (From reference 26, with permission.)

mamide. However, the features of the spectrum due to the CO group and the nitrogen atom are similar. The strongest band in the spectrum occurs at 1675 cm1 and is due to the CO stretching mode. Another prominent feature N stretching mode. The CH3 rocking mode at 1389 cm1 is assigned to the OC 1 is seen at 1095 cm and the O C N deformation at 659 cm1 . The infrared spectra of other aprotic solvents are also characterized by a strong band or bands associated with the polar group in the molecule. For example, dimethylsulfoxide has a strong absorption at 1057 cm1 due to the SO stretching mode. However, absorption in this region is complicated by several overlapping bands, one of which is due to DMSO dimers [31]. Propylene carbonate,

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LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 5.19 Ball-and-stick diagram of dimethylformamide together with its infrared spectrum in the liquid state. (From reference 26, with permission.)

which has a CO bond attached to a ﬁve-member heterocyclic ring, has a strong CO band at 1797 cm1 . In the case of nitrobenzene and nitromethane, the polar group is triatomic. In nitromethane, the strongest band in the infrared spectrum is at 1563 cm1 due to the asymmetric stretching mode of the –NO2 group. The symmetric stretch mode at 1378 cm1 is considerably weaker. The –NO2 group has other vibrational modes at lower frequencies.

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239

In summary, the infrared spectra of aprotic solvents are much simpler than those of protic solvents because of the absence of complications due to hydrogen bonding. On the other hand, the aprotic liquids are characterized by very high molecular dipole moments. As a result, dimers can be formed in which some of the molecules line up with their dipole vectors in an antiparallel fashion [30]. This can lead to the spectra being complicated by additional spectral features due to solvent aggregates. Nevertheless, the prominent band due to the polar group in these solvents can be used to probe solute–solvent interactions in non-aqueous media. This subject is discussed in more detail in the next section.

5.8 Infrared Spectroscopy of Non-Electrolyte Solutions When a polar solute such as acetonitrile is dissolved in another organic solvent, the shift in the frequency of the polar group may be used to monitor solute– solvent interactions. This type of study is especially interesting because it demonstrates clearly the importance of short-range chemical interactions in determining the properties of the solution. A number of studies of the IR spectra of polar molecules dissolved in a variety of solvents both polar and non-polar have been carried out [32]. In the case of acetonitrile, the C N stretching frequency (n2 band) is shifted to higher values in solvents which are stronger Lewis acids. Many of these solvents are protic and interact with the electronegative C N group via hydrogen bonding. The electron density in this bond shifts toward the Lewis acid. Since the electron density is associated with an antibonding orbital, the net eﬀect is to make the C N bond stronger, and a blue shift, that is, a shift to higher frequencies is observed. A plot of the shift of the n2 band for both acetonitrile and deuterated acetonitrile (CD3CN) as a function of the solvent’s acceptor number AN is shown in ﬁg. 5.20 for 12 solvents which are more acidic than acetonitrile. An excellent correlation between n2 and the acceptor number AN is found except for formamide (F) and N-methyl formamide (NMF). In the latter cases, the solvents are both strong Lewis acids and strong Lewis bases. Corresponding results are obtained when acetonitrile is dissolved in solvents which are stronger Lewis bases. In this case, there is a good correlation between n2 and the solvent’s donor number DN, with the solvent shifts being negative (ﬁg. 5.21). These experiments point out the importance of Lewis acid–base properties in determining the nature and strength of short-range interactions in solution. The strong Lewis bases interact with the CH3 group, which is at the positive end of the molecular dipole. This interaction aﬀects the C N group, so that more electron density is associated with this part of the molecule. As a result, a red shift occurs, that is, negative values of n2 are observed. Solvent-induced frequency shifts have also been studied for dimethylsulfoxide (DMSO), which is a strong Lewis base. Most solvents behave as Lewis acids and interact with DMSO via the electron density on the SO group. This interaction leads to a red shift for this band which is as large as 56 cm1 in the presence of acetic acid [34]. The magnitude of the shift correlates well with the solvent’s acceptor number giving the relationship nðSOÞ ¼ 1080 1:375AN

ð5:8:1Þ

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LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 5.20 Plot of n2 for CH3CN () and CD3CN () in more acidic solvents against the solvent’s acceptor number AN. The best straight lines were estimated without the data for F and NMF. The data for CD3CN have been shifted vertically by 20 cm1 for clarity. (From reference 33, with permission.)

Fig. 5.21 Plot of n2 for CH3CN () and CD3CN () in more basic solvents against the solvent’s donor number DN. The best straight lines were estimated without the data for F and NMF. The data for CD3CN have been shifted vertically by 10 cm1 for clarity. (From reference 33, with permission.)

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241

For this system, removal of electron density from the SO bond results in its being weakened as one would expect without knowing the detailed molecular orbital structure. Acetone provides another interesting system for studying intermolecular interactions. The basicity of this solvent, whose DN is 17.0, is close to that of water. On the other hand, it is a very weak Lewis acid with an AN equal to 12.5. Thus, one expects the CO stretching frequency n3 to shift with change in the acidity of the solvent in which acetone is dissolved. A good correlation is found between the frequency of this band in 22 solvents and the solvent’s acceptor number AN, with the frequency shifting in the red direction as AN increases. The equation describing the correlation is n3 ¼ 1718 0:286AN ð5:8:2Þ with a correlation coeﬃcient equal to 0.876. Interaction of the acetone molecule with a Lewis acid results in a weakening of the CO bond due to withdrawal of electron density. The correlation is weakened by the eﬀects of acetone–acetone interactions which are stronger in solvents which are weak Lewis acids [29]. The eﬀects of intermolecular interactions are seen much more clearly for the n17 mode which is the skeletal asymmetric stretch. Values of the frequency of this vibration are plotted against solvent AN in ﬁg. 5.22. The equation describing the linear correlation is ð5:8:3Þ n17 ¼ 1216 þ 0:339AN with a correlation coeﬃcient of 0.984 [30]. This mode is relatively unaﬀected by acetone dimerization, and shifts in the blue direction with increase in solvent acidity. Other systems in which solvent-induced frequency shifts have been studied include benzonitrile, cyclopentanone, and tetramethylurea [32]. These studies all

Fig. 5.22 Plot of the frequency of the asymmetric stretching band of the molecular skeleton (n17 ) of acetone as a dilute solute in various solvents against the acceptor number of the solvent, AN.

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LIQUIDS, SOLUTIONS, AND INTERFACES

demonstrate the importance of speciﬁc solvation eﬀects in determining intermolecular interactions in these systems. Vibrational spectroscopy provides a valuable tool for probing these interactions in liquid solutions.

5.9 Infrared Spectroscopy of Electrolyte Solutions Ions are stabilized in electrolyte solutions by the dipolar character of the solvent molecules. Thus, one expects the ion–dipole interaction to result in a shift in electron density along the direction of the dipole vector. This, in turn, should result in a change in frequency of the vibrational modes associated with the molecular dipole. As has been seen earlier, there is a strong band in the spectrum of aprotic molecules which is associated with the electronegative group. Changes in the frequency of this group allow one to monitor interactions of the solvent molecules with cations. Interactions with anions are more diﬃcult to see in the vibrational spectrum of these systems but can be assessed under some circumstances. In the case of protic solvents, hydrogen bonding is important in the solvation of anions. Thus, features of the IR spectrum associated with this aspect of solvent structure are aﬀected by the anion in the system. Another way of assessing cation–solvent interactions is to examine the vibrational characteristics of the bond formed between the cation and the negative end of the solvent molecule’s dipole. This bond is mainly electrostatic in character and therefore is much weaker than intramolecular covalent bonds. As a result the infrared band corresponding to vibration of the metal–solvent bond is normally observed in the far infrared at frequencies less than 500 cm1 . Some results obtained for the alkali metal cations in four aprotic solvents are summarized in table 5.8. It is clear that the frequency of the metal–solvent band decreases with increase in the size of the alkali metal, that is, with decrease in the ﬁeld due to the charge on the ion. When ion pairing is absent, the value of the frequency is independent of the nature of the anion. This is deﬁnitely the case in DMSO, which strongly solvates cations because of its high donor number. In addition, the relatively high permittivity of this solvent reduces the inﬂuence of long-range ion–ion interactions. Because instrumentation capable of working in the far infrared is not commonly available, studies of interactions between polar solvent molecules and ions have more commonly involved intramolecular vibrational modes in the solvent

Table 5.8 Frequency of Cation–Solvent Band, n/cm1 [G1] Solvent Cation

AC

DMSO

NMP

PC

Liþ Naþ Kþ Rbþ Csþ

425 195 140

429 199 153 124 109

398 204 140 106

397 186 144 115 112

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243

molecule. In the case of cations, vibrations associated with the polar group at the negative end of the molecular dipole can be studied. Thus, cation–solvent interactions can be assessed by following changes in the C N stretching frequency in acetonitrile, in the SO stretching frequency in dimethylsulfoxide, or in the CO stretching frequency in acetone. The eﬀect of dissolving LiClO4 in acetonitrile (AcN) is illustrated in ﬁg. 5.23 on the basis of a diﬀerence spectrum. The lower spectrum in this ﬁgure shows the absorbance features of pure AcN in the region from 600 to 3400 cm1 . The upper spectrum was obtained by subtracting the spectrum of pure AcN from the spectrum of an AcN solution containing 0.7 M LiClO4. As a result the diﬀerence spectrum has both positive-going features corresponding to new bands in the electrolyte solution, and negative-going features corresponding to bands whose intensity decreased in the electrolyte solution. The most important changes are 1 those associated with the C N stretching mode at 2253 cm . In the diﬀerence spectrum a negative-going band is observed at this frequency, indicating that the concentration of free acetonitrile has decreased in the electrolyte solution. These AcN molecules are now coordinated to Li+ ions through the negative charge density on the N atom and have their C N stretching frequency shifted in the blue direction to 2280 cm1 . Negative-going bands are also observed at 918 cm1 (C C stretching mode) and 748 cm1 (C C N bending overtone). Spectral properties associated with the CH3– deformation modes at 1375 and 1453 cm1 are also aﬀected by the addition of electrolyte. Perchlorate ion has its own vibrational spectral features which appear at 625 and 1100 cm1 . By following the concentration dependence of the negative-going band at 2253 cm1 , one can determine the average number of AcN molecules which are coordinated to the metal ion in solution. Experiments show that the integrated intensity of this band is linear in the electrolyte concentration over reasonable concentration ranges. Interpretation of the negative slope of these plots requires that the extinction coeﬃcient of free acetonitrile molecules be determined in a

Fig. 5.23 Diﬀerence spectrum for an acetonitrile solution containing 0.7 M LiClO4 obtained by subtracting the spectrum of pure acetonitrile from that of the solution, together with the spectrum of pure acetonitrile.

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LIQUIDS, SOLUTIONS, AND INTERFACES

suitable inert medium. This was done by observing the intensity of the C N stretching mode for AcN dissolved in carbon tetrachloride as solvent [35, 36]. In this way, the average number of solvent molecules coordinated to a single cation was determined. Results for several monovalent and divalent cations in acetonitrile are summarized in table 5.9. It is clear from the results in table 5.9 that the average coordination number nc of solvent molecules to a cation varies signiﬁcantly with cation nature. This variation is attributed to a corresponding variation in the extent of ion pairing. If the coordination number is less than the maximum value for a given cation, then the value of nc can be used to estimate the average number of anions associated with a cation, and thus, the average charge on the cationic species. The maximum number of solvent molecules which can coordinate to a cation is usually considered to be six, but for small cations it can be four. The observed spectral shift for the C N stretching mode n2 varies considerably with cation nature. It is always in the blue direction, indicating that the C N bond is stronger in the presence of the positive electron-withdrawing ﬁeld of the cation. This observation suggests that the electron density withdrawn from the C N system involves antibonding orbitals. The variation in n2 with cation is usually attributed to variation in the cation’s electrical ﬁeld. The latter quantity is usually estimated from the ratio of the cationic charge zi e0 to its radius ri . However, a reasonable correlation between n2 and zi e0 =r is not obtained without considering the eﬀects of ion pairing. On the other hand, if the ﬁeld is estimated using the eﬀective average cationic charge, zi0 , estimated from the average coordination number nc an acceptable correlation is obtained (see ﬁg. 5.24). Cations also aﬀect other solvent bands for AcN including the combination band at 2293 cm1 (n3 þ n4 ), the C C stretching mode at 918 cm1 (n4 ), and 1 the C C N deformation mode at 746 cm (2n8 ). Changes in the intensity of these bands with electrolyte concentration may also be used to assess ion pairing in acetonitrile solutions [35, 36]. Anions aﬀect the features of the AcN spectrum to a much smaller extent [37]. Any change in the spectrum is usually associated with CH3– symmetrical stretch-

Table 5.9 Average Coordination Number for Acetonitrile to the Metal Cation and Frequency Shift of the n2 Band ( C N stretch) from IR Spectra of Metal Perchlorate Solutions in Acetonitrile [36] Cation

Average Coordination Number nc

Frequency Shift n2

Li+ Na+ Mg2+ Ca2+ Sr2+ Ba2+ Zn2+ Cd2+ Pb2+

3.7 3.8 3.4 4.9 5.9 4.8 5.7 5.4 5.0

23 13 42 24 42 12 41 32 11

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245

Fig. 5.24 Plot of the shift of the C N stretching frequency of acetonitrile, n2 observed in solutions containing various metal perchlorates against the eﬀective ﬁeld of the metal cation, z0i =ri [36].

ing mode at 2944 cm1 (n1 ) and the corresponding antisymmetrical mode at 3002 cm1 (n5 ). Polyatomic anions such as ClO4– and NO3– have very little eﬀect on this region of the spectrum. On the other hand, monoatomic ions such as the halides, and linear polyatomic ions such as NCS aﬀect this region of the spectrum strongly. This suggests that the anion is interacting with the methyl group at the positive end of the molecular dipole, and that the extent of interaction depends on the electrical ﬁeld associated with the anion [37]. Electrolyte eﬀects on the vibrational spectra of polar solvents have been studied in a number of other solvents including acetone, pyridine, dimethylsulfoxide, dimethylformamide, tetramethylsulfone, and nitromethane [37]. These studies provide valuable information on the nature of solvation of ions in these systems. In addition, they can be used to assess ion pairing and its variation with electrolyte nature and concentration.

5.10 Ultraviolet–Visible Spectroscopy and Solvatochromic Effects Ultraviolet-visible spectroscopy usually involves electronic transitions in which valence electrons in molecules and polyatomic molecules move to higher unﬁlled orbitals. This type of spectroscopy is well known in organic chemistry and often involves the p electrons in groups such as CC, CO, NN, and NO. The electronic transitions taking place are p ! p transitions, and in molecules with heteroatoms, n ! p transitions, that is, from nonbonding orbitals for electrons on atoms such as oxygen to an antibonding p orbital. Electronic transitions also occur between s bonding orbitals and s antibonding orbitals but the associated

246

LIQUIDS, SOLUTIONS, AND INTERFACES

energy is usually quite high, so that the absorption is observed in the vacuum ultraviolet at a wavelength less than 150 nm. The discussion in this section is mainly concerned with species containing a transition metal ion and the involvement of its d electrons in electronic transitions. The simplest description of electronic energy levels in a transition metal complex is based on crystal ﬁeld theory, but a more detailed description requires the application of molecular orbital theory. When the metal ion is surrounded by six ligands it is in an octahedral ﬁeld, and the d orbitals are no longer energetically degenerate. The three orbitals with electron density directed away from the axes of the reference coordinate system, namely, dxy , dyz , and dxz have lower energy than those with electron density directed along these axes, namely, dx2 y2 and dz2 . The energy diﬀerence between the three lower levels (t2g orbitals) and the two higher ones (eg orbitals) is called the crystal ﬁeld splitting energy and is designated as 0 (see ﬁg. 5.25). The value of 0 depends both on the nature of the metal ion, on the number of d electrons, and on the nature of the ligands. On the basis of a large number of spectroscopic studies the splitting energy has been determined for a wide variety of complexes. The variation in 0 with metal ion nature is illustrated with ions formed from the elements in the ﬁrst transition series of the periodic table in table 5.10 for the case that the ligands are six water molecules. The energy associated with the electronic transition varies from a minimum of 9.3 meV for Mn2+ to a maximum of 26.0 meV for Mn3+. Considerable variation in 0 is also found with the nature of the ligand. In this case 0 increases in the series I < Br < SCN < Cl < N 3 < OH < H2 O < NCS < NH3 < NO2 < CN < CO. The sequence is known as the spectrochemical series. Another common coordination number observed in transition metal complexes is four. When the ligands are situated as far away from each other as possible, one obtains a tetrahedral conﬁguration. In this case the ligands interact repulsively with the dxy , dyz and dxz orbitals, which have their electron density directed

Fig. 5.25 Crystal ﬁeld splitting of the d orbitals in a transition metal ion in an octahedral ﬁeld and a tetrahedral ﬁeld due to surrounding ligands (Lewis bases).

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247

Table 5.10 Crystal Field Data for the Complex Ions M(H2O)nþ 6 from the First Transition Series Ion

d Electrons

Ti3+

Ground State for Free Ion

Octahedral Field Conﬁguration

0 =cm1

t12g

20,100

1

2

3+

V

2

3

F

t22g

19,950

V2+

3

4

F

t32g

12,100

Cr3+

3

4

F

t32g

17,400

Cr2+ Mn3+

4 4

5

D 5 D

t32g e1g t32g e1g

14,000 21,000

Mn2+

D

5

6

S

t32g e2g

7,500

3+

5

6

S

t32g e2g

14,000

Fe2+

Fe

6

5

D

t42g e2g

9,350

3+

Co

6

5

D

t62g

20,760

Co2+

9,200

7

4

F

t52g e2g

2+

Ni

8

3

F

t62g e2g

8,500

Cu2+

9

2

D

t62g e3g

12,000

between the axes of the x, y, z coordinate system. As a result, the d orbitals are split into two degenerate sets with the dx2 y2 and dz2 orbitals (e orbitals) having lower energy than the dxy , dyz , and dxz orbitals (t2 orbitals). The energy diﬀerence is designated as t (see ﬁg. 5.25). Another conﬁguration with four ligands is the square planar geometry. As the name implies, the ligands lie in a plane centered on the transition metal ion. This system is favored by metal ions with a d 8 conﬁguration and has a more complicated distribution of d orbital energy levels than either the octahedral or tetrahedral systems. Other geometries observed for transition metal complexes include a linear one with two ligands, and a trigonal one with three ligands. One normally thinks of the electronic spectroscopy of transition metal complexes as involving d ! d transitions. In fact, many of the spectra observed for these systems do involve the promotion of d electrons from a lower lying level to one which is higher because of the crystal ﬁeld splitting [38]. However, there are also charge transfer transitions from an orbital associated with a ligand to an empty one associated with the metal (LMCT) or from a d orbital on the metal to an empty orbital associated with the ligand (MLCT). These are more frequent when there are p bonds in the ligands of the transition metal complex. Solvatochromic eﬀects occur then the color of a solute depends on the nature of the solvent in which it is dissolved. Two such systems were discussed in section 4.9. The ﬁrst is the betaine dye, 4-(2,4,6-triphenylpyridinium)-2,6-diphenylphenoxide, which is used to deﬁne the Dimroth–Reichardt parameter ET . This molecule undergoes a p ! p transition in the visible region whose wavelength is very solvent dependent. In the strongly acidic solvent water it is equal to 453 nm which corresponds to an energy gap of 264 kJ mol1 . In the very weakly acidic solvent diphenylether, the wavelength for the same transition is 810 nm. This corresponds

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LIQUIDS, SOLUTIONS, AND INTERFACES

to an energy gap and value of ET equal to 148 kJ mol1 . The change in the color of the dye with solvent nature is clearly related to the charge transfer which accompanies the p ! p transition as illustrated in ﬁg. 4.12. Since the negatively charged O atom in the ground state is stabilized to an extent which depends on solvent acidity, the ET parameter gives an excellent way of estimating this molecular property of the solvent. Solvatochromic eﬀects related to solvent acidity have been found for several transition metal complexes. Burgess [39] studied the visible spectra of bis-(2,2 0 bipyridyl)biscyanoiron(II), bis-(1,10-phenanthroline)biscyanoiron(II), and related iron complexes in a wide range of solvents, both polar and non-polar. In most cases the wavelength of maximum absorption depends on solvent acidity and is linear in the solvent’s acceptor number. The structure of the phenanthroline complex is shown in ﬁg. 5.26. It is clear that the solvent interacts with the negative charge density on the two cyano groups so that the complex probes the ability of the solvent to act as a Lewis acid. Iron(II) is a d 6 system and the ligand ﬁeld splitting of the d orbitals is large for these systems. Thus, all six d electrons are in the low-lying t2g orbitals. Burgess [38] reported the frequency for an MLCT band, that is, for charge transfer from a t2g orbital on the metal to a p orbital on the bipyridyl or phenanthroline ligand. As the acidity of the solvent increases and its interaction with the cyano ligands becomes stronger, the splitting between the t2g and p orbitals increases. At the same time the frequency observed at maximum absorption shifts in the blue direction. For example, the phenanthroline complex absorbs at 634 nm in hexamethylphosphoramide, an extremely weak Lewis acid, so that the solution is blue in color. In ethanol, a moderate Lewis acid, maximum absorption occurs at 557, and in triﬂuoroacetic acid, a very strong Lewis acid, at 395 nm. The relationship between the frequency (cm1 ) at maximum adsorption and the Gutmann acceptor number AN is nmax ¼ 107:18AN þ 14,340

ð5:10:1Þ

on the basis of data in 12 polar solvents ðr ¼ 0:980Þ [40]. A similar study with bis(2,20 bipyridyl)biscyanoruthenium(II) and related complexes has also been carried

Fig. 5.26 Chemical structure of bis-(1,10phenanthroline)biscyanoiron(II). Solvent molecules can interact with the electron density on the cyano groups.

SPECTROSCOPIC STUDIES OF LIQUID STRUCTURE AND SOLVATION

249

out [41]. These results show the same sensitivity to solvent acidity seen in the earlier work with iron complexes. Another group of transition metal complexes behave as solvatochromic indicators of solvent basicity. The best known of these is Cu(II) N; N; N 0 ; N 0 -tetramethylethylene diamine acetoacetonate or Cu(II)(tmen)(acac) (see ﬁg. 5.27). In this complex the two chelates are located in a plane so that the resulting geometry is approximately square planar. The transition of maximum frequency corresponds to a d electron associated with the oﬀ-planar orbitals dxz and dyz being promoted to the dx2 y2 orbital which has its electron density in the plane containing the ligands. As the strength of interaction of the solvent molecules in the axial positions with the metal ion increases, the eﬀective geometry of the complex changes from square plane through tetragonal to octahedral. At the same time the splitting between the lowest occupied and the highest empty d orbital decreases so that the frequency associated with the d ! d transition shifts in the red direction. Values of the wavelength associated with the transition for the polar solvents considered in chapter 4 are given in table 4.10 where they are designated as Bsc , the solvatochromic basicity. Considerable variation in wavelength is observed, from a minimum of 522 nm in nitrobenzene, a weakly basic solvent to 633 nm in hexamethylphosphoramide which is a strong Lewis base. For this system, the wavelength correlates strongly with solvent basicity as assessed by the Gutmann donor number (see equation (4.9.2)). Another group of complexes exhibiting solvatochromic properties are Ru(II) and Ru(III) ammine complexes containing one or more other ligands with p bonds [40, 41]. A simple example is the Ru(II) pentammine pyridine cation, which has an MLCT band at 407 nm in acetonitrile. This transition corresponds to a d electron in a t2g orbital on the metal being transferred to a p orbital on pyridine. When the solvent is changed to a more basic one such as dimethylsulfoxide, the wavelength increases. As a result there is a correlation between the wavelength for the MLCT band and the Gutmann donor number DN, namely, lSC ¼ 390:9 þ 1:698DN

ð5:10:2Þ

The correlation is very strong but the experimental study was limited to aprotic solvents. One can argue that correlations such as equations (4.9.2) and (5.10.2) are not physically reasonable in that they are based on the wavelength of the electronic transition related to the solvatochromic eﬀect, not its frequency. The frequency is

Fig. 5.27 Chemical structure of Cu(II)N,N,N0 ,N0 -tetramethylethylenediamine acetoacetonate. Solvent molecules can interact as Lewis bases with the axial sites in the complex as shown.

250

LIQUIDS, SOLUTIONS, AND INTERFACES

directly proportional to the associated energy diﬀerence, which in turn reﬂects the strength of the interaction of the solvent as a Lewis base with the transition metal complex. Thus, the frequency is the more relevant parameter to consider in assessing the relation between the solvatochromic eﬀect and the Gutmann donor number. In fact, equally strong correlations are found between the frequency and the DN for both of these systems. The fact that linear correlations are found with both frequency and wavelength, which is related to the reciprocal of the frequency, reﬂects the fact that the net change in either parameter with respect to its average value is small. In this case the reciprocal of one quantity is approximately linear in that quantity over a limited range on the basis of a Taylor’s series expansion. Equations (4.9.2) and (5.10.2) are used here because the parameters BSC and lSC can then be used as empirical measures of solvent basicity which increase with increase in basicity. The examples of electronic transitions in transition metal complexes discussed here have emphasized solvatochromic eﬀects and often involve intramolecular electron transfer. The role of the solvent in electron transfer is an important aspect of these processes and is discussed in more detail in chapter 7. Many electronic transitions do not display the type of solvent eﬀects considered here but depend instead on bulk solvent properties. The systems discussed above interact in a speciﬁc way with the solvent usually as a result of a signiﬁcant change in the dipole moment of the molecule or the complex ion as a result of the electronic transition. There are many other interesting aspects of this area of spectroscopy which have been discussed in more detailed treatments [38, 43].

5.11 Concluding Remarks The discussion in this chapter has only considered representative examples of the application of spectroscopy to the elucidation of the structure of liquids and solutions. Raman spectroscopy is complementary to infrared spectroscopy and has been used just as frequently to study solute–solvent, solute–solute, and solvent–solvent interactions in solution. Electron spin resonance spectroscopy involves the spins of unpaired electrons in molecules and ions and their interaction with a magnetic ﬁeld. It has been used to study hydrogen bonding and hydrophobic interactions in aqueous solutions. Mossbauer spectroscopy involves the nuclear transitions that result from the absorption of g-rays by certain nuclei. It is used to study the coordination spheres of the corresponding ions in solution. Dielectric spectroscopy is used to determine the dielectric properties of liquids and solutions at microwave frequencies. As shown in chapter 4, this technique gives information about molecular reorientation in an alternating electrical ﬁeld. Ultrasonic absorption spectrometry is a study of the density changes which occur when sound energy is passed through a liquid or solution and provides information about structural relaxation times. Electron diﬀraction studies are related to neutron and X-ray diﬀraction. Electrons are diﬀracted by the local electrical potential in the sample, which depends on the spatial conﬁguration of the nuclei and density distribution of the associated electrons. It is complementary to X-ray diﬀraction because it is more sensitive to the distribution of light atoms

SPECTROSCOPIC STUDIES OF LIQUID STRUCTURE AND SOLVATION

251

in the liquid sample. More information about all these techniques, including those discussed in more detail in this chapter, are available in the reviews in reference [G1]. The important contribution of spectroscopic studies is structural information about liquids and solutions at the atomic and molecular level. Most of the information discussed in earlier chapters is thermodynamic in nature and does not provide a microscopic description of the system. When the spectroscopic details of intermolecular interactions are available, one is able to understand thermodynamic results in much greater detail. Spectroscopic experiments were developed mainly in the latter half of the twentieth century. As a result of the detailed information they provide, understanding of the structure of condensed phases is now at the same level as our understanding of the much simpler gas phase.

General References G1. Dogonadze, R. R.; Kalman, E.; Kornyshev, A. A.; Ulstrup, J. The Chemical Physics of Solvation, Part B: Spectroscopy of Solvation; Elsevier: Amsterdam, 1986. G2. Marcus, Y. Ion Solvation; Wiley-Interscience: New York, 1986. G3. Burgess, J. Metal Ions in Solution; Ellis Horwood: Chichester, 1978. G4. Drago, R. S. Physical Methods in Chemistry, 2nd ed.; Saunders, Fort Worth, TX, 1992. G5. Nakamoto, K. Infrared and Raman Spectra of Inorganic and Coordination Compounds, 5th ed.; Wiley Interscience, New York, 1997.

References 1. Enderby, J. E.; Neilson, G. W. Rep. Prog. Phys. 1981, 44, 593. 2. Blum, L.; Narten, A. H. Adv. Chem. Phys. 1976, 34, 203. 3. Palinkas, G.; Radnai, T.; Hajdu, F. Z. Naturforsch. 1979, 35a, 107. 4. Magini, M.; Licheri, G.; Paschina, G.; Piccaluga, G.; Pinna, G. X-Ray Diﬀraction of Ions in Aqueous Solutions: Hydration and Complex Formation; CRC Press: Boca Raton, FL, 1988. 5. Ohtaki, H.; Radnai, T. Chem. Rev. 1993, 93, 1157. 6. Radnai, T.; Kalman, E.; Pollmer, K. Z. Naturforsch. 1984, 39a, 464. 7. Tamura, Y.; Spohr, E.; Heinziger, K.; Palinkas, G.; Bako, I. Ber. Bunsen-Ges Phys. Chem. 1992, 96, 147. 8. Enderby, J. E. Chem. Soc. Rev. 1995 24, 159. 9. Neilson, G. W.; Enderby, J. E. J. Phys. Chem. 1996, 100, 1317. 10. Tromp, R. H.; Neilson, G. W.; Soper, A. K. J. Chem. Phys. 1992, 96, 8460. 11. Powell, D. H.; Neilson, G. W.; Enderby, J. E. J. Phys. Condens. Mater 1993, 5, 5723. 12. Hindman, J. C. J. Chem. Phys. 1962, 36, 1000. 13. Davies, J.; Ormonroyd, S.; Symons, M. C. R. Trans. Faraday Soc. 1971, 67, 3465. 14. Deverell, C.; Richards, R. E. Mol. Phys. 1996, 10, 551. 15. Templeman, G. J.; van Geet, A. L. J. Am. Chem. Soc. 1972, 94, 5578. 16. Bloor, E. G.; Kidd, R. G. Can. J. Chem. 1972, 50, 3926. 17. Deverell, C.; Richards, R. E.; Mol. Phys. 1969, 16, 421. 18. Desnoyers, J. E.; Jolicoeur, C. In Comprehensive Treatise of Electrochemistry; Conway, B. E., Bockris, J. O’M., Yeager, E., eds.; Plenum: New York, 1983; Vol. 5. 19. Covington, A. K.; Newman, K. E. In Modern Aspects of Electrochemistry; Bockris, J. O’M., Convway, B. E., eds.; Plenum: New York, 1977; Vol. 12, Chapter 2. 20. Coetzee, J. F.; Sharpe, W. R. J. Solution Chem. 1972, 1, 77. 21. Popov, A. I. Pure Appl. Chem. 1975, 41, 275.

252 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43.

LIQUIDS, SOLUTIONS, AND INTERFACES

Gutmann, E.; Wychera, E. Inorg. Nucl. Chem. Lett. 1966, 2, 257. Stengle, T. R.; Pan, Y.-C. E.; Langford, C. H. J. Am. Chem. Soc. 1972, 94, 9037. Spaziante, P. M.; Gutmann, V. Inorg. Chim. Acta 1971, 5, 273. Mayer, U.; Gutmann, V.; Gerger, W. Monatsh. Chem. 1975, 106, 1235. Hansen, D. L., Ed. Sprouse Collection of Infrared Spectra, Book IV, Common Solvents: Condensed Phase, Vapor Phase and Mass Spectra; Sprouse Scientiﬁc Systems: Paoli, PA. Lin-Vien, D.; Colthrup, N. B.; Fateley, W. G.; Grasselli, J. G. The Handbook of Infrared and Raman Characterstici Frequencies of Organic Molecules; Academic Press: New York, 1991. Smith, C. H.; Thompson, R. H. J. Mol. Spectrosc. 1972, 42, 227. Perelygin, I. S. Fluid Phase Equilib. 1996, 116, 395. Cha, D. K.; Kloss, A. A.; Tikanen, A. C.; Fawcett, W. R. Phys. Chem. Chem. Phys. 1999, 1, 4785. Fawcett, W. R.; Koss, A. A. J. Chem. Soc., Faraday Trans. 1996, 92, 3333. Fawcett, W. R. In Quantitative Treatments of Solute/Solvent Interactions; Politzer, P., Murray, J. S., eds.; Elsevier: Amsterdam, 1994; Chapter 6. Fawcett, W. R.; Liu, G.; Kessler, T. E. J. Phys. Chem. 1993, 97, 9293. Fawcett, W. R.; Kloss, A. A. J. Phys. Chem. 1996, 100, 2019. Fawcett, W. R.; Liu, G. J. Phys. Chem. 1992, 96, 4231. Fawcett, W. R.; Liu, G.; Kloss, A. A. J. Chem. Soc., Faraday Trans. 1994, 90, 2697. Perelygin, I. S. In Ionic Solvation; Krestov, G. A., Ed.; Nauka: Moscow, 1987; Chapter 3. Lever, A. B. P. Inorganic Electronic Spectroscopy, 2nd ed.; Elsevier: Amsterdam, 1984. Burgess, J. Spectrochim. Acta 26A, 1970, 1369, 1957. Soukup, R. W.; Schmid, R. J. Chem. Educ. 1985, 62, 459. Timpson, C. J.; Bignozzi, C. A.; Sulliman, B. P.; Kober, E. M.; Meyer, T. J. J. Phys. Chem. 1996, 100, 2915. Curtis, J. C.; Sullivan, B. P.; Meyer, T. J. Inorg. Chem. 1983, 22, 224. Murrell, J. N. The Theory of Electronic Spectra of Organic Compounds; Wiley: New York, 1963.

Problems 1. The overall pair correlation function for acetonitrile is obtained by X-ray and neutron diﬀraction experiments. List the component pair correlation functions which make up GðrÞ in each experiment. Devise a strategy for separating these based on isotopic substitution. 2. Write down the pair correlation functions which are involved in a neutron diﬀraction study of aqueous LiNO3 solutions. Devise a strategy for separating these using isotopes. 3. Estimate the fraction of 7 Li atoms in each of the four quantum states when this atom is in a magnetic ﬁeld of 5 T at 258C. 4. The equation relating the chemical shift for

23

Na to the DN is

dð23 Naþ Þ ¼ 16:78 0:510DN Estimate the chemical shift for 23 Na in nitrobenzene, benzonitrile, and butyronitrile using the DN data in table 4.10.

SPECTROSCOPIC STUDIES OF LIQUID STRUCTURE AND SOLVATION

253

5. Examine the importance of isotope eﬀects on the IR spectrum of acetonitrile. Estimate the frequencies of the symmetric and asymmetric CH3 stretching modes in deuterated acetonitrile given that they are 2945 and 3003 cm1 in 13 15 normal acetonitrile. What is the frequency of the C N stretch in CH3 C N? 6. The following frequencies have been observed for the C N stretch in various nitriles: 2253 cm1 2247 cm1 2228 cm1

Acetonitrile Propionitrile Benzonitrile

Can these results be accounted for quantitatively using the simple harmonic oscillator model? 7. In the NMR spectroscopy of liquids, the spin lattice relaxation time falls in the range 102 to 102 s. This inﬂuences the width of the NMR adsorption line. Estimate the corresponding minimum width of the line using the relationship n ¼

1 2pT1

What is the source of this relationship? 8. The correlation between the wavelength of the solvatochromic absorption by the Cu complex shown in ﬁg. 5.27 is lSC ¼ 525:2 þ 2:89DN where DN is the Gutmann donor number. Re-express this equation in terms of the energy hn associated with the transition. Show that the energy is also approximately a linear function of DN for the energy range in question.

6

Non-Equilibrium Phenomena in Liquids and Solutions

Lars Onsager was born in 1903 in Oslo, Norway, where he grew up and attended school. In 1920 he entered the Norwegian Institute of Technology in Trondheim to study chemical engineering, graduating in 1925. After spending several months with Debye in Zurich, he became interested in the theory of electrolytes and developed the model for electrolyte conductivity which he later published with Debye. In 1928 he moved to the United States and held positions at Johns Lars Onsager Hopkins and Brown Universities. He moved to Yale University in 1933, where he was appointed Gibbs Professor of Theoretical Chemistry in 1945, and remained there until he retired. Onsager is best known for his work in the area of irreversible thermodynamics where he developed the reciprocal relations which are now known after him. He submitted this work as a doctoral thesis to his alma mater in Trondheim but it was not accepted. Clearly his work in this area of physical chemistry was very advanced for its time. Onsager also worked in other areas of physical chemistry including colloids, dielectrics, order–disorder transitions, metals and superﬂuids, hydrodynamics, and fractionation theory. He received the Nobel Prize in chemistry in 1968 for his discovery of the reciprocal relations. His work was also recognized by many other awards both in the United States and internationally. He died in 1976.

6.1 Non-Equilibrium Processes Are Usually Complex The topics considered up to this point have involved liquids and solutions at equilibrium. Attention is now turned to systems which are not at equilibrium, and the processes which occur spontaneously in such systems. The physical phenomena involved can be quite complex, so that the task faced in early experiments was to separate the various processes and understand the physical properties of the system which govern them. Consider what happens when a beaker of pure isothermal water is placed on a hot plate. The water near the bottom rises in temperature and a temperature gradient is set up. As a result heat ﬂows from the bottom of the beaker, producing 254

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255

a gradual increase in temperature in the water at a given height above the bottom. In addition, the temperature varies with distance, being highest at the bottom and lowest at the top. Eventually, the temperature of the water in the beaker is uniform and equal to that of the hot plate, assuming that the water does not boil. However, the ﬂow of heat is not the only process resulting from the heat source. The density of the hot water is less than that of the cold water, so that a convection process is set up in order to achieve uniform density. Convection results in cold water moving down into the hot region so that the ﬂow of water molecules assists the ﬂow of heat. The changes which occur in this system cannot be understood without considering both processes. A system undergoing an irreversible change involving an electrolyte is electrolysis in an electrochemical cell. When current ﬂows between two copper electrodes in an aqueous solution of CuSO4, the charge in solution is carried by migration of Cu2þ ions moving in one direction and SO2 4 ions moving in the opposite. At the cathode, the incoming Cu2þ ions are reduced to metallic copper, thereby lowering the concentration of these ions in the electrode’s vicinity. At the anode, Cu metal is oxidized to produce Cu2þ ions in the solution, so that the local concentration of cations is increased. The changes in ionic concentrations near each electrode result in the establishment of concentration proﬁles. As a result, diﬀusion of ions occurs as well as migration in the electrical ﬁeld. If the density of the solution is signiﬁcantly changed near the electrode, convection is also present so that three irreversible processes contribute to the mass transfer, and thus, to the total current observed in the electrolysis experiment. In general, non-equilibrium processes are a function of both position in the system and time. A convenient way of discussing these processes is in terms of ﬂuxes, that is, the ﬂow of matter or energy through unit area per second in a given direction. The ﬂux is a vector quantity which depends on the speciﬁc location in the system, and it is also usually a function of time. When the ﬂux is independent of time, the non-equilibrium process is called a steady-state process. Irreversible thermodynamics is used to relate the ﬂuxes in the non-equilibrium system to the forces which give rise to them. As will be seen in the next section in this chapter, this subject gives a very useful way of relating these quantities when there is more than one of each, as illustrated in the examples discussed above. In this chapter, attention is focused on mass transfer processes in isothermal systems, especially in electrolyte solutions. Heat transfer is not discussed, but the methods used to treat this problem are very similar to those used to analyze diﬀusion problems. Chemical reactions in solutions are another example of non-equilibrium processes. These are discussed in detail in chapter 7.

6.2 The Thermodynamics of Irreversible Processes Onsager [1] developed an approach to the study of non-equilibrium processes which is based on elementary thermodynamic concepts. This provides a useful background to the subjects discussed in this chapter and helps show the connection between various processes occurring in a system undergoing an irreversible change.

256

LIQUIDS, SOLUTIONS, AND INTERFACES

The fundamental quantity describing the irreversible process is the ﬂux vector Ji for species i. In the case of mass transfer, it describes a ﬂow in a given direction in space in units of moles or grams crossing unit area per second. It can also be described as the product of the local concentration ci times the velocity vi at which molecules or ions are moving. Thus, Ji ¼ c i v i

ð6:2:1Þ

The latter deﬁnition is conceptually helpful when the concentration is changing with position in the system. One may write a similar relationship for an energy ﬂux, Je , that is J e ¼ re v e

ð6:2:2Þ

where re is the energy density in joules per cubic meter and ve , its velocity in a given direction. The change in the ﬂux with position in the system may be related to the time derivative of the local concentration by applying Gauss’ theorem. Consider a system with volume V and surface area A. Suppose that substance i is ﬂowing out of this volume. Then the rate of substance i leaving in moles per second can be found by integrating the ﬂux Ji over the surface area A, so that ð dn i ¼ Ji dA ð6:2:3Þ dt A where dni =dt is the rate of loss of the substance ni in moles per second. This can be related to the divergence of the ﬂow, which is deﬁned in vector notation as divJi ¼

@Ji @Ji @Ji þ þ @x @y @z

ð6:2:4Þ

Physically, a positive divergence of the ﬂow at a given point x; y; z in the system means that substance i is ﬂowing out of that point, that is, the point is a source. On the other hand, if the divergence is negative, substance i is ﬂowing into the point and it is a sink. By integrating div Ji over the volume of the system, one can determine the net change in mass with respect to time for the system. Thus, ð ð divJi dV ¼ Ji dA ð6:2:5Þ V

A

This is Gauss’ theorem stated with respect to mass transfer. The ﬁnal part of the analysis involves relating the change in the number of moles of component i to the changes in its concentration within the system. Assuming conservation of mass and no chemical reactions, this means that ð dni @ci ¼ dV ð6:2:6Þ dt @t V

Combining equations (6.2.3)–(6.2.6), one obtains ð ð ð dni @ci ¼ dV ¼ Ji dA ¼ divJi dV dt @t V

A

V

ð6:2:7Þ

257

NON-EQUILIBRIUM PHENOMENA IN LIQUIDS AND SOLUTIONS

It follows that @ci ¼ div Ji @t

ð6:2:8Þ

This is an important result connecting the variation in concentration with time at a given point in the system to the divergence of the ﬂux at the same point. It is called the equation of continuity with respect to mass. The cause of the ﬂux in the irreversible process is called a force. This term is not used in the strict Newtonian sense but instead it can refer to any source of perturbation. Forces resulting in ﬂuxes include a temperature gradient, a concentration gradient, a density gradient, and a gradient in electrical potential. These forces are vectors and are represented by the general symbol Xi . If concentration is varying in the x direction in a system, the gradient is @ci =@x. In order to keep the discussion general, the gradient is written in vector notation so that variations in ci in all possible directions can be considered. Thus, Xc ¼ rci ¼

@ci @c @c iþ ijþ ik @x @y @z

ð6:2:9Þ

where Xc is the force due to the concentration gradient and i, j, and k are unit vectors in the x; y, and z directions respectively. The force due to a temperature gradient is XT ¼ rT

ð6:2:10Þ

that due to a gradient in electrical potential X ¼ r

ð6:2:11Þ

and so on. It is obvious that forces deﬁned in this way have diﬀerent units. The general equation relating forces and ﬂuxes from Onsager’s theory is Ji ¼

n X

Lik Xk

ð6:2:12Þ

k¼1

where Lik is the phenomenological coeﬃcient relating force Xk to ﬂux Ji . This linear relationship only holds when the departure from equilibrium is not large. The exact limitations for equation (6.2.12) can only be determined experimentally. It is usually found that each force leads to a diﬀerent ﬂux. Thus, if there are two forces, there are two ﬂuxes and four phenomenological coeﬃcients. A system with three forces has three ﬂuxes and nine phenomenological coeﬃcients. Fortunately, these coeﬃcients are not all unique, as will be seen below. The example of the beaker of water on a hot plate is now considered with respect to equation (6.2.12). There are two forces in this system, namely, a temperature gradient XT , and a density gradient, Xr . Thus, one may write, JT ¼ LTT XT þ LTr Xr

ð6:2:13Þ

Jr ¼ LrT XT þ Lrr Xr

ð6:2:14Þ

and

258

LIQUIDS, SOLUTIONS, AND INTERFACES

where LTT , LTr , LrT and Lrr are the phenomenological coeﬃcients. JT and Jr are the ﬂuxes for the temperature and density, respectively. LTT is directly related to the coeﬃcient of thermal conductivity for water, and Lrr , to the viscosity of water. The coeﬃcients LrT and LTr are related to the eﬀect one process has on the other. According to one of Onsager’s theorems, the cross-coeﬃcients are equal to one another if the forces are deﬁned appropriately. This is an example of an Onsager reciprocal relation. In this case, only three phenomenological coeﬃcients are required to describe the irreversible process in the system. The easiest way to achieve an appropriate deﬁnition is to deﬁne the force as the gradient of the energy associated with the process. This conclusion is illustrated in more detail in this chapter. Some of the phenomenological coeﬃcients relating forces and ﬂuxes are already familiar from less general treatments of the subject. For example, the phenomenological coeﬃcient relating a concentration gradient and a mass transfer ﬂux is the diﬀusion coeﬃcient. Other phenomenological coeﬃcients are related to the ionic mobility, the coeﬃcient of thermal conductivity, and the solvent viscosity. These are discussed in more detail later in this chapter. EXAMPLE

Compare the phenomenological coeﬃcient for mass transfer deﬁned in terms of the concentration gradient with that based on a Newtonian force deﬁned for the gradient of the chemical potential. The traditional relationship between the ﬂux of species i and its concentration gradient is Ji ¼ Di rci

ð6:2:15Þ

where Di is the diﬀusion coeﬃcient, that is, the traditional phenomenological coeﬃcient. The negative sign expresses the fact that the ﬂux vector is pointed in the opposite direction to that in which the concentration gradient increases. If rci has units of mol m4 and Ji , units of mol m2 s1 , the units of the diﬀusion coeﬃcient are m2 s1 . In order to deﬁne a Newtonian force, one must relate the concentration gradient to an energy gradient. The easiest way of achieving this is to use the chemical potential of species i, which is the thermodynamic measure of the species’ ability to do useful work. The chemical potential of species i may be written as mi ¼ mii0 þ RT ln ai

ð6:2:16Þ

where mi0 is the standard chemical potential and ai , the activity of species i. The gradient of the chemical potential is rmi ¼ RT r ln ai

ð6:2:17Þ

Ji ¼ Li RT r ln ai

ð6:2:18Þ

Now one may write

where Li is the Newtonian phenomenological coeﬃcient for mass transfer. The units of rmi are J m1 mol1 or N mol1 . Thus the units of Li are mol2 m2 N1 s1 .

NON-EQUILIBRIUM PHENOMENA IN LIQUIDS AND SOLUTIONS

259

Another important aspect of Onsager’s theory is the principle of microscopic reversibility. This is usually discussed with respect to the detailed balancing of chemical reactions. When it is considered with respect to the phenomenological coeﬃcients, it leads to the reciprocal relations discussed above, namely that Lik ¼ Lki

ð6:2:19Þ

In other words, the matrix giving the phenomenological coeﬃcients for the system is symmetric. Another important part of this theory is that one can estimate the entropy generated in an irreversible process. Onsager showed that the local rate of entropy production per unit volume is y¼

1 @S 1 X ¼ J X V @t T i i i

ð6:2:20Þ

This relationship can only be applied when the forces are appropriately deﬁned. In the case of mass transfer, if the force is measured in J m1 mol1 and the ﬂux in mol m2 s1 , the rate of entropy production is obtained in J K1 m3 s1 . Onsager’s theory provides an elegant background for the consideration of nonequilibrium processes. It is especially helpful when more than one mechanism of irreversibility is present. This subject has only been brieﬂy introduced here. More details can be found in monographs devoted to this area of physical chemistry [G1, 2].

6.3 The Viscosity of Liquids Viscosity is the property of a ﬂuid which characterizes its resistance to ﬂow. It is often measured by timing the ﬂow of a liquid through a cylindrical tube under the inﬂuence of gravity. In order to understand the deﬁnition of the viscosity, consider a ﬂuid ﬂowing between two large plane parallel plates (ﬁg. 6.1). The velocity of the ﬂuid in the direction of the ﬂow, vx , varies with position. It is at its maximum midway between the plates and decreases to zero between each plate on the basis of experimental observation. Now imagine that the ﬂuid is made up of horizontal layers which are parallel to the plates. The movement of one layer with respect to another is retarded by a frictional force which is related to the ﬂuid’s viscosity. The origin of this friction is clearly intermolecular forces. In order to deﬁne the viscosity, one imagines a cylinder of thickness dx and radius r located on the central axis of the tube. The retarding force is proportional to the surface area of the cylinder involved in the ﬂow and to the gradient of the

Fig. 6.1 Schematic diagram of a liquid ﬂowing between two parallel plates. The arrows indicate the magnitude of the velocity and the direction of the ﬂow.

260

LIQUIDS, SOLUTIONS, AND INTERFACES

velocity in the r-direction (see ﬁg. 6.1). This leads to Newton’s law of viscosity, which is Fx ¼ ZA

dvx dr

ð6:3:1Þ

where Fx is the force in the x-direction, dvx =dr, the velocity gradient in the r-direction, Z, the viscosity, and A, the area of contact. The outer area of the cylinder for the example considered is 2pr dx. In the case of laminar ﬂow, the velocity on the outer surface of the cylinder is constant. This means that the retarding force due to the viscous drag is exactly counterbalanced by the force driving the liquid forward, which usually is a pressure diﬀerential in the tube. In the classical experiments used to measure liquid viscosity the pressure diﬀerential is established by gravity. If the pressure gradient is dP=dx, then the force in the x-direction on the cylinder is Fx ¼ pr2

dP dx

ð6:3:2Þ

where the minus sign takes into account the fact that the pressure drops as x increases. Equating these forces with the appropriate value for the contact area, one obtains dvx r dP ¼ dr 2Z dx

ð6:3:3Þ

If the radius of the tube is r0 , integration of equation (6.3.3) gives the result that vx ¼

1 2 dP ðr r20 Þ 4Z dx

ð6:3:4Þ

This shows that the forward velocity is a parabolic function of position in the tube varying from zero at the wall where r is equal to r0 , to the maximum value at the center of the cylindrical tube, where vx is equal to r20 ðdP=dxÞ=ð4ZÞ. It is emphasized that equation (6.3.4) is only applicable to laminar or non-turbulent ﬂow in a cylindrical tube. When a liquid ﬂows under diﬀerent geometrical boundary conditions, the relationship between the ﬂux and the force is not the same. From equation (6.3.1), the SI units of viscosity are N s m2 or Pa s. In the older literature, the viscosity is cited in cgs units, that is, in dyn s cm2 . 1 dyn s cm2 is called a poise. Since one Newton is equal to 105 dynes, it follows that one poise is equal to 0.1 Pa s. At room temperature, liquids typically have viscosities of 1 cP, which corresponds to 1 mPa s. Another important relationship used to describe the movement of a heavier solid object through a ﬂuid was derived by Sir George Stokes. In the case that the object is a sphere, the viscous drag or force on it in the direction opposite to its movement is F ¼ 6pZrv

ð6:3:5Þ

where r is the radius of the moving sphere. When a heavier solid sphere, for instance, a marble, is dropped into a viscous liquid, its velocity, which should continuously increase under the force of gravity alone, is observed to decrease and

NON-EQUILIBRIUM PHENOMENA IN LIQUIDS AND SOLUTIONS

261

eventually become constant. At this point, the gravitational force corrected for buoyancy is exactly balanced by the viscous drag. This leads to the following relationship: vmax ¼

2r2 g ðr r1 Þ 9Z s

ð6:3:6Þ

Here vmax is the maximum velocity, g, the acceleration due to gravity, rs , the density of the solid, and r1 , the density of the liquid. Stokes’ law is important in relating the movement of ions and molecules dissolved in liquids to other transport properties. In order to connect the above expressions to Onsager’s theory, it is necessary to extend equations such as (6.3.1) to three dimensions. This equation shows that a force in one direction leads to velocity changes in the other two spatial directions. A general treatment of ﬂuid ﬂow requires that one identify the components of the pressure tensor associated with the force which leads to ﬂuid ﬂow. The force vector F has, in general, a component in each of the three Cartesian directions. The component in the x-direction, Fx , gives rise to three pressure components, one in the same direction, Pxx , and two shear components, Pxy and Pxz . Six more pressure components are obtained from the force components in the y- and zdirections, Fy and Fz . As a result, there is a second-rank pressure tensor Pij with nine components. Analysis on the basis of classical dynamics leads to the conclusion that the oﬀ-diagonal elements, Pij and Pji , are equal. As a result, six distinct elements of this tensor must be determined to deﬁne it. The second important step is to write an equation of continuity in terms of momentum. Deﬁning the momentum density as ¼ rv

ð6:3:7Þ

where r is the local density and v, the local velocity, the momentum equation of continuity is @ ¼ r J @t

ð6:3:8Þ

where J is the ﬂux of momentum. J is also a second-rank tensor so that the relationship between and F can only be obtained by considering the tensorial nature of the problem. As solution of the problem was obtained many years ago and resulted in the Navier–Stokes equation [3]. This equation is the basis of the subject of hydrodynamics. An excellent introduction to this topic can be found in the monograph by McQuarrie [3]. Values of the viscosity for the polar liquids discussed in chapter 4 are collected in table 6.1. A large variation in viscosity is seen, varying from 0.3 mPa s for acetone to 10 mPa s for tetramethylenesulfone. The viscosity is a direct reﬂection of intermolecular forces, which obviously change considerably in strength in this group of liquids. Debye [5] considered the viscosity in more detail in this regard and pointed out that there should be a connection with the relaxation time tD observed when a polar liquid is exposed to a high frequency alternating electrical ﬁeld (see chapter 4). This parameter also reﬂects the eﬀect of intermolecular forces when a molecule or group of molecules undergo rotational diﬀusion. Debye

262

LIQUIDS, SOLUTIONS, AND INTERFACES

Table 6.1 Values of the Viscosity and the Enthalpy Associated with Its Temperature Dependence for Selected Polar Solvents at 258C [4] Solvent

Viscosity Z=mPa s

Enthalpy HZ =kJ mol1

0.890 0.545 1.083 1.956 2.593 3.302 1.65

2.09 9.3 12.5 17.2 18.6 18.6 12.1

0.303 0.341 1.237 0.549 0.927 0.802 1.996 3.245 1.666 1.795 0.610 0.410 2.513 10.29 1.395

5.9 6.1 12.0 8.5 9.4 8.5 13.6 15.0 — 12.7 7.5 6.6 14.9 20.0 —

Protic 1. 2. 3. 4. 5. 6. 7.

W MeOH EtOH PrOH BuOH F NMF

Aprotic 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

AC AcN BzN BuN DMA DMF DMSO HMPA NMP NB NM PrN PC TMS TMU

developed a model for tD in which it was imagined that one polar molecule which is assumed to be a sphere undergoes rotational diﬀusion in a liquid continuum whose intermolecular forces are evaluated by the bulk viscosity Z. On this basis, he derived the relationship tD ¼

4pZr3m kB T

ð6:3:9Þ

in which rm is the radius of the molecule. If this is applied to estimate the value of tD for the solvents listed in table 6.1, the result obtained for aprotic solvents is of the correct magnitude, but always too high. EXAMPLE

Estimate the Debye relaxation time for nitromethane at 258C given that its viscosity is 0.61 mPa s and assuming that its molecular diameter is 431 pm.

NON-EQUILIBRIUM PHENOMENA IN LIQUIDS AND SOLUTIONS

263

On the basis of equation (6.3.8), tD ¼

4pð6:1 104 Þð2:155 1010 Þ3 ¼ 1:86 1011 s 1:38 1023 298:2

ð6:3:10Þ

This estimate of 19 ps is much larger than the experimental estimate of 4 ps, which is obtained by applying the Debye model to dielectric relaxation data for the pure solvent (see table 4.2). However, an approximate relationship between tD and Zr3m is found when data for more solvents are considered, as shown below. The fact that estimates of tD are the correct order of magnitude suggests that there is an approximate relationship between tD and Z, at least for liquids which obey the Debye relaxation model (see section 4.5). This point is illustrated in ﬁg. 6.2 where tD is plotted against Zr3m with logarithmic scales for the aprotic solvents listed in table 6.1. A very good correlation is found, conﬁrming that equation (6.3.9) is approximately correct. On the basis of a least-squares ﬁt for 14 solvents, the relationship is ln tD ¼ 5:4 103 þ 0:91 lnðZr3m Þ

ð6:3:11Þ

with a correlation coeﬃcient of 0.961. Addition of the data for the protic solvents results in a much poorer ﬁt. This is not surprising because the value of tD for these liquids reﬂects the rotational diﬀusion of hydrogen-bonded clusters, not that of single molecules as assumed in the simple Debye model. The simplest model for the temperature dependence of the viscosity is that proposed by Arrhenius for a reaction rate constant which leads to the equation

Fig. 6.2 Plot on logarithmic scales of the Debye relaxation time tD for aprotic solvents against the product Zr3m , where Z is the viscosity and rm , the molecular solvent radius.

264

LIQUIDS, SOLUTIONS, AND INTERFACES

HZ Z ¼ A exp RT

ð6:3:12Þ

where HZ is the activation enthalpy associated with viscous ﬂow and A, a preexponential factor. Values of HZ are also summarized for the polar solvents considered here in table 6.1. Since the temperature dependence of tD is assumed to have the same form (equation (4.7.1)) the analysis presented in ﬁg. 6.2 allows one to relate HD and HZ for aprotic solvents. Assuming that rm is independent of temperature, the relationship is HD ¼ 0:91HZ

ð6:3:13Þ

Comparison of the data in tables 4.2 and 6.1 shows that this is approximately correct. The discussion here has been directed to the viscosity of pure liquids. When an electrolyte or other solute is added, the viscosity changes. Many data exist in the literature for the viscosity of solutions. This subject is of special importance to engineers interested in the ﬂow properties of solutions. The discussion here is meant to be only a brief introduction to the basic concepts. In the following sections, attention is turned to the transport of solute species in solutions.

6.4 Isothermal Diffusion in Solutions As seen in the earlier discussion of Onsager’s theory (section 6.2), in the classical approach diﬀusion is a response to a concentration gradient. If species i is diﬀusing then the ﬂux of i is given by Ji ¼ Di rci

ð6:4:1Þ

where Di is the diﬀusion coeﬃcient of i. This relationship is known as Fick’s ﬁrst law. It is most readily applied to describe a steady-state situation in which the ﬂux Ji is independent of time. Very often, the mass transfer problem involves conditions in which the local concentration is also a function of time. By applying the equation of continuity with respect to mass one obtains the result @ci ¼ rJi ¼ Di r2 ci @t

ð6:4:2Þ

This is known as Fick’s second law; it states that the time derivative of the concentration at a given point in space is equal to the diﬀusion coeﬃcient times the Laplacian of the concentration. In deriving this equation it has been assumed that the diﬀusion coeﬃcient is independent of concentration. In a simple situation the variation of concentration only occurs in one direction in space. Then Fick’s ﬁrst law becomes Jxi ¼ Di

@ci @x

ð6:4:3Þ

where Jxi is the component of the ﬂux in the x-direction. In one dimension, Fick’s second law is

NON-EQUILIBRIUM PHENOMENA IN LIQUIDS AND SOLUTIONS

@ci @2 c ¼ Di 2i @t @x

265

ð6:4:4Þ

This is a second-order partial diﬀerential equation whose solution gives the concentration of species i as a function of position and time, that is, of x and t. As seen earlier, the dimensions of the diﬀusion coeﬃcient in SI units are m2 s1. Typical diﬀusion coeﬃcients in aqueous solutions are the order of 1 109 m2 s1 . On the basis of Onsager’s theory the quantity leading to the ﬂux, that is, the force should have the dimensions of Newtons. On the basis of classical thermodynamics this is most directly achieved by writing Fick’s laws in terms of the chemical potential rather than the concentration. This means that Fick’s ﬁrst law becomes Ji ¼ Li rmi

ð6:4:5Þ

where rmi is the gradient of the chemical potential of species i, and Li , the phenomenological coeﬃcient relating the ﬂux to the force for the diﬀusion process. In general, the chemical potential can be related to the activity of species i, and thus to its concentration using the equation mi ¼ mi0 þ RT ln ai ¼ mi0 þ RT ln yi ci

ð6:4:6Þ

where ai is the activity on the concentration scale, yi , the activity coeﬃcient and mi0 , the standard chemical potential. Combining equations (6.4.5) and (6.4.6), one obtains Ji ¼ Li RT r ln ai ¼

Li RT rai ai

ð6:4:7Þ

If the solution is ideal (yi ¼ 1) or if the activity coeﬃcient is constant for the range of concentrations involved in the experiment, then this equation becomes Ji ¼

Li RT rci ci

ð6:4:8Þ

It follows that the relationship between the Onsager phenomenological coeﬃcient and the traditional diﬀusion coeﬃcient is Li ¼

Di ci RT

ð6:4:9Þ

In many cases, for example, electrolyte solutions, the eﬀects of solution nonideality must be considered. Then equation (6.4.7) can be written as Ji ¼ Li RTr ln ci Li RT r ln yi ¼

Li RT L RT r ln yi rci i rc ci ci r ln ci i

ð6:4:10Þ

It follows that the diﬀusion coeﬃcient depends on the concentration of the diﬀusing species according to the relationship Di ¼ Di0 þ Di0

r ln yi r ln ci

ð6:4:11Þ

266

LIQUIDS, SOLUTIONS, AND INTERFACES

where Di0 is the diﬀusion coeﬃcient in the ideal solution. In one dimension this equation has the form Di ¼ Di0 þ Di0

d ln yi d ln ci

ð6:4:12Þ

EXAMPLE

The diﬀusion coeﬃcient of Naþ is 1:34 109 m2 s1 at inﬁnite dilution in water at 258C. Estimate its value for a Naþ concentration of 0.1 M in a solution of the same ionic strength. Use the extended Debye–Hu¨ckel theory to estimate the concentration dependence of the activity coeﬃcient assuming that the ion size parameter is 400 pm. The activity coeﬃcient for the Na+ ion is given by ln yi ¼

ADH z2i I 1=2 ADH ci1=2 ¼ 1 þ BDH aI 1=2 1 þ BDH ac1=2 i

ð6:4:13Þ

where I is the ionic strength, and the Debye–Hu¨ckel constants are ADH ¼ 1:174 M1=2 and BDH ¼ 3:288 M1=2 nm1 . The derivative of this quantity with respect to ci is d ln yi 0:5ADH ci1=2 0:5ADH BDH a ¼ þ 1=2 dci ð1 þ BDH aci1=2 Þ2 1 þ BDH aci

ð6:4:14Þ

d ln yi 0:5ADH c1=2 0:5ADH BDH aci i ¼ þ 1=2 d ln ci 1 þ BDH aci ð1 þ BDH aci1=2 Þ2

ð6:4:15Þ

Thus,

Substituting into equation (6.4.15) for the case that ci is 0.1 M, the value of d ln yi =d ln ci is 0:093. Now, on the basis of equation (6.4.12), the diﬀusion coeﬃcient for Na+ in the 0.1 M solution is 1.22 109 m2 s1 . In practice, conditions in a diﬀusion experiment can be maintained such that the activity coeﬃcient does not change signiﬁcantly with electrolyte concentration. For electrolyte solutions, this is done by adding an inert electrolyte whose ions are not involved in the mass transfer process. Then, it can be assumed that the ionic strength is constant and independent of position in the system. Otherwise, the mass transfer problem becomes extremely diﬃcult to solve and requires information about the diﬀusion coeﬃcient as a function of concentration.

6.5 Linear Diffusion from a Wall In this section, a simple example of the solution of a mass transfer problem is presented. When one solves diﬀerential equations, it is very important to state the boundary conditions clearly. In the present example, diﬀusion takes place in the

NON-EQUILIBRIUM PHENOMENA IN LIQUIDS AND SOLUTIONS

267

positive x-direction into a solution, but half of the space is ﬁlled with a wall extending in the minus x-direction for all values of y and z. The diﬀusing material or solute dissolves into the solution and its concentration at the wall is maintained constant and equal to c0 . As a result the value of the concentration varies with distance from the wall, x and with time, t. Since the system does not reach a steady state, the partial diﬀerential equation to be solved is @c @2 c ¼D 2 @t @x

ð6:5:1Þ

c ¼ cðx; tÞ

ð6:5:2Þ

where It is also assumed that the diﬀusion coeﬃcient D is independent of location in the solution (x > 0). The boundary conditions with respect to x are cð0; tÞ ¼ c0

ð6:5:3Þ

cð1; tÞ ¼ 0

ð6:5:4Þ

and The second condition implies that diﬀusion does not occur for a very long time, so that the concentration far from the wall is zero. The boundary condition with respect to time is cðx; 0Þ ¼ 0

ð6:5:5Þ

which states that the solute is not present anywhere in the solution initially. The solution to equation (6.5.1) is obtained using the method of Laplace transforms. Since there are two independent variables in the present problem, this means that Laplace transformation must be performed twice, once with respect to the variable t, and then again with respect to the variable x. The Laplace transform of the concentration with respect to time is given by (see appendix A) 1 ð

cðx; sÞ ¼

cðx; tÞest dt

ð6:5:6Þ

0

The variable s now contains the information about the time dependence of the problem. Applying Laplace transformation to the derivative on the right-hand side of equation (6.5.1), one obtains d 2c ¼ dx2

1 ð

0

@2 c st e dt @x2

ð6:5:7Þ

Laplace transformation of the left-hand side is more diﬃcult and requires use of the product rule. In the present case, this gives dðc est Þ ¼ est Thus, one obtains

@c s est c @t

ð6:5:8Þ

268

LIQUIDS, SOLUTIONS, AND INTERFACES 1 ð

0

1 1 ð @c st st e dt ¼ ce þ s c est dt ¼ sc @t 0

ð6:5:9Þ

0

Now the partial diﬀerential equation has been transformed into an ordinary diﬀerential equation in c, namely D

d2 c ¼ sc dx2

ð6:5:10Þ

At this point, Laplace transformation is carried out with respect to x using the relationship 1 ð

c~ðu; sÞ ¼

cðx; sÞeux dx

ð6:5:11Þ

0

The most diﬃcult part of the solution involves transformation of the derivative on the left-hand side of equation (6.5.10) which requires that the product rule be used twice: 1 ð

0

1 1 ð d2 c ux dc ux dc ux e e dx e dx ¼ þ u dx dx dx2 0 0

¼

1 1 ð dcð0; sÞ þ uc eux þ u2 c eux dx dx 0

ð6:5:12Þ

0

dcð0; sÞ ucð0; sÞ þ u2 c~ ¼ dx Laplace transformation of the right-hand side gives 1 ð

sc eux dx ¼ s c~

ð6:5:13Þ

0

The result can be expressed as Du2 c~ DAu DB ¼ s c~

ð6:5:14Þ

A ¼ cð0; sÞ

ð6:5:15Þ

where

and B¼

d c; ð0; sÞ dx

ð6:5:16Þ

Equation (6.5.14) is rearranged to obtain an expression for c~: c~ ¼ where

DAu þ DB Au þ B ¼ 2 Du2 s u v2

ð6:5:17Þ

NON-EQUILIBRIUM PHENOMENA IN LIQUIDS AND SOLUTIONS

v2 ¼

s D

269

ð6:5:18Þ

By using the method of partial fractions this result may be expressed as c~ ¼

a b þ uþv uv

ð6:5:19Þ

where the constants a and b may be found by equating the numerators of equations (6.5.17) and (6.5.19) when equation (6.5.19) is in complex form. In other words aðu vÞ þ bðu þ vÞ ¼ Au þ B

ð6:5:20Þ

From the coeﬃcients of u, aþb¼A

ð6:5:21Þ

vðb aÞ ¼ B

ð6:5:22Þ

and from the constant term,

At this point, the inverse Laplace transformation of equation (6.5.19) is carried out. In order to do this, one must consult a table of Laplace transforms such as the one given in appendix A. Operating on each term individually, one obtains a L1 ¼ a evx ð6:5:23Þ uþv and L

1

b ¼ b evx uv

ð6:5:24Þ

As a result, the general expression for cðx; sÞ is cðx; sÞ ¼ aevx þ bevx

ð6:5:25Þ

Now, the boundary conditions with respect to x are applied so that the solution speciﬁc to the problem addressed can be found. When x is inﬁnitely large, the concentration goes to zero (equation (6.5.4)). The only way that this result is obtained is by setting the constant b equal to zero. When x is equal to zero, the concentration is constant and equal to c0 (equation (6.5.3)). Laplace transformation of this boundary condition with respect to time gives 1 ð

0

c0 est dt ¼

1 c0 st c e ¼ 0 ¼ cð0; sÞ s s 0

ð6:5:26Þ

By comparing this result with equations (6.5.15) and (6.5.21), it is clear that a selfconsistent solution has been obtained. Thus, one may now write cðx; sÞ ¼

c0 vx e s

ð6:5:27Þ

270

LIQUIDS, SOLUTIONS, AND INTERFACES

The ﬁnal step is to perform inverse Laplace transformation on cðx; sÞ to obtain cðx; tÞ. Using the table of Laplace transformations, inverse transformation of equation (6.5.27) results in x cðx; tÞ ¼ c0 erfc ð6:5:28Þ ð4DtÞ1=2 where erfc is the error function complement. The error function of the parameter z is deﬁned as ðz 2 2 erfðzÞ ¼ 1=2 ey dy p

ð6:5:29Þ

0

This function corresponds to the area under the normal error distribution curve from its maximum value to a value z. The error function has a value of 0 when z is zero, and a value of one when z is inﬁnity. The error function complement is simply deﬁned as erfcðzÞ ¼ 1 erfðzÞ

ð6:5:30Þ

This function goes from one to zero. The ﬁnal result in equation (6.5.28) shows that the concentration falls from its initial value of c0 at the wall in a manner determined by the error function and the diﬀusion coeﬃcient D. Typical concentration proﬁles estimated at various values of t are shown in ﬁg. 6.3. Obviously, the concentration increases at a given distance from the wall as time increases. It is also useful to calculate the ﬂux of the diﬀusing species using Fick’s ﬁrst law: Jðx; tÞ ¼ D

@cðx; tÞ @x

ð6:5:31Þ

Fig. 6.3 Typical concentration proﬁles estimated for various times for diﬀusion from a semi-inﬁnite wall with D ¼ 1 109 m2 s1 .

NON-EQUILIBRIUM PHENOMENA IN LIQUIDS AND SOLUTIONS

271

This can be done by diﬀerentiating equation (6.5.28) with respect to x, and requires that the ﬁrst derivative of the error function be estimated. On the basis of equation (6.5.29), 2 dz d erfðzÞ 2 ¼ 1=2 ez dx dx p

ð6:5:32Þ

d erfc½x=ð2ðDtÞ1=2 Þ expðx2 =ð4DtÞÞ ¼ dx ðpDtÞ1=2

ð6:5:33Þ

! 1=2 D x2 Jðx; tÞ ¼ c0 exp pt 4Dt

ð6:5:34Þ

1=2 D Jð0; tÞ ¼ c0 pt

ð6:5:35Þ

It follows that

Thus,

At the wall, the ﬂux is

This is an important result used in electroanalytical chemistry. The importance of the boundary conditions in obtaining a solution to Fick’s equation is apparent from the example presented here. This equation has been solved for many diﬀerent situations, including cylindrical and spherical geometries [G2, G3]. It has often been applied to mass transfer problems in electroanalysis and important examples can be found in monographs devoted to this subject [G4].

6.6 The Electrochemical Potential When mass transfer occurs in electrolyte solutions, the component ions can move independently of one another. Because of the ionic charge, the Gibbs energy of an ion depends not only on its local concentration but also on the local electrostatic potential. It is also necessary to consider circumstances in which changes in the local potential are brought about by application of an external ﬁeld. Thus, it is important to be able to deal with the chemical and electrostatic contributions to the Gibbs energy separately. The thermodynamic tool for achieving this is the electrochemical potential. The electrochemical potential of charged species i, m~ ai is deﬁned as the work done when this species is moved from charge-free inﬁnity to the interior of a homogeneous phase a which carries no net charge [6]. The opposite process is known as the work function and is familiar for the case of removal of an electron from a metal. In fact, the work function for single ions in electrolyte solutions can also be measured experimentally, as described in detail in chapter 8. This means that the electrochemical potential is an experimentally determinable quantity. However, separation of the electrochemical potential into chemical and electrostatic contributions is arbitrary, even though it is conceptually very useful.

272

LIQUIDS, SOLUTIONS, AND INTERFACES

A very helpful way of understanding the contributions to m~ ai has been described by Parsons [6]. It is imagined that phase a, for example, an electrolyte solution, is a sphere isolated in free space (ﬁg. 6.4). This sphere is divided into two regions, a central spherical region where the distribution of ions and solvent molecules reﬂects the homogeneous nature of the bulk, and a thin spherical shell at the surface which contains the interfacial double layer and any excess surface charge associated with phase a. The double layer at the aqueous electrolyte solution|air interface is due to any net orientation of water molecules at the surface, and to the fact that hydrated cations and anions may have diﬀerent distances of closest approach due to diﬀering degrees of hydration. Electrolyte solutions do not normally possess excess charge, but this could easily be the case if phase a were a metal. Now, the two parts of phase a are separated (ﬁg. 6.4) and the process of introducing a charged species to each part repeated. The work done in the case of the homogeneous phase a is the chemical potential mai . On the other hand, the work done in moving the charged particle through the surface shell into the empty interior is associated with the electrostatic characteristics of phase a and is written as zi Ffa . fa is called the inner potential of phase a, and cannot be measured for an isolated phase. This fact can be understood from the point of view that one is only able to measure a potential diﬀerence in electrostatics. One is also not able to measure the chemical potential of a single charged species, for example, a single ion. On the other hand, one is able to determine the chemical potential of a neutral electrolyte in which the charges on the individual ions add to zero. As a result of the thought experiment described above, the expression for the electrochemical potential of species i in phase a is m~ ai ¼ mai þ zi Ffa

ð6:6:1Þ

For a simple 1–1 electrolyte MX, the electrochemical potential of the cation is m~ aM ¼ maM þ Ffa

ð6:6:2Þ

m~ aX ¼ maX Ffa

ð6:6:3Þ

and that of the anion

Fig. 6.4 Schematic representation of the division of the electrochemical potential of species i in phase a into chemical and electrical contributions.

NON-EQUILIBRIUM PHENOMENA IN LIQUIDS AND SOLUTIONS

273

The electrochemical potential of the electrolyte is m~ aMX ¼ m~ aM þ m~ aX ¼ maM þ maX ¼ maMX

ð6:6:4Þ

Thus, the distinction between the electrochemical potential and the chemical potential is lost when the properties of the electrolyte as a whole are considered. The development of the thermodynamic properties of the electrolyte are those discussed earlier in section 3.6. The inner potential is an important property of individual phases. Much more will be said about this property when the interface between two phases is discussed in chapter 8. For the moment, fa is regarded as a property of phase a which is the same throughout the phase with a value deﬁned with respect to charge-free inﬁnity. On the other hand, in the case of an electrolyte solution, the local electrostatic potential varies from point to point due to the presence of discrete charges on the ions. Thus the electrostatic potential is more positive at a cation and more negative at an anion. These ﬂuctuations occur about the average value, fa . This can be seen more clearly by writing out the chemical potential of ion i in terms of its concentration ci and activity coeﬃcient yi . Thus, from equation (6.6.1) m~ ai ¼ mai0 þ RT ln yi ci þ zi Ffa

ð6:6:5Þ

The activity coeﬃcient is directly related to the work done to create the ionic atmosphere around ion i, so that this equation can also be written as m~ ai ¼ mai0 þ RT ln ci þ zi Fai

ð6:6:6Þ

ai

is the micropotential or local potential experienced by this ion (see where section 3.8). The relationship between fa and ai is ai ¼ fa þ ia

ð6:6:7Þ

where ia is the discreteness-of-charge potential. On the basis of this deﬁnition, ia is the diﬀerence between the local potential and its average value which arises because the charge in the solution is localized on ions. The discreteness-of-charge potential may be expressed in terms of two contributions as follows: a a zi F ia ¼ zi F self þ zi F atm

ð6:6:8Þ

a self

is the self-potential of the ion which depends on its radius and the dielectric a , is the self atmosphere potential due to the properties of the medium, and atm ionic cloud formed around ion i. The latter quantity is directly related to the activity coeﬃcient yi as follows: a RT ln yi ¼ zi F ;atm

ð6:6:9Þ

Since the self-potential of ion is not included in the deﬁnition of yi , the standard states in equations (6.6.5) and (6.6.6) are not the same (see section 3.8). The concept of the electrochemical potential is also helpful in analyzing changes in the Gibbs energy of dipoles and multipoles when the electrical environment is changed. In the case of a dipole, the electrostatic energy is determined by the local ﬁeld Ee . The local ﬁeld can change due to a change in environment or

274

LIQUIDS, SOLUTIONS, AND INTERFACES

the imposition of the ﬁeld from an external electrical source. A well-known example of an environmental change occurs when a dipolar molecule is adsorbed at an interface. Under these circumstances the local ﬁeld can be quite diﬀerent, so that the electrostatic contribution to the Gibbs energy changes signiﬁcantly. The electrochemical potential of a dipolar molecule may be written as m~ ai ¼ mai0 þ RT ln ai þ pi Ee

ð6:6:10Þ

where ai is the activity of molecule i, pi , its dipole moment, and Ee , the local ﬁeld. The electrostatic energy is expressed as a vector dot product because the angle between the vectors must be known in order to estimate it. An example of the application of dipolar electrochemical potentials is given in chapter 10. Discussion of non-equilibrium processes involving ions in terms of the micropotential is especially helpful because it focuses attention on the fact that major source of non-ideality in these systems is electrical in character. The arbitrary nature of the separation of the electrochemical potential into chemical and electrical contributions has often been pointed out in the literature. In fact, chemical interactions are fundamentally electrical in nature. However, the formal separation discussed here is conceptually important. Its usefulness becomes clear when one tackles problems related to the movement of ions in electrolyte solutions under the inﬂuence of concentration and electrostatic potential gradients. These problems are discussed in the following section.

6.7 The Conductivity of Electrolyte Solutions Electrolyte solutions contain ions which can move in response to a gradient in electrical potential. The transport properties of these systems are important in devices such as batteries and in living systems. The movement of ions in solution is very diﬀerent from the movement of electrons in metallic conductors, and it is important to understand the fundamental laws which govern the conductivity of electrolyte solutions. Ions move according to the classical laws of physics, whereas the movement of electrons is quantal. The force governing the movement of ions is the local gradient of the electrostatic potential, ri . If eﬀects due to the presence of an ion at a given point in the solution and other ions surrounding it are ignored, and only the macroscopic potential gradient is considered, then ri can be replaced by rf. Local electrostatic eﬀects are generally considered to be the major source of non-ideality in electrolyte solutions at equilibrium. In the classical treatment of conductivity all eﬀects due to non-ideality are included in the phenomenological coeﬃcients, namely, the mobility and the molar conductance discussed below. The ﬂux of ion i is normally written in terms of the current density, ii . Recalling that ﬂux has units of moles per square meter per second, the corresponding current density is given by ii ¼ zi FJi

ð6:7:1Þ

where zi is the valence of ion i. The total current density which is observed experimentally is the sum of the contributions from each ion:

NON-EQUILIBRIUM PHENOMENA IN LIQUIDS AND SOLUTIONS

i¼

X

zi FJi

275

ð6:7:2Þ

i

Since cations and anions move in opposite directions under the inﬂuence of an electrical ﬁeld, the vector JM for a cation is in the opposite direction to that for an anion JX . This means that the ﬂux Ji multiplied by the ionic valence zi has the same sign for all ions; as a result each ion contributes to the solution conductivity in such a way that it becomes larger. The phenomenological coeﬃcient relating the current density and the macroscopic potential gradient is the conductivity due to the ion, ki . Thus, one has the general relationship ii ¼ ki rf

ð6:7:3Þ

The minus sign reﬂects the fact that a positive ion moves in the direction opposite to the increase in electrical potential. If rf is positive, the electrical potential increases in a speciﬁed direction, for instance the x-direction, but a cation moves in the opposite direction. This equation is a special form of Ohm’s law written in terms of the current density and the potential gradient, rather than the current and the potential diﬀerence. The experimentally measured quantity is the conductivity ke , which is the sum of the contributions from each ion in solution: X ke ¼ ki ð6:7:4Þ i

In order to measure the conductivity of an electrolyte solution, a special cell is constructed in which the two electrodes of accurately known dimensions are placed at a known distance from one another (ﬁg. 6.5). The electrodes are usually made of an inert material such as platinum. If a d.c. voltage is applied to such a cell, no current ﬂows until a certain minimum value is reached. Below this minimum, no reactions occur at the electrodes, and the electrode|solution interface behaves as a capacitor. When d.c. current ﬂows, reactions occur at each electrode and ions move in the electrolyte solution. The nature of the reactions depends on the nature of the electrolyte but certainly the component ions are involved in these reactions. For this reason, d.c. experiments are not used in the precise determination of solution conductivity. On the other hand, if an a.c. voltage of low amplitude is applied to the cell, its conductivity may be determined in the absence of any net change in the composition of the solution. Traditionally, this experiment was carried out by making the conductance cell one arm of an a.c. Wheatstone bridge designed to measure resistance.

Fig. 6.5 Schematic diagram of a conductivity cell (a) and its impedance representation as an equivalent circuit (b). Capacitances C1 and C2 are at the electrode|solution

276

LIQUIDS, SOLUTIONS, AND INTERFACES

The relationship between the solution resistance Rs and the conductivity ke is ke ¼

d Rs A

ð6:7:5Þ

where d is the separation between the plates of the conductivity cell and A is the area of the electrodes. The units of ke are 1 m1 . In order to estimate ke from the experimental value of the resistance, the cell parameters d and A must be carefully determined. In practice, the ratio d=A is found by calibrating the conductivity cell with a solution of known conductivity. Another important parameter related to electrolyte solution conductivity is the transport number ti . This is deﬁned as the fraction of the current carried by species i. On the basis of equation (6.7.4), it is clear that ti is given by ti ¼

ki ke

ð6:7:6Þ

The classical way of measuring transport numbers is in a Hittorf cell, which is illustrated schematically in ﬁg. 6.6. The cell consists of three compartments designated L (left), M (middle), and R (right). In the present example the two electrodes are composed of pure copper, and the electrolyte is an aqueous solution of Cu(NO3)2. Direct current ﬂows through the cell, and the amount of charge which passes through the system, q, is measured by the coulometer in the circuit. For the example chosen, the reaction at electrode L, which is the cathode, is Cu2þ þ 2e !Cu

ð6:7:7Þ

This is accompanied by the transport of Cu2þ ions into L and transport of NO 3 ions out of L. The net change in number of moles of Cu2þ ion in compartment L is

Fig. 6.6 Hittorf cell for determining transport numbers with associated circuitry. Circuit element C is a coulometer. The three compartments are labeled L (left), M (middle), and R (right).

NON-EQUILIBRIUM PHENOMENA IN LIQUIDS AND SOLUTIONS

nL ðCu2þ Þ ¼

q t q t q þ þ ¼ 2F 2F 2F

277

ð6:7:8Þ

Since there are only two kinds of ions, the fractions tþ and t must add to unity. The number of moles of NO 3 ion moving out of L is nL ðNO 3Þ¼

t q F

ð6:7:9Þ

Since each nitrate ion has a charge of 1, the number of moles is obtained by dividing the charge moved by the Faraday constant F. Thus, the total change in the number of moles of Cu(NO3 Þ2 is t q=ð2FÞ with the requirement of electroneutrality being maintained. The reaction at the anode R is the opposite, namely Cu!Cu2þ þ 2e

ð6:7:10Þ

In this compartment, Cu2þ ion is transported out and NO 3 ion moves in. Analysis of the next changes leads to the result nR ðCu2þ Þ ¼

nR ðNO t q 3Þ ¼þ 2 2F

ð6:7:11Þ

The net change in the middle compartment is zero. By analyzing the contents of each compartment after passing a known charge q through the system, the transport numbers tþ and t can be calculated. Two other quantities are commonly used in discussing the transport properties of electrolytes. The ﬁrst is the molar conductivity of ion i, which is deﬁned as li ¼

ki ci

ð6:7:12Þ

where ci is the concentration of the ion. In order to obtain reasonable dimensions for li , the concentration is expressed in mol m3 so that li has units of 1 m2 mol1 . As the concentration of ion i increases one expects the conductivity of the solution to increase as well. To a ﬁrst approximation, this increase should be proportional to the change in concentration, so that li is a constant. In fact, li decreases with increase in electrolyte concentration as a result of the eﬀects of non-ideality. This observation is discussed in detail in the following section. Another way of presenting information about the concentration dependence of ki is in terms of the equivalent conductance deﬁned as l i ¼

ki jzi jci

ð6:7:13Þ

One equivalent of ion i is the amount which contains one mole of univalent charge. This quantity is often found in the older literature. Another important parameter used to discuss the transport properties of electrolytes is the ionic mobility, ui . This is the velocity of the ion in a unit electrical ﬁeld. It can be found from the molar conductivity by dividing it by the charge on a mole of ions:

278

LIQUIDS, SOLUTIONS, AND INTERFACES

ui ¼

li jzi jF

ð6:7:14Þ

Dimensional analysis shows that ui has units of m2 V1 s1 in the SI system. It is also useful to relate ui to the conductivity of the solution. Combining equations (6.7.4), (6.7.12), and (6.7.14), one obtains X ke ¼ jzi jF ui ci ð6:7:15Þ i

EXAMPLE

The molar conductance of 0.5 M MgCl2 in water is 165.52 cm2 mol1 1 at 258C. The cationic transport number is 0.308 at this concentration. Estimate the molar conductances of the individual ions. Also estimate the ionic mobilities. The molar conductance for Mg2þ is given by lðMg2þ Þ ¼ 0:308 165:52 ¼ 50:98 cm2 mol1 1 For the Cl ion, lðCl Þ ¼

ð1 0:308Þ 165:62 ¼ 57:27 cm2 mol1 1 2

Using equation (6.7.14), the mobility of Mg2þ is uðMg2þ Þ ¼

50:98 ¼ 2:64 104 cm2 s1 V1 2 96;485

and that of Cl is uðCl Þ ¼

57:27 ¼ 5:94 104 cm2 s1 V1 96;485

The Mg2þ ion is much more strongly solvated by water molecules than Cl . Thus, it moves with more water molecules and its mobility is much less. The relationship between ionic conductivity and Onsager’s theory can now be presented in terms of the electrochemical potential. By expressing the force leading to the transport of ions in terms of the gradient of m~ i , one ﬁnds important relationships between the diﬀusion coeﬃcients of the ions, and the molar conductivity and mobility. Furthermore, when the force has the correct Newtonian units, one is also in a position to calculate the rate of entropy production. On the basis of the thermodynamics of irreversible processes, the relationship between the ﬂux of ion i and the force rm~ i is Ji ¼ Li rm~ i

ð6:7:16Þ

where Li is the appropriate phenomenological coeﬃcient. Using equation (6.6.5), this can be written as L RT r ln yi 1þ ð6:7:17Þ rci Li zi Frf Ji ¼ i ci r ln ci

NON-EQUILIBRIUM PHENOMENA IN LIQUIDS AND SOLUTIONS

279

The factor in front of rci is simply the diﬀusion coeﬃcient Di (equation (6.4.10)). Now, using equations (6.7.1) and (6.7.3), the contribution to the ﬂux from the potential gradient may be expressed in terms of the molar conductivity. The result is k ð6:7:18Þ Ji ¼ Di rci i rf zi F Since ki ¼ jzi jF ui ci

ð6:7:19Þ

the ﬂux of ion i can also be written as Ji ¼ Di rci

z i ui c i rf jzi j

ð6:7:20Þ

Comparing equations (6.7.17) and (6.7.20), one ﬁnds that Li ¼

Di ci uc ¼ i i RT jzi jF

ð6:7:21Þ

In the limit of inﬁnite dilution, where the eﬀects of non-ideality are absent, one obtains the Einstein relationship between these parameters for a single ion: Di0 ¼

RT u jzi jF i0

ð6:7:22Þ

The subscript ‘‘0’’ designates the values of Di and ui in the inﬁnitely dilute limit. A typical mobility for an ion in water is 5 108 m2 s1 V1 . For a monovalent ion, the corresponding diﬀusion coeﬃcient at 258C is 1:3 109 m2 s1 . The diﬀusion coeﬃcient of an individual ion may also be related to its molar conductance. Making use of equation (6.7.14), one may write li0 ¼ jzi jui0 F ¼

z2i F2 Di0 RT

ð6:7:23Þ

For the whole electrolyte, the limiting molar conductivity is i0 ¼ nþ lþ0 þ n l0 ¼

F2 ðn z2 D þ n z2 D0 Þ RT þ þ þ0

ð6:7:24Þ

where nþ and n are the stiochiometric numbers of the cation and anion, respectively. This is known as the Nernst-Einstein equation. Another important relationship relevant to the mobility of ions may be derived from Stokes’ law. If it is imagined that the ion is a sphere moving through a solvent of viscosity Z at a constant velocity vi , then the force due to the electrical ﬁeld is exactly counterbalanced by the force due to the viscous drag. The electrical force is the ionic charge zi e0 times the electrical ﬁeld E. Thus, using equation (6.3.5), one may write jzi je0 E ¼ 6pZri vi

ð6:7:25Þ

280

LIQUIDS, SOLUTIONS, AND INTERFACES

The ratio of vi to E is simply the ionic mobility ui so that jz je ui ¼ i 0 6pZri In terms of the diﬀusion coeﬃcient, one obtains kT Di ¼ 6pZri

ð6:7:26Þ

ð6:7:27Þ

This relationship is known as the Stokes–Einstein equation. Strictly speaking it should only be applied at inﬁnite dilution to monoatomic ions. However, in practice it is applied to more complex ions and at ﬁnite ionic strengths. If the diﬀusion coeﬃcient for the ion is measured experimentally, an eﬀective radius for the ion can be estimated using the viscosity of the pure solvent. EXAMPLE

Using the data given in the previous example and in table 6.1, estimate the diﬀusion coeﬃcients for Mg2þ + and Cl in 0.5 M MgCl2 . Estimate also their eﬀective Stokes radii. The mobility of Mg2þ is 2:64 104 cm2 s1 V1 . The diﬀusion coeﬃcient is equal to 0:0257 ¼ 3:39 106 cm2 s1 DðMg2þ Þ ¼ 2:64 104 2 Using equation (6.7.27), the Stokes radius for this ion at 258C is kT 4:11 1021 ¼ 6pZDi 6p 8:9 104 3:39 106 104 ¼ 7:24 1010 m ¼ 724 pm

ri ¼

For Cl , the diﬀusion coeﬃcient is DðCl Þ ¼ 5:94 104 0:0257 ¼ 1:527 105 cm2 s1 The Stokes radius is ri ¼

4:11 1021 ¼ 1:60 1010 m ¼ 160 pm 6p 8:9 104 1:527 109

The results for the Stokes radii are reasonable but they should have been obtained using the viscosity of the solution, not that of the pure solvent. Unfortunately, the necessary data are usually not available. At ﬁnite concentrations, the ﬂuxes of the ions of the electrolyte are not completely independent of one another because of ion–ion interactions. This can be seen clearly when this phenomenon is examined within the context of Onsager’s treatment of ion transport. It is assumed in the following that the electrolyte MX contains two univalent ions. The ﬂux of the cation is JM ¼ LMM rm~ M LMX rm~ X

ð6:7:28Þ

where rm~ M and rm~ X are the gradients of the electrochemical potentials of the cation and anion, respectively, LMM , the phenomenological coeﬃcient describing

NON-EQUILIBRIUM PHENOMENA IN LIQUIDS AND SOLUTIONS

281

the ﬂux of the cation resulting from rm~ M and LMX , that describing the ﬂux of the cation resulting from rm~ X . The cross-coeﬃcient LMX is expected to be much smaller than the principal coeﬃcient LMM , but both must be considered in a general treatment. For the anionic ﬂux, the corresponding expression is JX ¼ LXM rm~ M LXX rm~ X

ð6:7:29Þ

where LXX is the principal phenomenological coeﬃcient for the anion, and LXM , the cross-coeﬃcient. Because of the principle of microscopic reversibility, the coeﬃcients LMX and LXM are equal. This means that three phenomenological coeﬃcients must be determined in order to obtain the transport properties of the system. In terms of traditional treatments, the three quantities determined experimentally at a given electrolyte concentration are the diﬀusion coeﬃcient of the electrolyte and the mobilities of each ion. By examining the experiments used to determine each of these quantities on the basis of the Onsager treatment, the relationships among LMM , LMX , LXX , and the traditional parameters can be obtained. In a conductivity experiment, the conductance is determined in the absence of a concentration gradient by using an a.c. potential gradient rfac so that the force on ion i is zi F rfac . Thus, the ionic ﬂuxes in this experiment are JM ¼ zM LMM F rfac zX LMX F rfac

ð6:7:30Þ

Jx ¼ zM LMX F rfac zX LXX F rfac

ð6:7:31Þ

and

Using equation (6.7.2) the a.c. current density for a 1–1 electrolyte is iac ¼ FJM FJX ¼ F2 ðLMM 2LMX þ LXX Þrfac

ð6:7:32Þ

Thus, according to equations (6.7.3) and (6.7.4), the conductivity of the solution is k ¼ F2 ðLMM 2LMX þ LXX Þ

ð6:7:33Þ

Expressions may also be written for the transport numbers tM and tX using the above analysis: tM ¼

FJM LMM LMX ¼ iac LMM 2LMX þ LXX

ð6:7:34Þ

and tX ¼

FJX LXX LMX ¼ iac LMM 2LMX þ LXX

ð6:7:35Þ

As required by the deﬁnition of the transport numbers, tM and tX add to unity. When diﬀusion occurs in an electrolyte solution in the absence of an external electrical ﬁeld, local electrical neutrality must be maintained. When rf is zero, the ionic ﬂuxes are JM ¼ LMM rmM LMX rmX and

ð6:7:36Þ

282

LIQUIDS, SOLUTIONS, AND INTERFACES

JX ¼ LMX rmM LXX rmX

ð6:7:37Þ

The net electrical current in the solution must be zero so that FJM FJX ¼ 0

ð6:7:38Þ

ðLMM LMX ÞrmM ¼ ðLXX LMX ÞrmX

ð6:7:39Þ

It follows that

The gradient of the chemical potential of the electrolyte is given by rme ¼ rm~ M þ rm~ X ¼ rmM þ rmX

ð6:7:40Þ

Combining equations (6.7.39) and (6.7.40), one ﬁnds after some straightforward algebra that rmM ¼

LXX LMX rm LMM 2LMX þ LXX e

ð6:7:41Þ

rmX ¼

LMM LMX rm LMM 2LMX þ LXX e

ð6:7:42Þ

and

Now, using equations (6.7.36) and (6.7.37), one ﬁnds that JM ¼ JX ¼

LMM LXX L2MX rm LMM 2LMX þ LXX e

ð6:7:43Þ

The ﬂux of each ion is also equal to the overall ﬂux of the electrolyte Je when electroneutrality is maintained. The relationship between this quantity and rce is Je ¼ De rce

ð6:7:44Þ

where De is the overall diﬀusion coeﬃcient of the electrolyte. The relationship between rme and rce is rme ¼ RTr ln ae ¼

RT

rc ce e e

ð6:7:45Þ

where

e ¼ 1 þ

r ln ye r ln ce

ð6:7:46Þ

This leads to the ﬁnal result that De ¼

RT LMM LXX L2MX

e ce LMM 2LMX þ LXX

ð6:7:47Þ:

Equations (6.7.33), (6.7.45), and (6.7.34) or (6.7.35) form a system of three equations in three unknowns which may be solved for the individual phenomenological coeﬃcients LMM , LMX , and LXX . The results are LMM ¼

De ce ke t2M þ 2 RT e F

ð6:7:48Þ

NON-EQUILIBRIUM PHENOMENA IN LIQUIDS AND SOLUTIONS

LMX ¼

283

De ce ke tM tX RT e F2

ð6:7:49Þ

De ce ke t2X þ 2 RT e F

ð6:7:50Þ

and LXX ¼

The non-ideality of the electrolyte solution leads to complex relationships between the phenomenological coeﬃcients and the transport parameters measured experimentally. These only become simple in the limit of inﬁnite dilution where LMX goes to zero and e to unity. For example, under these conditions equation (6.7.47) reduces to De0 ¼

RT LMM LXX ce LMM þ LXX

ð6:7:51Þ

Since the diﬀusion coeﬃcients for the individual ions can be deﬁned from LMM and LXX using equation (6.7.21), one may write De0 ¼

DM0 DX0 DM0 þ DX0

ð6:7:52Þ

where DM0 and DX0 are the diﬀusion coeﬃcients for Mþ and X , respectively, at inﬁnite dilution. This equation was ﬁrst derived by Nernst. As can be seen from the above development, study of the transport properties of electrolyte solutions led scientists in the late nineteenth and early twentieth centuries to think about these systems on a microscopic scale. Important connections between the movement of ions under the inﬂuence of thermal and electrical eﬀects were made by Einstein. This was all brought together in an elegant way by Onsager. An important question faced by those involved with these studies is whether the electrolyte is completely dissociated or not. The answer to this question can be found be examining both the equilibrium and non-equilibrium properties of electrolyte solutions. The latter aspect turns out to be more revealing and is discussed in more detail in the following section.

6.8 Experimental Studies of Conductivity The study of electrolyte conductivity dates from the latter half of the nineteenth century when it was realized that simple salts dissociate into ions in aqueous solution. Furthermore, when a current ﬂows in such a solution, the cations move to a large extent independently of the anions, resulting in two separate vectorial components with positive charge moving in one direction and negative charge in the opposite. These studies showed that the molar conductivity varies with electrolyte concentration. In addition, it was possible to distinguish two types of behavior, namely, that characteristic of strong electrolytes, and that characteristic of weak electrolytes. Many of the careful early experiments were carried out in the laboratory of Kohlrausch who showed that the conductivity of solutions of dilute strong elec-

284

LIQUIDS, SOLUTIONS, AND INTERFACES

trolytes was linear in the square root of the electrolyte concentration. The Kohlrausch result may be expressed as ¼ 0 ke c1=2 e

ð6:8:1Þ

where 0 is the molar conductance at inﬁnite dilution and ke , the slope of the relationship between and c1=2 e . Examples of the conductivity behavior of several electrolytes are shown in ﬁg. 6.7. In all cases, the conductivity decreases as the concentration increases. This can be understood as an eﬀect of the ionic environment on the motion of individual ions. Three of the electrolytes shown in ﬁg. 6.7, namely, NaCl, KCl, and sodium acetate (NaC2 H3 O2 ) show the behavior predicted by equation (6.8.1). Thus, the conductivity is linear in the square root of the concentration in the limit of low concentrations so that these systems can be classiﬁed as strong electrolytes. Extrapolation to zero concentration is easily carried out, so that values of 0 can be estimated. In the case of acetic acid, a weak electrolyte, the behavior is quite diﬀerent and the value of 0 is impossible to obtain on the basis of the Kohlrausch plot. The second important parameter obtained directly from experiment is the transport number for one of the ions of the electrolyte. On the basis of the analysis in section 6.7, the cationic transport number for a 1–1 electrolyte can be written as tþ ¼

lþ uþ ¼ l þ þ l u þ þ u

ð6:8:2Þ

where li is the molar conductivity of ion i, and ui , its corresponding mobility. If it is assumed that the individual ionic molar conductivities are also given by an equation similar to equation (6.8.1), then one may write

Fig. 6.7 Plots of the molar conductivity against the square root of the electrolyte concentration for three strong electrolytes (*) and one weak electrolyte (^). The data for the strong electrolytes were ﬁtted to a straight line at lower concentrations in order to obtain the limiting molar conductivity 0 .

NON-EQUILIBRIUM PHENOMENA IN LIQUIDS AND SOLUTIONS

li ¼ li0 1

ki c1=2 e li0

!

285

ð6:8:3Þ

where ki is the constant describing the concentration dependence of li for ion i. Substituting this expression for each molar conductivity into (6.8.2), the following expression is obtained for the concentration dependence of tþ at low concentrations: tþ ¼ tþ0 þ kt c1=2 e

ð6:8:4Þ

tþ0 is the transport number in the limit of inﬁnite dilution, and kt , the constant describing the concentration dependence of tþ . The sign of kt can be positive or negative depending on the relative values of the corresponding coeﬃcients for the constituent ions, kþ and k . Since the transport numbers tþ and t add to unity, the corresponding equation for t is t ¼ t0 kt c1=2 e

ð6:8:5Þ

Typical transport number data for 1–1 electrolytes are shown as a function of c1=2 e in ﬁg. 6.8. It is clear that these data can easily be extrapolated to obtain the value of the transport number at inﬁnite dilution. Having obtained the limiting conductance for an electrolyte and the limiting transport number for one of its ions, one may calculate limiting molar conductivities for the individual ions. Examination of experimental data for strong electrolytes shows that lþ0 for a single cation, say Naþ , is independent of the nature of the anion. Similarly, for a series of electrolytes with a common anion, say Cl , the value of l0 is independent of the nature of the cation. Thus, by selecting contributions from the appropriate ions one can calculate the limiting ionic conductance for any electrolyte whose constituent ions are listed in table 6.2. For

Fig. 6.8 Plots of the cationic transport number 258C against the square root of the electrolyte concentration for three strong electrolytes in water.

286

LIQUIDS, SOLUTIONS, AND INTERFACES

Table 6.2 Limiting Molar Conductivities of Single Ions in Water at 258C [7]* Cation

li0 =cm2 mol1 1

Hþ Liþ Naþ Kþ Rbþ Csþ Agþ NHþ 4 Mg2þ Ca2þ Sr2þ Ba2þ Cd2þ Zn2þ Ni2þ Pb2þ

349.8 38.68 50.10 73.50 77.81 77.26 61.90 73.55 106.1 119.0 118.9 127.3 108.0 107.0 108.0 140.0

Anions OH F Cl Br I NO 3 ClO 4 C2 H3 O 2 N3 CN SCN SO2 4 CO2 3 C2 O24 Fe(CN)3 6 Fe(CN)4 6

li0 =cm2 mol1 1 198.3 55.4 76.35 78.14 76.84 71.46 67.36 40.90 69.5 78.0 66.5 160.04 138.6 148.3 302.7 442.0

*In the older literature, the limiting conductance is usually given per equivalent of the ion rather than per mole as cited here. The equivalent conductance is equal to the molar conductance divided by the number of charges on the ion. Thus, the equivalent conductance for the divalent cations listed above is equal to the molar conductance divided by two.

example, in the case of Zn(ClO4 )2 , there is a contribution of 107.0 cm2 mol1 2 from the Zn2þ ion, and a contribution of 134.7 cm2 mol1 1 from the two perchlorate ions. These add to a value of 0 equal to 241.7 cm2 mol1 1 for the electrolyte. The same data can be used to estimate the limiting ionic conductivity for a weak electrolyte. In the case of acetic acid the result is 390.7 cm2 mol1 1 . Examination of the data for this system presented in ﬁg. 6.7 shows that the value of changes markedly at low concentrations, so that 0 clearly could not be estimated on the basis of the Kohlrausch plot. The limiting molar conductance is a fundamental property of ions in electrolyte solutions. It can be used to estimate the ionic mobility (equation (6.7.14)) and the diﬀusion coeﬃcient of the ion (equation (6.7.22)) in the limit of inﬁnite dilution. When data are available for ions in a given series, trends can be seen in the values of li0 . In the case of the alkali metal cations, lþ0 increases with increase in cation atomic number up to Rbþ , the values for Rbþ and Csþ being approximately the same. A similar trend is seen for the halide ions for which l0 increases in the series F < Cl < Br , and then decreases slightly for I . These results are usually interpreted in terms of the extent to which the ion is solvated in water. Since Liþ is the smallest crystallographic radius in the alkali metal series, it is the most strongly hydrated and carries some surrounding water molecules when it moves. On the other hand, the large Csþ ion is only weakly hydrated, and therefore, it can move more rapidly through the electrolyte solution. The concepts that ions move independently and that the individual ionic limiting conductivities can be used to estimate the limiting conductivity of an electro-

NON-EQUILIBRIUM PHENOMENA IN LIQUIDS AND SOLUTIONS

287

lyte were major achievements in developing an understanding of the properties of electrolyte solutions. However, many studies were also made with weak electrolytes which are only partially dissociated. The concentration behavior of these systems, for example, acetic acid (see ﬁg. 6.7), is clearly diﬀerent. When a simple 1–1 electrolyte is only partially dissociated one has the equilibrium MX Ð Mþ þ X

ð6:8:6Þ

whose equilibrium constant is Ka ¼

aM aX y2 a2 ce ¼ aMX yMX ð1 aÞ

ð6:8:7Þ

ai is the activity of species i, yi , the corresponding activity coeﬃcient, ce , the concentration of the weak electrolyte and a, the fraction of the electrolyte dissociated. Deﬁning Kc as the equilibrium constant in terms of concentration, one may write Kc ¼

yMX Ka a 2 ce ¼ 1a y2

ð6:8:8Þ

Kc will be constant only in the limit of very low concentrations where the activity coeﬃcients are unity. The second assumption regarding weak electrolytes is that the fraction dissociated can be directly obtained from the ionic conductivity, that is, a¼

0

ð6:8:9Þ

Combining equations (6.8.8) and (6.8.9), and writing the result in a linear form, one obtains 1 1 c ¼ þ e 0 Kc 20

ð6:8:10Þ

This is the Ostwald dilution law for weak electrolytes. Plots of data for two weak electrolytes, namely acetic acid and ammonium hydroxide, according to Ostwald’s law are shown in ﬁg. 6.9. These plots give reasonable straight lines with somewhat more scatter in the case of NH4OH. The value of 0 for acetic acid from the intercept is 388.5 cm2 mol1 1 , which agrees quite well with the value calculated from the limiting ionic conductivities in table 6.2 (390.7 cm2 mol1 1 ). The value of Kc estimated from the slope is 7:8 105 M, which is close to the value found in tabulations of acidity constants. In the case of NH4OH, the estimate of 0 from the intercept is 115.9 cm2 mol1 1 , which is very diﬀerent from the value obtained from limiting conductances (271.4 cm2 mol1 1 ). The estimate of Kc from the slope is 7:8 105 M compares with 1:8 105 M from basicity constant tabulations. These calculations show the weakness of the Ostwald analysis. Since 0 is much larger than the measured conductances, the error associated with estimating it from its reciprocal in the plots shown is very large. Any error in 0 is then reﬂected in the estimate of the Kc which comes from the reciprocal of the slope, equal to Kc 20 . An additional problem comes from the neglect of the activity coeﬃcients involved

288

LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 6.9 Plot of 1/ at 258C against ce for two weak electrolytes, namely, acetic acid (^) and NH4 OH (*).

in the thermodynamic association constant Ka . Robinson and Stokes [7] have described an iterative technique for extracting values of 0 and Ka , which includes consideration of activity coeﬃcients for the constituent ions on the basis of the Debye–Hu¨ckel law. This technique is useful when highly precise conductivity data are available. Many studies of electrolyte conductivity have been carried out [7]. This work certainly helped to conﬁrm modern ideas about electrolyte solutions. One aspect of the behavior of strong electrolytes which was initially not well understood is the fact that their molar conductance decreases with increase in concentration. Although this is now attributed to ion–ion interactions, early work by Arrhenius [8] ascribed the decrease in all electrolytes to partial dissociation. However, it is clear from the vast body of experimental data that one can distinguish two types of behavior for these systems, namely, that for strong electrolytes and that for weak electrolytes, as has been illustrated here. The theory of the concentration dependence of the molar conductance of strong electrolytes was developed earlier this century and is discussed in detail in the following section.

6.9 The Debye–Onsager Model for Conductivity In chapter 3, it was shown that the Debye–Hu¨ckel theory for ion–ion interactions is able to account for solution non-ideality in very dilute systems. The same model forms the basis for understanding the concentration dependence of the conductance observed for strong electrolytes. Thus, Onsager [9] showed in 1927 that the limiting conductance law for 1–1 electrolytes has the form ¼ 0 ðB1 0 þ B2 Þc1=2 e

ð6:9:1Þ

NON-EQUILIBRIUM PHENOMENA IN LIQUIDS AND SOLUTIONS

289

where B1 and B2 are constants whose signiﬁcance is discussed below. The success of the Debye–Onsager theory is that it is able to account quantitatively for the behavior observed empirically by Kohlrausch at low concentrations (equation (6.8.1)). The inﬂuence of the interionic forces is due to two phenomena, namely, the electrophoretic eﬀect and the time-of-relaxation eﬀect. The net ionic atmosphere around a given ion carries the opposite charge and therefore moves in a direction opposite to the central ion. The ﬁnal result is an increase in the local viscosity, and retardation of the central ion. This is called the electrophoretic eﬀect. The time-ofrelaxation eﬀect is also related to the fact that the ionic atmosphere around a given ion is moving and therefore disrupted from its equilibrium conﬁguration. It follows that the ionic atmosphere must constantly be re-formed from new counter ions as the ion under observation moves through the solution. The net eﬀect is that the electrical force on each ion is reduced so that the net forward velocity is smaller. The derivation of an expression for the electrophoretic eﬀect is based on the Debye–Hu¨ckel theory described earlier in section 3.8. On the basis of the Boltzmann distribution law, the local concentration of ion i at a distance r from a central reference ion j is cri ¼ c i expðzi fÞ

ð6:9:2Þ

c i

where is the average concentration of ion i, zi , its valence, and , the local potential. Expanding the exponential as an inﬁnite series in zi f, the following expression is obtained for the deviation of the local concentration from its average value: cri c i ¼ c i

1 X ð1Þn n¼1

n!

ðzi fÞn

ð6:9:3Þ

In keeping with the Debye–Hu¨ckel theory, it is assumed that the electrostatic energy of an individual ion is small with respect to kB T so that only the ﬁrst term in the expansion need be considered; as a result cri c i ¼ zi c i f

ð6:9:4Þ

If fi is the force acting per mole on ion i in a spherical shell which is at a distance r from the reference ion, and the shell has a thickness dr , then the incremental force on the ions of type i in the shell is dFi ¼ 4pzi c i fr2 fi dr

ð6:9:5Þ

For a binary electrolyte with one type of cation and anion, the incremental force is dF ¼ 4pfr2 ½zþ c þ fþ þ z c f dr

ð6:9:6Þ

Using Stokes’ law (equation (6.3.5)) this can be related to an incremental change in the velocity of ion j. Thus, dvj ¼

2 f r½z c f þ z c f dr 3Z j þ þ þ

ð6:9:7Þ

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LIQUIDS, SOLUTIONS, AND INTERFACES

where j is the local potential due to ion j and Z is the viscosity of the solution. The net change in velocity is found by integrating this expression from the boundary of ion j at r ¼ a to inﬁnity. Thus, 1 ð

vj ¼

2fj ½z c f þ z c f r dr 3Z þ þ þ

ð6:9:8Þ

a

The expression obtained for j in Debye–Hu¨ckel theory (equation (3.8.26)) is ka zj e0 ekr e j ¼ ð6:9:9Þ 4p"0 "s r 1 þ ka where k is the Debye–Hu¨ckel reciprocal distance (equation (3.8.13)). For simplicity the derivation is restricted here to symmetrical zz electrolytes. Substituting equation (6.9.9) into equation (6.9.8) for the case that the central ion is a cation, 1 ka ð z2 fe0 c e e ðf f Þ ekr dr vþ ¼ 6pe0 es Z 1 þ ka þ

ð6:9:10Þ

a

or, after integration, vþ ¼

z2 fe0 c e ðfþ f Þ 6pe0 es Z kð1 þ kaÞ

ð6:9:11Þ

For a symmetrical zz electrolyte the ionic strength is equal to z2 c e . Recalling that the Debye–Hu¨ckel reciprocal distance k is given by 2NL e0 f I 1=2 k¼ ð6:9:12Þ e0 es the equation for the electrophoretic eﬀect on the velocity of a cation may be further simpliﬁed to obtain vþ ¼

kðfþ f Þ 12pNL Zð1 þ kaÞ

ð6:9:13Þ

The corresponding equation for the case that the central ion is an anion is v ¼

kðf fþ Þ 12pNL Zð1 þ kaÞ

ð6:9:14Þ

The forces fþ and f are given by the product of the charge per mole of ions and the local ﬁeld. The latter is equal to the external applied ﬁeld E plus the relaxation ﬁeld E. The relaxation ﬁeld arises because of the asymmetry in the ionic atmosphere caused by the motion of the ion with respect to its atmosphere. Thus, for the cation fþ ¼ NL z þ e0 ðE þ EÞ ¼ zþ FðE þ EÞ

ð6:9:15Þ

f ¼ z FðE þ EÞ

ð6:9:16Þ

and for the anion

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NON-EQUILIBRIUM PHENOMENA IN LIQUIDS AND SOLUTIONS

Combining these results with equations (6.9.13) and (6.9.14), one obtains vþ ¼ v ¼

ze0 kðE þ EÞ 6pZð1 þ kaÞ

ð6:9:17Þ

This result may be interpreted as showing that the ionic atmosphere has a charge z, opposite to that of the ion on which attention is focused. Furthermore, the atmosphere has an eﬀective radius equal to a þ 1=k. The motion of the atmosphere is in the opposite direction to that of the ion whose velocity is given by Stokes’ law. The theory of the relaxation eﬀect makes use of the equations of hydrodynamics applied with consideration of the eﬀects of interionic forces. It is the most diﬃcult part of the description of non-equilibrium processes in electrolytes, and the details of the derivation are not given here. The result obtained for the relative value of the relaxation ﬁeld is E jzþ z je20 q k ¼ E 12pe0 es kB T ð1 þ q1=2 Þ ð1 þ k0 Þ

ð6:9:18Þ

where q¼

jzþ z j lþ0 þ l0 ðzþ þ jz jÞ ðzþ l0 þ jz jlþ0 Þ

ð6:9:19Þ

In the case of symmetrical electrolytes for which zþ ¼ jz j, the parameter q is equal to 1/2. The limiting ionic mobility for ion i, ui 0 is reduced by an amount ui (relx) as a result of the ﬁeld relaxation eﬀect. Thus, ignoring the electrophoretic eﬀect one may write E ui ¼ ui0 1 þ ð6:9:20Þ E where the term in E=E gives the decrement in ui due to the ﬁeld relaxation eﬀect. From equation (6.9.17), the decrement in ui due to the electrophoretic eﬀect is vi zi e0 k E 1þ ¼ ui ðelphÞ ¼ ð6:9:21Þ 6pZð1 þ kaÞ E E Combining equations (6.9.20) and (6.9.21), one obtains

zi e0 k E 1þ ui ¼ ui0 6pZð1 þ kaÞ E

ð6:9:22Þ

The expression for E=E (equation (6.9.18)) is now substituted into equation (6.9.22) with the result that ! zi e0 jzþ z je20 ui0 q k ð6:9:23Þ þ ui ¼ ui0 6pZ 12p"0 "s kB T ð1 þ q1=2 Þ 1 þ ka where the second-order term in E=E has been neglected.

292

LIQUIDS, SOLUTIONS, AND INTERFACES

Table 6.3. Values of the Debye-Onsager Constants B1 and B2 for 1–1 Electrolytes Together with the Viscosity of Water in the Temperature Range 0–508C T/8C

Z=mPa s

B1 =L1=2 mol1=2

B2 =cm2 L1=2 1 mol3=2

0 10 20 25 30 40 50

1.792 1.307 1.002 0.890 0.798 0.653 0.547

0.2208 0.2236 0.2272 0.2292 0.2315 0.2362 0.2416

29.71 40.91 53.65 60.57 67.78 83.39 100.3

The ﬁnal result is obtained by multiplying each mobility ui by the Faraday to obtain the individual ionic equivalent conductances li . The values of li for the ions in the binary electrolyte are added so that I1=2 ¼ Fðuþ þ u Þ ¼ 0 ðB1 0 þ B2 Þ ð6:9:24Þ 1 þ BDH aI1=2 where !1=2 jzþ z je20 q 2NL e20 B1 ¼ ð6:9:25Þ 12p"0 "s kB T ð1 þ q1=2 Þ "0 "s kB T and !1=2 NL ðzþ jz jÞe20 2NL e20 B2 ¼ 6pZ "0 "s k B T

ð6:9:26Þ

The Debye–Onsager constants B1 and B2 depend explicitly on temperature, and also on the solvent parameters es and Z, which themselves are temperature dependent. If a factor of 1000 is included in the terms under the square root sign so that the ionic strength may be expressed in moles per liter, then B1 and B2 may be written as B1 ¼

jzþ z je20 q B 12p"0 "s kB T ð1 þ q1=2 Þ DH

ð6:9:27Þ

NL ðzþ þ jz jÞe20 BDH 6pZ

ð6:9:28Þ

and B2 ¼

where BDH is the Debye–Hu¨ckel constant deﬁned in equation (3.8.34). Values of B1 and B2 for water together with its viscosity in the temperature range from 0 to 508C are given in table 6.3. EXAMPLE

Estimate the molar conductivity of 0.1 M HCl at 258C given that its limiting molar conductivity is 426.2 cm2 1 mol1 and assuming an ionic size parameter a of 0.45 nm.

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293

The calculation is based on equation (6.9.24) and requires estimation of the Debye–Hu¨ckel factor I 1=2 =ð1 þ BDH a I 1=2 ), where the value of BDH is given in table 3.8; the ionic strength for this system is equal to its concentration so that I1=2 0:316 ¼ 0:2153 mol 1=2 L1=2 ¼ 1 þ BDH aI1=2 ð1 þ 3:288 0:45 0:316Þ

ð6:9:29Þ

The estimate of is ¼ 426:2 ð0:2292 426:2 þ 60:57Þ 0:2153 ¼ 392:1

ð6:9:30Þ

Thus, the molar conductance is 392.1 cm2 1 mol1 . Application of the Debye–Onsager model to conductivity data for NaCl in water is illustrated in ﬁg. 6.10. When the size parameter a is set equal to 0.4 nm, an excellent ﬁt of theory to the experimental data is obtained for concentrations up to 0.5 M (I1=2 ¼ 0:7 M1=2 ). At higher concentrations the experimental data fall below the theoretical curve indicating that the model does not describe the decrease in with concentration properly in this range. A variety of reasons can be cited for this failure and many of them have been discussed with respect to the Debye–Hu¨ckel theory for the equilibrium properties of electrolyte solutions. They include the failure to describe properly the thickness of the ionic atmosphere and the strength of ion–ion interactions at high ionic strength. Attempts have been made to extend the concentration range over which equation (6.9.24) is valid by adding extra terms in the concentration. The interested reader is referred to more extensive discussions of this subject [7, 10] for further details. In the limit of very dilute solutions where ion size eﬀects may be neglected, equation (6.9.24) becomes

Fig. 6.10 Plot of the equivalent conductance of NaCl in water according to equation (6.9.1) against the square root of the ionic strength at 258C. The solid curve shows the prediction of the Debye–Onsager equation with a ¼ 0:4 nm, and the straight line, the prediction of this model in the limit that ion size eﬀects may be neglected.

294

LIQUIDS, SOLUTIONS, AND INTERFACES

¼ 0 ðB1 0 þ B2 ÞI1=2

ð6:9:31Þ

This is identical with equation (6.9.1) in the case of 1–1 electrolytes. The predictions of the limiting law for the NaCl system are also shown in ﬁg. 6.10. It is valid for concentrations up to 0.01 M. The success of the theory is clear from this result. First of all, it conﬁrms that plots of against the square root of ionic strength provide a valid route for determining 0 , the equivalent conductance in the limit of inﬁnite dilution. In addition, it explains why the slope of the plot in the dilute solutions regime depends on the nature of the electrolyte. The above theory can also be applied to account for the concentration dependence of transport numbers, especially in dilute solutions. Since the transport number can be deﬁned as a ratio of the equivalent conductance of the given ion to the total ionic conductance (equation (6.7.6)), it is clear that a non-linear relationship can be derived describing the concentration dependence using equations (6.9.23) and (6.9.24). In summary, Onsager’s extension of the Debye–Hu¨ckel theory to the nonequilibrium properties of electrolyte solutions provides a valuable tool for deriving single ion properties in electrolyte solutions. Examination of the large body of experimental data for aqueous electrolyte solutions helped conﬁrm the model for a strong electrolyte. In more recent years, these studies have been extended to non-aqueous solutions. Results in these systems are discussed in the following section.

6.10 Transport Phenomena in Non-Aqueous Solutions In the development of the theory of ionic conductance it has been shown that the viscosity of the solvent is an important parameter determining ionic mobility. Initially, conductivity data were only available in water so that attention was focused on the eﬀects of ionic size, structure, and charge in determining mobility and its concentration dependence. More recently, data have become available in a wide variety of non-aqueous solvents [11, 12], that is, in media with a wide range of permittivities and viscosities. On the basis of these data one may examine in more detail the role of solvent viscosity in determining the transport properties of single ions. Values of the limiting ionic molar conductance for selected monovalent cations and anions are summarized in tables 6.4 and 6.5, respectively. By combining the Stokes–Einstein equation (equation (6.7.27)) with equation (6.7.23), the following expression for the limiting ionic molar conductance is obtained: li0 ¼

z2i F2 6pNL ZrST

ð6:10:1Þ

where Z is the viscosity of the pure solvent and rST , the eﬀective Stokes radius of the ion. Rearranging, one obtains the following result for the Stokes radius: rST ¼

z2i F2 6pNL Zli0

ð6:10:2Þ

NON-EQUILIBRIUM PHENOMENA IN LIQUIDS AND SOLUTIONS

295

Table 6.4 Limiting Ionic Conductances for Monovalent Cations in Various Solvents at 258C li0 =cm2 1 mol1 Solvent

Liþ Naþ Kþ

Csþ TEAþ TBAþ

38.7 39.6 17.1 8.8 8.3 14.7

50.1 45.2 20.5 10.5 9.9 18.3

73.5 52.4 23.3 12.1 12.4 19.0

77.3 69.0 26.4 13.8 13.5 21.2

32.7 60.5 29.2 15.1 10.4 21.2

19.5 38.9 19.6 10.9 6.6 —

72.1 69.5 — 20.3 26.1 11.5 5.3 13.9 14.5 53.9 8.3 4.3 14.5

74.8 76.8 19.6 25.8 30.0 14.1 5.5 14.6 15.9 56.7 9.3 3.6 16.1

78.9 85.1 23.5 25.2 31.6 14.7 5.7 15.2 17.4 59.7 11.2 4.0 15.7

84.4 87.3 — 27.9 35.4 15.6 6.2 — — — 12.3 4.3 16.9

91.4 85.9 24.7 32.9 36.2 17.3 9.1 18.3 16.1 48.1 13.2 3.9 22.1

67.3 62.4 17.2 22.8 26.6 11.4 6.1 — 11.5 34.7 9.0 2.8 15.0

Protic 1. 2. 3. 4. 6. 7.

W MeOH EtOH PrOH F NMF

Aprotic 8. 9. 10. 12. 13. 14. 15. 16. 17. 18. 20. 21. 22.

AC AcN BzN DMA DMF DMSO HMPA NMP NB NM PC TMS TMU

The product li0 Z is known as the Walden product and is the focus of the discussion in the present section. To a ﬁrst approximation, li0 Z should be independent of solvent nature if the ion moves in the solution without any accompanying solvent molecules. However, some variation in li0 Z is seen on the basis of an analysis of the data in tables 6.4 and 6.5 using the viscosities reported in table 6.1[13]. Plots of the Stokes radius for the Naþ and TEAþ cations estimated using equation (6.10.2) in 18 diﬀerent solvents are shown as a function of the eﬀective radius of the solvent molecule, rs , in ﬁg. 6.11. Quite a large change in solvent size is involved for the solvents considered, from the very small water molecule with a radius of 137 pm to the very large hexamethylphosphoramide molecule with a radius of 394 pm. The results are quite striking: the Stokes radius of the TEAþ cation is approximately constant and equal to 270 pm, whereas the Stokes radius of the Naþ cation clearly increases with solvent size. The line shown for the Naþ data in this ﬁgure was drawn with unit slope and with an intercept on the y-axis of 102 pm, that is, the crystallographic radius of the Naþ ion. Although there is considerable scatter about the line, it is clear that the most important factor determining the size of the moving ion is the size of the solvent molecules

Table 6.5 Limiting Ionic Conductances for Monovalent Anions in Various Solvents at 258C li0 =cm2 1 mol1 Solvent

Cl

Br

I

76.4 52.1 22.0 10.4 17.3 53.8

78.1 56.4 24.1 12.2 17.5 53.4

76.8 62.6 27.2 13.4 16.8 51.1

ClO 4 BPh4

Protic 1. 2. 3. 4. 6. 7.

W MeOH EtOH PrOH F NMF

67.4 70.6 31.8 16.3 16.7 51.6

— 36.5 20.0 10.7 6.3 —

AC 105.1 115.9 113.0 115.5 AcN 99.2 100.7 102.5 103.7 DMA 45.9 43.2 41.8 42.8 DMF 53.8 53.4 51.1 51.6 DMSO 24.2 23.8 23.6 24.0 HMPA 19.7 18.9 17.2 16.0 NMP 26.5 28.2 27.0 27.2 NB 22.4 22.0 21.3 21.5 NM 61.9 62.3 63.1 65.3 PC 20.2 18.9 18.5 18.6 TMS 9.3 8.9 7.2 6.7 TMU — 30.1 28.8 28.4

66.0 58.0 — — 10.7 6.1 12.2 10.8 32.5 8.4 — —

Aprotic 8. 9. 12. 13. 14. 15. 16. 17. 18. 20. 21. 22.

Fig. 6.11 Plots of the Stokes radii rST for Naþ and TEAþ estimated using equation (6.10.2) against the eﬀective solvent radius rs . The data for TEAþ have been shifted vertically by 300 pm for the sake of clarity.

NON-EQUILIBRIUM PHENOMENA IN LIQUIDS AND SOLUTIONS

297

which accompany it. The scatter can be attributed to several factors, one of them being the obvious fact that not all solvent molecules are perfectly spherical. However, other factors also play a role. For example, the three solvents EtOH, AcN, and NM have radii which are close to 216 pm. The Stokes radius for Naþ in NM is smallest (237 pm), that in AcN intermediate (313 pm), and that in EtOH largest (371 pm). This follows the increase in solvent donicity. Since NM is a very weak Lewis base, the Naþ ion holds fewer solvent molecules on the average when it moves, and therefore has a smaller radius. On the other hand, EtOH is a stronger Lewis base so that the moving ion has a larger radius reﬂecting the fact that more solvent molecules move with it. TEAþ is a much larger ion than Naþ with a tetrahedral structure and an approximate radius of 337 pm. For this cation, the Stokes radius is independent of solvent nature with an average value equal to 270 pm. This result indicates the eﬀective size of the ion when it moves as an unsolvated species. The results for the Stokes radii of anions are more complex and are considered separately for protic and aprotic solvents. In the case of aprotic solvents, the value of rST decreases in the aprotic solvents with increase in solvent radius (ﬁg. 6.12). Since the interaction between the anion and solvent is weak, the number of solvent molecules which move with the ion decreases with increase in solvent size. This trend is weaker for the larger ClO 4 ion than for the smaller Cl ion, demonstrating that the Cl anion is more strongly solvated in solvents of the smallest molecular size. In the case of protic solvents, hydrogen bonding is involved in anion solvation. Data are available in only a few solvents so that a detailed analysis is not possible at present. The results presented here demonstrate clearly that the solvent radius is the important parameter to be considered in discussing the solvent dependence of the

Fig. 6.12 Plots of the Stokes radii for Cl and ClO 4 ions in aprotic solvents estimated using equation (6.10.2) against the eﬀective solvent radius rs . The data for ClO 4 have shifted vertically by 300 pm for the sake of clarity.

298

LIQUIDS, SOLUTIONS, AND INTERFACES

Stokes radius or of the Walden product. Ion size and structure both determine the extent to which the ion moves with associated solvent molecules.

6.11 Proton Transport Phenomena The proton is formed from the lightest element in the periodic table and has the smallest ionic size. It also has a special relationship to the most common polar solvent, namely, water. These two facts combine to give this ion the highest mobility in water, approximately 10 times that of the Liþ ion, which is the next larger monoatomic cation. In order to understand these observations one must develop a picture of how the proton is incorporated in the structure of protic solvents. Using the analysis of the thermodynamic properties of monoatomic ions in water presented earlier (section 3.5), the Gibbs energy of solvation of the proton at inﬁnite dilution and 258C is 1104 kJ mol1 . On the basis of the MSA, using the parameters for monoatomic cations given in table 3.5, the corresponding radius of the Hþ ion is 13 pm. Thus, the proton radius is close to one-sixth of the radius of the Liþ ion (74 pm). This in turn means that the electrical ﬁeld due to the Hþ ion is approximately six times larger than that due to the Liþ ion. As a result, the interaction between the proton and surrounding solvent dipoles is extremely large compared to ions of more usual dimensions, that is, with radii greater than 100 pm. In the case of water, a covalent bond is formed by interaction of the proton with the lone pair electron density on an adjacent water molecule. The existence of the hydronium ion, H3 Oþ , was demonstrated using infrared spectroscopy by Falk and Gigue`re [14]. It has also been observed in the solid state in compounds such as HClO4 H2 O using X-ray crystallography. In dilute aqueous solutions at equilibrium, the hydronium ion is solvated by other water molecules. Considering the structure of water, the most probable local species corresponds to H9 Oþ 4. The mechanism by which the hydronium ion moves in water is very diﬀerent than that by which most other ions move. As shown in ﬁg. 6.13, the conductance process involves the transfer of a proton between two adjacent water molecules with the proton in a hydrogen bond between these molecules. Because of the relative orientation of the water molecules with respect to one another, the net motion of the proton is not exactly in the direction of the electrical ﬁeld but rather in zig-zag motion about this direction. This feature of the net forward motion can also be described in terms of rotation of the water molecule to which the proton is transferred so that the orbitals on this molecule are favorably oriented with respect to the direction of the ﬁeld. The net eﬀect is that the value of l0 at 258C is 349.8 cm2 1 mol1 , that is, much higher than any other monovalent ion. The hydroxyl ion moves by a similar mechanism, which involves proton transfer in the opposite direction of the electrical ﬁeld. The molar conductance of the OH ion is smaller (197.6 cm2 1 mol1 ) simply because this ion occupies more space in the water structure. In addition, the ﬁeld due to OH is much less than that due to Hþ because of the much larger size of the former ion. Anomalous proton transfer also occurs in the lower alcohols and their mixtures with water.

NON-EQUILIBRIUM PHENOMENA IN LIQUIDS AND SOLUTIONS

299

Fig. 6.13 Schematic diagram illustrating the mechanism for proton and hydroxyl ion conductance in water.

This also reﬂects the fact that proton transfer in a solvent such as methanol involves the species CH3 OHþ 2 . However, in the alcohols there is only one site for hydrogen bond formation, whereas the water molecule has two. Thus orientational eﬀects are more important in acidiﬁed alcohol systems, so that the enhancement of proton mobility with respect to other monovalent cations is not as great as it is in water. An important question regarding proton transfer in water is whether it occurs by a classical mechanism or by quantum-mechanical tunneling. This problem can be elucidated by comparing the rates of proton and deuteron transfer. In this way it was concluded that transfer occurs by both mechanisms [15]. Another way of examining the mechanism of proton transfer is by comparing its temperature dependence with that of other monoatomic cations. Plots of the molar conductance of three cations, namely, Hþ , Liþ , and Kþ , on a logarithmic scale are shown as a function of reciprocal temperature in the range 5–558C in ﬁg. 6.14. Excellent linear plots are found from which an energy barrier associated with the process may be calculated. Using a simple Arrhenius expression, the temperature dependence of li for a small temperature interval is given by li ¼ Al exp

Ul RT

ð6:10:3Þ

where Al is the pre-exponential factor, and Ul , the activation energy associated with the conductance process. From the data presented in ﬁg. 6.14, the value of Ul is 10.0 kJ mol1 for the proton; on the other hand, it is 16.3 kJ mol1 for Liþ , and 14.2 kJ mol1 for Kþ . This type of analysis certainly oversimpliﬁes the physical picture of proton transport because several mechanisms are involved. However, it clearly shows that the net eﬀect for the proton is that the average activation barrier is much less than that for other cations.

300

LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 6.14 Plot of the molar conductance for Hþ , Liþ , and Kþ ions against the reciprocal temperature in the range 5–558C.

The study of proton transfer in solutions is a rich area of research [16] which has only been brieﬂy introduced by the present discussion. It is obviously important not only in conduction processes but also in acid–base processes in solution. More information about these reactions is given in chapter 7.

6.12 Concluding Remarks The discussion in this chapter has emphasized the Onsager approach to non-equilibrium phenomena. This provides an excellent background for understanding the way in which the ﬂuxes observed experimentally often depend on several forces in the system. Early experimental work dealt mainly with electrolyte solutions. Conductivity and diﬀusion measurements in these systems were very important in developing a microscopic picture of the structure of electrolyte solutions. Electrolyte solutions are non-ideal, and the non-ideality is reﬂected in non-equilibrium processes by the concentration dependence of the phenomenological coeﬃcients. The work by Debye and Onsager has shown that the concepts developed to understand non-ideality in electrolyte solutions at equilibrium are applicable to the same systems, when they are not at equilibrium. The Debye–Onsager theory described here is currently being extended to include a more careful description of ionic atmosphere eﬀects on the basis of the MSA which was introduced in chapter 3. Mass transfer and heat transfer are important subjects in engineering. Whole monographs have been devoted to solutions of the pertinent diﬀerential equations for a variety of boundary conditions [G2, G3]. Only one example is worked out in this chapter to give an idea of what is involved mathematically. Mass transfer problems make up an important part of electroanalytical chemistry. In a typical experiment the current which ﬂows in the electrochemical cell which is not at

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301

equilibrium is used to determine the amount of analyte. This species reacts at the working electrode and the observed current depends not only on the concentration of the analyte but also on the mass transfer conditions by which it reaches the reaction site at the electrode. Much more can be found about this subject and the associated mass transfer problems in monographs devoted to electroanalysis [G4]. General References G1. Katchalsky, A.; Curran, P. F. Non-Equilibrium Thermodynamics in Biophysics; Harvard University Press: Cambridge, MA, 1967. G2. Jost, W. Diﬀusion in Solids, Liquids, Gases; Academic Press: New York, 1960. G3. Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; Wiley: New York, 1960. G4. Galus, Z. Fundamentals of Electrochemical Analysis; Ellis Horwood: New York, 1994.

References 1. Onsager, L. Phys. Rev. 1931, 37, 405; 1931, 38, 2265. 2. de Groot, S. R. Thermodynamics of Irreversible Processes; North-Holland: Amsterdam, 1952. 3. McQuarrie, D. A. Statistical Mechanics; Harper and Row; New York, 1976. 4. Riddick, J. A.; Bunger, W. B.; Sakano, T. K. Organic Solvents, 4th ed.; Wiley: New York, 1986. 5. Debye, P. Polar Molecules; Chemical Catalogue Co.: New York, 1929. 6. Parsons, R. Modern Aspects of Electrochemistry; Bockris, J. O’M., Conway, B. E., Eds.; Butterworths: London, 1954; Chapter 3. 7. Robinson, R. A.; Stokes, R. H. Electrolyte Solutions, 2nd ed; Butterworths: London, 1959. 8. Arrhenius, S. Z. Phys. Chem. 1887, 1, 631. 9. Onsager, L. Phys. Z. 1927, 28, 277. 10. Justice, J. C. In Comprehensive Treatise of Electrochemistry; Conway, B. E., Bockris, J. O’M., Yeager, E., eds.; Plenum Press: New York, 1983; Vol. 5, Chapter 3. 11. Karapetyan, Yu. A.; Eychis, V. N. Physico-chemical Properties of Non-Aqueous Electrolyte Solutions (in Russian); Khimia: Moscow, 1989. 12. Barthel, J.; Neueder, R.; Schro¨der, P. Electrolyte Data Collection; Dechema: Frankfurt, 1992; Vol. XII, Part 1; 1993; Vol. XII, Part 1a; 1994; Vol. XII, Part 1b; 1996; Vol. XII, Part 1c. 13. Fawcett, W. R. Mol. Phys. 1998, 95, 507. 14. Falk, M.; Gigue`re, P. Can. J. Chem. 1957, 35, 1195. 15. Conway, B. E.; Bockris, J. O’M., Linton, H. J. Chem. Phys. 1956, 24, 834. 16. Lengyel, S.; Conway, B. E. In Comprehensive Treatise of Electrochemistry; Conway, B. E., Bockris, J. O’M., Yeager, E., Eds.; Plenum Press: New York, 1983; Vol. 5, Chapter 5.

Problems 1. The viscosity of DMSO is 1.996 mPa s at 258C. Estimate its Debye relaxation time assuming a molecular diameter equal to 491 pm. Compare your answer with the experimental value, which is 18.9 ps. 2. Fick’s law for diﬀusion to a sphere of radius r0 is written

302

LIQUIDS, SOLUTIONS, AND INTERFACES

" # @cðr; tÞ @2 cðr; tÞ 2 @cðr; tÞ ¼D þ @t r @r @r2

Solve this expression for cðr; tÞ with the conditions cðr; 0Þ ¼ c ; cðr0 ; tÞ ¼ 0ðt > 0Þ; lim cðr; tÞ ¼ c ðr ! 1Þ Show that the ﬂux J at r ¼ r0 follows the expression

1 1 J ¼ Dc ¼ r0 ðpDtÞ1=2 (Hint: By making the substitution vðr; tÞ ¼ r cðr; tÞ in Fick’s equation and in the boundary conditions, the problem becomes essentially the same as that for linear diﬀusion.) 3. Using the limiting ionic conductances recorded in table 6.4 for the alkali metal ions in acetonitrile, estimate their diﬀusion coeﬃcients and Stokes radii. 4. The following parameters apply to mass transfer in 0.01 M NaCl: D ¼ 1:545 105 cm2 s1 ¼ 118:53 cm2 1 mol1 tþ ¼ 0:3918 Assume that the coeﬃcient e given in equation (6.7.46) can be estimated using the Debye–Hu¨ckel theory with an ion size parameter equal to 0.42 nm. Calculate values for the phenomenological coeﬃcients LMM , LXX , and LMX using equations (6.7.48)–(6.7.50). 5. The following data are reported for the conductivity of solutions of chloroacetic acid in water at 258C. Conc./M

/cm2 1 mol1

1.101104 3.027104 5.899104 1.323103 2.821103 3.812103 7.462103 1.4043102

362.10 328.92 295.58 246.15 197.14 177.98 139.85 109.00

Determine the acidity constant for this acid and its limiting ionic conductivity. 6. Determine the best value of the ion size parameter for LiNO3 given the following data obtained at 188C. At this temperature the Debye–Hu¨ckel

NON-EQUILIBRIUM PHENOMENA IN LIQUIDS AND SOLUTIONS

303

BDH is 3.274 M1=2 nm1 , and the Debye–Onsager constants are B1 ¼ 0:2264 M1=2 and B2 ¼ 50:99 cm2 M1=2 1 mol1 . ce /M

=cm2 1 mol1

0.001 0.005 0.01 0.05 0.1 0.5 1.0

92.9 90.3 88.6 82.7 79.2 68.0 60.8

7. Estimate the missing data points for Csþ in table 6.4 using a linear correlation between the Stokes radius and the solvent radius as illustrated in ﬁg. 6.11. 8. The following data were obtained for ZnSO4 in water at 258C. ce /M

=cm2 1 mol1

2104 5104 1103 2103 5103 0.01 0.02 0.05 0.1

242.0 234.0 225.0 212.0 190.0 169.0 146.4 121.4 105.0

(a) Determine the limiting ionic conductance assuming that ZnSO4 is a strong electrolyte (b) Determine this parameter assuming that ZnSO4 can be considered a weak electrolyte due to strong ion pairing. (c) Comment on these results using the parameters given in table 6.2.

7

Chemical Reaction Kinetics in Solution

Rudolph (Rudy) Marcus was born in 1923 at Montreal, Quebec, Canada, where he also grew up and went to school. He entered McGill University to study chemistry, obtaining a BSc in 1943 and a Ph.D. in 1946. His doctoral research under Professor Carl Winkler was in the area of chemical kinetics. He then spent two years as a post-doctoral fellow at the National Research Council in Ottawa, Canada where he worked in the photochemistry laboratory of E.W.R. Steacie. As a young student, Marcus had strong Rudolph Arthur Marcus interests in both mathematics and chemistry. In order to develop these interests further he joined the research group of Professor Oscar Rice at the University of North Carolina in 1949. There he worked on the theory of unimolecular reactions becoming one of the contributors to what is now known as RRKM (Rice–Ramsperger– Kassel–Marcus) theory. In 1951, Marcus joined the faculty of the Polytechnic Institute of Brooklyn as Assistant Professor. It was during his years in Brooklyn that he developed the now famous theory of electron transfer. He has subsequently held faculty positions at the University of Illinois and, at California Institute of Technology where he presently holds the Arthur Amos Noyes Chair of Chemistry. His work on the theory of chemical kinetics has also involved proton and atom transfer reactions as well as reactions at interfaces. In 1992, Marcus was awarded the Nobel Prize in Chemistry for his contributions to the theory of electron transfer reactions in chemical systems. He has received numerous other honors including the Peter Debye award of the American Chemical Society and foreign membership in the Royal Society (London).

7.1 What Time Scales Are Involved for Chemical Reactions in Solution? The kinetics of chemical reactions were ﬁrst studied in liquid solutions. These experiments involved mixing two liquids and following the change in the concentration of a reactant or product with time. The concentration was monitored by removing a small sample of the solution and stopping the reaction, for example, 304

CHEMICAL REACTION KINETICS IN SOLUTION

305

by rapidly lowering the temperature, or by following a physical property of the system in situ, for example, its color. Although the experiments were initially limited to slow reactions, they established the basic laws governing the rate at which chemical changes occur. The variables considered included the concentrations of the reactants and of the products, the temperature, and the pressure. Thus, the reacting system was examined using the variables normally considered for a system at equilibrium. Most reactions were found to be complex, that is, to be made up of several elementary steps which involved one or two reactants. As the fundamental concepts of chemical kinetics developed, there was a strong interest in studying chemical reactions in the gas phase. At low pressures the reacting molecules in a gaseous solution are far from one another, and the theoretical description of equilibrium thermodynamic properties was well developed. Thus, the kinetic theory of gases and collision processes was applied ﬁrst to construct a model for chemical reaction kinetics. This was followed by transition state theory and a more detailed understanding of elementary reactions on the basis of quantum mechanics. Eventually, these concepts were applied to reactions in liquid solutions with consideration of the role of the non-reacting medium, that is, the solvent. An important turning point in reaction kinetics was the development of experimental techniques for studying fast reactions in solution. The ﬁrst of these was based on ﬂow techniques and extended the time range over which chemical changes could be observed from a few seconds down to a few milliseconds. This was followed by the development of a variety of relaxation techniques, including the temperature jump, pressure jump, and electrical ﬁeld jump methods. In this way, the time for experimental observation was extended below the nanosecond range. Thus, relaxation techniques can be used to study processes whose half lives fall between the range available to classical experiments and that characteristic of spectroscopic techniques. The experimental techniques for studying fast reactions provided a means of studying fundamental processes in solution that were previously considered to be ‘‘instantaneous.’’ These include electron and proton transfer reactions. Proton transfer is the elementary step involved in acid–base reactions, which are so important in classical analytical chemistry. On the other hand, electron transfer is the elementary step involved in redox reactions. The theory of electron transfer is especially well developed and is discussed in detail below. This chapter is mainly concerned with fast reactions and the experimental methods used to study them. Other than the relaxation techniques already mentioned, spectroscopic methods applied to the study of elementary reactions are outlined. Especially important in this regard are laser methods which are able to probe fundamental processes in solution in the femtosecond time range.

7.2 Fundamental Concepts The experimental study of chemical kinetics traditionally involves the measurement of the concentration of a reactant or a product of the reaction as a function of time in a homogeneous system. When the experimental data are plotted, one

306

LIQUIDS, SOLUTIONS, AND INTERFACES

usually ﬁnds that the slope of the plot, that is, the rate of reaction, decreases as time increases. On the basis of a number of diﬀerent experiments, the conclusion was reached that the rate of reaction could be related to the concentration of one reactant in a fairly simple way. Thus, for the irreversible reaction A ! products

ð7:2:1Þ

the rate of reaction is often given by the general expression

dcA ¼ kr cnA dt

ð7:2:2Þ

where dcA =dt is the rate of reaction with respect to the reactant A, cA, its concentration, n, the order of the reaction, and kr, the rate constant. Typical values of n are 1 corresponding to a ﬁrst-order reaction and 2, corresponding to a second-order reaction. However, other values are found experimentally including fractional values such as 1/2 and 3/2. In these cases, a complex mechanism is involved. Experimental determination of the reaction order can be made in several ways [G1, G2]. Given values of cA as a function of time, one can estimate dcA/dt by numerical diﬀerentiation. Then, the reaction order can be estimated on the basis of a plot of the logarithm of the reaction rate against the logarithm of the reactant concentration. Thus, from equation (7.2.2) dcA ln ¼ ln kr þ n ln cA ð7:2:3Þ dt The order of the reaction is estimated from the slope of the plot and the rate constant from the intercept. Since estimation of the rate of reaction from the experimental data involves ampliﬁcation of experimental error, an improved estimate of the rate constant is usually sought using an integrated version of the general expression for the rate of reaction once it is known. For a ﬁrst-order reaction (n ¼ 1), integration of equation (7.2.2) gives the result ln cA ¼ ln cA0 kr t

ð7:2:4Þ

where cA0 is the initial concentration of reactant A. It follows that a plot of ln cA against time t should be linear, and that the rate constant kr can be determined from the slope of the plot. If the reaction is second order (n ¼ 2), the relationship resulting from integration of equation (7.2.2) is 1 1 ¼ þ kr t cA cA0

ð7:2:5Þ

Thus, a plot of 1/cA should be linear in time t with a slope equal to the rate constant kr. Once the order of reaction is known, equation (7.2.2) can be integrated to obtain a relationship between cA and t. Using this relationship the data may be analyzed further to obtain the rate constant kr . Most chemical reactions involve more than one reactant. For example, for an irreversible reaction involving two reactants such as aA þ bB ! products

ð7:2:6Þ

CHEMICAL REACTION KINETICS IN SOLUTION

307

the rate of reaction can be written as

1 dcA 1 dcB n ¼ ¼ kr cm A cB a dt b dt

ð7:2:7Þ

where a and b are the stoichiometric coeﬃcients for reactants A and B, respectively, and m and n, the corresponding orders with respect to each of these reactants. The overall order is m þ n. The simplest case for this general system occurs when the reaction is ﬁrst order in A, and ﬁrst order in B with an overall order of two. When the stoichiometric coeﬃcients are also unity one can write

dcA dc ¼ B ¼ k r cA cB dt dt

ð7:2:8Þ

Integration of this equation to give an expression for cA or cB as a function of time is very simple in two limiting cases. If the initial concentration of A is equal to that of B, then cA is equal to cB for all values of time and equation (7.2.8) reduces to equation (7.2.2) for the case that n ¼ 2. The integrated result is then given by equation (7.2.5). The other limiting case obtains when the initial concentration of one reactant is much greater than that of the other, so that the change in concentration of the dominant reactant is negligible. If reactant B is in excess, then integration of equation (7.2.8) gives the result ln cA ¼ ln cA0 kr cB t

ð7:2:9Þ

By performing experiments with the concentration of one reactant much smaller than the others, the order with respect to the minority component is easily found. This is called the method of isolation. Integration of equation (7.2.8) for the general case of non-equal concentrations of A and B is best carried out using a variable x which is equal to the amount of A or B which has been consumed. Thus, the diﬀerential equation is rewritten as

dcA dc dx ¼ kr ðcA0 xÞðcB0 xÞ ¼ B¼ dt dt dt

ð7:2:10Þ

where cA0 and cB0 are the initial concentrations of A and B, respectively. Using the method of partial fractions this becomes

dx 1 dx dx ¼ ¼ kr dt ð7:2:11Þ ðcA0 xÞðcB0 xÞ cB0 cA0 cA0 x cB0 x After integration, one obtains c x c ln B0 ¼ ln B0 þ ðcB0 cA0 Þkr t cA0 x cA0 or

ln

cB c ¼ ln B0 þ ðcB0 cA0 Þkr t cA cA0

ð7:2:12Þ

ð7:2:13Þ

This equation describes the time dependence of the two concentrations cA and cB for all cases except the case that cA0 is equal to cB0.

308

LIQUIDS, SOLUTIONS, AND INTERFACES

Many elementary reactions are reversible so that an analysis of the kinetics must consider both the forward and reverse processes. A simple example is a reversible reaction consisting of two ﬁrst-order reactions, which is described as (B A+ ð7:2:14Þ If the rate constant for the forward step is k1 and that for the backward step, k1, then the rate of consumption of A is

dcA ¼ k1 cA k1 cB dt

ð7:2:15Þ

On the basis of mass balance, cA þ cB ¼ cA0

ð7:2:16Þ

where cA0 is the initial concentration of A. Thus, equation (7.2.15) can be rewritten as

dcA ¼ k1 cA k1 cA0 þ k1 cA dt

ð7:2:17Þ

This can be rearranged to give the diﬀerential equation ðk1 þ k1 ÞdcA ¼ ðk1 þ k1 Þ dt ðk1 þ k1 ÞcA k1 cA0

ð7:2:18Þ

After integration and further rearrangement, the relationship giving cA as a function of time is cA ¼

k1 cA0 þ k1 cA0 exp½ðk1 þ k1 Þt k1 þ k1

ð7:2:19Þ

When the system is at equilibrium (t ¼ 1), the concentration of A is cAe ¼

k1 cA0 k1 þ k1

ð7:2:20Þ

cBe ¼

k1 cA0 k1 þ k1

ð7:2:21Þ

and that of B,

The equilibrium constant in terms of concentration is Ke ¼

cBe k ¼ 1 cAe k1

ð7:2:22Þ

This is an important result linking chemical kinetics and thermodynamics. The concentration change occurring for a ﬁrst-order reaction with a rate constant of 1 s1 is illustrated in ﬁg. 7.1. At 7 s, the concentration of the reactant falls below one-thousandth of its original value. It continues to fall, reaching concentrations which are presumably too small to be detected experimentally. When the reaction is reversible with a reverse rate constant one hundred times slower than the forward reaction (k1 ¼ 0:01 s1), the concentration of the reactant reaches its equilibrium value in approximately 10 s. In this case, the behavior of the concentration against time is noticeably diﬀerent.

CHEMICAL REACTION KINETICS IN SOLUTION

309

Fig. 7.1 Plots of the logarithm of the concentration of reactant A against time for a ﬁrstorder reaction with a rate constant equal to 1 s1 and a reversible ﬁrst-order reaction with the same forward rate constant and a reverse rate constant equal to 0.01 s1. The initial concentration of A is 1 M.

Another important feature of many mechanisms is sequential steps. This is illustrated here with the simple example of two ﬁrst-order reactions, one following another, that is k1

k2

A ! B ! C

ð7:2:23Þ

If the rate constant for the ﬁrst reaction is k1, then the concentration of A is given by cA ¼ cA0 ek1 t

ð7:2:24Þ

where cA0 is its initial concentration. The diﬀerential equation giving the concentration of B is dcB ¼ k1 cA k2 cB dt

ð7:2:25Þ

where k2 is the rate constant for the second step. Substituting equation (7.2.24) into equation (7.2.25) and multiplying through by the factor ek2t, one obtains ek2 t

dcB ¼ k1 cA0 eðk2 k1 Þt k2 cB ek2 t dt

ð7:2:26Þ

d ðc ek2 t Þ ¼ k1 cA0 eðk2 k1 Þt dt B

ð7:2:27Þ

or

This is now easily integrated with the result cB ¼

k1 cA0 k1 t ðe ek2 t Þ k2 k1

ð7:2:28Þ

310

LIQUIDS, SOLUTIONS, AND INTERFACES

where the initial concentration of B is assumed to be zero. The concentration of C is now found by subtraction: cC ¼ cA0 cA cB ¼ cA0 cA0 ek1 t

k1 cA0 k1 t ðe ek2 t Þ k2 k1

ð7:2:29Þ

A plot of the concentrations of A, B, and C against time is shown in ﬁg. 7.2 for the case that the rate constant for the second step is much greater than that of the ﬁrst (k1 k2 ). Under these conditions the concentration of B does not reach very high values during the course of the reaction. For smaller values of k2, the intermediate B reaches higher concentrations which represent a signiﬁcant fraction of the number of moles of the reactant A at the beginning of the experiment (cA0). When the concentration of B is much smaller than that of A and C, so that any change in its concentration may be neglected, it can be assumed to be in a steady state. The steady-state assumption is an important approximation used in the analysis of the kinetics of complex reactions which results in much more simple rate expressions. In the present case, the steady-state approximation for the reactive intermediate B is dcB ¼0 dt

ð7:2:30Þ

It follows from equation (7.2.25) that cBss ¼

k 1 cA k 1 ¼ cA0 ek1 t k2 k2

ð7:2:31Þ

For the product C

Fig. 7.2 Plots of the concentration of A, B, and C against time for a consecutive reaction scheme with irreversible ﬁrst-order processes. The solid lines show the results based on the exact solution assuming k1 ¼ 1 s1 and k2 ¼ 100 s1; the points were calculated on the basis of the steady-state approximation (equations (7.2.31) and (7.2.32)).

CHEMICAL REACTION KINETICS IN SOLUTION

cCss ¼ cA0 cA0 ek1 t

k1 c ek1 t k2 A0

311

ð7:2:32Þ

When the steady-state estimates of cB and cC (equations (7.2.31) and (7.2.32)) are compared with the exact values (equations (7.2.28) and (7.2.29)) it is apparent that two conditions are involved, namely, k2 k1 and k2 t 1. Values of cB and cC obtained on the basis of the steady-state approximation are also shown in ﬁg. 7.2. It is clear that the steady-state values provide excellent estimates of the concentrations of these species at longer times. However, when k2 t < 1, the steady-state estimates of cB and cC are incorrect. The examples of reversible and consecutive reactions presented here give a very modest introduction to the subject of reaction mechanisms. Most reactions are complex, consisting of more than one elementary step. An elementary step is a unimolecular or bimolecular process which is assumed to describe what happens in the reaction on a molecular level. In the gas phase there are some examples of termolecular processes in which three particles meet simultaneously to undergo a reaction but the probability of such an event in a liquid solution is virtually zero. A detailed list of the elementary steps involved in a reaction is called the reaction mechanism. In concluding this section, the reaction mechanism involved in a very important solution reaction, namely, enzyme catalysis, is considered. In this reaction, the enzyme E coordinates with a substrate species S to form a reactive intermediate ES. A very small fraction of the intermediate reacts to form product P releasing the enzyme for further catalysis. The majority of the reactive intermediate reverts to the substrate and enzyme. Thus, the mechanism can be described as k1

E þ S ! ES

ð7:2:33Þ

k1

ð7:2:34Þ

k2

ð7:2:35Þ

ES ! E þ S and ES ! P þ E

Assuming that the concentration of the reactive intermediate is given the steadystate approximation, one has dcES ¼ 0 ¼ k1 cE cs k1 cES k2 cES dt

ð7:2:36Þ

Solving for the concentration of ES at steady state, one obtains cESss ¼

k 1 cE cS k1 þ k2

ð7:3:37Þ

Since the enzyme is a catalyst its total concentration does not change so that dðcE þ cES Þ dcE ¼ ¼0 dt dt

ð7:2:38Þ

The rate of the overall reaction can be expressed in terms of the disappearance of substrate S,

312

LIQUIDS, SOLUTIONS, AND INTERFACES

dcS ¼ k1 cE cS k1 cES dt

ð7:2:39Þ

or in terms of the formation of product P, dcP ¼ k2 cES dt

ð7:2:40Þ

Combining equations (7.2.37) and (7.2.40), one obtains for the steady state dcP k1 k2 cE cS ¼ dt k1 þ k2

ð7:2:41Þ

The steady-state concentration of enzyme may be related to its initial concentration as follows: cEss ¼ cE0 cESss ¼ cE0

k1 cE cS k1 þ k2

ð7:2:42Þ

where cE0 is the initial enzyme concentration. Simplifying, one ﬁnds that cEss ¼

ðk1 þ k2 ÞcE0 k1 þ k2 þ k1 cS

ð7:2:43Þ

Thus, the expression for the rate of reaction becomes dcP k1 k2 cS cE0 ¼ dt k1 þ k2 þ k1 cS

ð7:2:44Þ

Since the steady-state approximation requires that k2 k1 , this expression simpliﬁes to

dcS dc k k c c ¼ P ¼ 1 2 S E0 dt dt k1 þ k1 cS

ð7:2:45Þ

This result was ﬁrst derived by Michaelis and Menten [1]. This section has provided only a brief introduction to the principal concepts in chemical kinetics. Most of the discussion which follows in this chapter relates to fast reactions in solution. Before describing the experimental techniques used to study fast reactions, the important types of solution reactions are outlined and their properties discussed.

7.3 General Types of Solution Reactions Important solution reactions are well known from the study of quantitative analysis in analytical chemistry. These include acid–base reactions, redox reactions, and complex formation reactions. Often the reaction is so fast that the system is considered to come ‘‘instantaneously’’ to equilibrium, for example, in the acid– base reaction involved in a titration. In fact, any of these reactions has a ﬁnite rate whose kinetics can be determined using modern experimental techniques. The above reactions may be described in a more fundamental way in terms of the elementary act involved in the reaction. For example, the reaction between an acid and a base involves proton transfer so that it is better described as a proton

CHEMICAL REACTION KINETICS IN SOLUTION

313

transfer reaction. Redox reactions involve electron transfer between at least two components in the system and are better described as electron transfer processes. Complex formation reactions are really ligand exchange reactions. Fundamental processes such as electron and proton transfer are of great interest to theoreticians, and have been studied in great detail experimentally. Some of the important properties of fundamental solution reactions are now outlined. Bimolecular reactions in the gas phase and in liquid solutions diﬀer in important ways. In the gas phase the progress of the reaction is described in terms of the collisions which occur between the two reactants A and B. In the liquid solution the same reactants are surrounded by solvent molecules, and the approach of these species to each other is controlled by the laws of diﬀusion. If A and B collide in solution, it is highly probable that they undergo numerous successive collisions before reaction occurs. Such a set of successive collisions is referred to as an encounter. The fact that the surrounding solvent molecules can transfer energy to or from A and B during the course of a solution reaction is important in developing the theory of reactions in condensed phases. A consequence is that the couse of the reaction is described in terms of the Gibbs energy of the system rather than by the potential energy, as is the case for gas phase reactions. More is said about this aspect of solution reactions in the presentation of transition state theory in section 7.4 In a bimolecular solution reaction, the reactants A and B diﬀuse to a point close to one another at which reaction is possible. This process is called formation of the precursor complex. At this point, rearrangement of bond lengths and bond angles in the two reactants, and of the surrounding solvent molecules, can occur to form an activated complex or transition state between the reactants and products. As one would expect, the nature of this process depends on the speciﬁc reaction involved. It is the focus of the development of the theory of the elementary step in the reaction and the associated energy requirements. In some cases it has been studied experimentally using very fast laser spectroscopic techniques which provide time-resolved information about the elementary step in the femtosecond range. In the case of very fast bimolecular reactions, the contributions of both the diﬀusion process and the rate of decomposition of the precursor complex must be considered in analyzing the experimental observations. The overall reaction mechanism can be described as A þ B ! AB AB ! C þ D

kdiff

ð7:3:1Þ

kr

ð7:3:2Þ

where A and B are the reactants, AB, the precursor complex, and C þ D, the products. The rate constant for formation of the precursor complex, kdiﬀ, depends on the diﬀusion coeﬃcients of A and B. Noyes [2] showed that the overall rate constant, taking into account the diﬀusion process, is knet ¼

kdiff kr kdiff þ kr

ð7:3:3Þ

314

LIQUIDS, SOLUTIONS, AND INTERFACES

In the case of slow reactions for which kr kdiff , the overall rate constant knet is equal to the rate constant for transformation of the precursor complex, kr. However, for very fast reactions, kr is greater than kdiﬀ, so that the overall rate constant is close to kdiﬀ. Therefore, it is important to understand that there is an upper limit on reaction rates that can be measured experimentally for diﬀusion controlled processes. In section 7.5 it is shown that kdiﬀ for molecular reactants is given by kdiff ¼ 4pNL ðDA þ DB Þa

ð7:3:4Þ

where DA and DB are the diﬀusion coeﬃcients for A and B in the given medium, and a, the distance between their centers in the precursor complex. Assuming that DA and DB are each equal to 109 m2 s1 and a, 500 pm, the estimate of kdiﬀ is 8106 m3 mol1 s1 or 8109 M1 s1. For reactions involving ions, the estimate is larger or smaller depending on whether the ions attract one another, or repel one another, respectively. The rate of the diﬀusion process is often comparable to the rate of electron and proton transfer reactions in solution. Therefore, it must be considered in analyzing the experimental data. The distinctive features of the elementary step involved in the important classes of solution reactions are now considered in more detail. A. Electron Transfer Reactions Electron transfer reactions are classiﬁed as simple or complex on the basis of the number of electrons transferred and whether or not chemical bonds are broken or formed in the overall reaction. An example of a complex electron transfer reaction is 2þ 2 3þ S2 O2 8 þ 2FeðH2 OÞ6 ! 2SO4 þ 2FeðH2 OÞ6

ð7:3:5Þ

In this reaction two electrons are transferred and the peroxide bond in the peroxydisulfate anion is broken. It is possible that this reaction actually involves three elementary steps. In the ﬁrst step an electron is transferred from Fe(H2O)62þ to S2O82 to form a radical anion S2O83 . The radical then breaks . The ﬁnal step involves the up into the anion SO2 4 and the radical anion SO4 transfer of the second electron to the radical anion SO4 . An example of a simple electron transfer reaction is 2þ 3þ 2þ CrðH2 OÞ3þ 6 þ FeðH2 OÞ6 !FeðH2 OÞ6 þ CrðH2 OÞ6

ð7:3:6Þ

In this case, one electron is transferred from Fe(II) to Cr(III), and no bonds are broken or formed. The reaction is also classiﬁed as heteronuclear, since the reactants belong to diﬀerent redox couples. This means that the overall reaction involves a net change in the standard Gibbs energy, rG . There is considerable interest also in homonuclear reactions such as 3þ 3þ 2þ FeðH2 OÞ2þ 6 þ FeðH2 OÞ6 ! FeðH2 OÞ6 þ FeðH2 OÞ6

ð7:3:7Þ

These systems are studied using radioactive isotopes designated by the asterisk (*) and provide an example of an electron transfer reaction for which rGo is zero.

CHEMICAL REACTION KINETICS IN SOLUTION

315

Much of the early work on electron transfer reactions focused on systems involving transition metal ions. Many redox couples based on the Fe(III)/ Fe(II), Co(III)/Co(II), and Cr(III)/Cr(II) can be prepared in the laboratory by changing the ligands coordinated to the central metal ion. From these studies it is possible to distinguish between two types of reactions which would otherwise be classiﬁed as simple. A classic example from the work of Taube [3] is the reaction 2þ CoðNH3 Þ5 Cl2þ þ CrðH2 OÞ2þ 6 ! CoðNH3 Þ5 H2 O þ CrðH2 OÞ5 Cl

ð7:3:8Þ

In this reaction the Cl ligand is transferred from the substitution-inert coordination sphere of the Co(III) complex to the substitution-inert coordination sphere of the Cr(III) product. On the other hand, the hexaquo Cr(II) complex is substitution-labile. Thus, it was postulated that the Cl ligand is shared between Cr(II) and Co(III) in the transition state for the reaction in which one water ligand is displaced from the substitution-labile Cr(II) reactant. Accordingly, the activated complex is [(NH3)5Co . . . Cl . . . Cr(H2O)5]4+. After electron transfer, the Co(II) product is substitution-labile so that the Cl ligand remains with the substitutioninert Cr(III) product. This type of electron transfer reaction follows the inner sphere mechanism. The bridging Cl ion reduces the repulsion between the cations involved in electron transfer and thus lowers the energy required to reach the transition state. It may also provide orbitals of the appropriate symmetry to delocalize metal–ion electrons and thereby provide a continuous path of good orbital overlap between the two metal ions. Much more about the inner sphere mechanism of electron transfer can be found in the work of Henry Taube [3]. In the outer sphere mechanism of electron transfer the ligands around each reactant remain intact in the passage through the transition state. This mechanism is expected when both reactants are substitution-inert. Formation of the activated complex may involve a change in bond lengths and bond angles within the reactants but it does not involve the sharing of ligands. The theory of electron transfer by the outer sphere mechanism has been developed in some detail and is discussed later in section 7.8. An important aspect of this theory concerns the relative velocities of the electron being transferred and the nuclei in the reactants. The transfer of the electron from an orbital on one reactant to an orbital on the other takes place in a time which is the order of femtoseconds ( 1015 s). Nuclear motion is much slower and is characterized by the time associated with intramolecular vibration, that is, 100 fs ( 1013 s). As a result, the nuclei in the reactants are virtually motionless during the electron jump. This results in the Franck– Condon restriction for electron transfer which was originally formulated in the study photochemically induced electronic transitions in molecules. It follows that the Gibbs energy of the system remains constant during electron transfer. In addition, the spatial conﬁguration of the nuclei is the same in the product system as it was in the reactant system. This conﬁguration is not the same as the low energy equilibrium conﬁguration of the nuclei in either system and is usually achieved as a result of random thermal motion. This description of the formation of the transition state for electron transfer is the basis for the Marcus model of electron transfer [4]. The Gibbs energy barrier involved in an electron transfer reaction is shown in ﬁg. 7.3. As a result of random thermal processes the reactants achieve the neces-

316

LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 7.3 Gibbs energy proﬁle for the formation of an activated complex 6¼ in a bimolecular heteronuclear reaction (A) and a bimolecular homonuclear reaction (B).

sary nuclear conﬁguration for electron transfer. This is the transition state for the reaction, designated as 6¼. As stated above, the electron is transferred very rapidly with respect to the time taken to achieve the required nuclear conﬁguration. After electron transfer the system relaxes to the product state. For a heteronuclear reaction the Gibbs energy of the ﬁnal state P is lower than that of the initial state R (ﬁg. 7.3A). If the reaction is homonuclear (see equation 7.3.7) then the Gibbs energy of the ﬁnal state P is equal to that of the initial state R (ﬁg. 7.3B). Rate constant data for several homonuclear electron transfer reactions involving transition metal complex ions in water are summarized in table 7.1. The striking feature of the results is that the rate constants vary over a very wide range from a low 2 107 M1 s1 to a high of 4 108 M1 s1 . Since these reactions are all assumed to occur by an outer sphere mechanism, the observations show that the potential energy barrier shown in ﬁg. 7.3 varies signiﬁcantly with the nature of the reactants. According to the theory of electron transfer (section 7.8), the magnitude of the barrier depends on the energy required to

Table 7.1 Kinetic Data for Homonuclear Electron Transfer Reactions Involving Transition Metal Complex Ions in Water at 258C [5]* Reaction [Cr(H2O)6]3þ=2þ [V(H2O)6]3þ=2þ [Fe(H2O)6]3þ=2þ [Co(H2O)6]3þ=2þ [Ru(H2O)6]3þ=2þ [Ru(NH3)6]3þ=2þ [Ru(en)3]3þ=2þ [Ru(bpy)3]3þ=2þ [Co(en)3]3þ=2þ [Co(bpy)3]3þ=2þ

Rate Constant ket =M1 s1

Ionic Strength M

2 107 1:5 102 4.2 3.3 60 3:2 103 2:8 104 4:2 108 7:7 105 18

1.0 2.0 0.55 0.5 1.0 0.1 0.1 0.1 1.0 0.1

*The chelates involved in the last four reactions are ethylenediamine (en) and bipyridine (bpy).

CHEMICAL REACTION KINETICS IN SOLUTION

317

rearrange the ligands and solvation atmosphere around the reactants that must occur prior to the actual transfer of the electron. However, there is another important feature of these reactions which leads to variation in the rate constants. Since the reactions involve highly charged ions the observed rate constants are very dependent on ionic strength. Thus, a comparison of kinetic data for diﬀerent reactions should be made at the same ionic strength. An even better way of examining these data is to use estimates of the rate constants obtained in the limit of zero ionic strength. More details of all of the important features of electron transfer reactions are given in section 7.8. The other bimolecular processes which are considered here involve the movement of atoms and ions. The lightest ion is the proton, and its reactions are discussed in the next section. B. Proton Transfer Reactions According to the Brønsted deﬁnition an acid is a proton donor, and a base is a proton acceptor. It follows that acid–base reactions can also be called proton transfer reactions. Thus, the reaction between acid HA and base B is written as HA þ B( +A þ HBþ

ð7:3:9Þ þ

where A is the conjugate base formed from the acid HA and HB , the conjugate acid formed from the base B. A well-known example is the reaction between HCl and NH3: HCl þ NH3 ( +Cl þ NHþ 4

ð7:3:10Þ

Usually these reactions have been studied in water or in other protic solvents such as the alcohols. Thus, the acid–base properties of the solvent are important in determining the relative strength of acids and bases which are solutes in water. This leads to the deﬁnition of two other types of proton transfer reaction, namely, the protolysis reaction, HA þ H2 O( +H3 Oþ þ A

ð7:3:11Þ

B þ H2 O( +BHþ þ OH

ð7:3:12Þ

and the hydrolysis reaction,

Finally, there is the neutralization reaction, H3 Oþ þ OH ( +2H2 O

ð7:3:13Þ

The thermodynamics of these reactions is well understood and is documented in terms of the acidity and basicity constants of the corresponding acids and bases. On the other hand, their kinetics were only determined after relaxation techniques for studying very fast reactions became available. The neutralization reaction is one of fastest reactions known and its rate may be estimated, assuming that its kinetics are only controlled by diﬀusion of the two reactants to the reaction site (see section 7.5). When proton transfer reactions are studied in protic solvents such as water the role of the solvent can be very important. For example, in a neutral solution at

318

LIQUIDS, SOLUTIONS, AND INTERFACES

pH 7, the reaction between HCl and NH3 is described by equation (7.3.10). If the pH is low, the predominant reactant is H3O+ which can be formed by protolysis, so that the corresponding reaction is H3 Oþ þ NH3 ( +NHþ 4 þ H2 O On the other hand, if the pH is high, OH dominant reaction is

ð7:3:14Þ

is formed by hydrolysis and the

OH þ HCl( +Cl þ H2 O

ð7:3:15Þ

This means that a complete analysis of this system involves determination of ten rate constants: two for direct proton transfer, two for the protolysis reaction, two for the hydrolysis reaction, and four for proton transfer reactions such as (7.3.14) and (7.3.15). The relationship among these reactions is shown in ﬁg. 7.4 for a general reaction involving the acid HA and the base B in water. The theory for the elementary act of proton transfer is well developed [6]. It involves three steps, the ﬁrst of which is formation of the precursor complex [AH B]: HA þ B ! ½AH B

ð7:3:16Þ

This step involves work which can have an electrostatic component if the reactants are charged, and includes an orientational component resulting from the requirement that HA have a speciﬁc orientation with respect to B. The second step is the actual proton transfer which involves intramolecular and solvent reorganization to form the transition state for proton transfer: ½AH B ! ½A H B

ð7:3:17Þ

Finally, there is the disorientation step in which the complex formed after proton transfer resolvates as the products: ½A HB ! A þ HBþ

ð7:3:18Þ

The energetics of each of these steps have been described in the Marcus model for proton transfer [6]. Proton transfer is a bimolecular process so that formation of the transition state involves an energy barrier of the type shown in ﬁg. 7.3. Kinetic data [7] for some acid–base reactions involving the solvated proton H3O+ are summarized in table 7.2. The forward reaction is always very fast and close to the diﬀusion-controlled limit. On the other hand, the reverse reaction is much slower as required by the relationship between the acidity constant for the

Fig. 7.4 Reaction scheme for proton transfer between the acid HA and base B in water illustrating the role of protolysis (P) and hydrolysis (H) reactions.

319

CHEMICAL REACTION KINETICS IN SOLUTION

Table 7.2 Kinetic Data for Some Simple Acid–Base Reactions in Water at 258C [7] Rate Constants Forward kf =M1 s1 Backward kb =s1

Reaction H3 Oþ þ OH Ð 2H2 O H3 Oþ þ F Ð HF þ H2 O H3 Oþ þ HS Ð H2 S þ H2 O H3 Oþ þ HCO 3 Ð H2 CO3 þ H2 O H3 Oþ þ HCOO Ð HCOOH þ H2 O H3 Oþ þ CH3 COO Ð CH3 COOH þ H2 O H3 Oþ þ NH3 Ð NHþ 4 þ H2 O

1:4 1011 1:0 1011 7:5 1010 4:7 1010 5 1010 4:5 1010 4:3 1010

2:5 105 7 107 4:3 103 8 106 8:6 106 7:8 106 25

equilibrium Ka and the two rate constants kf and kb . Similar results are found when data for a collection of reactions between OH1 and a weak acid are considered [8]. The forward rate constant is very fast, whereas the backward reaction reﬂects the basicity constant for the weak base in the equilibrium. Since ions are involved in these reactions, medium eﬀects are important and must be considered in a complete analysis of the experimental data. Many more details can be found about the kinetics of proton transfer reactions [8]. These often involve speciﬁc chemical features of the reactants. Since much of the focus in this chapter is on experimental methods, these reactions are not discussed further. However, the use of NMR spectroscopy to study proton transfer kinetics is considered in section 7.9. C. Ligand Exchange Reactions The kinetics of formation of complex ions is the subject of considerable interest in inorganic chemistry [9]. If the discussion is limited to octahedral complexes, then the type of reaction being considered is z ( ðnzÞ MðH2 OÞnþ þ H2 O 6 þ L +MðH2 OÞ5 L

ð7:3:19Þ

The ligand exchanging with the water molecule can be a neutral molecule such as NH3 (z ¼ 0) or monovalent anions such as the halides (z ¼ 1). The interesting observation is that the rate of the reaction is practically independent of the nature of the ligand but very much dependent on the nature of the metal ion. This has led to the hypothesis that the mechanism of ligand exchange involves a reactive intermediate with less than six ligands: nþ ( MðH2 OÞnþ 6 +MðH2 OÞ5 þ H2 O

ð7:3:20Þ

The reactive intermediate then recombines with a water molecule or with the or to give the product ligand Lz to reform the reactant MðH2 OÞnþ 6 MðH2 OÞ5LðnzÞþ . This type of mechanism in which a reactant intermediate involving fewer than the normal number of ligands or bonds is formed is called the SN1

320

LIQUIDS, SOLUTIONS, AND INTERFACES

mechanism in organic chemistry. The rate of the reaction is a reﬂection of the strength of the M–OH2 bond. In order to characterize the ligand exchange process more quantitatively, studies have been made of the kinetics of water exchange for a wide variety of metal ions 17 nþ ( þ H2 O ½MðH2 OÞ6 nþ þ H17 2 O+½MðH2 OÞ5 ðH2 OÞ

ð7:3:21Þ

By using water molecules labelled with 17O in the solvent one is able to follow the kinetics of the reaction. One popular technique for measuring the kinetics of reaction (7.3.21) is 17O NMR line-broadening experiments. The exchange reaction is usually quite fast but has a wide range of rate constants. In terms of the ﬁrstorder reaction (7.3.20), the rate constant is small and close to 1 s1 when the metal ion is Al3þ , but very fast and greater than 109 s1 when the metal ion is Csþ . In the case of ions from the main group elements, the interaction between the metal ion and the water ligands is mainly electrostatic in character. In this case, plots of the logarithm of the rate constant against z2i r1 i , where zi is the cationic valence and ri its radius, are linear [10]. For the alkali metal ions, the exchange rate is very fast, with the rate constant falling in the range 108–109 s1 . For the divalent alkaline earth metal ions, the range of observed rate constants is much wider, varying from 103 s1 for Be2þ to 109 s1 for Ba2þ . For the transition metal ions other factors play an important role in determining the rate of ligand exchange, especially the ligand ﬁeld stabilization energy. Thus, a cation such as Cr3þ which has a high charge and three unpaired d electrons in the t2g orbitals (see ﬁg. 5.25) has a very low ligand exchange rate ( 106 s1 ). For this reason Cr3þ is substituently inert. When ligand ﬁeld stabilization energy is lost by adding an electron to the eg orbitals in chromium to form Cr2þ , the rate of ligand exchange increases dramatically to 109 s1 . Thus, Cr2þ is a substitutionally labile cation. Values of the rate constants for water exchange for the common metal ions of the ﬁrst transition series are given in table 7.3. The radii of these ions do not vary greatly so that the

Table 7.3 Rate Constants for Water Exchange for Common Ions of the First Row of the Transition Metals [9] Ion V2þ Cr3þ Cr2þ Mn2þ Fe3þ Fe2þ Co2þ Ni2þ Cu2þ Zn2þ

Number of d Electrons

Rate Constant kex =s1

3 3 4 5 5 6 7 8 9 10

90 3 106 8 109 3:1 107 3 103 3:2 106 2:4 106 3:5 104 8:0 109 3 107

CHEMICAL REACTION KINETICS IN SOLUTION

321

water exchange rate depends mainly on the number of d electrons and the associated ligand ﬁeld stabilization energy. According to the above mechanism (equation (7.3.20)) the reactive intermediate in the ligand exchange process is formed in a unimolecular step. The height of the Gibbs energy barrier in this step decreases as the rate of reaction (7.3.20) increases in the forward direction. On the other hand, there is no barrier for the reverse reaction in which the octahedral complex is reformed. The Gibbs energy proﬁle associated with a unimolecular reaction is illustrated in ﬁg. 7.5. The stable complex has an average energy equal to that at the minimum on the curve. As a result of thermal ﬂuctuations the energy can reach a value suﬃciently high to form the activated complex designated by the 6¼ sign. The diagram also shows that there is no energy barrier associated with the products in the activated state returning to form the reactant. The water exchange process described here gives the essential details about the ligand exchange process for octahedral complexes. There are many other interesting details which become apparent when one considers a wider variety of ligands and complexes with diﬀerent coordination numbers. Details can be found in monographs devoted to this subject [9, 11]. D. Atom Transfer Reactions The atom transfer reaction can be written in general as A þ BC ! AB þ C

ð7:3:22Þ

A is a base or nucleophile which attacks the species BC, forming a new molecule AB and the species C. A well-known example from organic chemistry is the reaction Cl þ CH3 I ! CH3 Cl þ I

ð7:3:23Þ

This reaction is also called a displacement reaction and it follows the SN2 mechanism. The attacking nucleophile can be an ion or a neutral molecule. Obviously the solvent plays an important role in the kinetics of the reaction when ions are involved. The formation of the transition state for reaction (7.3.23) is shown in ﬁg. 7.6. The electrophilic reactant is the simple molecule methyl iodide, which has tetrahedral geometry with respect to the carbon atom. As the attacking chloride ion

Fig. 7.5 Gibbs energy diagram for the formation of an activated complex in a unimolecular reaction step.

322

LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 7.6 Formation of the transition state in an atom transfer reaction between Cl and CH3I.

approaches the methyl group, the C–H bonds are distorted. The transition state consists of a ﬁve-atom species with the C–H bonds all perpendicular to the linear Cl–C–I portion of the molecule. As the reaction proceeds to completion the distance between the carbon and iodine atoms increases until the product state is reached. Extensive studies have been made of solvent eﬀects on atom transfer reactions involving ions [12]. In the case of reaction (7.3.23), the rate constant decreases from 250 M1 s1 in N-methylpyrrolidinone to 3 106 M1 s1 in methanol. This eﬀect can be attributed to solvation of the anionic reactant Cl and the anionic transition state [12]. Since the reactant is monoatomic, its solvation is much more important. It increases signiﬁcantly with solvent acidity leading to considerable stabilization of the reactants. As a result the potential energy barrier increases and the rate decreases with increase in solvent acidity. As shown in ﬁg. 7.7, this leads to an approximate linear relationship between the logarithm of the rate constant and the solvent’s acceptor number AN, an empirical measure of solvent acidity (see section 4.9). Most of the results were obtained in aprotic solvents which have lower values of AN. The three data points at higher values of AN are for protic solvents.

Fig. 7.7 Plot of the logarithm of the rate constant for reaction (7.3.23) in various solvents against the solvent’s acceptor number AN (table 4.10).

CHEMICAL REACTION KINETICS IN SOLUTION

323

The distinction between atom transfer reactions involving neutral and ionic nucleophiles is important in developing the theory of these processes [G3]. Actually, proton transfer is also an ionic atom transfer reaction. However, it is treated separately because of the special relationship that the proton has to the most common solvent, namely, water. In addition, because of its very low mass, proton tunnelling must be considered in proton transfer processes. The Gibbs energy proﬁle involved in formation of the transition state for atom transfer is that shown in ﬁg. 7.3. When the attacking nucleophile is the same as the leaving group, then the proﬁle is that for a homonuclear bimolecular reaction. This type of system can be studied using radioactive isotopes. The above examples illustrate the important types of solution reactions, especially from the point of view of analytical chemistry. But it is not an exhaustive list. For example, another type of solution reaction which is of interest to theorists is the isomerization reaction [G3]. In this chapter, the theory of the electron transfer process is considered in detail. Some of the results from this treatment can also be applied to other bimolecular processes. More details follow in the later sections of this chapter.

7.4 Temperature Effects and Transition State Theory The eﬀect of temperature on reaction kinetics was an important aspect of experimental studies dating from the earliest work. The common observation is that the rate constant for the reaction increases with increase in temperature. This was considered on a quantitative basis by Arrhenius, who derived the following empirical expression: kr ¼ Ar exp

Ea RT

ð7:4:1Þ

In this equation, Ar is the pre-exponential factor, which was assumed to be temperature independent, and Ea , the activation energy. Further development of the theory of the temperature dependence of the rate constant was based on reactions in the gas phase. The simple picture is that two molecules which can react, collide with a relative kinetic energy given by their individual velocities. If this relative kinetic energy is greater than a certain minimum value called the ‘‘activation energy,’’ then reaction can occur. If it is less than the minimum value, then reaction does not occur and the two molecules behave as if they had undergone an elastic collision. In the Arrhenius model, the number of successful collisions is estimated using the Boltzmann factor, expðEa =ðRTÞ), which gives the fraction of collisions with relative kinetic energy greater than Ea . On the basis of this discussion, the preexponential factor Ar should really be identiﬁed with a collision frequency. Methods of calculating this quantity are well known on the basis of the kinetic theory of gases. Using this theory it is clear that the preexponential factor does depend on temperature. However, the temperature dependence is much less than that expressed explicitly in the Boltzmann factor. For this reason, it is more diﬃcult to determine experimentally.

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LIQUIDS, SOLUTIONS, AND INTERFACES

Transition state theory was also developed as a means of rationalizing rate constants for gas phase reactions and their temperature dependence. It is most directly applied to bimolecular reactions and is based on three fundamental postulates for reactions in solution: 1. One can conceive of a Gibbs energy surface which gives the Gibbs energy of the reacting system as a function of the distances between the atoms in both the reactants and products. There exists a most probable path describing the conﬁguration of the atoms and surrounding solvent molecules in the system and its Gibbs energy as it changes from reactants to products. This path is called the reaction coordinate. The point where the Gibbs energy on this path is a maximum is called the transition state (see ﬁg. 7.3). 2. The chemical species which exists at the transition state X6¼ is in equilibrium with the reactant state. 3. The rate of reaction is equal to the product of the concentration of X6¼ and frequency n6¼ with which it decomposes to the product state.

On the basis of these postulates, a bimolecular reaction may be described as +X6¼ A þ B(

ð7:4:2Þ

6¼

X ! products

ð7:4:3Þ

dcA dc ¼ B ¼ k6¼ cX6¼ dt dt

ð7:4:4Þ

The rate of reaction is

where k6¼ is the rate constant for reaction (7.4.3). Since this reaction is unimolecular, the rate constant has the units of s1 and is equivalent to the frequency of passage through the transition state n6¼ . On the basis of the equilibrium process in which X6¼ is formed, its concentration may be determined from the equilibrium constant for reaction (7.4.2). For solution processes, the equilibrium constant is a 6¼ y 6¼ cX6¼ K6¼ ¼ X ¼ X ð7:4:5Þ aA aB yA yB c A c B where ai is the activity of species i, and yi , its activity coeﬃcient on the concentration scale. Combining equations (7.4.4) and (7.4.5), the rate of reaction becomes

dcA dc y y y y ¼ B ¼ k6¼ A B K6¼ cA cB ¼ n6¼ A B K6¼ cA cB dt dt yX6¼ yX6¼

It follows that the rate constant is given by y y k6¼ ¼ n6¼ A B yX6¼

ð7:4:6Þ

ð7:4:7Þ

The parameters n6¼ , K6¼ , and yX6¼ are deﬁned within the context of the transition state model, whereas yA and yB are quantities which can be obtained from equilibrium thermodynamics. In order to estimate the rate constant, the equilibrium constant is written down in terms of the partition functions of the components of equilibrium (7.4.2):

CHEMICAL REACTION KINETICS IN SOLUTION

K6¼ ¼

q6¼ qA qB

325

ð7:4:8Þ

where qi is the partition function of component i. Now, one component of q6¼ , the partition function for the activated complex or transition state, corresponding to a weak or ‘‘loose’’ vibration state, is factored out. This vibration occurs at a frequency n6¼ and is the process which leads to decomposition of the transition state to form the products of the reaction. The expression for the corresponding vibrational partition function is h6¼ 1 ð7:4:9Þ q6¼v ¼ 1 exp kB T Since the vibration is weak, n6¼ is small and the ratio hn6¼ =ðkB TÞ is much less than unity. As a result the exponential term in equation (7.4.9) can be replaced by the ﬁrst two terms in its series expansion, so that h6¼ 1 kB T q6¼v ¼ 1 1 þ ¼ ð7:4:10Þ hn6¼ kB T Thus the partition function for the activated complex becomes q6¼ ¼

kB T q hn6¼ 6¼r

ð7:4:11Þ

where q6¼r is the remainder of the partition function for the activated complex accounting for all other degrees of freedom. The equilibrium constant can now be written as K6¼ ¼

kB T q6¼r k T ¼ B Ke6¼ hn6¼ qA qB hn6¼

ð7:4:12Þ

where Ke6¼ is the eﬀective equilibrium constant obtained after factoring out the contribution from one degree of vibrational motion. The expression for the rate constant in the solution is k T yA yB e K ð7:4:13Þ kr ¼ B h yX6¼ 6¼ The frequency associated with decomposition of the activated complex is kB T=h. At room temperature this corresponds to 6:2 1012 s1 . In the thermodynamic approach to the transition state model the equilibrium constant Ke6¼ is related to the standard Gibbs energy change for process (7.4.2). Thus, one writes RT ln Ke6¼ ¼ 6¼ G ¼ 6¼ H þ T6¼ S

ð7:4:14Þ

where 6¼ G is the standard Gibbs energy change, 6¼ H , the corresponding enthalpy change, and 6¼ S , the standard entropy change associated with reaction (7.4.2). One should remember that these quantities are obtained for an equilibrium constant deﬁned in terms of concentration with the standard state being 1 mol L1 .

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LIQUIDS, SOLUTIONS, AND INTERFACES

By combining equations (7.4.7), (7.4.13), and (7.4.14), the expression for the rate constant for a reaction in solution is obtained: kB T yA yB 6¼ G exp kr ¼ ð7:4:15Þ h yX6¼ RT Alternatively, one may write 6¼ kB T yA yB S 6¼ H exp kr ¼ exp h yX6¼ R RT

ð7:4:16Þ

Comparing this result with the Arrhenius equation (equation (7.4.1)) it is clear that the major part of the temperature dependence of the rate constant is associated with the exponential term containing the standard enthalpy of activation, 6¼ H . An approximate relationship between the Arrhenius activation energy Ea and the standard enthalpy of activation 6¼ H can be found by taking the derivative of ln kr with respect to 1=T at constant pressure using equation (7.4.15). Neglecting the temperature derivatives of the activity coeﬃcients, one obtains

@ ln kr h kB T2 6¼ S 6¼ G 6¼ H þ RT ð7:4:17Þ þ ¼ ¼ kB T h R @ð1=TÞ p RT R Comparing this result with the corresponding derivative obtained from equation (7.4.1) it follows that Ea ¼ 6¼ H þ RT

ð7:4:18Þ

for reactions in solution. When the rate of reaction is studied as a function of pressure, the standard volume of activation can be found. Neglecting the pressure dependence of the activity coeﬃcients, on the basis of equation (7.4.15), @ ln kr @ lnð6¼ G =ðRTÞÞ 6¼ V ð7:4:19Þ ¼ ¼ @P @P T RT T EXAMPLE

The following rate constants are reported for the reaction between p-bromodimethylaniline and methyliodide in nitrobenzene: Temperature/8C

Rate Constant/M1 s1

15.0 24.8 40.1 60.0

8:24 106 1:80 105 5:55 105 2:13 104

Estimate the Arrhenius activation energy, the pre-exponential factor, and the enthalpy and entropy of activation at 258C. Ignore any eﬀects due to solution non-ideality. The data are ﬁtted using least squares to the Arrhenius equation in the form

CHEMICAL REACTION KINETICS IN SOLUTION

327

Ea RT

ð7:4:20Þ

6:969 103 T

ð7:4:21Þ

ln kr ¼ ln Ar The result is ln kr ¼ 12:46

The value of the activation energy is given by Ea ¼ 6:97 103 8:314 ¼ 58:0 kJ mol1

ð7:4:22Þ

The pre-exponential factor is Ar ¼ expð12:456Þ ¼ 2:57 105 M1 s1

ð7:4:23Þ

From equation (7.4.18), the enthalpy of activation is 6¼ H ¼ Ea RT ¼ 57:9 2:5 ¼ 55:4 kJ mol1

ð7:4:24Þ

The Gibbs activation energy is calculated from equation (7.4.15) ignoring the activity coeﬃcients. From equation (7.4.20) the rate constant at 258C is 1:82 105 M1 s1 . Thus, ! 5 1:82 10 ð7:4:25Þ 6¼ G ¼ RT ln ¼ 100:1 kJ mol1 6:2 1012 Now, 6¼ S ¼

6¼ H 6¼ G ð55:4 100:1Þ 1000 ¼ 150 J K1 mol1 ¼ 298:2 T

ð7:4:26Þ

The entropy of activation is negative because formation of the activated complex involves the combination of two molecules. Estimation of the activation parameters on the basis of fundamental theoretical considerations is a formidable task for solution reactions. Because of the surrounding solvent molecules which solvate the reactants A and B in a bimolecular process, there are many degrees of freedom which must be considered in evaluating the partition functions qA , qB , and q6¼r . Moreover, the number of degrees of freedom often changes when the transition state is formed. For example, consider the case that A and B are two highly charged cations (reaction (7.3.6)). Each cation is strongly solvated so that the internal degrees of freedom associated with both the cation and surrounding solvent molecules must be considered in evaluating qA and qB . The charge on the transition state, zA þ zB , is much higher. As a result, more solvent molecules are probably solvating the activated complex than the total number solvating A and B separately. Thus, formation of the transition state is accompanied not only by a change in the energy of the system, but also by a change in entropy. This example illustrates the importance of considering the Gibbs energy in discussing the course of the reaction through the transition state. There is considerable interest in comparing the reaction rate for a given reaction in solution with that in the gas phase [G3]. Assuming that the rate of decomposition of X6¼ is the same in both phases, the ratio of the rate constants is

328

LIQUIDS, SOLUTIONS, AND INTERFACES

" # s ð6¼ Gs 6¼ Gg Þ ksr yA yB K6¼ yA yB ¼ ¼ exp yX6¼ Kg6¼ yX6¼ RT kgr

ð7:4:27Þ

where kir is the rate constant in phase i, Ki6¼ , the equilibrium constant to form the activated complex in phase i and 6¼ Gi , the corresponding standard Gibbs activation energy. If consideration is limited to reactions not involving ions in nonpolar solvents, the activity coeﬃcients can be neglected. Furthermore the Gibbs activation energy in solution can be expressed as follows: 6¼ Gs ¼ 6¼ Gg þ 6¼ Gsolv 6¼

Gsolv

6¼

Gs .

ð7:4:28Þ 6¼

Gsolv

where is the solvent contribution to In other words, is the contribution to the work done due to the presence of the solvent in bringing the reactants A and B from sites in the solution which are very far apart up to the transition state site where they are in the same solvent cage. The expression for the rate constant in solution is then 6¼ Gsolv ksr ¼ kgr exp ð7:4:29Þ RT An eﬀective way of estimating 6¼ Gs and 6¼ Gg is to carry out molecular dynamics calculations for the system being studied. In this way, cumbersome quantum-mechanical calculations involving the reacting system in the solvent are avoided. From a qualitative point of view, an atom exchange reaction involving neutral reactants in a non-polar solvent is expected to be faster in the solvent than in the gas phase. This is due to the fact that two neutral species form one species in the transition state. Very few experimental studies have been carried out; one example of a reaction which has been studied is H þ D2 ! HD þ D

ð7:4:30Þ

This reaction is faster in solution than in the gas phase by approximately a factor of 40. Theoretical calculations agree with the experimental result [G3]. Transition state theory has also been applied in quite another way to reactions in solution. Reaction (7.4.2) can be described as a reaction precursor equilibrium which is characterized by the diﬀusion of A and B to a position close enough so that reaction can take place. This pre-equilibrium has an equilibrium constant which can also be thought of as a collision frequency. The expression for the rate constant is then G ð7:4:31Þ kr ¼ ZB exp RT where ZB is the collision frequency for the bimolecular process in solution and G is the standard Gibbs activation energy for reaction (7.4.3). G diﬀers from 6¼ G in that it is calculated using the total partition functions for A, B, and X6¼ ; on the other hand, 6¼ G has the contribution from one degree of vibrational motion in the activated complex factored out. Electron transfer kinetics in solution are usually analyzed on the basis of equation (7.4.31). A more complete description of this subject as well as methods of estimating ZB are given later in sections 7.8 and 7.10.

CHEMICAL REACTION KINETICS IN SOLUTION

329

Medium eﬀects are very important for solution reactions. They are dealt with quantitatively through the activity coeﬃcient ratio which appears in equation (7.4.15). Thus, if the nature of the non-reacting components in the solution is changed, one can expect to observe a change in the rate constant. The easiest way of dealing with these eﬀects is to keep the nature of the medium constant when other aspects of the system being studied are changed. On the other hand, there is an interest in studying the eﬀects of the medium on solution reactions. This subject is considered in more detail in section 7.10.

7.5 Diffusion-Controlled Rapid Reactions The fastest reactions in solution are limited in their reaction rates by mass transfer. For most bimolecular reactions this limitation arises because the two reactants must diﬀuse together in order to form a reactant pair. Therefore it is interesting to estimate the magnitude of the rate constant for a diﬀusion-controlled reaction and to compare it with experimentally determined rate constants for very fast reactions. The treatment presented here applies to bimolecular reactions involving either molecules or ions. Consider a bimolecular reaction A þ B ! products

ð7:5:1Þ

which takes place every time these species come into contact. In order to solve the mass transfer problem, it is imagined that only A moves and B is stationary. On the basis of the analysis in section 6.7, the ﬂux of species A is z u c JA ¼ DA ;cA A A A ;( ð7:5:2Þ jzA j where ;cA is the gradient of the concentration of A, ;( the gradient of the electrostatic potential, DA , the diﬀusion coeﬃcient of A, uA , its mobility, and zA , its charge. The second term obviously accounts for electrostatic eﬀects which are only important in the case of reactions involving ions. In the case of reactions involving molecules, the second term disappears. Using the relationship between the diﬀusion coeﬃcient and mobility of an ion (equation (6.7.21)), the above equation may be written as z Fc ð7:5:3Þ JA ¼ DA ;cA þ A A ;( RT It is convenient to re-express the electrostatic term in terms of the electrostatic energy UeA , where UeA ¼ zA F Thus, the equation for the ﬂux of A may be written as c JA ¼ DA ;cA þ A ;UeA RT

ð7:5:4Þ

ð7:5:5Þ

The problem is now solved, assuming that diﬀusion is a spherically symmetrical process and that it occurs at a steady state. Furthermore, the reactants

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are assumed to be spherical with radii rA and rB . The ﬂux which is a constant independent of angular direction becomes dcA cA dUeA þ JA ¼ DA ð7:5:6Þ dr RT dr Introducing the function

U CðrÞ ¼ cA exp eA RT one may write

ð7:5:7Þ

UeA dCðrÞ JA ¼ DA exp dr RT

ð7:5:8Þ

The fact that both A and B move in reality is now dealt with by using DA þ DB as the mutual diﬀusion coeﬃcient; this eﬀectively accounts for the motion of the coordinate system which is now deﬁned with respect to B. Thus, UeA dCðrÞ ð7:5:9Þ JA ¼ ðDA þ DB Þ exp RT dr The number of molecules moving through the surface of a sphere of radius r can now be calculated: dnA UeA dCðrÞ 2 2 ¼ 4pr JA ¼ 4pr ðDA þ DB Þ exp ð7:5:10Þ dt RT dr At a steady state, dnA =dt is a constant so that solution of the diﬀerential equation is relatively easy. The rate of reaction is found by integrating equation (7.5.10) in the form dnA dt

1 ð

expðUeA =ðRTÞÞ dr ¼ r2

a

cðA

4pðDA þ DB Þ dCðrÞ

ð7:5:11Þ

0

In the limit that r equals a the molecules A and B are in contact, that is, a ¼ rA þ rB

ð7:5:12Þ

In this case, species A reacts with species B, and the concentration of A is zero. At very large distances, the concentration of A is its bulk value, cA . After integration, one obtains dnA ¼ 4pðDA þ DB Þ a y cA ð7:5:13Þ dt where 1 ð expðUeA =ðRTÞÞ dr ð7:5:14Þ y1 ¼ a r2 a

The factor a is introduced into the fraction y, so that it is deﬁned as a dimensionless quantity. Equation (7.5.13) applies to one molecule of B. By multiplying by the concentration of B, cB , one obtains an expression for the reaction rate:

CHEMICAL REACTION KINETICS IN SOLUTION

dnA ¼ 4pðDA þ DB Þ a y cA cB dt

331

ð7:5:15Þ

The second-order rate constant is kr ¼ 4pNL ðDA þ DB Þ a y

ð7:5:16Þ

where Avogadro’s constant NL is introduced so that this quantity has units of m3 mol1 s1 when DA and DB have units of m2 s1 and a, units of meters. When the two reactants are molecules, the electrostatic energy UeA is zero and the fraction y is equal to one. Then, the expression for kr is kr ¼ 4pNL ðDA þ DB Þ a

ð7:5:17Þ

When the two reactants are ions, the value of y is most easily estimated for the case that the solution is very dilute and the electrostatic energy is given by UeA ¼

zA zB e20 NL a ¼ r 4pe0 es r

ð7:5:18Þ

where e0 is the electronic charge, es , the relative permittivity of the solvent, and e0 , the permittivity of free space. Integration of equation (7.5.14) gives the ﬁnal result 1 a y ¼ ea=a 1 ð7:5:19Þ a Finally, if the bimolecular reaction involves the same species, namely, A þ A ! products

ð7:5:20Þ

the rate of reaction is given by

dcA ¼ 2kr c2A dt

ð7:5:21Þ

In this case the expression for the diﬀusion-limited bimolecular rate constant is kr ¼ 4pNL DA a y

7:5:22Þ

EXAMPLE

Estimate the diﬀusion-limited rate constant for the recombination of I radicals in carbon tetrachloride at 258C given that the radius of the radical is 200 pm, and its diﬀusion coeﬃcient, 4:12 109 m2 s1 . The reaction is ð7:5:23Þ I þI ! I2 Since the reactants are neutral, electrostatic eﬀects are negligible and y is equal to one. The collision distance a is equal to 400 pm. Thus, kr ¼ 4p 6:022 1023 4:12 109 4:0 1010 ¼ 1:27 107 m3 mol1 s1

ð7:5:24Þ

The experimentally observed value for kr is 8:2 106 m3 mol1 s1 . The fact that the experimental value is smaller than that predicted theoretically suggests that there may be a small activation energy associated with this process.

332

LIQUIDS, SOLUTIONS, AND INTERFACES

EXAMPLE

Estimate the diﬀusion-limited rate constant for recombination of H3O+ and OH to form water at 258C given that the diﬀusion coeﬃcients for these species are 9:34 109 m2 s1 and 5:23 109 m2 s1 , respectively, and assuming that the eﬀective radius for each species is 250 pm. The reaction is H3 Oþ þ OH ! 2H2 O

ð7:5:25Þ

This reaction is accelerated by the attractive Coulombic force between the ions. The electrostatic parameter is zA zB e20 NL 1 ð1:602 1019 Þ2 6:022 1023 ¼ 4pe0 es RTa 4p 8:854 1012 78:5 8:3145 298:2 5 1010 ð7:5:26Þ ¼ 1:429 The electrostatic factor y is y¼

1:429 ¼ 1:879 expð1:429Þ 1

ð7:5:27Þ

The estimate of the rate constant is kr ¼ 4p 6:022 1023 14:57 109 5 1010 1:879 ¼ 1:04 108 m3 mol1 s1

ð7:5:28Þ

The experimental value of kr is 1:4 108 m3 mol1 s1 . The theoretical estimate is low probably because the model of the reactants as hard spheres is inappropriate for this system. This follows from the fact that the reacting ions have a special relationship to the solvent, water. Although the treatment described here is approximate it certainly gives the correct order of magnitude for the rate constants of very fast reactions in solution. Methods of improving these estimates have been considered by Noyes [2]. In the following section experimental methods of studying very fast reactions based on relaxation techniques are considered.

7.6 Relaxation Techniques for Rapid Reactions Relaxation techniques were developed by Eigen and coworkers [13, 14] as a means of studying very fast reactions in solution. The experiment involves applying a stress rapidly to a system at equilibrium and following the time lag required to establish new equilibrium concentrations. For example, in the temperature jump method, the temperature of the system can be changed by several degrees in a time as short as 106 s by dissipating the energy stored in a high-voltage capacitor in a solution which contains a conducting electrolyte. Spectrophotometric or conductometric methods are then used to follow the relaxation of the components of the equilibrium to new concentrations which reﬂect the value of the equilibrium constant at the higher temperature. Then, the proﬁle of concentration against

CHEMICAL REACTION KINETICS IN SOLUTION

333

time is analyzed to ﬁnd the rate constants characterizing the reactions taking place. In the following section some simple examples are considered to illustrate how the relaxation data are analyzed. Some important experimental techniques used are also outlined. A. Analysis of the Experimental Data As a simple example, consider the equilibrium in a homogeneous reaction kf

AþB ÐC kb

ð7:6:1Þ

which is characterized by a forward rate constant kf and a backward rate constant kb . At equilibrium cCe k ¼ Ke ¼ f cAe cBe kb

ð7:6:2Þ

The initial concentrations of these components which are present before the stress is applied to the system are cA0, cB0, and cC0. Then at any moment during the relaxation one may write x ¼ cA cAe ¼ cB cBe ¼ cCe cC

ð7:6:3Þ

where x is the extent that the instantaneous concentrations diﬀer from their new equilibrium values. The rate of reaction is

dcA dx ¼ ¼ kf cA cB kb cC dt dt

ð7:6:4Þ

This equation is now written in terms of the variable x: dx ¼ kf ðcAe þ xÞðcBe þ xÞ kb ðcCe xÞ dt ¼ kf cAe cBe þ kf xðcAe þ cBe Þ þ kf x2 kb cCe þ kb x

ð7:6:5Þ

From equation (7.6.2) kf cAe cBe kb cCe ¼ 0

ð7:6:6Þ

2

Furthermore, the term in x may be neglected because the departure from equilibrium is not large. The ﬁnal expression for the rate of reaction is dx ¼ ½kf ðcAe þ cBe Þ þ kb x dt

ð7:6:7Þ

This has the same form as the expression derived for a simple ﬁrst-order reaction with the quantity in the square brackets as the ﬁrst-order rate constant. The usual practice in relaxation kinetics is to deﬁne the reciprocal of this quantity as the relaxation time t for the reaction. Thus, t ¼ ½kf ðcAe þ cBe Þ þ kb 1

ð7:6:8Þ

By varying the equilibrium concentration of one reactant while holding the concentration of the other high and approximately constant, the two rate constants

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LIQUIDS, SOLUTIONS, AND INTERFACES

Table 7.4 Relaxation Times for Equilibrium Reactions Involving One Step Reaction

Relaxation Time/t

kf

ðkf þ kb Þ1

AÐ B kb

kf

ð4kf cAe þ kb Þ1

2A Ð B kb

kf

½kf ðcAe þ cBe Þ þ kb 1

AþB Ð C kb kf

AþB Ð CþD kb

½kf ðcAe þ cBe Þ þ kb ðcCe þ cDe Þ1

which contribute to t can be separated. Relaxation times for several other mechanisms involving one step are summarized in table 7.4. If these reactions do not correspond to equilibria but actually are irreversible, then the relaxation time is found by setting kb equal to zero. Most chemical reactions involve more than one elementary step. As a result there is more than one relaxation time associated with the reacting system. In the case of fast reactions it is necessary to extract information about each step. Although the mathematical analysis of complex systems can be quite involved, it is important to understand how such an analysis is made. The following example involves a two-step mechanism in which both steps are reversible. Consider a bimolecular process k1f

A þ B Ð AB k1b

ð7:6:9Þ

in which the reactive intermediate AB is formed. The forward rate constant for this reaction is k1f and the backward rate constant, k1b. The species AB can undergo an intramolecular rearrangement k2f

AB Ð C k2b

ð7:6:10Þ

to form the ﬁnal product C. The second step is characterized by a forward rate constant k2f and a backward rate constant k2b. Now one deﬁnes variables which relate to the departure of each concentration from its equilibrium value. They are xA ¼ cA cAe ¼ cB cBe xAB ¼ cAB cABe xC ¼ cC cCe

ð7:6:11Þ ð7:6:12Þ ð7:6:13Þ

Mass conservation requires that xA þ xAB þ xC ¼ 0 The rate of reaction of A is

ð7:6:14Þ

CHEMICAL REACTION KINETICS IN SOLUTION

dcA ¼ k1f cA cB k1b cAB dt

335

ð7:6:15Þ

In terms of the variables xA and xAB this becomes

dxA ¼ k1f ðxA þ cAe ÞðxA þ cBe Þ k1b ðxAB þ cABe Þ dt

ð7:6:16Þ

When this expression is simpliﬁed by neglecting terms in x2 and using the equilibrium condition k1f cAe cBe ¼ k1b cABe

ð7:6:17Þ

one obtains

dxA ¼ k1f ðcAe þ cBe ÞxA k1b xAB dt

ð7:6:18Þ

The expression for the rate of formation of the intermediate AB is dcAB ¼ k1f cA cB k1b cAB k2f cAB þ k2b cC dt

ð7:6:19Þ

On the basis of equilibrium (7.6.10) k2f cABe ¼ k2b cCe

ð7:6:20Þ

Making use of equations (7.6.11)–(7.6.14), the expression for the rate of formation xAB is dxAB ¼ ½k1f ðcAe þ cBe Þ k2b xA ðk1b þ k2f þ k2b ÞxAB dt

ð7:6:21Þ

Equations (7.6.18) and (7.6.21) comprise linear ﬁrst-order diﬀerential equations in the variables xA and xAB. For simplicity they may be rewritten as dxA ¼ a11 xA þ a12 xAB dt

ð7:6:22Þ

dxAB ¼ a21 xA þ a22 xAB dt

ð7:6:23Þ

where

a11 ¼ k1f ðcAe þ cBe Þ

ð7:6:24Þ

a12 ¼ k1b

ð7:6:25Þ

a21 ¼ k1f ðcAe þ cBe Þ þ k2b a22 ¼ k1b þ k2f þ k2b

ð7:6:26Þ ð7:6:27Þ

This system of equations may be solved by several methods including Laplace transformation and matrix algebra. The mathematical details have been given by Connors [G2]. The solutions to these equations may be expressed in terms of the relaxation times t1 and t2 for step 1 and step 2, respectively. The sum of their reciprocals is given by 1 t1 ð7:6:28Þ 1 þ t2 ¼ a11 þ a22 and the product by

336

LIQUIDS, SOLUTIONS, AND INTERFACES

11 t1 2 ¼ a11 a22 a12 a21

ð7:6:29Þ

Using the deﬁnitions of the coeﬃcients given above these become 1 t1 1 þ t2 ¼ k1f ðcAe þ cBe Þ þ k1b þ k2f þ k2b 1 t1 1 t2 ¼ k1f ðcAe þ cBe Þðk2f þ k2b Þ þ k1b k2b

ð7:6:30Þ ð7:6:31Þ

Thus, each relaxation time depends on all four rate constants. The analysis of the experimental data to obtain the two relaxation times is equivalent to the analysis of two consecutive ﬁrst-order reactions. If the values of t1 and t2 are quite diﬀerent, this is not a problem and can be accomplished using standard methods of data analysis [G2]. If t1 and t2 are close to one another, separate determination of these quantities is very diﬃcult. Typically, the ratio of the relaxation times must be greater than three to estimate them separately. In this regard, some variation in the relative values of t1 and t2 can be achieved by changing the values of cAe and cBe. Further analysis of the data involves determining t1 and t2 for a range of values of the sum cAe þ cBe. On the basis of equations (7.6.30) and (7.6.31), the individual rate constants can then be found using linear plots of t11 þ t21 against cAe þ cBe and (t1t2)1 against the same quantity. One limiting case of interest occurs when t1t2. This corresponds to reaction (7.6.9) coming rapidly to equilibrium with the second step (7.6.10) being much slower. Under these circumstances, the terms in k2f and k2b in the expressions for a21 and a22 disappear. The reaction is then termed decoupled because the kinetic processes determining the ﬁrst relaxation can be completely separated from those determining the second. The above system provides one example of the treatment of a relaxation process for a complex reaction involving a two-step process. Other systems involving two or more steps can be treated by this method, and details can be found elsewhere [G2, 14]. B. Some Experimental Methods The temperature jump (T-jump) method is the most widely used relaxation technique. It is based on the temperature dependence of the equilibrium constant for the reaction. For a one-step process such as reaction (7.6.1), this is given by the van’t Hoﬀ equation, that is, @ ln Ke H ¼ @ð1=TÞ R

ð7:6:32Þ

where H is the standard enthalpy change for the reaction. The experiment usually involves a specially designed cell with a small volume of solution. The temperature jump is accomplished via the joule heating which occurs when a capacitor is discharged through the solution, which must be electrically conducting. Typical temperature changes are between 1 and 10 C, and these can be achieved in times as short as 10 ns. Although this technique was originally used with aqueous electrolyte solutions, it can also be applied to systems with much lower conductivity using microwave and laser heating techniques. A typical T-

CHEMICAL REACTION KINETICS IN SOLUTION

337

jump cell designed for use with an electrical discharge is shown in ﬁg. 7.8. Monochromatic light passes through the cell and out to a detector in which the absorption transient is recorded. The relaxation time is determined by analyzing the absorption data as a function of time. EXAMPLE

A system with an equilibrium constant of 5 105 L mol1 at 25 C is subjected to a temperature jump of 5 C. Calculate the new equilibrium constant given that the standard enthalpy change for the reaction is 10 kJ mol1 . On the basis of equation (7.6.32) H 1 1 ln Ke ðT2 Þ ln Ke ðT1 Þ ¼ R T2 T1 ð7:6:33Þ 10,000 1 1 2 ¼ 6:65 10 ¼ 8:314 303:2 298:2 The equilibrium constant at 30 C is 5:3 105 L mol1 corresponding to a 7% percent change. The pressure jump (P-jump) relaxation technique employs a sudden change in pressure for a reversible reaction which has associated with it a signiﬁcant volume change. In this case the change in the equilibrium constant with pressure is given by @ ln Ke V ¼ @P RT

ð7:6:34Þ

The sample is mounted in a ﬂexible cell which is placed in a pressurized vessel at a high pressure, say, 60 atm. The pressure is suddenly reduced by rupturing a diaphragm so that it drops to the atmospheric value. Unfortunately, the value of V associated with most reactions is rather small, so that the equilibrium is only slightly perturbed by the pressure change.

Fig. 7.8 Experimental setup with a temperature-jump cell designed for Joule heating by means of an electrical discharge.

338

LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 7.9 Illustration of the time dependence of the forcing function, for example, pressure, and the concentration of one reactant in (A) a step relaxation experiment and (B) an experiment with a periodic perturbation.

The electrical ﬁeld-jump method is applied to reactions involving ions and dipoles. The ﬁeld induces a shift in the solution equilibrium in the direction of producing more ions, or in the orientation of dipoles. Thus, when a high electrical ﬁeld is applied to a solution containing a weak electrolyte, the extent of dissociation increases. The method has been used to study metal ion complex formation reactions and acid–base reactions. Ultrasound experiments rely on the periodic oscillations which result in a solution when a high-frequency sound wave is passed through it. Sound waves produce periodic oscillations in pressure and temperature which result in a periodical perturbation in a chemical equilibrium (see ﬁg. 7.9). In aqueous solutions, the pressure change is more important, whereas in non-aqueous solutions the temperature change is predominant. When the frequency of the sound wave is high enough, the chemical equilibrium cannot respond quickly enough and a phase shift is observed between the oscillations in the concentration of one component and the oscillations in pressure and/or temperature. Using transducers, these oscillations can be converted to electrical signals and precisely followed using lock-in ampliﬁers or similar electronic instrumentation.

7.7 Laser Spectroscopy and Femtochemistry in Solutions The use of short light pulses to study rapid chemical reactions in solution dates from the work of Norrish and Porter [15]. In a ﬂash photolysis experiment a light

CHEMICAL REACTION KINETICS IN SOLUTION

339

pulse of short duration is used to irradiate the solution, where it is absorbed by molecules or ions. The resulting excited species loses its excess energy either by returning to the ground state via ﬂuorescence or some other process, or by a chemical reaction which produces radicals and other intermediates of high energy. In the latter case, further reactions of these species are monitored using a less energetic ‘‘spectroﬂash’’ ﬁred a short time after the initial photolysis ﬂash. This procedure is repeated several times with diﬀerent delay times between the photolyzing ﬂash and the spectroﬂash. In this way the kinetics of the reactive intermediate can be elucidated. In the original ﬂash photolysis experiment a high-voltage capacitor was discharged through a ﬂash photolysis lamp. The typical energy involved about 1000 J which was dissipated within a few microseconds. This kind of experiment is carried out at present using lasers. Using these devices one is able to resolve chemical events at times less than one picosecond. This allows the experimenter to monitor changes which occur during times for which the vibrational motion in molecules is essentially frozen. These exciting developments have led to the coining of the new word femtochemistry, which refers to chemistry occurring in the femtosecond time scale. In this section the important features of an experiment involving ultrafast pulsed lasers are brieﬂy outlined. Then applications of these experiments to reactions in solution are described. A. Experimental Aspects In order to discuss this topic a brief introduction to the properties of common lasers is given. The word laser is an acronym for light ampliﬁcation by stimulated emission of radiation. A molecule can absorb a quantum of light when the associated energy corresponds exactly to a transition from the ground state in the molecule to an available excited state. Under normal circumstances the excited molecule decays back to the ground state by some process which may be radiative, involving the emission of light, or non-radiative, involving the generation of heat. Photon emission occurring this way is called spontaneous emission. On the other hand, if the system with molecules in the excited state is irradiated with an external light source corresponding in energy to that which would be emitted, each molecule can relax by emitting another photon of the same frequency. Under these circumstances the probability of emission is enhanced by the presence of similar photons and the emission occurs preferentially in the same direction as the external source. This leads to a population inversion with more molecules in the excited state than in the ground state. This type of emission is called stimulated emission and is the principal feature of laser operation. Moreover, since the emission occurs preferentially in the direction of the applied beam, this beam is thereby ampliﬁed in intensity. The stimulated photons, have the same phase and polarization as the incident photons, so that the laser radiation is termed coherent. The above discussion implies that a laser may be designed with a absorbing system involving two quantum states, namely, the ground state and the excited state. However, an analysis of the rate at which molecules reach the excited state from the ground state with respect to the rate at which molecules in the excited

340

LIQUIDS, SOLUTIONS, AND INTERFACES

state return to the ground state by emission shows that a population inversion is not possible with a two-state system. In order to realize a population inversion a multilevel system is required. An example of a three-level system which is capable of lasing action is shown in ﬁg. 7.10. The ground state system at energy E1 is pumped to the excited state E2 . It rapidly undergoes a non-radiative transition to a lower energy level E3 The lasing action involves the emission of a quantum of light corresponding to the energy diﬀerence E3 E1 . Other multiple level lasing schemes have been used to design functioning lasers [15]. The three essential elements of a laser are the resonator containing the gain medium, a pumping source that excites the gain medium, and mirrors which direct the light beam back and forth through the gain medium (see ﬁg. 7.11). One of these mirrors is not completely reﬂective, so that part of the laser beam escapes giving the output light from the laser system. The gain medium can be a solid, a liquid solution, or a gaseous mixture. One the ﬁrst lasers made use of a solid ruby rod which consists of Al2O3 doped with a minor Cr3þ impurity. This system outputs light at a wavelength of 694.3 nm. Other solid-state systems make use of Nd3þ or Ti3þ as the gain medium. Popular lasers with a gas as the gain medium include the He–Ne laser, and the CO2 laser. Of course, the output frequency depends on the characteristics of the gain medium and its associated spectroscopic properties. The energy source used to initiate lasing action is often a high-intensity light source which is used to excite the gain medium. In the case of gas phase lasers, an electrical discharge can be used to provide the energy to pump the gas molecules into the excited state involved in laser action. The properties of the laser cavity are important in achieving the necessary population inversion with more molecules in

Fig. 7.10 Transitions and energy levels in a three-level laser.

CHEMICAL REACTION KINETICS IN SOLUTION

341

Fig. 7.11 Schematic diagram of the essential components of a laser.

the excited state than in the ground state. Thus, the design of the resonating system by which most of the light travels back and forth between the two mirrors shown in ﬁg. 7.11 is an important feature of the system. Lasers are operated in two modes, on a continuous-wave basis, or in a pulsed mode. In the continuous mode the laser is pumped steadily at a rate suﬃcient to supply the emergent laser energy and the heat losses due to the non-radiative processes. The pulsed laser is of interest for kinetic studies because it allows a pulse of optical energy to be applied to the experimental system over a known period of time. In this mode, the laser is pumped with a pulsed energy source, and a shutter is used to control the release of energy in the laser. The shutters may be mechanical or electrooptical, and a variety of pulsing techniques have been developed. A self-mode-locked Ti:sapphire laser is a popular system for ultrafast spectroscopy. It can be tuned to operate over a wide range of frequencies in the visible, it has high average power output, and it can deliver ultrashort pulses down to 10 fs in duration. More details about the generation of ultrashort light pulses can be found elsewhere [16–18]. Having delivered optical energy to the experimental system using an ultrafast pulse, the next part of the experiment involves following the changes which occur over a very short period of time. This is called ultrafast time-resolved spectroscopy. One of these is pump-probe spectroscopy. After the system has been pumped to an excited optical state it is probed by a series of light pulses which are used to follow the response of the system as a function of time. For example, the pump-pulse system may be used to excite a simple molecule from its ground electronic level to an excited electronic level where a bond in the molecule is ruptured. The probe pulses can follow the absorption of light by one of the fragments of the molecule. On the other hand, the probe pulses can be used to induce ﬂuorescence in one of the fragments, and the ﬂuorescence is followed as a function of time. The probing pulse may be the same frequency as the pumping pulse, or it may have a diﬀerent frequency depending on the design of the experiment. The time delay between probe and pulse is obtained by altering the path length for the two light signals in the optical system. When the experimental system emits light after the initial pumping pulse, quite diﬀerent techniques can be used to obtain a time-resolved spectrum of the sample emission. The simplest of these is time-correlated single photon counting. The time resolution of this technique is limited by the design of the photon detectors. Two other methods used in emission spectroscopy are the streak camera and

342

LIQUIDS, SOLUTIONS, AND INTERFACES

ﬂuorescence up-conversion. Details of these experiments which are beyond the scope of this monograph can be found elsewhere [16–18]. Experimental studies of solution reactions using pulse lasers have blossomed since the 1980s. Important examples are considered in the following discussion. B. Time-Resolved Studies of Solvation Very interesting studies have been made of the eﬀects of solvation on the timeresolved ﬂuorescence of certain dye molecules dissolved in polar solvents. The solute molecules are often characterized by a non-polar ground state and a dipolar excited electronic state. As a result, when the excited state is formed by the absorption of a quantum of light, the molecule ﬁnds itself in a solvation environment with which it is not at equilibrium. With increasing time, the solvent restructures responding to the new charge distribution in the excited state. This results in a lowering of the energy of the excited state with respect to the ground state. As a result the ﬂuorescence spectrum changes with time with its peak frequency shifting to lower values. This shift in the emission spectrum in the red direction is called a Stokes shift (SS). Analysis of the spectral features as a function of time is used to obtain molecular details of the solvent restructuring process. The ﬂuorescence spectra are analyzed to obtain the correlation function C(t) deﬁned as CðtÞ ¼

nm ðtÞ nm ð1Þ nm ð0Þ nm ð1Þ

ð7:7:1Þ

where nm(t), nm(0), and nm(1) are the frequency of the maximum on the ﬂuorescence spectra observed at time t, zero, and inﬁnity, respectively. The time dependence of C(t) is related to the relaxation process of the solvent environment around the dipolar excited state. A theoretical model for this process has been derived in which the dye molecule in its excited state is considered to be a point dipole embedded in a dielectric continuum [19]. In this case the correlation function C(t) is an exponentially decaying function of the form t ð7:7:2Þ CðtÞ ¼ exp tL where tL is the longitudinal relaxation time of the solvent (see section 4.7). Studies carried out in a wide variety of solvents show that this simple model is approximately correct. Results obtained in a variety of solvents with the coumarin dye C152 (ﬁg. 7.12) are summarized in table 7.5. It is clear from these results that there is an approximate correlation between the value of the average relaxation time tS and the longitudinal relaxation time tL (see section 4.7). However, careful examination of the time-dependent Stokes shift (TDSS) data reveals that CðtÞ is described by more than one relaxation process. This is not diﬃcult to understand, considering that many aprotic solvents are found to have more than one relaxation process. This can be attributed to the formation of dimers and other aggregates in these liquids. This is especially the case in aprotic solvents such as DMSO and PC, which have very high dipole moments. Alcohols also have multiple relaxation processes due to the presence

CHEMICAL REACTION KINETICS IN SOLUTION

343

Fig. 7.12 Structure of the coumarin dye C152.

of hydrogen bonding (see section 4.7). The most important point is that the Debye model which is applied to the analysis of the dielectric relaxation data is a macroscopic model. What is really needed to interpret the spectroscopic data is a molecular model for the dielectric properties of the solvents. The example given is only one of several which have been reported for solvation phenomena studied at very short times after a molecule absorbs a quantum of light [17, 18]. These studies have only become possible with the advent of highspeed laser technology. C. Ultrafast Photoisomerization Reactions Stilbene is a molecule which exists in two isomeric states because of the large energy barrier associated with rotation around the central double bond (see ﬁg. 7.13). It has been known for some time [22] that when stilbene is excited to its ﬁrst singlet state by absorption of UV light, it rapidly ﬂuoresces to the ground state but with some molecules in the other isomeric form. Thus, if cis-stilbene is irradiated with ultraviolet light, one obtains approximately 50% trans-stilbene and

Table 7.5 Average Relaxation Times from Time-Dependent Stokes Shifts (TDSS) for Coumarin C152 in Various Solvents Together with the Longitudinal Relaxation Time Determined by Dielectric Relaxation Spectroscopy [20] Solvent MeOH AC AcN BzN DMF DMSO PC PrN

Longitudinal Relaxation Time* tL /ps

Average Relaxation Time from TDSS ts =ps

1.0 4.0 0.3 0.2 5.7 0.7 1.5 2.4 0.4

6.2 — 0.68 0.52 5.0 1.0 1.2 2.4 0.85

*The values of tL for aprotic solvents with more than one relaxation process are averages; two values are given for methanol, which has three relaxation processes. (See section 4.7 and [21].)

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LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 7.13 Isomerization of stilbene.

50% cis-stilbene. The kinetics of the photoisomerization reaction is so rapid that it can only be resolved using a pump-pulse laser technique. These experiments allow one to study the role that the solvent plays in the kinetics of the isomerization process. Detailed studies of this photo-initiated reaction have led to a picture of the energetics associated with the isomerization reaction (ﬁg. 7.14). When cis-stilbene reaches the excited singlet state, it rapidly loses energy by twisting about the central double bond. It then returns to the ground state from the energy minimum by ﬂuorescence and gives either the trans or cis isomer, depending on how the molecule leaves the twisted form to return to a low-energy conﬁguration in the ground state. For example, when cis-stilbene is irradiated at 316 nm emitted light can be observed at 475 nm when the experiment is carried out at 20 C in decane. The ﬂuorescence decay occurs over a few picoseconds and in this case is characterized by relaxation time t of 1.27 ps [23]. Studies have been made of the

Fig. 7.14 Energy proﬁles for twisting of the stilbene molecule in the ground state S0 and ﬁrst excited singlet state S1 .

CHEMICAL REACTION KINETICS IN SOLUTION

345

dependence of t on solvent nature, temperature, and the nature of substituents on the stilbene molecule [22]. Fluorescence decay times measured in several polar solvents are summarized in table 7.6. It is apparent there is a correlation between the viscosity and t, longer decay times being observed in more viscous solvents. This eﬀect is related to the friction of the solvent molecules surrounding stilbene when it undergoes rotation in the excited state. Solvent friction in solution kinetics is of considerable theoretical interest (see section 7.10). In order to determine the activation parameters associated with photoisomerization, decay times have been measured in a series of solvents with the same structure. The temperature of the experiment is varied so that the solvents all have the same viscosity. A plot of the logarithm of t against the reciprocal of the absolute temperature using data obtained in this way is known as an isoviscosity plot. Some data for the isomerism of cis-stilbene in the alkanes are given in table 7.6. The decay times observed in these solvents are longer than those obtained in polar solvents indicating that a larger energy barrier is involved. The isoviscosity plot of these data gives an activation energy of 4.6 kJ mol1. The corresponding result for trans-stilbene is 14.6 kJ mol1 [22]. These results are consistent with the energy proﬁle for the excited state shown in ﬁg. 7.14. When the same reaction is studied in polar solvents, namely, a series of nalkanenitriles, the barrier is signiﬁcantly lower (10.8 kJ mol1). This suggests that the excited state has some polar character. Much more has been learnt about this simple unimolecular reaction on the basis of the ultrafast laser studies. It is clear from the data presented in table 7.6 that the study could not be carried out without the ability to resolve the spectral data on a time scale less than 1 ps. This kind of experiment has been applied to several important solution reactions and represents the forefront of modern research in this area.

Table 7.6 Cis-stilbene Fluorescence Decay Times in Organic Solvents for Emission at 475 nm [23] Solvent

Temperature T/8C

Viscosity Z/mPA s

Decay Time t/ps

20 20 20 20 20

0.38 2.20 0.59 2.38 4.18

0.38 0.63 0.50 0.80 0.94

7 31 64 87

0.78 0.78 0.78 0.78

1.58 1.23 1.02 0.92

Polar Acetonitrile Dimethyl sulfoxide Methanol* 2-Propanol n-Pentanol Non-polar Octane Decane** Dodecane Tetradecane *Measured at 465 nm. **Measured at 450 nm.

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LIQUIDS, SOLUTIONS, AND INTERFACES

7.8 The Theory of Homogeneous Electron Transfer The theory of electron transfer is the most developed of that for any reaction in solution. In addition this type of reaction has been studied extensively by experiment. As a result many of the predictions made by theory have been addressed in the laboratory. As was seen from earlier discussion, simple electron transfer may proceed by two mechanisms, inner sphere and outer sphere. The inner sphere mechanism involves both electron and atom transfer. The discussion here is limited to the outer sphere mechanism, which involves the transfer of a single electron without the formation or breaking of any chemical bonds. Within the context of transition state theory there are two problems to be addressed in developing a model to estimate the rate constant for electron transfer. One is the estimation of the Gibbs energy of activation for the process; the second is estimation of the pre-exponential factor. The model used to describe the activation barrier is considered ﬁrst. Consider the simple electron transfer process Aþ þ B ! A þ B þ

ð7:8:1Þ

+A Aþ þ e( þ B þ e( +B

ð7:8:2Þ

involving the redox couples ð7:8:3Þ

When the reactants are at the site where electron transfer occurs, rearrangement of the bond lengths and angles within the reactants must take place to achieve a conﬁguration intermediate between that of the reactants and that of the products. In addition, there must be some rearrangement of the solvation atmosphere 2þ around charged reactants. For example, if CrðH2 OÞ3þ 6 and FeðH2 OÞ6 are the reactants, the length of the metal ion–ligand bonds must change. In the case of CrðH2 OÞ3þ 6 this means that the bonds are lengthened to achieve a structure inter2þ 2þ mediate between that of CrðH2 OÞ3þ 6 and CrðH2 OÞ6 . In the case of FeðH2 OÞ6 , the length of the metal ion–ligand bonds decreases to achieve a conﬁguration closer to that of FeðH2 OÞ3þ 6 . This process is called inner sphere reorganization. At the same time there must be a readjustment of the solvation atmosphere around these highly charged ions. According to the Born model (section 3.4), the Gibbs energy of ion solvation depends on the square of the ionic charge. The solvation atmosphere must readjust to achieve a solvation conﬁguration intermediate between that of reactants and product. This process is called outer sphere reorganization. Both inner sphere and outer sphere reorganization contribute to the energy barrier associated with electron transfer. These processes occur randomly due to the thermal ﬂuctuations in the system. According to the Frank–Condon restriction, electron transfer is only possible when this, yet unspeciﬁed, intermediate conﬁguration is achieved. In order to visualize the energy barrier between reactants and products, it is assumed that each system can be represented as a classical harmonic oscillator along the reaction coordinate. This is illustrated in ﬁg. 7.15. The left-hand parabola gives the Gibbs energy of the reactants and the right-hand parabola, that of

CHEMICAL REACTION KINETICS IN SOLUTION

347

Fig. 7.15 Gibbs energy barrier for an outer sphere electron transfer process with a standard Gibbs energy change of G(A). Interaction of the surfaces for reactants and products leads to electronic coupling and creates an energy gap 2Jr at the point where they intersect (B).

the products. There are three important characteristics of this diagram. The ﬁrst is that the curvature of the two parabolas is assumed to be the same. This is equivalent to assuming that the work done to achieve a given non-equilibrium conﬁguration of the reactant system is the same as that in the product system when measured from the parabola’s minimum. This is clearly an approximation which may not be valid for some systems. A second important feature of the diagram is the distance between the minima of the two parabolas. This is related to the relative positions of the two reactants before reaction and determines the height of the energy barrier along the reaction coordinate. As the distance between the minima increases the value of the Gibbs energy at the point where they intersect increases. This corresponds to an increase in the Gibbs energy of activation for the reaction. The third feature of the energy barrier diagram shown in ﬁg. 7.15 is related to the point where the Gibbs energy surfaces meet. Electron transfer is not possible if the individual parabolas remain intact. Electronic coupling of the energy surfaces is necessary if the system is to pass from the reactant state to the product state. This is illustrated as a fusing of the lower portions of the two parabolas at the intersection point to create a gap of height 2Jr between the lower and upper portions of the curve. The magnitude of Jr determines whether the reaction is adiabatic or diabatic (non-adiabatic). An adiabatic reaction is one for which the electronic coupling is strong. This means that every reactant system which reaches the activated state, that is, the intersection point of the two parabolas, passes through to the product state if solvent friction is absent. If the electronic coupling is weak for some reason, so that Jr kT, then the reaction is diabatic and the passage from the reactant state to products is less probable. An important factor involved in determining the degree of reaction adiabacity is the distance to which the species A and B, which exchange an electron, can approach one another. If this distance is large, then the degree of electronic coupling is small and the reaction is diabatic. An important example of this type of electron transfer occurs in biochemistry when one of the redox centers is embedded in a large biomolecule,

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LIQUIDS, SOLUTIONS, AND INTERFACES

for example, in a protein. On the other hand, electron transfer reactions involving 2þ simple transition metal systems such as CrðH2 OÞ3þ 6 =CrðH2 OÞ6 are expected to be adiabatic because the reactants can approach one another to distances the order of 500 pm. The estimation of a rate constant involves determination of the Gibbs activation energy and the pre-exponential factor. An important contribution to the latter is the Coulombic work which is involved in bringing charged reactants to the reaction site where the precursor complex is formed. The Gibbs activation energy and pre-exponential factor are the central components of the transition state model. Methods of estimating the Gibbs activation energy are discussed ﬁrst of all. A. Estimation of the Gibbs Activation Energy Using the properties of the parabolas shown in ﬁg. 7.15, one can determine a relationship between the Gibbs energy of activation and the shape and position of the parabolas on the reaction coordinate. The equation giving the Gibbs energy of the reactant system is GR ¼

kf ðq qR Þ2 2

ð7:8:4Þ

where kf is the force constant describing the curvature of the parabola, q, the position on the reaction coordinate, and qR, the position of the minimum for the reactant system. The corresponding equation for the product system is GP ¼

kf ðq qP Þ2 þ G 2

ð7:8:5Þ

where qP is the position of the minimum for the parabola corresponding to the products and G , the standard Gibbs energy change for the reaction. In the case of homonuclear reactions, G is zero. At the intersection point where q is equal to qC, the values of GR and GP are equal so that kf ðqC qR Þ2 kf ðqC qP Þ2 ¼ þ G 2 2

ð7:8:6Þ

Solving this equation for qC, one obtains qC ¼

qP þ qR G þ 2 kf ðqP qR Þ

The Gibbs energy at the intersection point can now be calculated:

2 k ðq qR Þ2 kf qP qR G GR ðqC Þ ¼ f C ¼ þ 2 2 2 kf ðqP qR Þ

ð7:8:7Þ

ð7:8:8Þ

In order to simplify the last equation the reorganization energy, El, is introduced, where El ¼

kf ðqP qR Þ2 2

ð7:8:9Þ

CHEMICAL REACTION KINETICS IN SOLUTION

349

El is the energy that the reacting system must be given to reach the position qP where there is a minimum on the Gibbs energy surface for the products. Substituting equation (7.8.9) into equation (7.8.8) and simplifying, one obtains G ¼ GR ðqC Þ GR ðqR Þ ¼

ðEl þ G Þ2 4El

ð7:8:10Þ

This is the fundamental relationship in electron transfer theory relating the Gibbs activation energy to the reorganization energy El and the standard Gibbs energy change for the reaction G . There are several points which should be noted about equation (7.8.10). First, although the theory is developed here within the context of the transition state model, the estimate of the Gibbs energy of activation is made without factoring out the degree of freedom associated with passage through the transition state. Thus, G is similar to the quantity deﬁned in equation (7.4.31). Second, the rounding oﬀ of the barrier due to electronic coupling was neglected in deriving equation (7.8.10). If this is not a negligible eﬀect, it is easily accounted for by subtracting JR from the right-hand side of equation (7.8.10). Finally, when the reaction is homonuclear, so that G is zero, G ¼

El 4

ð7:8:11Þ

As was discussed above, the reorganization of the reacting system required to reach the transition state is made up of two components, namely, an inner sphere contribution and an outer sphere contribution. It follows that G ¼ Gis þ Gos

ð7:8:12Þ

where Gis is the inner sphere reorganization energy, and Gos , the corresponding quantity for outer sphere reorganization. Details regarding estimation of these quantities are given later in this section. According to equation (7.8.10) the Gibbs activation energy is a quadratic function of the driving force for the reaction, namely, G . This has important consequences when the driving force is large in magnitude. As G increases, one eventually reaches a value at which it is equal to El. Under these circumstances the reaction proceeds without a Gibbs energy barrier. Such a reaction is called activationless. Thus, if one moves the right-hand parabola down with respect to the one on the left, one eventually reaches a position where the righthand parabola crosses the left-hand parabola at its minimum (ﬁg. 7.16). If the value of G is even greater then the right-hand parabola intersects the lefthand one on its left side. Clearly the Gibbs activation energy is again positive in spite of the large driving force for the reaction. The region in which the rate of reaction decreases with further increase in G is called the Marcus inverted region. The variation in G with the standard Gibbs energy change G is shown in ﬁg. 7.17 for typical values of the parameters involved. An interesting demonstration of the inverted Marcus region has been made with a reaction involving photo-induced back electron transfer within a radical ion pair. Thiophene (diaminophenothiazin-5-ium chloride) is a dye molecule which may be excited to the triplet state using UV light. In the triplet state the

350

LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 7.16 Conﬁguration of the Gibbs energy surfaces for reactants and products for (A) an activation less process and (B) a process in the Marcus inverted region.

thiophene cation is strongly electrophilic and forms an ion pair with an donor molecule in solution as follows: 3

THþ þ D ! 3 ½TH D þ

ð7:8:13Þ

where 3THþ is the triplet state of the thiophene cation and D, the donor molecule. When this reaction is carried out in buﬀered methanolic solution, the ion pair disappears in two reactions [24]. One is a dissociation step which breaks up the ion pair and gives thiophene in the ground state:

Fig. 7.17 Plot of the Gibbs activation energy G against the Gibbs energy barrier G for an electron transfer reaction with El equal to 100 kJ mol1.

351

CHEMICAL REACTION KINETICS IN SOLUTION 3

½TH D þ ! TH þ D þ

ð7:8:14Þ

The other reaction involves electron transfer within the ion pair with subsequent dissociation: 3

½TH D þ ! THþ þ D

ð7:8:15Þ

The rate constant for the electron transfer step has been measured in the presence of at least eight diﬀerent donors with increasingly positive redox potentials. These data are summarized in table 7.7. As the driving force for the reaction increases the rate constant increases, reaches a maximum, and then decreases again. When the rate of reaction is a maximum, there is no barrier to electron transfer and the process is activationless. This condition is reached for a rate constant of 1 109 s1 . For higher driving forces, the electron transfer reaction occurs in the Marcus inverted region. Other examples of this behavior have been described in the literature. They give strong conﬁrmation of the model for electron transfer presented here. An important task in the theory of electron transfer is estimation of the reorganization energy El. As discussed earlier, this consists of inner sphere and outer sphere contributions. These are now considered in more detail. B. The Inner Sphere Reorganization Energy The model for the inner sphere reorganization was originally based on simple electron transfer reactions involving octahedral inorganic complexes. For a homonuclear electron exchange reaction such as MðH2 OÞz1 þ MðH2 OÞzn ( +MðH2 OÞzn þ MðH2 OÞz1 n n

ð7:8:16Þ

the work required to achieve a common value l* of the M–OH2 distance in each reactant is given by [5] nk nk ð7:8:17Þ w ¼ A ðl lA Þ2 þ B ðl lB Þ2 2 2 Table 7.7 Kinetic Data for Reaction (7.8.15) with Various Donor Molecules Together with the Standard Potentials for Oxidation of the Donor Molecule to its Cation Radical [24]

Donor Molecule N; N; N0 ; N0 -Tetramethyl-P-phenyldiamine p-Phenylenediamine p-Aminodiphenylamine 1-Naphthylamine Diphenylamine 9,10-Dimethylanthracene 1,2,4-Trimethoxybenzene 9-Methylanthracene

Electron Transfer Rate Constant 107 ket =s1

Standard Redox Potential E /V

3.3 3.3 15.0 100 95 42 28 17

0.16 0.18 0.27 0.54 0.83 1.05 1.12 1.16

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LIQUIDS, SOLUTIONS, AND INTERFACES

where ki and li are the force constant and equilibrium bond length, respectively, in reactant i, and n, the number of ligands. When this work is minimized with respect to l*, one ﬁnds that l ¼

kA lA þ kB lB kA þ kB

ð7:8:18Þ

Substitution of this expression for l * into equation (7.8.17) leads to the result w ¼

nkA kB ðlB lA Þ2 2ðkA þ kB Þ

ð7:8:19Þ

For this type of system, the entropy associated with inner sphere reorganization is negligible [5]. Therefore, the work can be equated to the corresponding change in Gibbs energy, so that Gis ¼ NL

nkA kB ðlB lA Þ2 2ðkA þ kB Þ

ð7:8:20Þ

where NL is the Avogadro constant. For a more general reaction, the following formulae have been given by Marcus [4]: Gis ¼ NL m2 E li

ð7:8:21Þ

1 G þ ðc GP c GR Þ m¼ 2 2El

ð7:8:22Þ

where

and E li ¼

X kjR kjP ðlj Þ2 k þ k jR jP j

ð7:8:23Þ

c GP and c GR are the Coulombic work terms for the reactants and products, respectively. They are discussed in more detail below in the section dealing with formation of the precursor complex. kjR and kjP are the force constants of the jth vibrational coordinate in a species when it participates as a reactant and product, respectively, and lj is the associated change in bond length. The latter quantities allow one to calculate the work associated with inner sphere reorganization. EXAMPLE

Calculate the inner sphere reorganization energy for the homonuclear electron 3þ transfer between Fe(H2O)2þ 6 and Fe(H2O)6 given that the force constant for 1 the Fe(II)–OH2 bond is 149 N m and its equilibrium length 221 pm, and that the corresponding quantities for the Fe(III) complex are 416 N m1 and 205 pm, respectively. The bond length required in these species to achieve electron transfer is l ¼

149 221 þ 416 205 ¼ 209 pm 149 þ 416

ð7:8:24Þ

CHEMICAL REACTION KINETICS IN SOLUTION

353

This result shows that the required bond length is closer to that in the Fe(III) complex than in the Fe(II) complex. The inner sphere reorganization energy is 6 149 416ð205 221Þ2 1024 ¼ 50:7 kJ mol1 2ð149 þ 416Þ ð7:8:25Þ The above example illustrates the estimation of Gis for a homonuclear reaction involving octahedral transition metal complexes. Most systems are more complicated. However, if the details of the changes in bond lengths and bond angles which accompany electron transfer are known, estimates of Gis can be made. Quantum-mechanical calculations are often required to obtain the necessary information. Gis ¼ 6:023 1023

C. The Outer Sphere Reorganization Energy Estimation of the outer sphere reorganization energy is based on a continuum description of the solvent around the reacting species and therefore is based on the Born model of ionic solvation. Marcus [25] has described a two-step charging process in which the non-equilibrium polarization of the dielectric medium is produced in a reversible manner. The two reactants A and B with radii rA and rB and charges zA and zB are imagined to be conducting spheres whose centers are separated by a distance a at the moment of electron transfer. In step 1, the charge on each reactant is changed from zi to zi0 , a value intermediate between what it has as reactant and what it has as product. At the same time the surrounding solvent molecules readjust their orientation to accommodate the change in charge. Since the reversible work is estimated on the basis of a continuum model, it depends on the static permittivity of the solvent es. In step 2, the charge on each reactant is changed back from the intermediate value zi0 to the original value zi , holding the solvent molecules ﬁxed. The medium can only respond to this change through its polarizability. It follows that the reversible work associated with this step depends on the optical permittivity of the solvent medium, eop. The Gibbs energy required to produce the non-equilibrium polarization of the solvent in the transition state for transfer of one electron was shown by Marcus [4, 25] to be e2 1 1 1 1 1 ð7:8:26Þ þ Gl0 ¼ 0 4pe0 2rA 2rB a eop es where e0 is the permittivity of free space, and e0, the fundamental electronic charge. The term involving the reciprocal relative permittivities is called the Pekar factor. Since eop is so much smaller than es, the contribution of the rapid polarization of the medium at optical frequencies dominates, especially in solvents with high static permittivities such as water. The outer sphere contribution to the Gibbs energy of activation is equal to Gos ¼ NL m2 Gl0 where the fraction m is given by equation (7.8.22).

ð7:8:27Þ

354

LIQUIDS, SOLUTIONS, AND INTERFACES

EXAMPLE

Estimate the outer sphere reorganization energy for the homonuclear electron 3þ transfer between FeðH2 OÞ2þ 6 and FeðH2 OÞ6 assuming that the radii of the reactants are 353 and 337 pm, respectively, and that the Coulombic work terms CGR and CGP are zero. In the transition state the reactants are assumed to be in contact, so that a is equal to rA þ rB. The values of the radii of the aquo complexes are assumed to be 132 pm larger than the M–O bond lengths to account for the contribution of the water molecules to the overall radius. The distance factor in equation (7.8.26) is 1 1 1 1 1 1 þ ¼ 1:45 103 pm1 þ ¼ 2rA 2rB a 706 674 690

ð7:8:28Þ

The Pekar factor is 1 1 1 1 ¼ 0:550 ¼ eop es 1:776 78:46

ð7:8:29Þ

Thus, the outer sphere contribution to the Gibbs reorganization energy is Gl0 ¼

ð1:602 1019 Þ2 1:451 109 0:550 ¼ 1:840 1019 J molec1 4p 8:854 1012 ð7:8:30Þ

For a homonuclear reaction G is zero. Since the Coulombic work terms are neglected in this calculation, the factor m is equal to 1/2. It follows from equation (7.8.27) that the outer sphere contribution to the Gibbs activation energy is equal to Gos ¼

6:022 1023 1:840 1019 ¼ 27:7 kJ mol1 4

ð7:8:31Þ

This is considerably less than the corresponding inner sphere contribution but certainly not negligible. On the basis of the analysis described by Marcus [25], the Gibbs energy Gl0 may be written as the diﬀerence between a Born solvation term at static frequencies and one at optical frequencies. Thus, from equation (7.8.26) e20 1 e20 1 Gl0 ¼ ð7:8:32Þ 1 1 es eop 4pe0 Ra 4pe0 Ra where 1 1 1 1 ¼ þ Ra 2rA 2rB a

ð7:8:33Þ

The Born model certainly overestimates the solvation term at static frequencies. Using the MSA to account for the eﬀects of dielectric saturation, equation (7.8.32) can be rewritten as e20 1 e20 1 Gl0 ¼ 1 1 ð7:8:34Þ es eop 4pe0 ðRa þ ds Þ 4pe0 Ra

CHEMICAL REACTION KINETICS IN SOLUTION

355

where ds is the MSA parameter which corrects the ionic radii to give a more realistic estimate of solvation energies (see section 3.5). The parameter ds depends mainly on the nature of the solvent but also on whether the ion is positive or negative. This modiﬁcation of the estimates of Gl0 has been used to discuss experimentally observed diﬀerences in the rates of one-electron oxidation and reduction of the same molecule [26]. Other modiﬁcations of the original Marcus model have been suggested [27]. Many reactants are not spherical in shape and are better approximated as ellipsoids. In this case a much more complex expression for the eﬀective distance R is obtained which depends on the length of the two axes which describe the shape of the ellipsoid. Another improvement in the model is to describe each reactant as a dielectric cavity with ﬁxed charges located within it. In this case, the calculation of Gl0 requires a description of the charge distribution within the reactants and an estimate of the local permittivity in the dielectric cavity. Attempts have been made to test the Marcus estimate of Gl0 experimentally. One way of doing this is to study the variation in electron transfer rate constant with solvent nature. This results in a change in the Pekar factor and thus in Gl0 . More will be said about these experiments in section 7.10. D. The Pre-Exponential Factor The pre-exponential factor is the most diﬃcult part of the rate constant to estimate. Although fairly good estimates can be made of the Gibbs activation energy using concepts based on transition state theory, this theory is not applicable to estimation of the pre-exponential factor in a condensed medium. Factors that need to be considered in a successful model include the electronic coupling parameter J and the role of solvent friction in determining the rate of crossing the Gibbs energy barrier. These features were included in a model developed by Zusman [28]. The pre-exponential factor may be expressed as A ¼ Kp net

ð7:8:35Þ

where Kp is the equilibrium constant for formation of the encounter complex between the two reactants, and net is a frequency associated with electron transfer. The parameter Kp can be estimated from the Eigen–Fuoss model, which was originally derived to estimate the ion pair formation constant (see section 3.10). One of the ions is assumed to have a radius a which is equal to the distance between the reactants at the reaction site. Other ions of the redox couple which are within or on a sphere of radius a can react. The estimate of Kp also takes into consideration any electrostatic work done to bring the reactants to the reaction site. The equation for Kp in units of M1 is Kp ¼

w 4000pNL a3 el exp 3 RT

ð7:8:36Þ

where wel is the electrostatic work and NL , the Avogadro constant, the distance a being expressed in meters. In inﬁnitely dilute solutions the electrostatic work is

356

LIQUIDS, SOLUTIONS, AND INTERFACES

wel ¼

zA zB NL e20 8pe0 es a

ð7:8:37Þ

where zA and zB are the valences associated with the reactants A and B, e0 , the electronic charge, es , the relative permittivity of the solvent, and e0 , the permittivity of free space. At ﬁnite solution concentrations, the shielding eﬀect of the ionic atmosphere should be considered in estimating wel . This can be done using the extended Debye–Hu¨ckel theory or the MSA. EXAMPLE

Estimate the equilibrium constant for formation of the encounter complex between the Fe3þ and Fe2þ ions in water at inﬁnite dilution and 25 C, assuming that the distance of closest approach is 690 pm. The electrostatic work is equal to wel ¼

3 2 6:022 1023 ð1:602 1019 Þ2 ¼ 7:697 kJ mol1 8p 8:854 1012 78:46 6:9 1010

The value of Kp is 4000p 6:022 1023 ð6:9 1010 Þ3 7:697 103 exp Kp ¼ 3 8:3145 298:2 ¼ 0:0372 M1

ð7:8:38Þ

!

ð7:8:39Þ

The value of Kp for reactions in which wel is zero is close to unity. When the reactants are charged with the same sign the repulsion results in a lower value of Kp. In the estimate of wel made at zero ionic strength, the repulsive eﬀect is overestimated, and the estimate of Kp is too low. The frequency associated with electron transfer, net, is estimated by quantum mechanics and depends on the degree of reaction adiabacity as measured by the coupling parameters, Jr and the reorganization energy, El. In the case of adiabatic reactions it also depends on the longitudinal relaxation time of the solvent, tL. A general expression for net is nd ð7:8:40Þ net ¼ 1 þ ga where nd is the frequency associated with a diabatic electron transfer process, and ga, the adiabacity factor. When ga 1, the value of net is nd/ga, which is the frequency associated with an adiabatic electron transfer. From quantum mechanics, the expression for nd is nd ¼

4p2 J2r NL hð4pEl RTÞ1=2

ð7:8:41Þ

8p2 J2r tL NL hEl

ð7:8:42Þ

and that for ga, ga ¼

When Jr is large and the reaction adiabatic, the expression for net is

357

CHEMICAL REACTION KINETICS IN SOLUTION

net ¼

1=2 nd 1 El ¼ ga tL 16pRT

ð7:8:43Þ

EXAMPLE

Estimate the electron transfer frequency for a reaction occurring in water at 25 C for which El is 314 kJ mol1, assuming that Jr is 0.1 kJ mol1. Repeat the calculation for the case that Jr is 5 kJ mol1. Compare these estimates with the value obtained for an adiabatic electron transfer. The estimate of the diabatic frequency when Jr is 100 J mol1 . nd ¼

4p2 1002 6:022 1023 6:626 1034 ð4p 314 103 8:3145 298:2Þ1=2

¼ 1:00 1010 s1

(7.8.44)

The value of tL in water at 25 C is 0.4 ps (see section 4.7). Thus the adiabacity parameter is equal to ga ¼

8p2 1002 4 1013 ¼ 2:5 103 6:022 1023 6:626 1034 314 103

ð7:8:45Þ

Since ga is very small, the electron transfer frequency net is equal to the diabatic frequency, that is, to 1:0 1010 s1 . When Jr is 5000 J mol1 , the corresponding value of nd is 2:50 1013 s1 . The adiabacity parameter also increases to 6.3. According to equation (7.8.40) the electron transfer frequency is now net ¼

2:50 1013 ¼ 3:42 1012 s1 1 þ 6:3

ð7:8:46Þ

For an adiabatic reaction 1 314 103 net ¼ 4 1013 16p 8:3145 298:2

!1=2 ¼ 3:97 1012 s1

Thus, when Jr ¼ 5000 J mol1, the value of net (7.8.40) is very close to the adiabatic value.

ð7:8:47Þ

estimated by equation

Now, the rate constant for electron transfer may be calculated. All of the examples considered in this section have related to the reaction between Fe3þ and Fe2þ , so that the rate constant for this process is estimated now. EXAMPLE

Estimate the rate constant for electron transfer between Fe3þ and Fe2þ in water at 25 C, assuming that Jr is 5 kJ mol1 and using the results obtained above. The inner and outer sphere reorganization energies for this reaction are 50.7 and 27.7 kJ mol1, respectively. This gives a value of El equal to 314 kJ mol1. The lowering of the activation barrier due to interaction of the energy proﬁles

358

LIQUIDS, SOLUTIONS, AND INTERFACES

for the reactants and products is considered in estimating the Boltzmann factor. Thus, the Gibbs energy barrier is G ¼ 50:7 þ 27:7 5:0 ¼ 73:4 kJ mol1 and the Boltzmann factor is ! G 73:4 103 ¼ exp exp ¼ 1:39 1013 RT 8:3145 298:2

ð7:8:48Þ

ð7:8:49Þ

The pre-exponential factor is Kp net ¼ 0:0372 3:42 1012 ¼ 1:27 1011 M1 s1

ð7:8:50Þ

Thus, the electron transfer rate constant is ket ¼ 1:27 1011 1:39 1013 ¼ 0:018 M1 s1

ð7:8:51Þ

This is two orders of magnitude smaller than the experimental value of 4.2 M1 s1 (table 7.1). If the eﬀects of ionic strength on Kp were considered the theoretical estimate would be higher and closer to the experimental value. The calculations presented here show that many diﬀerent factors must be considered in estimating the rate constant. Nevertheless, electron transfer theory is remarkably successful in describing this elementary solution reaction. Theory has gone much further than described here, especially in developing the quantummechanical description of electron transfer. More details can be found in recent reviews [29, 30]. There are other related topics which have not been discussed in this section. They include, for example, photo-induced electron transfer [30], and the Marcus cross-relation [5].

7.9 NMR Spectroscopy and Chemical Exchange Reactions When a proton is transferred from one molecule to another in solutions, it usually ﬁnds itself in a diﬀerent magnetic environment. As a result, if ﬁnite amounts of both reactant and product are present in solution at equilibrium, the proton can produce two lines in the NMR spectrum, one corresponding to the proton donor and the other to the proton acceptor. A second feature of this system relates to the rate with which the proton is exchanged between the donor and acceptor. If the frequency with which the proton is transferred is comparable to the radio frequency associated with the NMR spectrometer, then the lines corresponding to the two species are broadened. The extent of line broadening can be used to determine the rate constants associated with the exchange process [31]. The most direct explanation of line broadening due to exchange processes is in terms of the Heisenberg uncertainty principle. If t is the average lifetime associated with the proton in one location, then the uncertainty in the frequency of the corresponding absorption line n is given by (2pt)1. Of course, the principle involved here is general and applies to all spectroscopies, including UV-visible

CHEMICAL REACTION KINETICS IN SOLUTION

359

and infrared spectroscopies. However, the frequencies associated with the latter experiments are the order of 1013 to 1015 s1. This is much higher than the frequency associated with most chemical exchange processes. As a result, any line broadening due to chemical exchange is too small to be seen experimentally. On the other hand, an NMR spectrometer operating at 100 MHz has a frequency which is the order of 106 times smaller. Under these conditions the relaxation time associated with chemical exchange is often shorter than that of relaxation processes due to interaction of the radio frequency ﬁeld with the magnetic nuclei in the system. In the NMR experiment, the sample is exposed to a magnetic ﬁeld B0 which is imagined to act along the z-axis (see section 5.3). As a result, some of the magnetic nuclei are in energy levels higher than the lowest level. In the case of protons, there are two levels (m ¼ 12), and the relative population at equilibrium is given by the Boltzmann fraction, which depends on the magnitude of B0 (equation (5.35)). The system is now exposed to a rotating magnetic ﬁeld B1 in the (x; y)plane that is produced by a radio-frequency wave. The frequency of this ﬁeld is swept over a small range which encompases the Larmor frequencies which characterize the nuclei being studied. When the radio frequency corresponds exactly to the Larmor frequency of a given nucleus, transitions occur and the equilibrium Boltzmann distribution is altered. Interactions with ﬂuctuating local magnetic ﬁelds in the sample cause relaxation. The relaxation in which the system returns to equilibrium is characterized by two processes, namely, spin–lattice relaxation which has a relaxation time T1 and spin–spin relaxation with a characteristic time T2. T1 corresponds to relaxation of Mz, the component of the magnetization along the z-axis in the direction of the ﬁeld B0. On the other hand, T2 corresponds to relaxation of Mx and My, the components of the magnetization in the (x; y)plane (see ﬁg. 5.7). For small molecules in solution, local ﬁeld ﬂuctuations, which are caused by molecular motions, are fast and occur in the time range 1–100 ps; thus, they are considerably faster than the time characterizing the rotating magnetic ﬁeld B1 (10 ns). Under these circumstances T1 and T2 are equal. In viscous media or in solids where T1 and T2 are longer, the relationship between them is more complex. In order to obtain quantitative relationships characterizing the time dependence of Mx, My, and Mz, one must solve the Block equations, which are described in more detail elsewhere [G2, G4]. A more molecular description of spin–lattice and spin–spin relaxation is given in section 5.3. When a chemical exchange reaction occurs in solution, it can lead to a broadening of the related NMR peaks, as described in the following discussion. Consider ﬁrst of all a proton exchange reaction between protons in diﬀerent environments denoted as HA and HB: HA( +HB

ð7:9:1Þ

Since the protons are in diﬀerent environments they are expected to produce two distinct lines in the NMR spectrum. The intensities of these lines reﬂect the relative concentrations of the protons at equilibrium. On the basis of the theory of chemical relaxation (section 7.6), the relaxation time associated with proton exchange is

360

LIQUIDS, SOLUTIONS, AND INTERFACES

t ¼ ðkf þ kb Þ1

ð7:9:2Þ

Here, kf and kb are the rate constants associated with the forward and backward reactions in equilibrium (7.9.1). The experiment is usually designed so that the total proton concentration is very small with respect to the concentration of other solution components, for example, the solvent. Then the expression for the relaxation time can be rewritten as 1 1 1 ¼ þ t tf tb

ð7:9:3Þ

where tf ¼

1 kf

ð7:9:4Þ

tb ¼

1 kb

ð7:9:5Þ

and

are the relaxation times associated with the forward and backward processes, respectively. Equation (7.9.3) can be rearranged as tt ð7:9:6Þ t¼ f b tf þ tb The fraction of molecules present in the HA environment at any time is tf xA ¼ ð7:9:7Þ tf þ tb and that in the HB environment is xB ¼

tb tf þ tb

ð7:9:8Þ

At this point it is necessary to relate the relaxation time for the chemical process to the spin–spin relaxation process which is always present when the system absorbs energy from the radio frequency wave. Both processes lead to broadening of the NMR absorption peak. If Ti is the total relaxation time for process i, it may be estimated from T2 and ti using the relationship 1 1 1 ¼ þ Ti T2 ti

ð7:9:9Þ

The relaxation time is now related to the peak width at half height, (n1/2)i as follows: ðn1=2 Þi ¼

1 pTi

ð7:9:10Þ

Thus, from equation (7.9.9) ðn1=2 Þi ¼ ðn1=2 Þi þ

1 pti

ð7:9:11Þ

CHEMICAL REACTION KINETICS IN SOLUTION

361

where (1=2 Þi is the peak width at half height for process i in the absence of chemical exchange. If (1=2 Þi can be determined in an experiment without the system undergoing chemical exchange, then the value of ti can be calculated from the increase in peak width. Further analysis requires consideration of the frequencies of the two lines corresponding to HA and HB, and how the system behaves under conditions of slow and fast chemical exchange. If these frequencies are designated as nA and nB then one may deﬁne a frequency n0 which is a weighted average depending on the fractions of protons present as HA and HB: n 0 ¼ xA n A þ x B n B

ð7:9:12Þ

n0 falls in between nA and nB and can be used as a point of reference to discuss precession of the magnetic vectors of the protons at the two sites. At site A, the magnetic vector is imagined to precess at a frequency n0 nA ; at B, it precesses in the opposite direction with a frequency nB n0 . Now the eﬀect of the chemical exchange on the appearance of the NMR spectrum can be considered in greater detail (see ﬁg. 7.18). Four representative situations are considered from a qualitative point of view, namely, very slow, moderately slow, moderately fast, and very fast exchange processes. When the chemical exchange process is very slow, the relaxation times tf and tb are long. This means that protons at sites A and B precess many times before they exchange. Thus, there is time for the absorption of energy from the radio frequency ﬁeld B1 and sharp lines are seen in the spectrum at nA and nB (ﬁg. 7.18 (A)). When the exchange process is moderately slow, two eﬀects are seen (ﬁg. 7.18 (B)). The lines for the protons at sites A and B are broader, and the positions of

Fig. 7.18 NMR spectra for chemical exchange between two symmetrical sites for the cases of (A) very slow, (B) moderately slow, (C) moderately fast, and (D) very fast exchange processes.

362

LIQUIDS, SOLUTIONS, AND INTERFACES

the maxima shift. The band broadening can be attributed to the uncertainty principle, which plays a more important role as the lifetime of the proton in a given state decreases. At the same time, the position of the band changes due to a corresponding change in the average environment of each proton. The value of the relaxation time t can be found from the peak separation nA nB . If ðnA nB Þ0 is the peak separation in the absence of exchange broadening (t ¼ 1), then " #1=2 A nB 1 ¼ 1 2 2 ðnA nB Þ0 2p t ðnA nB Þ20

ð7:9:13Þ

If the process is symmetrical (tf ¼ tb), then the relaxation time in each direction is equal to t/2. In writing equation (7.9.13) it is assumed that the absorption peaks have a Lorentzian shape. In addition, the line-broadening eﬀects illustrated in ﬁg. 7.18 are those for a symmetrical process. Eventually, as the exchange process becomes faster, the two bands coalesce to form one peak at frequency n0. The relaxation time for this condition, tc is given by tc ¼

1 21=2 pðn

A

nB Þ 0

ð7:9:14Þ

Further increase in the exchange rate results in sharpening of the band at n0. When this process is very fast a sharp line is observed at frequency n0 whose width is determined by spin–spin relaxation in the absence of chemical relaxation (t ¼ 0). Under these circumstances a nucleus in site A cannot precess to a signiﬁcant extent before it leaves A to enter site B. From the point of view of the rotating frame of reference deﬁned with respect to n0, the proton is essentially stationary. An important method of observing the eﬀects of chemical exchange illustrated in ﬁg. 7.18 is to change the temperature of the system. At low temperatures, the exchange is very slow. As the temperature increases, the exchange speeds up and the various stages from very slow to very fast are seen. It is important to understand what time range for kinetic experiments is accessible by NMR. The frequency diﬀerence n which can be resolved in 1H NMR is of the order of 10–1000 Hz. When the exchange process is very slow, the relaxation time t is very much greater than n1 (103–0.1 s); for a very fast process t is much less than n1. Thus, values of t in the range 104–1 s can be measured using line-broadening experiments. Other methods are used to determine rates which cannot be measured by line broadening experiments. One of these is based on observation of the absorption signal intensity at one of the sites in an exchanging system, while absorption at the other site is saturated by using a second radio frequency ﬁeld tuned at the required frequency. During the irradiation the absorption intensity falls oﬀ to a new steady-state value due to transfer from the irradiated site. The rate constant for the decay process can be used to calculate the lifetime of the observed state. More information about this technique, which is called spin relaxation transfer, and other methods used to determine the kinetics of exchange processes using NMR, can be found in monographs devoted to dynamic NMR spectroscopy [32, 33].

CHEMICAL REACTION KINETICS IN SOLUTION

363

1

H NMR has been used extensively to study the kinetics of proton transfer reactions in water and other solvents. In the case of the protolysis reaction HA þ S( +A þ HSþ

ð7:9:15Þ

þ

where S represents the solvent, and HS , its protonated form (lyonium ion), the relaxation time is given by t ¼ ½kf ðcHAe þ cSe Þ þ kb ðcAe þ cHSe Þ1

ð7:9:16Þ

Normally, the equilibrium solvent concentration cSe is much higher than the concentrations of the other species involved in reaction (7.9.15). Furthermore, the experiment is designed with the concentrations cHAe and cAe suﬃciently small that the relaxation times associated with proton exchange can be measured in the experiment. In addition, the equilibrium concentration of the lyonium ion cHSe is much less than that of the conjugate base A. As a result, equation (7.9.16) becomes t ¼ ðkf cSe þ kb cAe Þ1

ð7:9:17Þ

Since the solvent concentration is normally constant, the term kf cSe is treated as a ﬁrst-order rate constant with units of s1. Also the concentration of A is a known parameter in the experiment, so that the value of kb as a second-order rate constant can be extracted from the relaxation time tb associated with the back reaction. The same approach is used to determine the kinetics of hydrolysis reactions. The general hydrolysis reaction is written as B þ SH( +BHþ þ S

ð7:9:18Þ

where B is the Brønsted base and SH, the solvent which can donate a proton. Under conditions that the equilibrium concentration of B is very small with respect to that of the solvent, and that of the lyanate ion S is very small with respect to that of BHþ, the expression for the relaxation time is t ¼ ðkf cSHe þ kb cBHe Þ1

ð7:9:19Þ

Data for some protolysis and hydrolysis reactions in water and methanol are summarized in table 7.8. The rate constants for the backward reactions are all very fast, whereas those for the forward reactions depend very much on the nature of the reaction. When these data are combined with those in table 7.2 a more complete description of proton transfer reactions in protic solvents is available. It is clear that the solvent plays an important role in these processes. A very eﬀective way of studying a chemical exchange process is to determine line widths as a function of temperature. This has been applied with success to the determination of the kinetics of ligand exchange processes [31]. For example, the kinetics of exchange of acetonitrile (AcN) as a ligand in [Ni(AcN)6]2þ with acetonitrile as solvent has been determined by examining the temperature dependence of the 1H NMR spectra of the ligand [34]. The quantity determined experimentally is 1 1 1 Y¼ ð7:9:20Þ xX T2 T2;0

364

LIQUIDS, SOLUTIONS, AND INTERFACES

Table 7.8 Kinetic Data Obtained by 1H NMR for Proton Transfer Reactions in Water and Methanol at 25 C [32] Reaction

kf cse =s1

kb =M1 s1

6 1011 24.5 4.7 2:1 1011 1:1 106

— 4:3 1010 3:0 1010 — 1:7 1010

17:6 1010 0.7 1:85 1010

— 6 109 —

Water H3 Oþ þ H2 O Ð H2 O þ H3 Oþ þ NHþ 4 þ H2 O Ð NH3 þ H3 O þ ðCH3 Þ3 NH þ H2 O Ð ðCH3 Þ3 N þ H3 Oþ OH þ H2 O Ð H2 O þ OH ðCH3 Þ3 N þ H2 O Ð ðCH3 Þ3 NHþ þ OH Methanol þ CH3 OHþ 2 þ CH3 OH Ð CH3 OH þ CH3 OH2 ðCH3 Þ3 NHþ þ CH3 OH Ð ðCH3 Þ3 N þ CH3 OHþ 2 CH3 O þ CH3 OH Ð CH3 OH þ CH3 O

*For totally symmetrical reactions, kf ¼ kb :

where xX is the mole fraction of AcN bound to the transition metal cation, in this case, Ni2þ, T2, the relaxation time determined from the width at half-height for the 1H peaks in the NMR spectrum, and T2,0, the same quantity determined in pure AcN. The results of this experiment are shown in ﬁg. 7.19 as a plot of Y against the reciprocal of the absolute temperature in the range 230–420 K. Four regions are clearly distinguishable in which the variation in y with 1/T is approximately linear. The two regions with steep slopes are connected with the ligand exchange process, and the ﬁrst and last region with variation in the spin–spin relaxation time with temperature. Each of these parts of the plot in ﬁg. 7.19 is now discussed in more detail. In region I where the temperature is very low, the ligand exchange process is so slow that it does not contribute to the line broadening. The mechanism of line broadening involves T2,0, that is, spin–spin relaxation related to the protons in the acetonitrile molecule. As the temperature increases, this process speeds up and the line width decreases. By extrapolating the line drawn through the data obtained in this region, estimates of T2,0 at higher temperature are obtained.

Fig. 7.19 Plot of the parameter Y as a function of 1=T for an exchange process in regions of (I) negligible exchange, (II) moderately slow exchange, (III) moderately fast exchange, and (IV) very fast exchange.

CHEMICAL REACTION KINETICS IN SOLUTION

365

As the temperature increases, the ligand exchange process begins to contribute to the observed line width. This results in the increase in y observed in region II. In fact, the parameter y in this region is simply given by Y¼

1 tx

ð7:9:21Þ

where tX is the relaxation time for ligand exchange. The activation parameters for this process may be determined from the slope of the linear plot in region II. EXAMPLE

Given that the value of tX is 0.76 ms at 288.2 K and 0.19 ms at 308.2 K, estimate the enthalpy and entropy of activation for the ligand exchange process. Neglecting medium eﬀects, the relationship between tX and the activation parameters is 6¼ 1 kB T S 6¼ H ¼ exp ð7:9:22Þ exp R RT tx h The ratio of the values of tX measured at the two temperatures is

0:76 308:2 6¼ H 6¼ H ¼ exp þ 0:19 288:2 308:2R 288:2R

ð7:9:23Þ

Solving this equation, one obtains a value of the enthalpy of activation equal to 49.0 kJ mol1. When this result is substituted into equation (7.9.17), the result for the entropy of activation is 15.1 J K1 mol1. The maximum line width for the system being considered occurs at 313 K. At this temperature the two lines being followed in the NMR spectrum coalesce. Further increase in temperature results in a decrease in line width as illustrated experimentally in region III. The parameter y in this region is given by Y¼

1 þ ðoÞ2 tx T2X

ð7:9:24Þ

where T2X is the relaxation time of the proton in the ligand and o ¼ 2pðnX nS Þ

ð7:9:25Þ

nX being the chemical shift for the proton in the ligand, and nS, the value in the bulk solvent. Over most of the region the main contribution to Y comes from the second term, which is proportional to (o)2. In region IV, the exchange is so fast that the broadening eﬀects related to exchange disappear. This means that o is zero and the parameter Y is given by the reciprocal of the spin–spin relaxation time T2. This type of experiment has been used succesfully to determine the kinetic parameters associated with ligand exchange reactions. Another application of NMR relaxation experiments is the study of intramolecular processes. The classic example is rotation about the C–N bond in amides. Since this bond is partially double in character, the rotation requires energy and occurs at a ﬁnite rate. In addition, it results in exchange of protons in diﬀerent chemical environments. For the case of N–N dimethylacetamide

366

LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 7.20 Equivalent structures for N,N-dimethylacetamide illustrating the partial double bond character of the C–N bond.

(DMA), the protons on the methyl groups attached to the nitrogen atom are either cis- or trans to the carbonyl group (ﬁg. 7.20). As a result, the room temperature 1H NMR spectrum has two lines corresponding to the diﬀerent environments. At higher temperatures, rapid internal rotation makes the protons of the two methyl groups equivalent and only one peak is observed. Analysis of the spectral data leads to a value of 50 kJ mol1 for the energy barrier associated with the exchange process. The examples chosen here have emphasized the use of 1H NMR. However, other nuclei can be examined, including 13C, 14N, and 17O. Thus, in the case of ligand exchange processes involving acetonitrile, one can use 13C NMR with an isotope-enriched sample [G4]. In addition, the discussion here has been limited to a few examples of kinetic processes which have been studied using NMR spectroscopy. Closely related experimental techniques are electron paramagnetic resonance and Raman spectroscopies, which are discussed more fully by Strehlow [G4].

7.10 Medium Effects in Solution Reactions Since the components of a reaction in solution are surrounded by other species, most of which do not react, the nature of the intermolecular interactions experienced by the reactants and products can have an important inﬂuence on the Gibbs energy surfaces, and thus, on the kinetics of the reaction. The most important nearest neighbor in most cases is the solvent. Thus, the study of solvent eﬀects on the rate and mechanism of solution reactions has been an important area of experimental research. Interactions with other solute components may also be important. This is especially true for reactions involving ions. Since the equilibrium properties of the medium are involved in determining these eﬀects they are called static medium eﬀects. However, there is another type of relevant phenomenon which is related to the dynamic properties of a condensed phase. Since the movement of molecules with respect to one another is required for the reaction to take place, the local viscosity of the system can also inﬂuence the rate of reaction. This property is also related to local intermolecular forces. Eﬀects which depend on local viscosity have also been studied experimentally and are known as dynamic medium eﬀects. In the following discussion, important examples of both static and dynamic medium eﬀects are described and discussed with respect to data in the literature.

CHEMICAL REACTION KINETICS IN SOLUTION

367

A. Static Medium Effects As pointed out in section 7.4, the medium eﬀect is described within transition state theory in terms of the activity coeﬃcient ratio in the expression for the rate constant. Thus, on the basis of equation (7.4.13), the rate constant for a simple bimolecular reaction may be written as kr ¼ kr 0

y A yB yx6¼

ð7:10:1Þ

where kr 0 is the rate constant in the absence of medium eﬀects, yA and yB, the activity coeﬃcients of the reactants A and B on the molarity scale, and yx6¼ , the activity coeﬃcient of the activated complex. The most direct method of changing the medium is to change the solvent in which the reaction is carried out. The reactions studied are usually limited to organic and organometallic systems for reasons of solubility. Solvent eﬀects have been studied for a wide variety of reactions [35] including electron transfer [36, 37], proton transfer [8, 32], and atom transfer reactions [12]. In the case of electron transfer reactions, the role of the solvent is expressed directly in the outer sphere Gibbs activation energy Gos . This depends on the Pekar factor and thus on the static and optical values of the solvent’s relative permittivity (section 7.8C), and has been investigated in detail for these reactions [36]. However, dynamic medium eﬀects may also be important for electron transfer reactions and are expressed through the longitudinal relaxation time tL in the pre-exponential factor (section 7.8D). For this reason, solvent eﬀects in electron transfer reactions are discussed later in this section. Proton transfer reactions have not only been studied in water but also in the alcohols and other protic solvents [8, 32]. As one would expect, the Gibbs energy barrier and the rate constants for these processes depend on strength of the intramolecular forces which aﬀect the proton in each environment. Solvent eﬀects for atom transfer reactions were illustrated earlier in this chapter (section 7.3D). In the example chosen, an anion is involved in a SN2 reaction. As a result the solvent’s acidity plays a role in determining the Gibbs energy barrier for the process. Another well-known example of a medium eﬀect in solution kinetics is the eﬀect of ionic strength on reactions between ions [38, 39]. This is normally treated using the extended Debye–Hu¨ckel theory to estimate the activity coeﬃcients in equation (7.10.1). The activity coeﬃcient for species i is given by ln yi ¼

ADH z2i I1=2 1 þ BDH ai I1=2

ð7:10:2Þ

ADH and BDH are the Debye–Hu¨ckel constants (see section 3.8) and I is the ionic strength. zi is the charge on species i and ai, the ion size parameter. If it is assumed that the individual values of ai can be replaced by an average value, a, and that z6¼ is equal to zA þ zB, then ln

yA yB 2ADH zA zB I1=2 ¼ yx6¼ 1 þ BDH aI1=2

ð7:10:3Þ

368

LIQUIDS, SOLUTIONS, AND INTERFACES

It follows that ln kr ¼ ln kr 0 þ

2ADH zA zB I1=2 1 þ BDH aI1=2

ð7:10:4Þ

so that ln kr should be a linear function of I1/2 in the limit of very low ionic strengths. Plots of ln(kr/kr 0) against I1/2 for a series of reactions with diﬀerent values of zAzB are shown in ﬁg. 7.21. In the case that the ions have the same sign (zAzB> 0), considerable reaction acceleration is observed when the ionic strength increases. This is clearly a result of the screening eﬀect of the inert electrolyte on the repulsion between the reactants. When the ions have opposite signs, the eﬀect is reversed and the observed rate constant decreases as ionic strength increases. The slopes of the lines drawn in ﬁg. 7.21 reﬂect the change in the value of zAzB. The range over which the extended Debye–Hu¨ckel theory is applicable is limited to ionic strengths equal to 0:1 M. This range can be extended signiﬁcantly when the MSA model is used to estimate the activity coeﬃcients [40]. The study of static medium eﬀects is obviously important because it provides direct information regarding the factors which inﬂuence the Gibbs energy barrier for the reaction. These factors are important in deciding the mechanism of complex reactions. As a result, medium eﬀects are often used by the synthetic chemist to determine the ﬁnal product of a reaction in solution. As already pointed out, the equilibrium environments are not the only factor that must be considered in assessing the overall medium eﬀect. During the course of the reaction these environments change, and molecules and ions move with respect to one another. The friction involved in this movement can also play a role in determining the rate constant, as is discussed now.

Fig. 7.21 Plots of log10 (kr/kr0) against the square root of the ionic strength for ionic reactions of various types in water: (A) CoðNH3 Þ5 Br2þ þ Hg2þ ðzA zB ¼ 4Þ; (B) S2 O2 8 þ I ðzA zB ¼ 2Þ; (C) COðOC2 H5 ÞN : NO2 þ OH ðzA zB ¼ 1Þ; (D) [Cr(urea)6]3þ þ H2O (), CH3COOC2H5 þ OH (), (zAzB ¼ 0); (E) H3Oþ þ Br þ H2O2 (zAzB ¼ 1); (F) CoðNH3 Þ5 Br2þ þ OH ðzA zB ¼ 2Þ; and (G) Fe2þ þ CoðC2 Þ3 4 (zAzB ¼ 6). The lines are drawn with slopes equal to 2zAzB. (From reference G1, with permission.)

CHEMICAL REACTION KINETICS IN SOLUTION

369

B. Dynamic Medium Effects Dynamic medium eﬀects in solution kinetics were ﬁrst recognized by Kramers [41]. He treated the problem on the basis of the Langevin equation [42] according to which the velocity of the reactants along the reaction coordinate and the friction of the surrounding medium play a role. Details of Kramers’ theory are not given here but an introduction to this subject can be found elsewhere [G3]. The parameters involved in quantitatively assessing the dynamic solvent eﬀect are the frequency associated with the shape of the barrier of the transition state n6¼ and a friction parameter z6¼ which is related to solvent viscosity. From the solution of the Langevin equation, Kramers recognized three important regimes. For moderate friction, the system behaves as if dynamic solvent eﬀects were absent. This means that the equilibrium conditions assumed in the standard formulation of transition state theory are satisﬁed and the reactants move freely over the potential energy barrier to the product state. As solvent friction increases, the second regime is reached. As a result of collision with the viscous solvent molecules, the crossing of the Gibbs energy barrier is not always successful. Thus, the barrier may be crossed several times before the system succesfully reaches the product state. This means that the eﬀective barrier crossing frequency is less than n6¼ with a corresponding decrease in the rate constant. The magnitude of the eﬀect depends not only on the magnitude of the friction but also on the shape of the barrier. The third regime corresponds to systems with very low solvent friction. Since the reactants are now in an environment in which the solvent appears to be absent, an equilibrium distribution of reactant states is not maintained, and transition state theory is no longer valid. The results of Kramers’ theory [41] have been elaborated in detail by Hynes [G3]. A brief introduction to this subject is given here. As a result of dynamic friction the observed rate constant kr can fall bellow the transition state value kTST, that is, kr ¼ kT kTST

ð7:10:5Þ

where kT is the Kramers transmission coeﬃcient. The rate constant according to transition state theory is given by 6¼ G kTST ¼ nR exp ð7:10:6Þ RT where nR is a frequency associated with the Gibbs energy surface for the reactants and6¼ G , the standard Gibbs energy of activation. According to equation (7.4.13), the expression for nR is nR ¼

kB T yA yB h yX6¼

ð7:10:7Þ

Static medium eﬀects appear in equation (7.10.6) through nR and 6¼G , both of which depend on solvent properties. After solving the Langevin equation, the following expression is obtained for kT:

370

LIQUIDS, SOLUTIONS, AND INTERFACES

"

z6¼ kT ¼ 1 þ 4pn6¼

2 #1=2 z6¼ 4pn6¼

ð7:10:8Þ

z6¼ is the friction parameter related to the viscosity of the medium, and n6¼, a frequency associated with the Gibbs energy barrier. If one imagines that the Gibbs energy barrier has a parabolic shape (see ﬁg. 7.3), then n6¼ can be calculated from the equation describing the parabola. The exact expression for z6¼ depends on the nature of the reaction but it is clear that it must have units of s1 because kT is a dimensionless fraction. In the case of moderate friction, z6¼/4pn6¼ 1 and there is no dynamic solvent eﬀect. On the basis of equation (7.10.8), kT is equal to unity and the rate constant kr is given by its transition state theory value, kTST. However, if solvent friction is large, then z6¼/4pn6¼ 1, and the expression for the transmission coeﬃcient is !1=2 z6¼ z6¼ 16p2 n26¼ 2pn6¼ kT ¼ 1þ ¼ ð7:10:9Þ 2 4pn6¼ 4pn6¼ z6¼ z6¼ In this limit, the dynamic solvent eﬀect can also be treated as a diﬀusion process using the Smoluchowski equation. This emphasizes the fact that the friction leads to multiple crossings of the Gibbs energy barrier. The diﬀusion coeﬃcient associated with this process is given by D6¼ ¼

kB T n6¼ z6¼

ð7:10:10Þ

where m6¼ is the reduced mass of the reacting system. Expressing the transmission coeﬃcient in terms of D6¼, one obtains kT ¼

2pn6¼ m6¼ D6¼ kB T

ð7:10:11Þ

Hynes [G3] has discussed the application of this theory to a wide variety of solution reactions, including electron transfer reactions, atom transfer reactions involving both neutral and charged species, and isomerization reactions. The following discussion is limited to electron transfer processes. An important parameter in the Hynes treatment of the dynamic solvent eﬀect is the frequency associated with motion of the reactant system in the corresponding potential energy well. On the basis of the dielectric continuum model, this frequency is given by 2es þ e1 kB T 1=2 nR ¼ ð7:10:12Þ 6 2 gK e1 Irot where es and e1 are the static and high-frequency values of the solvent’s relative permittivity, respectively, gK, the Kirkwood correlation parameter (see section 4.4), and Irot, the eﬀective moment of inertia of an individual solvent molecule [29]. Values of nR for selected solvents which approximately follow the Debye equation with a single relaxation process are given in table 7.9 together with the solvent molecule’s moment of inertia and the reciprocal of the longitudinal relaxa-

CHEMICAL REACTION KINETICS IN SOLUTION

371

Table 7.9 Reactant System Characteristic Frequencies in Selected Debye Solvents Estimated Using Equation (7.10.12) Together with the Solvent Molecule’s Effective Moment of Inertia and the Reciprocal of the Longitudinal Relaxation Time Solvent Acetone Acetonitrile Benzonitrile Dimethyl sulfoxide Nitromethan

1047 Irot =kg m2

1 t1 L =ps

nR =ps1

62.51 9.1 303.9 93.13 9.0

3.4 5.0 0.17 0.47 4.5

1.3 4.6 0.52 0.91 4.5

1 tion time t1 L . It is apparent that nR is close to tL in magnitude and follows the 1 same trend, varying from 0.1 ps in slow solvents to 5 ps1 in fast solvents such as acetonitrile. These results are all smaller than the gas phase transition state frequency kB T=h, which is equal to 6.2 ps1 at room temperature.

EXAMPLE

Estimate the characteristic reaction frequency nR in acetonitrile given that the eﬀective moment of inertia of this molecule is 9:1 1047 kg m2. The ratio (kB T=Irot Þ1=2 at room temperature is !1=2 kB T 1=2 4:115 1021 ¼ ¼ 6:72 1012 ð7:10:13Þ Irot 9:1 1047 The dielectric parameters for acetonitrile are es ¼ 35.9 and e1 ¼ 2.26 (table 4.2). The Kirkwood correlation parameter is gK ¼ 1.18 (table 4.4). The dielectric factor in equation (7.10.12) may now be estimated: 1=2 2es þ e1 1=2 2 35:9 þ 2:26 ¼ ¼ 0:685 ð7:10:14Þ 6p2 gK e1 6ð3:1416Þ2 1:18 2:26 The value of nR is nR ¼ 0:685 6:72 1012 ¼ 4:60 ps1

ð7:10:15Þ

The longitudinal relaxation time for acetonitrile given a Debye relaxation time of 3.2 ps is tL ¼

2:26 3:2 ¼ 0:20 ps 35:9

ð7:10:16Þ

Its reciprocal value is 5.0 ps1. This value is very close to that of nR. Hynes [43] has discussed dynamic solvent eﬀects for electron transfer reactions and described the role of solvent friction for both diabatic and adiabatic reactions. In the case of diabatic reactions the rate is strongly dependent on the coupling between the energy surfaces for the reactants and products as expressed through the parameter Jr (see section 7.8D). When Jr is very small, dynamic

372

LIQUIDS, SOLUTIONS, AND INTERFACES

solvent eﬀects are normally not important and the pre-exponential factor is given by equations (7.8.35) and (7.8.40). The latter equation is derived from quantum mechanics for the case that movement from the reactant energy surface to the product surface at the intersection point is the rate-determining step [44]. As interaction between the two energy surfaces increases, the character of the reaction changes from diabatic to adiabatic. This interaction aﬀects the shape of the cusp-shaped barrier associated with the transition state and thus the value of n6¼. In the case of a simple Debye solvent the friction parameter is given by z6¼ ¼ 4p2 n2R tL

ð7:10:17Þ

The expression for the Kramers transmission coeﬃcient for adiabatic electron transfer is therefore n6¼ ð7:10:18Þ kT ¼ 2pn2R tL Hynes [43] also derived a relationship between n6¼ and nR for this type of reaction, which is n6¼ pEl 1=2 ¼ ð7:10:19Þ nR 4RT where El is the reorganization energy for electron transfer (equation (7.8.9)). Substituting this into equation (7.10.18) the expression for kT becomes 1=2 1 El kT ¼ ð7:10:20Þ nR tL 16pRT The corresponding expression for the adiabatic pre-exponential frequency for electron transfer is 1=2 1 El net ¼ kT nR ¼ ð7:10:21Þ tL 16pRT This is identical with the expression derived by Zusman [28] which was discussed in section 7.8D. EXAMPLE

Estimate the frequency associated with the transition state barrier for an adiabatic electron transfer in acetonitrile assuming that El is 100 kJ mol1. The ratio n6¼/nR is estimated using equation (7.10.19): !1=2 n6¼ p 100 103 ¼ ¼ 5:63 ð7:10:22Þ nR 4 8:3145 298:2 Using the value of nR for acetonitrile given in table 7.9, the frequency associated with the transition state is n6¼ ¼ 5:63 4:60 ¼ 26:0 ps1

ð7:10:23Þ

Hynes [43] has also discussed the dynamic solvent eﬀect for solvents with more complex dielectric relaxation behavior. In the following it is assumed that the

CHEMICAL REACTION KINETICS IN SOLUTION

373

simple Debye model is suﬃcient to describe the dynamic medium eﬀect. When one compares the ﬁnal expression for diabatic and adiabatic reactions, one sees that the solvent eﬀect enters in quite diﬀerent ways. For a diabatic electron transfer reaction the rate constant can be expressed as Ad G exp kd ¼ ð7:10:24Þ RT ðE RTÞ1=2 where the standard Gibbs energy of activation G depends on solvent nature through the outer sphere contribution Gos , and Ad accounts for the solventindependent terms in the pre-exponential factor (see equation (7.8.41)). The outer sphere contribution Gos depends on the Pekar factor, 1/eop 1/es (see equation (7.8.26)). Because of the dominating eﬀect of the exponential term, it is reasonable to expect ln ket to be linear in the Pekar factor for diabatic reactions with small values of Gis . On the other hand, if the electron transfer reaction is adiabatic, the dependence of the rate constant on solvent nature is quite diﬀerent: 1=2 E G 1 ka ¼ Aa tL exp ð7:10:25Þ RT RT In this equation, Aa gives the solvent-independent portion of the pre-exponential factor. Separation of the solvent contributions to the exponential and pre-exponential terms in the rate constant is much more diﬃcult for adiabatic reactions. However, if the outer sphere contribution to G is small, then the contribution from t1 L in the pre-exponential factor dominates. Under these circumstances a plot of the logarithm of the electron transfer rate constant ln ket should be linear in ln tL. Grampp and Jaenicke [36] have carried out detailed studies of solvent eﬀects on the kinetics of electron transfer for a variety of organic redox couples. The kinetic data were obtained by electron spin resonance (ESR) line-broadening experiments for very fast reactions with rates close to the diﬀusion-limited value. Some of the reactions were clearly diabatic, and others adiabatic on the basis of the solvent eﬀect. An example of a diabatic reaction is the oxidation of tetramethyl-p-diaminobenzene (TMDAB) to its cation radical. Kinetic data were obtained in seven solvents with Pekar factors ranging from 0.27 (chloroform) to 0.53 (acetonitrile). The Pekar factor gP is deﬁned as 1 1 gP ¼ ð7:10:26Þ eop es For the TMDAB, the logarithm of the electron transfer rate constant ln ket is clearly a linear function of the Pekar factor gP (ﬁg. 7.22). On the other hand, a plot of ln ket against ln tL for the same data yields a weak linear correlation with a positive slope (ﬁg. 7.23). This correlation is the opposite to that expected; in other words, the rate constant should decrease with increase in tL if the reactions were adiabatic. The estimate of Gis for this reaction is 6.4 kJ mol1; the value of Gos depends on the nature of the solvent but it is close to Gis in most solvents considered. The reason that the reaction is diabatic is clearly due to the weak

374

LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 7.22 Plots of the logarithm of the electron transfer rate constant for oxidation of TMDAB () and TTF () against gP , the Pekar factor of the solvent [36].

interaction between the Gibbs energy surfaces for reactants and products which results in a small value of the interaction parameter Jr . Quite diﬀerent behavior is found for the electron transfer reaction between tetrathiofulvalene (TTF) and its cation radical. For this system, there is no correlation between ln ket and the Pekar factor g (ﬁg. 7.22). However, there is a strong correlation between ln ket and ln tL. These results are consistant with an adiabatic reaction with a dynamic solvent eﬀect. The estimate of Gis for this

Fig. 7.23 Plots of the logarithm of the electron transfer rate constant for oxidation of TTF (^) and TMDAB () against the logarithm of the solvent’s longitudinal relaxation

time [36].

CHEMICAL REACTION KINETICS IN SOLUTION

375

reaction is 4.9 kJ mol1. The corresponding value of Gos is 9g. Since g is close to 0.5 for many solvents, the outer sphere reorganization energy is close to the inner sphere value. For this system, interaction between the Gibbs energy surfaces is stronger, so that the reaction follows an adiabatic path. When the reaction parameters are such that it is neither strongly adiabetic or strongly diabetic, analysis of the solvent eﬀect is more diﬃcult. Dynamic solvent eﬀects have been studied for other solution reactions, including proton transfer reactions, isomerizations, and time-resolved ﬂuorescence studies. This is an important area of modern research involving ultrafast kinetic experiments in solution.

7.11 Linear Gibbs Energy Relationships As the study of solution kinetics advanced, data were accumulated for speciﬁc types of reactions in which the nature of one of the reactants was changed in a systematic way. An example is a reaction involving an acid which is studied for a series of acids of varying acidity constant. As the body of data increased in size it was observed that the rate constant for the reaction was often a linear function of an equilibrium constant associated with the overall process, for example, the acidity constant. This correlation is called the Brønsted relationship and can be written as ln kr ¼ ln kr 0 þ bB ln Ke

ð7:11:1Þ

where kr is the rate constant, Ke, the equilibrium constant, bB, the Brønsted slope, and kr 0, the rate constant when the equilibrium constant is unity. An early description of such a relationship was given by Brønsted and Pederson [45] who presented data for the base-catalyzed decomposition of nitramide (H2N2O2). They found that the logarithm of the rate constant for this reaction is linear in the acidity constant for the conjugate acid HB of the base B involved. Their data, which are shown in ﬁg. 7.24, demonstrate that the linear relationship between the two experimental parameters holds over a very wide range. The slope of this line, 0.78, gives the Brønsted coeﬃcient bB. For this system the rate constant increases with increase in basicity of the anion involved in the reaction. Although the Brønsted relation was ﬁrst discussed with respect to proton transfer reactions, it has been found to apply to other types of reactions including electron and atom transfer reactions. Equation (7.11.1) implies that there is also a linear relationship between the Gibbs activation energy for the reaction and the standard Gibbs energy change for the associated equilibrium. Thus, one may also write 6¼ G ¼ 6¼ G0 þ bB G

ð7:11:2Þ

where 6¼ G is the standard Gibbs energy of activation, G , the standard Gibbs energy change for the equilibrium, and 6¼ G0 , the standard Gibbs energy of activation when G is zero. The relationship of the Brønsted coeﬃcient to the fundamental parameters of an electron transfer reaction is easily derived on the

376

LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 7.24 Plot of the logarithm of the rate constant for the decomposition of nitramide against the logarithm of the basicity constant for the base used as catalyst [45].

basis of equation (7.8.10). For a series of electron transfer reactions in which the reorganization energy El is independent of G, one obtains bB ¼

d6¼ G 1 G ¼ þ dG 2 2El

ð7:11:3Þ

When the reaction is homonuclear (G ¼ 0), the Brønsted coeﬃcient is exactly equal to 1/2. Equations similar to equation (7.8.10) have also been used to estimate the Brønsted slope for proton and atom transfer reactions [46]. Another linear Gibbs energy relationship which is used extensively in organic chemistry is the Hammett equation [47]. This equation is used to relate the rate constants and equlibrium constants for reactions of meta- and para-substituted benzene derivatives. The Hammett equation for the rate constants of a given reaction series is written as log10 kX ¼ log10 k0 þ sX rH

ð7:11:4Þ

where kX is the rate constant for the benzene derivative with substituent X, k0, the rate constant for the unsubstituted compound, sX, the substituent parameter, and rH, the reaction parameter. The values of sX are based on the dissociation constants of substituted benzoic acids, so that sX ¼ log10 KaX log10 Ka0

ð7:11:5Þ

where KaX is the acidity constant for the benzoic acid with substituent X and Ka0, that for unsubstituted benzoic acid. Some values of sX for typical substituents in the meta and para positions are given in table 7.10. When sX is negative, the substituents are electron donating and push electron density into the benzene ring. As a result, the –COOH group becomes more electronegative and is able to hold on to the proton more easily.

CHEMICAL REACTION KINETICS IN SOLUTION

377

Table 7.10 Hammett Substituent Parameters [48] Position Substituent Group OH OCH3 CH3 H F Cl Br I CN NO2

Meta

Para

þ0.12 þ0.115 0.07 0 0.34 0.37 0.39 0.35 0.56 0.71

0.37 0.27 0.17 0 0.06 0.23 0.23 0.18 0.66 0.78

Substituents with positive values of sX are electron withdrawing and have the opposite eﬀect. The benzoate anion is stabilized with respect to the acid and the acidity constant is higher. An example of application of the Hammett relation is shown in ﬁg. 7.25. Data for the rate constant for bromination of substituted benzeneboronic acids in 20% aqueous acetic acid show an excellent linear correlation with the Hammett sX parameter for both para and meta substituents. The reaction parameter rH for this system is –4.52. The Hammett equation has been shown to successfully describe kinetic and equilibrium data for over 300 aromatic reactions [49]. Following the original work of Hammett [47] several other linear Gibbs energy relationships have been derived. The most important of these is the Taft relationship [50] which allows consideration of aliphatic as well as aromatic systems and also of steric eﬀects. The Taft equation has the same form as equation (7.11.4) but the substituent constants are deﬁned in another way. More information about this relationship and other linear Gibbs energy relationships can be found in the monograph by Hammett [48]. Obviously, linear Gibbs energy relationships are quite useful because of their predictive power. However, it must be remembered that they are based on fundamental concepts which originate in the theory of elementary reactions. Thus, the experimental study of these relationships has helped elucidate the energy proﬁles involved in the basic processes which occur in solutions.

7.12 Concluding Remarks The study of reaction kinetics in solution has seen tremendous advances since the 1960s. This progress has been recognized by four Nobel Prizes. In 1967, Manfred Eigen, Ronald Norrish, and Sir George Porter received the Nobel Prize for their studies of extremely fast reactions aﬀected by disturbing a solution equilibrium by

378

LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 7.25 Plot of the logarithm of the rate constant for bromination of a substituted benzeneboronic acid relative to that of the unsubstituted acid, log10(kr/kr0) against the Hammett substituent parameter, sX.

means of very short pulses of energy. Their work was carried out before lasers were used in this type of study. Henry Taube received the Nobel Prize in 1983 for his studies of the mechanism of electron transfer reactions, especially those involving metal complexes. In 1992, Rudy Marcus was given the Nobel Prize for his work on the theory of electron transfer reactions. The most recent Nobel Prize was awarded in 1999 to Ahmed Zewail for his studies of the transition states of chemical reactions using femtosecond spectroscopy. Resolution of chemical events on such short time scales emphasizes the great progress made in this area of chemistry. The discussion in this chapter has focused on fundamental types of solution reactions including electron transfer, proton transfer, atom transfer, and ligand transfer reactions. The theory of these processes is of great fundamental interest and has been discussed in detail for electron transfer processes. Of course, not all processes studied in solution could be covered in this chapter. A notable example is oscillation reactions, which have also attracted considerable attention. The remarkable progress made in this ﬁeld clearly has been the result of both experimental and theoretical advances. As theory asked new questions about fundamental processes, new and more powerful experiments were designed to answer them. As a result, this ﬁeld now has developed a molecular-level understanding of the dynamics of reactions in condensed phases. General References G1. Laidler, K. J. Chemical Kinetics, 3rd ed.; Harper and Row: New York, 1987. G2. Connors, K. A. Chemical Kinetics, The Study of Reaction Rates in Solution; VCH Publishers: New York, 1990.

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G3. Hynes, J. T. In Theory of Chemical Reaction Dynamics; Baer, M., Ed.; CRC Press: Boca Raton, FL, 1985; Vol. 4, Chapter 4. G4. Strehlow, H. Rapid Reactions in Solution; VCH: Weinheim, 1992.

References 1. Michaelis, L.; Menten, M. L. Biochem. Z. 1913, 49, 333. 2. Noyes, R. M. Progress in Reaction Kinetics; Pergamon Press: Oxford, 1961; Vol. 1, p 129. 3. Taube, H. Angew. Chem., Int. Ed. Engl. 1984, 23, 329. 4. Marcus, R. A. J. Chem. Phys. 1965, 43, 679. 5. Sutin, N. Prog. Inorg. Chem. 1983, 30, 441. 6. Marcus, R. A. J. Phys. Chem. 1968, 72, 891; in Proton Transfer, Symposia of the Faraday Society, No. 10, 1975, p 60. 7. Hague, D. N. Fast Reactions; Wiley-Interscience: New York, 1971. 8. Bell, R. P. The Proton in Chemistry; Cornell University Press: Ithaca, NY, 1973. 9. Basolo, F.; Pearson, R. G. Mechanisms of Inorganic Reactions, A Study of Metal Complexes in Solution, 2nd ed.; Wiley: New York, 1967. 10. Kruger, H. Chem. Soc. Rev. 1982, 11, 227. 11. Sykes, A. G. Kinetics of Inorganic Reactions; Pergamon: Oxford, 1996. 12. Parker, A. J. Chem. Rev. 1969, 69, 1. 13. Eigen, M.; De Maeyer, L. In Investigation of Rates and Mechanisms of Reaction, 3rd ed.; Hammes, G. G., Ed.; Wiley-Interscience: New York, 1974; Vol. VI, Part II, Chapter 3. 14. Bernasconi, C. F. Relaxation Kinetics; Academic Press: New York, 1976. 15. Norrish, R. G. W.; Porter, G. Nature 1949, 164, 658. 16. Andrews, D. L. Lasers in Chemistry, 3rd ed.; Springer: Berlin, 1997. 17. Fleming, G. R. Chemical Applications of Ultrafast Spectroscopy; Oxford University Press: New York, 1986. 18. Simon, J. D. Ultrafast Dynamics of Chemical Systems; Kluwer Academic Publisher: Dordrecht, 1994; Acc. Chem. Res. 1988, 21, 128. 19. Van der Zwan, G.; Hynes, J. J. J. Phys. Chem. 1985, 89, 4181. 20. Maroncelli, M. J. Mol. Liq. 1993, 57, 1. 21. Fawcett, W. R. Chem. Phys. Lett. 1992, 199, 153. 22. Waldeck, D. H. Chem. Rev. 1991, 91, 415. 23. Todd, D. C.; Fleming, G. R. J. Chem. Phys. 1993, 98, 269. 24. Grampp, G.; Hetz, G. Ber. Bunsen-Ges. Phys. Chem. 1992, 96, 198. 25. Marcus, R. A. In Special Topics in Electrochemistry; Rock, P. A., ed.; Elsevier: Amsterdam, 1997; p 161. 26. Fawcett, W. R.; Blum, L. Chem. Phys. Lett. 1991, 187, 173. 27. German, E. D.; Kuznetsov, A. M. Electrochim. Acta 1981, 26, 1595. 28. Zusman, L. D. Chem. Phys. 1980, 49, 295. 29. Heitele, H. Angew. Chem., Int. Ed. Eng. 1993, 32, 359. 30. Kapturkiewicz, A. Adv. Electrochem. Sci. Eng. 1997, 5, 1. 31. Drago, R. S. Physical Methods in Chemistry; Saunders: Philadelphia, 1997; Chapter 8. 32. Jackman, L. M.; Cotton, F. A. Dynamic Nuclear Magnetic Resonance Spectroscopy; Academic Press: New York, 1975; Chapter 15. 33. Sandstro¨m, J. Dynamic NMR Spectroscopy; Academic Press: New York, 1982. 34. Ravage, D. K.; Stengle, T. R.; Langford, C. H. Inorg. Chem. 1967, 6, 1252. 35. Burgess, J.; Pelizetti, E. Prog. React. Kinet. 1992, 17, 1. 36. Grampp, G.; Jaenicke, W. Ber. Bunsen-Ges. Phys. Chem. 1991, 95, 904. 37. Sanchez-Burgos, F.; Moya, M. L.; Galan, M. Prog. React. Kinet. 1994, 19, 1. 38. Brønsted, J. N. Z. Phys. Chem. 1922, 102, 169. 39. Bjerrum, N. Z. Phys. Chem. 1924, 108, 82.

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40. 41. 42. 43. 44. 45. 46. 47. 48.

Fawcett, W. R.; Tikanen, A. C.; Henderson, D. J. Can. J. Chem. 1997, 75, 1649. Kramers, H. A. Physica 1940, 7, 284. McQuarrie, D. A. Statistical Mechanics; Harper and Row: New York, 1976. Hynes, J. T. J. Phys. Chem. 1986, 90, 3701. Levich, V. G. Adv. Electrochem. Eng. 1965, 4, 249. Brønsted, J. N.; Pederson, K. Z. Phys. Chem. 1923, 108, 185. Marcus, R. A. J. Phys. Chem. 1968, 72, 891. Hammett, L. P. J. Am. Chem. Soc. 1937, 59, 96. Hammett, L. P. Physical Organic Chemistry, 2nd ed.; McGraw-Hill: New York, 1970. 49. Jaﬀe, H. H. Chem. Rev. 1953, 53, 191. 50. Taft Jr., R. W. J. Phys. Chem. 1960, 64, 1805.

Problems 1. Consider the mechanism k1

A P B k1

k2

B ! C where all reactions are ﬁrst order. Derive an exact expression for the concentration of B as a function of time. What is expression for cB if it is assumed to be in a steady state? Calculate cB as a function of time using each expression assuming k1 ¼ 10 s1, k–1 ¼ 1 s1, and k2 ¼ 0.01 s1. 2. The relaxation time for the reaction Hþ þ OH ( + H2 O is 3.5105 s at pH ¼ 7 and 25 C. What are the rate constants for the forward and reverse reactions? 3. The following rate constants were measured for the reaction between dimethylaniline and methyliodide in nitrobenzene. Temperature T=8C

Rate Constant kr =L mol1 s1

25 40 60 80

8.39 21.0 77.2 238

Determine the Arrhenius activation energy and the pre-exponential factor. Use these results to estimate the enthalpy and entropy of activation at 25 C. 4. The following data were obtained for the rate constant of a second-order reaction as a function of pressure:

CHEMICAL REACTION KINETICS IN SOLUTION

Pressure P/atm

Rate Constant kr =L mol1 s1

1 272 544 816 1088

9:30 104 11:13 104 13:1 104 15:3 104 17:9 104

381

Estimate the volume of activation for the reaction. 5. For the rate of proton transfer between MeOHþ 2 and methanol, and between MeO and methanol, Grunwald gives the result 10 rate ðL mol1 s1 Þ ¼ 8:8 1010 ½MeOHþ 2 þ 1:85 10 ½MeO

What are the second-order rate constants for these processes? 6. The dehydration of acetaldehyde has been studied in the presence of various weak acids in aqueous acetone at 25 C. The following table gives the name of the weak acid, its acidity constant, KA, and the rate constant for the catalyzed reaction, kr. Analyze these data to obtain the parameters of the Brønsted relation. Acid Diethylketoxine Hydroquinone Resorcinol Nitromethane o-Chlorophenol Benzoylacetone(enol) p-Nitrophenol 2,4,6-Trichlorophenol Propionic Acetic Phenylacetic Diphenylacetic Glycolic Dichloroacetic

kr/M1min1

KA/M

0.104 0.013 0.026 0.00084 0.112 0.0088 0.52 1.53 18.0 19.2 33.0 44.2 56.6 773

2:5 1013 4:5 1011 1:55 1010 5:8 1010 3:2 109 5:9 109 6:75 108 3:9 107 1:35 105 1:76 105 4:88 105 1:15 104 1:54 104 5:0 102

7. A common type of enzyme inhibition occurs when an inhibitor combines with a reaction intermediate to form an inactive complex. This mechanism is E þ S ! ES ES ! E þ S ES þ I ! ESI ESI ! ES þ I ES ! E þ P

382

LIQUIDS, SOLUTIONS, AND INTERFACES

Assuming ES and ESI are at a steady state, where I is the inhibitor, derive an expression for the rate of formation of product P. 8. Consider the following data for the pKa’s of anilinium ions: pKa Substituent

meta

para

H Br Cl CN F I CH3 O CH3 NO2

4.60 3.58 3.52 2.75 3.57 3.60 4.23 4.72 2.47

4.60 3.86 3.98 1.74 4.65 3.78 5.34 5.10 1.00

Determine the Hammett parameter rH for this equilibrium process using the Hammett substituent parameters given in table 7.10.

8

Liquids and Solutions at Interfaces

Alexander Naumovich Frumkin was born in 1895 in Kishinev, Moldova, and grew up and went to school in Odessa, Ukraine. After ﬁnishing middle school in 1912, he studied in Western Europe at Strasbourg and Bern for two years. In 1915, he passed the Diploma exam at the Physics and Mathematics Faculty of Novorossijski University in Odessa and began his study of electrocapillary phenomena. His doctoral dissertation entitled ‘‘Electrocapillary Phenomena and Electrode Alexander Naumovich Frumkin Potentials’’ was published in 1919. He continued to teach and carry out research in Odessa until 1922 when he was invited to join the Karpov Institute of Physical Chemistry in Moscow. In 1928– 29, he spent a sabbatical year at the University of Wisconsin, and on return to Moscow became vice-director of the Karpov Institute. In 1930, he was appointed Professor of Chemistry at Moscow State University where he founded the Department of Electrochemistry. He founded the Institute of Electrochemistry at Moscow in 1958; this became during his lifetime the most important center for electrochemical research in the world. In his early work, Frumkin resolved the problem arising from the earlier work of Nernst regarding the source of the potential drops within an electrochemical cell. He demonstrated the importance of the point of zero charge of a metal electrode in determining adsorption processes and reaction kinetics at the electrode solution interface. His experimental work in interfacial electrochemistry helped consolidate fundamental ideas in the ﬁeld and demonstrate their relationship to other important areas of physical chemistry. Frumkin received many honors for his scientiﬁc work both in the Soviet Union and internationally. He died while attending a scientiﬁc conference at Tula, Russia, in 1976.

8.1 The Molecular Environment at the Interface Is Different Than in the Bulk When the properties of liquids and solutions are considered, attention is normally focused on the bulk of the phase, and the properties of the system at its boundaries are ignored. Signiﬁcant eﬀects are associated with the region near the surface 383

384

LIQUIDS, SOLUTIONS, AND INTERFACES

of a liquid phase and an understanding of these is an important part of solution chemistry. As a simple example, consider a beaker of pure water at room temperature in a closed inert environment. As has been seen in the consideration of liquid structure, the properties of water are strongly inﬂuenced by hydrogen bonding between neighboring molecules, and to a lesser extent by dipole–dipole interactions. As an observer at the molecular level, one would ﬁnd that the molecules near the boundaries of the water phase have diﬀerent properties. There are two boundaries in this system, the water | air interface and the water | glass interface. At the water | air interface, the important feature is the termination of intermolecular interactions, so that molecules must adjust to an environment where the number of nearest neighbors is reduced. At the water | glass interface, water molecules meet the components of glass, a supercooled liquid with silicon dioxide as the major component. Interaction between water and silicon dioxide is diﬀerent from interaction among water molecules. It is clear that the molecular environment at these interfaces is very diﬀerent than it is in the bulk. As a result, local properties are diﬀerent. Now imagine that the water in the beaker is dispersed as a fog, that is to say, as many very small droplets for which the ratio of surface area to volume is much larger than for the water in the beaker. It is obvious that the thermodynamic properties of the fog, a colloidal system, are very diﬀerent from those of the water as a macrosystem in a beaker. In order to create the fog considerable work must be done to form a system with a much larger surface area. This means that the Gibbs energy of a fog containing the same number of water molecules as the beaker of water is much higher. It follows that the fog is a non-equilibrium system in which the water droplets grow in size as a result of collisions. An understanding of the properties of liquids and solutions at interfaces is very important for many practical reasons. Some reactions only take place at an interface, for example, at membranes, and at the electrodes of an electrochemical cell. The structural description of these systems at a molecular level can be used to control reactions at interfaces. This subject entails the important ﬁeld of heterogeneous catalysis. In the discussion which follows in this chapter the terms surface and interface are used interchangeably. There is a tendency to use the term surface more often when one phase is in contact with a gas, for example, in the case of solid | gas and liquid | gas systems. On the other hand, the term interface is used more often when condensed phases are involved, for example, for liquid | liquid and solid | liquid systems. The term interphase is used to describe the region near the interface where the structure and composition of the two phases can be diﬀerent from that in the bulk. The thickness of the interphase is generally not known without microscopic information but it certainly extends distances corresponding to a few molecular diameters into each phase. In the present chapter, the properties of interfaces involving liquids and solutions are described from the point of view of thermodynamics, and also at a molecular level. Examples are given for the liquid | gas, liquid | liquid and liquid | solid interfaces. The electrical properties of interfaces are also considered and their relevance to processes such as the extraction of an ion from an electrolyte solution is discussed. As will be seen, a knowledge of interfacial properties and

LIQUIDS AND SOLUTIONS AT INTERFACES

385

reactions is extremely important to the understanding of many processes in living systems and in the environment.

8.2 The Interfacial Tension of Liquids When one considers that a molecule at an interface or surface has fewer nearest neighbors than a molecule in the bulk of the solution, it is easy to understand that the interfacial molecule has diﬀerent thermodynamic properties. If one wishes to increase the area of a liquid, one must bring molecules from the bulk to the surface, thereby breaking intermolecular bonds and doing work. The work done to increase the surface area by unit amount is the interfacial or surface tension. Mathematically, the reversible work dw is given by dw ¼ g dA

ð8:2:1Þ

where g is the interfacial tension and dA the change in area. The units of interfacial tension are J m2 . This is often expressed as N m1 . The interfacial tension is obviously an important property of a liquid because it gives a direct indication of the magnitude of intermolecular forces. As a result of interfacial tension a liquid which is not in contact with another condensed phase, such as a water droplet in air, assumes the shape which has minimum area. It turns out that this shape is a sphere. As a result, there are no elliptical or square water droplets! By maintaining a spherical shape, the area-to-volume ratio, and the number of molecules at the surface are their lowest possible values. One is not surprised by this fact on the basis of experience. Some experimental methods for measuring interfacial tension involve the contact of three phases, usually glass and air, together with the liquid being studied. This means that one must understand the properties of three interfaces, namely, the liquid | air, the liquid | glass, and the glass | air interface, in order to interpret the experimental results. Since the forces of cohesion in the liquid, that is, the intermolecular forces, are not necessarily the same as the forces of adhesion between the glass and the liquid, the interface is usually curved. Furthermore, the properties of curved interfaces are diﬀerent than those of ﬂat interfaces. As a result, the contact angle formed between the glass and the liquid must be examined carefully. Other methods, for example, the measurement of drop proﬁle, and the maximum bubble pressure method, depend only on the interfacial tension between the two phases in contact. The formation of a contact angle is illustrated in ﬁg. 8.1. The forces on a molecule near the interface between the glass wall and the water are shown as vectors with a longer vector fa for the adhesive force and a shorter vector fc for the cohesive force, indicating that adhesive forces are stronger than cohesive forces. When these two vectors are added, the resulting vector points away from the air | water interface at an angle determined by the relative magnitudes of fa and fc . In order for the system to be at equilibrium, this interface must be perpendicular to the resultant vector fT . Thus, the interface is curved and makes a contact angle y with the wall, as shown in ﬁg. 8.1.

386

LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 8.1 Schematic diagram of the contact angle formed between a glass wall and a liquid where the adhesive force on a molecule near the air | water interface and wall is fa and the cohesive force with the bulk, fc . The vector sum of these forces is fT .

When the contact angle is less than 90 , the liquid is said to wet the wall. In the case of glass and water this angle is close to 0 , indicating very strong adhesive forces at this interface. In the case of mercury and glass, the contact angle is 140 , demonstrating that adhesive forces are much weaker than cohesive forces. As a result, the meniscus for a mercury column in a glass tube is convex rather than concave. Now, consider again a spherical liquid drop. Because of the curvature of the interface, there is a pressure diﬀerence between the inside and outside of the drop. This diﬀerence exists because of the interfacial tension, which tends to reduce the area of the liquid system, so that equilibrium is maintained with a higher pressure inside the drop than the atmospheric pressure outside. If the radius of the drop is r, its surface area is 4pr2 . The incremental work dws done in increasing the radius by dr is dws ¼ 8pgrdr

ð8:2:2Þ

This is counterbalanced by mechanical work dwm done on the basis of the volume change which occurs under a pressure diﬀerence, Pi Patm , where Pi is the internal pressure in the drop and Patm is the external atmospheric pressure. Thus, dwm ¼ ðPi Patm Þ dV ¼ 4pr2 ðPi Patm Þdr

ð8:2:3Þ

At equilibrium, dws is equal to dwm so that Pi Patm ¼

2g r

ð8:2:4Þ

This is the Young–Laplace equation applied to a spherical surface. A more general form of this equation is used when the curvature of the interface is not spherical [G1]. EXAMPLE

Calculate the increase in pressure inside a water droplet at 25 C given that its radius is 0.1 mm. The surface tension of water at 25 C is 71.8 mJ m2. From equation (8.2.4), P ¼

2 71:8 ¼ 1:44 106 m Pa ¼ 1440 Pa 1 104

ð8:2:5Þ

LIQUIDS AND SOLUTIONS AT INTERFACES

387

Thus, the pressure drop is 1440 Pa or 0.014 bar. This is not a negligible pressure diﬀerence. Two diﬀerent methods for measuring surface tension are now considered. The ﬁrst is based on the fact that a liquid which wets glass rises in a glass capillary when this is immersed in a beaker of the liquid. The height of the liquid in the capillary can be related to the interfacial tension at the liquid | air interface. The meniscus is concave on the liquid side, indicating that the pressure inside the liquid in the capillary is less than that in the atmosphere. This experiment is discussed here for the more general case of two ﬂuids a and b, as illustrated in ﬁg. 8.2. In the speciﬁc example a corresponds to air and b to water, but the analysis could be applied to any two ﬂuids. The meniscus is assumed to be part of a sphere of radius R whose center is also indicated in the ﬁgure. Geometrical analysis shows that r ð8:2:6Þ R¼ cos y where r is the radius of the capillary tube and y, the contact angle. The diﬀerence in pressure across this interface is given by 2g P ¼ ð8:2:7Þ R Now consider the pressure exerted through distance h by phase a outside of the capillary, where h is height of the rise of ﬂuid b in the capillary. The pressure is given by Pa ¼ ra gh ð8:2:8Þ where ra is the density of phase a and g, the acceleration due to gravity. For equilibrium the same pressure drop must occur inside of the capillary. This pressure is Pb ¼ rb gh

2g R

Combining equations (8.2.6)–(8.2.9), one obtains the result that r g¼ ðr ra Þgh 2 cos y b

Fig. 8.2 Schematic diagram of the rise of ﬂuid b with respect to ﬂuid a in a narrow glass capillary. The distance h is measured from the bottom of the meniscus in the capillary.

ð8:2:9Þ

ð8:2:10Þ

388

LIQUIDS, SOLUTIONS, AND INTERFACES

This result is general and can be applied to any two ﬂuids, provided the meniscus has a spherical shape. The diﬃculty in using this technique arises when the contact angle is not known or when the capillary is not a perfect cylinder. This means that the experimental system is usually calibrated using a system of known interfacial tension. Other problems arise when the meniscus is not perfectly spherical. These have been discussed in detail by Adamson [G1]. EXAMPLE

Estimate the height of benzene in a glass capillary with a diameter of 0.01 cm at 20 C given that g is 28.96 mJ m2 and the contact angle is zero. The densities of benzene and air at this temperature are 0.8785 and 0.0014 g cm3, respectively. The acceleration due to gravity is 9.807 m s2. Applying equation (8.2.9), one obtains h¼

2 28:86 203 ¼ 6:7 102 m 104 ð0:8785 0:0014Þ 10þ3 9:807

ð8:2:11Þ

Thus the benzene rises a distance of 6.7 cm in the capillary. Another method for measuring g is the maximum bubble pressure technique, which has the advantage that its results are independent of the contact angle. This technique as applied to studies of the mercury | liquid solution interface is illustrated schematically in ﬁg. 8.3. The capillary is bent into a U-shape so that the mercury emerges in an upward direction. As the pressure on the mercury is increased, a bubble emerges from the end of the capillary. The sequence of shapes assumed by the mercury bubble is such that it is always a section of a sphere, its radius going to a minimum when it is exactly a hemisphere. At this point the pressure is a maximum, further increase in pressure resulting in dislodgment of a mercury drop. According to the Young–Laplace equation,

Fig. 8.3 Schematic diagram of a maximum bubble pressure apparatus used to study the mercury | electrolyte solution interface. The pressure applied to the mercury in reservoir R is measured as a function of the shape of the emerging mercury bubble at the capillary tip T. The pressure is a maximum when this shape is hemispherical. The experiment is carried out for diﬀerent voltages V applied between the reference electrode and the mercury reservoir.

LIQUIDS AND SOLUTIONS AT INTERFACES

Pmax ¼

2g r

389

ð8:2:12Þ

where r is the radius of the capillary at its oriﬁce. Estimation of the pressure involves consideration of the height of solution above the end of the capillary. In practice, the apparatus is calibrated with a system of known interfacial tension. This technique is conveniently adapted to computer-controlled experiments and has been used to study the interfacial tension of the mercury | solution interface as a function of electrical potential [1]. The electrical aspects of these experiments are discussed in detail in chapter 10. Values of the interfacial tension at the liquid | air interface for the polar solvents considered in chapter 4 are summarized in table 8.1. There is a very large variation in g from a high of 71.81 mJ m2 for water to a low of 21.90 mJ m2 for ethanol. The high value for water is a clear indication of the importance of hydrogen bonding in determining its physical properties. Formamide, which can form two hydrogen bonds per solvent monomer, also has a high value

Table 8.1 Values of the Interfacial Tension and Its Temperature Coefﬁcient at the Liquid–Air Interface for Selected Polar Solvents at 25 C [2] Solvent

Interfacial Tension g=mJ m2

Temperature Coeﬃcient ð@g=@TÞ=mJ K1 m2

71.81 22.30 21.90 23.30 24.16 58.15 38.7

0.155 0.093 0.088 0.069 0.079 0.793 0.59

22.68 27.56 38.46 26.79 33.15 36.42 42.98 33.8* 40.7 42.43 36.48 25.82 41.39 35.5**

0.123 0.122 0.106 0.100 0.147 0.122 0.121 — — 0.118 0.168 0.110 0.110 —

Protic 1. 2. 3. 4. 5. 6. 7.

W MeOH EtOH PrOH BuOH F NMF

Aprotic 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

AC AcN BzN BuN DMA DMF DMSO HMPA NMP NB NM PrN PC TMS

*Measured at 30 C. **Measured at 20 C.

390

LIQUIDS, SOLUTIONS, AND INTERFACES

of g. In the case of the alcohols, which can form only one hydrogen bond per monomer, the values of g are much lower. Considerable variation in g is also seen among the aprotic solvents. The highest value is found for dimethylsulfoxide, which is highly dimerized in the bulk. On the other hand, a rather low value is found for acetone. A proper ranking of these data would require knowledge of the area occupied by a single molecule at the interface on the basis of its molecular size and orientation. Then, the interfacial tension data could be re-evaluated as work done per molecule to take it from the bulk to the interface. If g is measured at constant temperature and pressure in a one component system, then it is equal to the Gibbs energy of the interface, Gs . A more precise deﬁnition of Gs is given in the following section, but the idea that work done can be equated to Gibbs energy is familiar from thermodynamics. It follows that the temperature derivative of g can be related to the entropy of the interface as follows: @Gs @g ð8:2:13Þ ¼ Ss ¼ @T @T Thus, the values of @g/@T recorded in table 8.1 give the entropy change when the area of the interface is increased by 1 m2. In all cases, Ss is positive, indicating that order is lost when molecules are moved from the bulk of the solution to the interface. This is a result of the fact that intermolecular bonds are broken in the change of environment. The largest entropy change is found for formamide due to its strongly hydrogen-bonded structure. Much smaller entropies are found for water and the alcohols. In the case of the aprotic solvents, the entropies fall in the range from 0.1 to 0.2 mJ K1 m2. In order to assess these parameters in more detail, the entropy per molecule must be estimated on the basis of the area occupied by each molecule. In assessing the meaning of the parameters g and @g=@T, one eventually asks the question ‘‘What is the interface and how thick is it?’’ The interface certainly has the thickness of the layer of molecules at the termination of the liquid phase but the disruption of normal liquid structure may extend somewhat further into the bulk. Since the thickness of the interface is not known, it is diﬃcult to give a molecular interpretation of the thermodynamic properties of this region. However, eﬀective thermodynamic conventions have been developed for discussing interfacial properties. These are outlined in detail in the following section.

8.3 The Thermodynamics of Fluid Interfaces Fluid interfaces are those systems in which the two phases forming the interface are mobile. Thus, they include liquid|liquid and liquid|gas interfaces. The example chosen here to illustrate the application of thermodynamics to these interfaces is a closed system containing both gas and liquid phases consisting of two components, propanol and water. The liquid phase is designated b and the gas phase a. Now imagine a microprobe moving up through b to the interface (see ﬁg. 8.4). The concentration of the alcohol is constant in this phase and equal to cbA . In the gas phase, its concentration is much lower. Near the interface the alcohol concentration rises before it falls to the lower gas phase value caA . This observation is

LIQUIDS AND SOLUTIONS AT INTERFACES

391

Fig. 8.4 (a) A two-component system with liquid (b) and gas (a) phases; (b) concentration of the alcohol component as a function of vertical position near the interface.

attributed to the hydrophobic nature of the alkyl group in propanol which likes to be out of the aqueous environment as much as possible. Under normal circumstances where the surface area/volume ratio is small, any depletion of propanol from the bulk liquid is negligible. Thus, a change in bulk concentration due to the preference of propanol for the interface would not be observed experimentally. However, if the area/volume ratio is increased, for example, by converting phase b to an aerosol, the alcohol could be suﬃciently depleted from the bulk that its concentration would change. Two conventions exist in the treatment of interfacial thermodynamics. The ﬁrst, due to Gibbs [G1], treats the interface as a region with no volume. Since the interphase is actually a region of ﬁnite width extending over several molecular diameters, this convention is equivalent to associating all interfacial inhomogeneities with a plane placed arbitrarily in the interphase. The choice of position of the dividing surface is often made so that the surface excess of one component is arbitrarily deﬁned to be zero. Suppose that the dividing surface is placed at position xs as shown in ﬁg. 8.4, and that positions xb and xa are suﬃciently far within the bulk phases that the interface does not inﬂuence local properties. In the Gibbs model, the properties of b are assumed to be uniform right up to the plane at xs . Similarly, the properties of a are assumed to be uniform up to this plane. This means that inhomogeneities associated with the interphase are now attributed to a plane. As a result, one may deﬁne the surface excess for propanol, A as follows: xða xðs b x A ¼ ðcA cA Þdx þ ðcxA caA Þdx ð8:3:1Þ xb

xs

where cxA is the concentration of propanol at position x. This deﬁnition is arbitrary because it depends on where one places the dividing plane xs . The surface excess has units of mol m2 when the concentration is expressed in mol m3. A similar expression can be written for the surface excess of water, namely, xða xðs b x W ¼ ðcW cW Þdx þ ðcxW caW Þdx ð8:3:2Þ xb

xs

cbW ,

where cxW is the local water concentration, the average concentration in phase b, and caW , the corresponding value in phase a. If the dividing plane xs is moved,

392

LIQUIDS, SOLUTIONS, AND INTERFACES

both A and W change but a simple expression relates the two values. As a result, one may only determine relative surface excesses and not their absolute values, as shown in more detail below. The second convention, due to Guggenheim [G1], deﬁnes a surface phase s whose width is suﬃcient to include the inhomogeneities associated with the surface. Placing the boundaries at xa and xb , the surface excesses may be deﬁned as follows: xða ns ð8:3:3Þ W ¼ cxW dx ¼ W A xb

and xða

A ¼

cxA dx ¼

nsA A

ð8:3:4Þ

xb

nsW

nsA

and are the number of moles of water and alcohol, respectively, in the surface phase, and A is the area of the interface. The volume of the surface phase Aðxa xb Þ is ﬁnite but arbitrary, its value depending on the width of the inhomogeneous region. Just as with the Gibbs convention, one can only measure relative surface excesses, not absolute values. This aspect of the surface excess is clariﬁed further below. The fundamental thermodynamic expression for the inﬁnitesimal increase in Gibbs energy for a multicomponent system which can exchange matter with its surroundings was derived earlier on the basis of the ﬁrst and second laws of thermodynamics (equation (1.3.21)): X dG ¼ SdT þ VdP þ mi dni ð8:3:5Þ i

mi is the chemical potential of species i and the other symbols have their usual meanings. This expression was derived assuming that the only work done on the system is mechanical. If work is also done to change its area A, then the expression becomes X mi dni þ g dA ð8:3:6Þ dG ¼ SdT þ VdP þ i

It follows that the interfacial tension g is given by @G g¼ @A T;P;ni

ð8:3:7Þ

Two points should be noted about equation (8.3.6). First, the number of components considered in the summation is that which can be added to the system in an independent way. Thus, if NaCl is a component, Naþ is not considered separately from Cl, since the concentrations of the two ions are exactly equal. However, local concentrations near the interface may not be equal. If one considers Naþ and Cl as independent components the total number of variables in equation (8.3.6) increases. In the case of charged components, the chemical potential mi must be replaced by the electrochemical potential m~ i . It also follows

LIQUIDS AND SOLUTIONS AT INTERFACES

393

that the number of degrees of freedom in the system is exactly the same because of the condition of electroneutrality. Second, one is normally interested in the Gibbs energy change associated with the surface phenomena. These are diﬃcult to detect when the surface to volume ratio is small. Following the Gibbs convention, one deﬁnes a surface phase s of zero volume, and contributions to dG due to perfectly homogeneous a and b phases and a s phase with which all inhomogeneity is associated. Thus, dG ¼ dGa þ dGb þ dGs

ð8:3:8Þ

Now, the right-hand side of equation (8.3.5) is divided up to obtain the contributions from phases a, b, and s under the condition that temperature and pressure are constant throughout the system. In addition, at equilibrium the chemical potential of each component is the same in each phase. The total entropy of the system is S ¼ Sa þ Sb þ Ss

ð8:3:9Þ

Remembering that the surface phase has no volume in the Gibbs convention, V ¼ Va þ Vb

ð8:3:10Þ

Finally, any incremental change in the number of moles of component i can be divided into changes in each phase so that dni ¼ dnai þ dnbi þ dnsi

ð8:3:11Þ

The following equations can now be written: dGa ¼ Sa dT þ Va dP þ

X

mi dnai

ð8:3:12Þ

i

dGb ¼ Sb dT þ Vb dP þ dGs ¼ Ss dT þ

X

X

mi dnbi

ð8:3:13Þ

i

mi dnsi þ g dA

ð8:3:14Þ

i

The ﬁnal equation, which describes the thermodynamic properties of the surface phase, is the one which is important for the present analysis. Now, an integration is performed on equation (8.3.14) at constant temperature to ﬁnd the Gibbs energy of the surface phase. The result is X Gs ¼ mi nsi þ gA ð8:3:15Þ i

Taking the total diﬀerential and comparing with equation (8.3.14), one obtains X nsi dmi þ A dg ¼ 0 ð8:3:16Þ i

394

LIQUIDS, SOLUTIONS, AND INTERFACES

This result is known as the Gibbs adsorption isotherm and is the starting point for analyzing the properties of the surface phase at constant temperature and pressure. It is more commonly written as X dg ¼ i dmi ð8:3:17Þ i

where i is the surface excess equal to the number of moles of component i in the surface phase divided by the area of the surface. In the case of the propanol–water system discussed above, the Gibbs adsorption isotherm is dg ¼ A dmA þ W dmW

ð8:3:18Þ

Since it is impossible to change the chemical potential of A without also changing that of W, one cannot measure either A or W independently. The practical signiﬁcance of this can be seen using the Gibbs–Duhem relationship, according to which xA dmA þ xW dmW ¼ 0

ð8:3:19Þ

where the xi ’s are mole fractions in the bulk of the solution. Accordingly, one may write x ð8:3:20Þ dg ¼ A W A dmA ¼ ðWÞ A dmA xW ðWÞ A is the surface excess of alcohol relative to that of water. The relative value is approximately equal to the absolute one when W xA xW . When the solution is suﬃciently dilute such that Henry’s law holds for the solute, then dmA ¼ RT d ln cA

ð8:3:21Þ

where cA is the concentration of the solute, namely, propanol in the present example. Its surface excess is then given by 1 @g A ¼ ð8:3:22Þ RT @ ln cA where it has been assumed that the relative nature of A may be neglected in the dilute solution. EXAMPLE

In the concentration range 0.01–0.03 M, the interfacial tension of solutions of 3-phenylpropanoic acid (PBA) at 21 C is given by the equation g ¼ 67:87 567:3cA

ð8:3:23Þ

where g is given in mJ m2 and cA is the concentration of PBA. Calculate its surface excess at cA ¼ 0:02 M. From the relationship given @g @g ¼ cA ¼ 0:02 567:3 ¼ 11:35 mJ m2 @ ln cA @cA

ð8:3:24Þ

LIQUIDS AND SOLUTIONS AT INTERFACES

395

The surface excess is A ¼

1 @g 11:35 ¼ 4:64 103 mmol m2 ¼ RT @ ln cA 8:3145 294:2

ð8:3:25Þ

It is clear that determination of the change of interfacial tension with solute concentration in dilute solutions provides a way to ﬁnd the surface excess of the solute. This method has been successfully applied to many two-component systems involving molecules. Application to electrolyte solutions is more complicated but is discussed further below. Before doing so, more details about the electrical aspects of interfaces are outlined.

8.4 The Electrical Aspects of Interfaces When the thermodynamic properties of charged components in condensed phases are considered it is convenient to evaluate the chemical contributions to the Gibbs energy separately from the electrostatic contributions. This leads to the deﬁnition of the electrochemical potential of charged species i and the inner potential fa of the phase a in which species i is located (see section 6.6). It was emphasized in the earlier discussion that the separation of the electrochemical potential into chemical and electrostatic contributions is arbitrary, but conceptually very useful. The inner potential arises as a result of the hypothetical work done to take a charged species from inﬁnity across the double layer at the surface of phase a under circumstances where this species experiences no chemical interactions in the bulk. The double layer is described as that region at the surface of phase a in which there may be a potential drop due to free charge on the surface, unequal adsorption of ions of opposite charge, or net orientation of dipoles [G4]. The inner potential is a quantity which cannot be measured experimentally, but only estimated on the basis of a suitable model. The inner potential is further subdivided into the outer potential ca and the surface potential wa , that is, fa ¼ ca þ wa

ð8:4:1Þ

The outer potential is due to the free or excess charge on the surface of phase a and can be measured experimentally. The surface potential is due to the dipolar distribution of charge at the interface due to the unequal adsorption of ions and orientation of molecular dipoles. It cannot be measured experimentally. Since these quantities are deﬁned with respect to the process of bringing a charged species from inﬁnity into the phase, the surface potential is positive when the positive end of the dipolar charge points toward the center of the solution and the negative end toward charge-free inﬁnity. The method of measuring the outer potential of phase a can be described on the basis of a conceptual experiment in which a test charge is brought up to the phase, which bears a net charge q. Assuming that the phase is an isolated sphere of radius r, the potential experienced by the test charge is

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c¼

q q ¼ 4pe0 d 4pe0 ðr þ xÞ

ð8:4:2Þ

where d is its distance from the center of the sphere, and x, its distance from the surface of the sphere. When the distance is very small with respect to the radius r, c is constant and equal to q=ð4pe0 rÞ. For example, if r is 1 cm, c is constant to more than 1% for values of x less than 105 m. For very small values of x, other factors need to be taken into consideration, for example, redistribution of the charge on the surface of the sphere due to the so-called image eﬀect. As a result the ideal distance from the surface at which one should measure c is 1 mm [G2, G4]. When the outer potential of phase a is zero, it is convenient to deﬁne another quantity, namely the real potential. Recalling that the electrochemical potential for species i in phase a is given by m~ ai ¼ mai þ zi Ffa ¼ mai þ zi Fðca þ wa Þ

ð8:4:3Þ

the real potential of species i in phase a is deﬁned to be the electrochemical potential when the phase is not charged: aai ¼ m~ ai ðca ¼ 0Þ ¼ mai þ zi Fwa

ð8:4:4Þ

This quantity can also be measured. For example, the electronic work function for metals discussed in section 8.5 is the negative value of the real potential for an electron in the metal. Since the discussion here concerns interfaces, it is also important to consider the potential diﬀerences which arise between the boundaries of two phases a and b. The inner potential diﬀerence, which is known as the Galvani potential diﬀerence, can be deﬁned as a b f

¼ fb fa

ð8:4:5Þ

The direction in which the potential diﬀerence is measured must be deﬁned so that the i operator gives the initial phase on the left-hand side and the ﬁnal phase on the right-hand side. Of course, one could choose to deﬁne the potential diﬀerence in the opposite direction and write b a f

¼ fa fb

ð8:4:6Þ

In general, the Galvani potential diﬀerence cannot be measured because the individual inner potentials cannot be measured. However, there is one important exception to this rule which is of great practical importance. When the phases a and b have the same chemical composition, a b f can be found from the diﬀerence in electrochemical potentials for a common species i in the phases. Thus, m~ bi m~ ai ¼ ðmbi þ zi Ffb Þ ðmai þ zi Ffa Þ ¼ zi Fa b f

ð8:4:7Þ

This principle is used every time one measures the electromotive force of an electrochemical cell. In this case the potential measuring device determines the diﬀerence between the electrochemical potentials of electrons in two pieces of the same metal, for example, in two copper wires. The classical device for doing this measurement is a Poggendorf compensation potentiometer but a modern in-

LIQUIDS AND SOLUTIONS AT INTERFACES

397

strument is a very high impedance electronic voltmeter based on operational ampliﬁers. The diﬀerence in outer potentials between two phases a and b is known as the Volta potential diﬀerence and is deﬁned as a b c

¼ cb ca

ð8:4:8Þ

Since each outer potential may be determined as described above, the Volta potential diﬀerence may also be found experimentally. A method for measuring aib c is discussed later in this chapter. It is important to keep in mind that, when two condensed phases a and b come into contact, the potentials associated with the isolated phases can change. When a and b contain a common charged species i, that is, an electron or an ion, contact of these phases results in the ﬂow of charge in order to establish equilibrium with respect to species i. This is accompanied by a movement of free charge on the surfaces of a and b which are exposed to air or vacuum. The ﬁrst process leads to a change in the Galvani potential diﬀerence a b f on contact of the two condensed phases. The second process, which involves a change in the free charge at the interfaces of a and b with air or vacuum, results in a change in the Volta potential diﬀerence a b c. Concrete examples of these eﬀects are discussed in more detail later with respect to the metal | solution and the liquid | liquid interfaces. The relationship between the Galvani potential diﬀerence, the Volta potential diﬀerence, and the surface potentials of two condensed phases in contact is illustrated in ﬁg. 8.5. Both of the condensed phases a and b are in contact with an inert gas or vacuum. The outer potential ca is the potential diﬀerence between charge-free inﬁnity and a point just outside of but close to phase a; cb is the corresponding quantity for phase b. The Volta potential diﬀerence is simply the diﬀerence between these quantities. wa and wb give the surface potentials of phases a and b, respectively. They are measured from vacuum into the condensed phase. The Galvani potential diﬀerence a b f is deﬁned within the condensed phases. Since the sum of the potential drops around a loop must be zero, one obtains a b f

wb cb þ ca þ wa ¼ 0

Fig. 8.5 Illustration of the deﬁnitions of the outer potentials, ca and cb, the surface potentials, wa and wb, and the Galvani potential diﬀerence a b for two condensed phases a and b in contact.

ð8:4:9Þ

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LIQUIDS, SOLUTIONS, AND INTERFACES

or a b f

¼ a b c þ wb wa

ð8:4:10Þ

This is the fundamental relationship between the quantities which describe the electrical properties of the interface between two condensed phases and the gas phase. The electrical properties of systems containing charged species are very important in achieving an understanding of how they behave at interfaces. In this chapter, attention is focused on two such systems, namely, metals and electrolyte solutions, that is, the components of electrochemical cells. In the following section, the properties of electrons in metals and the work required to extract an electron are examined in more detail.

8.5 The Work Function for Electrons in Metals A well-known property of metals is their ability to carry an electrical current by means of the movement of the electronic cloud associated with the metal atoms. Most metals are solids at room temperature and only a few, such as mercury, are liquid. Because of their importance in the design of electrochemical cells, their electronic properties are considered here in some detail. The work function is deﬁned as the minimum work required to withdraw an electron from a metal under the conditions that the metallic phase bears no net charge. On the basis of the earlier discussion, one may write m m m Wm el ¼ ael ¼ mel þ Fw

ð8:5:1Þ

where Wm el is the work function for an electron in metal m. Thus the electronic work function is the negative value of the real potential of the electron am el . The chemical potential of the electron depends on the bulk structure of the metal, whereas the surface potential wm depends on the arrangement of the metal atoms at the metal | vacuum interface. In the case of single crystals, the variation in Wm el observed when diﬀerent crystal faces are exposed at this interface is due to a corresponding variation in wm. Because the electronic cloud within the metal tends to spill out over the edge of the crystal lattice, wm is a positive quantity. Thus, a test charge moving from vacuum into the metal experiences ﬁrst the negative end of a surface dipole, due to the overspilt electron cloud, and then the positive end, due to the metal atoms in the lattice at the interface. The energy of electrons in metals is made up of two contributions, namely, potential energy and kinetic energy. The potential energy contains contributions from the bulk and from the surface. The bulk contribution arises from the coulombic interaction of the electronic cloud with the positive charge of the metal ion cores. The surface contribution arises from the dipole at the surface due to electron overspill, as described above. The kinetic energy of the electrons can be estimated by the Sommerfeld model [G3] using a quantum-mechanical description of the translational motion of electrons in the lattice system. On the basis of the Pauli exclusion principle, two electrons of opposite spin occupy each energy level

LIQUIDS AND SOLUTIONS AT INTERFACES

399

corresponding to increasing kinetic energy. The highest ﬁlled level at 0 K is called the Fermi level, and its energy is eF ¼

h2 ð3p2 rel Þ2=3 8p2 mel

ð8:5:2Þ

where rel is the electron density, mel , the electron mass, and h, the Planck constant. The kinetic energy is a positive destabilizing contribution to the overall energy of the system, whereas the potential energy U is a stabilizing negative energy. On the basis of this description, the work function is given by Wm el ¼ U eF ¼ UB Us eF

ð8:5:3Þ

where U B is the bulk contribution to the potential energy and U s, the surface contribution. These energies are shown schematically in ﬁg. 8.6. When the electrochemical approach to describing the work function (equation (8.5.1)) is compared to the physical approach (equation (8.5.3)), the following relationships are obtained. The chemical potential of the electron is related to bulk properties so that mm el ¼ UB þ eF

ð8:5:4Þ

In the same way, the surface potential energy is related to the surface potential so that U s ¼ Fwm

ð8:5:5Þ

When the metal phase is charged by connecting it to a voltage source, the potential energy U is changed but the kinetic energy eF is not changed. When the charge is positive, U shifts down, indicating that more work must be done to extract an electron. On the other hand, a negative charge results in an upward shift in U, corresponding to easier removal of an electron. In order to determine the electronic work function the metal must be ultrapure with no contamination on the surface. The biggest problem in this regard is the fact that most metals have a surface oxide layer under ambient conditions. A clean metallic surface can be formed by sputtering the metal in a vacuum chamber, and techniques are available for growing thin metal ﬁlms of known crystallographic orientation. Otherwise, the metal surface may be cleaned by positive ion bombardment techniques at very low pressures. Values of the electronic work function for polycrystalline metal surfaces are summarized in table 8.2. The lowest

Fig. 8.6 Energy proﬁle at the metal/vacuum boundary showing the separate contributions of the potential and kinetic energy of electrons to the work function.

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LIQUIDS, SOLUTIONS, AND INTERFACES

Table 8.2 Values of the Electronic Work Function for Polycrystalline Metal Surfaces* Metal Ag Al Au Ba Be Bi Ca Cd Ce Co Cr Cs Cu Fe Ga Hf Hg K

Wm el =eV 4.30 4.25 4.30 2.49 3.92 4.40 2.80 4.10 3.20 4.41 4.58 1.81 4.40 4.31 3.96 3.53 4.52 2.22

Metal In Ir La Li Mg Mn Mo Na Nb Nd Ni Os Pb Pd Pr Pt Rb Re

Wm el =eV 3.80 4.70 3.30 2.38 3.64 3.83 4.30 2.35 3.99 3.07 4.50 4.70 4.00 4.80 2.54 5.32 2.16 5.00

Metal Rh Ru Sb Sc Sm Sn Sr Ta Te Th Ti Tl U V W Zn Zr

Wm el =eV 4.75 4.60 4.08 3.30 2.70 4.38 2.35 4.12 4.73 3.41 3.95 4.00 3.74 4.12 4.54 4.24 3.90

*As given by Trassatti [G3] and Fomenko [3].

work function is found for cesium, the heaviest commonly available alkali metal. On the other hand, high values of Wm el are observed for the precious metals gold, palladium, and platinum. As stated earlier, the exact value depends on the potential and kinetic energy of the electron in the metal (equation (8.5.3)). Quantum-mechanical models have been developed for the electrons in sp metals, the best known of these being the jellium model [4]. Sp metals are systems such as the alkali and alkaline earth metals in which d electrons are not involved as valence electrons. For these systems, reasonable estimates of mm el can be made; these are summarized in table 8.3 for the alkali metals. It is interesting that this

Table 8.3 Experimental Value of the Work Function (Welm ) and Theoretical Estimates of the Chemical Potential of an Electron (mm el ), the Surface Potential (wm), the Fermi Kinetic Energy (eF), and the Potential Energy of Electrons (U) in the Alkali Metals [G3] Metal Li Na K Rb Cs

Wm el =eV

mm el =eV

wm =V

eF =eV

U=eV

2.38 2.35 2.22 2.16 1.81

1:65 1:95 2:00 1:94 1:80

0.73 0.40 0.22 0.22 0.01

4.72 3.23 2.12 1.85 1.58

7:10 5:58 4:34 4:01 3:39

LIQUIDS AND SOLUTIONS AT INTERFACES

401

quantity does not change in a monotonic fashion but reaches its minimum value at potassium. By subtracting mm el from the value of the work function, one obtains an estimate of the surface potential wm. This quantity is always positive, reﬂecting the fact that electrons spill out of the metal at the metal | gas interface. The surface potential is largest for Li and smallest for Cs. The values of the Fermi energy eF and the depth of the potential energy, U also increase monotonically with increase in the number of electrons in the alkali metal. These results reﬂect the fact that loss of one electron from Cs is much less important than the loss of one electron from Li. The fact that electrons tend to spill out from the metal reﬂects the mobility of the electron gas within these systems. The distance involved in the formation of the surface dipole is approximately 100 pm. The exact value of wm depends on the crystallographic structure of the metal at the surface and reﬂects the density of metal atoms associated with this structure. For metals with a low melting point, the data for the polycrystalline surface recorded in table 8.2 give the best indication of surface properties. Since the metal atoms are relatively mobile at room temperature, the system tends to maintain the properties described in the jellium model at the surface. For metals with high melting points, it is possible to cut single crystals so that surfaces with diﬀerent packing densities of metal atoms are exposed. In the case of face-centered cubic systems such as metallic gold, the packing density increases for the low Miller indices in the order (110) < (100) < (111). The electronic work function increases in the same direction, being lowest for Au(110) and highest for Au(111). Since the bulk structure of gold is independent of its surface structure, the value of mm el is constant. Thus, the change in work function reﬂects a change in surface potential wm so that wm has its largest value for Au(111), which also has the highest density of gold atoms at the surface. Polycrystalline surfaces result from a mixing of all possible crystal orientations. From an energetic point of view, the low index faces discussed above predominate. The work function of the polycrystalline surface reﬂects a weighted average of the work functions for each crystallographic orientation. In the case of facecentered cubic systems, it falls between that of the (110) and (100) single crystal surfaces. Since polycrystalline metals are involved in most practical applications, the eﬀect of surface structure on the work function is not discussed further here. More information on this topic is available in reviews by Trasatti [G3, 5]. The concept of the electronic work function is very familiar to physicists and chemists and is discussed in elementary textbooks. A much less familiar concept is that one may measure experimentally the work required to remove an ion from an electrolyte solution. This is also discussed in this chapter but before doing so, the properties of liquids and solutions at gas interfaces are considered.

8.6 The Liquid | Gas Interface and the Adsorption Isotherm Much of the early work in interfacial chemistry involved studies of the liquid | gas interface, especially for aqueous solutions. The techniques described earlier,

402

LIQUIDS, SOLUTIONS, AND INTERFACES

namely the capillary rise experiment and the maximum bubble pressure experiment are easily applied to measure the interfacial tension of liquid solutions. Aqueous systems have been studied extensively for both electrolytes and organic compounds as solutes. Before discussing the interfacial properties of aqueous solutions it is useful to consider the structure of pure water at its interface with air. Although the surface potential of water cannot be measured experimentally, indirect estimates of wW on the basis of a variety of experiments lead to the conclusion that it has a small positive value, of the order of 0.1 V. Of course, all of these estimates involve at least one extrathermodynamic assumption. The positive value of wW suggests that there is a net small orientation of water dipoles with their oxygen atoms pointing to the gas phase. As a result, when a positive test charge approaches the surface of pure water it experiences a small positive dipole with its positive end pointing into the solution. This result probably reﬂects the strong eﬀect of hydrogen bonding, which prevails right up to the interface, so that a few water molecules in the last monolayer are attached to other molecules below by two hydrogen bonds, whereas most are attached by only one. The orientation of these few would be such that the molecular dipole vector is approximately perpendicular to the interface. When an electrolyte is added to water, a small increase in surface tension is observed [6]. This is illustrated for several 1–1 electrolytes in ﬁg. 8.7. The Gibbs adsorption isotherm for these systems can be written as dg ¼ e dme þ W dmW

ð8:6:1Þ

Fig. 8.7 Plots of the surface tension of electrolyte solutions g relative to that of pure water g0 against the molar concentration of the electrolyte [6]. The ordinate scale is correct for the NaCl system. It has been shifted vertically for the KCl, CsI, and KCNS systems by 0.01, 0.02, and 0.03 units, respectively.

LIQUIDS AND SOLUTIONS AT INTERFACES

403

where e is the surface excess of the electrolyte, W , that of water, and me and mW, the corresponding chemical potentials. As described earlier the relative surface excess of the electrolyte is given by @g x ðWÞ e ¼ ¼ e e W ð8:6:2Þ @me T;P xW where xe and xW are the mole fractions of the electrolyte and water, respectively. Since the slope of a plot of g against RT ln ae is positive, the surface excess ðWÞ is e negative, indicating that ions are excluded from the region of the interface. It should also be noted that over most of the concentration range considered xe is much less than xW , so that the relative surface excess is approximately equal to its absolute value e . Since it is not possible to vary the activities of the cation and anion separately, the individual surface excesses of the two ions cannot be determined. It follows that these experiments give no information about the distribution of charge at the interface. In order to understand the above results, one must remember that the electrolyte is eﬀectively not present in the gas phase. This is due to the fact that the relative permittivity of the gas is much smaller than that of water. On the basis of simple electrostatic concepts using the method of images, an ion of charge zi e0 located at a distance x from the interface is repelled by an image charge located in the gas phase at a distance 2x with a charge equal to q ¼ zi e0

eW ea eW þ ea

ð8:6:3Þ

where ew is the relative permittivity of the aqueous phase and ea, that of the gas phase. Since ea is very much less than ew, the image charge is approximately equal to the ion’s charge. The force between the ion and its image is Fim ¼

z2i e20 eW ea 2e þe 16pe0 eW x W a

ð8:6:4Þ

This is a repulsive force which increases as 1=x2 as the ion gets closer to the interface. A model based on these concepts was developed by Onsager and Samaras [6] to estimate values of e for electrolyte solutions. This model is reasonably successful for dilute electrolyte solutions. Most studies of the solution | gas interface have involved organic compounds, for example, organic alcohols and acids. These dissolve in water to some extent as a result of the polar group. However, the alkyl group in the molecule is hydrophobic, so that the solute molecules are always found in excess at the solution | air interface. The accumulation of the organic molecule means that its surface excess is positive so that the interfacial tension of the solution decreases as the concentration of the organic molecule increases. An example of data for the straight chain alcohols in the series n-butanol to n-octanol is shown in ﬁg. 8.8. The results are presented in terms of the surface pressure , which is deﬁned as the interfacial tension, with respect to that for the pure solvent, that is, ¼ g0 g

ð8:6:5Þ

404

LIQUIDS, SOLUTIONS, AND INTERFACES

The experimental results [7] show that the surface pressure is positive and increases with alcohol concentration. This indicates that the alcohol accumulates at the water | air interface. Furthermore, the surface pressure increases with increase in the alkyl chain length at constant bulk concentration, demonstrating that the extent to which the alcohol accumulates at the interface depends on the alkyl chain length, or molecular hydrophobicity. The Gibbs adsorption isotherm for the alcohol–water system can be written as dg ¼ d ¼ A dmA þ W dmW The relative surface excess of the alcohol is given by @g x ¼ ¼ A A W ðWÞ A xW @mA T;P

ð8:6:6Þ

ð8:6:7Þ

Visual examination of the data in ﬁg. 8.8 shows that the slope of the plot of against the logarithm of the alcohol concentration approaches a constant at the highest concentrations considered. This demonstrates that the relative interfacial excess approaches a maximum value, AM ðWÞ . The maximum interfacial excess corresponds to a situation in which the alcohol molecules are arranged at the interface in such a way that further increase in the interfacial excess is not possible. Analysis of the concentration dependence of A ðWÞ allows one to determine the form of the adsorption isotherm for the alcohol. The process of adsorption of the alcohol at the aqueous interface can be described by the equation Ab þ rWad ÐAad þ rWb

ð8:6:8Þ

Fig. 8.8 Plots of the surface pressure for the water | air interface against alcohol concentration for aqueous solutions of butanol (C4), pentanol (C5), hexanol (C6), heptanol (C7), and octanol (C8) [7].

LIQUIDS AND SOLUTIONS AT INTERFACES

405

where the subscript ‘‘ad’’ designates molecules at the interface, and the subscript ‘‘b,’’ molecules in the bulk of the solution; r is the number of water molecules displaced in the adsorption of one alcohol molecule. In order to derive the adsorption isotherm, one writes down the thermodynamic conditions for equilibrium in terms of the chemical and electrochemical potentials of the individual species. For the molecules in the solution, the chemical potentials are mA ¼ mA þ RT ln aA

ð8:6:9Þ

mW ¼ mW þ RT ln aW

ð8:6:10Þ

and where mA and mW are the standard chemical potentials, and aA and aW , the activities of the alcohol and water, respectively. Since the electrostatic environment at the interface is very diﬀerent at the interface than in the solution, the electrochemical potentials of the adsorbed molecules must be considered. This requires knowing the average dipole moment of the interfacial molecule in a direction perpendicular to the interface, and also the local electrical ﬁeld in the same direction. For the alcohol molecules, the electrochemical potential at the interface can be written ad; ad; m~ A ¼ mA þ RT ln A þ hPA iEe

ð8:6:11Þ

is the standard chemical potential at the interface, A , the absolute where surface excess of alcohol, hPA i, the average dipole moment perpendicular to the interface, and Ee, the local ﬁeld experienced by the alcohol dipoles in the same direction. The corresponding expression for interfacial water molecules is ad;o mA

ad; m~ ad A ¼ mW þ RT ln W þ hpW iEe

ð8:6:12Þ

On the basis of equilibrium (8.6.8), one obtains the expression RT ln A rRT ln W ¼ Gad þ RT ln aA rRT ln aW þ ðhpA iEA rhpW iÞEe ð8:6:13Þ where ad; Gad ¼ mad; A þ rmw mA rmW

ð8:6:14Þ

If it is assumed that alcohol can completely cover the interface replacing all of the water molecules originally present, then one may deﬁne a maximum surface excess for alcohol, Am , which is given by rAm ¼ rA þ W

ð8:6:15Þ

As a result the condition for equilibrium becomes ln A r lnðAm A Þ ¼ ln Bad aA þ ðhpA i rhpW iÞ

Ee RT

ð8:6:16Þ

where ln Bad ¼

Gad þ r ln r r ln aw RT

ð8:6:17Þ

406

LIQUIDS, SOLUTIONS, AND INTERFACES

RT ln Bad is the eﬀective standard Gibbs adsorption energy and includes the activity of water, which is approximately constant, provided the alcohol solutions are dilute. A thorough analysis of the electrostatic contribution in equation (8.6.16) is a diﬃcult problem in statistical thermodynamics. An approximate solution to this problem is to assume that this contribution is proportional to the surface excess of the alcohol. In the present case the assumption is reasonable because the hydrophobicity of the alkyl groups causes the alcohol molecules to adopt a similar orientation with respect to one another at the interface. Then equation (8.6.16) can be rearranged as ln A r lnðAm A Þ ¼ ln Bad a Aad

A Am

ð8:6:18Þ

where Aad is the interaction coeﬃcient. This equation is known as the Flory– Huggins isotherm and has often been used to analyze the adsorption of molecules at interfaces when the adsorbate diﬀers in size from the solvent [8]. When Aad is positive, interaction between adjacent adsorbed molecules is repulsive and larger bulk concentrations are required to reach maximum coverage. The interaction term can also be attractive (Aad < 0). Under these conditions, phase transformation is possible at the interface. When interactions between the adsorbed species are absent (Aad ¼ 0), the isotherm becomes ln A r lnðAm A Þ ¼ ln Bad aA

ð8:6:19Þ

In the early literature the adsorption isotherm was usually written with a simpler coverage term, namely, with r equal to unity. This practice reﬂected work at the solid | gas interface, where the adsorption process does not involve replacement of one molecule by another. When size eﬀects are ignored, equation (8.6.18) has the form A Aad A ¼ Bad aA exp ð8:6:20Þ Am A Am This equation is known as the Frumkin isotherm. Isotherms with more complex versions of the coverage term have also been derived [8]. Plots of the fraction A =Am against the logarithm of the activity of the adsorbate in the bulk of the solution are shown in ﬁg. 8.9. The data were calculated assuming that r is one (equation (8.6.20)), and Bad is 1000, with various values of Aad =Am . The curves all have the sigmoidal shape expected on the basis of the coverage factor Am A . When interactions between the adsorbed molecules are absent (Aad ¼ 0), the major part of the change in A takes place for a change in bulk adsorbate activity by a factor of 400. When repulsive interactions are present a larger change in bulk adsorbate concentration is required to achieve the same coverage at the interface. Application of the general adsorption isotherm to experimental data requires the determination of four parameters, namely, r, Am ; Aad , and Bad . The value of Am is often apparent from the experimental data. If the size factor r is also known, equation (8.6.18) is rearranged so that Aad and Bad can be determined from a linear plot.

LIQUIDS AND SOLUTIONS AT INTERFACES

407

Fig. 8.9 Plot of the fraction A =Am against the logarithm of adsorbate activity aA according to the Frumkin isotherm (equation (8.6.20)) with the interaction parameter Aad =Am equal to 0, 2, and 5.

ln

A r lnðAm A Þ ¼ ln Bad Aad A aA Am

ð8:6:21Þ

A plot of the function on the left hand side of equation (8.6.21) against A is linear with a slope equal to Aad =Am and intercept on the ordinate axis equal to ln Bad. Most often the parameter r is not known a priori on the basis of molecular size. Information about molecular orientation is also required for the adsorbate. In the case of aqueous solutions, water molecules may act in clusters because of hydrogen bonding, so that the eﬀective size of water at interfaces may be larger than that estimated on the basis of one water molecule. Plots of data generated on the basis of equation (8.6.18) but plotted according to equation (8.6.21), assuming that r ¼ 1, are shown in ﬁg. 8.10. When the value of r is indeed unity, the plot is linear with a slope equal to Aad and an intercept on the ordinate axis equal to ln Bad . When r is not unity, the plots are curved. The best way to determine r involves calculating the function on the left-hand side of equation (8.6.21) for a reasonable range of r values, and in this way searching for the value of r which gives the best straight line for a plot of the left-hand side against A =Am . Many diﬀerent isotherms have been derived for molecular adsorption at interfaces, including the liquid | gas, liquid | liquid, and liquid | solid interfaces. Some of these isotherms can be shown to be approximate versions of the general isotherm derived here. Although the changes in interfacial composition can be followed by measuring the change in interfacial tension, these experiments provide no direct information about the electrical changes at the interface. These are especially important for

408

LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 8.10 Plot of the function lnðA =aA Þ lnðAm A Þ against A =Am using data generated with equation (8.6.18) assuming Bad ¼ 1000; Aad ¼ 2, and Am ¼ 5 molec. nm2. The value of r is 1, 1.5, or 2.0 for each curve as indicated.

electrolyte solutions in which the microscopic description at the interface can be diﬀerent for the cation and anion. More details about these systems are given in the next section.

8.7 Experimental Measurement of the Volta Potential Difference at Interfaces A Volta potential diﬀerence usually arises between two condensed phases when they come into contact. Because these phases have diﬀerent chemical properties, charge may be redistributed at their interface with air, even though this interface carried no net charge before contact. Measurement of the Volta potential diﬀerence is possible in a high-impedance electrochemical cell which contains an air gap. The potential drop across this cell is determined under conditions for which the potential drop across the air gap is zero, and is known as the compensation potential diﬀerence. A schematic diagram illustrating the experimental determination of the Volta potential diﬀerence between two metals, a and b, is given in ﬁg. 8.11. For this system, electrons are the species common to each phase. The experiment involves an arrangement in which there is an air gap between a and b. On the other side of b, there is a second portion of phase a which is designated a 0 , so that the system can be described as a j air j b j a 0

ð8:7:1Þ

Here, the vertical bars represent phase boundaries according to the usual conventions for describing electrochemical cells (see chapter 9). A variable dc potential

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LIQUIDS AND SOLUTIONS AT INTERFACES

Fig. 8.11 Experimental setup for determining the Volta potential diﬀerence between two metals a and b. The compensation potential is determined with potentiometer P by moving a with respect to b until there is zero current in galvanometer G.

may be applied between the metals a and a 0 at the ends of the cell. The ﬁeld strength in the space between a and b is zero if no current ﬂows in the galvanometer when phase a is moved with respect to phase b. Under these conditions, the Volta potential diﬀerence between the adjacent surfaces of a and b must be zero. Recalling that the Galvani potential diﬀerence between two phases can be measured when they have the same chemical composition, this experiment gives 0

0

Fa a 0 f ¼ Ffa Ffa ¼ m~ ael m~ ael ¼ m~ ael m~ bel

ð8:7:2Þ

0

Because of the electronic equilibrium between phases a and b, the electrochemi0 cal potentials of electrons in these phases, mael and mbel , are equal. Furthermore, the measurement of a a 0 f is made under conditions that a b c is zero, so that Fa a 0 f ¼ mael Fwa mbel þ Fwb ¼ Wbel Wael

ð8:7:3Þ

Thus, under these circumstances, the measured Galvani potential diﬀerence is equal to the diﬀerence in the electronic work functions of the metallic phases a and b. At the interface between a 0 and b where there is electronic equilibrium, 0

m~ ael ¼ m~ bel

ð8:7:4Þ

so that one may write 0

0

mael Ffa ¼ mbel Ffb

ð8:7:5Þ

Rearranging this equation to obtain an expression for b a 0 c, one ﬁnds that 0

0

0

0

Fb a 0 c ¼ Fðca cb Þ ¼ mael Fwa mbel þ Fwb ¼ W bel W el

ð8:7:6Þ

Combining equations (8.7.3) and (8.7.6) the ﬁnal result is c f ¼ a a 0 f ¼ b a 0 c ¼

W bel W ael F

ð8:7:7Þ

where c f refers to the compensation potential diﬀerence. Thus, this experiment gives the Volta potential diﬀerence between two diﬀerent metals in contact, which is related in a very simple way to the diﬀerence in work functions for the two metals.

410

LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 8.12 Schematic diagram of the Kenrick apparatus used to determine the Volta potential diﬀerence between two liquids a and b. Liquid a (Hg in this example) ﬂows down the center of tube T in a stream of droplets, whereas b (0.1M HCl) ﬂows down the walls.

Another important experiment used to measure the Volta potential diﬀerence between two liquids was described by Kenrick* [9] and is illustrated here for the mercury | electrolyte solution interface (see ﬁg. 8.12). An air gap between the two liquids is established in a cylindrical tube T. The mercury emerges from a central reservoir in a stream of small droplets which ﬂow down the center of the tube. On the other hand, the solution, namely 0.1 M HCl, ﬂows down its walls. The experiment is designed so that the two liquids ﬂow suﬃciently rapidly that no charge can be built up on their surfaces. As a result the Volta potential diﬀerence across the air gap is zero. The HCl solution is part of an electrochemical half-cell connected to a calomel electrode. The total cell may be described as Cu j Hg j Hg2 Cl2 j 0:1 MHCl j Hg 00 j air j 0.1 M HCl 0 j Hg2 Cl2 j Hg 0 j Cu 0 ð8:7:8Þ Identical calomel electrodes are on each side of the cell, but that on the left-hand side is also in contact with the pure mercury which ﬂows down the center of the cylindrical tube. The purpose of the calomel electrodes is to establish an equilibrium between electrons in mercury and chloride ions in solution through the process Hg2 C12 þ 2e Ð 2Hg þ 2C1

ð8:7:9Þ

Since these are the only charged species in the equilibrium, any change in the electrochemical potential of the electron in mercury will be reﬂected in a change in the electrochemical potential of the Cl in solution. The Galvani potential diﬀerence between the two copper leads, which is measured experimentally, is given by *Frank Kenrick was a professor in the Department of Chemistry, University of Toronto from 1900 until 1944.

411

LIQUIDS AND SOLUTIONS AT INTERFACES 0

~ Cu ~ Hg ~ Hg Fc f ¼ FCu Cu 0 f ¼ m~ Cu el m el ¼ m el m el

0

ð8:7:10Þ

The electrochemical potentials of the electrons in the Cu and Hg in contact are equal because of their electronic equilibrium. Furthermore, because of the calomel equilibrium on the left-hand side of the cell, ~ sC1 mHg2 C12 þ 2m~ Hg el ¼ 2mHg þ 2m

ð8:7:11Þ

A similar equation can be written for the calomel system on the right-hand side, which involves Hg 0 and the HCl 0 solution, which has the same concentration as the solution on the left-hand side. It follows from equation (8.7.11) that the compensation potential diﬀerence also measures the diﬀerence between the electrochemical potentials of the Cl ions in the two HCl solutions: 0

0

~ Hg ~ sCl m~ sCl ¼ Ffs þ Ffs Fc f ¼ m~ Hg el m el ¼ m

0

ð8:7:12Þ

The object of the experiment is to measure the Volta potential diﬀerence between Hg 00 and the HCl solution with which it is in contact. As described above, any Volta potential diﬀerence between the surfaces of Hg 00 and the HCl 0 solution on the other side of the air gap is eliminated by streaming these liquids in the central cylinder of the apparatus. In addition, nitrogen gas is passed up the gap in the cylinder to prevent condensation of water vapor on the mercury droplets. Since the resistance of this system is very high it is important that the device used to measure the potential diﬀerence have an even higher resistance, so that charge does not ﬂow from the experimental cell to the measuring device. In order to understand the connection between the observed cell potential diﬀerence and s Hg 00 c, equation (8.7.12) is rewritten as 0

00

00

c f ¼ fs fHg þ fHg fs

ð8:7:13Þ

Hg 00

where the term f has been added and subtracted on the right-hand side. Since the Volta potential diﬀerence at the air gap between Hg 00 and solution s 0 has been eliminated: 00

0

cHg cs ¼ 0

ð8:7:14Þ

Thus, equation (8.7.13) may also be written as 0

00

00

00

c f ¼ ws wHg þ fHg fs ¼ cHg cs ¼ Hg 00 s c s

ð8:7:15Þ 0

where it has been assumed that the surface potential w is equal to ws . This assumption is perfectly reasonable because the solutions have exactly the same composition. It follows that the Galvani potential diﬀerence between the copper leads to cell (8.7.8) is equal to the Volta potential diﬀerence from the mercury streaming in the center of the system to the solution with which it is in contact. In a slightly diﬀerent conﬁguration, the Kenrick apparatus can be used to measure the Volta potential diﬀerence between two miscible liquids. Such a cell is .

Cu j Hg j Hg2 Cl2 j 0:1M HCl .. S j air j 0:1M HCl 0 j Hg2 Cl2 j Hg 0 j Cu 0

ð8:7:16Þ

412

LIQUIDS, SOLUTIONS, AND INTERFACES .

The three vertical dots .. represent a liquid junction. The exact nature of the construction of this junction is not given here, but it is used to prevent mixing of solution S with the 0.1 M HCl solution. More details about liquid junctions and methods of forming them are given in chapter 9. The other aspect about the liquid junction that must be considered is that, in general, there is a Galvani potential drop across it. In the Kenrick experiment, solution S streams down the center of the vertical tube shown in ﬁg. 8.12 and the reference solution, 0.1 M HCl, ﬂows down the walls of the tube, thereby establishing zero potential drop across the air gap. On the basis of the analysis given above, the potential drop across the cell under these circumstances is given by 0

c f ¼Cu Cu 0 f ¼ fr fr 0

0

¼ fr fs þ fs fr ¼ wr ws fs fr ¼ s r c

ð8:7:17Þ

where the superscript ‘‘r’’ refers to the reference solution, 0.1 M HCl, ‘‘r 0 ,’’ to the reference solution, 0.1 M HCl 0 , and ‘‘s,’’ to the solution S. This system provides a convenient means of studying changes in the surface potential of a solution associated with adsorption of one of its components at the solution | air interface. Rewriting equation (8.7.17) as c f ¼ s r f wr þ w s

ð8:7:18Þ

where s r f is the liquid junction potential between the two solutions, the compensation potential diﬀerence provides a means of following changes in the surface potential of solution S; ws , provided that s r f is known or is constant. A. The Surface Potential at the Solution | Gas Interface The change in surface potential at the air interface has been studied for a variety of electrolyte solutions in water [6, 10]. The results are reported as w where w ¼ ws wr

ð8:7:19Þ

wr is the surface potential of a reference solution which is usually the inﬁnitely dilute system. The experimentally observed values of c f were corrected for the liquid junction potential on the basis of theories discussed in chapter 9. Typical values obtained by Randles are shown in ﬁg. 8.13. For most simple electrolytes, w is positive and increases in magnitude with increase in electrolyte concentration. The surface potential depends strongly on the nature of the anion and only weakly on the nature of the cation when this is an alkali metal ion. One can identify three physical phenomena which lead to the observed values of w. First, an ionic double layer can be established if the distance of closest approach for cations and anions to the interface is not the same. Second, if one of the components of the solution has a dipole moment, it may assume a preferred orientation at the interface, thereby giving rise to dipolar potential drop. Finally, the presence of the solute can change the orientation of water molecules at the interface from that present in the pure solvent. The fact that w is usually positive is evidence that the anion approaches the surface more closely than the cation. This is not diﬃcult to understand given that anions are more weakly solvated than

LIQUIDS AND SOLUTIONS AT INTERFACES

413

Fig. 8.13 Change in the surface potential at the aqueous solution | air interface with respect to that for the inﬁnitely dilute solution plotted against electrolyte concentration.

cations in water. As a test charge approaches the interface it ﬁrst experiences an anionic excess. This leads to a cationic excess further in the solution, with the result that the net charge in the double layer at the interface is zero (see ﬁg. 8.14). Since the surface potential is measured from the gas phase into the liquid solution phase, it is positive. The change in the surface potential going from 2 M KCl to 2 M KI is 50 mV. This reﬂects the fact that the iodide ion is more weakly solvated than the chloride ion, so that the former establishes a greater potential diﬀerence at the interface. KF is the only electrolyte for which w is negative. This demonstrates that the ﬂuoride anion is more strongly solvated by water molecules than the chloride ion which is in the reference solution. Values of w have also been determined for solutions of simple inorganic acids. The surface potential change for the acid is always signiﬁcantly larger

Fig. 8.14 Schematic diagram showing the distribution of Kþ and I ions at the solution | gas interface for an aqueous solution of KI.

414

LIQUIDS, SOLUTIONS, AND INTERFACES

than that for the potassium salt. For example, w is 2 mV for KCl at a concentration of 1 M, whereas it is 23 mV for 1 M HCl. The results suggest that the hydronium ion is oriented at the interface with the hydrogen atoms directed into the solution and the negative electron density on the oxygen atom directed toward the gas phase. The hydronium ion contributes in two ways to the surface potential, both as a cation and as a dipole, so that the net electrostatic eﬀect is larger. The eﬀect of organic solutes on the surface potential has been studied in some detail [G2]. In the cases of short-chain alcohols and acids, and simple ethers, the value of w is negative and much larger than that found for inorganic systems. This result is interpreted as evidence that water-soluble organic molecules are oriented at the interface with the hydrophobic part, for example, the alkyl group, pointing into the gas phase. The polar group interacts with the water molecules, which can form hydrogen bonds with electronegative atoms like oxygen and nitrogen. As a result, a test charge entering the solution from the gas phase experiences the electrostatic eﬀect of a dipolar layer with its positive end pointing to the gas phase and its negative end into the solution. Some typical values of w are given in table 8.4. The value of w increases with increase in the concentration of the organic solute and reaches a limiting value wm in the same way as the surface excess. This limit is independent of the length of the alkyl chain but it does depend on the nature of the functional group. For example, in the case of alcohols wm is approximately 350 mV, whereas it is about 550 mV for dialkylethers. The shape of the dependence of w on solute concentration suggests that there may be a simple relationship between w and the surface excess . This is illustrated for the 1-propanol–water system by the plot in ﬁg. 8.15. The simple linear relationship demonstrates that the alcohol molecule is adsorbed in a monolayer, and that the orientation of individual molecules at the interface does not change with coverage. The relationship between w and is not always simple and depends on the solubility of the organic solute. When this is sparingly soluble or insoluble, a linear relationship is not observed. In this case, the experimental observations are attributed to change in the orientation of the organic molecules in a monolayer at the interface with the extent of ﬁlling of the monolayer.

Table 8.4 Typical Values for the Change in Surface Potential at the Aqueous Solution | Air Interface After Addition of an Organic Solute Organic Solute

Electrolyte

1 M CH3OH 1 M C2H5OH 1 M C3H7OH 1 M HCOOH 1M CH3COOH 1M (CH3)2CO 0.5M (C2H5)2O 1M C2H5NH2

0.01 M KCl 0.01 M KCl 0.005 M H2SO4 — — 0.005 M H2SO4 0.005 M H2SO4 —

Surface Potential Change w=mV 100 230 220 40 285 380 480 310

LIQUIDS AND SOLUTIONS AT INTERFACES

415

Fig. 8.15 Plot of the change in surface potential w against surface excess for 1-propanol in water at the solution | air interface [11].

When the orientation of the organic molecule at the interface does not change with coverage, a simple relationship exists between w and , namely w ¼

NL p cos y es e0

ð8:7:20Þ

NL is the Avogadro constant, p, the dipole moment of the adsorbate molecule, y, the angle that the dipole vector makes with the direction perpendicular to the interface, es the relative permittivity of the solvent, and e0 , the permittivity of free space. Assuming that es is 6 in the surface region, the data in ﬁg. 8.15 yield an estimate for p cos y of 1.4 debyes. This result is reasonable when compared to the dipole moment of 1-propanol (1.7 debyes). It is clear that equation (8.7.20), which shows that w is proportional to , can only be applied if the orientation of the organic molecules at the interface does not change with coverage. Studies of w have been used to make estimates of the absolute value of wW , the surface potential at the interface between pure water and air. Since this quantity cannot be determined experimentally, such estimates necessarily involve an extrathermodynamic assumption. The most convincing evidence about the sign of wW and thus, the orientation of water at the interface, comes from experiments designed to study the temperature coeﬃcient dwW =dT. The cell used was Cu j Hg j Hg2 Cl2 j 1:0 M KCl; H2 O ðT0 Þ j 0:01 M KCl ðT0 Þ j air j 0:01 M KCl ðT1 Þ j 0:01 M KCl ðT0 Þ j 1:0 M KCl ðT0 ÞjHg2 Cl2 j Hg0 j Cu0

ð8:7:21Þ

where the temperature of the KCl solution on the right-hand side of the air gap, T1 , was varied with respect to the temperature of the other components of the cell (T0 ). All of the liquid junction potentials in this cell can be estimated except that

416

LIQUIDS, SOLUTIONS, AND INTERFACES

between the two KCl solutions with diﬀerent temperatures. Assuming that this is negligible, analysis of the thermodynamics of this cell leads to the result that dc f dwW ¼ dT dT

ð8:7:22Þ

Experimental data were collected for several values of T1 in the vicinity of room temperature and gave the result that dwW =dT is 2:7 104 V K1 . The fact that the temperature coeﬃcient is negative shows that the surface potential wW is positive [10]. As temperature increases, thermal disorder results in a decrease in wW which eventually goes to zero at the critical temperature (647 K). Various methods for estimating wW were critically reviewed by Randles [10]. Although it is diﬃcult to make a precise estimate, it is clear from the experimental work that the surface potential of water is positive and approximately equal to 80 mV at 25 C. Surface potentials have also been estimated for a variety of non-aqueous solvents [12, 13]. Some of these estimates are given in table 8.5. The results are mainly based on compensation potential measurements in cells with an aqueous solution on one side of the air gap and the non-aqueous solution on the other. Details about these measurements and the extrathermodynamic assumption involved are given in the original literature [12, 13]. It is especially interesting that wS for the polar non-aqueous systems is always negative, so that water is unique with a positive value for its surface potential. The negative surface potential for non-aqueous solvents indicates that the hydrocarbon part of the molecule points into the gas phase, whereas the dipolar functional group points into the solution. Thus, a test charge passing from the gas phase into solution experiences a dipole potential diﬀerence which is negative. In the case of an alcohol such as methanol, the polar–OH group prefers to remain in solution where it is involved in hydrogen bonding. For aprotic systems like acetonitrile, the polar C N group is involved in strong dipole–dipole interactions with other solvent molecules. Although one can be quite conﬁdent that the sign given to the values of wS is correct, there is a large uncertainty in the magnitude. Estimation of surface potentials is closely related to the experimental measurement of the real potentials Table 8.5 Estimates of the Surface Potential for Some Polar Solvents [12, 13] Solvent Water Methanol Ethanol Acetone Acetonitrile Dimethylformamide Dimethylsulfoxide Nitrobenzene

Surface Potential wS =mV 80 230 300 390 100 260 250 160

LIQUIDS AND SOLUTIONS AT INTERFACES

417

for individual ions in solution. This is an important part of the study of the properties of electrolyte solutions. It is discussed in detail in the following section. B. The Work Function for Ions in Solution The work function for ions is deﬁned with respect to the process in which one mole of ions in the ideal standard state of unit concentration in a given solvent are transferred to charge-free inﬁnity in vacuum as unsolvated ions under conditions that the solution bears no net charge. This process diﬀers considerably from that for electrons in metals, which involves removal of electrons from the Fermi level in the metal to charge-free inﬁnity. An important aspect of the latter process is that it involves the deﬁnition of zero on the physical potential scale. Thus, an electron at rest in vacuum is deﬁned to have zero potential energy on that scale. Using the above deﬁnition, the work function for an ion is based on the process ionðsolnÞ!ionðgasÞ

ð8:7:23Þ

~ is; W si ¼ m~ g; i m

ð8:7:24Þ

and is given by

where m~ xi is the electrochemical potential of ion i in phase x, and the superscript ‘‘’’ refers to the standard state. Since the solution phase carries no net charge (cs ¼ 0), and the inner potential of the gas phase can be set equal to zero, one can write W si ¼ mig; mis; zi Fws0 ¼ mig; as; i

ð8:7:25Þ

where ws0 is the surface potential of the solution, and as; i , the standard real potential of ion i in solution. In order to determine ais; one must be able to measure the Volta potential diﬀerence between the solution and a metal electrode which is in electrochemical equilibrium with ion i. This point is illustrated in detail in the following discussion. The standard real potential for the H+ ion can be determined using the following cell: Cu j Hg j air j x m HCl j Pt, H2 j Cu 0

ð8:7:26Þ

When the compensation potential is determined using the Kenrick apparatus, the mercury streams down the center of the vertical tube and the HCl solution down its walls so that the Volta potential diﬀerence across the air gap is eliminated. In this system both the concentration of HCl and the pressure of hydrogen gas can be varied. It diﬀers from cells (8.7.8) and (8.7.16) in that no attempt is made to balance the charge carriers on opposite sides of the cell. The Galvani potential diﬀerence measured between the two copper leads can be related to the electrochemical potentials of electrons in the mercury and platinum as follows: 0

~ Cu ~ Hg ~ Pt Fc f ¼ FCu Cu 0 f ¼ m~ Cu el m el el ¼ m el m Pt Pt Hg ¼ mHg el mel þ F f f

ð8:7:27Þ

418

LIQUIDS, SOLUTIONS, AND INTERFACES

In order to proceed further, one must relate the electronic equilibria in the metallic terminals of the cell to the electrochemical equilibrium in the hydrogen halfcell. This equilibrium is ( 1 H2 ðgasÞ Hþ ðsolnÞ þ e ðPtÞ + 2

ð8:7:28Þ

so that the condition for equilibrium is m~ sHþ þ m~ Pt el ¼

mgH2

ð8:7:29Þ

2

where the superscript ‘‘s’’ designates the solution phase. Expanding each of the electrochemical and chemical potentials and rearranging, one can show that Fðfs fPt Þ ¼ G RT ln aHþ þ

RT ln pH2 2

ð8:7:30Þ

where

G ¼

g; mH 2

2

s; Pt; mH þ mel

ð8:7:31Þ

Combining equations (8.7.27) and (8.7.30) for the case that the pressure of the hydrogen gas is 1 bar, one may write Fc f ¼ Fðfs fHg Þ þ RT ln aHþ G

0

ð8:7:32Þ

where 0

G ¼

mg; H2 2

Hg; s; mH þ mel

ð8:7:33Þ

Since the Volta potential diﬀerence across the air gap between the mercury and the aqueous solution is maintained at zero, this result may also be written as Fc f Fðws wHg Þ þ RT ln aHþ G

0

ð8:7:34Þ

Equation (8.7.34) shows that the compensation potential depends on the hydrogen ion activity in the electrolyte solution. Strictly speaking, single ion activities are not subject to experimental determination. However, if the solution is suﬃciently dilute, the single ion activity aHþ may be estimated by the mean activity of the electrolyte a to a good approximation. Cell (8.7.26) was studied in detail by Farrell and McTigue [14] using 13 diﬀerent molalities of HCl in the range 0.001–0.02 m. In this range, it is legitimate to assume that the hydrogen ion activity aHþ is equal to the mean ionic activity a , which is easily determined experimentally. Thus, it is possible to determine c f ðRT=FÞ ln a from the experimentally measured compensation potential. A plot of this quantity against the square root of the ionic strength is shown in ﬁg. 8.16. It is clear that this plot is linear at lower concentrations, so that the intercept, which gives the value of c f ðRT=FÞ ln a in the limit of zero ionic strength, can easily be found. From ﬁg. 8.16 the result is 56:3 mV. Keeping in mind that ws is equal to ww in the inﬁnitely dilute solution, equation (8.7.34) can be rewritten as

LIQUIDS AND SOLUTIONS AT INTERFACES

419

Fig. 8.16 Plot of c ðRT=FÞ ln a for cell (8.7.26) against the square root of the ionic strength due to HCl, I1/2.

s; W Fc f0 RT ln a ¼ mH þ mHg; FwHg þ þ Fw el s; Hg; ¼ aH þ þ ae

mg; H2

mg; H2 2

ð8:7:35Þ

2

s; aH þ

is the standard real potential for the hydrogen ion in an ideal inﬁnitely where dilute solution, and aHg; , the standard real potential for an electron in mercury. e Thus, the Kenrick experiment carried out with cell (8.7.26) gives the Gibbs energy change for the reaction: 1 H ðgasÞ!Hþ ðsolnÞ þ e ðHgÞ 2 2

ð8:7:36Þ

In order to estimate the work function for the H+ ion, two other pieces of information are required. The ﬁrst is the work function for an electron in mercury from table 8.2, which is equal to 4.50 eV (434.2 kJ mol1). The second piece of information is the work associated with the dissociation and ionization of molecular hydrogen in the gas phase, corresponding to the reaction 1 H ðgasÞ!Hþ ðgasÞ þ e ðgasÞ 2 2

ð8:7:37Þ

The energy required for this process is 1517.1 kJ mol1 [15]. Using these data, the Gibbs energy associated with the reaction Hþ ðsolnÞ!Hþ ðgasÞ

ð8:7:38Þ

is 1088.3 kJ mol1. This is precisely the work function for the H+ ion in aqueous solution. Having obtained the work function for one ion it is now a simple matter to extract work functions for other ions provided the necessary thermodynamic data

420

LIQUIDS, SOLUTIONS, AND INTERFACES

Table 8.6 Work Functions for Simple Monovalent Ions at Inﬁnite Dilution in Water at 25 C [14] Ion

Wi =kJ mol1

H+ Li+ Na+ K+ Rb+ Cs+ F Cl Br I

1088 509 409 336 314 283 436 319 305 258

for the appropriate electrolytes are available. For example, in the case of HCl, the Gibbs energy for the process HClðsolnÞ!Hþ ðgasÞ þ Cl ðgasÞ

ð8:7:39Þ

1

is equal to 1405.2 kJ mol . Comparing equations (8.7.38) and (8.7.39), it is clear that the work function for the Cl ion is 316.9 kJ mol1. By choosing electrolytes with one ion for which the work function is known and the other for which it is not known, the list of values of W si may be extended. A summary of results for monovalent monoatomic ions is given in table 8.6. It is interesting to compare values of the standard real potential for a given ion in table 8.6 with values of the absolute Gibbs energy of solvation given in table 3.3. On the basis of equation (8.4.4), s aiS; ¼ W si ¼ mS; i þ zi Fw0

ð8:7:40Þ

miS;

where is the standard Gibbs energy for ion i in solvent S, which is equal to the standard Gibbs energy of solvation, s G. This comparison gives a method of obtaining an independent value of s G for the hydrogen ion using the estimate of the surface potential of water discussed earlier. Thus, s GHþ ¼ 1088 96:5 0:08 ¼ 1096 kJ mol1

ð8:7:41Þ

1

This result is 8 kJ mol more positive than the estimate discussed in section 3.5 (1104 kJ mol1). Considering that both estimates involve extrathermodynamic assumptions the agreement is quite good. Values of the work function for simple ions have also been measured in nonaqueous solutions [16]. These experiments involve measuring the compensation potential in a cell such as Ag j AgCl j NaCl, W j air j NaCl, S j AgCl j Ag 0

ð8:7:42Þ

The compensation potential for this system is 0

c f ¼ fAg fAg

ð8:7:43Þ

LIQUIDS AND SOLUTIONS AT INTERFACES

421

Because of the equilibrium at the Ag | AgCl electrodes, namely, + Ag þ Cl AgCl þ e (

ð8:7:44Þ

the inner potentials of the silver electrodes at each end of the cell can be related to the electrochemical potentials of the chloride ion in each solution. Thus, 0

~ SC1 Fc f ¼ FfAg FfAg ¼ m~ W C1 m

ð8:7:45Þ

m~ W C1

where is the electrochemical potential of the chloride ion in the aqueous solution, and m~ SC1 , that in the non-aqueous solution involving solvent S. The Volta potential diﬀerence across the air gap is eliminated, for the example, by using the Kenrick experiment (ﬁg. 8.12). Then, equation (8.8.26) can be rewritten as W S s W S Fc f ¼ mW C1 Fw mC1 þ Fw ¼ aC1 aC1

ð8:7:46Þ

where miC1 is the chemical potential of the chloride ion in phase i, and aiC1 its real potential in the same medium. Since aW C1 is known, this experiment gives the real potential of the Cl ion in the non-aqueous solvent S. Now, thermodynamic data for solvation of whole electrolytes, for example, HCl, can be used to extract real potentials for other ions. Values of the work function for the hydrogen ion in several non-aqueous media are summarized in table 8.7 (W SHþ ¼ aSHþ Þ: In assessing the change in W si with solvent nature, one should keep in mind that both the Gibbs energy of solvation and the surface potential contribution change with solvent nature. The change in the ﬁrst contribution can be assessed using data for the Gibbs energy of transfer which are given in chapter 4. On the other hand, one may estimate the Gibbs solvation energy using an estimate of the surface potential term which gives a small contribution to the real potential. EXAMPLE

The real potential of the Na+ ion in methanol is 435.6 kJ mol1. Estimate the Gibbs energy of solvation of Na+ ion in methanol using the surface potential for this solvent given in table 8.5. Compare this result with that obtained from the Gibbs energy of solvation of Na+ ion in water (424 kJ mol1) and the Gibbs energy of transfer of Na+ ion from water to methanol (8.2 kJ mol1).

Table 8.7 Values of the Work Function for the Proton, WHS þ , in Different Polar Solvents Solvent Water Methanol Ethanol Formamide Acetone Acetonitrile

W SHþ =kJ mol1 1088 1113 1111 1102 1118 1077

422

LIQUIDS, SOLUTIONS, AND INTERFACES

The Gibbs solvation energy for ion i in solvent S is given by s Gi ¼ mSi ¼ aSi zi FwS

ð8:7:47Þ

The surface potential for methanol is 0.23 V. Therefore s GNaþ ¼ 435:6 þ 96:5 0:23 ¼ 413:4 kJ mol1

ð8:7:48Þ

The value of the Gibbs energy of solvation estimated from the data in water together with the Gibbs energy of transfer is s GNaþ ¼ 424 þ 8:2 ¼ 416 kJ mol1

ð8:7:49Þ

The estimates diﬀer by 3 kJ mol1. Considering the fact that extrathermodynamic assumptions are involved in both routes used to obtain s GNaþ , the agreement is quite good. Values of aSi are available for a number of ions in several non-aqueous solvents [16]. These data provide a route to estimating the Gibbs solvation energy for individual ions when the estimates of wS given in table 8.5 are used to calculate the work done in transporting the ion across the double layer at the solution–air interface. The Kenrick experiment and other methods for determining the work function for individual ions in solution provide valuable information about ion solvation. This subject is of great interest in the development of realistic models for electrolyte solutions and in understanding their properties at the microscopic level.

8.8 The Metal | Solution Interface The metal | solution interface has special signiﬁcance in electrochemistry and electroanalysis. The usual electrochemical cell contains two such interfaces, and an understanding of its properties is important in describing the physical chemistry of these systems. When a metal comes into contact with a polar liquid, a redistribution of charge occurs at the newly formed interface. At the metal | air interface, the electrons spill out of the metal atom lattice, giving rise to a positive surface potential, wm . At the metal | solution interface, this spillover is not the same because the same electrons encounter the electron density associated with the solvent molecules. Similar considerations apply to the liquid phase where the molecules usually adopt a special orientation at the liquid | air interface giving rise to the surface potential of the solvent or the solution, ws . This orientation may change upon contact with the metal phase. Finally, there is usually a redistribution of charge between the two phases. The exact nature of this process depends on the composition of the two phases, as will be seen from the following discussion. The electrostatic properties of the metal | solution interface are expressed in terms of the Galvani potential diﬀerence, s m f, where s m f

¼ fm fs

ð8:8:1Þ

LIQUIDS AND SOLUTIONS AT INTERFACES

423

fm is the inner potential of the metal, and fs that of the solution. The Galvani potential diﬀerence cannot be measured experimentally. On the basis of the above discussion, it is made up of three contributions, so that it can also be written as s m f

¼ s f þ g m g s

ð8:8:2Þ

where s f is the potential drop due to the net free charge at the interface, gm , the dipolar potential in the metal phase, and gs , that in the solution. gm at the interface between the condensed phases is the analog of wm at the metal | gas interface; in general, these quantities are not equal. The same comments apply to gs and ws . The value of the Galvani potential diﬀerence may be changed by changing the composition of the metal phase, for example, by using amalgams of varying composition, or by changing the composition of the solution. In this way one can achieve conditions for which there is no net charge at the interface, so that s f is zero and s m f

s ¼ gm 0 g0

ð8:8:3Þ

This is the potential of zero charge, an important point of reference for this type of interface. In equation (8.8.3), the subscript ‘‘0’’ designates the speciﬁc conditions which are present at this reference potential. On the basis of ﬁg. 8.5, the relationship between the Galvani potential diﬀerence between the metal and the solution, and the Volta potential diﬀerence just outside of these phases where they form an interface with the gas phase is s m f

¼ s m c þ wm ws

ð8:8:4Þ

Only the Volta potential diﬀerence s m c can be measured experimentally. For example, using the Kenrick experiment illustrated in ﬁg. 8.12, one can measure the value of s m c between liquid metals like mercury and electrolyte solutions. In the case of solid metals, the cell used to measure the compensation potential can have the same conﬁguration as that for the Kenrick experiment (cell (8.7.9)). However, the air gap is formed between a metal disk placed a small distance above the solution phase. The potential diﬀerence between these phases is eliminated using ionizing alpha radiation from a radioactive substance placed in the gap. The end result is that the Volta potential diﬀerence is obtained for the cell. Typical values of s m c0 for the metal | solution interface at the PZC are given in table 8.8. The Volta potential diﬀerence is negative, indicating that the metal | air interface is negative with respect to the solution | air interface. Combining equations (8.8.3) and (8.8.4), one obtains s m c0

m s s ¼ gm 0 w ðg0 w Þ

ð8:8:5Þ

Thus the Volta potential diﬀerence at the PZC is related to the dipole potential diﬀerences at the metal | solution, metal | air, and solution | air interfaces. EXAMPLE

Estimate the Galvani potential diﬀerence at the Hg | aqueous solution interface at the PZC given that the surface potential at the Hg | air interface is 2.2 V. The surface potential at the metal | air interface may be estimated by a variety of techniques, including the jellium model for sp metals. In the case

424

LIQUIDS, SOLUTIONS, AND INTERFACES

Table 8.8 Values of the Volta Potential Difference at the Metal | Water Interface for Different Metals at the Point of Zero Charge (PZC) Metal

Volta Potential Diﬀerence W m c0 =V 0.295 0.425 0.495 0.248 0.365 0.335 0.365 0.485

Bi Cd Ga Hg In Sn Tl Zn

of mercury, it is estimated from thermodynamic data for the Gibbs energy of solvation of metals in mercury [G2]. The estimate of the surface potential of water is 0.08 V. Thus from equation (8.8.4) s m f0

¼ 0:25 þ 2:2 0:08 ¼ 1:9 V

ð8:8:6Þ

This potential diﬀerence is such that electrons tend to remain in the metal phase and do not easily go over to the solution phase. The value of s m f0 depends greatly on the value of wm , which in turn depends on the nature of the metal. The Volta potential diﬀerence has also been estimated at the mercury | solution interface for diﬀerent polar solvents. The results obtained (table 8.9) demonstrate that interaction between mercury and the solvent varies greatly with solvent

Table 8.9 Values of the Volta Potential Difference at the Mercury | Solution Interface for Different Solvents at the PZC Solvent Water Methanol Ethanol Acetone Dimethylformamide Dimethylsulfoxide

S Hg c0 =V

0.248 0.52 0.45 0.62 0.68 0.69

LIQUIDS AND SOLUTIONS AT INTERFACES

425

nature. It is clear that the dipolar contribution to interfacial properties is an important part of its electrostatic description. Further study of the metal | solution interface reveals two limiting cases in terms of their thermodynamic properties. One case occurs when the solution contains cations formed from the metal and corresponds to electrochemical equilibrium. In general, this process can be described as +M Mzþ þ ze (

ð8:8:7Þ

where z is equal to the oxidation state of the metal cation. The thermodynamic condition for equilibrium is m~ si þ zm~ m el ¼ mm

ð8:8:8Þ m~ m el ,

m~ si

that for the where is the electrochemical potential of the cation in solution, electron in the metal, and mm , the chemical potential of the metal. The electrochemical potential of the cation is given by s m~ si ¼ ms; i þ RT ln ai þ zFf

ð8:8:9Þ

mis;

is the standard chemical potential of this species and, ai , its activity. where The corresponding equation for electrons is m; m m~ m el ¼ mel Ff

ð8:8:10Þ

where it is assumed that the electron concentration in the metal cannot be changed signiﬁcantly. Therefore, the condition for equilibrium leads to the result that s m f ¼

G RT ln ai þ zF zF

ð8:8:11Þ

where m; G ¼ mm ms; i zmel

ð8:8:12Þ

Equation (8.8.11) is a form of the Nernst equation for equilibrium (8.8.7). However, it is expressed in terms of the Galvani potential diﬀerence which cannot be measured experimentally. As will be seen in the discussion in chapter 9, this problem is overcome in practice by deﬁning a convenient reference half reaction whose value of G is arbitrarily set equal to zero. If reaction (8.8.7) is driven in either the forward or reverse direction by passing a current, the potential s m f does not change provided there is no signiﬁcant change in the ionic activity ai . For this reason, the electrode | solution interface is termed non-polarizable. In practice, the forward and reverse reactions of the equilibrium (8.8.7) occur at ﬁnite rates. The other limiting case for the metal | solution interface occurs when there is no ﬂow of charge across the interface, so that it behaves as a capacitor. A simple example is the Hg | aqueous KC l interface. By connecting the mercury to an external d.c. source, the charge density on its surface can be changed without current ﬂowing. However, the potential drop s f changes in response to the change in charge density at the interface. This leads to a corresponding change in s m f (equation (8.8.2)), This limiting form of the metal | solution interface is called polarizable and is characterized by electrostatic equilibrium. In practice,

426

LIQUIDS, SOLUTIONS, AND INTERFACES

there are limits over which the value of s m f can be changed. For the example given, if the mercury electrode is made too negative, the K+ ion is reduced by the reaction Kþ þ Hg þ e !K; Hg

ð8:8:13Þ

to form potassium amalgam. On the other hand, if the mercury potential is driven too far in the positive direction, mercury is oxidized in the reaction 2Hg þ 2Cl !Hg2 Cl2 þ 2e

ð8:8:14Þ

in which insoluble mercury(I) chloride is formed. If either of these reactions occurs, the polarizable nature of the interface is lost. However, there is an interval of approximately two volts over which the system remains polarizable. The metal | solution interface both in its polarizable and non-polarizable forms is extremely important in electroanalysis and in practical electrochemical devices. The discussion in this section has focused on its fundamental electrical properties. These systems are considered in much more detail in chapter 9, which deals with electrochemical equilibria, and chapter 10, which is concerned with polarizable interfaces and the electrical double layer.

8.9 The Liquid | Liquid Interface Many examples exist of interfaces formed between two immiscible liquids. A wellknown one is the interface between a long-chain hydrocarbon, for example, dodecane, and water, which is commonly known as the oil | water interface. Dodecane and water are immiscible because the hydrocarbon phase is nonpolar. Liquid |liquid interfaces are also formed between water and organic liquids with polar groups such as octanol and heptanoic acid, which also have rather long hydrocarbon chains. The polar liquid nitrobenzene, which has a relative permittivity of 35, is also immiscible with water. Another well-known system is the mercury | polar liquid interface. This has been studied extensively, especially for aqueous electrolyte solutions. However, the mercury | polar liquid interface is also an example of a metal | solution interface which was considered in the previous section. The discussion here is limited to liquids with relative dielectric permittivities falling in the range 1–200, and systems which have poor conductivities as pure liquids. In deﬁning the interfacial tension of liquids in section 8.2, the discussion was kept general in terms of two ﬂuids a and b. When one of these ﬂuids is air, then the interfacial tension has the values which are given in table 8.1 for a collection of polar liquids. A relationship between the interfacial tension between two immiscible liquids a and b, and that at the liquid | air interface for each pure liquid can also be deﬁned. These quantities are connected by the work of adhesion wab, which is the work required to separate the two phases so that an air interface is formed at both a and b. Thus, if gab is the interfacial tension at the a | b interface, that is, the work required to increase the area of this interface by unit amount, then

LIQUIDS AND SOLUTIONS AT INTERFACES

427

Table 8.10 Interfacial Tension Between Water and Immiscible Organic Liquids at Room Temperature [G1] gab / mJ m2

Interface Water Water Water Water Water Water Water

| | | | | | |

benzene chloroform carbon tetrachloride nitrobenzene octane heptanoic acid octanol

gab þ wab ¼ ga þ gb

34.4 33.3 43.8 24.7 49.8 5.3 7.3

ð8:9:1Þ

where ga and gb are the interfacial tensions at the a|air and b|air interfaces, respectively. Some typical values of gab for interfaces in which water is one component are given in table 8.10. A wide range of values are found reﬂecting the way in which the organic system interacts with the water phase at the interface. There has been considerable interest in recent years in electrochemical processes at liquid|liquid interfaces involving water and an immiscible organic liquid [17, 18]. When the organic liquid is suﬃciently polar, as is nitrobenzene (NB), an electrolyte may be dissolved in both liquids, so that they become electrically conductive. This system is often referred to as an interface involving two immiscible electrolyte solutions, or ITIES. Electrolytes commonly used in NB include tetraphenylarsonium tetraphenylborate (TATB), and tetra-n-butylammonium tetraphenylborate (TBATB). LiCl is often used as a background electrolyte in water. Electrolytes such as TATB and TBATB are virtually insoluble in water, whereas LiCl does not dissolve in NB. Thus, when the NB|water interface is formed the electrolytes remain almost entirely in their original phases. Such a system behaves as an ideally polarizable interface over a limited potential range. Thus, the application of a potential diﬀerence between two electrodes, one placed in the aqueous phase and the other in the NB, results in the accumulation of charge at the interface. When the electrolyte is soluble in both phases, an equilibrium is established in which it is partitioned between the two phases. At equilibrium, the chemical potential of the electrolyte is equal in the two liquids, that is, NB mW e ¼ me

ð8:9:2Þ

where mie is the chemical potential of the electrolyte in phase i. The electrolyte activities in the two phases are related by the expression ! aNB e RT ln meNB; ¼ tr Ge ð8:9:3Þ ¼ mW; e aW e

428

LIQUIDS, SOLUTIONS, AND INTERFACES

X; where aX the corresponding stane is the electrolyte activity in phase X, and me dard chemical potential, and tr Ge the standard Gibbs energy of transfer from water to the non-aqueous phase. The factors controlling the partition equilibrium can be seen more clearly when the equilibrium condition for the individual ions is considered. For the sake of simplicity, a simple 1–1 electrolyte MX is considered. Equating the electrochemical potentials of the cation in the two phases, one obtains W W NB; NB mW; þ RT ln aNB ð8:9:4Þ M þ Ff M þ RT ln aM þ Ff ¼ mM i þ where mW; M is the standard Gibbs energy of cation M in phase i, aM , its activity, i and f , the inner potential of phase i. The Galvani potential diﬀerence at the water nitrobenzene interface is ! tr GM RT aW M þ ln NB ð8:9:5Þ W NB f ¼ F F aM

The Galvani potential diﬀerence may also be expressed in terms of the equilibrium for the anion, which leads to ! tr GX RT aNB X þ ln W ð8:9:6Þ W NB f ¼ F F aX The Galvani potential diﬀerence may not be found experimentally because single ion quantities are not subject to experimental determination. However, because of the requirement of electroneutrality, the concentration of cations and anions in each phase is equal. Furthermore, in the limit of dilute solutions, the activities of these ions are equal in each phase. Thus, by equating (8.9.5) and (8.9.6), one obtains for dilute electrolyte concentrations ! aNB G þ tr GX G ¼ tr e ð8:9:7Þ ln W ¼ tr M 2RT 2RT a On the basis of the thermodynamics of electrolytes, equation (8.9.7) is the equivalent of equation (8.9.3). Because the partition equilibria observed at the liquid | liquid interface are relevant to interfacial phenomena in speciﬁc ion electrodes and biological membranes, there is an interest in determining single ion quantities associated with transfer of an ion from water to the non-aqueous phase. This quantity can only be estimated from experimental data after making an extrathermodynamic assumption. One common assumption discussed earlier in section 4.8 is the so-called TATB assumption, according to which tr GTAþ ¼ tr GTB ¼

tr GTATB 2

ð8:9:8Þ

where tr GTATB is the standard Gibbs energy of transfer of the whole electrolyte, that is, tetraphenylammonium tetraphenylborate. By determining the partition ratio in dilute solutions for a series of electrolytes, values of the single ion standard Gibbs energy of transfer may be estimated. Some typical values for the water | nitrobenzene interface are given in table 8.11. In the case of the mono-

LIQUIDS AND SOLUTIONS AT INTERFACES

429

Table 8.11 Standard Gibbs Energy of Transfer for Monovalent Ions from Water to Nitrobenzene at 25 C (TATB Assumption) [18] Cation Liþ Naþ Kþ Rbþ Csþ TMAþ TEAþ TBAþ TAþ

tr G =kJ mol1 38.2 34.2 23.4 19.4 15.4 3.4 5.7 24.0 35.9

Ion

tr G =kJ mol1

F Cl Br I ClO4 BF4 NO3 IO4_ TB

44.0 31.4 28.4 18.8 8.0 11.0 24.4 6.9 35.9

TMAþ ¼ tetramethylammonium; TEAþ ¼ tetraethylammonium; TBAþ ¼ tetrabutylammonium.

atomic ions, tr G is large and positive, demonstrating that the ion is much less soluble in NB than in water. However, there is a trend for tr G to decrease as the radius of the ion increases, as one would expect. Negative values of tr G are found for most of the tetraalkylammonium cations and the tetraphenyl ions in the reference electrolyte TATB. In the study of the interface with two immiscible electrolyte solutions (ITIES), considerable attention has been focused on the estimation of the Galvani potential diﬀerence at the water | oil interface on the basis of a reasonable extrathermodynamic assumptions. The discussion of these estimates is often made in terms of the ionic distribution coeﬃcient, which is deﬁned on the basis of equations (8.9.5) and (8.9.6). Generalizing this equation for the a | b interface at which ion i with charge zi is transferred, one may write mia; mib; RT aai þ ln f ¼ a b zi F zi F abi

! ð8:9:9Þ

The limiting ionic distribution coeﬃcient is deﬁned as

Kab i

ma; mib; ¼ exp i RT

! ð8:9:10Þ

so that the expression for the Galvani potential diﬀerence may also be written as ! a RT ab ai ln Ki b ð8:9:11Þ a b f ¼ zi F ai For a simple 1–1 electrolyte one obtains

430

LIQUIDS, SOLUTIONS, AND INTERFACES

RT aaM ln Kab a b f ¼ M b F aM or RT Kab M ln f ¼ a b 2F Kab X

!

!

RT aaX ln Kab ¼ X b F aX

RT aa abX ln M þ 2F abM aaX

! ð8:9:12Þ

! ð8:9:13Þ

where the subscripts M and X refer to contributions from the cation and anion, respectively. In dilute solutions where the activity of the cation is equal to that of the anion, this expression becomes ! RT Kab M ln ab ð8:9:14Þ a b f ¼ 2F KX One method of estimating a b f is based on the measurement of cell potential diﬀerences [19]. Consider the cell . . . . . . . Hg j Hg2 Cl2 j sat0 d KCl .. .. MX in W .. MX in b .. .. TEAPic in b .. .. TEAPic in 1 2 3 4 W j Hg2 Pic2 j Hg0 ð8:9:15Þ

where Pic stands for picrate (2,4,6-trinitrophenolate). The measured potential drop across the cell is Hg Hg 0 f1 and the important Galvani potential drops at each liquid | liquid junction have been numbered. The Galvani potential drop of interest is that at position 2. Those at positions 1, 3, and 4 are designed to be very small. The second cell considered is ..

.

..

10

20

30

Hg j Hg2 Cl2 j sat0 d KCl .. .. TEAPic in W .. TEAPic in b .. .. TEAPic in W jHg2 Pic2 j Hg0 ð8:9:16Þ In this cell all liquid | liquid junction potentials are negligible, so that the measured potential Hg Hg 0 f2 reﬂects the diﬀerence in potentials of the two reference electrodes which are common to each cell. As a result W b f

ﬃ

Hg

Hg0 f1 Hg Hg0 f2

ð8:9:17Þ

The Galvani potential diﬀerence or distribution potential measured in this way agrees well with that calculated from values of the standard Gibbs energies of transfer of the component ions, which have been estimated on the basis of solubility data. EXAMPLE

Estimate the Galvani potential diﬀerence for KI at the water | nitrobenzene interface using the data for the standard Gibbs energy of transfer given in table 8.11. For the Kþ cation, the value of tr G is 23.4 kJ mol1. The corresponding limiting ionic distribution coeﬃcient is

Kab ¼ exp Kþ

LIQUIDS AND SOLUTIONS AT INTERFACES

23:4 ¼ 7:96 105 8:3144 0:2982

For the I anion, tr G is 18.8 kJ mol1 and 18:8 Kab ¼ 5:09 104 ¼ exp I 8:3144 0:2982

431

ð8:9:18Þ

ð8:9:19Þ

On the basis of equation (8.9.14), the Galvani potential diﬀerence is ! 0:0257 7:96 105 ln ¼ 0:048V ð8:9:20Þ a b f ¼ 2 5:09 104 The negative result reﬂects the fact that the NB phase is negative with respect to the water phase. Since the Kþ cation is more diﬃcult to solvate in nitrobenzene than the I anion, the value of a b f shows that electroneutrality is maintained in the system by attracting Kþ cations into NB and at the same time repelling I anions from it. The above calculations are only valid in the absence of ion pairing. Since ion pairing is strong in NB, the observed values of a b f are expected to change signiﬁcantly with concentration of the electrolyte. The above discussion applies to a non-polarizable interface with a common electrolyte MX distributed in two immiscible phases a and b: . MX; a .. MX; b

ð8:9:21Þ

Non-polarizability implies that the Galvani potential diﬀerence a b f does not change when the activity of MX is changed on one side of the interface. Addition of MX to phase a results in the transfer of the ions M+ and X through the interface into phase b in order to maintain the distribution equilibrium as deﬁned in equation (8.9.13), and constancy of a b f. The rate of the ion transfer processes is ﬁnite so that re-establishment of equilibrium is a kinetic process which is of fundamental interest in itself. One may also design interfaces which behave as ideally polarizable systems. In this case, one or more of the component ions is essentially present in only one solvent. In order to illustrate this situation, a system with a common anion but two diﬀerent cations is considered ﬁrst: . M1 X; a .. M2 X; b

ð8:9:22Þ

Anion X dissolves easily in both phases, and its concentration in each is determined by the total cation concentration. The distribution coeﬃcient for M1, Kab M1 , is a very small number, indicating that this cation is predominantly in the a phase. On the other hand, the distribution coeﬃcinet for M2, Kab M2 is a very large number because this cation is predominantly in the b phase. The Galvani potential diﬀerence is determined by the anion X so that ! a RT ab aX ln KX b ð8:9:23Þ a b f ¼ F aX

432

LIQUIDS, SOLUTIONS, AND INTERFACES

In this system the anion X is called the potential determining ion. Assuming that the anion can be added to or removed from phase a by adding the electrolyte M1X, the activity of X remains approximately constant because the electrolyte concentration in phase b does not change. Thus, changes in aaX are directly reﬂected in changes in a b f according to equation (8.9.12). As a result, the Galvani potential diﬀerence is linear in ln aaX . Under these circumstances the potential changes can be used to detect changes in anion activity as is done with a speciﬁc ion electrode. If aaX is very low or very high, changes occur in abX , so that a b f is no longer linear in ln aaX . EXAMPLE

An ITIES is formed between water and nitrobenzene (NB) using electrolytes with the picrate (Pic) anion at the same activity (0.1 M). The electrolyte in water is Li Pic and that in NB, tetraphenylarsonium (TAþ) picrate. Given that the standard Gibbs energy of transfer of Pic is –4.6 kJmol1 [18], estimate the Galvani potential diﬀerence at the interface. Use the data in table 8.11 to estimate the activity of Liþ in NB. On the basis of equations (8.9.5) and (8.9.6), one may write for ion i W NB f

¼

tr Gi RT aw þ ln i zi F aNB zi F i

ð8:9:24Þ

Applying this equation to the Pic ion W NB f

¼

4:6 ¼ 0:048 V 96:485

ð8:9:25Þ

Thus, the NB phase is negative with respect to the W phase. From table 8.11, the Gibbs energy of transfer for Liþ is 38.2 kJ mol1. From equation (8.9.24) W ln aNB Liþ ¼ ln aLiþ

tr GLiþ F f RT W NB RT

ð8:9:26Þ

38:2 ¼ lnð0:1Þ þ 38:92 0:048 ¼ 15:84 8:314 0:2982 The activity of Liþ in the NB is 1:3 107 M. Although Liþ is attracted to the NB phase by the fact that it is negative with respect to water, the activity is very small because of the large positive value of tr GLiþ Another important system is one with a common electrolyte in both phases, and a second electrolyte in which one ion is soluble in only one phase. Such a system is . MX; MY; a .. MX; b

ð8:9:27Þ

where the anion Y is present in phase a only. The Galvani potential diﬀerence in this system is called the Donnan potential diﬀerence since its characteristics are similar to those observed at membranes which are permeable to some ions but not to others. The Galvani potential diﬀerence is determined by both the distribution of the cation Mþ and the anion X, and is given by equation (8.9.13). The condition of electroneutrality in phase a gives

LIQUIDS AND SOLUTIONS AT INTERFACES

433

caM ¼ caX þ caY

ð8:9:28Þ

cbM ¼ cbX

ð8:9:29Þ

whereas in phase b,

Thus, for this system equation (8.9.13) becomes " # a

ab a b RT KM yM yX RT cX þ caY þ ln ln f ¼ a b 2F 2F caX KXab ybM yaX

ð8:9:30Þ

where yxi is the activity coeﬃcient of ion i in solvent x. Under conditions that caY >> caX , the Donnan or Galvani potential diﬀerence is linear in the logarithm of the concentration of the anion X in the a phase. This system also has analytical applications in which a b f is used to determine the concentration of X in the a phase. The liquid | liquid interface is ideally polarizable when the electrolyte in phase a is insoluble in b, and that in b, insoluble in a. Such a system may be designated as . M1 X1 ; a .. M2 X2 ; b

ð8:9:31Þ

Under these circumstances the distribution coeﬃcient for M1X1 is zero, and that for M2X2, inﬁnity. It follows that the Galvani potential diﬀerence is not deﬁned for this interface. However, the thermodynamics of the interface can be derived on the basis of the Gibbs adsorption isotherm. The discussion in this section has focused on liquid | liquid interfaces for systems in which electrolytes are soluble in both liquid phases. Of course, there are many examples of systems in which electrolytes are not present in at least one phase. Examples include some of the systems listed in table 8.10 in which the organic phase has such a low permittivity that electrolytes are essentially insoluble. One interesting subclassiﬁcation of such systems includes those in which an organic phase may be spread on a polar liquid such as water to form a surface ﬁlm with the thickness of a monolayer. These systems are considered in more detail in the following section.

8.10 Surface Films on Liquids When an insoluble liquid a is placed at the liquid | air interface of another liquid b, the ﬁrst liquid can spread out as a thin ﬁlm which becomes monomolecular if the area of liquid b is suﬃciently large. Everyone has experienced the colors which are seen when oil is spread on water in a muddy puddle on the road. The colors result from the interference of the light rays reﬂected from the oil | air interface with those reﬂected at the water | oil interface. The interference occurs because the oil ﬁlm is very thin. The properties of thin ﬁlms are especially interesting when their thickness corresponds to one molecule. The ﬁrst question regarding thin ﬁlms is how they spread on the surface of the substrate liquid b (see ﬁg. 8.17). In order for this to occur, the Gibbs energy

434

LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 8.17 Schematic diagram illustrating the interface present when a small amount of liquid a is present on substrate liquid b as a lens at the air interface.

change associated with the spreading process must be negative. As liquid a spreads on substrate b, the area of a exposed to air increases and that of b decreases. At the same time, the area of the a | b interface increases. Since the interfacial tension is the incremental work done to increase the area of a given interface, the net work associated with spreading is Ws ¼ gab þ ga gb

ð8:10:1Þ

where gab is the interfacial tension between the two liquids. If Ws is negative then spreading is a spontaneous process. EXAMPLE

Examine the spreading of benzene on water, given that the interfacial tension of pure benzene is 28.18 mJ m2, using the data in tables 8.1 and 8.10. From table 8.1, the interfacial tension of pure water is 71.81 mJ m2, and from table 8.10, the interfacial tension between water and benzene is 34.4 mJ m2. Thus, the work of spreading is Ws ¼ 34:4 þ 28:2 62:2 ¼ 0:4 mJ m2

ð8:10:2Þ

As a result, a small amount of benzene placed on the surface of water rapidly spreads. However, this simple calculation is naı¨ ve in that it ignores the eﬀect of benzene on the interfacial tension of pure water. When the benzene and water are in contact, a very small amount of benzene dissolves in the water to yield a saturated solution. Since the benzene is hydrophobic it accumulates at the water | air interface and causes a signiﬁcant lowering of the interfacial tension of the aqueous phase. After equilibrium is reached, gb is 62.2 mJ m2. The work of spreading of benzene on water saturated with benzene is thus Ws ¼ 34:4 þ 28:2 62:2 ¼ 0:4 mJ m2

ð8:10:3Þ

Since Ws is positive, benzene does not spread on the saturated solution. As a result, when benzene is introduced at the water | air interface it initially spreads on the water. However, the water soon becomes saturated with benzene. When the interfacial tension of pure water falls to its equilibrium value for the solution, the benzene layer on the surface retracts to a lens as shown in ﬁg. 8.17. The phenomenon described with benzene on water occurs for most organic liquids of lower molecular weight. However, most experimental work with unimolecular ﬁlms has involved molecules of higher molecular weight involving at least ten carbon atoms. These substances are often solids at room temperature and are introduced to the water surface as a solution in a volatile organic solvent. For example, a unimolecular ﬁlm of octadecanoic acid can be spread on water by

LIQUIDS AND SOLUTIONS AT INTERFACES

435

introducing it as a solution in pentane. The pentane evaporates rapidly, leaving the C18 acid on the water surface as a unimolecular ﬁlm. Methods for studying unimolecular ﬁlms usually involve the Langmuir ﬁlm balance (ﬁg. 8.18), which allows one to study the surface pressure due to the ﬁlm, , as a function of the area available, A. In this experiment the trough is ﬁlled with water and the surfactant is introduced into the section of the trough with the larger area. The area available to the surfactant ﬁlm is varied by moving the sweep barrier. At the same time, the surface pressure is determined by the torsion wire connected to the ﬂoating barrier on the right-hand side of the trough. The torsion wire incorporates a pressure transducer which allows very precise determination of the surface pressure . In order to obtain data of high quality the purity of the water and surfactant must be carefully controlled. In addition, temperature must be kept constant, since the relationship between and A is sensitive to this parameter. Typical surface pressure against area data obtained for monolayers of a longchain carboxylic acid, namely, pentadecanoic acid, spread on acidiﬁed water, are shown in ﬁg. 8.19. Considerable change in the characteristics of the –A isotherms is seen with change in temperature. Extensive study of monomolecular ﬁlms has led to the deﬁnition of several states for the two-dimensional system [G1, 20]. A gaseous ﬁlm is one for which the A product is constant for ﬁxed temperature and corresponds to a situation where the area occupied per molecule in the ﬁlm is very large, so that the molecules do not interact with one another. It is clear that none of the isotherms in ﬁg. 8.19 corresponds to a gaseous state. However, such a state is possible at a temperature higher than those studied. A liquid expanded state, designated L1, is one in which the molecules can interact but are not well organized. This state is present for the pentadecanoic acid ﬁlm at 35.2 C for areas greater than 0.30 nm2 molec1 (see ﬁg. 8.19). The A product is not constant in this region but instead exhibits the properties of a two-dimen-

Fig. 8.18 Sketch of a Langmuir ﬁlm balance. The trough (1) is completely ﬁlled with water or an aqueous solution; the ﬁlm is conﬁned between a movable barrier (2) and a ﬁxed ﬂoating barrier (3) which is attached to the posts of the pressure transducer system by gold foil barriers (4); surface pressure is detected by movement of mirror (5) attached between a torsion wire (6) and the ﬂoating barrier (3); the calibration arm (7) and torsion wire control (8) are used to relate mirror position to surface pressure.

436

LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 8.19 Surface pressure against area isotherms for pentadecanoic acid monolayers on acidiﬁed water (pH ¼ 2). (From reference G1, with permission.)

sional non-ideal gas. In some cases the isothermal data can be ﬁtted to a twodimensional version of the van der Waals equation: a ð8:10:4Þ þ vw2 ðA bvw Þ ¼ RT A Here aVW and bVW are two-dimensional van der Waals constants describing the isotherm. Below 0.3 nm2 molec1, there is a sharp change in the slope of the –A plot corresponding to transition to an intermediate ﬁlm. In this state the monomers begin to assemble with the polar carboxylic acid group in the aqueous phase and the hydrophobic hydrocarbon tail pointing into the air phase. Further compression of the ﬁlm leads to expulsion of water molecules between the monomers and their alignment as a liquid condensed (L2) ﬁlm (see ﬁg. 8.20). The –A isotherm shown at 17.9 C is characteristic of an L2 ﬁlm showing a discontinuity

Fig. 8.20 Schematic diagram showing the structural changes in a unimolecular ﬁlm going from the liquid expanded state (L1) through the intermediate state (I) to the liquid condensed state (L2). The hydrophilic head group is designated by the circle at the end of a long hydrophobic hydrocarbon chain.

LIQUIDS AND SOLUTIONS AT INTERFACES

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in its two-dimensional compressibility for the range of areas studied. At high surface pressures the liquid condensed state becomes a solid-state ﬁlm for which the compressibility is negligibly small (dA=d ! 0). Studies of monolayer ﬁlms have included alcohols and carboxylic acids with hydrocarbon chains of varying length in the range C10–C26 [20, 21]. In the liquid condensed and solid states the –A isotherms have similar shapes at high surface pressures. These data show that the monolayer in the self-assembled state occupies an area of 0.205 nm2 per monomer. This result is independent of chain length and provides evidence that the monolayer consists of a close-packed structure with all molecular units oriented with their hydrocarbon chains perpendicular to the interface. Under these circumstances, strong attractive van der Waals forces are present between the hydrocarbon tails. As a result, formation of the solid-state ﬁlm can be irreversible, so that the ﬁlm does not break up when the surface pressure is decreased. Unimolecular ﬁlms provide a rich area for study in interfacial chemistry which has only been brieﬂy introduced here. Other investigations include the kinetics of ﬁlm formation, the viscosity of the monolayer, and measurements of surface potential changes [G1]. Unimolecular ﬁlms in condensed format may be removed from the surface of the liquid on which they are formed using a glass plate or other suitable solid substrate. The hydrophilic ends of the molecules in the ﬁlm remain in contact with the solid surface, and the hydrophobic ends, with the air. This type of unimolecular ﬁlm, which is called a Langmuir–Blodgett ﬁlm, is also the subject of considerable research interest.

8.11 Spectroscopy at Liquid Interfaces Spectroscopic studies of liquid interfaces provide important information about the composition and structure of the interfacial region. Early work was mainly carried out at the solid | liquid interface and involved techniques such as neutron and X-ray diﬀraction, and reﬂection FTIR spectroscopy. More recently, powerful techniques have been developed to study the liquid | liquid and liquid | gas interfaces. These studies are especially important because of their relevance to biological systems such as cell membranes. The techniques described here are secondharmonic generation (SHG) and vibrational sum frequency spectroscopy (VSFS). They are both second-order non-linear optical techniques which are speciﬁc to the interfacial region. Since the second-order eﬀects involve signals of low intensity, they rely on high-power lasers. When electromagnetic radiation interacts with a medium it induces a polarization P which is proportional to the electrical ﬁeld E due to the radiation: P ¼ wð1Þ E

ð8:11:1Þ

(1)

The proportionality constant w is called the linear electrical susceptibility. It is related in a simple way to the permittivity of the medium (see equation (4.5.3)). Thus, wð1Þ ¼ e0 ðe 1Þ

ð8:11:2Þ

438

LIQUIDS, SOLUTIONS, AND INTERFACES

where e is the relative permittivity of the medium and e0, the permittivity of free space. In general, both w(1) and e are functions of frequency. In the gas phase, the susceptibility is related to the polarizability aP of the molecules in the gas. Equation (8.11.1) is a basic relationship in linear optics and the starting point for deriving the fundamental equations of spectroscopy. When a molecule in the gas phase experiences a very large electromagnetic ﬁeld, the resulting polarization in no longer a linear function of the ﬁeld. In this case, the polarization is given by P ¼ aP E þ bP E2 þ gP E3 þ . . .

ð8:11:3Þ

The coeﬃcient bP is called the second-order non-linear polarizability or the hyperpolarizability. The corresponding equation in a condensed medium is P ¼ wð1Þ E þ wð2Þ E2 þ wð3Þ Eð3Þ þ . . . (2)

ð8:11:4Þ

(3)

w and w are the non-linear contributions to the susceptibility. Non-linear optical eﬀects become possible with high-powered lasers. SHG and VSFS depend on w(2), the second-order non-linear susceptibility of the medium at an interface. More details about these two methods are given below. A. Second-Harmonic Generation Optical second-harmonic generation involves the non-linear conversion of two photons of frequency to a single photon of frequency 2. This process requires a non-centrosymmetric medium, one example being the interface between two centrosymmetric media. Since only a few molecular layers are involved in the symmetry breaking at the interface, the SHG process is a highly interface-selective optical probe. SHG arises from polarization of the molecules at the interface and depends on their second-order non-linear susceptibility. Thus, if the amplitude of the electrical ﬁeld due to the incident light is E(), the polarization responsible for the SHG is given by Pð2Þ ð2nÞ ¼ wð2Þ E2 ðnÞ

ð8:11:5Þ

The value of w(2) is related to the molecular hyperpolarizability and depends on the electronic distribution in a given molecule. Only those molecules which have an inherent asymmetric electron density distribution, or an asymmetric distribution induced by adsorption, are capable of yielding an interfacial second-harmonic response. The value of w(2) may be calculated for both molecules and metals given their electronic properties [22]. In the case of molecules it depends on the number of electrons in the system, and therefore, is proportional to their surface concentration. For metals, w(2) is directly related to the surface concentration of free electrons. The intensity of the second-harmonic signal is proportional to the square of w(2). SHG experiments may also be used to determine molecular orientation at interfaces. By determining the polarity of the SH signal with respect to that of the incident light one may determine the independent components of w(2). In general, w(2) is a third-rank tensor with 27 elements. When the composition of

LIQUIDS AND SOLUTIONS AT INTERFACES

439

the monolayer is the same in all directions away from the normal to the interface ð2Þ ð2Þ (z-direction), only three of these are unique, namely, wð2Þ xzx , wzxx , and zzz These may be related to the corresponding elements of the hyperpolarizability tensor, and the orientation of the molecule determined [22, 23]. A typical experimental conﬁguration for SHG studies at a liquid | liquid interface is shown schematically in ﬁg. 8.21. A nanosecond or picosecond pulsed laser is the source of the input signal. It is polarized and ﬁltered to ensure that any extraneous second-harmonic light is removed. Typical power densities at the interfaces are in the range 105–108 W cm2. Conversion eﬃciencies are normally very small, and on the order of 1 photon in 1014. The light emitted at 2 is analyzed with respect to polarization in order to obtain information about the orientation of the molecules whose non-linear susceptibility give rise to the SHG signal. Early studies were carried out at the liquid | gas interface [22, 23]. Castro et al. [24] studied the adsorption of p-propyl-phenol from aqueous solutions at the air interface as a function of phenol concentration in the bulk. They showed that the square root of the second-harmonic intensity plotted against bulk phenol concentration followed a Langmuir isotherm with a standard Gibbs energy of adsorption equal to –24.3 kJ mol1. Similar results were obtained for other alkylphenols and alkylanilines. In other work with phenols, the orientation of phenol at the water | ð2Þ air interface was determined by studying the phase of the xzx component of the susceptibility. As expected, the OH was oriented toward the water phase [25] so that it could participate in the hydrogen-bonded structure of water. The same conclusion was reached for p-bromophenol and p-nitrophenol. SHG has also been used to study adsorption at the liquid | liquid interface [22, 23]. 2(-N-octadecylamino)naphthalene-6-sulfonate (ONS) is an anionic surfacant which is adsorbed at the water | 1,2-dichloroethane (DCE) interface [26]. The long alkyl chain prefers to be in the relatively non-polar DCE medium with the charged sulfonate group in the water at the interface. If the interface is polarized in an electrochemical cell, the change in the ONS surface excess with charge at the polarizable DCE | water interface can be followed using the SHG signal due to the surfactant. The other important area where SHG has been applied is to the study of adsorption at the metal | gas and metal | electrolyte solution interfaces [22, 23].

Fig. 8.21 Schematic diagram of the experimental apparatus used to study secondharmonic generation at a liquid | liquid interface. (From reference 23, with permission.)

440

LIQUIDS, SOLUTIONS, AND INTERFACES

The SHG at metal surfaces is almost always dominated by the nonlinear polarizability of the metal electron cloud at the interface. This technique is especially important in electrochemistry and has been used to study the adsorption of ions, molecules, and metal atoms at electrodes. B. Vibrational Sum Frequency Spectroscopy Sum frequency spectroscopy (SFS) provides a powerful tool for obtaining the vibrational spectrum of the molecules at the interface between two condensed phases or one condensed phase and a gas. It is a non-linear optical technique in which a photon of light with a frequency 1 in the visible region interacts with a photon of light with a frequency 2 in the infrared region to give a photon with frequency 1 þ 2 [27, 28]. By scanning the infrared light source over a range of frequencies i2, the vibrational spectrum is obtained as 1 þ i2. The relationships between the intensity of the incoming and outgoing optical signals are similar to those for second-harmonic generation. Thus, if P(2)(1 þ 2) is the second-order non-linear polarization due to the ﬁelds of the incoming light, one may write Pð1Þ ðn1 þ n2 Þ ¼ wð2Þ ðn1 þ n2 ÞEðn1 ÞEðn2 Þ

ð8:11:6Þ

where w (1 þ 2) is the non-linear susceptibility at the frequency 1 þ 2, and E(i), the ﬁeld due to radiation at frequency i. The square root of the VSFS 1=2 intensity, ISF , is proportional to the number density of molecules at the interface. When the frequency of the infrared beam is equal to the vibrational frequency of a molecule in the interface, a very large resonance enhancement of the SF signal is possible. Only vibrational modes that are both IR and Raman active are SF active. On the other hand, molecules or vibrational modes that possess an inversion center are not SF active. The resulting infrared spectrum of the interface provides information about what is at the interface. It can also be used to determine molecular orientation when the incoming and outgoing optical signals are polarized. In VSF spectroscopy, two pulsed laser beams, one of ﬁxed frequency in the visible and another of variable frequency in the IR region, overlap at the interface. Since the fraction of incident photons undergoing the SF process is very small, pulsed lasers are used in order to obtain detectable SF signals. Short pulses in the picosecond or femtosecond range are optimum. Details regarding operation of the lasers for diﬀerent optical geometries have been given by Richmond [28]. Typical geometries for VSFS in the reﬂection mode at the liquid | liquid and liquid | air interfaces are shown in ﬁg. 8.22. VSFS has been used to carry out studies of water structure at the aqueous solution | air and water | organic liquid interfaces [27, 28]. Hydrogen bonding plays an important role in determining water structure at the air interface and is responsible for the positive dipole potential found at this interface (see section 8.7.A). The VSF spectrum of water at the air interface shows two features, a fairly sharp band centered at 3700 cm1 and a broad band appearing between 3000 and 3500 cm1 (ﬁg. 8.23). The latter feature is also seen in bulk water and reﬂects the eﬀects of hydrogen bonding on the symmetrical and asymmetrical stretching modes of water. The high-energy band is due to free OH bonds which do not (2)

LIQUIDS AND SOLUTIONS AT INTERFACES

441

Fig. 8.22 Experimental geometries used in the reﬂection mode for VSFS at the liquid | liquid and liquid | air interfaces. The critical angle for the SF signal is yc.

participate in the hydrogen-bonded structure. These results conﬁrm that the water molecules at the surface are strongly hydrogen-bonded so that there is a net excess of water dipoles with their negative ends pointing to the gas phase. SHG studies of the water | air interface [23] also led to the conclusion that the net orientation of water is with the positive end of the molecular dipole pointing to the interior of the solution at an angle of 70 from a line perpendicular to the interface. Signiﬁcant changes in the VSF spectra of water are seen at the water | carbon tetrachloride interface (ﬁg. 8.23). The band due to free–OH bonds is shifted to lower energies by 40 cm1. This suggests attractive interaction between the free– OH oscillators and the organic phase. In addition, the spectral characteristics due to the hydrogen-bonded water molecules are signiﬁcantly modiﬁed. The intensity seen at the low-frequency region at the air interface has largely disappeared at the organic interface, suggesting that hydrogen bonding at the interface is much weaker in the presence of the organic phase. A detailed analysis of these spectra with resolution into component bands has been given by Richmond [27, 28]. Stanners et al. [29] used VSFS to study the structure of the alcohols at the liquid | air interface. For low-molecular-mass alcohols, namely, methanol to butanol, the alcohol molecule is oriented with the hydroxy group in the liquid and the alkyl group pointing into the gas phase. This orientation is attributed to the eﬀects of hydrogen bonding involving the hydroxy group and other alcohol molecules in the bulk. The infrared data involved both the –OH and –CH stretch-

442

LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 8.23 Vibrational sum frequency spectra of water at the air | water and carbon tetrachloride | water interfaces [28]. The spectrum for the air | water interface has been shifted vertically by 0.6 units for the sake of clarity.

ing modes. These results are consistent with the negative surface potential estimated for the lower alcohols (see table 8.5). Richmond [27, 28] has also used VSFS to study surfactants at the water | air and water | organic interfaces. These studies have provided information not only about the extent of adsorption but also about changes in the orientation of the adsorbate molecules with coverage. Other experimental work has included studies of solvent molecule orientation at the electrode | solution interface [30]. VSFS is obviously a powerful technique for obtaining speciﬁc information about the composition and structure of molecules at interfaces. Because it provides the vibrational spectrum, it is more advantageous than SHG.

8.12 Concluding Remarks The discussion in this chapter has focused on the properties of liquids at interfaces. A related area of contemporary research is the study of solid | gas interface. The solid surface is quite diﬀerent in that atomic or molecular components of a solid are relatively motionless compared to those of liquid. For this reason it is easier to deﬁne a plane associated with a well-deﬁned solid surface. The approach to studying adsorption on solids has been more molecular with the development of sophisticated statistical mechanical models. On the other hand, the study of liquid | gas and liquid | liquid interfaces has been much more macroscopic in approach with a ﬁrm connection to classical thermodynamics. As the understanding of liquids has improved at the molecular level using contemporary statistical mechanical tools, these methods are being applied now to ﬂuids at interfaces. Emphasis has been placed on the electrical aspects of interfaces in this chapter. This is especially important in examining the properties of electrolyte solutions at

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443

interfaces and leads to the experimental determination of ionic work functions. The concepts of the inner potential and surface potential are important in deﬁning a single electrode potential in an electrochemical cell. Although these quantities cannot be measured experimentally they are important in developing a connection with fundamental theoretical treatments of the transfer of charge from and into a condensed phase. A great variety of diﬀraction and spectroscopic techniques have been developed to study the interfacial region. Most of these have been applied at solid interfaces with the probing electromagnetic radiation approaching the interface from the gas or liquid phase. Now SHG and VSFS provide powerful techniques for obtaining molecular level information at the liquid | liquid and liquid | gas interface. The concepts developed in this chapter are very important in understanding the properties of interfaces in electrochemical cells and in living organisms. In the following chapters these subjects are developed in more detail, ﬁrst for the case of electrochemical equilibria, and then for the case of electrostatic equilibria.

General References G1. Adamson, A. W. Physical Chemistry of Surfaces, 5th ed.; Wiley-Interscience: New York, 1990. G2. Damaskin, B. B.; Petrii, O. A. Introduction to Electrochemical Kinetics, 2nd ed.; Vysshaja Shkola: Moscow, 1983; Part I (in Russian). G3. Trasatti, S. In Comprehensive Treatise of Electrochemistry; Bockris, J. O’ M.; Conway, B. E., Yeager, E., eds.; Plenum Press: New York, 1980; Volume 1, Chapter 2. G4. Parsons, R. In Modern Aspects of Electrochemistry; Bockris, J. O’M., ed.; Butterworths: London, 1954; Vol. 1, Chapter 3. G5. Somorjai, G. A. Introduction to Surface Chemistry and Catalysis; Wiley-Interscience: New York, 1994.

References 1. Schiﬀrin, D. J. J. Electroanal. Chem. 1969, 23, 168. 2. Riddick, J. A.; Bunger, W. B.; Sakano, T. K. Organic Solvents, 4th ed.; Wiley: New York, 1986. 3. Fomenko, V. S. The Emission Properties of Materials (in Russian); Naukova Dumka: Kiev, 1981. 4. Appelbaum, J. A.; Hamann, D. R. Rev. Mod. Phys. 1976, 48, 479. 5. Trasatti, S. In Advances in Electrochemistry and Electrochemical Engineering; Gerischer, H., Tobias, C. W., eds.; Wiley: New York, 1967; Vol. X, Chapter 4. 6. Randles, J. E. B. Adv. Electrochem. Electrochem. Eng. 1963, 3, 1. 7. Posner, A. M.; Anderson, J. R.; Alexander, A. E. J. Colloid Sci. 1952, 7, 623. 8. Levine, S. J. Colloid Interface Sci. 1971, 37, 619. 9. Kenrick, F. B. Z. Phys. Chem. 1896, 19, 625. 10. Randles, J. E. B. Phys. Chem. Liquids 1977, 7, 107. 11. Frumkin, A. N.; Damaskin, B. B. Pure Appl. Chem. 1967, 15, 263. 12. Parsons, R.; Rubin, B. T. J. Chem. Soc., Faraday Trans. 1, 1974, 70, 1636. 13. Koczorowski, Z.; Zagorska, I.; Kalinska, A. Electrochim. Acta. 1989, 34, 1857. 14. Farrell, J. R.; McTigue, P. J. Electroanal. Chem. 1982, 139, 37. 15. Bard, A. J.; Parsons, R.; Jordan, J. Standard Potentials in Aqueous Solution; Marcel Dekker: New York, 1985.

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16. Case, B.; Parsons, R. Trans. Faraday Soc. 1967, 63, 1224. 17. Girault H. H.; Schiﬀrin, D. J. In Electroanalytical Chemistry; Bard, A. J., ed., 15, 1, 1989. 18. Vanysek, P. Lect. Notes Chem. 1985, 39, 1. 19. Koryta, J.; Stulik, K. Ion Selective Electrodes, 2nd ed.; Cambridge University Press: Cambridge, 1983; Chapter 2. 20. Harkins, W. D. The Physical Chemistry of Surface Films; Reinhold Publishing: New York, 1952. 21. Boyd, G. E. J. Phys. Chem. 1958, 62, 536. 22. Corn R. M.; Higgins, D. A. Chem. Rev. 1994, 94, 107. 23. Eisenthal, K. B. Chem. Rev. 1996, 96, 1343. 24. Castro, A.; Bhattacharyya, K.; Eisenthal, K. B. J. Chem. Phys. 1991, 95, 1310. 25. Kemnitz, K.; Bhattacharyya, K.; Hicks, J. M.; Pinto, G. R.; Eisenthal, K. B.; Heinz, T. F. Chem. Phys. Lett. 1986, 131, 285. 26. Higgins, D. A.; Corn, R. M. J. Phys. Chem. 1993, 97, 489. 27. Watry, M. R.; Brown, M. G.; Richmond, G. L. Appl. Spectrosc. 2001, 55, 321A. 28. Richmond, G. L. Annu. Rev. Phys. Chem. 2001, 52, 357. 29. Stanners, C. D.; Du, Q.; Chin, R. P.; Cremer, P.; Somorjai, G. A.; Shen, Y.-R. Chem. Phys. Lett. 1995, 232, 407. 30. Baldelli, S.; Mailhot, G.; Ross, P. N.; Somorjai, G. A. J. Am. Chem. Soc. 2001, 123, 7697.

Problems 1. Given the following data for the interfacial tension at the water | air interface, determine the entropy of formation of this interface at 20 C. T / C

g / mJ m2

10 15 20 25 30

74.22 73.49 72.75 71.97 71.18

2. The following data have been obtained for the interfacial tension of benzene– trimethylpentane (TMP) mixtures at 30 C: Mole Fraction of TMP xTMP

Interfacial Tension g=mJ m2

0.000 0.186 0.274 0.378 0.483 0.583 0.645 0.794 1.0

27.53 23.40 22.47 21.21 20.29 19.70 19.32 18.74 17.89

LIQUIDS AND SOLUTIONS AT INTERFACES

445

A simple equation for the surface tension of mixtures is g ¼ g1;0 x1 þ g2;0 x2 bx1 x2 where g1,0 is the surface tension of pure component 1, g2,0, that of component 2, and x1 and x2 , the corresponding mole fractions. Determine the best value of b for the methanol–water interfacial tension data. How well does the model ﬁt the data? 3. Using the data given in problem 2 and assuming an ideal solution, estimate the relative surface excesses of benzene and TMP for mole fractions of 0.2, 0.4, 0.6, and 0.8. Plot the results against the mole fraction of benzene. 4. The following data for the surface tension of aqueous n-hexyl alcohol solutions were obtained at 12 C. Alcohol Concentration 103 M

Surface Pressure =mJ m2

0.62 0.81 1.25 1.72 2.50 3.43 4.90 6.86 9.80

2.3 2.5 3.9 5.7 7.9 9.4 13.4 16.3 19.4

Use a numerical method from appendix C to determine the relative surface excess of n-hexyl alcohol at c ¼ 1:25; 2:5, and 5 103 M. Assume that Henry’s law holds for the solute in the given concentration range. Express the results in units of mol m2 and molec nm2. 5. Consider a mercury sphere, radius r ¼ 0:01 cm, carrying a charge density of 10 pC cm2 in vacuo. The potential experienced by unit test charge e0 at a distance x from the metal surface is

1 q e0 r c¼ 4 e0 r þ x 2ðx2 þ 2xrÞ where q is the charge on the sphere and e0 ¼ 1:6 1019 C. What is the origin of the second term on the right-hand side of the above equation? Calculate c for decadic increments in x in the range 107–1 cm. Over what range of x is c equal to the outer potential of the mercury drop? 6. On the basis of the MSA the chemical potential of a monovalent cation is N e2 1 1 mi ¼ 0 0 1 es ri þ s 8 e0

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LIQUIDS, SOLUTIONS, AND INTERFACES

where ri is the ion’s radius and s, the MSA parameter for cation in water. Plot the work functions for the alkali metal cations from table 8.6 against ri þ

s assuming the s is 49 pm and using the Shannon–Prewitt radii from table 3.1. Estimate the surface potential of water from the intercept of best straight line through the data. Does the slope of the plot have the value expected from the MSA? 7. Estimate the Galvani potential diﬀerence for NaCl at the water | 1,1- dichloroethane interface given that the standard Gibbs energy of transfer of Naþ is 29 kJ mol1 and that for Cl, 58 kJ mol1. 8. An insoluble lipopolysaccharide forms an insoluble monolayer on water. Use the following data to estimate the molecular mass of this compound assuming ideal behavior: c=mg m2

=mN m1

0.06 0.09 0.10 0.14 0.17 0.23

10.6 16.4 20.4 25.9 34.3 50.0

9. Two capillary tubes with inside radii of 0.600 and 0.400 mm are inserted in a liquid with a density of 0.901 g mL1. The diﬀerence in the capillary rises in the two tubes is 1.00 cm. Estimate the surface tension of the liquid assuming zero contact angle and an air density of 0.001 g mL1.

9

Charge Transfer Equilibria at Interfaces

Walther Nernst was born in Briesen, Prussia (now in Poland) in 1864. He grew up in Graudentz and went on to study physics and mathematics at the Universities of Zurich, Berlin, and Graz. He completed his doctoral thesis in 1887 at the University of Wurzburg. He then joined Professor Ostwald at the University of Leipzig, where he started his work in physical chemistry. At this time Arrhenius and van’t Hoﬀ were also working in the Ostwald group. In 1894, he moved to Walther Hermann Nernst Go¨ttingen where he founded the Institute of Physical Chemistry and Electrochemistry. His ﬁnal appointment was as Professor of Chemistry at the University of Berlin from 1905. Nernst contributed in a signiﬁcant way to the development of the theory of electrolyte solutions and electrochemical cells. His name is connected to the well-known equation relating the electrode potential to the activities of the components involved in the electrode reaction. His scientiﬁc work was not limited to electrochemistry and in the latter part of his career he focused on the properties of matter at very low temperatures. This resulted in an early version of the Third Law of Thermodynamics. He was awarded the Nobel Prize in Chemistry for his work in this area in 1920. After retirement in the 1930s he lived in his country home near Berlin, where he died in 1941.

9.1 Electrochemical Equilibria Occur at a Wide Variety of Interfaces Processes in which charge is transferred from one phase to another at an interface make up an important class of interfacial reactions. Well-known examples are the reactions which occur at the electrodes of an electrochemical cell. These are electron transfer reactions, oxidation taking place at one electrode and reduction at the other. The early study of electrochemical cells provided valuable thermodynamic information about the redox processes occurring in them. When an electrochemical cell is a source of energy, for example, a battery, chemical energy is converted to electrical energy. When electrical energy is driven into an electro447

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LIQUIDS, SOLUTIONS, AND INTERFACES

chemical cell from an external source, electrode reactions producing products of commercial interest are possible. Thus the general subject discussed here is of considerable practical importance. Another important class of interfacial charge transfer processes occurs at the membrane | solution interface. Some solute species can move into the membrane phase, whereas others cannot. When ions are involved in membrane selectivity, a potential drop is established at the interface. Ion transfer processes at membranes are extremely important in living organisms and form the basis for the functioning of the nervous system. Membranes are also involved in ion selective electrodes such as the ubiquitous pH electrode. These electrodes are often used in modern analytical techniques based on potentiometry. In the present chapter, the relationship between the electrode potential and the activity of the solution components in the cell is examined in detail. The connection between the Galvani potential diﬀerence at the electrode solution interface and the electrode potential on the standard redox scale is discussed. This leads to an examination of the extrathermodynamic assumption which allows one to deﬁne an absolute electrode potential. Ion transfer processes at the membrane | solution interface are then examined. Diﬀusion potentials within the membrane and the Donnan potentials at the interface are illustrated for both liquid and solid state membranes. Speciﬁc ion electrodes are described, and their various modes of sensing ion activities in an analyte solution discussed. The structure and type of membrane used are considered with respect to its selectivity to a particular ion over other ions. At the end of the chapter, emphasis is placed on the deﬁnition of pH and its measurement using the glass electrode.

9.2 Electrochemical Cells The classical description of an electrochemical cell is that it consists of two electrodes immersed in an electrolyte solution. The simplest type of electrode is a metal, so that a simple cell involves two metal | solution interfaces. However, experimental study of the electrochemical cell always involves other interfaces which are introduced when the electrodes are connected to the leads of the potential measuring device. When the cell is a source of energy, an electrical current ﬂows in an external circuit as a result of a chemical reaction at each electrode. There are important conventions regarding the presentation of information about these systems. These are outlined in detail below. Consider the following cell as an example: Cu j Pt j H2 ð1 barÞ j HCl; H2 O j AgCl j Ag j Cu0

ð9:2:1Þ

This is called the diagram of the cell and includes all of the components required to make experimental measurements of its properties. The single vertical bar represents a phase boundary. The two electrodes in the cell are the hydrogen electrode and the silver | silver chloride, which are illustrated in ﬁg. 9.1. The electrolyte solution is 1 M HCl in water. Each electrode is connected to a copper wire which is used as a lead to connect to the measuring device, for example, a voltmeter with a very high input impedance. Since the wires at the ends of the cell

CHARGE TRANSFER EQUILIBRIA AT INTERFACES

449

Fig. 9.1 Sketches of half-cells corresponding to (a) a metal | metal ion system, (b) the hydrogen electrode, and (c) the silver | silver chloride electrode.

have the same chemical composition, the Galvani potential diﬀerence between them can be measured. . Two other symbols are used in cell diagrams. They are a dotted vertical bar (..) .. .. and a double dotted vertical bar (. .). The ﬁrst represents a liquid junction between miscible liquids, and the second, a double liquid junction in which the liquid junction potential has been eliminated to a good approximation, for example, by using a salt bridge. Liquid junctions are formed in practice using fritted glass disks and gels such as agar. They permit ion transport but greatly reduce mechanical mixing of the two solutions. The properties of liquid junctions are examined later in this chapter. The Galvani potential diﬀerence across the cell is called the EMF (electromotive force) and is given the symbol E. Furthermore, it is always measured as the potential of the right-hand electrode with respect to the left-hand one. Thus, 0

E ¼ Cu Cu0 f ¼ fCu fCu

ð9:2:2Þ

The term ‘‘electromotive force’’ is no longer used but the abbreviation EMF is still considered a valid way of referring to E. If the terminals of the cell are connected by a wire a spontaneous current ﬂows with electrons moving from the left-hand terminal to the right. This is accompanied by a chemical reaction in the cell. For cell (9.2.1), the reaction is H2 þ AgCl ! 2 HCl þ 2Ag

ð9:2:3Þ

When a spontaneous current ﬂows the potential diﬀerence between the terminals is no longer equal to E. When the potential diﬀerence is measured under conditions of zero current in the external circuit, it has important thermodynamic signiﬁcance. Speciﬁcally, the reactions at each electrode are at equilibrium. For cell (9.2.1), the equilibria are ( 2H2 2Hþ þ 2e + ( AgCl þ e + Ag þ Cl

ð9:2:4Þ ð9:2:5Þ

450

LIQUIDS, SOLUTIONS, AND INTERFACES

By convention, these equilibria are always written with the electrons on the lefthand side. In order to obtain the cell reaction (9.2.3), the reaction at the left-hand electrode is subtracted from that at the right-hand electrode using multiplying factors such that the net number of electrons in the overall reaction is zero. It should be noted that the Gibbs energies of the electrons in each electrode are not equal when the cell has an EMF. This means that the cell is capable of doing electrical work when current from the cell ﬂows in an external circuit. By measuring E under conditions of zero current, one can calculate the reversible electrical work which is available from the cell. In addition, the reversible work is equal to the Gibbs energy change associated with reaction (9.2.3). Thus, one may write E¼

G nF

ð9:2:6Þ

where n is the number of electrons involved in the cell reaction. For reaction (9.2.3), n is equal to two. The classical experimental method for measuring E involves a Poggendorf compensation potentiometer. In this experiment the voltage due to the electrochemical cell is exactly balanced by an opposing voltage source in the potentiometer (ﬁg. 9.2). Under conditions of balance, no current ﬂows from the cell, and the voltage can be related to the reversible electrical work available from the system. If the opposing voltage is slightly smaller, a current ﬂows from the electrochemical cell, which then acts as an energy source. On the other hand, if the opposing voltage is slightly greater, current is forced into the cell so that the electrode reactions take place in a direction opposite to the spontaneous direction. In a modern measurement of E, an electronic voltmeter with a very high input impedance is used. Nevertheless, the classical technique with the Poggendorf method is very helpful in illustrating the thermodynamic signiﬁcance of the EMF because it involves an exact balance between the electrochemical source and a variable d.c. source in the potentiometer device, thereby demonstrating that the cell properties are being measured under reversible conditions. The value of G and thus E for reaction (9.2.3) is determined by subtracting the chemical potentials of the reactants from those for the products:

Fig. 9.2 Schematic diagram of a Poggendorf compensation potentiometer. The contact position C on slidewire AB is moved until no current ﬂows in galvanometer G. Calibration of AB, for which position is related to voltage, permits determination of the EMF of cell X.

CHARGE TRANSFER EQUILIBRIA AT INTERFACES

G ¼ 2mHCl þ 2mAg mH2 2mAgCl

451

ð9:2:7Þ

Each chemical potential for a component in solution can be expressed in terms of its standard value plus a term in its activity; for gas phase components which are not at high pressure, the activity is equal to the partial pressure of the gas. Thus, expanding these terms in equation (9.2.7) one obtains G ¼ 2mHCl þ 2RT ln aHCl þ 2mAg mH2 RT ln PH2 2mAgCl

ð9:2:8Þ

This may also be written as

G ¼ G þ 2RT ln

aHCl

! ð9:2:9Þ

P1=2 H2

where G , the standard Gibbs energy change, is deﬁned as G ¼ 2mHCl þ 2mAg mH2 2mAgCl

ð9:2:10Þ

In the case of pure solids such as Ag and AgCl the chemical potential is identical to the standard chemical potential at 25 C and 1 bar pressure. For solutions, the standard state of the solute is unit activity at the same temperature and pressure. In the case of electrolytes as solutes, the activity ae is deﬁned on the concentration (molarity) scale, and the standard state is the hypothetical ideal state of unit molarity for which the activity coeﬃcient ye is unity. Under these circumstances, the activity of the solvent, which does not appear explicitly in equation (9.2.9), is also unity to a good approximation when the solvent is water. For gases the standard state is a pressure of 1 bar (105 Pa) at 25 C. In the older literature the standard pressure was 1 atm (101,325 Pa). In data compilations appearing after 1982, the standard state of 1 bar and 25 C is always used for gases [G3]. By combining equations (9.2.6) and (9.2.9), one obtains the Nernst equation for cell (9.2.1), that is, ! ! RT aHCl RT aHþ aCl ln 1=2 ¼ E ln ð9:2:11Þ E¼E F F pH P1=2 H 2

2

where E ¼

G 2F

ð9:2:12Þ

In writing these equations it is recognized that reaction (9.2.3) involves two electrons. The value of G from the relevant thermodynamic data is –42.84 kJ mol–1. The corresponding standard EMF E is 0.222 V. Nernst equations may also be written for the equilibria (9.2.4) and (9.2.5). These equations are written in terms of the EMF of a cell containing the electrode in question and a standard hydrogen electode (SHE). The latter is a deﬁned point of reference which is obtained when a hydrogen electrode is operated with a H2 pressure of 1 bar and unit Hþ activity in the electrolyte solution. On this basis the Nernst equation for the hydrogen electrode is

452

LIQUIDS, SOLUTIONS, AND INTERFACES

EHþ =H2 ¼

RT ln EHþ =H2 2F

PH2 a2Hþ

! ð9:2:13Þ

and that for the silver | silver chloride electrode EAgCl=Ag ¼ EAgCl=Ag

RT ln aCl F

ð9:2:14Þ

Equation (9.2.11) can be obtained by subtracting equation (9.2.13) from equation (9.2.14). In other words, Ecell ¼ EAgCl=Ag EHþ =H2

ð9:2:15Þ

It is emphasized that experiments only provide values of the EMF of the whole cell, and the polarity of one electrode with respect to the other. The arbitrary reference system, that is, the SHE, is introduced in order to establish a table of standard potentials for common electrodes. Since the standard potential for this system is zero by deﬁnition, EHþ =H2 0

ð9:2:16Þ

It follows that the standard EMF of cell (9.2.1) can be used to determine the standard potential of the silver | silver chloride electrode. On the basis of the analysis presented for cell (9.2.1), EAgCl=Ag ¼ 0:222 V

ð9:2:17Þ

It is important to recognize that a SHE cannot be constructed in the laboratory on the basis of experimental data. This follows from the fact that single ion activities cannot be measured experimentally. Referring to cell (9.2.1), the activity of the Hþ ion is, in general, diﬀerent from that of the Cl ion for a given concentration. One is only able to measure the mean activity a of the HCl electrolyte by experiment. Thus, although there exists a solution in which the Hþ ion activity is unity, the corresponding concentration of HCl is not known. However, there is no problem in determining the standard EMF of cell (9.2.1) experimentally as shown in section 9.5A. The standard potentials for the individual reactions are deﬁned from this experimental result. The properties of a cell with a zinc electrode and a silver | silver chloride electrode are now analyzed: Cu j Zn j ZnCl2 ; H2 O j AgCl j Ag j Cu0

ð9:2:18Þ

Since electrons ﬂow spontaneously from the left-hand electrode to the right-hand one when these terminals are connected externally, the spontaneous cell reaction is Zn þ 2AgCl ! ZnCl2 þ 2Ag

ð9:2:19Þ

The Gibbs energy change is G ¼ G þ RT ln aZnCl2

ð9:2:20Þ

where aZnCl2 is the activity of the ZnCl2 electrolyte in the aqueous solution. The corresponding equation for the EMF is

CHARGE TRANSFER EQUILIBRIA AT INTERFACES

E ¼ E

RT ln aZnCl2 2F

453

ð9:2:21Þ

where it is recognized that reaction (9.2.19) involves two electrons. The standard Gibbs energy change for this reaction is –190.08 kJ mol-1, and the corresponding standard EMF, 0.985 V. The individual equilibria at each electrode when no current is drawn from the cell are Zn2þ þ 2e ( + Zn AgCl þ e ( + Ag þ Cl

ð9:2:22Þ ð9:2:23Þ

The Nernst equation for the zinc electrode is EZn2þ =Zn ¼

EZn2þ =Zn

RT 1 ln 2F aZn2þ

ð9:2:24Þ

and that for the silver | silver chloride electrode is equation (9.2.14). Equation (9.2.21) can be obtained by subtracting equation (9.2.24) from equation (9.2.14). Since the standard potential for the silver | silver chloride half-cell is 0.222 V, it follows that the standard potential of the zinc half-cell is 0.222 V 0.985 V, that is, 0:763 V. Because the cell diagram describes a system from which thermodynamic data may be obtained, cells can be added or subtracted to obtain new cells. For example, subtracting cell (9.2.1) from cell (9.2.18), one obtains the cell ..

Cu j Zn j ZnCl2 ; H2 O .. .. HCl; H2 O j H2 ð1 bar) j Pt j Cu0

ð9:2:25Þ

The cell reaction for this system is Zn þ 2Hþ ! Zn2þ þ H2

ð9:2:26Þ

and the standard EMF is 0.763 V. This is also the negative value of the standard potential for the zinc half-cell. Cell (9.2.25) contains a special liquid junction which prevents mixing of the two electrolyte solutions containing HCl and ZnCl2 . It is important to note that the liquid junction has been designed to eliminate any associated potential drop. In general, liquid junctions do have an associated potential drop which must be considered in assessing the properties of the system. Liquid junction potentials are considered in detail in section 9.7. Cells like (9.2.25) may be used to determine the standard potentials for a series of metal|metal ion half cells with the general reaction Mnþ þ ne ( +M

ð9:2:27Þ

In these experiments the SHE may be regarded as a reference electrode, and the standard EMF is equal to the negative value of the standard potential for the metal | metal ion equilibrium when the electrolyte concentration corresponds to unit activity. Thus, for the cell ..

Cu j M j Mnþ ; H2 O .. .. Hþ ; H2 O j H2 (1 barÞ j Pt j Cu0 E ¼ ESHE EMnþ =M ¼ EMnþ =M

ð9:2:28Þ ð9:2:29Þ

because ESHE is zero by deﬁnition. The Nernst equation for equilibrium (9.2.27) is

454

LIQUIDS, SOLUTIONS, AND INTERFACES

EMnþ =M ¼

EMnþ =M

RT 1 ln nF aMnþ

ð9:2:30Þ

Values of the standard potentials for some metal | metal ion half-reactions are given in table 9.1. These cover a very wide range from –3.05 V for the strongly reducing metal Li to þ1.4 V for gold, which is a very stable ‘‘noble’’ metal. Three factors determine the value of E . One is the Gibbs energy of solvation of the metal ion in water. As has been seen from the discussion in chapter 3, this depends on the radius of the ion and the properties of water as a solvating medium. The second factor is the Gibbs energy of the metal, which depends on its structure as a solid. Finally, the ionization potential of the metal plays a role as well. Metal | metal ion reactions are examples of type I electrode reactions. This name derives

Table 9.1 Standard Potentials for Some Metal-Metal Ion Half-Reactions in Aqueous Solution at 25 C* Reaction

Standard Potential E =V

( Li Liþ e + ( Cs Cs þ e +

3.040

( Rb Rbþ þ e + (K Kþ þ e +

2.943

( Ba Ba2þ þ 2e + 2þ ( Sr Sr þ 2e +

2.906

( Ca Ca þ 2e + þ ( Na þ e + Na

2.869

( Mg Mg2þ þ 2e + ( Al Al3þ þ 3e +

2.357

( Mn Mn2þ þ 2e + 2þ ( Cr Cr þ 2e +

1.182

( Zn Zn2þ þ 2e + 2þ ( Fe þ 23 + Fe

0.762

( Cd Cd2þ þ 2e + ( Co Co2þ þ 2e +

0.402

( Ni Ni þ 2e + 2þ ( Sn Sn þ 2e +

0.236

( Pb Pb2þ þ 2e + 2þ ( Cu Cu þ 2e +

0.127

þ

2þ

2þ

( Ag Agþ þ e + ( Hg Hg2þ þ 2e + 3þ

Au

( Au þ 3e +

3.027 2.936 2.899 2.715 1.676 0.902 0.409 0.282 0.141 +0.339 +0.799 +0.852 +1.52

*Based on data in references [G3] and [1].

CHARGE TRANSFER EQUILIBRIA AT INTERFACES

455

from the fact that only one equilibrium is involved in the electrode process, and that the electrode itself takes part in this equilibrium. Another example of a type I reaction is an amalgam electrode. In such a system the metal which is in equilibrium with its ion in solution is dissolved in mercury. The metal activity in the liquid amalgam may be varied by varying its concentration in the amalgam. An example of such an electrode is the system Zn; Hg j Zn2þ ; H2 O

ð9:2:31Þ

Zn2þ þ 2e ( + ZnðHgÞ

ð9:2:32Þ

in which the half-reaction is

The Nernst equation for this electrode is EZnðHgÞ ¼ EZnðHgÞ

RT a ln Zn 2F aZn2þ

ð9:2:33Þ

where aZn is the activity of Zn metal in the amalgam. In this system both aZn and aZn2þ may be varied to change the electrode potential. Redox processes are examples of interfacial equilibria in which the electrode is not chemically involved, but acts only as a source or sink for electrons. A simple example of such an electrode is Pt j Fe3þ ; Fe2þ ; H2 O ð9:2:34Þ for which the half-reaction is Fe3þ þ e ( + Fe2þ

ð9:2:35Þ

In this system, platinum is the inert source or sink for electrons. The Nernst equation for the electrode process is EFe3þ =Fe2þ ¼ EFe3þ =Fe2þ

RT aFe2þ ln F aFe3þ

ð9:2:36Þ

There are many examples of redox processes in which all components are in the electrolyte solution. An example of a complex reaction which is used in analytical 2þ chemistry is the reduction of MnO in acidic solution. The half-reaction 4 to Mn is þ 2þ ( MnO þ 4H2 O 4 þ 8H þ 5e + Mn

for which the Nernst equation is EMnO4 =Mn2þ ¼

EMnO =Mn2þ 4

RT aMn2þ ln 5F aMnO4 a8Hþ

ð9:2:37Þ ! ð9:2:38Þ

Another example of a redox electrode is a gas electrode such as the hydrogen electrode discussed above. In this system a constant concentration of molecular hydrogen is maintained in the electrolyte solution, which also contains the hydrogen ion Hþ. In order to achieve these conditions, hydrogen gas at constant pressure is bubbled over the inert platinum electrode at the point where it contacts the solution (ﬁg. 9.1). Other gas electrodes involve molecular oxygen and chlorine.

456

LIQUIDS, SOLUTIONS, AND INTERFACES

A type II electrode is one in which two equilibria are involved in determining the electrode process. These systems often involve a metal electrode coated with an insoluble salt. A well-known example is the silver | silver chloride electrode discussed above. The solid silver chloride ensures that the surrounding solution is saturated with sparingly soluble AgCl. The Ag metal is in equilibrium with Agþ ion in solution: Agþ þ e ( + Ag

ð9:2:39Þ

The activity of the silver ion is controlled by the solubility equilibrium + Agþ þ Cl AgClcry (

ð9:2:40Þ

These two processes add to give the overall electrode equilibrium expressed by equation (9.2.23). As a result the silver | silver chloride responds to the activity of the chloride ion in solution. Another well-known example of a type II electrode is the calomel system: Hg j Hg2 Cl2 j Cl ; H2 O

ð9:2:41Þ

for which the half-reaction is Hg2 Cl2 þ 2e ( + 2Hg þ 2Cl

ð9:2:42Þ

The Nernst equation for the calomel electrode is Ecal ¼ Ecal

RT ln a2Cl 2F

ð9:2:43Þ

This electrode also responds to the chloride ion activity in solution. The corresponding electrode responding to the sulfate ion activity in solution is Hg j Hg2 SO4 j SO2 4 ; H2 O

ð9:2:44Þ

These half-cells are often used to make very stable and reliable reference electrodes. In the same way one may deﬁne a type III electrode process. Such a system involves three equilibria, one involving the metal | metal ion system and two equilibria in solution. Examples of type III electrodes are systems involving complex ions undergoing two complex formation steps.

9.3 The Thermodynamic Basis of the Nernst Equation The Nernst equation for single electrodes was introduced above on the basis of a cell in which the second electrode is the SHE. Since each electrode reaction in the cell is at equilibrium the Gibbs energy change for each process must be zero. It follows that the Nernst equation for a single electrode process cannot be derived in the same way as that for the overall cell reaction. Moreover, the overall cell reaction can be written in terms of uncharged species so that the thermodynamics for the process can be developed on the basis of the chemical potential of each species involved. However, a single electrode reaction necessarily involves charged

CHARGE TRANSFER EQUILIBRIA AT INTERFACES

457

species. Therefore, the thermodynamics must be derived using electrochemical potentials (section 6.6). In order to illustrate the thermodynamics of the Nernst equation for a single electrode, the following system is considered: Cu j Zn j ZnCl2 ; H2 O j AgCl j Ag j Cu0

ð9:3:1Þ

By choosing electrodes which are reversible to the cation and anion in the electrolyte, a liquid junction in the cell is avoided. The equilibrium process at the lefthand electrode is Zn2þ þ 2eðZnÞ ( + Zn

ð9:3:2Þ

The condition for equilibrium is ~Zn2þ þ 2~ m mZn el ¼ mZn

ð9:3:3Þ

~Zn ~Zn2þ is the electrochemical potential of the Zn2þ ions in solution, m where m el , that of the electrons in the Zn electrode, and mZn , the chemical potential of metallic zinc. Now each term is written out in the usual way: for the Zn2þ ion, one obtains ~Zn2þ ¼ mZn2þ þ RT ln aZn2þ þ 2Ffs m

ð9:3:4Þ

where mZn2þ is the standard chemical potential of the Zn2þ ion, aZn2þ , its activity, and fs , the inner potential of the solution; the expression for the electron is Zn; ~Zn m FfZn el ¼ mel

ð9:3:5Þ

where mZn; is its standard chemical potential, and fZn , the inner potential of the el Zn electrode; ﬁnally, for the zinc metal mZn ¼ mZn

ð9:3:6Þ

where mZn is the standard chemical potential of Zn. The last result follows from the fact that the Zn is a pure substance. Substituting equations (9.3.4)–(9.3.6) into the equilibrium condition (equation (9.3.3)), and rearranging, one obtains the result s Zn f

¼ fZn fs ¼

GZn2þ =Zn 2F

þ

RT ln aZn2þ 2F

ð9:3:7Þ

GZn2þ =Zn is the standard chemical potential change associated with equilibrium (9.3.2), that is, GZn2þ =Zn ¼ mZn mZn2þ 2mZn; el

ð9:3:8Þ

Equation (9.3.7) relates the Galvani potential diﬀerence at the Zn | solution interface to the activity of the Zn2þ ion in solution. However, since neither s Zn f nor aZn2þ can be measured experimentally, and GZn2þ =Zn contains contributions which are also not available from experiment, equation (9.3.7) is only of theoretical interest. On the other hand, it contains the essential elements of the Nernst equation for this single electrode system. The equilibrium process at the right-hand electrode in this cell involves

458

LIQUIDS, SOLUTIONS, AND INTERFACES

AgCl þ eðAgÞ ( + Ag þ Cl

ð9:3:9Þ

The condition for equilibrium is ~Ag ~Cl mAgCl þ m el ¼ mAg þ m

ð9:3:10Þ

~Ag ~Cl , the elecwhere m el is the electrochemical potential for the electrons in Ag, m trochemical potential for chloride ions in solution, and mAgCl and mAg , the chemical potentials for the pure substances AgCl and Ag metal, respectively. The electrochemical potential of the electron is given by Ag; ~Ag FfAg m el ¼ mel

ð9:3:11Þ

where mAg; is the standard chemical potential for this species, and fAg , the inner el potential of the silver electrode. For the Cl ion in solution, the corresponding expression is ~Cl ¼ mCl þ RT ln aCl Ffs m

ð9:3:12Þ

where mCl is the standard chemical potential, and aCl , its activity. The equation for the Galvani potential diﬀerence associated with equilibrium (9.3.9) is obtained after substituting equations (9.3.11) and (9.3.12) into equation (9.3.10), which leads to s Ag f

¼ fAg fs ¼

GAg=AgCl RT ln aCl F F

ð9:3:13Þ

where GAg=AgCl ¼ mAg þ mCl mAgCl mAg; el

ð9:3:14Þ

The results obtained for the Galvani potential diﬀerences s Zn f and s Ag f deﬁne the fundamental electrical properties of the electrode | solution interfaces in the electrochemical cell (9.3.1). However, as pointed out earlier, they cannot be measured experimentally. Further analysis of the properties of this system requires that the potential drops at all phase boundaries be considered. In fact, there are four phase boundaries in cell (9.3.1) and therefore four Galvani potential diﬀerences. If the Galvani potential diﬀerence obtained by experiment is Cu Cu0 f , then the following relationship holds: Cu Cu0 f

¼ Cu Zn f þ Zn s f þ s Ag f þ Ag Cu0 f

ð9:3:15Þ

The contributions involving the Cu metal contacts at each end of the cell can be determined by recognizing the electronic equilibria which exist between the electrons in each metal. On the left-hand side, ( eðZnÞ eðCuÞ + ð9:3:16Þ so that ~Cu ~Zn m el ¼ m el

ð9:3:17Þ

~Cu where m el is the electrochemical potential of the electrons in Cu. Writing each electrochemical potential as a standard chemical potential together with the elec-

459

CHARGE TRANSFER EQUILIBRIA AT INTERFACES

trical contribution depending on the inner potential of the given phase, equation (9.3.17) leads to the result Cu Zn f

¼ fZn fCu ¼

mZn; mCu; el el F F

ð9:3:18Þ

Similarly for the equilibrium eðAgÞ ( + eðCu0 Þ

ð9:3:19Þ

on the right-hand side of the cell, 0

Ag

Cu0

Cu0

f¼ f

f

Ag

mCu ; mAg; el ¼ el F F

ð9:3:20Þ

Substituting equations (9.3.18) and (9.3.20) into equation (9.3.15), one obtains mZn; mCu; el el 0 þ f ¼ Cu Cu F F

0

melCu ; mAg; el f þ f þ Zn s s Ag F F

ð9:3:21Þ

Since the composition of the Cu wires at0 the ends of the cell is identical, the standard chemical potentials mCu; and melCu ; are equal. Thus, with consideration el of equation (9.2.2) relating the cell EMF and Cu Cu0 f, equation (9.3.21) simpliﬁes to ! ! Zn; mAg; m E ¼ Cu Cu0 f ¼ s Ag f el s Zn f el ð9:3:22Þ F F where the ﬁrst term in the square brackets only contains contributions from the right-hand electrode in the cell and the second, from the left-hand electrode. This is an important result from a theoretical point of view, but in practice it does not permit separation of the EMF into contributions from each electrode. It is important to note that the fact that copper contacts are used to measure the Galvani potential diﬀerence for the cell is not reﬂected in the terms involved in equation (9.3.22). This means that the cell EMF does not depend on the nature of the metal used to measure the Galvani potential diﬀerence. EXAMPLE

Consider the cell Ag0 j Zn j ZnCl2 ; H2 O j AgCl j Ag

ð9:3:23Þ

in which the Galvani potential diﬀerence is measured between two pieces of silver. Show that the value of Ag0 Ag f is equal to that obtained for cell (9.3.1) which is given by equation (9.3.22). The expressions for s Zn f and s Ag f are those derived for cell (9.3.1) (equations (9.3.7) and (9.3.13)). At the Ag0 | Zn interface, 0

Ag0 Zn f

¼f

Zn

f

Ag0

mZn; mAg ; ¼ el el F F

ð9:3:24Þ

The experimentally measured Galvani potential diﬀerence is given by

460

LIQUIDS, SOLUTIONS, AND INTERFACES Ag0 Ag f

¼ Ag0 Zn f þ Zn s f þ s Ag f

Substituting in equation (9.3.24), this simpliﬁes to ! 0 ! ; mAg mZn; el el s Zn f Ag0 Ag f ¼ s Ag f F F

ð9:3:25Þ

ð9:3:26Þ

Since the silver wires at each end of the cell are identical, melAg; is equal to and equation (9.3.26) is the same as equation (9.3.22).

0 ; , mAg el

On the basis of the above analysis one may identify an absolute potential associated with each electrode. In the case of the silver | silver chloride electrode the absolute potential is a EAg=AgCl ¼ s Ag f

mAg; el þ Kabs F

ð9:3:27Þ

where Kabs is a constant used to deﬁne the absolute potential scale. Kabs cannot be determined by thermodynamic means. But other methods have been found to estimate Kabs so that the electrode potential can be related to other quantities measured on the absolute scale, for example, the Fermi level of the electrons in the metal. For the zinc electrode, a EZn 2þ =Zn ¼ s Zn f

mZn; el þ Kabs F

ð9:3:28Þ

The EMF is the diﬀerence between the absolute potentials a a EZn E ¼ EAg=AgCl 2þ =Zn

ð9:3:29Þ

As usual, the EMF is obtained by measuring the potential of the right-hand electrode with respect to that of the left-hand electrode. Although the thermodynamic analysis has given results with clear contributions from each of the electrodes, the observed EMF cannot be separated into these contributions by experiment. As was seen in the previous section, the solution to this problem has been to choose the SHE as a reference and to quote the standard potentials of all other half-reactions with respect to this point on the redox potential scale. In order to illustrate the application of this concept using absolute electrode potentials, the following cell is considered: Cu j Pt; H2 j HCl; H2 O j AgCl j Ag j Cu0

ð9:3:30Þ

The equilibrium process for the hydrogen electrode is 2Hþ þ 2eðPtÞ ( + H2

ð9:3:31Þ

Analysis of the thermodynamics of cell (9.3.30) using the methods developed above gives the following expression for the absolute potential of the hydrogen electrode: EHa þ =H2 ¼ s Pt f

melPt; þ Kabs F

ð9:3:32Þ

CHARGE TRANSFER EQUILIBRIA AT INTERFACES

where GHþ =H2 RT aH ln 2 2 f ¼ s Pt 2F 2F aH þ

461

! ð9:3:33Þ

and GHþ =H2 ¼ mH2 2mHþ 2melPt;o

ð9:3:34Þ

Under standard conditions at which the activities of the hydrogen gas and hydrogen ion in solution are both unity, the absolute potential of this system is EHa;þ =H ¼ s Pt f0 2

mH melPt; m þ þ Kabs ¼ 2 þ H þ Kabs F 2F F

ð9:3:35Þ

s Pt f0

is the value of the Galvani potential diﬀerence deﬁned in equation (9.3.33) for unit H2 gas pressure and unit Hþ ion activity. As emphasized above, the standard potential of the hydrogen electrode is deﬁned to be zero on the conventional scale of standard electrode potentials. On the basis of equation (9.3.35) there are two factors which cannot be evaluated thermodynamically in the expression for EHa;þ =H . First, the value of Kabs is not 2 known. Second, the standard chemical potential of a single ion such as the Hþ ion cannot be determined experimentally. Nevertheless, attempts have been made to estimate EHa;þ =H on the basis of extrathermodynamic assumptions. One approach 2 to determining this parameter is examined in the following section.

9.4 The Absolute Electrode Potential Although estimation of absolute electrode potentials falls outside of the realm of thermodynamics, it is of interest to theoreticians, especially those involved with the quantum-mechanical description of solids and liquids. These quantities allow one to deﬁne an absolute potential scale which is referenced to charge-free inﬁnity. For example, when semiconductors are used as electrodes, it is helpful to compare the absolute potential for a redox couple in solution with that of the Fermi level in the electrode. Trasatti [2] has described the methods used to estimate the absolute electrode potential on the basis of suitable extrathermodynamic assumptions. The method presented here is the one which gives an estimate which can be related to the potential scale used by physicists. Moreover, the resulting estimates of the absolute values of the standard electrochemical potential are based on experimentally measured quantities. The analysis is illustrated here for cell (9.3.30), which contains a hydrogen electrode. An air gap is introduced into the cell, so that the solutions surrounding each electrode are separated. The resulting cell is Cu j Pt; H2 j HCl; H2 O j air j HCl; H2 O j AgCl j Ag j Cu0

ð9:4:1Þ

If the Volta potential gap across the air gap is maintained at zero, then the potential drop across the cell is a compensation potential as described in section 8.7. Thus, one may write

462

LIQUIDS, SOLUTIONS, AND INTERFACES

c f ¼

Cu Cu0 f

¼ Cu Pt f þPt s f þs a f þa s f þs Ag f þAg Cu0 f ð9:4:2Þ

where s represents solution and a, the air. Since the Volta potential diﬀerences at the gap are zero, s a c

¼ a s c ¼ 0

ð9:4:3Þ

¼ a s f ¼ ws

ð9:4:4Þ

Thus, it is anticipated that s a f s

where w is the surface potential of the solution. Applying the analysis presented in section 9.3 for the remaining terms in equation (9.4.2), one obtains ! ! mAg; melPt; s el S þ w s Pt f þw ð9:4:5Þ c f ¼ s Ag f F F The two terms in square brackets may be identiﬁed with the absolute potentials of the silver | silver chloride and hydrogen electrode, respectively. For the standard hydrogen electrode

EHa;þ =H ¼s Pt fo 2

mH mPt;p m þ el þ w s ¼ 2 þ H þ ws F 2F F

ð9:4:6Þ

Kabs ¼ ws

ð9:4:7Þ

It is clear from this analysis that

Furthermore, the standard real potential of the hydrogen ion is Hþ ¼ mHþ þ Fws

ð9:4:8Þ

Thus, the standard potential of the hydrogen electrode is EHa;þ =H ¼ 2

mH2 Hþ F 2F

ð9:4:9Þ

Both quantities needed to estimate EHa;þ =H may be obtained from experiment. 2 In order to estimate EHa;þ =H the value of mH2 must be obtained with respect to 2 the same reference point as that used for Hþ . The real potential is deﬁned as the work done to move the Hþ ion from inﬁnity into the bulk of the solution phase. Thus, estimation of mH2 involves calculation of the Gibbs energy change associated with formation of molecular hydrogen from unsolvated protons in the gas phase according to the reaction 2Hþ g þ 2eg ! H2

ð9:4:10Þ

This process involves recombination of the proton and electron to given a H atom, Hþ g þ eg ! H g

ð9:4:11Þ

followed by the reaction of hydrogen atoms to give molecular hydrogen in the gas phase, 2H g ! H2

ð9:4:12Þ

CHARGE TRANSFER EQUILIBRIA AT INTERFACES

463

From the ionization potential for the H atom, the Gibbs energy change for reaction (9.4.11) is 1312.0 kJ mol1 [G3, 1]. The Gibbs energy change associated with recombination of H atoms is 406:5 kJ mol1 . Thus, the Gibbs energy change associated with reaction (9.4.10) is 3030:5 kJ mol1 . It follows from equation (9.4.9) that the absolute value of the standard potential for the hydrogen electrode is 1088 3030:5 þ ¼ 4:42 V 96:485 2 96:485

EHa;þ =H ¼ 2

ð9:4:13Þ

The absolute potential of the hydrogen electrode may be estimated by another route using the well deﬁned properties of a polarizable mercury electrode [2]. Since the metal only acts as a source or a sink for electrons for the hydrogen electrode reaction, one could use the half-cell Hg; H2 j Hþ

ð9:4:14Þ

to set up a standard hydrogen electrode. Although this system is thermodynamically the same as that using a Pt electrode, it is not used in practice because equilibrium (9.3.31) is established slowly on mercury. Ignoring the kinetic problem, thermodynamic analysis for this system gives the result that EHa;þ =H ¼ s Hg f0 2

melHg; þ ws F

ð9:4:15Þ

where s Hg f0 is the standard Galvani potential diﬀerence at the Hg | solution interface and mHg; , the standard chemical potential for an electron in mercury. el The Galvani potential diﬀerence may be divided into a Volta potential diﬀerence, which can be measured, and a diﬀerence in surface potentials, which cannot be measured. Thus, EHa;þ =H ¼ s Hg c0 þ wHg 2

mHg; el F

ð9:4:16Þ

where s Hg c0 is the Volta potential diﬀerence at the standard potential, and wHg, the surface potential of the mercury. Recalling that the work function for an electron in mercury is Hg; WHg þ FwHg el ¼ mel

ð9:4:17Þ

equation (9.4.16) becomes EHa;þ =H ¼ s Hg c0 þ 2

WHg el F

ð9:4:18Þ

From Volta potential diﬀerence measurements at the metal | solution interface discussed in section 8.8, the value of s Hg c at the potential of zero charge (PZC) of mercury is 0:248 V. However, the PZC is 0.192 V negative of the standard potential of the SHE. Thus, the value of the Volta potential diﬀerence at the standard potential of the hydrogen electrode is 0:056 V. Using the work function for an electron in Hg (4.50 eV), the estimate of EHa;þ =H is 2

EHa;þ =H ¼ 0:056 þ 4:50 ¼ 4:44 V 2

ð9:4:19Þ

464

LIQUIDS, SOLUTIONS, AND INTERFACES

The ﬁnal estimate of EHa;þ =H obtained by averaging the values from the two 2 results is EHa;þ =H ¼ 4:43 0:01 V

ð9:4:20Þ

2

The uncertainty in this estimate is quite realistic in terms of what is known about the parameters involved. The above analysis is easily extended to other half-cell reactions. Much of the data required to do the necessary calculations has been collected for cells involving aqueous solutions [1] and can also be found in thermodynamic tables published by the National Bureau of Standards in Washington [G3]. In practice, standard potentials are always used on the conventional scale because no extrathermodynamic assumptions are involved in their calculation. Any of these quantities can be converted to the absolute scale by adding the estimate of the absolute potential of the SHE, that is, 4.43 V, to the conventional value of the standard potential.

9.5 Experimental Studies of Electrochemical Cells The experimental study of electrochemical cells contributed in an important way to the acquisition of thermodynamic data for electrolyte solutions. Experimental methods for precisely measuring the EMF of these systems were well established in the ﬁrst half of the twentieth century [3]. As a result, many data were obtained for electrolyte activities and for important solution equilibria. Some examples of the experiments involved in these studies are given in this section. A. Measurement of Electrolyte Activity Coefﬁcients When an electrochemical cell is set up without liquid junctions it can be used to study the thermodynamic properties of the electrolyte involved. This requires that one electrode be reversible to the cation of the electrolyte and the other to the anion. A simple example is the cell Pt; H2 j HCl; H2 O j Hg2 Cl2 j Hg

ð9:5:1Þ

which was studied in detail by Hills and Ives [4]. The cell reaction for this system is H2 þ Hg2 Cl2 ! 2HCl þ 2Hg

ð9:5:2Þ

When the hydrogen pressure is 1 bar, the EMF is given by E ¼ E

RT RT ln aHCl ¼ E ln aHþ aCl F F

ð9:5:3Þ

where E is the standard EMF, and aHþ and aCl are the activities of the individual ions. A summary of the results obtained by Hills and Ives as a function of HCl concentration is given in table 9.2. As the concentration of HCl increases, the activity term on the right-hand side of equation (9.5.3) becomes more negative and the EMF of the cell decreases. This is clearly seen from the experimental data recorded in table 9.2.

CHARGE TRANSFER EQUILIBRIA AT INTERFACES

465

Table 9.2 Experimental Results for Cell (9.5.1) Molality of HCL 103 mc

EMF E=V

Activity Coeﬃcient g

Molality of HCl 102 me

EMF E=V

Activity Coeﬃcient g

1.6077 3.0769 5.0403 7.6938 10.947 13.968 18.871

0.60080 0.56825 0.543665 0.522675 0.50532 0.49339 0.47870

0.9568 0.9416 0.9276 0.9140 0.9002 0.8903 0.8772

2.5067 3.7690 5.1645 6.4718 7.5081 9.4276 11.9304

0.46490 0.44516 0.42994 0.41906 0.41187 0.40088 0.38948

0.8639 0.8436 0.8278 0.8163 0.8095 0.7984 0.7876

The experimental results for cell (9.5.1) can be used to determine the mean activity coeﬃcient of the electrolyte HCl. Expressing the individual ionic activities in terms of the mean activity coeﬃcient and the electrolyte molality, equation (9.5.3) can be rewritten as E ¼ E

2RT ln g me F

ð9:5:4Þ

In order to extract the activity coeﬃcient g at each concentration one must ﬁrst determine the standard EMF E . Taking the term ln me , which is known, to the left-hand side, one may write Eþ

2RT 2RT ln me ¼ E ln g F F

ð9:5:5Þ

In the limit of very dilute solutions one may assume that ln g is given by the limiting Debye–Hu¨ckel law and write Eþ

2RT 2RT 0 ln me ¼ E þ ADH m1=2 e F F

ð9:5:6Þ

0

where ADH is the Debye–Hu¨ckel constant, and the ionic strength has been expressed in terms of the molality. This expression shows that the quantity on in the limit of very dilute solutions. the left-hand side should be linear in m1=2 e A plot of the data according to equation (9.5.6) is shown in ﬁg. 9.3. It is apparent that it would be very diﬃcult to determine E on the basis of this plot, which shows curvature even in the lowest concentration range. Under these circumstances it is better to use the extended Debye–Hu¨ckel model to perform the necessary extrapolation. Then equation (9.5.5) becomes 0

Eþ 0

2RT 2RT ADH m1=2 e ln me ¼ Eo þ F F 1 þ B0DH ame1=2

ð9:5:7Þ

where BDH is the second Debye–Hu¨ckel constant and a, the ion size parameter. In order to use this equation the ion size parameter must be chosen so that a linear plot is obtained for data obtained up to 0.05 M. When values of a in the range 0.2–0.9 nm are tested, the best linearity is obtained for a equal to 0.6 nm. The corresponding results are also shown in ﬁg. 9.3. Extrapolation to zero

466

LIQUIDS, SOLUTIONS, AND INTERFACES

0

0

Fig. 9.3 Plot of E þ ð2RT=FÞ ln me against ADH me1=2 =ð1 þ BDH ame1=2 Þ for cell (9.5.1) using the data recorded in table 9.2. The value of a is zero for the results designated by (^) and 0.6 nm for those designated by (*).

molality gives a value of E equal to 0.2680 V. The relevant values of g are then easily calculated at each molality using equation (9.5.5). These are recorded in table 9.2. Activity coeﬃcients may be determined using electrochemical cells in a straightforward manner, provided there are electrodes available which are reversible to the cation and anion of the electrolyte. For example, one may determine the activity coeﬃcients for aqueous solutions of ZnSO4 using the cell Zn j ZnSO4 ; H2 O j Hg2 SO4 j Hg

ð9:5:8Þ

Electrodes of the second kind are available for many anions. In the case of cations, one may use the pure metal or the metal amalgam. The latter choice is convenient for reactive metals such as the alkali and alkaline earth metals. Of course, amalgams containing very reactive metals such as sodium also react with water. However, special techniques have been worked out so that activity coeﬃcient measurements can also be carried out with amalgams involving these metals [5]. B. Determination of Equilibrium Constants Electrochemical cells can also be used to determine other thermodynamic parameters such as equilibrium constants. For example, the solubility product for the sparingly soluble salt AgCl may be determined by comparing the properties of the silver | silver chloride electrode (9.2.23) with those of the silver | silver ion electrode (9.2.39). The potentials of these electrodes are equal when they are in a saturated solution of AgCl, that is, when the activities of these ions are those given by equilibrium (9.2.40). Therefore, under these conditions

CHARGE TRANSFER EQUILIBRIA AT INTERFACES

E ¼ EAgþ =Ag

RT 1 ln F aAgþ

¼ EAg=AgCl

RT ln aCl F

467

ð9:5:9Þ

Rearranging, one obtains ln Ksp ¼ lnðaAgþ aCl Þ ¼

F EAg=AgCl EAgþ =Ag RT

ð9:5:10Þ

The tabulated standard potentials are 0.799 V for the Ag | Agþ electrode and 0.222 V for the silver | silver chloride electrode. It follows that Ksp is equal to 1.76 10-10. Since a saturated aqueous solution of AgCl is very dilute, this result may be used on either the concentration or the molality scale, and the activity coeﬃcients set equal to one. Thus, cAgþ cCl ¼ Ksp ¼ 1:76 1010

ð9:5:11Þ

Electrochemical cells can also be used to determine the dissociation constants of weak electrolytes. This application is illustrated with the cell Pt; H2 j HAðm1 Þ; Na Aðm2 Þ; NaClðm3 Þ; H2 O j Hg2 Cl2 j Hg

ð9:5:12Þ

which can be used to determine the dissociation constant of the weak acid HA. Since the left-hand electrode is reversible to the Hþ ion and the right-hand one to the Cl ion, the EMF of the cell is given by E ¼ E

RT ln aHþ aCl F

ð9:5:13Þ

where E is equal to 0.268 V for this system (see cell (9.5.1)). The dissociation of the weak acid is described by the equilibrium HA ( + Hþ þ A

ð9:5:14Þ

with an equilibrium constant given by Ka ¼

aH þ aA aHA

ð9:5:15Þ

Substituting equation (9.5.15) into equation (9.5.13) and rearranging, one obtains E E þ

RT mHA mCl RT RT gHA gCl ln ln Ka ln ¼ m A gA F F F

ð9:5:16Þ

All of the quantities on the left-hand side are known if one assumes that the molality of the undissociated acid HA is equal to its stoichiometric molality. This means that any dissociation of the weak acid is neglected in estimating the left-hand side. On the other hand, the quantities on the right-hand side are unknown. The activity coeﬃcient of the Cl ion and the anion A from the acid are functions of ionic strength and are approximately equal to one another. Thus, one expects the term involving the activity coeﬃcients to depend weakly on the ionic strength. Analysis of the data to extract Ka involves plotting the function on the left-hand side against ionic strength and extrapolating to zero ionic strength where all activity coeﬃcients are unity. Thus, the intercept of such a plot yields the value of the acidity constant Ka .

468

LIQUIDS, SOLUTIONS, AND INTERFACES

Data obtained in a study of the dissociation of propanoic acid are given in table 9.3. In the experiment, the ionic strength was varied by approximately an order of magnitude with the concentration of the three solute components being kept equal. A plot of the function on the left-hand side of equation (9.5.16) against ionic strength demonstrates that the variation in the ordinate variable is not large, but it shows a slight tendency to increase with increase in ionic strength (see ﬁg. 9.4). Extrapolation to zero ionic strength gives a value of 0.2884 V from which the estimate of Ka is 1:33 105 . The above technique was applied by Harned and his colleagues to determine acidity constants for a variety of weak acids in both water and in water–nonaqueous solvent mixtures [3]. It may also be used to determine the self-dissociation constant of water. In the case of moderately weak acids the extrapolation procedure requires a more careful consideration of the contribution of Hþ to the ionic strength. More details can be found in the monograph by Harned and Owen [3]. C. Concentration Cells Another important area of experimental investigation involves concentration cells. These devices derive their EMF from the fact that a given cell component has a diﬀerent concentration in two parts of the cell. A simple example of a concentration cell is one involving the electrodes. For example, one may set up a cell with two amalgam electrodes containing the same metal at diﬀerent concentrations: Znða1 Þ; Hg j ZnSO4 ; H2 O j Znða2 Þ; Hg

Table 9.3 Experimental Results for Cell (9.5.12) for the Case of Propanoic Acid at 25 C (mHA ¼ mNaA ¼ mNaCl ) Molality of Propanoic Acid mHA

EMF E=V

0.004899 0.005918 0.01281 0.01605 0.01643 0.01715 0.01811 0.02220 0.02555 0.02648 0.03113 0.03179 0.03179 0.03569

0.69318 0.68809 0.66846 0.66252 0.66211 0.66098 0.65961 0.65461 0.65082 0.64983 0.64561 0.64561 0.64518 0.64224

ð9:5:17Þ

CHARGE TRANSFER EQUILIBRIA AT INTERFACES

469

Fig. 9.4 Plot of Ecor , the function on the left-hand side of equation (9.5.16) against ionic strength I using the data in table 9.3. Extrapolation of the straight line ﬁtted by least squares to I ¼ 0 allows one to estimate the acidity constant Ka for propanoic acid.

The equilibrium at each electrode is + Zn; Hg Zn2þ þ Hg þ 2 (

ð9:5:18Þ

The Nernst equation for this process is EZnðHgÞ ¼ EZnðHgÞ

RT aZnðiÞ ln 2F aZn2þ

ð9:5:19Þ

where aZnðiÞ is the activity of zinc in electrode i and aZn2þ the activity of Zn2þ ion in solution. Subtraction of the Nernst equation for the left-hand electrode from that for the right-hand electrode gives E¼

RT aZnð1Þ ln 2F aZnð2Þ

ð9:5:20Þ

E is positive when the zinc activity in the left electrode is higher than that in the right electrode. Study of these systems shows that the amalgam phase behaves as a non-ideal solution. Thus, activity coeﬃcients for amalgam solutions can be determined using cells such as (9.5.17). In these experiments, the amalgam concentration is changed from very dilute, for which the activity coeﬃcient can be assumed to be unity, to more concentrated. Another type of concentration cell involves gas electrodes with the same gas at diﬀerent pressures. An example involving the hydrogen electrode is Pt; H2 ðP1 Þ j HCl; H2 O j Pt; H2 ðP2 Þ

ð9:5:21Þ

On the basis of the Nernst equations for this system (equation (9.2.13)) the expression for the EMF of the cell is

470

LIQUIDS, SOLUTIONS, AND INTERFACES

E¼

RT aH2 ðP1 Þ ln 2F aH2 ðP2 Þ

ð9:5:22Þ

where aH2 ðPi Þ is the activity of hydrogen gas at pressure Pi . E is positive when P1 is greater than P2 . The third type of concentration cell involves two electrolyte solutions of different concentrations. Such a cell can be set up with or without transference. The simpler system is one without transference and it is considered ﬁrst. It can be constructed by placing two cells of identical format but diﬀerent electrolyte concentrations in opposition to one another. An example based on cell (9.5.8) is Zn j ZnSO4 ðm1 Þ; H2 O j Hg2 SO4 j Hg j Hg2 SO4 j ZnSO4 ðm2 Þ; H2 O j Zn0 ð9:5:23Þ This cell has potential drops at three locations, namely, at each electrode | solution interface, and across the central electrode. For this reason it is easier to determine the cell EMF on the basis of the individual Galvani potential drops which make up the observed Galvani potential diﬀerence Zn Zn0 f. The equilibrium at each electrode is Zn2þ þ 2e ( + Zn

ð9:5:24Þ

The Galvani potential diﬀerence at the electrode | solution interface is s Zn f

¼

GZn2þ =Zn 2F

þ

RT ln aZn2þ 2F

ð9:5:25Þ

The function of the central electrode is to facilitate the exchange of sulfate ions between the two solutions. The equilibrium associated with a single mercury | mercury sulfate electrode is Hg2 SO4 þ 2e ( + 2Hg þ SO2 4

ð9:5:26Þ

When two of these systems are put back-to-back, the net equilibrium process is ( 2 SO2 4 ðm1 Þ + SO4 ðm2 Þ

ð9:5:27Þ

that is, sulfate ions in solution 1 are in equilibrium with sulfate ions in solution 2. The Galvani potential diﬀerence between the two solutions is s1 s2 f

¼ fs2 fs1 ¼

RT aSO2 4 ð2Þ ln 2F aSO2 ð1Þ

ð9:5:28Þ

4

Now, one can calculate the EMF of the system using the relationship E ¼ Zn Zn0 f ¼ Zn s1 f þ s1 s2 f þ s2 Zn0 f

ð9:5:29Þ

Combining equation (9.5.25) written for each electrode | solution interface with equation (9.5.28), one obtains E¼

RT aZn2þ ð2Þ aSO2 RT aZnSO4 ð2Þ 4 ð2Þ ln ln ¼ 2F aZn2þ ð1Þ aSO2 ð1Þ 2F aZnSO4 ð1Þ 4

ð9:5:30Þ

CHARGE TRANSFER EQUILIBRIA AT INTERFACES

471

On the basis of this result, it is clear that the EMF is positive when the activity of ZnSO4 in the left-hand compartment is less than that in the right-hand compartment. It is helpful to analyze the functioning of cell (9.5.23) when it performs electrical work under reversible conditions. This means that zero current ﬂows in the external circuit and that no net changes in concentration occur in the two electrolyte solutions. The reaction which occurs at the left-hand electrode which is in contact with the dilute solution is Zn ! Zn2 þðm1 Þ þ 2e

ð9:5:31Þ

The opposite reaction occurs on the right-hand electrode Zn2 þðm2 Þ þ 2e ! Zn

ð9:5:32Þ

The net process at the central electrode is transfer of sulfate ions from solution 2 to solution 1: 2 SO2 4 ðm2 Þ ! SO4 ðm1 Þ

ð9:5:33Þ

By adding these reactions, one obtains the overall process ZnSO4 ðm2 Þ ! ZnSO4 ðm1 Þ

ð9:5:34Þ

This result shows clearly how the EMF derives from the diﬀerence in concentration between the two electrolyte solutions. An electrolyte concentration cell may also be set up with a liquid junction between two electrolyte solutions with diﬀerent concentrations. An example based on the Zn | Zn2þ system is .

Zn j ZnSO4 ðm1 Þ; H2 O .. ZnSO4 ðm2 Þ; H2 O j Zn0

ð9:5:35Þ

The liquid junction between the two electrolyte solutions is formed physically in a device which allows electrical contact between the two solutions but prevents their mixing in a signiﬁcant way. One popular device is a fritted glass disk. When this type of barrier is used there is negligible transport of the components of one solution to the other. However, since the electrolyte ﬁlls the pores of the frit from both sides, electrical contact is established, and one is able to measure the ability of the electrochemical cell to do electrical work under conditions of zero external current. The electrode equilibria in cell (9.5.35) are exactly the same as those in cell (9.5.23). However, the process at the liquid junction is very diﬀerent from that at the central Hg | Hg2 SO4 electrode in cell (9.5.23). In order to maintain electroneutrality, both the cation and anion must be involved in the equilibrium at the liquid junction. The process at this junction is complicated by the fact that the individual ions move with diﬀerent mobilities. An easy way of analyzing the operation of this cell is to consider the cell reactions under reversible conditions. At the left-hand electrode Zn2þ ions enter solution 1 according to reaction (9.5.31). In order to maintain electroneutrality in solution 1, Zn2þ ions move out to solution 2, and SO2 4 ions move in the opposite direction from solution 2 to solution 1. If the fraction of the current across the liquid junction carried by the Zn2þ ions is tþ , then transport of Zn2þ is described as

472

LIQUIDS, SOLUTIONS, AND INTERFACES

tþ Zn2þ ðm1 Þ ! tþ Zn2þ ðm2 Þ

ð9:5:36Þ

In a similar way, since t is the fraction of the current carried by SO2 4 ions 2 t SO2 4 ðm2 Þ ! t SO4 ðm1 Þ

ð9:5:37Þ

In both equations it is assumed that the transport numbers tþ and t are independent of concentration. Finally, the reaction occurring at the right-hand electrode is given by equation (9.5.32). The net cell process is obtained by adding equations (9.5.31), (9.5.32), (9.5.36), and (9.5.37): 2þ 2 ð1 tþ ÞZn2þ ð2Þ þ t SO2 4 ð2Þ ! ð1 tþ ÞZn ð1Þ þ t SO4 ð1Þ

ð9:5:38Þ

Since there are only two ions in solution, the fractions tþ and t add to give unity, and equation (9.5.38) simpliﬁes to t ZnSO4 ðm2 Þ ! t ZnSO4 ðm1 Þ

ð9:5:39Þ

Since the net charge associated with this process on a molar scale is two moles of electrons, the expression for the EMF is E¼

aZnSO4 ðm2 Þ RT t ln 2F aZnSO4 ðm1 Þ

ð9:5:40Þ

Comparison of the results for cells (9.5.23) and (9.5.35) reveals that the EMF of the latter is smaller by the fraction t . This is a result of the potential drop which is present across the liquid junction separating the two solutions. Since cell (9.5.35) has the same metal at either end, the observed EMF can be written as a sum of Galvani potential diﬀerences. The result is given by equation (9.5.29), where s1 s2 f now represents the Galvani potential diﬀerence across the liquid junction. From equation (9.5.25), the Galvani potential drops at the two electrodes are Zn s1 f ¼

GZn2þ =Zn 2F

RT ln aZn2þ ðm1 Þ 2F

ð9:5:41Þ

RT ln aZn2þ ðm2 Þ 2F

ð9:5:42Þ

and s2 Zn f

GZn2þ =Zn 2F

þ

Subtracting equations (9.5.41) and (9.5.42) from equation (9.5.40), the following expression is obtained for the liquid junction potential: aZn2þ ðm2 Þ aSO2 ðm2 Þ RT aZn2þ ðm2 Þ RT s2 s1 4 t ln ln s1 s2 f ¼ f f ¼ 2F aZn2þ ðm1 Þ aSO2 ðm1 Þ 2F aZn2þ ðm1 Þ 4 ð9:5:43Þ aSO2 ðm2 Þ aZn2þ ðm2 Þ RT RT 4 t ln t ln ¼ þ 2F þ aZn2þ ðm1 Þ 2F aSO2 ðm1 Þ 4

Because single ion activities cannot be measured experimentally, this result shows that the liquid junction potential cannot be determined exactly by thermodynamic measurements. However, if the solutions are both suﬃciently dilute so that the single ion activities can be replaced by their mean values, then one obtains

473

CHARGE TRANSFER EQUILIBRIA AT INTERFACES s 1 s 2 f

¼

aðm2 Þ RT ðt t Þ ln 2F þ aðm1 Þ

ð9:5:44Þ

The liquid junction potential may also be expressed in terms of the EMF of the total cell. Dividing equation (9.5.44) by equation (9.5.40) one easily obtains the result t tþ E ð9:5:45Þ s1 s2 f ¼ 2t Another example of a concentration cell with transference is the system .

Ag j AgCl j KClðm1 Þ; H2 O .. KClðm2 Þ; H2 O j AgCl; Ag0

ð9:5:46Þ

The equilibrium at each electrode is given by equation (9.3.9) for the silver | silver chloride system. Reversible operation of this cell leads to a decrease in the KCl activity in the left-hand compartment and an increase in the right-hand one. At the same time there is transport of Cl ions across the liquid junction from left to right, and transport of Kþ ions in the opposite direction. The net cell reaction for exchange of one electron at each electrode is tþ KCl ðm1 Þ ! tþ KCl ðm2 Þ

ð9:5:47Þ

If the transport number is independent of electrolyte concentration, the expression for the EMF is aKClðm1 Þ RT t ln E¼ ð9:5:48Þ F þ aKClðm2 Þ Because of the direction of the spontaneous cell reactions it is clear that the solution in the left-hand compartment is more concentrated than that in the right-hand one. Further analysis of cell (9.5.46) leads to the following expression for the liquid junction potential: aðm1 Þ RT ðt t Þ ln ð9:5:49Þ s1 s2 f ¼ F þ aðm2 Þ This diﬀers from that derived for cell (9.5.35) because the diﬀerence in the nature of the electrode reactions. In cell (9.5.35) the reactions involve cations, whereas in cell (9.5.46) anions are involved. The above expressions were derived for systems in which the transport numbers are independent of electrolyte concentration. In reality, these parameters depend on ionic strength, as was discussed earlier in section 6.6. Under these circumstances the EMF can be calculated by assuming that the liquid junction is made up of an inﬁnite number of junctions of inﬁnitesimally small width. In addition, it is assumed that the activity change across each minijunction is inﬁnitesimally small. Under these circumstances, the cell potential can be calculated on the basis of an integral. In the case of cell (9.5.46), the expression for the liquid junction potential becomes ð1 RT ðtþ t Þd ln a s1 s2 f ¼ F 2

ð9:5:50Þ

474

LIQUIDS, SOLUTIONS, AND INTERFACES

Obviously, estimation of s1 s 2 f requires that the variation of tþ with electrolyte activity be known. A similar expression can be written for cell (9.5.35). The results obtained for the liquid junction potential show that its magnitude for a system with the same electrolyte on either side depends on the relative values of the transport numbers for the cation and anion. For an electrolyte like KCl for which tþ and t are approximately equal, the value of s1 s2 f is very small. On the other hand, if the liquid junction is formed between two HCl solutions, the value of s1 s2 f is considerably larger because the transport number for the Hþ cation is larger than that for the Cl anion. The general subject of liquid junction potentials is important in electrochemistry. These junctions often involve diﬀerent electrolytes on either side of the boundary in which case the potential diﬀerence is more diﬃcult to estimate as described later in this chapter. The experimental cells considered in this section demonstrate that only those systems which do not have a liquid junction can be treated exactly by thermodynamics. However, most cells used in electrochemistry have a liquid junction, and therefore can only be treated approximately. This is especially true for the cells used to perform electroanalysis.

9.6 Electrochemical Cells for Electroanalysis In the simple cells without liquid junctions considered in the previous section, one electrode was reversible to a cation in the electrolyte solution and the other to an anion in the same solution. In a cell designed for electroanalysis, the system must respond to only one ion in the electrolyte. A well-known example of such a system is the pH meter, which is designed to respond to the activity of the hydrogen ion in a test solution. The electrode that gives the response is called the indicator electrode. The other electrode, which is designed to give a constant response independent of the composition of the solution, is called the reference electrode. A constant response is achieved by isolating the reference electrode solution from the test solution via a liquid junction. Thus, the general construction of a cell used for potentiometry in electroanalysis is given by reference electrode j reference electrolyte solution test electrolyte solution j indicator electrode

.. .

ð9:6:1Þ

The reference electrode system is kept constant so that the composition of the cell up to the liquid junction does not change in the experiment. When the composition of the test electrolyte solution is changed, the cell EMF changes. These changes can be related to the concentration of a potential determining ion in the test solution via a calibration procedure. The system is also designed to keep the liquid junction potential constant and as small as possible. Thus, the observed potential drop across the cell can be written as Ecell ¼ Eind Eref þ flj

ð9:6:2Þ

where Eind is the potential of the indicator electrode, Eref , that of the reference electrode, and flj , the liquid junction potential.

CHARGE TRANSFER EQUILIBRIA AT INTERFACES

475

Many diﬀerent indicator electrodes may be constructed on the basis of the systems discussed in section 9.2. For example, a pH electrode could be constructed using the hydrogen electrode with hydrogen gas bubbling over a platinum electrode immersed in the test solution. However, this system is very inconvenient because it involves handling a gas which is explosive when ignited. Indicator electrodes responding to metal ions can be constructed using the corresponding metal. For example, a zinc electrode responds to the zinc ion activity in aqueous solution. However, such a system can only be used over a limited pH range. If the test solution is too acidic, the zinc metal dissolves due to a chemical reaction with the acid and the analytical result is too high. If the test solution is too basic, zinc hydroxide is formed. This may precipitate on the electrode and change the nature of the electrode process. In modern electroanalysis the classical electrode systems are usually avoided in favor of speciﬁc ion electrodes. These devices make use of membrane selectivity to obtain a response which depends on the activity of a given ion. Speciﬁc ion electrodes are discussed in detail later in this chapter. As stated earlier, the reference electrode in a cell used for electroanalysis is designed so that its potential is independent of the composition of the test solution. There are several general properties that reference electrodes should have in order to be useful in analysis: (1) they should be reversible with an electrode potential which is independent of time and reproducible; (2) they should have a small temperature coeﬃcient; (3) they should be ideally non-polarizable with negligible eﬀects from the ﬂow a small current through the system; and (4) they should be easily constructed. The most commonly used reference electrodes are those based on on the mercury calomel system and the silver | silver chloride system. The electrolyte most commonly used in these systems is KCl. Relevant parameters for commonly used reference electrodes are given in table 9.4. The calomel electrode is based on reaction (9.2.42). Insoluble mercury(I) chloride (calomel) lies on top of the liquid mercury phase and ensures that the solution is saturated with Hg2þ 2 . The potential of the electrode depends on the chloride ion activity as follows: Ecal ¼ Ecal

RT RT ln a2Cl ¼ Ecal ln aCl 2F F

ð9:6:3Þ

The most common electrolyte for this reference electrode is saturated KCl. This choice ensures that the liquid junction potential between the reference electrode Table 9.4 Potentials of Common Reference Electrodes at 25 C on the SHE Scale Electrode Ag | AgCl | sat’d KCl Hg | Hg2 Cl2 | sat’d NaCl (SSCE) Hg | Hg2 Cl2 | sat’d KCl (SCE) Hg | Hg2 Cl2 | 0.1 M KCl Hg | Hg2 SO4 | sat’d K2 SO4

Potential/V 0.197 0.2360 0.2412 0.3337 0.64

476

LIQUIDS, SOLUTIONS, AND INTERFACES

compartment and the indicator electrode compartment is small. In some cases the KCl is replaced by NaCl. The latter choice is made when the test solution contains þ the ClO 4 anion or another anion which forms sparingly soluble salts with the K cation. The calomel electrode is also used with lower concentrations of KCl when contamination of the test solution by Cl ion is a problem. The calomel electrode is easily assembled and provides excellent reproducibility. Since mercury is a liquid, this electrode must be kept in an orientation which maintains contact between the mercury and the metallic lead to the external connection. The other common reference electrode is the silver | silver chloride system which is based on reaction (9.2.23). This electrode may be made by anodizing a silver wire in an electrolyte containing Cl ions. This results in a deposit of AgCl on the metal surface so that a solution saturated in AgCl is maintained when the electrode is brought into contact with an aqueous solution. This electrode suﬀers from two disadvantages with respect to the calomel system. The ﬁrst is that the chemical potential of the silver depends somewhat on its metallurgical history. This in turn has a small eﬀect on the actual standard potential of the system. The second problem with this system is that silver chloride is photoreactive and degenerates with continued exposure to light. Thus, commercially available Ag | AgCl electrodes are encased in bakelite or other suitable material to prevent exposure of the system to light. On the other hand, this reference electrode may be used in any orientation in the experimental system. It is incorporated in most commercial speciﬁc ion electrodes as an internal reference. As already discussed, the standard hydrogen electrode (SHE) is the chosen reference half-cell upon which tables of standard electrode potentials are based. The potential of this system is zero by deﬁnition at all temperatures. Although this reference electrode was often used in early work in electrochemistry, it is almost never seen in chemical laboratories at the present time. It is simply too awkward to use because of the requirement for H2 gas at 1 bar pressure and safety considerations. The remaining feature of the electroanalytical cell is the liquid junction potential. As an example, consider the junction which arises in a cell used to determine the pH of 0.01 M HNO3 solution with a saturated calomel electrode (SCE) as reference: .

Sat0 d KCl; H2 O .. 0:01 M HNO3 ; H2 O

ð9:6:4Þ

Since the electrolytes on either side of the junction are diﬀerent, there is a tendency for the KCl on the left side of the junction to diﬀuse to the right-hand compartment. At the same time, the HNO3 on the right side diﬀuses to the lefthand compartment. The diﬀusional pressure of each ion is proportional to the product of the ionic concentration times its mobility. Thus the mass transfer process is dominated by movement out of the concentrated KCl solution. Since the mobilities of the Kþ cation and Cl anion are almost the same, the mass transfer process can proceed without any tendency to develop a potential diﬀerence. On the other hand, the mobility of the Hþ cation is signiﬁcantly greater than that of the NO 3 anion in the solution on the right-hand side. As a result a positive potential is established on the left-hand side of the junction with respect to the

CHARGE TRANSFER EQUILIBRIA AT INTERFACES

477

right. This potential drop acts to keep the solution electrically neutral by retarding mass transfer of the Hþ. For the conditions chosen for junction (9.6.4) mass transfer of KCl dominates so that the total potential drop at the junction is small, that is, the order of a few millivolts. It must be remembered that the models used to estimate flj are approximate. The only test of their approximate validity is the fact that diﬀerent models give estimates which are not too diﬀerent from one another. The above example illustrates the reason why saturated KCl is usually used in the calomel and silver | silver chloride electrodes. As a result, the liquid junction potential is kept small. Nevertheless, there is an interest in estimating the value of the liquid junction potential for the general case. This subject is considered in more detail in the following section.

9.7 The Liquid Junction Potential Liquid junctions are found in almost all electrochemical cells used in electroanalysis. In general, there is a potential drop across the liquid junction and it is important to be able to evaluate it. Because of the diﬀerent electrolyte compositions and concentrations involved, the liquid junction is associated with an irreversible mass transfer process. In this section, methods of estimating the potential drop due to the liquid junction are outlined. The case of a liquid junction between electrolyte solutions of the same composition was examined earlier for electrochemical cells with transport (section 9.5). This situation is now re-examined using Onsager’s method for dealing with mass transfer. The system considered is .

c1 MX .. c2 MX

ð9:7:1Þ

where MX is a 1–1 electrolyte at two concentrations c1 and c2 separated by a suitable liquid junction barrier. According to the Onsager treatment, the ﬂux of either ion, Ji is given by equation (6.7.16), that is, Ji ¼ Li rm~ i ¼ Li RT r ln ai Li zi Frf

ð9:7:2Þ

where m~ i is the electrochemical potential of the ion, ai its activity, zi , its valence, f, the inner potential, and Li , the phenomenological coeﬃcient connecting the ﬂux to the gradient of the electrochemical potential. In order to keep matters simple, it is assumed that the ﬂux takes place in one direction only, namely, along the x direction. This means that the gradients of the ionic activity and the electrical potential can be replaced by their derivatives with respect to x. Both ions move in the same direction from the solution of higher concentration to that of lower concentration. If the concentration c1 is higher than c2 then the ions on the left-hand side move across the liquid junction from the left to the right. The movement is governed by the law of electroneutrality which requires that the local concentrations of cation and anion be the same throughout the system. This condition is met by requiring that the local ﬂux of the two ions be equal, that is,

478

LIQUIDS, SOLUTIONS, AND INTERFACES

JM ¼ JX

ð9:7:3Þ

If the ions move in the x direction only, this means that LM RT

d ln aM df d ln aX df ¼ LX RT LM F þ LX F dx dx dx dx

ð9:7:4Þ

where LM and LX are the phenomenological coeﬃcients for the cation and anion, respectively. This equation may be rearranged to obtain an expression for the potential gradient df=dx: df RT d ln aM d ln aX ¼ LM LX ð9:7:5Þ dx FðLM þ Lx Þ dx dx The phenomenological coeﬃcients may be related to the diﬀusion coeﬃcients or to the mobility of the ions (equation (6.7.21). In the present case, they can be related to the ionic transport numbers because tM ¼

LM LM þ LX

ð9:7:6Þ

tX ¼

LX LM þ LX

ð9:7:7Þ

and

Another simpliﬁcation is possible if the individual ionic activities aM and aX are replaced by their mean value a . Strictly speaking, this simpliﬁcation is only valid in dilute solutions. Equation (9.7.5) now becomes df RT d ln a ¼ ðt t Þ dx F þ dx

ð9:7:8Þ

Integrating from solution 1 to solution 2, one obtains ð2 RT d ln a ðtþ t Þ 1 2 f ¼ f2 f1 ¼ F dx

ð9:7:9Þ

1

If the transport numbers are independent of solution composition, the ﬁnal result is 1 2 f

¼

RT a ðc Þ ðtþ t Þ ln 2 F a ðc1 Þ

ð9:7:10Þ

This result was obtained earlier on the basis of an analysis of the potential drops within concentration cells with transport (equations (9.5.44) and (9.5.49)). In practice, liquid junctions are usually much more complex than the example just considered. However, it is clear that the potential drop arises because of the requirement that the solution be locally electroneutral. For a liquid junction involving diﬀerent ions of varying charges this requirement may be expressed as X z i Ji ¼ 0 ð9:7:11Þ i

CHARGE TRANSFER EQUILIBRIA AT INTERFACES

479

Applying equation (9.7.2) for the case that mass transfer occurs only in the xdirection, the expression for the potential gradient is P df RT ¼ dx F

i

d ln afi zi Li dx P 2 zi Li

ð9:7:12Þ

i

The transport number of species i can be found from the fraction of the current carried by that species when the concentration gradient is zero. The current carried by ion i is ii ¼ zi FJi

ð9:7:13Þ

so that the transport number for this ion is z FJ ti ¼ P i 2 i zi FJi

ð9:7:14Þ

i

Using equation (9.7.2) with d ln ai =dx set equal to zero, one obtains z2 L t i ¼ Pi 2 i zi Li

ð9:7:15Þ

i

It follows that the potential gradient can now be written as df RT X ti d ln ai ¼ dx F i zi dx

ð9:7:16Þ

By integrating this equation, one obtains an expression for the liquid junction potential: ð2 RT X ti d ln ai 1 2 f ¼ f2 f1 ¼ F zi i

ð9:7:17Þ

1

Before the value of 1 2 f can be estimated the concentration proﬁles associated with each ion must be determined. When diﬀerent electrolytes are present on either side of the boundary, the electrolyte distribution is time dependent. This means that an exact thermodynamic solution to the problem is not possible. The solution to the problem given here is a steady-state solution, that is, the solution appropriate to a system in which mass transfer is occurring but under conditions that the liquid junction potential is independent of time. The porous diaphragms described earlier are examples of junctions which meet this condition. There are two well-known solutions to equation (9.7.17), one by Planck [6] and the other by P. Henderson [7]. The latter solution is more often used in practice and therefore is presented here. There are three important assumptions made in obtaining the Henderson equation for the liquid junction potential. First of all, it is assumed that the concentration of each ion changes linearly from the value that it has in the solution on

480

LIQUIDS, SOLUTIONS, AND INTERFACES

the left of the junction, ci (1) to the value that it has on the right-hand side, ci ð2Þ. For a junction with a total thickness d, this means that x ci ¼ ci ð1Þ þ ½ci ð2Þ ci ð1Þ ¼ ci ð1Þ þ si x ð9:7:18Þ d where si is the slope of the linear dependance. The second assumption is that the activity coeﬃcients do not change in the region of the liquid junction. This means that the derivative in ln ai may be replaced by the corresponding derivative in ln ci . Finally, the mobility of each ion ui is assumed to be independent of solution composition in the junction. The latter two assumptions lead to considerable simpliﬁcation of the problem, but are clearly in disagreement with existing experimental data. Writing equation (9.7.16) in terms of ionic concentration, one obtains df RT X zi ui ci d ln ci P 2 ¼ ð9:7:19Þ dx F i zi ui ci dx i

The sum in the denominator can be written as X z2i ui ci ð1Þ þ z2i ui si x ¼ ALJ þ BLJ

ð9:7:20Þ

i

where ALJ ¼

X

z2i ui ci ð1Þ

ð9:7:21Þ

i

and BLJ ¼

X

ð9:7:22Þ

z2i ui si

i

Thus, equation (9.7.19) becomes

df RT X z i ui dci ¼ dx F i ALJ þ BLJ dx

As a diﬀerential equation, one may write RT X zi ui si dx df ¼ F i ALJ þ BLJ Integrating from solution 1 to solution 2, one obtains RT X zi ui si ALJ þ BLJ d f ¼ ln 1 2 F i BLJ ALJ Substituting the expressions for si , ALJ , and BLJ , the ﬁnal result is 1 0P 2 P zi ui ½ci ð2Þ ci ð1Þ zi ui ci ð2Þ RT i C B i P 2 ln@P A 1 2 f ¼ F zi ui ½ci ð2Þ ci ð1Þ z2i ui ci ð1Þ i

This is the Henderson equation.

i

ð9:7:23Þ

ð9:7:24Þ

ð9:7:25Þ

ð9:7:26Þ

CHARGE TRANSFER EQUILIBRIA AT INTERFACES

481

The equation for the liquid junction potential is much simpler when 1–1 electrolytes with one common ion are present on either side of the junction. For example, if the same anion is present in both solutions, the liquid junction can be designated as .

c1 M1 .. Xc2 M2 X The sums in the Henderson equation can now be written as X zi ui ½ci ð2Þ ci ð1Þ ¼ uM2 cM2 ð2Þ uM1 cM1 ð1Þ uX ½cX ð2Þ cX ð1Þ

ð9:7:27Þ

ð9:7:28Þ

i

X i

z2i ui ½ci ð2Þ ci ð1Þ ¼ uM2 cM2 ð2Þ uM1 cM1 ð1Þ þ uX ½cX ð2Þ cX ð1Þ X

ð9:7:29Þ

z2i ui ci ð2Þ ¼ uM2 cM2 ð2Þ þ uX cX ð2Þ

ð9:7:30Þ

z2i ui ci ð1Þ ¼ uM1 cM1 ð1Þ þ uX cX ð1Þ

ð9:7:31Þ

i

X i

The expression for the liquid junction potential is 1 2 f

¼

RT ½uM2 cM2 ð2Þ uM1 cM1 ð1Þ uX cX ð2Þ þ uX cX ð1Þ F ½uM2 cM2 ð2Þ uM1 cM1 ð1Þ þ uX cX ð2Þ uX cX ð1Þ

ln

uM2 cM2 ð2Þ þ uX cX ð2Þ uM1 cM1 ð1Þ þ uX cX ð1Þ

ð9:7:32Þ

When the concentrations c1 and c2 are the same, the expression for the liquid junction potential is even more simple: 1 2 f

¼

RT M2 X ð2Þ ln F M1 X ð1Þ

ð9:7:33Þ

where ij is the equivalent conductance of electrolyte ij. This equation is known as the Lewis–Sargent equation. If the liquid junction involves a common cation with equal electrolyte concentrations such as .

c1 MX1 ; H2 O .. c1 MX2 ; H2 O

ð9:7:34Þ

the corresponding equation is 1 2 f:

¼

RT MX2 ð2Þ ln F MX1 ð1Þ

ð9:7:35Þ

EXAMPLE

Estimate the liquid junction for the boundary .

0:1 M NaCl .. 0:01 M HCl

ð9:7:36Þ

using the data in table 6.2. Repeat the calculation for the case that the concentration of HCl is 0.1 M.

482

LIQUIDS, SOLUTIONS, AND INTERFACES

Since ratios are involved in the Henderson equation, the molar conductivities may be used directly instead of the mobility P zi ui ½ci ð2Þ ci ð1Þ 0:01 349:8 0:1 50:1 þ 0:09 76:35 i P ¼ 0:639 ¼ 2 zi ui ½ci ð2Þ ci ð1Þ 0:01 349:8 0:1 50:1 0:09 76:35 i

ð9:7:37Þ P i P i

z2i ui ci ð2Þ z2i ui ci ð1Þ 1 2 f

¼

0:01 349:8 þ 0:01 76:35 ¼ 0:337 0:1 50:1 þ 0:1 76:35

¼ 0:0257 ð0:639Þ lnð0:337Þ ¼ 0:018 V

ð9:7:38Þ ð9:7:39Þ

The potential in the right-hand solution is more negative than that in the lefthand solution. This slows down movement of the most mobile ion, namely Hþ, in moving from right to left; in addition, the second most mobile ion, namely, Cl is also retarded in its motion from left to right. When the concentration of the two electrolytes is the same the more simple Lewis–Sargent equation may be used: 349:8 þ 76:35 ¼ 0:031 V ð9:7:40Þ f ¼ 0:0257 ln 1 2 50:1 þ 76:35 The increase in HCl concentration results in an increase in the liquid junction potential. These estimates could be improved by using a model in which the actual conductivities at each concentration are used rather than the values at inﬁnite dilution. The Henderson equation may also be used to illustrate the principle involved in keeping the liquid junction potential small. Consider the junction between a very concentrated or saturated electrolyte solution and a dilute electrolyte solution: .

saturated M1 X1 .. dilute M2 X2

ð9:7:41Þ

Neglecting the terms in the sums which are due to M2 X2 , the Henderson equation becomes uM2 cM2 ð2Þ þ uX2 cX2 ð2Þ RT uX1 cX1 ð1Þ uM1 cM1 ð1Þ ln ð9:7:42Þ f ¼ 1 2 F uM1 cM1 ð1Þ þ uX1 cX1 1ð1Þ uM1 cM1 ð1Þ þ uX1 cX1 ð1Þ The ratio in the logarithm term gets larger as the concentration diﬀerence increases. However, when the mobilities of the cation and anion in the concentrated solution are the same, the sum in front of the logarithm is zero so that 1 2 f is zero. As stated earlier, the electrolyte most commonly used for which this condition is approximately met is KCl. Another suitable system is concentrated sodium formate. Values of the liquid junction potential between 4.2 M KCl and more dilute solutions of the same electrolyte estimated by equation (9.7.32) are shown in table

CHARGE TRANSFER EQUILIBRIA AT INTERFACES

483

Table 9.5 Liquid Junction Potentials Between 4.2 M KCl (Solution 1) and More Dilute Solutions of HCl and KCl (Solution 2) Estimated by the Henderson Equation (9.7.32) Concentration of the More Dilute Solution

Liquid Junction Potential 1 2 f=mV

0.01 M KCl 0.1 M KCl 1.0 M KCl 0.01 M HCl 0.1 M HCl 1.0 M HCl

3.0 1.8 0.7 3.0 4.6 14.1

9.5. 4.2 M KCl corresponds to a saturated solution. The liquid junction potential varies from approximately 3 mV for 0.01 M KCl to 1 mV for 1 M KCl on the basis of the Henderson estimate. Because the mobility of the Kþ ion is slightly less than that of the Cl ion, small positive values for 1 2 f are obtained. HCl is an example of an electrolyte in which the cation has a much higher mobility than that of the anion. For this electrolyte the estimate of 1 2 f for the dilute 0.01 M HCl solution is the same as that for 0.01 M KCl. As the HCl concentration increases, the estimate of the liquid junction potential also increases, reaching a value of 14 mV for 1.0 M HCl. These results demonstrate that the liquid junction potential is an important feature of the electroanalytical system. The most eﬀective way of dealing with this feature is to calibrate the indicator electrode by making measurements of E over the concentration range where analysis is being carried out. In some cases one wishes to reduce the liquid junction between two dilute electrolytes as much as possible. In this case, a salt bridge is used to connect the two solutions. This can be quite simply a U-shaped tube ﬁtted with fritted glass disks at each end and ﬁlled with saturated KCl through a central tap. Thus, the salt bridge involves two liquid junctions each of which is small. The junction between the two solutions is described as ..

dilute M1 X1 .. .. dilute M2 X2

ð9:7:43Þ

where the double dotted vertical bars represent the salt bridge. Some eﬀort has been made to estimate liquid junction potentials experimentally, especially for systems containing the chloride anion. A suitable cell for these measurements is .

Ag j AgCl j c1 M1 Cl .. c2 M2 Cl j AgCl0 j Ag0

ð9:7:44Þ

The EMF of this cell is E ¼

aCl ð1Þ RT ln þ 1 2 f F aCl ð2Þ

ð9:7:45Þ

484

LIQUIDS, SOLUTIONS, AND INTERFACES

Table 9.6 Liquid Junction Potentials Estimated from Cell (9.7.44) and from the Lewis–Sargent Equation (9.7.33) 1 2 f=mV

Solution 1 0.1 M KCl 0.1 M NaCl 0.1 M LiCl 0.1 M NaCl 0.01 M KCl 0.01 M NH4 Cl

Solution 2

Experiment

Lewis–Sargent Equation

0.1 M HCl 0.1 M HCl 0.1 M HCl 0.1 M KCl 0.01 M HCl 0.01 M HCl

26.8 33.1 34.9 6.4 25.7 27.0

28.5 33.4 36.1 4.9 27.5 27.5

The ﬁrst term on the right-hand side of equation (9.7.45) may be estimated, assuming that the single ion activity is equal to the mean ionic activity, which is known from studies of electrolyte solutions. Values of the liquid junction potential estimated using the Lewis–Sargent equation are compared with experimental estimates in table 9.6. For most of the systems studied the two quantities agree to within 1 mV. Thus, this equation, although it is approximate, gives acceptable results. It should be stressed that the Henderson model recognizes that the system is not at equilibrium, and instead assumes that it is in a steady state. In addition, it is not the only model which was developed to investigate liquid junctions. The design of the liquid junction is an important aspect of obtaining reproducible experimental results. More information about this aspect can be found in the monograph by Koryta and Sˇtulik [8].

9.8 Membrane Potentials and the Donnan Effect The electrical aspects of membrane phenomena have long been of interest in science, and attracted the attention in the nineteenth century of famous physical chemists including Gibbs, Nernst, Planck, and Ostwald. Originally, this interest was connected to electrical phenomena in biological systems. In the present discussion attention is focused on membranes used in speciﬁc ion electrodes. A membrane can be either a liquid or a solid. Its electrical properties arise when it allows transport of an ion of one charge but not that of another. Membranes are usually suﬃciently thick that one can distinguish an inside region and two outer boundary regions which are in contact with electrolyte solutions. Two types of membranes are considered here: (1) membranes of solid and glassy materials; (2) liquid membranes with dissolved ion-exchanging ions or neutral ion carriers (ionophores). In fact all of these membranes are involved in ion exchange. It is important to understand how this process aﬀects the potentials which develop in the system at both sides of the membrane. The membrane system may be represented as

CHARGE TRANSFER EQUILIBRIA AT INTERFACES

Electrolyte1 j L

Membrane

R j Electrolyte 2

485

ð9:8:1Þ

where L and R represent the regions just inside the membrane on the left and right, respectively. In solid membranes these regions are usually charged because they contain a local excess of positive or negative charges. For this reason they are called space charge regions. They correspond to the electrical double layer which is formed in liquid electrolyte solutions at interfaces such as those involved in system (9.8.1). The total potential drop across the membrane may now be divided up as follows: 1 2 f

¼ 1 L f þ L R f þ R 2 f

ð9:8:2Þ

where 1 L f is the Donnan potential associated with ion exchange on the lefthand side of the membrane, R 2 f, the Donnan potential associated with the same process on the right-hand side, and L R f, the diﬀusion potential due to the concentration gradient within the membrane. Each of these contributions must be considered in order to understand how the membrane is used in electroanalysis. The functioning of a membrane is illustrated for the case that it can exchange cations with solutions 1 and 2. On the other hand, anions cannot enter the membrane. Furthermore it is assumed that the solvent in the solutions, usually water, also does not enter the membrane. This situation is not always valid. Estimation of the membrane potential is more complex when the solvent enters the membrane due to osmotic eﬀects. The cation involved in membrane transport is designated Mþ and is assumed to be present in both solutions, so that the system can now be described as c1 MX; H2 O j L

Membrane

R j c2 MY; H2 O

ð9:8:3Þ

In general, the chemical potentials of the cation are not equal in a given solution and in the membrane. At equilibrium the electrochemical potentials of the cation must be equal in the two phases. As a result, a potential diﬀerence called the Donnan potential is established at each interface. Moreover, the concentration of the cation on the left-hand side of the membrane is not always the same as that on the right and the cation diﬀuses from the location of high concentration to the one where it is lower. The non-equilibrium diﬀusion process gives rises to a diﬀusion potential. Of course, there are many membranes into which anions may enter and cations are excluded. Some important examples will be discussed in the following presentation. It turns out that basic principles of the functioning of membranes in electroanalysis can be illustrated very well using the example of the glass membrane which is involved in the pH electrode. This is discussed in detail in the following section. Then, the functioning of liquid membranes is considered in the next section. A. Membranes of Solid and Glassy Materials These membranes function as ion-selective media when the cations and anions in the solid material have very diﬀerent conductivity. For example, in a crystal of lanthanium ﬂuoride (LaF3 ), the cation and anion have approximately the same conductivity because they have approximately the same size and are packed

486

LIQUIDS, SOLUTIONS, AND INTERFACES

together compactly in the crystal lattice. However, the mobility of F with respect to that of La3þ can be increased by doping LaF3 with a small amount of EuF3 . In the case of a glass membrane, the structure is determined essentially by a supercooled solution of silicon dioxide (SiO2 ). By doping SiO2 with sodium oxide (Na2 O), a small cation, namely, Naþ, is introduced into the disordered structure, which is made up of covalently bonded silicon and oxygen atoms. The mobile species is this type of glass is the Naþ cation. In the following discussion, a monovalent cation Mþ is assumed to be the mobile species. Initially, the simplest situation is considered in which the solutions on either side of the membrane contain only this cation, without any interfering cations. An equilibrium is established between cation Mþ in the solution and in the membrane, which can be described as + Mþ ðmembraneÞ Mþ ðsolutionÞ (

ð9:8:4Þ

Equating the electrochemical potentials of the cation in the solution and the membrane, one can write for the left-hand side of the membrane (see diagram 9.8.3), 1; L þ RT ln aMð1Þ þ Ff1 ¼ mL; mM M þ RT ln cMðLÞ þ Ff

ð9:8:5Þ

1; þ where mM and mL; M are the standard chemical potentials of the cation M in solution 1 and the left-hand side of the membrane, respectively, aMð1Þ , the activity of Mþ in the solution, cMðLÞ , its concentration in the membrane on the left side, and f1 and fL , the inner potentials in solution 1 and the left side of the membrane, respectively. Thus, the potential diﬀerence on the left side due to ion exchange is 1 L f

¼

RT RT aMð1Þ ln Kex ln Mþ F F cMðLÞ

ð9:8:6Þ

L; m1; M mM RT

ð9:8:7Þ

where ln Kex M ¼

The activity coeﬃcient of the cation within the membrane is assumed to be constant independent of location, so that it is not explicitly written in the equilibrium expression. Kex M is the equilibrium constant for the ion exchange process. This potential diﬀerence is called the Donnan potential. It arises essentially because the activities of the ion Mþ are not the same in the solution and in the membrane. A similar equation can be written for the ion exchange equilibrium on the right-hand side, namely 2; R; mM þ RT ln aMð2Þ þ Ff2 ¼ mM þ RT ln cMðRÞ þ FfR

ð9:8:8Þ

The corresponding Donnan potential is 2 R f

where

¼

RT RT aMð2Þ ln Kex ln Mþ F F cMðRÞ

ð9:8:9Þ

CHARGE TRANSFER EQUILIBRIA AT INTERFACES

ln Kex M ¼

2; R; ðmM mM Þ RT

487

ð9:8:10Þ

The ion exchange equilibrium constant is the same as that for the process on the left because the standard potentials in the solution and in the membrane are 1; L; R; independent of location (mM ¼ m2; M and mM ¼ mM Þ: The diﬀusion potential can be estimated on the basis of the Henderson equation (9.7.26). Since only one ion is moving in the membrane, the sums in this equation have only one term and the diﬀusion potential is uM cMðRÞ cMðRÞ RT uM ðcMðRÞ cMðLÞ Þ RT ln ln ¼ ð9:8:11Þ L R f ¼ F uM ðcMðRÞ cMðLÞ Þ F uM cMðLÞ cMðLÞ where uM is the mobility of ion M in the membrane. Now using equation (9.8.2) the total potential drop across the membrane is aMð1Þ aMð2Þ cMðRÞ RT RT RT RT RT ex ex ln K ln ln K ln ln f ¼ þ 1 2 M M F F F F F cMðLÞ cMðRÞ cMðLÞ aMð2Þ RT ln ¼ F aMð1Þ ð9:8:12Þ This result shows that the membrane potential is related to the ratio of the activities of cation Mþ in the two solutions in a very simple way. It illustrates how a properly chosen membrane can be used to relate the activity of the ion, which can be transported, on one side of the membrane to that on the other. The derivation presented here gives a greatly oversimpliﬁed picture of the actual situation in a membrane with ﬁxed ion exchange sites. Use of the Henderson equation implies that the diﬀusion process in the membrane has reached a steady state with a linear concentration distribution. This is certainly not the case for solid membranes such as glass which are thick with respect to the diﬀusion length of the ion. Moreover, it is probably not valid to assume that the ionic mobility is independent of position in the membrane. More complex models for the membrane potential have been developed but they lead to essentially the same result. More details can be found in monographs devoted to this subject [9]. The above analysis ignores the possibility that more than one cation in the test solution can exchange with cations in the membrane. This is usually not the case so that operation of an ion-selective electrode requires that the operator be aware of interference from other species. For example, the pH electrode also responds to Naþ ions so that a non-Nernstian response is obtained when the Hþ ion concentration is very low, that is, at high pH, and the Naþ ion concentration is high. The eﬀects of an interfering cation are now assessed for the case that there are two monovalent cations Mþ and Nþ in the test solution which can interact with the membrane and which have the same charge. The membrane system is now described as c1 MX; H2 O j L

Membrane

R j c2 MY; c3 NY; H2 O

ð9:8:13Þ

488

LIQUIDS, SOLUTIONS, AND INTERFACES

where the test solution (solution 2) now also contains the electrolyte NY at concentration c3 . The nature of the anions on this side of the membrane is not important except in as much as they aﬀect the activity of the cations Mþ and Nþ. The principal eﬀect of the interfering cation Nþ is that it may replace the ion to be determined, Mþ, from the membrane. The corresponding exchange selectivity equilibrium is Mþ(membrane) þ Nþ(solution) ( + Mþ(solution) þ Nþ(membrane) with an equilibrium constant Ksel equal to aMð2Þ cNðRÞ Ksel ¼ cMðRÞ aNð2Þ

(9.8.14)

ð9:8:15Þ

where aið2Þ is the activity of ion i in the solution and ciðRÞ , its concentration on the right-hand side of the membrane. In writing the equilibrium constant this way, the eﬀects of non-ideality within the membrane have been ignored. Alternatively, one could include the activity coeﬃcients, which are assumed to be constant because the membrane concentrations are small. Thus, in writing equation (9.8.15) the activity coeﬃcients for the ions in the membranes are included in the deﬁnition of Ksel . Now it is assumed that there are a ﬁxed number of cation sites within the membrane on the right-hand side so that one can deﬁne a total concentration cTðRÞ such that cMðRÞ þ cNðRÞ ¼ cTðRÞ Substituting for cNðRÞ from equation (9.8.15), one obtains cMðRÞ aNð2Þ ¼ cTðRÞ cMðRÞ þ Ksel aMð2Þ

ð9:8:16Þ

ð9:8:17Þ

so that cMðRÞ ¼

aMð2Þ cTðRÞ aMð2Þ þ Ksel aNð2Þ

ð9:8:18Þ

The corresponding expression for cNðRÞ is cNðRÞ ¼

Ksel aNð2Þ cTðRÞ aMð2Þ þ Ksel aNð2Þ

ð9:8:19Þ

The expressions for the Donnan potentials at each side of the membrane may now be written. The Donnan potential on the left-hand side, 1 L f, is given by equation (9.8.6) because the concentration of the interfering ion Nþ is zero at this side of the membrane. On the other hand, the Donnan potential on the right-hand side is given by aMð2Þ þ Ksel aNð2Þ RT RT ex ln K ln ð9:8:20Þ f ¼ þ 2 R M F F cTðRÞ The diﬀusion potential must be estimated considering transport of both ions through the membrane. The interfering ion Nþ diﬀuses from the right-hand side where its concentration is cNðRÞ , to the left-hand side where it is zero. The situation

CHARGE TRANSFER EQUILIBRIA AT INTERFACES

489

for the ion being determined, Mþ, is the same as it was in the absence of the interferant. From the Henderson equation one obtains uM cMðRÞ þ uN cNðRÞ RT uM cMðRÞ uM cMðLÞ þ uN cNðRÞ ð9:8:21Þ f ¼ ln L R F uM cMðRÞ uM cMðLÞ þ uN cNðRÞ uM cMðLÞ Only ions of the same charge are involved in diﬀusion so this simpliﬁes to cMðRÞ þ uN cNðRÞ =uM RT ln ð9:8:22Þ L R f ¼ F cMðLÞ Now equations (9.8.18) and (9.8.19) are used to relate cMðRÞ and cNðRÞ to the total concentration of cation sites on the right-hand side of the membrane. After substituting these expressions and simplifying, one obtains ! aMð2Þ cTðRÞ þ uN Ksel aNð2Þ cTðRÞ =uM RT ln ð9:8:23Þ L R f ¼ F cMðLÞ ðaMð2Þ þ Ksel aNð2Þ Þ The total potential drop across the membrane is now estimated by adding the two Donnan potentials and the diﬀusion potential. The result is aMð2Þ þ Ksel aNð2Þ RT RT aMð1Þ RT RT ex ex ln KM þ ln ln KM ln 1 2 f ¼ cMðLÞ cTðRÞ F F F F ! aMð2Þ cTðRÞ þ uN Ksel aNð2Þ cTðRÞ =uM RT ln ð9:8:24Þ F cMðLÞ ðaMð2Þ þ Ksel aNð2Þ Þ This simpliﬁes to

aMð2Þ þ uN Ksel aNð2Þ =uM RT ln 1 2 f ¼ aMð1Þ F

ð9:8:25Þ

The parameters determining the response of the membrane to the interfering ion are the equilibrium constant Ksel for the ion exchange process (9.8.14) and the mobilities of the two ions uM and uN . Since these quantities are all assumed to be constants, equation (9.8.25) can be further simpliﬁed to ! 0 aMð2Þ þ Ksel aNð2Þ RT ln ð9:8:26Þ 1 2 f ¼ F aMð1Þ where 0

Ksel ¼

uM Ksel uN

ð9:8:27Þ

Equation (9.8.26) is known as the Nikolsky equation [10]. More complex versions of this equation have been derived for the case that the interfering ion has a diﬀerent charge than the ion being determined, or there is more than one interfering ion. The eﬀect of the interfering ion on the membrane potential is illustrated in ﬁg. 9.5. The membrane potential is estimated for a constant activity of Mþ on the lefthand side ðaMð1Þ ¼ 0:1 MÞ and constant activity of the interfering ion Nþ on the

490

LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 9.5 Plot of the membrane potential 1 2 for a system responding to monovalent cation Mþ against the logarithm of the cation’s activity log10 aM for various activities of an 0 interfering cation Nþ with a selectivity constant Ksel of 10-7: (*) aN ¼ 0.001; (&) 0.01; and (~) 0.1 M.

right-hand side ðaNð2Þ ¼ 0:001; 0:01, and 0.1 M) with a typical selectivity con0 stant (Ksel ¼ 10-7 ). The activity of Mþ on the right-hand side is varied over a to the ratio of wide range. When aMð2Þ is high, the membrane potential responds 0 Mþ activities on its two sides in a Nernstian fashion because Ksel aNð2Þ << aM ð2Þ. The membrane potential is approximately given by equation (9.8.12) and it can be used in an analytical application to determine the activity of Mþ. However, when 0 aMð2Þ is very low so that aM ð2Þ << Ksel aNð2Þ , the membrane potential is constant and given by ! 0 Ksel aNð2Þ RT ln ð9:8:28Þ 1 2 f ﬃ F aMð1Þ Provided aNð2Þ and aMð1Þ are known, the0 value of 1 2 f in this region can be used to determine the selectivity constant Ksel for the interfering ion Nþ. When the mobile ion in the solid membrane is an anion, the basic equations giving the membrane potential diﬀer with respect to the sign of the contributing terms. The equilibrium giving rise to the Donnan potential is now + X (membrane) X (solution) (

ð9:8:29Þ

The equations derived above apply to this system with the charge now being –1 for a monovalent anion. This aﬀects the potential terms in equation (9.8.5) and those which follow. The ﬁnal result for the potential drop across the membrane when a monovalent anion is the mobile species is 1 2 f

¼

RT aXð2Þ ln F aXð1Þ

ð9:8:30Þ

CHARGE TRANSFER EQUILIBRIA AT INTERFACES

491

There are other types of solid membranes which are used in ion-selective electrodes. An important group are those based on silver halides and sulﬁdes. In these systems the ionic conductor is Agþ. These membranes function in the same manner as other ionic solids in which one ion has a very diﬀerent conductivity than the other. Thus, the above analysis is also applicable to these systems. B. Liquid Membranes There are two important types of liquid membranes used in analytical applications. One type involves an ion-exchanging system dissolved in a hydrophobic solvent, usually of low permittivity. The other type makes use of an ionophore or neutral complexing agent dissolved in a similar solvent. In both systems, Donnan potentials are established on either side of the membrane but a diﬀusion potential is absent because of the mobility of the solute in the liquid phase. More details about the functioning of these membranes are given in this section. Examples of ions and ionophores which can be used in liquid membranes are shown in table 9.7. These species all have a strong hydrophobic character and can be used to design membranes which respond to either cations or anions. By using Table 9.7 Ions and Ionophores Used in Liquid Junction Membranes Formula

Primary Determinand

(a) Ion-exchanging ions Ca2þ

ClO 4

NO 3

(b) Ionophores

Kþ

Mg2þ

Ba2þ

492

LIQUIDS, SOLUTIONS, AND INTERFACES

liquid membranes, ion-speciﬁc electrodes responding to a wide variety of ions become possible. The ion-exchanging system consists of an electrolyte MA dissolved in a hydrophobic solvent S. In the example considered here, the cation Mþ is also present in the aqueous solutions on either side of the membrane. On the other hand, the anion A is very hydrophobic, so that its concentration in the aqueous phases is very small. Thus the membrane system responding to cation Mþ can be described as c1 MX; H2 O j cM MA; S j c2 MY; H2 O

ð9:8:31Þ

where c1 and c2 are the electrolyte concentrations in the aqueous phases, and cM , the electrolyte concentration in the membrane. Donnan potentials are established at each interface as a result of the exchange of cation Mþ between the aqueous and membrane phases. The distribution equilibrium involving the electrolyte MA in the membrane phase is MA (in membrane) ( + MA (in aqueous solution)

(9.8.32)

From the condition of equilibrium m m w; w w mm; MA þ RT ln aM aA ¼ mMA þ RT ln aM aA

ð9:8:33Þ

where aiðmÞ is the activity of ion i in the membrane, and ai (w), its activity in water. As a result, one may deﬁne the transfer coeﬃcient from the membrane to water as aMðwÞ aAðwÞ Ktr ð9:8:34Þ MA ¼ aMðmÞ aAðmÞ where

Ktr MA ¼ exp

w; mm; MA mMA RT

ð9:8:35Þ

Because of the hydrophobic character of the anion A , Ktr MA is very small, and w; << mMA . The Donnan potentials are established as a result of exchange of monovalent cations Mþ between the membrane and the aqueous phases. If the activity of Mþ is constant and fairly high on the left-hand side of the membrane then the corresponding Donnan potential can be written as m; mMA

1 m f

¼

RT RT aMð1Þ ln Kex ln Mþ F F aMðmÞ

ð9:8:36Þ

where 1; m; RT ln Kex M ¼ mM m M

ð9:8:37Þ

þ Kex M is the exchange equilibrium constant for the cation M and the other symbols have their usual meanings. In writing the equation this way it is assumed that there is no diﬀusion potential, the activity of Mþ being uniform throughout the membrane. The corresponding equation on the right-hand side is

CHARGE TRANSFER EQUILIBRIA AT INTERFACES 2 m f

¼

RT RT aMð2Þ ln Kex ln Mþ F F aMðmÞ

493

ð9:8:38Þ

where 2; m; RT ln Kex M ¼ mM mM

ð9:8:39Þ

2; However, the standard potentials for the cation, m1; M and mM , are equal since both solutions are aqueous. As a result the exchange constant Kex M is the same on both sides of the membrane. Now equation (9.8.38) is subtracted from (9.8.36) to obtain the potential drop across the membrane, 1 2 f

¼

RT aMð1Þ ln F aMð2Þ

ð9:8:40Þ

This simple result shows that the membrane potential is directly related to the logarithm of the ratio of the activities of Mþ in the two aqueous solutions. Liquid membranes with an ionophore L function via complex formation between the ionophore and an ion in the aqueous phase. When this ion is a monovalent cation Mþ the membrane phase contains the cation in the form MLþ, that is, completely complexed with the ionophore. The other components of the membrane are a hydrophobic solvent S which constitutes the majority of the liquid phase and a hydrophobic anion A . The concentration of free cation Mþ within the membrane is very small but must be considered in assessing the Donnan equilibria on each side. The description of the membrane system is c1 MX; H2 O j cM MLþ ; A ; S j c2 MY; H2 O

ð9:8:41Þ

The complexation equilibrium is Mþ ðmembraneÞ þ LðmembraneÞ ( + MLþ ðmembraneÞ

ð9:8:42Þ

with a stability constant KML given by KML ¼

aMLðmÞ aMðmÞ aLðmÞ

ð9:8:43Þ

KML is very high by design so that the activity of free Mþ within the membrane is extremely small. The system is often designed so that the activity of free ionophore L is also very small. The equilibrium determining the Donnan potential is given by equation (9.8.4). Assuming that there is no diﬀusion potential in the membrane, the Donnan potentials on either side of the membrane are given by the same equations obtained for the ion-exchanging system, namely, equations (9.8.36) and (9.8.38). In addition, the expression for the membrane potential is the simple result given by equation (9.8.40). Interfering ions can also aﬀect the membrane potential for liquid membranes. If it is assumed that solution 2 contains cation Nþ which can exchange with the cation Mþ in the liquid membrane, the system becomes c1 MX; H2 O j cM MA; S j c2 MY; c3 NY; H2 O

ð9:8:44Þ

Ion exchange equilibrium (9.8.14) is established at the second boundary. An expression for the membrane potential is easily derived using the procedure set

494

LIQUIDS, SOLUTIONS, AND INTERFACES

out above for a solid membrane. However, in the case of a liquid membrane the diﬀusion potential within the membrane can be neglected because the mobilities of Mþ and Nþ are the same to a good approximation. The expression for the Donnan potential between solution 1 and the membrane is still given by equation (9.8.36). On the basis of the analysis given for solid-state membranes, the Donnan potential between solution 2 and the membrane is aMð2Þ þ Ksel aNð2Þ RT RT ex ln KM þ ln ð9:8:45Þ 2 m f ¼ F F aMðmÞ Thus, the membrane potential is 1 2 f ¼

aMð1Þ RT ln F aMð2Þ þ Ksel aNð2Þ

ð9:8:46Þ

This has the same form as the Nikolsky equation (9.8.26) derived for solid-state membranes. The simpliﬁed analysis presented here fails for ion-exchange systems when the activity of the exchanging ion, Mþ in the test solution, falls to very low values. The nature of the problem is easily understood on the basis of distribution equilibrium (9.8.32). The activity of the electrolyte MA which is in the aqueous phase is very small and can be neglected under most circumstances. However, when the activity of Mþ originally present in the aqueous solution is low and comparable to that which comes from the membrane, then both sources of Mþ aﬀect the Donnan potential at that interface. In other words, the ability of the membrane to respond in a linear way to the logarithm of the activity of Mþ in a given solution is limited tr by the value of Ktr MA . The smaller the value of KMA , the lower the range of linear response observed experimentally. When the activity of Mþ originally present in the aqueous solution falls below that which comes from the membrane, the Donnan potential is approximately constant reﬂecting the contribution from the distribution equilibrium. More details about this limitation of ion-exchanging liquid membranes and other problems can be found in the monograph by Koryta and Sˇtulik [8].

9.9 Ion-Selective Electrodes An ion-selective electrode (ISE) is a sensor with a membrane which is designed so that its potential indicates the activity of a speciﬁc ion in an electrolyte solution. The membrane may be a solid, either crystalline or glassy, or a liquid. The best known ISE is the glass electrode used to determine pH. The membrane not only develops a potential diﬀerence which responds to the unknown ionic activity in the test solution, but it also separates completely the test solution from the internal reference solution of the ISE. Three diﬀerent ISEs involving glass, crystalline, and liquid membranes are shown in ﬁg. 9.6. All of these systems involve an internal reference electrode. In each case, the membrane is at the bottom of the ISE, which is dipped into the test solution. More details about each of these electrodes are given in the following discussion.

CHARGE TRANSFER EQUILIBRIA AT INTERFACES

495

Fig. 9.6 Illustration of the structures of three ion selective electrodes (left to right): a glass electrode; an electrode with a crystalline membrane; and a liquid membrane electrode.

An ISE usually is constructed with an internal silver | silver chloride reference electrode. This means that the internal reference solution must contain the Cl ion together with the ion for which the membrane is selective, for example, Mþ. Thus, a typical format for the ISE is Ag j AgCl j a1 MCl; H2 O j membrane j test solution; a2 Mþ

ð9:9:1Þ

The potential of the silver | silver chloride electrode is EAg;AgCl ¼ EAg;AgCl

RT ln aCl ð1Þ F

ð9:9:2Þ

Since the internal reference solution is sealed within the ISE and cannot be changed, the potential of the silver | silver chloride electrode is constant, depending only on the activity of Cl in the internal reference solution (solution 1) and on temperature. The potential drop across the membrane is 1 2 f

¼

RT aMþ ð1Þ ln aMþ ð2Þ F

ð9:9:3Þ

Since aMþ ð1Þ is constant, the membrane potential depends only on the activity of the speciﬁc ion Mþ in the test solution. The potential of the ISE is EISE ¼ EAg;AgCl þ1 2 f ¼ EAg;AgCl

RT aCl ð1Þ aMþ ð1Þ ln F aMþ ð2Þ

ð9:9:4Þ

This can also be written as EISE ¼ EISE þ

RT ln aMþ ð2Þ F

ð9:9:5Þ

where EISE ¼ EAg;AgCl

RT ln aCl ð1Þ aMþ ð1Þ F

and aMþ ð2Þ is the activity of Mþ in the test solution (solution 2).

ð9:9:6Þ

496

LIQUIDS, SOLUTIONS, AND INTERFACES

In the case that the membrane responds to an anion X , the internal reference solution must contain that anion. As discussed in section 9.8, the sign of the membrane potential is then reversed. Extending consideration to both cations and anions of all charges, equation (9.9.5) can be written as EISE ¼ EISE þ

RT ln aið2Þ zi F

ð9:9:7Þ

where aið2Þ is the activity of the ion to which the membrane is selective in the test solution, and zi , its charge. The cell is completed with a reference electrode, for example, a saturated calomel electrode (SCE). The observed cell potential is then E ¼ EISE ESCE þ flj ¼ EISE þ

RT ln aið2Þ ESCE þ flj zi F

ð9:9:8Þ

The choice of the SCE ensures that the liquid junction potential between the ISE and the reference electrode is small, so that variation in E approximately follows variation in the logarithm of the activity of the selected ion in the test solution. Interfering ions cause problems with all ISEs. Usually these arise when the concentration of the analyte ion is very low and that of the interfering ion, very high. The expression for the cell potential in the presence of an interfering ion j is RT

ðz =z Þ ln ai þ Ksel aj i j ESCE þ flj E ¼ EISE þ ð9:9:9Þ zi F where Ksel is the selectivity coeﬃcient, aj , the activity of the interfering ion, and zi and zj , the valences of the analyte and interfering ions, respectively. This equation is clearly a generalized form of the result obtained earlier in section 9.8 (equation (9.8.20)). In many cases the valence of the interfering ion is the same as that of the analyte ion so that the exponent associated with the activity aj reduces to unity. It is important to characterize a given ISE with respect to interfering ions by determining the value of Ksel . A common method of doing this is to determine the value of the cell potential for a ﬁxed activity of the interfering ion but varying activity of the analyte ion. When the activity ai is the dominant term in the logarithm in equation (9.9.9), the cell potential is given by E ¼ EISE þ

RT ln ai ESCE þ flj zi F

ð9:9:10Þ

Thus a plot of E against ln ai is linear with a slope equal to RT=ðzi FÞ, provided that the liquid junction potential is constant. On the other hand, when ai falls to very low values so that the term in aj dominates, the cell potential is RT

ðz =z Þ ln Ksel aj i j ESCE þ flj E ¼ EISE þ ð9:9:11Þ zi F Under these conditions, E is a constant independent of ln ai . If one extrapolates the two linear sections to the point where they meet, one can estimate Ksel (see ﬁg. 9.7). If the value of ai at the intersection point is aix , then

CHARGE TRANSFER EQUILIBRIA AT INTERFACES

497

Fig. 9.7 Plot of E E against log10 ai for an ISE with an interfering ion activity of 0.1 and Ksel of 10–5. The straight lines through the linear sections of the curve meet at ai ¼ 10–6.

Ksel ¼

aix ðzi =zj Þ aj

ð9:9:12Þ

ISEs can function with interfering ions provided that the value of Ksel is less than 10–3. Successful operation of an ISE requires that the operator be aware of the properties of the membrane used in the electrode and of its limitations. Some important membranes used in these systems are now outlined. A. Glass Electrodes The glass electrode for determining pH is the best known and most widely used ISE. It was discovered in the late nineteenth century that a thin layer of glass which contains a relatively high Naþ content develops a membrane potential in response to diﬀerences in Hþ activity on either side. Further study of the properties of sodium glass revealed that an optimum composition consisted of 22 wt% Na2 O, 6 wt % CaO, and 72 wt % SiO2 . The structure of this system is determined to a large extent by the structure of a supercooled solution of SiO2 . By doping the SiO2 with sodium oxide a small cation is introduced into the disordered system which is made up predominately by covalently bonded silicon and oxygen atoms. The small conductivity of this membrane is due to movement of the Naþ ion (see ﬁg. 9.8). When the glass is immersed in water, a very thin layer (up to 100 nm) at the surface becomes hydrated. The hydration process involves the exchange of Naþ in the glass with protons in solution: þ þ þ Hþ soln þ Na Gl ¼ Nasoln þ H Gl

ð9:9:13Þ

498

LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 9.8 Cross-sectional view of a glass membrane containing two diﬀerent cations (shaded circles). Each Si atom (black) is bonded to four oxygen atoms (clear), three of which are in the same plane, with an additional oxygen atom above or below it.

The equilibrium constant is extremely large so that the mobile ions in the hydrated layer are essentially protons except at very high pH. Formation of this layer is essential for most pH electrodes if they are to function well. For this reason these electrodes are kept in distilled water when not in use. The membrane potential of a glass electrode is a result of an ion exchange process between protons in the solution and those at cation sites in the hydrated layer. Although protons are the mobile species in the hydrated layer, charge is carried by Naþ ions in the dry interior. The response to the pH of the test solution is linear for most systems between 2 and 12. In strongly acid solutions a positive deviation is observed and in strongly alkaline solutions the observed pH is too low. The exact extent of the deviations depend on the composition of the glass and to some extent on its history. The alkaline error is due to the fact that the system responds to alkali metal ions, each of which has a diﬀerent Ksel . When the Hþ ion activity is very low, the contribution from the alkali metal cations becomes signiﬁcant and the membrane response suggests that aHþ is higher than it really is. The response of the glass at high pH can be improved by modifying its composition, for example, by adding Li2 O. The electrode error at very low pH is much less well understood than that at very high pH. By altering the composition of the glass one can fabricate ISEs which are speciﬁc to diﬀerent alkali metal cations. The best known of these is the Naþ

CHARGE TRANSFER EQUILIBRIA AT INTERFACES

499

ISE. The glass membrane for this electrode has a composition of 11 wt % Na2 O, 18 wt % Al2 O3 and 71 wt% SiO2 . This system gives a linear response for Naþ activities down to 10–5 M. More information about these systems can be found elsewhere [8]. B. Crystalline Electrodes Crystalline membranes function by virtue of the fact that one ion in an ionic crystal may have signiﬁcant conductivity with respect to the others. These can be systems in which this ion is smaller than the others and can occupy interstitial positions in the crystal lattice. The best known ISE in this category is the F electrode in which the membrane is a single crystal of LaF3 . The pure crystal has a very low conductivity due to its regular structure. However, the conductivity of the crystals used in ISEs is increased by doping the crystal with a small amount of EuF3 . With 0.5 wt% EuF3 , the crystal structure is suﬃciently distorted that the doped crystal has a conductivity of 2 10-6 -1 cm-1. This system has a Nernstian response for F ion concentrations down to 10-6 M. It is widely used in the analysis of natural waters for ﬂuoride. The most important interferant is the OH ion, so the electrode cannot be used in basic solutions. The O¼ and OH ions are only slightly larger than F , so that they can enter the LaF3 crystal and eventually destroy the ISE. The OH ion is the only serious interferant for the F electrode. The construction of an ISE with a crystalline membrane is shown in ﬁg. 9.6(b). The membrane is located at the bottom of the electrode, where it comes into contact with the test solution. The solution in the reference compartment inside the electrode contains both the Cl ion, which establishes the potential of the Ag | AgCl reference electrode, and the ion to which the membrane is responding. For example, in the case of the F ISE, it also contains the ﬂuoride anion. Crystalline membranes for ISEs are also constructed with silver halides and sulﬁde. The mobile species in these materials is the Agþ cation. The ISE is usually designed to detect the anion in the silver salt. Thus, an ISE for the Cl anion makes use of AgCl in the membrane. The membrane is polycrystalline and is formed from precipitated AgCl. The functioning of these systems as anion indicators is more complicated than the F electrode because of the fact that the Agþ ion is the mobile species. In essence, the Donnan potential at the test solution interface is controlled by the activity of Agþ in the membrane and the solubility product of the silver halide salt. These systems are widely used for determining Cl , Br , I , and S¼ anions. ISEs for determining Pb2þ and Cd2þ are also made by mixing PbS or CdS, respectively with Ag2 S [8]. C. Electrodes with Liquid Membranes Commercial ISEs are made with liquid membranes involving both ion-exchanging systems and ionophores (see table 9.7). The solvent used is a hydrophobic liquid with a low relative permittivity. Examples include decane-1-ol, 5-phenyl pentan-2ol, octyl phtalate, tri-n-phenyl phosphate, and 2-nitro-p-cymene. Another important property of the solvent is that it have a low vapor pressure so that it is not

500

LIQUIDS, SOLUTIONS, AND INTERFACES

lost by evaporation. The electrode is constructed with a porous plastic membrane at the bottom (ﬁg. 9.6(c)). This becomes ﬁlled with the solution of the ion exchanger or ionophore in the hydrophobic solvent. The porous plastic is positioned such that it contacts the internal reference solution inside the ISE on one side and the external test solution on the other side. The internal reference solution contains an electrolyte in which there is both Cl for the Ag | AgCl reference electrode and the ion for which the ISE is selective. The most popular ISEs using liquid membranes are those for Ca2þ, Kþ, NO 3, 2þ . Ca ISEs are made with both ion-exchangers and with ionophores. and ClO 4 The alkyl phosphate anion shown in table 9.7 is an example of an ion-exchanging system. By using a long alkyl chain on the phosphate its solubility in water becomes extremely low. Two phosphate anions complex with one Ca2þ cation to form calcium dialkyl phosphate, which is soluble in the liquid membrane. This system gives a linear response to Ca2þ ion activity down to 10-6 M. The principal interfering ions are Pb2þ and other divalent cations. This ISE is often used to determine Ca2þ ion in blood serum and other biomedical applications. ISEs designed for polyatomic anions such as NO 3 and ClO4 make use of strongly hydrophobic cations as ion exchangers. Typical examples of suitable cations are shown in table 9.7. The nitrate electrode responds to other anions, especially perchlorate. It is widely used to test for nitrates in soil samples and agricultural products. A wide variety of ISEs for alkali metals and alkaline earth metals are available which make use of ionophores. The best known of these is the Kþ ISE. These systems have varying degrees of selectivity with respect to cations of the same group. Thus, the Kþ ISE is usually very selective with respect to Naþ but is less selective toward Csþ. Since this ISE is most often used in clinical applications, its poor selectivity toward alkali metal cations of higher atomic mass is not important because these cations are not typical components of body ﬂuids. Because the membranes used in these systems contain a liquid, greater care must be exercised in storing the ISE between experiments to prevent loss of the membrane system from the porous plastic in which it contacts the test solution. D. Composite Electrode Sensors Composite electrode sensors are based on an ISE which is separated from the test solution by another membrane. The second membrane either selectively separates a certain component of the analyte or modiﬁes it by a suitable reaction. The bestknown composite electrode sensors are those designed for speciﬁc gases, and with biomolecules, especially enzymes. The operation of a gas-sensing system and an enzyme electrode are now described brieﬂy. An Hþ ISE may be designed to detect gaseous carbon dioxide when it is isolated from the test solution by a second membrane. Operation of the sensor is based on the reaction + Hþ þ HCO CO2 þ H2 O ( 3

ð9:9:13Þ

which involves the formation of both Hþ and the hydrogen carbonate ion HCO 3. The gas-permeable membrane at the outer end of the sensor system is a thin

CHARGE TRANSFER EQUILIBRIA AT INTERFACES

501

microporous ﬁlm fabricated from a hydrophobic plastic. The pores of this ﬁlm allow passage of gas molecules but are repellant to water molecules and electrolytes. The solution inside the sensor contains a high concentration of the HCO 3 ion. As a result of equilibrium (9.9.13), there is a change in Hþ activity inside the sensor but a negligible change in the activity of the hydrogen carbonate ion. Thus, using equation (9.9.13) one may write aHþ ¼ Keq

aCO2 aHCO3

ð9:9:14Þ

where Keq is the equilibrium constant, and ai , the activity of species i. On the basis of equation (9.9.10) for an Hþ ISE E ¼ const þ

Keq RT RT RT ln aHþ ¼ const þ ln ln aCO2 þ F F F aHCO3

ð9:9:15Þ

Since the term involving the HCO 3 ion activity is approximately constant, the expression for the cell potential becomes E ﬃ const0 þ

RT ln aCO2 F

ð9:9:16Þ

where const0 includes all constant terms in equation (9.9.15). This result shows that the system responds to the CO2 activity when designed with the appropriate gas-permeable membrane. A variety of other gas-detecting sensors are also available. Sensors for NH3 and SO2 are similar to the CO2 system in that they make use of an Hþ ISE which responds to the acid–base reaction between the gas molecule and water. Other gas-sensing systems use other ISEs. For example, a sensor for H2 S can be based on the S¼ ISE. The sulﬁde anion is produced when H2 S dissolves in water by an acid-base reaction. The S¼ ISE then responds to the resulting activity of the sulﬁde anion. Another important category of composite electrode sensors is that of enzyme electrodes. In these systems, the analyte is brought into contact with an enzyme immobilized on the surface of the sensor. The analyte then undergoes a catalytic reaction to yield a species for which an ISE is sensitive, for example, ammonia, carbon dioxide, or Hþ ions. The best-known enzyme electrode is that used to analyze for urea in blood. The enzyme urease is immobilized in a polyacrylamide hydrophilic gel and ﬁxed at the bottom of a glass electrode whose characteristics make it an NHþ 4 ISE. Alternatively, the ISE can be a composite system designed to detect NH3 . In the presence of the enzyme, urea is hydrolyzed according to the reaction ðNH3 Þ2 CO þ 2H2 O þ Hþ ! 2NHþ 4 þ HCO3

ð9:9:17Þ

The NHþ 4 produced is then detected by either of the ISE systems mentioned þ above. Some problems exist for the NHþ 4 ISE because of interference by Na þ and K ions which are present in blood. Composite electrode sensors are an active area of research at the present time. The examples discussed give a general idea of the modiﬁcations required to

502

LIQUIDS, SOLUTIONS, AND INTERFACES

develop a composite ISE. Obviously, this type of development extends the range of substances which can be detected by ISEs in a signiﬁcant way.

9.10 p-Functions and the Deﬁnition of pH p-Functions are familiar ways of expressing very small quantities in chemistry. The best known p-function is the pH. The deﬁnition of pH is pH ¼ log10 aHþ

ð9:10:1Þ

þ

where aHþ is the activity of the H ion. Although pH is often thought of in terms of Hþ ion concentration, the exact deﬁnition becomes important in systems of higher ionic strength. Thus, the pH of a 10-4 M HCl solution is very close to 4.00. Since the ionic strength of the solution is also 10-4 M, the activity coeﬃcients of the ions are very close to unity. Thus, the Hþ ion activity is very close to its concentration. However, if the ionic strength of the system is increased by adding an electrolyte such as NaCl, the Hþ ion activity falls below its concentration. As a result, the pH of such a solution is greater than 4.0. p-Functions are used to express other ion activities. pCl refers to log10 aCl and pNa, to log10 aNaþ . They are also used to express equilibrium constants. Thus pKa ¼ log10 Ka

ð9:10:2Þ

Since logarithms are given to the base 10 it is easy to convert the p-function mentally to the actual activity to obtain an order of magnitude estimate. Since p-functions refer to the activity of one ion in an electrolyte solution and this solution contains at least two ions, there is a fundamental problem with its experimental measurement. Consider for example the measurement of the pH of an HCl solution using the cell glass electrode | x HCl, H2 O || SCE

(9.10.3)

The potential of this cell can be written as E ¼ Eglass ESCE þ

RT ln aHþ þ flj F

ð9:10:4Þ

where flj is the liquid junction potential between the test solution containing HCl and the SCE. The latter quantity cannot be deﬁned in an exact thermodynamic way, even though the liquid junction was chosen to keep flj as small as possible. Another problem with this system is the asymmetry potential which sometimes arises with the glass membrane. For example, if the bulb-shaped membrane was formed by glassblowing, the properties of the hydrated layer inside the electrode could be diﬀerent than those on the outside. This diﬀerence leads to a small potential drop across the membrane even when the activities of the Hþ ion are the same on both sides. Thus, the cell deﬁned by (9.10.3) cannot be used to determine pH without calibration using solutions of known pH. The problem can be examined in another way using a cell whose potential is exactly deﬁned by thermodynamics, namely the system

CHARGE TRANSFER EQUILIBRIA AT INTERFACES

Pt; H2 ð1 atmÞ j xHCl; H2 O j AgCl j Ag

503

ð9:10:5Þ

The potential of this cell is given by E ¼ ESHE EAg=AgCl þ

RT ln aHþ aCl F

ð9:10:6Þ

Although the concentrations of the Hþ and Cl ions are exactly equal, their activities are not necessarily equal. Furthermore, there is no way to determine the individual ion activities separately. As discussed earlier in this chapter, thermodynamic data for this cell are analyzed in terms of the mean ionic activity a and mean ionic activity coeﬃcient y . In order to deal with the practical problem of pH measurement, standard buﬀer solutions are used to calibrate the pH meter. These solutions have been carefully studied by the National Institute of Standards and Technology (NIST) in the USA using electrochemical cells designed without liquid junctions as much as possible. In this way, a very good estimate of the ‘‘true value’’ of the pH could be made. Selected results for three buﬀer solutions in the temperature range 5–45 C are given in table 9.8. In order to make pH measurements using cell (9.10.3), the operator ﬁrst calibrates the instrument using standard buﬀers which bracket the pH range where experimental results are expected. In this way, the pH measurements obtained using diﬀerent glass electrodes and diﬀerent reference electrodes are comparable. The experimental problem in deﬁning the pH of an aqueous solution exists for all p-functions related to ionic activity. The fundamental reason for the problem is that individual ionic activity coeﬃcients cannot be measured experimentally. A variety of extra thermodynamic assumptions are used to circumvent this problem. As might be expected, the exact nature of the assumption depends on the nature of the ion and the way it interacts with the solvent, which in most cases is water. Finally, the distinction between pH deﬁned in terms of Hþ ion activity and Hþ ion concentration is often ignored in practice. This is especially true in acid–base titrations. If a calibrated pH meter is used to carry out an acid–base titration, the

Table 9.8 Selected Values of NIST Standard Buffer Solutions in the Temperature Range 5–45 C pH

Temperature

0.05 m KH2 citrate

0.025 m KH2 PO4 + 0.025 m Na2 HPO4

5 15 25 35 45

3.840 3.802 3.776 3.759 3.750

6.951 6.900 6.865 6.844 6.834

0.025 m NaHCO3 + 0.025 m Na2 CO3 10.245 10.118 10.012 9.925 9.856

504

LIQUIDS, SOLUTIONS, AND INTERFACES

observed pH is the exact value. However, this distinction is usually ignored in calculating a titration curve. Keeping in mind the fact that the pH is varying over a wide range in a titration, neglect of the diﬀerence between the pH deﬁned in terms of activity and concentration has a negligible eﬀect on determination of the equivalence point. Thus, the features of the titration curve are much more easily determined, ignoring any small variation in activity coeﬃcients during the titration. This aspect of titrations and other problems associated with pH determination have been discussed in detail by de Levie [11].

9.11 Concluding Remarks The material presented in this chapter has focused on two types of electrochemical equilibria, namely, those at the electrodes in an electrochemical cell, and those at membrane interfaces. When the EMF of an electrochemical cell is exactly balanced by an opposing voltage source, one can measure the Gibbs energy change associated with the cell reaction. By studying the EMF as a function of temperature and pressure, one can measure the corresponding entropy and volume changes. Thus, one can determine all of the basic thermodynamic properties of the cell reaction on the basis of fairly simple experiments which can be carried out in a highly precise fashion. This type of study provided a signiﬁcant fraction of the activity coeﬃcient data for aqueous electrolyte solutions. Studies of the properties of electrochemical cells have also provided data for a variety of solution equilibria, including the dissociation constants of weak acids and bases, and solubility product data for sparingly soluble electrolytes. Much of the work in this area was carried out in the ﬁrst half of the twentieth century. The other important subject discussed in this chapter is ionic equilibria at membranes. The focus of this discussion was the membranes involved in ionselective electrodes. A variety of systems were considered, including both solidstate and liquid membranes. The study of ionic equilibria at membranes is very important for biological systems. Although biological membranes are often more complex structurally, the basic phenomena described here apply there as well. The thermodynamic analyses used in this chapter make use of the electrochemical potential. In this way the electrical aspects of the interfacial equilibria are clearly deﬁned. Earlier work on this problem, especially that by Volta and Nernst, had led to diﬀerent conclusions regarding the source of the EMF in an electrochemical cell [12]. This problem was resolved by Frumkin, essentially, by writing out the interfacial equilibria using electrochemical potentials. In this regard, all interfaces in the cell must be considered including those between different metals at the terminals of the cell. This was shown in the discussion of the thermodynamic basis of the Nernst equation. Finally, the systems discussed in this chapter have provided an excellent way of demonstrating how important the electrical aspects of equilibria involving charged species are. Modern research in this area involves the study of both the thermodynamic and kinetic aspects of electrochemical reactions. This is especially true for fuel cells, a very important area of contemporary research. In these systems the electrode reactions are complex and the kinetics of the individual

CHARGE TRANSFER EQUILIBRIA AT INTERFACES

505

steps are extremely important. Study of these systems requires an understanding of electrocatalysis and the mechanistic aspects of complex electrode reactions. General References G1. Newman, J. S. Electrochemical Systems; Prentice-Hall: Englewood Cliﬀs, NJ, 1973. G2. Laitinen, H. A.; Harris, W. E. Chemical Analysis, An Advanced Text and Reference, 2nd ed.; McGraw-Hill: New York, 1975. G3. NBS Tables of Chemical Thermodynamic Properties; J. Phys. Chem. Ref. Data 1982, 11 (Suppl. 2).

References 1. Parsons, R. In Standard Potentials in Aqueous Solutions; Bard, A. J., Parsons, R., Jordan, J., eds.; Marcel Dekker: New York, 1985; Chapter 2. 2. Trasatti, S. Pure Appl. Chem. 1986, 58, 955. 3. Harned, H. S.; Owen, B. B. The Physical Chemistry of Electrolytic Solutions, 2nd ed.; Reinhold: New York, 1950; Chapter X. 4. Hills, G. J.; Ives, D. J. G. J. Chem. Soc. 1951, 318. 5. MacInnes, D. A. The Principles of Electrochemistry; Dover Publications: New York, 1939; Chapter 8. 6. Planck, M. Ann. Phys. Chem. N.F. 1890, 39, 196; 1890, 40, 561. 7. Henderson, P. Z. Phys. Chem. 1907, 59, 118; 1908, 63, 325. 8. Koryta, J.; Sˇtulik, K. Ion Selective Electrodes, 2nd ed.; Cambridge University Press: Cambridge, 1983. 9. Eisenman, G. Glass Electrodes for Hydrogen and Other Cations, Principles and Practice; Marcel Dekker: New York, 1967. 10. Nikolsky, B. P. Zh. Fiz. Khim. 1937, 10, 495. 11. de Levie, R. Aqueous Acid-Base Equilibria and Titrations; Oxford Chemistry Primer; Oxford University Press: New York, 1999. 12. Damaskin, B. B.; Petrii, O. A. An Introduction to Electrochemical Kinetics, 2nd ed.; Vysokaya Shkola: Moscow, 1986; p 98.

Problems 1. Use the following data at 298.15 K for a cell whose cell diagram is Zn(s) | ZnCl2 (aq) | AgCl(s) | Ag(s) to determine the value of E . m/mol kg1

E=V

0.002941 0.007814 0.01236 0.02144 0.04242 0.09048 0.04242 0.09048 0.02211 0.04499

1.1983 1.1650 1.1495 1.1310 1.1090 1.0844 1.1090 1.0844 1.0556 1.0328

Then determine values of g at each molality.

506

LIQUIDS, SOLUTIONS, AND INTERFACES

2. The value of the acid-dissociation constant of acetic acid can be determined using the cell PtðsÞ j H2 ðgÞjHAcðaqÞ; NaAcðaqÞ; NaClðaqÞ j AgClðsÞ j AgðsÞ, where Ac represents the acetate ion. Use the following data at 0 C to determine the value of Ka for acetic acid at 0 C. Take E for the cell to be 0.23655 V at 0 C. mHAc =mol kg1

mNaAc =mol kg1

mNaCl =mol kg1

E/V

0.004779 0.012035 0.021006 0.049220 0.081010 0.090560

0.004599 0.011582 0.020216 0.047370 0.077960 0.087160

0.004896 0.012426 0.021516 0.050420 0.082970 0.092760

0.61995 0.59826 0.58528 0.56545 0.55388 0.55128

3. Use the Henderson equation to estimate the liquid junction potentials for the following systems; assume that the limiting molar conductivities given in table 6.2 can be used to calculate the ionic mobility. (a) 0.01 M HCl/0.01 M NaCl (b) 0.1 M HCl/0.1 M NaCl (c) 0.01 M KCl/1 M KCl (d) 0.1 M MgCl2 /0.1 M NH4 Cl 4. A calcium ISE is calibrated in a 0.5 M buﬀer solution at 25 C. This solution contains a constant activity of magnesium ion (0.2 M), the principal interferant. Use the following data to determine the standard potential for the cell, containing the ISE and the selectivity coeﬃcient for Mg2þ aCa2þ =M

E=V

0.1 10–2 10–3 10–4 10–5 10–6 10–7

94.3 65.1 36.8 15.3 8.7 7.7 7.5

5. EMF data have been obtained at 50 C for the cell Zn ðsat0 dÞ; Hg j 0:1 m ZnSO4 ; H2 O jZnðxÞ; Hg where x is the atomic fraction of Zn in Hg. The saturated amalgam contains 0.09608 atomic fraction of Zn and has a deﬁned zinc activity of 1.00. In other words the saturated solution of zinc in mercury is deﬁned to be the standard state. Use the following data to estimate the Zn activity and activity coeﬃcient on the atomic fraction scale as a function of its concentration.

CHARGE TRANSFER EQUILIBRIA AT INTERFACES

xZn

E=mV

0.09608 0.08389 0.07283 0.05609 0.04464 0.02918 0.01750 0.520

0.0 1.296 2.652 5.702 8.020 13.111 19.420 35.520

507

6. The standard potential of the Ag | AgCl | Cl electrode has been measured very carefully over a range of temperatures. The result may be summarized as Eo ¼ 0.23659 4.8564 10–

4

T 3.4205 10–5 T

2

þ 5.869 10–9 T3

where T is the Celsius temperature. Calculate the standard Gibbs energy and enthalpy of formation of Cl in water and its entropy at 25 C. 7. (a) Estimate the EMF of the cell Ag j AgCl j NaCl ða ¼ 0:01 mÞ; H2 O j Na; Hg j NaCl ða ¼ 0:025 mÞ; H2 O j AgCl j Ag (b) Estimate the EMF of the cell Ag | AgCl | NaCl (a ¼ 0:01 m), H2 O | NaCl (a ¼ 0:025 m), H2 O | AgCl | Ag given that the transference number of Naþ is 0.390. Use the Lewis–Sargent equation to estimate the liquid junction potential. 8. Consider the eﬀect of the distribution equilibrium (9.8.32) for an ion-exchanging membrane system designed to detect cation Mþ. The total activity of Mþ in the test solution is W W aW M;T ¼ aM;X þ aM;IE W where aW M;X is the activity due to the unknown analyte and aM;IE , the activity W W due to the ion exchange process. When aM;X >> aM;IE , the membrane behaves ideally and gives a Nernstian response to the unknown analyte W activity. However, when aW M;X falls to very low values so that aM;X << W aM;IE , the response is no longer Nernstian. Derive an expression for the Donnan potential at the test solution interface on the basis of the total activity of M, aW M;T which is valid over a wide range of activities. Use equation (9.8.34) to describe aW M;IE as a function of other activities in the test solution.

10

The Electrical Double Layer

Roger Parsons was born in 1926 in London, England, where he grew up and attended school except for his high school years, which he spent in Edmonton, Canada. He entered Imperial College, London in 1944 where he obtained a B.Sc. in 1947. At the same time he carried out doctoral research in the laboratory of Dr. John Bockris, and received a Ph.D. in 1948. After several more years at Imperial College, he moved to St. Andrews University in Scotland, and then Roger Parsons in 1954 to the University of Bristol, where he spent almost 25 years. In 1977, he became Director of the Laboratoire d’Electrochimie Interfaciale at the Centre Nationale de Recherche Scientiﬁque in Meudon, France. He returned to England in 1985 as Professor of Chemistry at the University of Southampton where he remained until his retirement in 1992. For 37 years, Parsons was editor of the Journal of Electroanalytical Chemistry and Interfacial Electrochemistry. Under his stewardship, this became the most important periodical in the ﬁeld. He contributed in signiﬁcant ways to several areas in interfacial electrochemistry, most importantly, to studies of the electrical double layer and electrode kinetics. He was elected a Fellow of the Royal Society, London in 1980. Parsons traveled widely during his career and was known in electrochemical laboratories all over the world. He received many international honors including the Palladium Medal of the Electrochemical Society, and the Prix Paul Pascal of the French Academy. With his rare combination of expertise, diplomacy, and dedication, he played an important role in bringing electrochemistry from its thermodynamic roots of the ninteenth century to the molecular science it is today.

10.1 The Electrical Double Layer Is an Example of Electrostatic Equilibrium In examining the properties of the metal | solution interface, two limiting types of behavior are found, namely, the ideal polarizable interface and the ideally nonpolarizable interface. In the former case, the interface behaves as a capacitor so that charge can be placed on the metal using an external voltage source. This leads to the establishment of an equal and opposite charge on the solution side. The 508

THE ELECTRICAL DOUBLE LAYER

509

total system in which charge is separated in space is called the electrical double layer and its properties are characterized by electrostatic equilibrium. An electrical double layer exists in general at any interface at which there is a change in dielectric properties. It has an important inﬂuence on the structure of the interface and on the kinetics of processes occurring there. The classical example of an ideally polarizable interface is a mercury electrode in an electrolyte solution which does not contain mercury ions, for example, aqueous KCl. The charge on the mercury surface is altered using an external voltage source placed between the polarizable electrode and non-polarizable electrode, for example, a silver | silver chloride electrode in contact with the same solution. Within well-deﬁned limits, the charge can be changed in both the negative and positive directions. When the mercury electrode is positively charged, there is an excess of anions in the solution close to the electrode. The opposite situation occurs when the electrode is negatively charged. An important point of reference is the point of zero charge (PZC), which occurs when the charge on the electrode is exactly zero. The properties of the electrical double layer in solution depend on the nature of the electrolyte and its concentration. In many electrolytes, one or more of the constituent ions are speciﬁcally adsorbed at the interface. Speciﬁc adsorption implies that the local ionic concentration is determined not just by electrostatic forces but also by speciﬁc chemical forces. For example, the larger halide ions are chemisorbed on mercury due to the covalent nature of the interaction between a mercury atom and the anion. Speciﬁc adsorption can also result from the hydrophobic nature of an ion. Thus, tetraalkylammonium ions, which are soluble in water, are speciﬁcally adsorbed at the mercury | water interface because of the hydrophobic nature of the alkyl groups. Speciﬁc adsorption of molecular solutes, such as the alcohols, occurs for the same reason. Double layers can also be formed at non-polarizable electrodes. If a metal electrode is placed in a solution containing its cation, the metal is either oxidized or metal ions in the solution are reduced, depending on the concentration of the electrolyte. For example, when a zinc electrode is placed in a highly concentrated solution of ZnSO4, some Zn2þ ions are chemisorbed at the metal surface, so that it acquires a net positive charge. The double layer established in solution has an excess of SO2 4 ions whose charge compensates the positive charge on the electrode. When the ZnSO4 concentration is very low, Zn2þ ions dissolve from the electrode leaving a net negative charge on the metal. The double layer in solution then contains an excess of Zn2þ ions. A null solution is one in which the concentration of metal ions is such that no reaction occurs when the metal electrode is introduced. Double layers are established at most other interfaces. An important example is the semiconductor | solution interface. In this case, a space charge region or double layer exists both in the semiconductor phase and in the liquid solution. Double layers are also formed at the liquid | liquid interface when both phases contain an electrolyte. In the present chapter, attention is focused on the double layer formed at the metal | solution interface. The roles of the solvent and the electrolyte in establishing double layer structure are discussed in detail. The relationship of the PZC to

510

LIQUIDS, SOLUTIONS, AND INTERFACES

fundamental properties of the metal is also considered. The adsorption of ions and molecules at the interface is described with special attention given to the role of the electrical variable used to establish interfacial properties.

10.2 The Thermodynamics of the Ideally Polarizable Interface The thermodynamic properties of an ideally polarizable interface are most easily examined by considering an electrochemical cell with one polarizable electrode and one non-polarizable electrode. An example of such a system is Hg j KCl in H2 O j Hg2 Cl2 j Hg 0

ð10:2:1Þ

The potential of the electrode on the right-hand side is determined by the equilibrium + 2Hg þ 2Cl Hg2 Cl2 þ 2e (

ð10:2:2Þ

and remains constant, provided the cell is not subjected to large sudden changes in the potential diﬀerence between the mercury phases at each end. The mercury | aqueous solution interface is polarizable within certain potential limits, so that any changes in the potential diﬀerence across the cell are reﬂected in changes in the potential of the polarizable mercury electrode. The basic thermodynamic equation describing the properties of the polarizable interface is the Gibbs adsorption isotherm (GAI, equation (8.3.17)), that is, X dg ¼ i dm~ i ð10:2:3Þ i

where g is the interfacial tension, i, the surface excess of component i, and m~ i , its electrochemical potential. The GAI is applied here to a system with charged components and therefore is written in terms of electrochemical potentials rather than chemical potentials. Of course, the electrochemical potential reduces to a chemical potential when species involved in the electrostatic equilibrium is not charged. Strictly speaking, equation (10.2.3) is only valid when both phases are liquids. The components of the metal (mercury) phase are mercury and excess electrons. In solution, there are water molecules, Kþ ions, and Cl ions. The ions are chosen as separate components, so that their individual surface excesses may be assessed. However, it is understood that their bulk concentrations cannot be diﬀerent, so that other constraints appear in the thermodynamic description. On the basis of the above, the GAI for the polarizable interface in cell (10.2.1) becomes mel þ Kþ d~ mKþ þ Cl d~ mCl þ W dmW dg ¼ Hg dmHg þ el d~

ð10:2:4Þ

where the subscript ‘‘el’’ stands for electron and ‘‘W’’ for water. Since mercury is a component whose chemical composition does not change in this system, the chemical potential of mercury is constant, and dmHg is zero. The electrochemical

THE ELECTRICAL DOUBLE LAYER

511

potential of the electrons in mercury depends on the inner potential of this phase, so that ~el ¼ mel FfHg m

ð10:2:5Þ

d~ mel ¼ FdfHg

ð10:2:6Þ

As a result, one obtains

The change in the electrochemical potential of the electrons may be related to the surface charge density on the polarizable mercury electrode, sm, as follows: mel ¼ Fel dfHg ¼ sm dfHg el d~

ð10:2:7Þ

On the solution side, the excess charge in the double layer, ss, is due to the relative values of the surface excesses of cations and anions, namely, Kþ and Cl . Thus, one may write ss ¼ sm ¼ FKþ FCl

ð10:2:8Þ

On the basis of the equilibrium present in the non-polarizable electrode (equation (10.2.2)), one may obtain a relationship between the electrochemical potential of the chloride ion and the inner potential of the mercury on the right-hand side of cell (10.2.1). Thus, 0

mHg2 Cl2 þ 2~ mHg msCl el ¼ 2mHg0 þ 2~

ð10:2:9Þ

It follows that 0

0

Hg 2d~ mHg ¼ 2d~ msCl el ¼ 2Fdf

ð10:2:10Þ

Now the terms in the GAI relating to the ions are re-expressed in terms of the chemical potential of the electrolyte me and the inner potential of the mercury in 0 the non-polarizable electrode fHg . First of all, because of electroneutrality, the chemical potential of the electrolyte in the solution is ~sKþ þ m ~sCl mse ¼ m

ð10:2:11Þ

Thus, the terms due to the ions may be rewritten as follows: Kþ d~ msKþ þ Cl d~ msCl ¼ Kþ dmse Kþ d~ msCl þ Cl d~ msCl s s ms ¼ Kþ dme þ m d~ msCl ¼ Kþ dme s d~ F Cl F

ð10:2:12Þ

Combining equations (10.2.4), (10.2.7), (10.2.10), and (10.2.12), one obtains

0 ð10:2:13Þ dg ¼ sm dfHg dfHg þ Kþ dme þ W dmW 0

The potential diﬀerence fHg fHg can be determined experimentally because it corresponds to a Galvani potential diﬀerence between two phases of identical composition. It is designated here as E, where the minus sign indicates that the non-polarizable electrode or reference electrode responds to the activity of the anion in solution. Furthermore, on the basis of the Gibbs–Duhem relationship xe dme þ xw dmw ¼ 0

ð10:2:14Þ

512

LIQUIDS, SOLUTIONS, AND INTERFACES

where xi is the mole fraction of component i. Thus, the ﬁnal two terms in equation (10.2.13) can be rewritten as x Kþ dme þ W dmW ¼ Kþ e W dme ¼ ðWÞ dme ð10:2:15Þ Kþ xW where ðWÞ is the relative surface excess of Kþ ions. In aqueous solutions for Kþ which the mole fraction of water is usually much greater than that of the electrolyte, the relative nature of the surface excess can usually be neglected. Thus, the ﬁnal form of the GAI becomes dme dg ¼ sm dE þ ðWÞ Kþ

ð10:2:16Þ

It is interesting to derive the GAI for the case that the non-polarizable electrode responds to the concentration of cations in solution. Such a cell is Cu j Hg j KCl in H2 O j K in Hg j Cu 0

ð10:2:17Þ

The equilibrium at the non-polarizable electrode is + KðHgÞ Kþ þ Hg þ e (

ð10:2:18Þ

If the concentration of K in the amalgam is not too high and there is no signiﬁcant reaction with water during the time frame of the experiment, the electrode on the right-hand side provides a non-polarizable system which responds to the Kþ ion activity in solution. The condition for the electrochemical equilibrium is ~Hg ~sKþ þ mHg þ m m el ¼ mKðHgÞ

ð10:2:19Þ

where the subscript ‘‘el’’ again denotes the electron. It follows that 0

KðHgÞ d~ msKþ ¼ d~ mHg ¼ F dfCu el ¼ F df

ð10:2:20Þ

On the basis of equation (10.2.11), one obtains msKþ þ Cl d~ msCl ¼ Kþ d~ msKþ þ Cl dmsel Cl d~ msKþ Kþ d~ s msKþ þ Cl dmsel ¼ m d~ F Thus, the GAI becomes

0 dg ¼ sm dfHg dfCu þ Cl dmse þ W dmW

0 ðWÞ s s ¼ sm dfCu dfCu þ ðWÞ Cl dme ¼ sm dEþ þ Cl dme

ð10:2:21Þ

ð10:2:22Þ

where Eþ is the potential of the polarizable Hg electrode with respect to the nonpolarizable electrode reversible to the cation in solution. When equations (10.2.16) and (10.2.22) are compared, it is clear that the surface excess of the anion appears in the GAI when the reference electrode is reversible to the cation. Thus, the result valid for any 1–1 electrolyte can be summarized as s dg ¼ sm dE þ ðWÞ dme

ð10:2:23Þ

THE ELECTRICAL DOUBLE LAYER

513

where the choice of a reference electrode reversible to one of the ions results in the surface excess of the other. The above result can be used to examine the dependence of interfacial tension on cell potential for a given electrolyte at ﬁxed concentration. Thus, @g ¼ sm ð10:2:24Þ @E mse As the potential drop across the cell increases in the positive direction, the charge on the polarizable electrode changes from negative values through zero to positive values. The interfacial tension is a maximum at the PZC and falls to lower values for both positive and negative charge densities. Typical data obtained for the Hg | aqueous solution interface are shown in ﬁg. 10.1. These data are often referred to as electrocapillary curves in reference to the fact that they were obtained experimentally using a capillary electrometer. The experiment is based on the capillary rise eﬀect discussed in section 8.3 and is considered further in the following section of this chapter. When interfacial tension data are obtained both as a function of cell potential and electrolyte concentration, one may extract information about the relative ionic surface excesses. Using cell (10.2.1), the relative cationic surface excess is given by @g @g ¼ ¼ ðWÞ ð10:2:25Þ þ @mse E RT@ ln ase E where ase is the activity of the electrolyte in solution. The surface excess can also be expressed in terms of charge density on the solution side of the double layer. Thus, for a 1–1 electrolyte,

Fig. 10.1 Experimental data for the interfacial tension of the mercury | aqueous electrolyte solution (1 M) interface as a function of the potential drop across the cell at 188C. The reference electrode was a calomel electrode with 1 M KCl.

514

LIQUIDS, SOLUTIONS, AND INTERFACES

sþ ¼ FðWÞ þ

ð10:2:26Þ

where sþ is the charge density due to the cations in the double layer. The charge due to anions is obtained by using the condition for electroneutrality: s ¼ ss sþ ¼ sm sþ This also gives the surface excess of anions, that is, s ðWÞ ¼ F

ð10:2:27Þ

ð10:2:28Þ

Thus, having obtained values of sm and one surface excess for given values of E and ase , the other surface excess is easily estimated. The above analysis shows how the GAI is applied to the simplest polarizable interface in contact with a 1–1 electrolyte. Other more complicated situations have been analyzed for systems with more complex electrolytes and molecular solutes. More details can be found in reviews by Mohilner [1] and Parsons [G4]. The essential feature of these analyses is that an equation is derived which relates the change in interfacial tension to the change in the potential of the polarizable electrode with respect to that of a non-polarizable electrode, and to the chemical potentials of the components of the solution. The discussion so far has been limited to liquid metal electrodes. At solid metal electrodes other factors come into play. The interfacial tension of a liquid is the work done to increase the area of the liquid interface by unit amount. The surface area of a solid may be increased in two ways. One way is by plastic deformation in which the number of surface atoms is increased, for example, by cleavage of the crystal. The other way is by deforming the crystal lattice in a deﬁned direction such that the interatomic lattice spacing changes but the number of surface atoms remains constant. This is called elastic deformation. The change in the internal energy of the surface, dUs , due to a general change in area is dUs ¼ g dAp þ AYmn demn

ð10:2:29Þ

where g is the interfacial tension, dAp , the change in area due to plastic deformation, Ymn , the surface stress in a direction deﬁned by the mn component of the surface stress tensor, and emn , the corresponding surface strain due to the elastic deformation. The analysis of the thermodynamics of the polarizable interface for solid metals is obviously much more complex than that for liquids. Nevertheless, it can be shown that the equation equivalent to (10.2.23) for the solid electrode system at constant temperature and pressure is " " # # @emn @emn ðWÞ d ¼ sm þ ðg Ymn Þ dE þ þ ðg Ymn Þ dmse @E mse @mse E ð10:2:30Þ where ð@emn =@E Þmse is the change in surface strain with potential drop across the system at constant electrolyte composition or electrostriction, and ð@emn =@mse ÞE , the change in surface strain with electrolyte concentration at constant potential drop. Under the usual experimental conditions involved in the study of the polarizable interface at solid metals, the two derivatives in the surface strain are neg-

THE ELECTRICAL DOUBLE LAYER

515

ligible [2]. It follows that equation (10.2.30) reduces to equation (10.2.23) for these conditions. Sometimes it is convenient to analyze interfacial thermodynamic data at constant electrode charge density sm rather than at constant cell potential diﬀerence E . Once the interfacial tension data at a given concentration have been diﬀerentiated with respect to E to obtain sm , one may calculate the function x which is given by x ¼ g þ sm E

ð10:2:31Þ

This is known as the Parsons function and is often used in the analysis of interfacial thermodynamic data. Taking the total derivative of x , one obtains dx ¼ dg þ sm dE þ E dsm

ð10:2:32Þ

After substitution of equation (10.2.23), the result is ðWÞ dx ¼ E dsm þ dmse

ð10:2:33Þ

Thus, the GAI is now expressed as a diﬀerential equation with the electrode charge density and the chemical potential of the electrolyte as independent variables. By cross-diﬀerentiating equations like (10.2.23) and (10.2.33), one obtains useful relations between partial derivatives for the thermodynamic properties of the polarizable interface. From equation (10.2.23), cross-diﬀerentiation leads to the result ! @ðWÞ @g @sm ¼ ¼ ð10:2:34Þ @E @mse @mse E @E s me

This relationship states that the change in charge density on the electrode with electrolyte concentration measured at constant cell potential diﬀerence is equal to the change in ionic surface excess with cell potential diﬀerence measured at constant electrolyte concentration. Cross-diﬀerentiation of equation (10.2.33) gives ! @ðWÞ @E ¼ ð10:2:35Þ @mse sm @sm s me

By extending this procedure one may obtain two more useful relationships. ðWÞ When the independent variables are E and , it is easily shown that ! s @sm @me ¼ ð10:2:36Þ ðWÞ @E @ E

On the other hand, when they are sm and ðWÞ , it follows that ! s @E @me ¼ @s @ðWÞ m ðWÞ

ð10:2:37Þ

sm

The procedures used to derive these equations are entirely analogous to those followed in the thermodynamics of bulk solutions (see section 1.3).

516

LIQUIDS, SOLUTIONS, AND INTERFACES

The above discussion illustrates that the interfacial tension, cell potential difference, electrode charge density, the ionic surface excesses, and the electrolyte activity are the important properties of the simple system considered. Another property which is studied experimentally is the diﬀerential capacity. Experimental methods for measuring the properties of a polarizable electrode are now considered and the analysis of the resulting data outlined.

10.3 The Experimental Study of the Double Layer The ﬁrst experiments to study polarizable interfaces were carried out in the nineteenth century with mercury electrodes. After invention of the dropping mercury electrode by Heyrovsky [3] it was realized that very precise data could be obtained for the Hg | solution interface when the components of the system were carefully puriﬁed. Experiments to measure precisely the interfacial capacity were soon developed. Now these techniques and related ones are applied to study other interfaces, including liquid j liquid and solid metal j liquid systems. Experimental study of the double layer is not limited to thermodynamics. A variety of spectroscopic methods have been applied to determine the structure and composition of the double layer. Two of these, namely, second-harmonic generation and vibrational sum frequency spectroscopy, have already been described in section 8.11. Other important techniques are based on the absorption of electromagnetic radiation when it is transmitted through or reﬂected at the interface. Finally, the scattering of X-rays and neutrons at interfaces has proven to be a valuable tool for obtaining atomic level information about the interface. In the following section some of these methods are outlined in more detail. A. Thermodynamic Methods On the basis of the GAI, it is clear that the interfacial tension g is the most important experimental quantity. Three methods are commonly used to determine g at liquid j liquid interfaces, namely, the capillary electrometer method, the maximum bubble pressure method, and the drop weight or drop time method. The capillary electrometer method ﬁrst used by Lippmann [4] is based on the capillary rise principle. The interface between the liquid Hg electrode and the solution is established in a ﬁne capillary, the position of the interface being determined by the interfacial tension (see ﬁg. 8.2). More speciﬁcally, g is given by g ¼ ðhm rm hs rs Þgr cosðy=2Þ

ð10:3:1Þ

where hm is the height of the mercury column, hs , the height of the solution above the meniscus, rm and rs , the densities of mercury and the solution, respectively, g, the acceleration due to gravity, r, the radius of the capillary and y, the contact angle. Providing y and r are constant this equation can be rewritten as g ¼ khc where the corrected height is given by

ð10:3:2Þ

THE ELECTRICAL DOUBLE LAYER

hc ¼ hm

hs rs rm

517

ð10:3:3Þ

and the constant k by ð10:3:4Þ

k ¼ grm r cosðy=2Þ 2

The interfacial tension is normally in the range 200–400 mJ m ; it follows that a mercury column of about 0.6 m can be supported with a capillary radius of 10 mm. In principle the method is absolute, but in practice the device is calibrated with a solution of known interfacial tension because of the diﬃculty in determining r. The meniscus is viewed with a microscope positioned at an optical ﬂat in the cell wall near the capillary. Further details regarding this method have been given by Payne [5]. Under certain circumstances, the contact angle between Hg and the solution can change with applied potential. For this reason, the maximum bubble pressure method which is independent of contact angle is preferred (see section 8.2). This technique is easily adapted to computer-controlled experiments. The ﬁnal method discussed here is the drop time technique. At the critical size when a mercury drop breaks from the end of a capillary placed in an electrolyte solution, the weight of the drop is balanced by the interfacial tension forces so that 2prg ¼ mg

ð10:3:5Þ

where r is the radius of the capillary and m, the mass of the drop. If the ﬂow rate of the mercury, fl , is constant then the mass may be related to the drop time t: m ¼ fl t

ð10:3:6Þ

It is generally recognized that equation (10.3.5) is an approximate description of the condition for drop break away. However, this method can be used to obtain high-quality interfacial tension data provided one considers the more detailed description of the related physics. More details can be found in the work of Barradas and Kimmerle [6]. Interfacial tension against electrode potential curves have a parabolic shape with a maximum value which depends on the nature and concentration of the electrolyte (see ﬁg. 10.1). Detailed results for the mercury j aqueous solution interface were initially reported by Gouy [7, G5]. Examination of these data for the alkali metal halides shows that the interfacial tension depends markedly on the nature of the electrolyte at positive potentials. On the other hand, the variation with electrolyte at negative potentials is rather small. It follows that the anions in the electrolyte strongly aﬀect the interfacial tension when they predominate in the double layer. By diﬀerentiating the interfacial tension data, one obtains the surface charge density on the metal sm (equation (10.2.24)). This quantity is positive at the most negative potentials; then it decreases in magnitude, goes through zero, and becomes negative in the positive potential range. The maximum value of g is the point of reference for a polarizable electrode, namely, the point of zero charge (PZC, see section 8.8). In the case of mercury electrodes, it is also called the

518

LIQUIDS, SOLUTIONS, AND INTERFACES

electrocapillary maximum (ECM). Some typical results for the mercury j solution interface are shown in ﬁg. 10.2. A very sensitive method of studying the properties of a polarizable interface is the measurement of its diﬀerential capacity, C. This is given by the second derivative of the interfacial tension with respect to electrode potential: ! @2 g @sm ¼ ð10:3:7Þ C¼ @E mse @E2 s me

Early experiments with mercury were carried out by Grahame [G1] using a precise capacitance bridge. The experimental setup is shown schematically in ﬁg. 10.3. An a.c. signal is applied to the cell as the fourth arm of a Wien bridge. The cell consists of two polarizable electrodes and an electrolyte solution. One polarizable electrode is a dropping Hg electrode of small area and the other, a large-area electrode, for example, a Hg pool. Thus, the impedance of the cell consists of two capacitors, corresponding to each electrode j solution interface, and a resistor, corresponding to the electrolyte solution, in series, that is, Zcell ¼

1 1 þ Rs þ joC1 joC2

ð10:3:8Þ

C1 is the capacitance at one interface, C2 , that at the other, Rs , the solution resistance, and o the angular frequency of the applied a.c. signal. Since the area of the second electrode is at least 1000 times that of the ﬁrst, its contribution to the impedance may be neglected so that Zcell ¼

1 þ Rs joC1

ð10:3:9Þ

Fig. 10.2 Plots of the charge on the polarizable Hg electrode for various 0.1 M electrolytes against electrode potential with respect to a calomel reference electrode with 1 M KCl at 258C [G5].

THE ELECTRICAL DOUBLE LAYER

519

Fig. 10.3 Schematic diagram of the capacitance bridge used to measure the diﬀerential capacity of the Hg electrode | solution interface.

Since the area of the dropping Hg electrode changes with time, the experimental system must include a timing device which allows one to balance the bridge at a precise interval after the birth of a new drop. Using the measured ﬂow rate of the Hg electrode and the time at which the measurement is made, one may calculate the area of the Hg drop. The value of C1 obtained from the balanced bridge and the electrode area are then used to calculate the speciﬁc diﬀerential capacitance of the interface. The d.c. potential of the polarizable interface is established in a second circuit using a non-polarizable reference electrode (see ﬁg. 10.3). The experiment is repeated at ﬁxed intervals of potential over the polarizable range of the dropping Hg electrode by changing the d.c. potential drop in the polarizing circuit by the appropriate amount. Although they are very precise, experiments carried out using the Grahame bridge are rather tedious because the bridge is balanced manually. In a modern version of this experiment the in- and out-of-phase components of the cell impedance are recorded as a function of time during the life of the Hg drop using a phase-sensitive detector. The speciﬁc capacity is then easily estimated from the impedance data using the ﬂow properties of the Hg electrode. Typical diﬀerential capacity data obtained at the mercury j aqueous solution interface using various electrolytes at a concentration of 0.1 M at 258C are shown in ﬁg. 10.4. The capacity of the interface depends markedly on the nature of the anion at positive potentials. These data demonstrate that the anion strongly

520

LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 10.4 Plots of the diﬀerential capacity of the Hg | aqueous solution interface for a variety of 0.1 M electrolytes using a calomel reference electrode (1 M KCl) at 258C [G5].

inﬂuences double layer properties when it predominates in the double layer. On the other hand, the capacity of the interface depends very little on the nature of the cation when these are simple monoatomic species [8] (ﬁg. 10.5). The presence of polyatomic hydrophobic cations, such as the tetraalkylammonium cations in solution, results in much higher capacities in the negative potential range.

Fig. 10.5 Plots of the diﬀerential capacity of the Hg | aqueous solution interface at negative potentials for a variety of 0.1 M chloride electrolytes using a calomel reference electrode (1 M KCl) at 258C [G5].

THE ELECTRICAL DOUBLE LAYER

521

The interfacial capacitance may also be measured at solid polarizable electrodes in an impedance experiment using phase-sensitive detection. Most experiments are carried out with single crystal electrodes at which the structure of the solid electrode remains constant from experiment to experiment. Nevertheless, capacity experiments with solid electrodes suﬀer from the problem of frequency dispersion. This means that the experimentally observed interfacial capacity depends to some extent on the frequency used in the a.c. impedance experiment. This observation is attributed to the fact that even a single crystal electrode is not smooth on the atomic scale but has on its surface atomic level steps and other imperfections. Using the theory of fractals, one can rationalize the frequency dependence of the interfacial properties [9]. The capacitance that one would observe at a perfect single crystal without imperfections is that obtained at inﬁnite frequency. Details regarding the analysis of impedance data obtained at solid electrodes are given in [10]. An important point of reference for interfacial thermodynamic data is the PZC. If only diﬀerential capacity data are available, this must be determined in an independent experiment. For liquid electrodes such as mercury, the PZC is conveniently measured using the streaming electrode technique [5]. The polarizable electrode is designed so that the liquid metal, for example, mercury, emerges from a specially designed capillary in small droplets with a short lifetime. The capillary is connected to a column of mercury on which the pressure may be increased using an inert gas such as nitrogen. The capillary is connected to a d.c. potential measuring device and its potential measured against a suitable reference electrode. The pressure on the mercury, and thus, the rate at which it streams from the capillary, is increased until the measured d.c. potential is constant and independent of further pressure increase. This value gives the PZC against the chosen reference electrode. The principle of the technique is to cause the area of the polarizable electrode to increase so rapidly that no signiﬁcant charge can accumulate at the interface. Values of the PZC at the Hg j solution interface are shown as a function of electrolyte concentration in ﬁg. 10.6. In the case of NaF, the PZC with respect to a constant reference electrode is independent of electrolyte concentration. However, in the cases of the other halides, the PZC moves to more negative potentials as the electrolyte concentration increases. The latter observation is considered to be evidence that the anion in the electrolyte is speciﬁcally adsorbed at the interface. Speciﬁc adsorption occurs when the local ionic concentration is greater than one would anticipate on the basis of simple electrostatic arguments. Anions such as Cl , Br , and I can form covalent bonds with mercury so that their interfacial concentration is higher than the bulk concentration at the PZC. Furthermore, the extent of speciﬁc adsorption increases with the atomic number of the halide ion, as can be seen from the increase in the negative potential shift. A more complete description of speciﬁc adsorption will be given later in this chapter. The PZC may also be seen on diﬀerential capacity curves when speciﬁc adsorption is absent and the electrolyte concentration is low ð 0:01 M). At this point the capacity of the diﬀuse part of the double layer is a minimum and can fall below that of the compact or inner layer. As a result the total double layer capacity may

522

LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 10.6 Point of zero charge (PZC) for the Hg | aqueous solution interface for several alkali metal halides as electrolytes at 258C using a calomel reference electrode (1 M KCl) plotted against electrolyte concentration.

also reach a minimum. This is the principal method of determining the PZC at solid electrodes. Once the PZC is known, the capacity data may be integrated to obtain the surface charge density: ðE sm ¼

C dE

ð10:3:10Þ

Ez

where Ez is the PZC on the chosen potential scale E. A complete analysis of interfacial thermodynamic data also requires that the value of the interfacial tension at the PZC be known. Then, the capacity data may be integrated twice to obtain the interfacial tension as a function of cell potential: ð ðE g ¼ gm

C dE

ð10:3:11Þ

Ez

gm is the value of g at the PZC or electrocapillary maximum. This means that gm must be measured in an independent interfacial tension experiment at each electrolyte concentration at which the diﬀerential capacity data were obtained. Unfortunately, these data are often not available. In the case of the study of the adsorption of molecular solutes at the polarizable interface, diﬀerent tactics are often employed in data analysis. Suppose a study of the adsorption of pyridine is carried out at the mercury j solution interface from an aqueous solution containing KCl. The Gibbs adsorption isotherm for this system using a reference electrode reversible to the Cl anion is dg ¼ sm dE þ ðWÞ þ dmKCl þ py dmpy þ W dmW

ð10:3:12Þ

THE ELECTRICAL DOUBLE LAYER

523

where py is the surface excess of pyridine, mpy , its chemical potential, and the other symbols have their usual meaning. If capacity–potential data are obtained at varying pyridine concentration (varying mpy ) holding the KCl activity constant, it is observed that the capacity is independent of mpy at suﬃciently negative or positive potentials. This is a result of desorption of the pyridine molecules from the interface when the electrical ﬁeld due to the charge on the electrode is suﬃciently high. Choosing a potential in this region as a point of reference, one may integrate the capacity data to obtain the change in interfacial tension g as follows: ð ðE g ¼ g g r ¼

C dE

ð10:3:13Þ

Er

where gr is the interfacial tension at the point of reference and Er , the potential at that point. More will be said about the analysis of data for molecular adsorption in section 10.10. At solid polarizable electrodes, the charge density on the electrode can be measured directly using chronocoulometry [11]. This has been applied successfully to the study of molecular adsorption at single crystal gold electrodes. As discussed above, a reference potential Er is chosen at which molecular adsorption is absent. The potential of the polarizable electrode is then stepped to a new value Ei and the current transient which results is recorded. By integrating this transient, one obtains the change in charge density between the two potentials: sm ¼ sm ðEi Þ sm ðEr Þ

ð10:3:14Þ

where sm ðEi Þ is the charge density at the new potential and sm ðEr Þ, that at the reference potential. The potential of zero charge Ez is determined in an independent capacity experiment. By stepping the potential to the PZC, one can determine the absolute value of the charge density on the electrode at the reference potential: smz ¼ sm ðEz Þ sm ðEr Þ ¼ sm ðEr Þ

ð10:3:15Þ

where smz is the change in charge density required to reach the PZC. These experiments are carried out as a function of potential over the polarizable range of the electrode, and as a function of the concentration of the molecular adsorbate, for example, pyridine. The data may then be integrated to obtain the change in interfacial tension with respect to the reference potential: ðE g ¼ g ¼ gr ¼

sm dE

ð10:3:16Þ

Er

More details about these experiments can be found in the work of Lipkowski [11]. Once a complete set of interfacial tension data have been assembled as a function of electrode potential and solution composition, they may be diﬀerentiated to obtain the relative surface excess. For example, if the solution contains a single electrolyte and the reference electrode is reversible to the anion in the

524

LIQUIDS, SOLUTIONS, AND INTERFACES

electrolyte, the relative surface excess of the cation is obtained by estimating the derivative in equation (10.2.25). In the case of molecular adsorption such as pyridine, the surface excess is found from the relative change in the interfacial tension at constant electrode potential and electrolyte activity (see equation (10.3.12)): ! @g ð10:3:17Þ py ¼ @mpy E ;mKCl

Many studies of ionic and molecular adsorption have been carried out using the techniques described here. They provided quantitative information regarding the extent of adsorption. Detailed analysis of these data led to an improved picture of the forces which control the extent of adsorption at polarizable interfaces. This subject is considered in more detail in the discussion of ionic adsorption in section 10.9 and molecular adsorption in section 10.10. B. Spectroscopic Methods In recent years a rich variety of spectroscopic techniques has been developed to study the properties of the metal j solution interface [12–14]. These techniques fall into all areas of traditional spectroscopy used to study the properties of bulk phases, and involve absorption, emission, and scattering phenomena. The methods which are particularly useful for investigating the composition and structure of the electrical double layer are considered in more detail here. Vibrational spectroscopy provides an excellent tool for examining interfacial properties. Experiments have been carried out using both the infrared and Raman techniques [12–14]. Discussion is limited here to Fourier transform infrared spectroscopy (FTIR) in the reﬂection mode. It is important to understand how the infrared radiation interacts with dipolar adsorbates at the interface. Consider an electromagnetic wave travelling in the (x; z)-plane, which strikes the interface located in the (x; y)-plane at an angle y with respect to the interface (see ﬁg. 10.7). The electrical ﬁeld vector associated with the wave can be resolved into two components, one oscillating in the ðx; zÞ-plane (the parallel or p-component)

Fig. 10.7 Sketch of the vector deﬁning the reﬂection of electromagnetic radiation in the (x, z)-plane at a solution | metal interface in the ðx, y)-plane. Dipolar adsorbates in the perpendicular (a) and parallel orientations (b) are shown with their corresponding electrostatic images in the metal phase.

THE ELECTRICAL DOUBLE LAYER

525

and the other, in the (y; zÞ-plane (the perpendicular or s-component). Because the substrate phase is a metal, electrostatic images are formed from the adsorbed dipoles as a result of rearrangement of the mobile electron density within the metal. When the dipole vector is perpendicular to the interface, its image points in the same direction so that the eﬀective dipole moment is larger. Dipoles in this orientation interact with the parallel or p-component of the incident radiation. When the frequency of the incident radiation corresponds to a vibration associated with the dipole, energy is absorbed which can be detected by examining the reﬂected radiation. On the other hand, if the dipole vector is parallel to the interface, the image dipole in the electrode acts in the opposite direction to that of the molecular dipole, so that the net eﬀect is zero dipole moment in the (x; y)-plane. Thus, there is no interaction with the s-component of the electrical ﬁeld vector, which oscillates perpendicular to the plane of incidence. It is clear that polarization of the infrared radiation incident on the interface is useful in determining the orientation of adsorbate molecules. The fact that dipolar components perpendicular to the interface are infrared active and parallel components are not is known as the surface selection rule [12, 13]. Because most solutions absorb infrared radiation in their bulk, the design of the electrochemical cell is an important consideration in interfacial reﬂection experiments. The optimum conﬁguration involves a very thin solution layer which is a few microns thick and is sandwiched between the optical window and the reﬂective electrode (ﬁg. 10.8). In order to achieve maximum sensitivity, the window has a hemispherical or triangular (prism) shape. Nevertheless, most radiation is absorbed in the bulk of the solution and the eﬀect of interfacially adsorbed molecules cannot be seen unless special steps are taken. One procedure is to polarize the light in a cyclical fashion between s- and p-polarized light. If adsorbate molecules interact with the p-polarized light, the intensity of the

Fig. 10.8 Thin-layer electrochemical cell used in SNIFTIR studies and reﬂection optics: (a) Teﬂon cap; (b) N2 inlet; (c) glass tube; (d) Teﬂon cell body; (e) port for reference electrode; (f) ceramic tube; (g) Pt wire counter electrode; (h) single crystal working electrode; (i) hemispherical ZnSe window; (k) focal plane for reﬂection optics; (m) instrument focal plane; and (n) folding mirrors (From reference 16, with permission.)

526

LIQUIDS, SOLUTIONS, AND INTERFACES

reﬂected beam also oscillates, being lower when p-polarized light predominates. The diﬀerence between the signal observed in the presence of p-polarization with respect to that observed in the presence of s-polarization gives the spectrum of the interfacial layer. This experiment is known as infrared reﬂection absorption spectroscopy, or IRRAS [15]. It is carried out with the polarizable electrode at constant potential. Another experimental strategy is to compare spectra observed at two diﬀerent potentials. One potential is chosen as a reference point at which interfacial conditions are fairly simple, for example, at which adsorption is absent. By subtracting the spectrum at the reference potential from the spectrum at another potential, one determines the changes which have occurred in the interfacial region. When this experiment is carried out with an FTIR spectrometer it is known as subtractively normalized FTIR spectroscopy, or SNIFTIRS [15]. SNIFTIR spectra obtained in a study of double layer structure on an Au electrode in a non-aqueous electrolyte solution are shown in ﬁg. 10.9 [17]. The electrolyte involved is Cr(DMSO)6 (ClO4 )3 in dimethylacetamide (DMA) as solvent. The cation in which DMSO is a ligand has a band at 924 cm1 due to the SO vibration in the DMSO molecule. The perchlorate anion absorbs at 1100 cm1 . The reference spectrum was recorded at 0.16 V against the ferrocene –ferrocinium internal reference. Each of the diﬀerence spectra shown in ﬁg. 10.9 was obtained by subtracting the reference spectrum from the spectrum observed at the more negative potential. As the potential moves in the negative direction, the band at 924 cm1 increases in intensity, demonstrating that the [Cr(DMSO)6]3+ cation is attracted into the double layer. At the same time a negative-going band is observed at 1100 cm1 ; this demonstrates that the ClO 4 anion is repelled from the double layer as the electrode potential moves negative of the PZC. The most interesting feature of these spectra is the bipolar band found in the region 1600–1640 cm1 . This is attributed to interfacial solvent

Fig. 10.9 SNIFTIR spectra obtained in the 1700–700 cm1 region for a 2 mM [Cr(DMSO)6 ] (ClO4 )3 solution in dimethylacetamide at an Au electrode. The reference potential was 0.16 V against an internal ferrocene–ferrocinium couple; spectra were recorded at more negative potentials at intervals of 100 mV (From reference 17, with permission.)

THE ELECTRICAL DOUBLE LAYER

527

molecules which undergo reorientation as the electrical ﬁeld in the double layer changes sign. When the electrode is positively charged the DMA molecules are adsorbed on the gold with the negative end of the molecular dipole pointing toward the metal (ﬁg. 10.10(A)). Because of the interaction of the CO group with the metal, the stretching frequency is red shifted and appears at 1605 cm1 . As the charge on the electrode is made more negative the DMA molecules reorient so that the molecular dipole is oriented with its negative end in the solution (ﬁg. 10.10(B)). Under these circumstances, the CO stretching frequency is closer to 1640 cm1 . The reorientation of the interfacial DMA molecules results in a bipolar band with a positive-going lobe at 1640 cm1 and a negative-going lobe at 1605 cm1 . SNIFTIR spectroscopy can also be used to obtain quantitative information about double layer composition [18]. The surface excess of ClO 4 anion was estimated for 0.1 M aqueous HClO4 solutions at single crystal Au electrodes as a function of potential in the region around the PZC. The reference spectrum was obtained at 0.69 V with respect to a Ag j Ag+ reference electrode, that is, well positive of the PZC. The negative-going band observed at 1109 cm1 , which is attributed to ClO 4 , was integrated to obtain an estimate of the change in surface , ClO4 , with electrode potential. Data were collected for excess of ClO 4 Au(111), Au(110), and Au(100) electrodes. Since the PZC depends on the crystallographic orientation of the Au single crystal, the value of ClO4 depends on the nature of the single crystal at constant electrode potential. However, when plotted against electrode charge density ClO4 is independent of the nature of the Au surface in the region 10 mC cm2 sm 10 mC cm2 (see ﬁg. 10.11). The integrated intensity may be related to the surface excess estimated by the Gouy– Chapman theory once the extinction coeﬃcient is determined. Excellent agreement between these quantities is obtained in the region around the PZC (ﬁg. 10.11). However, for charge densities greater than 10 mC cm2 , the integrated intensity is considerably greater than the Gouy–Chapman estimate. This discre-

Fig. 10.10 Schematic diagram indicating the orientation of dimethylacetamide at a positively charged polarizable electrode (A) and at a negatively charged electrode (B).

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LIQUIDS, SOLUTIONS, AND INTERFACES

1 Fig. 10.11 Integrated intensity A R=R of the negative-going ClO in 4 band at 1109 cm aqueous 0.1 M HClO4 at Au(hkl) plotted against electrode charge density. The solid line gives the Gouy–Chapman estimate of the relative surface excess of ClO 4 . The data are for Au(100) (*), Au (111) (!), and Au(110) (~). (From reference 18, with permission.)

pancy is attributed to speciﬁc adsorption of ClO 4 which replaces water molecules on the electrode surface in this region. X-ray methods provide an important route to determining interfacial structure when the metal electrode is a single crystal [13, 14]. One technique, namely, extended X-ray absorption ﬁne structure, or EXAFS, depends on the absorption of X-rays by the atoms in the interfacial region. The other method is surface Xray scattering or surface X-ray diﬀraction. Both methods require intense X-ray sources and therefore can only be carried out at a particle accelerator or synchrotron. The discussion here is limited to scattering experiments. The power of the scattering technique in determining the details of adsorption at a single crystal electrode is readily apparent for the case of monoatomic adsorbates, for example, the adsorption of a monoatomic anion, or the underpotential deposition of one metal on another. Since diﬀerent atoms have diﬀerent sizes and interact with one other in diﬀerent ways, the adsorption phenomenon does not necessarily involve one adsorbate atom A sitting on top of each metal atom M. However, if the surface structure does involve atom A depositing on each atom M (see ﬁg. 10.12(A)), the adlayer is termed commensurate with a p(1 1) surface structure. Another possibility is that the adsorbate atoms only occupy a fraction of the available sites but they do so with a regular pattern. Figure 10.12(B) shows a commensurate adlayer with a p(22) surface structure. Finally, the adlayer may be deposited in a regular fashion but without following a pattern dictated by the substrate metal; this is called an incommensurate adlayer (ﬁg. 10.12(C)). The structure of the adlayer with respect to the substrate is found experimentally by comparing the diﬀraction pattern for the adsorption layer with that for the substrate.

THE ELECTRICAL DOUBLE LAYER

529

Fig. 10.12 Three surface structures for an adlayer of atoms A (shaded) on a substrate of metal atoms M (*): (a) a commensurate adlayer with a pð1 1Þ structure; (b) a commensurate adlayer with a pð2 2Þ structure; and (c) an incommensurate adlayer.

Surface scattering experiments must be carried out with the X-ray beam reaching the interface at grazing incidence (see ﬁg. 10.13). Typically the X-ray beam makes an angle of less than 18 to the plane deﬁning the electrode surface. If the Xray beam travels in the (x; z)-plane, scattering is chieﬂy in the (x; y)-plane. The solution layer must be very thin, because the path length through the solution is large at small incident angles. Finally, the window material must be carefully chosen in order to avoid signiﬁcant absorption by this medium. X-ray surface diﬀraction has been applied in situ to study several processes at the electrode j solution interface [13, 14]. An important phenomenon in electrochemistry at Au is surface reconstruction in which a monolayer of atoms on the surface of a single crystal acquires a diﬀerent arrangement from that of the

Fig. 10.13 Thin layer electrochemical cell for X-ray scattering experiments.

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LIQUIDS, SOLUTIONS, AND INTERFACES

atoms in the crystal. Surface reconstruction is best known on Au(100) and is easily observed in ex-situ experiments. Using X-ray surface diﬀraction, reconstruction of the Au(111) surface was observed at potentials suﬃciently negative of the PZC in the presence of p speciﬁcally adsorbed Cs+ ions. In this region a ﬃﬃﬃ monolayer with a rectangular ( 3 22) structure was detected on the Au(111) surface [13]. The speciﬁc adsorption of halide anions has been studied on Au and Ag single crystals [14]. On Au(111), these ions form incommensurate hexagonal monolayers that compress as the electrode potential is changed in the positive direction [19]. However, on Ag(100), Br adsorption occurs at the hollow site formed by four metal atoms in a square pattern. This type of commensurate monolayer has a c(2 2) surface structure. These studies demonstrate the role of atomic surface structure in determining the extent of adsorption. Diﬀerences between adsorption on Ag(100) and Au(100) are explained in terms of diﬀerences in the strengths of the metal–halide bonds [14]. Another important area in which X-ray surface scattering is applied is the underpotential deposition of metals. Underpotential deposition is the phenomenon by which one metal deposits on another at a potential positive of its normal reduction potential. For example, Pb deposits at underpotentials on Ag. This is due to the fact that the Gibbs energy for formation of a Pb–Ag bond is less than that for formation of a Pb–Pb bond. Other metals which undergo underpotential deposition on Ag, Au, and Pt are Tl and Bi. On the basis of the electrochemistry observed in formation of the metal monolayers, there is good reason to expect that they are well ordered. Tl, Pb, and Bi all form an incommensurate monolayers on Au(111). On Au(100), Tl and Bi form an incommensurate monolayer with a c(2 2) surface structure [14]. On the other hand, underpotential deposition of Pb on Ag(111) leads to an incommensurate monolayer [13]. These studies demonstrate clearly that the nature of the monolayer formed depends on both the nature and structure of the substrate metal. As stated earlier, there are other spectroscopic tools which have been used to study the structure and composition of the electrical double layer. However, it is clear that detailed information can be obtained both at the atomic and molecular level on the basis of the two techniques discussed here.

10.4 The Structure of the Double Layer The development of microscopic models of the double layer began over 100 years ago with work of Helmholtz [20]. He assumed that the charge on the polarizable metal electrode is exactly compensated by a layer of ionic charge in solution located at a constant distance from the geometrical electrode j solution interface. The separation distance zd was assumed to have molecular dimensions. This simple model which gave rise to the term double layer is the equivalent of a parallel-plate capacitor with a capacitance given by C¼

eH e0 zd

ð10:4:1Þ

THE ELECTRICAL DOUBLE LAYER

531

where eH is the relative permittivity of the Helmholtz layer. Considering the diameter of a water molecule and adding to that the radius of a typical monoatomic ion, a reasonable estimate of zd is 400 pm. A typical capacity observed at the Hg j aqueous solution interface is 20 mF cm2 . The resulting value of eH is 9. This simple calculation shows that the properties of the solvent at the interface are very diﬀerent from those in the bulk of the solution where most polar solvents have considerably higher relative permittivities. The above model is naı¨ ve in so far as it assumes that all the charge on the solution side is located on one plane. Although the Helmholtz model satisﬁes the electrostatic requirement of charge neutrality at the interphase, it ignores the randomizing eﬀects of thermal motion. This was recognized independently by Gouy [21] and Chapman [22] who proposed that the double layer involves a diﬀuse layer of charge extending a ﬁnite distance into the solution. As one would expect, the average thickness of the diﬀuse layer depends on electrode potential and electrolyte concentration; as potential and concentration increase, the diﬀuse character of this region is reduced and the average thickness decreases. The Gouy–Chapman theory does not consider the ﬁnite size of the ionic constituents of the diﬀuse layer. It follows that this model predicts that the capacity of the double layer goes to inﬁnity as the ﬁeld due to the electrode charge increases in the positive or negative direction. As pointed out by Stern [23], this problem can be resolved by combining the Helmholtz model with the Gouy–Chapman model of the diﬀuse layer. In this case one recognizes that the ionic constituents of the diﬀuse layer are limited in their approach to the interface by their size. The plane of closest approach of the counter ion is located at distance xd from the geometrical boundary of the interface, and is called the outer Helmholtz plane (oHp). It follows that the capacity of the double layer is composed of a series combination of the capacity due to the Helmholtz or inner layer, Ci , and that due to the diﬀuse layer, Cd : 1 1 1 ¼ þ C Ci Cd

ð10:4:2Þ

In the absence of ionic speciﬁc adsorption, the properties of the inner layer are determined by those of the polarizable metal and of the solvent. Some small inﬂuence is also seen from the counter ion at the oHp. The latter eﬀect is attributed to changes in the size of the counter ion, which results in a change in the total thickness of the inner layer. At potentials well removed from the PZC, the diﬀuse layer capacity is quite large, so that it gives only a minor contribution to the observed experimental capacity. As noted above, ionic speciﬁc adsorption occurs when the total surface excess of a given ion is greater than that which can be accounted for on the basis of electrostatics. It arises because of the formation of a covalent bond between the given ion and the atoms of the metal electrode, or because the ion is hydrophobic and therefore accumulates at the interface. For these reasons it is assumed that a speciﬁcally adsorbed ion displaces solvent molecules from the monolayer of solvent solvating the electrode (see ﬁg. 10.14). Stern [23] treated speciﬁc adsorption using the Langmuir adsorption isotherm, thereby emphasizing that the adsorbed ions interact chemically with the polarizable metal. Grahame [G1] emphasized

532

LIQUIDS, SOLUTIONS, AND INTERFACES

that the speciﬁcally adsorbed ions are often closer to the electrode than the counter ions at the oHp. He introduced the term inner Helmholtz plane (iHp) to refer to the location of the charge centers of these ions. This picture of the solution side of the double layer is called the Gouy–Chapman–Stern–Grahame model and is shown schematically in ﬁg. 10.14. From the time of Grahame’s famous review [G1], published in 1947, until the early 1980s, the model of the double layer was based on the above picture. However, during this period of time, experiments were carried out on polarizable metals other than mercury which showed that the nature of the metal plays an important role in determining double layer properties [G3]. Experiments on soft polarizable metals such as bismuth, tin, cadmium, and zinc, on other liquid metals such as gallium and various amalgams, and on single crystal electrodes such as silver and gold showed that inner layer properties depend markedly on metal nature. Moreover, in the case of single crystals, it depends on the crystallographic orientation of the metal which is exposed to the solution. In addition, the PZC depends on the nature and crystal character of the metal. This was investigated in

Fig. 10.14 Schematic diagram of the double layer according to the Gouy– Chapman–Stern–Grahame model. The metal electrode has a net negative charge and solvated monatomic cations deﬁne the inner boundary of the diﬀuse layer at the outer Helmholtz plane (oHp). Monatomic anions are contact adsorbed on the electrode at the inner Helmholtz plane (iHp). Their presence is stabilized by the formation of images in the conducting electode so that each adsorbed ion establishes a dipole at the interface.

THE ELECTRICAL DOUBLE LAYER

533

detail by Trasatti [24], who emphasized the relationship between the PZC and the work function of the metal. On the basis of these studies, it was clear that the metal could not be treated simply as a structureless perfect conductor. Both the spatial arrangement of the surface atoms in the metal and their associated electron cloud play a role in determining double layer properties. In 1983, the jellium model was introduced to account for the eﬀects of the metal at the polarizable interface [25, 26]. In its simplest version, this model treats the metal lattice of positively charged ions as a constant positive background charge. The electrons are modeled as an inhomogeneous electron gas which interacts with the positive background. At the surface of the metal crystal, which is called the jellium edge, the positive background drops abruptly to zero. The properties of the metal are determined by the electronic density in this model. This aﬀects the electronic density proﬁle at the surface, where the electrons spill out of the atomic lattice over a distance of 100–200 pm. This leads to the surface potential wm at the metal surface, as discussed in section 8.5. The surface potential is an important contribution to the electronic work function of the metal which depends on its electronic density in a complex way [26]. When the metal comes in contact with an electrolyte solution, the extent of electron spillover changes. Water molecules interact to some extent with the electrophilic metal atom lattice through the electron density on the oxygen atom. As a result, the electron overspill at the PZC is less than it is at the metal j vacuum interface. Furthermore, the extent of this overspill changes when the electrode is polarized to potentials away from the PZC. When the electrode carries a positive charge, the electron overspill is reduced somewhat; when the electrode is negatively charged, it is extended. If the surface potential of the metal at the metal j solution interface is denoted as gm (see section 8.8), then it is clear that this quantity varies with electrode charge density sm . As a result there is a contribution to the interfacial capacity from the metal which is deﬁned as 1 @gm ¼ Cm @sm

ð10:4:3Þ

Recalling that wm is deﬁned for the movement of a test charge from vacuum into the metal, and deﬁning gm in the same way for movement of a test charge from solution into the metal, gm is a positive quantity at the PZC. As sm increases from negative values to positive ones, gm becomes smaller. As a result, Cm is a negative quantity. Its role in the overall picture of the electrical double layer is to introduce the polarizability of the metal s electronic cloud into the interfacial model. The net capacity of the inner layer with consideration of the metal is now 1 1 1 ¼ þ Ci Cm Csolv

ð10:4:4Þ

where Csolv is the contribution to the inner layer capacity from the solvent molecules. Since physically relevant capacities are always positive, the ﬁrst term on the right-hand side of equation (10.4.4) must be smaller in magnitude than the second one. As will be seen in the following sections of this chapter, the inner layer capacity Ci is the dominant contribution to the experimental capacity C over

534

LIQUIDS, SOLUTIONS, AND INTERFACES

most of the polarizable range of the electrode. The metal contribution is such that the observed capacity is higher than it would be at a structureless perfect conductor. The simple jellium model, in which the metal atom lattice is treated as a constant positive background charge, does not account for eﬀects observed at single crystal surfaces. The jellium model may be extended to take into account crystal structure by replacing the positive background charge with a lattice of a pseudopotentials. At each lattice site the electrons are subjected to a potential which is governed by the electronic repulsion from the atomic cores at short distances, and by Coulomb’s law at large distances. The electronic density proﬁle within the crystal and its surface using a simple form of the pseudopotential model is shown in ﬁg. 10.15. Within the crystal the electron density oscillates, reaching maxima in the vicinity of a metal atom (ion). The electron overspill at the surface depends on the surface plane which is exposed. Thus, the revised jellium model is able to account for the dependence of the work function and PZC of single crystal electrodes on the crystallographic orientation at the interface. The introduction of pseudopotentials represents an important extension of the jellium model. It is clear from the above discussion that three aspects of the electrical double layer must be considered in order to understand experimental observations and double layer phenomena. The ﬁrst of these is the role of the metal and its inﬂuence on double layer properties. The second aspect concerns the inner layer or region immediately next to the metal. In the simplest case, this region is occupied only by solvent molecules. If adsorption is present, then some of these molecules are replaced by ions or solute molecules. In many cases the inner layer plays a dominant role in determining interfacial capacity. Thus, considerable eﬀort has been expended to develop models for solvent structure in this region and adsorption.

Fig. 10.15 Distribution of electron density within a Ag lattice at the (111) surface on the basis of the jellium model with pseudopotentials located at the metal atom cores as indicated by the arrows [26]. The broken vertical line shows the position of the metal surface.

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535

Finally, the third aspect is the diﬀuse layer. This has traditionally been treated by the Gouy–Chapman theory. Recently, attempts have been made to develop an improved model of the diﬀuse layer which is analytical and considers the eﬀects of ﬁnite ion size and change in dielectric properties. These aspects of the double layer are now considered in more detail.

10.5 The Potential of Zero Charge and the Role of the Metal The PZC is an important point of reference in discussing the properties of a polarizable interface. Its location depends on the nature of all of the components which are at the interface, that is, on the metal, on the solvent used for the electrolyte solution, and on the nature and concentration of the solute components in this solution. Its importance was ﬁrst pointed out by Frumkin [G3] who was able to carry out the ﬁrst experiments at polarizable electrodes other than mercury. He showed that there is a fairly simple relationship between the PZC and the work function of the metal for a given solution composition and reference electrode. In this section the relationship is derived and its consequences illustrated with experimental data. Then a model which describes the role of the metal in interfacial properties, namely, the jellium model, is presented. A. The Potential of Zero Charge and the Metallic Work Function In order to derive the relationship between the PZC and the work function, a cell with the conﬁguration Cu j M j 0:1 M KCl; H2 O j AgCl j Ag j Cu 0

ð10:5:1Þ

is considered, where M is a polarizable metal. The most studied metal in this group is mercury but it also includes metals such as gold, tin, lead, and bismuth within a deﬁned potential range. Since the silver j silver chloride reference electrode is reversible to the anion of the electrolyte the potential of the cell is designated as E and is given by the Galvani potential diﬀerence between the copper wires at the terminals. Thus, 0

E ¼ Cu0 Cu f ¼ fCu fCu

ð10:5:2Þ

This can be further decomposed into four Galvani potential diﬀerences as follows: 0

E ¼ ðfCu fm Þ þ ðfm fs Þ þ ðfs fAg Þ þ ðfAg fCu Þ

ð10:5:3Þ

The ﬁrst and the last of these may be re-expressed in terms of the standard potentials of the electrons in the pure metal because of the electronic equilibria which exist at the metal j metal interfaces. As a result, ! mAg mm el E ¼ s m f s Ag f el ð10:5:4Þ F F

536

LIQUIDS, SOLUTIONS, AND INTERFACES

The ﬁrst term on the right-hand side relates to the properties of the polarizable electrode and the second to those of the reference electrode. By adding the surface potential of the solution ws to each term within the brackets, these terms may be related to the absolute potentials for the electrodes (see section 9.4). Thus, ! mAg mm el s s el þ w s Ag f þw E ¼ s m f F F ð10:5:5Þ mm el s a ¼ s m f þ w EAg=AgCl F where E aAg=AgCl is the potential of the Ag j AgCl electrode on the absolute scale. The focus of interest in this section is the Galvani potential drop s m f. As discussed in section 8.8, this can be resolved into three contributions, so that s m f

¼ fm fs ¼ s f þ gm gs

ð10:5:6Þ

s f is the potential drop due to the net free charge at the interface; g is the dipolar potential due to the metal phase, more speciﬁcally, to the electron overspill that occurs at the surface of the metal; ﬁnally, gs is the dipolar potential due to the solution phase which arises because of the orientation of solvent molecules at the interface due to their proximity to the metal, and because of the unequal distances of closest approach of the cations and anions to the interface. gs is deﬁned in the opposite direction to gm because the concept of the dipolar potential originates at the condensed phase j vacuum interface where the deﬁnition of the potential drop is always from vacuum to the condensed phase. The dipolar potential gm arises for the same reasons as the surface potential wm at the metal j vacuum interface. However, it is not the same because of the eﬀect that the proximity of the molecules and ions of the solution phase have on the electron overspill. At the PZC, the contribution s f is zero and equation (10.5.6) becomes m

s m f0

s ¼ gm 0 g0

ð10:5:7Þ

where the subscript ‘‘0’’ indicates the values which apply at this point of reference. Combining equations (10.5.5) with equation (10.5.7), one obtains the result s E0 ¼ gm 0 g0

mm el þ ws E aAg=AgCl F

ð10:5:8Þ

E0 is the PZC measured with respect to the silver j silver chloride reference electrode. The next step is to relate the PZC to the work function for metal M, namely, Wm el . The work function is given by m m Wm el ¼ mel þ Fw

ð10:5:9Þ

where wm is the surface potential of the metal. Now combining equations (10.5.8) and (10.5.9), one obtains E0 ¼

Wm el m s s a þ ðgm 0 w Þ ðg0 w Þ E Ag=AgCl F

ð10:5:10Þ

THE ELECTRICAL DOUBLE LAYER

537

This result shows that the relationship between the PZC and the work function is simple only when the dipole contributions are independent of metal nature. The contribution from the reference electrode is constant, provided the electrolyte solution remains unchanged when the polarizable metal is changed. As will be shown below, the dipole potential terms deﬁnitely depend on the nature of the metal. The PZC has been determined for a wide range of metals in diﬀerent aqueous electrolyte solutions. Examination of these data shows that it varies considerably with both the nature of the metal and of the electrolyte. Values of this quantity determined at the metal j water interface in the absence of ionic adsorption are summarized in table 10.1. Considerable interest has been expressed in the relationship between the PZC and the work function of the metal, W m el . When the SHE is used as reference, equation (10.5.10) becomes E0 ¼

Wm el m s s a þ ðgm 0 w Þ ðg0 w Þ E SHE F

ð10:5:11Þ

where E aSHE is the absolute potential of the SHE. The SHE is the most common reference point used to cite experimental values of the PZC but in fact any convenient reference electrode may be used. In any case, the point of reference must be cited. A plot of Ez , that is, E0 against W m el using data for the sp metals is shown in ﬁg. 10.16. A non-linear correlation between these quantities is obtained, the value of Ez becoming more positive with increase in W m el . These results show clearly that the terms describing the dipole potentials vary with the nature of the metal. The sp metals have low melting points, some of them being liquids at room temperature. Thus, the fact that the data are for the polycrystalline form is not terribly important. If they all interacted with water to the same degree, the plot of Ez against W m el should be linear with unit slope. In other words, one would expect the dipole potential terms to be constant. In fact, the slope of the plot in ﬁg. 10.16 Table 10.1 Point of Zero Charge with Respect to the SHE for Various Metals in Water in the Absence of Ionic Adsorption [27] Metal

Ez / V

Metal

Ez / V

Hg Other sp metals Bi Cd Ga In Pb Sb Sn Tl Zn

0.192

Single crystal sd metals Ag (100) Ag (110) Ag (111)

0.62 0.73 0.46

0.38 0.74 0.67 0.65 0.60 0.17 0.39 0.71 0.92

Au (100) Au (110) Au (111)

0.33 0.19 0.56

Cu (100) Cu (110) Cu (111)

0.54 0.69 0.20

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LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 10.16 Plot of Ez , the point of zero charge for sp metals in an aqueous solutions with respect to the SHE with a non-adsorbing electrolyte against W m el , the vacuum work function of the metal. The dotted line indicates unit slope.

is close to unity for the metals with the largest work functions, namely, mercury and tin. However, as the work function decreases the slope of the plot also decreases. This has been discussed in terms of the hydrophilicity of the metal. Thus, examination of double layer data has led to the conclusion that gallium is more hydrophilic than mercury [24, 28]. This means that more water molecules are adsorbed on the metal with the negative electron density on the oxygen atom interacting with it. As a result, the dipole term gs becomes more negative as the metal is changed in the series Hg < Ga < Zn. As a result, Ez becomes more positive in the same series than it would be if gs were constant. The magnitude of the eﬀect can be judged by drawing a straight line with unit slope through the point for Hg on the plot. Examination of the data shows that the eﬀect is quite large. Other relevant data are available for single crystals of the sd metals, namely, copper, gold, and silver. The data for the low-index faces of these systems show interesting trends with crystallographic orientation and with metal nature. As the density of atoms on the metal face increases both the work function and PZC move in the positive direction. In addition, for a given crystallographic orientation both parameters increase in the order Cu < Ag < Au. These trends demonstrate a corresponding decrease in the hydrophilicity of the metal with a given crystallographic orientation at the surface, and a corresponding decrease in the magnitude of the dipole potential gs . Some data are also available for the variation in the PZC with metal nature in non-aqueous solvents [27]. These results are much more limited in terms of the number of metals studied but they show the same trends that are found in the case of water.

THE ELECTRICAL DOUBLE LAYER

539

B. The Jellium Model for the Metal The role of the metal in double layer properties can be understood in greater detail when the system is examined on the basis of the jellium model. This model was developed to describe the electron gas within sp metals. It can be used to estimate several properties of interest, including the chemical potential of an electron in the metal, the extent of electron overspill, and the work function of the metal. More recently, it has been extended to describe metal surfaces in contact with polar solvents [26]. In its simplest form, the metal atoms in the metal are modeled as a uniform positive background for the electron gas, no consideration being given to their discrete nature and position in the metal lattice. The most important property of the system is the average electron density, hNel i, which depends on the number of metal atoms per unit volume and the number of valence electrons per atom, nv . Thus, if rm is the mass density of the metal, and Mm , its atomic mass hNel i ¼

N L nv r m Mm

ð10:5:12Þ

where NL is the Avogadro constant. Values of hNel i for some typical sp metals are given in table 10.2. The electronic work function depends on both the electron density and on the extent of electron overspill, which itself is related to the electron density. This suggests that there should be a correlation between the work function and the electron density. The correlation for the sp metals listed in table 10.2, which is approximately linear, is shown in ﬁg. 10.17. The deviations Table 10.2 Average Electron Density hNel i in Typical sp Metals at 258C Together with the Number of Valence Electrons nv , the Metallic Density rm , and the Pseudopotential Radius rc Metal

nv

rm =g cm3

hNel i/electrons nm3

rc /pm

Li K Cs

1 1 1

0.534 0.869 1.879

46.3 13.7 8.5

70 117 146

Mg Ca Ba

2 2 2

1.74 1.54 3.51

86.2 46.3 30.8

69 92 107

Zn Cd Hg

2 2 2

7.14 8.64 13.59

131.5 92.6 81.6

57 67 69

Al Ga In

3 3 3

2.70 5.90 7.30

133.9 152.9 114.9

59 63 72

Sn Pb

4 4

6.54 11.34

132.7 131.8

59 57

*Values of rc were taken mainly from S. Elliot, The Physics and Chemistry of Solids, Wiley, New York (1998).

540

LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 10.17 Plot of the metallic work function W m el against the average electron density hNel i for sp metals.

from perfect linearity can be attributed to the fact that the simple jellium model ignores the structure of the metal lattice. The description of the metal is improved considerably if metallic structure is introduced by accounting for the local attractive force of the metal atoms on the free electron gas. This corresponds to the jellium model with pseudopotentials. Each metal atom in the lattice is pictured as being surrounded by a spherical volume Vc in which electrostatic eﬀects may be ignored. Outside of the sphere the metal atom behaves as a point charge of charge number nv . Thus, it has a pseudopotential fc , where fc ¼

nv e0 4pe0 r

ðr rc Þ

ð10:5:13Þ

and rc is the radius corresponding to spherical volume Vc . The spacing of the metal atoms in a given direction in the crystal is dm , and depends on both the nature of the crystal lattice and on the direction. It follows that, when jellium with pseudopotential is adopted, two additional parameters are introduced, namely, rc and dm . The parameter dm may be calculated from the metallic density rm , provided that lattice structure and direction have been speciﬁed. The radius rc is estimated on the basis of a quantum-mechanical calculation and is given in table 10.2. In the present discussion, attention is focused on the properties of the electron gas at the surface of the metallic crystal. As far as the metal atoms are concerned the lattice terminates sharply at the surface, which is called the jellium edge in this model. The important feature of the electron cloud is that it spills out over the edge of the lattice to an extent which is determined by lattice properties such as rc and dm , and by electron gas properties such as hNel i. When the jellium model is

THE ELECTRICAL DOUBLE LAYER

541

applied, one can determine Nel ðxÞ, that is, the electron density as a function of position in the crystal and at its edge (x ¼ 0). Typical results with pseudopotentials for the individual metal atoms included in the model are shown in ﬁg. 10.15. The electron overspill at the jellium edge determines the surface potential wm of the metal, and has an inﬂuence on the metallic work function. In order to determine the electronic density proﬁle Nel ðxÞ the surface energy of the electronic plasma Us must be determined. This is written as a functional of the density proﬁle as follows: Us ½Nel ðxÞ ¼ Ubulk þ Uion þ Udip þ Upsp þ Urep

ð10:5:14Þ

where Ubulk gives the bulk contributions to the energy, Uion , that due to electrostatic interaction of the plasma with ions in solution, Udip , the corresponding term for solvent dipoles, Upsp , that portion of the surface energy which arises from the introduction of pseudopotentials in the model, and Urep , the repulsion of the electrons from the solution due to short-range forces [26]. Ubulk contains six diﬀerent contributions which may be estimated from the quantum-mechanical description of metals [26, 29]. They are the kinetic energy, the exchange energy, and the correlation energy of the plasma, its electrostatic self-interaction, its interaction with the positive background charge, and the inhomogeneity contribution. Expressions for each of these terms have been given by Smith [29]. Uion , Udip and Urep are described by expressions given by Schmickler and Henderson [30]. At this point one must choose a functional form for Nel ðxÞ. This choice is based on what is known about the physical description of the system. Several appropriate functions have been described in the literature, a choice which has met with considerable success being the following [30]: ajel sm ajel x 1 Nel ðxÞ ¼ hNel i e hNel i þ ðx < 0Þ e 2 0 ð10:5:15Þ ajel sm ajel x 1 hNel i e ¼ ðx > 0Þ 2 e0 These functions are only applied to the potential region near the PZC. One new parameter namely, ajel , which is the characteristic decay length for the jellium plasma, is introduced in these functions. Now, the surface energy Us (equation (10.5.14)) is minimized by introducing the functions (10.5.15) into the appropriate terms and varying ajel . It turns out that ajel depends only weakly on hNel i and has a value close to 23 nm1 for the jellium model without pseudopotentials. Now, the various surface properties may be calculated, including the surface energy Us , the surface potential wm , and the electronic work function Wm el . Attention is focused on wm in the following discussion. According to the jellium model as described above, the surface potential at the PZC is given by wm 0 ¼

e0 hNel i e0 a2jel

ð10:5:16Þ

542

LIQUIDS, SOLUTIONS, AND INTERFACES

Thus, the surface potential is positive, and since ajel is approximately constant, it is approximately proportional to the average electron density. In the case of mercury for which hNel i is 81.6 electrons nm3 , wm 0 is equal to 2.79 V. It is positive because it is measured from vacuum into the interior of the metal, which is positive with respect to the electron cloud at the metal surface. When the metal is charged, the electron overspill changes in response to the sign and magnitude of the charge density on the metal. If the metal is positively charged, the extent of electron overspills is reduced and wm is smaller in magnitude. On the other hand, if the metal is negatively charged, the electron overspill extends further from the metal and wm is larger. In order to extend the above treatment to the metal j solution interface, one must consider the eﬀect of the solvent molecules adsorbed on the metal on the electronic overspill. Because the solvent molecules are polarizable, an induced dipole moment is established in the solvent monolayer, which acts to reduce the extent of overspill. As a result, the dipolar potential due to the metal is reduced by a factor corresponding to the optical permittivity of the monolayer, eop . Recalling that this dipole potential is designated as gm , one has at the PZC gm 0 ¼

e0 hNel i eop e0 a2jel

ð10:5:17Þ

In the case of water, eop is 1.776, so that gm 0 is equal to 1.57 V. In assessing this result one should remember that the total surface potential has two opposing contributions, namely, gm and gs : gs is also large and contains the contribution from orientation of solvent dipoles in the monolayer at the interface. By measuring the change in gm with charge density on the metal, the metal’s contribution to the inner layer capacity can be estimated. Thus, 1 @gm e0 hNel i @ajel ¼ ¼ Cm @sm eop e0 a3jel @sm

ð10:5:18Þ

Cm is negative, reﬂecting the fact that gm decreases in magnitude as sm increases from negative to positive values. Leiva and Schmickler have estimated Cm to be 47:6 mF cm2 at the PZC for the Hg j aqueous solution interface in the absence of ionic adsorption [31]. The parameters discussed here have been estimated for a number of metals at which double layer studies have been carried out [26, 30]. However, their signiﬁcance to the experimental data cannot be properly assessed without considering also the contribution of the solvent monolayer to interfacial properties. This subject is considered later in section 10.7 in the discussion of the solvent’s role in the double layer.

10.6 The Gouy–Chapman Model of the Diffuse Double Layer The model of the diﬀuse layer which is still used today was developed by Gouy [21] and Chapman [22] in the early part of the twentieth century. It is based on a

THE ELECTRICAL DOUBLE LAYER

543

one-dimensional solution of the Poisson–Boltzmann equation to yield the electrostatic and thermodynamic properties of the diﬀuse layer. It preceded the much better-known solution of the same equation by Debye and Hu¨ckel who used it to develop the ﬁrst theory of electrolyte solutions thirteen years later (section 3.8). The fundamental assumptions of the two theories are the same: the ions are assumed to be point charges with no volume, and the ionic distribution is determined by the laws of electrostatics with consideration of the randomizing eﬀects of thermal motion. In addition, the solvent is considered to be a dielectric continuum with a uniform relative permittivity equal to that of the pure solvent, es. The polarizable electrode is assumed to be a homogeneous conductor in which the charge distribution is unaﬀected by the presence of discrete ions in the adjacent solution. As was seen earlier, the Poisson equation, which is a combined statement of Coulomb’s and Gauss’ laws, may be written as rz ð10:6:1Þ r2 ¼ es e0 where is the local potential at a given point in the medium, rz, the charge density due to ions at that point, and e0, the permittivity of free space. In the Gouy–Chapman (GC) solution of this equation, the variation in inner potential is only considered in a direction perpendicular to the interface. Variation in in the (x; y)-planes which are parallel to the interface is ignored. This is equivalent to considering the point charges in any given (x; y)-plane as being smeared out. It also means that the local potential calculated at any distance z from the interface is the average potential fz estimated without consideration of the discrete charge located on the corresponding (x; y)-plane. It follows that the Laplacian in can be replaced by the derivative d2 fz =dz2 . Thus, equation (10.6.1) becomes d2 fz rz ¼ 2 es e0 dz

ð10:6:2Þ

The Boltzmann part of the model involves development of an expression for the charge density rz. For any ion i, its electrochemical potential in the diﬀuse layer must be equal to that in the bulk of the solution, that is, ~si ~zi ¼ m m

ð10:6:3Þ

~si , ~zi is the electrochemical potential at position z in the diﬀuse layer, and m where m that in the bulk of the solution. Since variation in potential is only considered in ~zi is the z-direction, the expression for m ~zi ¼ mi þ RT ln czi þ zi Ffz m

ð10:6:4Þ

where mi is the standard chemical potential, czi , the concentration of ion i at position z, fz the average potential at position z, and zi, the charge number of ion i. The corresponding equation in the bulk of the solution is ~si ¼ mi þ RT ln csi þ zi Ffs m

ð10:6:5Þ

where csi is the bulk concentration of ion i and fs , the average potential in the bulk. In the following discussion, fs is set equal to zero so that all potentials in the

544

LIQUIDS, SOLUTIONS, AND INTERFACES

double layer are referred to that in the bulk. It is emphasized that the assumptions made in writing equations (10.6.4) and (10.6.5) are equivalent to neglecting the eﬀects of non-ideality due to the ionic atmosphere which is present on a given plane in the diﬀuse layer. In addition, by setting fs equal to zero, the eﬀect of nonideality in the bulk of the electrolyte solution is neglected. Combining equations (10.6.3)–(10.6.5), one obtains the result czi ¼ csi expðzi ffz Þ

ð10:6:6Þ

where f ¼ F=ðRTÞ. This is the Boltzmann equation for ion i in the diﬀuse layer. The charge density due to this ion at position z is rzi ðzÞ ¼ zi Fczi

ð10:6:7Þ

The total charge density is obtained by adding up the contributions from each type of ion: X X X rzi ðzÞ ¼ zi Fczi ¼ zi Fcsi expðzi ffz Þ ð10:6:8Þ rz ¼ i

i

i

Finally, the Poisson–Boltzmann equation for the diﬀuse layer is X zi Fcsi d2 fz ¼ expð zi ffz Þ e e dz2 s 0 i

ð10:6:9Þ

In order to solve this diﬀerential equation both sides are multiplied by 2 dfz/dz to obtain z X zi Fcsi dfz d2 fz d df 2 z df ð10:6:10Þ 2 ¼ ¼ 2 expð z ff Þ i dz dz dz dz2 es e0 dz i The equation to be solved is thus z 2 X 2zi Fcsi df d ¼ expð zi ffz Þdfz dz e e s 0 i

ð10:6:11Þ

This equation can now be integrated from position z in the diﬀuse layer to the bulk of the solution. The quantity dfz =dz is the electrical ﬁeld in the diﬀuse layer perpendicular to the interface. This ﬁeld goes to zero in the bulk of the solution. After integration, one obtains z 2 X df 2RTcsi ¼ ½expð zi ffz Þ 1 ð10:6:12Þ dz e e s 0 i This result can be used to estimate the ﬁeld at any position in the diﬀuse layer. It is the fundamental equation of the GC model and is used to derive all the equations which follow. In the case of a simple 1–1 electrolyte, equation (10.6.12) has a much simpler form which is often seen in presentations of this model. If the bulk concentration of the electrolyte is denoted as ce, one may write z 2 df 2RTce 4RTce ¼ ½expð ffz Þ þ expð ffz Þ 2 ¼ ½coshð ffz Þ 1 ð10:6:13Þ dz e0 es e0 es

THE ELECTRICAL DOUBLE LAYER

Using the identity coshð ffz Þ 1 ¼ 2 sinh2

z ff 2

the expression for the ﬁeld becomes z z dfz 8RTce 1=2 ff 2k ff ¼ ¼ sinh sinh dz e 0 es 2 2 f

545

ð10:6:14Þ

ð10:6:15Þ

where k is the Debye–Hu¨ckel reciprocal distance introduced in section 3.8. It should be noted that equation (10.6.13) is a quadratic equation and therefore has two roots. Only the negative root is physically admissible. Thus, equation (10.6.15) shows that when the potential fz is positive, the ﬁeld dfz/dz is negative. Equations (10.6.12) and (10.6.15) are now used to derive the other important results from the GC model. A. The Potential Drop Across the Diffuse Layer The GC theory is used most often to estimate the potential drop across the diﬀuse layer. This quantity is important in colloid phenomena and electrode kinetics, for example. Noting that the inner boundary of the diﬀuse layer is located at distance zd from the interface, the square of the electrical ﬁeld at this location is given by z 2 X i df 2RTcsi h ¼ expð zi ffd Þ 1 ð10:6:16Þ dz d es e0 i where fd is the potential drop across the diﬀuse layer. This is a non-linear equation in fd which can be solved by iterative methods, provided that the value of the ﬁeld at the inner boundary, (dfz/dz)d, is known. The ﬁeld is not a quantity which is obtained experimentally so that fd cannot be calculated without further analysis. In the absence of speciﬁc ion adsorption the potential drop across the diﬀuse layer may be related to the charge density on the electrode sm by means of Gauss’ law. For the present system the relationship is z df sm ¼ e0 es ð10:6:17Þ dz d It follows that the electrode charge density is related to the potential drop by the equation i X h s2m ¼ 2GC ci expðzi ffd Þ 1 ð10:6:18Þ i

where

GC ¼ ð2RTe0 es Þ1=2

ð10:6:19Þ

GC is called the Gouy–Chapman constant. In the case of a simple 1–1 electrolyte at a concentration of ce M, equation (10.6.18) simpliﬁes to

546

LIQUIDS, SOLUTIONS, AND INTERFACES

s2m ¼ 2 2GC ce ½coshð ffd Þ 1 or using equation (10.6.14) ffd sm ¼ 2AGC sinh 2

ð10:6:20Þ

! ð10:6:21Þ

where AGC ¼ GC ce1/ 2. If the concentration ce is expressed in M, GC is equal to 5.8687 mC cm2 in water at 25 C. This is the well-known result relating the electrode charge density to the potential drop across the diﬀuse layer for a 1–1 electrolyte in the absence of speciﬁc ionic adsorption. Since fd has the same sign as sm, only the positive root of equation (10.6.20) is physically admissible. An analytical equation may also be written for fd as a function of sm: 2 !1=2 3 2RT 4 sm s2m d 5 ln f ¼ ð10:6:22Þ þ þ1 2AGC F 4A2GC EXAMPLE

Estimate the potential drop across the diﬀuse layer for an electrode charge density of 10 mC cm2 and a 1–1 electrolyte concentration of 0.1 M at 25 C. The GC constant AGC is given by AGC ¼ 5:8687 ð0:1Þ1=2 ¼ 1:856 mC cm2

ð10:6:23Þ

The ratio sm / (2AGC) is sm 10 ¼ 2:694 ¼ 2AGC 2 1:856

ð10:6:24Þ

From equation (10.6.22) one obtains fd ¼ 2 0:0257 ln½2:694 þ ð2:6942 þ 1Þ1=2 ¼ 0:0883 V

ð10:6:25Þ

Thus the potential drop across the diﬀuse layer is 88.3 mV. Plots of the potential drop across the diﬀuse layer are shown as a function of electrode charge density in ﬁg. 10.18 for a 1–1 electrolyte with concentrations in the range 0.01–1 M. As electrolyte concentration increases, the absolute value of fd decreases for constant charge density sm. This demonstrates that the screening ability of the electrolyte increases with the increasing concentration. The GC results are compared in ﬁg. 10.18 with Monte Carlo calculations of Boda et al. [32]. These were carried out assuming that the electrolyte ions are hard spheres with a diameter of 300 pm in a dielectric continuum. The estimates of fd using the Monte Carlo technique fall below the GC estimates. They demonstrate the importance of including ﬁnite ion size in a model of the diﬀuse layer. B. The Differential Capacity of the Diffuse Layer The diﬀuse layer capacity is an important quantity often used in the analysis of double layer capacity data. In the absence of speciﬁc adsorption, the diﬀuse layer capacity is deﬁned as

THE ELECTRICAL DOUBLE LAYER

547

Fig. 10.18 Plots of the potential drop across the diﬀuse layer fd against the electrode charge density for a 11 electrolyte at the concentrations indicated. The ﬁlled points show the results of Monte Carlo calculations assuming an ionic diameter of 300 pm [32].

Cd ¼

dsm dfd

ð10:6:26Þ

On the basis of equation (10.6.18), the diﬀuse layer capacity is equal to Cd ¼

2GC X z fc expðzi ffd Þ 2sm i i i

ð10:6:27Þ

For a 1–1 electrolyte a much simpler expression may be derived on the basis of equation (10.6.21), ! ffd Cd ¼ fAGC cosh ð10:6:28Þ 2 Using the identity cosh

2

! ! d ffd 2 ff sinh ¼1 2 2

ð10:6:29Þ

and equation (10.6.21), the diﬀuse layer capacity may also be written as !1=2 s2m ð10:6:30Þ Cd ¼ fAGC 1 þ 2 4AGC This result demonstrates that Cd increases with electrode charge density but that it is approximately independent of electrolyte concentration when sm is large. EXAMPLE

Estimate the diﬀuse layer capacity for an electrode charge density of 10 mC cm2 and a 1–1 electrolyte concentration of 0.1 M at 25 C. Repeat the calculation at the PZC.

548

LIQUIDS, SOLUTIONS, AND INTERFACES

From the earlier example, under these conditions AGC is equal to 1.856 mC cm2 and sm / (2AGC), to 2.694. It follows from equation (10.6.30) that Cd ¼ 38:92 1:856ð1 þ 7:258Þ1=2 ¼ 207:6 mF cm2

ð10:6:31Þ

When sm is equal to zero, the result is Cd ¼ 38:92 1:856 ¼ 72:24 mF cm2

ð10:6:32Þ

These results demonstrate that Cd is usually much greater than the experimentally observed capacity. Only when the concentration is very low so that the GC constant AGC is also low, does Cd reach values which are less than the experimentally observed capacity. Plots of the diﬀuse layer capacity Cd against electrode charge density are shown in ﬁg. 10.19. At low concentrations, Cd rises sharply from its minimum value at the PZC and reaches values which are more than 10 times the experimentally observed capacity. Cd is low and close to the experimentally observed capacity only in the region of the PZC. Thus, in very dilute solutions the position of the minimum in the experimental capacity can be used to locate the PZC if ionic adsorption is absent. Using estimates of Cd based on equation (10.6.27) or (10.6.28), the experimental capacity C may be corrected for the diﬀuse layer contribution to obtain the capacity of the inner layer Ci. Since Cd is usually much greater than C, errors in the GC estimate of Cd are not important except in the vicinity of the PZC, where Cd falls to its lowest values. When the experimental capacity is obtained at constant electrode charge density and as a function of electrolyte concentration, the GC estimates of Cd are often used to determine whether ionic speciﬁc adsorption

Fig. 10.19 Plots of the diﬀuse layer capacity Cd according to the Gouy–Chapman model gainst the electrode charge density sm for a 11 electrolyte at the concentrations indicated.

THE ELECTRICAL DOUBLE LAYER

549

is present using equation (10.4.2). In this test, the experimental values of 1/C are plotted against the GC estimates of 1/Cd. If a plot with unit slope is obtained, it is concluded that the inner layer capacity Ci is independent of electrolyte concentration, and that ionic adsorption is absent. The value of Ci may be found from the intercept. These plots are known as Parsons–Zobel plots. Another application of the same plot is to determine surface roughness of a solid metal electrode. With these systems the apparent surface area is used to estimate the speciﬁc diﬀerential capacity C. This quantity may diﬀer from the true surface area because of surface roughness. In this case, the slope of the Parsons–Zobel plot is not unity, but equal to a value which reﬂects the ratio of the true surface area to its apparent value. Examination of experimental data for liquid metals such as mercury reveals that errors in the GC estimates of Cd are not important near the PZC. However, this simple model certainly overestimates Cd further from the PZC. According to Monte Carlo calculations [32], Cd reaches a maximum and then becomes negative when sm is suﬃciently large. Studies of the diﬀuse layer capacity using more detailed models are the subject of current research. C. The Potential Proﬁle in the Diffuse Layer The potential in the diﬀuse layer is determined as a function of distance from the oHp by integrating the ﬁeld as given by equation (10.6.12) or (10.6.15). An analytical expression is only obtained for the case of symmetrical electrolytes. The derivation presented here is for 1–1 electrolytes for which the result obtained below is most often applied. On the basis of equation (10.6.15) the diﬀerential equation to be solved to obtain the potential proﬁle in the diﬀuse layer for a 1–1 electrolyte is z ff 2k z csch dz ð10:6:33Þ df ¼ f 2 The solution to this equation is z f ð

z ðz ff 2k z csch df ¼ dz f 2

fd

ð10:6:34Þ

zd

where the lower limit for integration is the oHp, which is located at a distance zd from the geometrical metal j solution interface. Integration gives ! z ff ffd ln tanh ð10:6:35Þ ln tanh ¼ kðz zd Þ 4 4 This result may also be expressed as ! z ff ffd tanh tanh expðkðz zd ÞÞ 4 4

ð10:6:36Þ

550

LIQUIDS, SOLUTIONS, AND INTERFACES

When the argument of the tanh function is small, this function may be replaced by the argument. Under these circumstances the potential proﬁle is given by 4RT fd fz ¼ fd expðkðz zd ÞÞ ð10:6:37Þ F It follows that fd decays exponentially with distance from the oHp under these conditions. The potential proﬁle is the least reliable feature of the GC model. Certainly, Monte Carlo calculations in which the ions are represented as charged hard spheres in a dielectric continuum show that the GC potential proﬁle is seriously in error at high electrolyte concentrations. However, it is sometimes used at very low concentrations to obtain an approximate idea of potential variation in the diﬀuse layer.

D. The Ionic Surface Excesses in the Diffuse Layer In order to estimate the surface excess of any ion in the diﬀuse layer its concentration proﬁle must be integrated in the diﬀuse layer. Thus, a general deﬁnition of the surface excess is 1 ð

i ¼

1 ð

ðczi

csi Þdz

¼

zd

½expðzi ffz Þ 1dz

csi

ð10:6:38Þ

zd

In order to integrate this equation, one must make use of the general relationship for the ﬁeld in the diﬀuse layer; that is, equation (10.6.12). In the following derivations only 1–1 electrolytes are considered, so that the expression used for the ﬁeld is equation (10.6.15). The surface excess of the monovalent cation in a 1–1 electrolyte is given by fce þ ¼ 2k

ð0 fd

expðffz Þ 1 z df sinhð ffz =2Þ

ð10:6:39Þ

Factoring out exp(f fz =2) in the numerator, one obtains fc þ ¼ e k

ð0 f

" ! # ffz 2ce ffd z exp exp df ¼ 1 2 k 2

ð10:6:40Þ

d

This is usually expressed in terms of sdþ , the excess charge in the diﬀuse layer due to cations. Thus, sdþ

" ! # " ! # 2Fce ffd ffd exp ¼ Fþ ¼ 1 ¼ AGC exp 1 k 2 2

ð10:6:41Þ

THE ELECTRICAL DOUBLE LAYER

551

It is easily shown in a similar way for the monovalent anion that sd

" ! # " ! # 2Fce ffd ffd exp ¼ F ¼ 1 ¼ AGC exp 1 k 2 2

ð10:6:42Þ

Plots of sdþ and sd against electrode charge density are shown in ﬁg. 10.20. The surface excess of the ion whose charge is opposite to that on the electrode is large and somewhat smaller in magnitude than that of the electrode charge. As electrolyte concentration increases, the magnitude of this surface excess decreases. On the other hand, the surface excess of the ion with the same sign as that of the charge on the electrode is small in magnitude and approximately constant for low electrolyte concentration and high electrode charge densities. It has the same sign as the excess for the predominant ion, indicating that it is actually in deﬁcit in the diﬀuse layer. Thus, a negatively charged electrode causes an excess of cations and a deﬁcit of anions in the diﬀuse layer. As electrolyte concentration increases, the magnitude of this deﬁcit increases. In summary, since the surface excesses are obtained by integrating over the whole diﬀuse layer, the GC estimates are assumed to give reasonable values for these quantities. The estimates are often used in the analysis of double layer data involving ionic adsorption in the inner layer. Considerable eﬀort has been made in recent years to improve the GC model. Early work [33] was carried out at the primitive level with the solvent represented as a dielectric continuum and the ions as hard spheres. The integral equation approach was one method applied to this problem. This work was followed by Monte Carlo studies [32]. The general result of these studies is that the GC model overestimates the magnitude of the diﬀuse layer potential drop (see ﬁg. 10.18).

Fig. 10.20 Plots of the ionic surfaces excesses sdþ (*) and sd (^) in the diﬀuse layer against electrode charge density sm for a negatively charged electode. Data are shown for a 1–1 electrolyte at 0.1 and 1 M.

552

LIQUIDS, SOLUTIONS, AND INTERFACES

10.7 The Structure of the Inner Layer in the Absence of Adsorption When adsorption is absent, the inner layer is populated by solvent molecules only. The change in number and orientation of these molecules with electrode potential or electrode charge density is considered to be the main reason for the experimental variation in inner layer capacity, Ci. The ﬁrst system to be examined extensively in the laboratory was the Hg / aqueous solution interface with NaF as electrolyte [34]. At lower temperatures, the inner layer capacity curve possesses a maximum at a small charge density just positive of the PZC (ﬁg. 10.21). Minima are observed at sm ¼ 10 mF cm2 and at 11 mF cm2. At the extremes of polarization, the inner layer capacity rises. As the temperature increases, the central maximum disappears, and at suﬃciently high temperature the maximum and minimum at positive potentials are no longer apparent. This behavior has been interpreted as an indication of solvent reorientation, the maximum on the capacity curve corresponding to the charge density at which net polarization in the solvent layer is zero. As the electrode is made more negative or positive, the solvent dipoles tend to orient with their positive or negative ends, respectively, toward the metal. This leads to a drop in inner layer capacity. The increase in capacity at the extremes of polarization can be attributed to ionic speciﬁc adsorption. Signiﬁcant changes in inner layer behavior are observed when water is replaced by a non-aqueous solvent in which electrolytes are soluble. In the case of the Hg | methanol interface, a deep minimum is observed on the inner layer capacity curve at negative charge densities [35]. In the case of the amide solvents [35], a variety of behaviors is observed. The protic solvents, formamide, and N-methyl formamide, possess capacity maxima at negative charge densities (ﬁg. 10.22). The other fea-

Fig. 10.21 Inner layer capacity against electrode charge density for the Hg | aqueous solution interface in the presence of NaF for temperatures in the range 0–85 C: (*) 0 C; (~) 25 C; (!) 45 C; (^) 65 C; (g) 85 C [34].

THE ELECTRICAL DOUBLE LAYER

553

Fig. 10.22 Diﬀerential capacity of the inner layer against electrode charge density for the Hg | formamide (KPF6 as electrolyte), Hg | N-methylformamide (KClO4 as electrolyte) and Hg | dimethylformamide interfaces (KClO4) at 25 C [35].

tures of these curves are similar to those found for water; minima are found on either side of the maximum and the capacity increases at the extremes of polarization. On the other hand, the capacity curve for the aprotic solvent, dimethylformamide is relatively featureless. At positive charge densities, a maximum is observed; then, the capacity falls to very low values at negative charge, where there is a shallow minimum. This type of capacity curve is found for a great variety of aprotic solvents, including propylene and ethylene carbonate, acetonitrile, acetone, dimethylsulfoxide, and sulfolane [35]. Relatively simple models in which the solvent molecules are represented as hard spheres with embedded dipoles were developed after capacity data for the metal | solution interface became available in a variety of solvents. More recently the MSA has also been applied at the non-primitive level to explain inner layer properties, especially near the PZC. More details about these models are presented below. A. Solvent Monolayer Models for the Inner Layer The simplest model of solvent structure in the inner layer is that proposed by Watts-Tobin [36]. In this treatment, the inner layer is assumed to consist of a monolayer of solvent dipoles represented as hard spheres; the dipoles may assume one of two orientations, that is, with the electrode ﬁeld or against it (ﬁg. 10.23). By estimating the relative concentrations of the two orientations, the potential drop across the monolayer, and its diﬀerential capacity may be found as a function of electrode charge density. In order to calculate the electrochemical potential of a dipolar molecule, one must determine the local ﬁeld at the position of the molecule. In general, the local ﬁeld Ee is given by the sum of the ﬁeld due to the charge

554

LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 10.23 Models of the solvent monolayer at the electrode | solution interface according to (a) the two-state model with spherical solvent dipoles in either the ‘‘up’’ (") or ‘‘down’’ orientations (#); (b) the three-state model in which a third state with solvent dipoles parallel (!) to the interface has been added; (c) the cluster model with clusters in the ‘‘up’’ (%&%) and ‘‘down’’ (&%&) orientations, and single molecules ‘‘up’’ (") and ‘‘down’’ (#).

on the electrode and the reaction ﬁeld due to surrounding dipoles; in order to calculate the later contribution, dipole–dipole interactions must be estimated in detail. A simpler procedure is to relate the local ﬁeld to the total potential drop across the monolayer. Accordingly, Ee is equal to Ee ¼ d

m f zd

ð10:7:1Þ

where dmf is the potential drop across the inner layer and zd, its thickness which corresponds to the diameter of the spherical solvent molecules. The electrochemical potential of a molecule in the ‘‘up’’ orientation with its positive end toward the electrode may be written m~ u ¼ m0 þ Ur u þ kB T ln Nu þ pEe

ð10:7:2Þ

where m0 is the standard chemical potential, Ur u, the energy of non-electrostatic interaction between the ‘‘up’’ dipole and the metal, Nu, the number of ‘‘up’’ dipoles per unit area, and p, the magnitude of the dipole. Equation (10.7.2) is written in molecular rather than molar units. The corresponding equation for the ‘‘down’’ dipoles is m~ d ¼ m0 þ Ur d þ kB T ln Nd pEe

ð10:7:3Þ

~d and the relative amounts of the two species is obtained ~u ¼ m At equilibrium, m from the relation

kB T ln

THE ELECTRICAL DOUBLE LAYER

Nu ¼ Ur d Ur u 2pEe Nd 2p f ¼ Ur d Ur u d m zd

The partition function for the monolayer is deﬁned as X ai expðbpi Ee Þ ¼ au expðbpEe Þ þ ad expðbpEe Þ qm ¼

555

ð10:7:4Þ

ð10:7:5Þ

i

where ai ¼ expðbUr i Þ. If pi is the component of the dipole moment in a direction perpendicular to the interface from the solution toward the metal, then the concentration of either species is given by Ni ¼ NT ½ai expðbpi Ee Þ=qm

ð10:7:6Þ

NT is the total number of solvent dipoles per unit area in the monolayer. In the case of hexagonally close-packed spheres, it is given by the relationship 2 NT ¼ pﬃﬃﬃ 2 ¼ Nu þ Nd 3z d

ð10:7:7Þ

The potential drop across the monolayer is composed of a contribution due to the charge on the electrode and one due to the net polarization of the dipoles. In the Watts-Tobin model, this is written as d m f

¼

sm zd pðNu Nd Þ þ ei e0 ei e0

ð10:7:8Þ

where ei is the relative permittivity of the inner layer and e0, the permittivity of free space. The parameter ei is introduced in order to account for distortional polarization due to molecular polarizability. Rearranging equation (10.7.8), the charge density may be expressed as a function of dmf: sm ¼

ei e0 d ln qm d m f þ kB TNT zd dðd m fÞ

ð10:7:9Þ

where d ln qm p ¼ ½a expðbpEe Þ ad expðbpEe Þ dðd m fÞ qm zd kB T u

ð10:7:10Þ

Diﬀerentiating equation (10.7.9) with respect to dmf, one obtains an equation for the solvent contribution to the inner layer capacity : Csolv ¼

ei e0 d2 ln qm þ kB TNT zd dðd m fÞ2

ð10:7:11Þ

where

d2 ln qm p2 d ln qm 2 ¼ dðd m fÞ dðd m fÞ2 z2d k2B T2

ð10:7:12Þ

Csolv is a maximum when d ln qm/d (dm f) ¼ 0, that is when Nu ¼ Nd. On the basis of equation (10.7.4), this condition is met when dmf ¼ zd (Ur d – Ur u)/2p. A

556

LIQUIDS, SOLUTIONS, AND INTERFACES

plot of Csolv against sm for typical values of the parameters deﬁned in this model is shown in ﬁg. 10.24. Since only one extremum, the maximum, is predicted, the model is not capable of accounting for all the features on the experimental curves (ﬁgs. 10.21 and 10.22). However, the simple two-state model provides a reasonable explanation for the maximum found experimentally for most systems provided the other extrema are attributed to physical phenomena other than solvent reorientation. A more reasonable description of a solvent monolayer at the interface should include solvent dipoles in all possible orientations. However, if one is only interested in the potential drop due to the monolayer in a direction perpendicular to the interface, and chemical interactions are ignored, it would be suﬃcient to consider three states, namely, those which result from vector resolution of all possible orientations. These orientations are the two considered above, ‘‘up’’ and ‘‘down,’’ and a third orientation with the dipole vector parallel to the interface. The three-state model (ﬁg. 10.23) has been applied to describe the structure of aprotic solvents at charged interfaces [37]. If one extends the two-state model of Watts-Tobin, the electrochemical potential of the third state is given by m~ p ¼ m0 þ Urp þ kB T ln Np

ð10:7:13Þ

where Urp is the chemical (non-electrostatic) interaction between the ‘‘parallel’’ dipole and the metal, and Np , the number of ‘‘parallel’’ dipoles per unit area. By combining this equation with either equation (10.7.2) or (10.7.3), the relation between the concentration of ‘‘parallel’’ dipoles, and ‘‘up’’ or ‘‘down’’ dipoles, respectively, may be obtained. The condition for total coverage of the surface becomes

Fig. 10.24 A plot of inner layer capacity due to the solvent, Csolv against electrode charge density, sm according to the two-state model (a) and three-state model (b). The values of the parameters for the two-state model are zd ¼ 0.48 nm, p ¼ 3.0 debye (1.0 10–29 C m), ei ¼ 9.0, T ¼ 298 K, Ur1 ¼ 5kT, Ur2 ¼ 0; the parameters are the same for the three-state model except that zd ¼ 0.30 nm and Ur3 ¼ 7kT.

THE ELECTRICAL DOUBLE LAYER

NT ¼ Nu þ Nd þ Np It follows that the partition function of the monolayer is given by X qm ¼ ai expðb pi Ee Þ ¼ au expðb pEe Þ þ ad expðb pEe Þ þ ap

557

ð10:7:14Þ

ð10:7:15Þ

i

The equations for the potential drop dmf and inner layer capacity Ci are the same as those for the two-state model (equations (10.7.8) and (10.7.11)). If the fraction of the monolayer composed of dipoles in the ‘‘parallel’’ orientation is relatively large, the shape of the inner layer capacity curve changes. This follows from the fact that the numerical values of the partition function qm and the derivative d2 ln qm/[d(dm f)]2 are quite diﬀerent when the third component is included in the model. The latter quantity is now given by the equation

d2 ln qm 2 p2 d ln qm 2 ¼ ½a expðpE Þ þ a expðpE Þ ð10:7:16Þ u e d e dðd m fÞ z2d ½dðd m fÞ2 where au expðbpEe Þ þ ad expðbpEe Þ qm

ð10:7:17Þ

It follows that d2ln qm/[d(dm f)]2 is a minimum when dln qm/d(dm f) ¼ 0, that is, when Nu ¼ Nd. The inner layer capacity curve calculated with the parameters chosen previously for the two-state system and with a low value of Urp is also shown in ﬁg. 10.24. As predicted, a minimum occurs at the position of the maximum on the curve for the two-state system. At charge densities suﬃciently far from the minimum, maxima are observed. The three-state model is able to account for inner layer capacity curves in a variety of solvents such as methanol, ethylene carbonate, and dimethylformamide [35]. In the case of strongly structured solvents such as water, formamide, and Nmethyl formamide, an improved description of inner layer solvent structure is obtained when it is assumed the clusters of solvent molecules are present at the interface. A cluster is assumed to contain two or more molecules held together by hydrogen bonding such that the net dipole moment of the unit adsorbed at the interface is signiﬁcantly less than that of a monomer in the direction perpendicular to the interface. The clusters are assumed to have two orientations, one with the net dipole vector pointing toward the metal (‘‘up’’) and one with the net dipole vector in the opposite direction (‘‘down’’). Furthermore, they predominate in the inner layer at low electrical ﬁelds. The central maximum on the inner layer capacity curve occurs approximately at the charge density where the populations of clusters in the ‘‘up’’ and ‘‘down’’ orientations are equal. As the charge density is made more positive or negative, the concentration of ‘‘down’’ or ‘‘up’’ clusters, respectively increase, and the inner layer capacity decreases. Considering the fact that the electrochemical potential of a monomer changes much more with electrode charge density than that of a cluster, the concentration of monomers in a given orientation increases, with respect to that of the clusters as the charge increases. This leads eventually to an increase in capacity at the polarization extrema. Thus, capacity curves for strongly structured solvents are characterized by a maximum and two minima (see ﬁgs. 10.21 and 10.22). It has been shown

558

LIQUIDS, SOLUTIONS, AND INTERFACES

[38–40] that cluster models provide reasonable descriptions of the inner layer capacity curves for water, formamide, and N-methylformamide. The development of the theory with inclusion of dipole–dipole interactions is rather complex, since species of diﬀerent size populate the monolayer. The interested reader is referred to the original literature [38–40] for further details of these models. It should be noted that the method of dealing with dipole–dipole interactions in the monolayer presented here is that originally due to Watts-Tobin [36]. The chief criticism that one can make of the Watts-Tobin model is that it introduces an eﬀective relative permittivity for the monolayer, ei, to account for the eﬀects of distortional polarization. In a completely molecular description, these eﬀects can be dealt with by considering the polarizability of the solvent molecules [37, 41]. More detailed models for the role of hydrogen bonding at the interface have also been described [42].

10.8 The Speciﬁc Adsorption of Ions An ion is considered to be speciﬁcally adsorbed at a polarizable electrode when its concentration in the double layer is greater than one would expect on the basis of electrostatic considerations only. Pragmatically speaking, this deﬁnition of speciﬁc adsorption implies that the surface excess of the given ion is greater than that predicted by diﬀuse layer theory for a given electrode charge density and bulk concentration. Up to the present, all experimental data have been analyzed assuming the Gouy–Chapman model for the diﬀuse layer. On the basis of data in the literature, most anions are speciﬁcally adsorbed at Hg and other polarizable metals from aqueous solutions. On the other hand, most inorganic cations are not adsorbed, or else, they are weakly adsorbed at large negative charge densities. In the following discussion some important features of the experimental results for anion adsorption are presented. Emphasis is placed on data obtained at mercury electrodes in aqueous solutions. This simply reﬂects the fact that a large fraction of the existing data were obtained for these systems. This is followed by a presentation of the theory applied to ion adsorption at polarizable interfaces. A. Experimental Data for Anion Adsorption The presence of ionic speciﬁc adsorption may be conﬁrmed by examining the dependence of the PZC on electrolyte concentration. From the data presented in ﬁg. 10.6, it is seen that the PZC measured with respect to a constant reference electrode is approximately independent of electrolyte concentration when ionic adsorption is absent. The classic example of such a system is NaF in water at an Hg electrode. However, when anion adsorption occurs the PZC shifts in the negative direction with increase in electrolyte concentration. This is most pronounced for the I anion in the case of the halides. When cations are adsorbed, the PZC shifts in the positive direction. The interpretation of the shift of the PZC with electrolyte concentration can be put on a sound thermodynamic basis when the electrode potential is measured with a reference electrode reversible to one of

THE ELECTRICAL DOUBLE LAYER

559

the ions in solution (Eþ or E– potential scale). Details of the exact thermodynamic analysis have been given by Delahay [G2]. In order to determine the extent of ionic speciﬁc adsorption, several assumptions must be made in analyzing thermodynamic data for the polarizable interface. If anions are adsorbed, it is customary to assume that cations are not adsorbed in the same potential range. Furthermore, it is assumed that the surface excess of cations is equal to that estimated by Gouy–Chapman theory. In order to illustrate the method of data analysis, the cell Hg j x KI in H2 O j AgI j Ag

ð10:8:1Þ

is considered. Since the reference electrode is reversible to the anion of the electrolyte, the Gibbs adsorption isotherm is given by dg ¼ sm dE þ þ dme

ð10:8:2Þ

It is often more convenient to analyze the data at constant electrode charge density sm rather than constant potential E . Introducing the Parsons function x ¼ g þ smE–, one may write dx ¼ dg þ sm dE þ E dsm ¼ E dsm þ dme

ð10:8:3Þ

Noting that dme ¼ RT dln ae , the charge density due to cations is given by @x ð10:8:4Þ sþ ¼ Fþ ¼ f @ ln ae sm Experimentally, one measures the interfacial tension, g and surface charge density, sm as functions of electrode potential E and electrolyte activity ae. This is most commonly done by measuring the diﬀerential capacity C as a function of E and ae . The values of the potential of zero charge and interfacial tension at this potential are also required, so that one may integrate C, once to obtain sm , and a second time to obtain g. When interfacial tension data are not available, the second integration constant may be estimated in a potential region where ionic adsorption is absent using the Gouy–Chapman theory. Parsons’ function x is then calculated for constant sm as a function of salt activity. Numerical diﬀerentiation of these data according to equation (10.8.4) allows one to estimate sþ. On the basis of the electroneutrality condition, one may also calculate s (s ¼ sm sþ Þ. As noted above, further analysis of the data requires the assumption that the Kþ cations are not speciﬁcally adsorbed. Then, the surface excess of cations is located in the diﬀuse layer and equal to that predicted by Gouy–Chapman theory, sdþ . Combining equations (10.6.41) and (10.6.42), it is easily shown that the surface excess of anions in the diﬀuse layer is given by sd ¼

AGC sdþ sdþ þ AGC

ð10:8:5Þ

The charge due to speciﬁcally adsorbed anions is equal to sad ¼ s sd ¼ sm sm sdþ

AGC sdþ sdþ þ AGC

ð10:8:6Þ

560

LIQUIDS, SOLUTIONS, AND INTERFACES

In the presence of speciﬁc adsorption, the potential drop across the diﬀuse layer is given by 2 !1=2 3 2 2RT s þ s ðs þ s Þ ad m ad 5 ln4 m ð10:8:7Þ fd ¼ þ þ1 F 2AGC 4A2GC The extent of ionic adsorption may be classiﬁed as strong, moderate, or weak. As the electrode potential is made more positive, the charge on the electrode increases in the positive direction. At the same time, the surface excess of adsorbed anions increases. If the sum sm þ sad is always negative for electrolyte concentrations greater than 0.01 M, so that the potential drop across the diﬀuse layer is also negative, then the ionic adsorption is strong. Examples of anions which are strongly adsorbed on Hg are I , CN , and CNS . When the sum sm þ sad changes sign from positive to negative with increase in electrolyte concentration above 0.01 M for electrode charge densities positive of the PZC, then anion adsorption is moderate. Good examples of moderately adsorbed anions are NO 3 and Cl . In the case of weak adsorption, the potential drop across the diﬀuse layer remains positive for positive electrode charge densities. This is the most diﬃcult situation to analyze experimentally because the estimate of sad depends strongly on the GC model used to estimate the ionic surface excesses in the diﬀuse layer. An important aspect of analyzing the double layer data in the presence of speciﬁc adsorption is the determination of the dielectric properties of the inner layer. In the Grahame model for ionic adsorption [G1], the adsorbed ions are assumed to have their charge centers located on the inner Helmholtz plane (iHp). Furthermore, the iHp is closer to the electrode surface than the oHp. This is due to the fact that the adsorbed ions replace solvent molecules on the electrode surface, whereas the counter ions on the oHp do not. Another feature of the following treatment is that the charge on the adsorbed ions is assumed to be located on the iHp. Accordingly, the potential drop across the inner layer is given by d m f

¼

sm sm þ sad þ Kma Kad

ð10:8:8Þ

where Kma is the integral capacity of the region of the inner layer between the metal and the iHp, and Kad that between the iHp and oHp (see ﬁg. 10.14). Noting that the integral capacity of the inner layer is that resulting from two capacitors Kma and Kad in series, one may write 1 1 1 ¼ þ Ki Kma Kad It follows that equation (10.8.8) may be rewritten as m sad þ d m f ¼ Ki Kad

ð10:8:9Þ

ð10:8:10Þ

On the basis of this relationship, the potential drop across the inner layer is zero at the PZC in the absence of ionic adsorption. Under these circumstances fd is also

THE ELECTRICAL DOUBLE LAYER

561

zero. These conditions provide the deﬁnition of the rational potential scale used by Grahame [G1]. In order to estimate the inner potential of the electrode fm, one must measure the PZC in a system in which speciﬁc adsorption is absent, for instance, the aqueous NaF system against a given reference electrode, Er0 and subtract that potential diﬀerence from the potential of the polarizable electrode measured against the same reference electrode, Er ; thus, fm ¼ Er Er0

ð10:8:11Þ

Plots of dm f against sad for constant sm using data obtained for the KI system [43] are shown in ﬁg. 10.25. It is apparent that the plots are approximately linear. The slope of the plot gives the reciprocal of the integral capacity of the outer region of the inner layer (equation (10.9.10)); according to the above results, Kad varies from approximately 80 mF cm–2 at high positive electrode charge densities to 70 mF cm–2 at the more negative values. From the values of dm f in the limit that sad ¼ 0, one may calculate the integral capacity of the inner layer, Ki. This quantity varies from 34 mF cm–2 at high positive charge densities to 18 mF cm–2 at the most negative values. These results are very similar to those obtained in the absence of speciﬁc adsorption in the aqueous KF system [G1]. However, the analysis is clearly approximate, since both Ki and Kad vary with sm, and to a lesser extent with sad. In addition, the contribution of the dipole potential and its possible change with adsorbed ionic charge density are ignored in the above analysis. An alternative method of studying ionic speciﬁc adsorption involves the use of electrolyte solutions of constant ionic strength. An electrolyte with no adsorbing ions such as KF is added to one in which one of the ions adsorbs, for instance, KI

Fig. 10.25 Potential drop across the inner layer, m d , against the charge due to speciﬁcally adsorbed iodide ions for a mercury electrode in KI solution at 25 C. The plots are made at constant electrode charge density, the charge in mF cm–2 being shown adjacent to each curve [43].

562

LIQUIDS, SOLUTIONS, AND INTERFACES

to keep the total ionic concentration constant, and the relative amounts of the two electrolytes are varied. The corresponding electrochemical cell is Hg j xKI; ðI xÞKF in H2 O j Kþ specific electrode

ð10:8:12Þ

where I is the ionic strength. The Gibbs adsorption isotherm for this system is dg ¼ sm dEþ þ I dmKI þ F dmKF

ð10:8:13Þ

Introducing Parsons’ function, xþ ¼ g þ smEþ , one obtains dxþ ¼ Eþ dsm þ I dmKI þ F dmKF

ð10:8:14Þ

Since the ionic strength is held constant, one may assume to a ﬁrst approximation that the activity coeﬃcients of the individual ions do not depend on solution composition. Since the concentration of Kþ ions is constant, it follows that dmKI ¼ RT d ln aKI ¼ RT d ln cI

ð10:8:15Þ

dmKF ¼ RT d ln aKF ¼ RT d ln cF

ð10:8:16Þ

and

where cI and cF are the concentrations of iodide and ﬂuoride ions, respectively. These concentrations are related to the ionic strength in such a way that one may write cI d ln cI ð10:8:17Þ d ln cF ¼ d lnðI cI Þ ¼ I cI Thus, the Gibbs adsorption isotherm becomes cI RT d ln cI dxþ ¼ Eþ dsm þ I I cI F

ð10:8:18Þ

At a given electrode charge density, the surface excess of an anion in the diﬀuse layer is proportional to its bulk concentration. Since ﬂuoride ion is not adsorbed, the total surface excess F may be equated to that in the diﬀuse layer, dF . It follows that the surface excess of iodide in the diﬀuse layer is given by cI dI ¼ ð10:8:19Þ I cI F Thus, the Gibbs adsorption isotherm may be written dxþ ¼ Eþ dsm þ ad I ln cI sad d ln cI ¼ Eþ dsm f

ð10:8:20Þ

where ad I is the surface excess of speciﬁcally adsorbed iodide ions. Having determined values of xþ at constant sm for varying bulk iodide concentration, the adsorbed charge density sad is obtained by diﬀerentiation as follows: @xþ ð10:8:21Þ sad ¼ f @ ln cI sm The eﬀects of ionic strength on anion adsorption have been studied for a number of systems involving both anions and cations. When adsorption is strong,

THE ELECTRICAL DOUBLE LAYER

563

the magnitude of sad increases with increase in ionic strength. This is due to a decrease in the magnitude of fd (see equation (10.8.7)). In the case of strong anionic adsorption, fd is negative, so that the double layer becomes less repulsive for adsorbing anions as the ionic strength increases. This has been demonstrated for the adsorption of I , which has been studied at several ionic strengths [44, 45]. In the case of anions which are moderately adsorbed, a change in ionic strength can act in both directions. For example, in the case of NO 3 adsorption, when the d concentration is low, f is positive. Thus, an increase in ionic strength bulk NO 3 results in a decrease in the magnitude of fd so that the double layer becomes less attractive for anions. As a result, the magnitude of sad decreases with increase in ionic strength. On the other hand, fd is negative for higher NO 3 concentrations and the eﬀect of ionic strength is the same as that observed for the I system. The above method of determining ionic speciﬁc adsorption from solutions of constant ionic strength has been applied extensively for both cations and anions. One reason for its popularity is that the correction for the ionic excess in the diﬀuse layer is made without explicitly introducing the GC model. However, the assumptions made in deriving equation (10.8.20) can lead to serious errors in the estimation of sad, especially when adsorption is moderate or weak. The assumption subject to most criticism is that the individual ionic activity coeﬃcients do not vary with solution composition [45, 46]. In a system such as KI þ KF, it is clear that the average cationic atmosphere around an anion does not change greatly with solution composition; however, the anionic atmosphere around the Kþ cation does change as the relative amounts of KI and KF are changed. A correction for this eﬀect can be made on the basis of models for ionic activity coeﬃcients in electrolyte mixtures [45, 46]. Another problem arises in the case of weak or moderate adsorption [46]. When the sign of the potential drop across the diﬀuse layer changes with solution composition or electrode charge density, there is a change in the nature of the predominant ion at the oHp. Since diﬀerent ions have diﬀerent sizes, the position of the oHp also changes. Ionic size eﬀects are not considered in the GC model of the diﬀuse layer. Thus, use of the model based on equation (10.8.10) must consider the possibility that Kma and Kad vary with adsorbed charge density sad for constant charge density sm on the electrode [46]. The discussion in this section is based to a large extent on the theory of ionic adsorption developed by Grahame [G1]. In the next section, other factors involved in ionic adsorption and the underpotential deposition of metals are considered. B. Partial Charge Transfer and the Dipole Moment of the Adsorbate In Grahame’s model of speciﬁc adsorption [G1] the charge on the adsorbed ion is located at a discrete site on the ion, normally considered to be the center of the ion when it is monoatomic. However, speciﬁc adsorption is assumed to involve the formation of a covalent bond between the ion and the metal atoms of electrode. This has led to the suggestion that charge is transferred between the adsorbed ion and the electrode. The process of ionic adsorption is then described as

564

LIQUIDS, SOLUTIONS, AND INTERFACES z +Az Azb þ rWad þ lz e ( ad þ rWb

ð10:8:22Þ

Az is an ion with charge number z replacing r water molecules at the interface, W, a water molecule, and lz , the fractional charge involved in the adsorption process. The subscripts ‘‘b’’ and ‘‘ad’’ refer to the bulk of the solution and adsorption at the interface, respectively. The fraction lz can be both positive and negative. It cannot be measured by thermodynamic methods. An eﬀective way of assessing the interaction of an adsorbed ion with its environment is to compare its dipole moment in the adsorbed state with the corresponding value for the same ion adsorbed on a metal from the gas phase. When the ion is adsorbed on the conducting metal, it forms an image charge which is located in the metal at a distance ri from the interface, where ri is the radius of the ion (see ﬁg 10.14). Thus, in the gas phase, a dipole is established with a magnitude pad ¼ e0 ri

ð10:8:23Þ

In the case of the iodide ion with a radius of 206 pm, the value of pad for gas phase adsorption is 3.310-29 C m, or 10.0 debyes. For adsorption from solution, the dipole moment can be estimated when the charge density due to adsorbed ions sad is equal and opposite to the charge density on the metal sm. On the basis of equation (10.8.8), the potential drop across the inner layer under these circumstances is sad ð10:8:24Þ d m f ¼ Kma Noting that the diﬀuse layer potential drop fd is zero when sad is equal to sm, this is also the potential due to the dipoles established by the adsorbed ions and their images in the metal. The corresponding dipole moment of the adsorbed ions is ze e pad ¼ i 0 0 ð10:8:25Þ Kma From the work of Grahame [43], the value of Kma obtained from a study of iodide ion adsorption at Hg from solution of KI in water at the pzc is 47.6 mF cm–2. The resulting estimate of pad is 0.9 D for the iodide ion. This is much smaller than the value obtained for adsorption from the gas phase. Values of pad for other ions obtained in studies of ionic adsorption at polarizable metals are summarized in table 10.3. They are all much smaller than estimates of pad for the corresponding process from the gas phase. The low values of pad recorded in table 10.3 are generally attributed to strong screening of the ionic charge by the solvent and the electrons in the metal [47, 48]. First, the solvent dipoles surrounding the adsorbed ion are polarized so that they align with their dipoles opposite to the dipole created in the adsorption process (see ﬁg. 10.14). Second, the electronic charge in the metal which spills out into the solution to an extent which depends on the metal’s properties partially surrounds the adsorbed species and further screens the adsorbed charge. Partial charge transfer would further decrease the value of pad. However, on the basis of quantum-mechanical considerations it is not expected to be large for the halide and alkali metal ions considered in table 10.3. This conclusion is supported by the

THE ELECTRICAL DOUBLE LAYER

565

Table 10.3 Dipole Moments of Ions Adsorbed on Polarizable Electrodes from Aqueous Solution at 25 C [47, 48] Ion

Metal

Dipole Moment pad /debyes

Hg Hg Hg Hg Hg Ag(110) Au(311)

1.18 0.97 0.81 1.01 1.01 0.35 0.49

Anions Cl Br I N3 SCN Br Br Cations Kþ Rbþ Csþ Rbþ Csþ

Hg Hg Hg Ga Ga

1.21 1.22 1.18 0.27 0.27

observation that the magnitude of pad increases slightly for the halide ions as the extent of adsorption increases at more positive electrode charges [48]. If partial charge transfer were important, the value of pad should decrease as the metal becomes more electrophilic. Quite diﬀerent adsorption characteristics are formed in the case of the underpotential deposition of metal ions [49]. As described in section 10.2, underpotential deposition (UPD) is the process by which a metal ion adsorbs on a diﬀerent metal substrate at a potential more positive than that at which it is electrodeposited on itself. UPD occurs when there is a signiﬁcant diﬀerence between the work functions of the depositing metal phase M and the substrate metal S. Consider as an example the electrodeposition of Pb2þ on Au. The work function of polycrystalline Au is approximately 300 mV greater than that of polycrystalline Pb (see table 8.2). This also means that the PZC of the Au electrode is positive of that for Pb. As a result, Pb2þ adsorbs on Au more readily than it does on Pb. The adsorption process is accompanied by signiﬁcant charge transfer. In fact, an estimate of pad for Pb2þ on Au is close to zero, indicating that the cation is essentially discharged. Very interesting studies have been made of the pattern for UPD on single crystal metal substrates [49]. A commensurate pattern is usually observed for the formation of a partial monolayer. The theory of UPD and formation of the initial monolayer is an area of active research. Of course, the phenomenon of UPD is restricted to formation of a monolayer. Once this has formed, the deposition process reverts to one of metal ion Mnþ on metal M. By comparing the UPD process with anion-speciﬁc adsorption, the role of partial charge transfer in these processes is clariﬁed.

566

LIQUIDS, SOLUTIONS, AND INTERFACES

C. The Adsorption Isotherm The adsorption isotherm is derived here for process (10.8.22) for the case that partial charge transfer is negligible (z ¼ 0). The condition for equilibrium is ~A þ r~ ~ad mad m A þ rmW ¼ m W

ð10:8:26Þ

~ad m A

~A , the value is the electrochemical potential of the ion in the adsorption site, m ~ad in the bulk, m W , the electrochemical potential of water in the adsorption site, and mW, its chemical potential in the bulk. The electrochemical potential of the ion in the solution is given by equation (8.6.9). The electrostatic contribution is expressed through the activity coeﬃcient of the ion which is needed to relate its concentration to its activity. The chemical potential of the water molecules in the bulk is given by equation (8.6.10). Thus, the electrostatic contribution to this term is considered to be negligible. The electrochemical potential of the adsorbed ion is given by ad; ad ~ad m A ¼ mA þ RT ln A þ zA F

ð10:8:27Þ

is the standard chemical potential of the adsorbed ion, A is its surface excess, and ad , the micropotential experienced by the ions at the adsorption site. The micropotential ad diﬀers signiﬁcantly from the average potential fad in the plane on which the charge center of the ions is located (see ﬁg. 10.14). This is mainly because the adsorption site is so close to the conducting electrode. As a result, the eﬀects of images formed in the electrode and of the shielding due to the electron overspill must be taken into account. Finally, on the basis of the discussion regarding interfacial water in section 8.6, the electrochemical potential of water molecules at the adsorption site is

ad; mA

ad; ~ad m W ¼ mA þ RT ln W þ hpW iEe

ð10:8:28Þ

ad; is the standard chemical potential of an adsorbed water molecule, W , where mA the surface excess of water molecules, hpW i, the average dipole moment perpendicular to the interface, and Ee, the local ﬁeld experienced by these molecules. When the expressions for the individual electrochemical potentials are introduced into equation (10.8.26), the following result is obtained:

RT ln A r RT ln W ¼ Gad þ RT ln zA Rt ln aW zA Fad þ rhpw iEe ð10:8:29Þ where ad; ad; Gad ¼ mA þ r mW mA r mW

ð10:8:30Þ

The two terms on the left-hand side of equation (10.8.29) describe the entropic eﬀects resulting from the replacing of adsorbed ions by water molecules at the interface. On the basis of the analysis given in section 8.6, this may be re-expressed in terms of the maximum surface excess of adsorbed ions Am to obtain ln A r lnðAm A Þ r ln r ¼ ln A r ln W

ð10:8:31Þ

where r Am ¼ r A þ W

ð10:8:32Þ

THE ELECTRICAL DOUBLE LAYER

567

In the case that one water molecule is replaced by one ion in the adsorption process, the result simpliﬁes to ð10:8:33Þ ln A lnðAm A Þ ¼ ln A ln W In order to understand the dependence of ion adsorption on the electrical properties of the interface, the micropotential ad is expressed in terms of fad , the average potential on the adsorption plane or inner Helmholtz plane, plus a term accounting for the local departure of the potential from its average value: ad ¼ fad þ zad

ð10:8:34Þ

ad

z is called the discreteness-of-charge potential; it depends on the distance of the adsorption plane from the metal phase, and the number and location of ionic images in this phase. On the basis of Gauss’ law the average potential drop between the adsorption plane and the oHp is s þ sad fad fd ¼ m ð10:8:35Þ Kad Estimation of the discreteness-of-charge potential zad requires a detailed model of the interfacial region which gives its dielectric properties and the location of images in the conducting metal phase. According to the model developed by Levine [50], the value of zad is ðC þ Kad ÞKi sad zad ¼ d ð10:8:36Þ ðCd þ Ki ÞKma Kad Since the diﬀuse layer capacity Cd is usually much greater than the capacities observed in the inner layer, an approximate expression for zad is Ki s ð10:8:37Þ zad ¼ Kma Kad ad Under most circumstances, the discreteness-of-charge potential has a sign opposite to that of the average potential fad. This means that the electrostatic work done to adsorb an ion on the iHp is considerably less than the value estimated on the basis of the average potential on this plane. This result reﬂects the stabilizing inﬂuence of the images formed in the conducting metal which reduce the electrostatic repulsion between adsorbed ions with the same charge. After substituting in the expression derived for ad the adsorption isotherm may be written as ln A r lnðAm A Þ ¼ ln Bad aA zA ffd þ Aad sad

ð10:8:38Þ

G z f þ r ln r r ln aW þ rhpw iEe A sm RT Kad

ð10:8:39Þ

where ln Bad ¼ and Aad ¼

zA F Kma Ki Kad Kma

ð10:8:40Þ

RT ln Bad is the standard Gibbs energy of adsorption for a given charge density on the electrode. The last term in sm gives the most important contribution to the dependence of Bad on the electrical characteristics of the interface. Bad also

568

LIQUIDS, SOLUTIONS, AND INTERFACES

depends on the electrostatic contribution of the electrochemical potential of water (rhpW iEe ) but this contribution changes much less with sm than the last term. Finally, Aad is the interaction coeﬃcient which describes the repulsive interactions between adsorbed ions on the adsorption plane. It depends on the dielectric properties of the inner layer, the discreteness-of-charge component being an important factor in determining its magnitude. When experimental data are only available for low coverages (A Am Þ, the entropic term on the left-hand side of equation (10.8.38) may be simpliﬁed as follows: r lnðAm A Þ ¼ r ln Am

r A Am

Then the adsorption isotherm becomes s ln ad ¼ ln B0ad aA zA ffd þ A0ad sad zA F

ð10:8:41Þ

ð10:8:42Þ

where ln B0ad aA ¼ ln Bad r ln Am

ð10:8:43Þ

and A0ad ¼ Aad þ

r zA FAm

ð10:8:44Þ

In equation (10.8.42) which is known as the virial isotherm, the surface excess of adsorbed ions A is expressed in terms of their charge density sad using the charge on one mole of ions zA F. The virial isotherm describes the adsorption of monoatomic ions such as the halides on mercury and other metals. In order to test its applicability to experimental data it is plotted in the form sad ð10:8:45Þ ln þ zA ffd ¼ ln B0ad þ A0ad sad z A aA Accordingly, the quantity on the left-hand side of equation (10.8.45) should be a linear function of sad, the charge density due to adsorbed ions, for constant charge density on the electrode. Typical results obtained for the adsorption of iodide ion at Hg [43] are shown in ﬁg. 10.22. Linear plots are obtained for a wide variation in the bulk iodide ion concentration, in this case from 0.025 M to 1 M. The isotherm parameters are found from the slope and intercepts of these plots. For example, at sm ¼ 0 mC cm–2, the slope of the isotherm plot in ﬁg. 10.26 is 0.28 cm2 mC–1. For monoatomic ions like iodide, the value of Am is expected to be large, a reasonable estimate of zAFAm being 100 mC cm–2. As a result, the contribution of the term r/(zAFAm) in equation (10.8.44) is negligible. From the plots of dm f against sad at sm ¼ 0 mC cm–2 in ﬁg. 10.25, the estimates of Ki and Kad are 28.9 and 73.5 mF cm–2, respectively. On the basis of equation (10.8.40) the estimate of the interaction parameter Aad is 0.21 cm2 mC–1. This compares favorably with the experimental result. In assessing this isotherm and its application to experimental data it should be remembered that the model used to derive the isotherm is not based on a molecular description of the solvent at the

THE ELECTRICAL DOUBLE LAYER

569

Fig. 10.26 Plots of experimental data for the adsorption of iodide ion on Hg according to the virial isotherm using data obtained from solutions of varying ionic strength [46]. The iodide ion activity was set equal to the mean ionic activity. The electrode charge density in mC cm–2 is indicated adjacent to each plot.

interface. Further development of this topic should be based on a completely microscopic description of interfacial properties. Other adsorption isotherms have been used in the literature to analyze experimental data. However, it can usually be shown that they are limiting forms of the general isotherm derived here (equation (10.8.38)). For example, in early work the importance of the term in the diﬀuse layer potential (zAf fd) was not recognized. By using equation (10.8.7) for this contribution, the exact dependence of this term on sm and sad is obtained. Equation (10.8.7) can be simpliﬁed in limiting cases and the form of the isotherm without an explicit dependence on fd obtained.

10.9 The Adsorption of Molecules at Electrodes Water-soluble organic molecules are usually preferentially adsorbed at the metal | solution interface. Thus, small molecules such as methanol, acetone, and acetonitrile have a polar group which results in their being water soluble, and non-polar groups which prefer to be at the interface out of the aqueous environment. The accumulation of the organic solute at the interface results in a lowering of the interfacial tension to an extent which depends on the surface excess of the adsorbing compound. Typical results obtained for the adsorption of 2-butanol from aqueous solutions containing 0.1 M Na2SO4 are shown in ﬁg. 10.27. It is apparent from these results that the extent of adsorption depends on both the electrode potential and the concentration of 2-butanol. In this system, the PZC which occurs at the maximum value of the interfacial tension shifts in the positive direction with adsorption of the alcohol molecule. The exact features of the electrocapillary curve depend on the dipole moment of the adsorbing molecule and its orientation at the

570

LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 10.27 Interfacial tension of the Hg | aqueous solution interface against electrode potential Eþ for 0.1 M Na2SO4 with varying amounts of 2-butanol at 25 C: (*) 0 M, (^) 0.105 M, and (~) 1.00 M [51].

interface. As the electrode potential is made more positive or negative, and consequently, the electrical ﬁeld at the interface increased, the organic molecules at the interface are replaced by solvent molecules. As a result the interfacial tension is independent of solution composition at the extremes of polarization. The other common way of studying molecular adsorption is to measure the diﬀerential capacity of the electrode | solution interface. Typical capacity data obtained for a similar system, namely, t-butanol in water are shown in ﬁg. 10.28. In the potential region when the organic molecule is adsorbed, the capacity is signiﬁcantly reduced from the value observed in the absence of adsorbate. In addition, the extent of capacity lowering increases with increase in adsorbate concentration. Maxima are observed on the capacity curves in the potential regions where desorption of the organic molecules occurs. They result from the rapid change in surface coverage with potential in these regions. At high bulk concentrations the maxima often involve very high capacities which are frequency dependent. In this case the adsorption process is not at equilibrium, so that the measured capacity is not thermodynamically relevant. Experimental data relevant to molecular adsorption have been obtained for a wide variety of organic compounds, including alcohols, aldehydes, ketones, amines, and amides [G5]. Included are studies of liquids which are polar solvents themselves, for example, acetonitrile, acetone, ethylene carbonate, formamide, and methanol. Studies have also included organic acids which are weakly dissociated in water. A. Analysis of the Thermodynamic Data An ideal thermodynamic cell which could be used to study the adsorption of adsorbate A at mercury is the following:

THE ELECTRICAL DOUBLE LAYER

571

Fig. 10.28 Diﬀerential capacity of the Hg | aqueous solution interface against electrode potential ESCE for 0.1 M Na2SO4 with varying amounts of t-butanol at 25 C: (*) 0 M, (^) 0.108 M, and (~) 0.540 M [52].

Hg j xA; yNaF; H2 O j Hg2 F2 j Hg

ð10:9:1Þ

x is the concentration of the adsorbate A which is varied in order to study the eﬀect of bulk concentration on the interfacial surface excess; y is the concentration of the electrolyte whose activity is kept constant. The corresponding electrolyte concentration is kept reasonably high to provide electrical conductivity to the solution. For low values of x, the electrolyte concentration is constant. However, at higher concentrations of organic solute, the activity coeﬃcient of the electrolyte varies with organic solute concentration. Thus, in general, the concentration of the electrolyte must also be varied in order to keep its activity constant. It is also important that the ions of the electrolyte not adsorb on the electrode to a signiﬁcant extent. The condition for equilibrium at the polarizable interface is dg ¼ sm dE þ þ dme þ A dmA þ w dmw

ð10:9:2Þ

where E is the electrode potential with respect to the ﬂuoride reference electrode, þ , the surface excess of the Naþ cation, A, that of the organic adsorbate, w, that of water molecules, me, the chemical potential of the electrolyte, mA, that of the adsorbate, and mw, that of water. The experiments are carried out for constant me and the relative activities of the adsorbate and water changed so that xA dmA þ xw dmw ¼ 0

ð10:9:3Þ

where xA and xw are the mole fractions of adsorbate and water. It follows that the relative surface excess of adsorbate ðwÞ A can be found by measuring the change in interfacial tension with chemical potential of the adsorbate holding the electrode potential E and the electrolyte activity ae constant. Thus,

572

LIQUIDS, SOLUTIONS, AND INTERFACES

ðwÞ A

xA @g ¼ A w ¼ @mA E ;me xw

ð10:9:4Þ

In the following discussion the relative nature of the surface excess is neglected and the role of water in the thermodynamic analysis ignored. In dilute solutions, the eﬀects of the organic solute on the electrolyte activity can be ignored so that experiments can be carried at constant electrolyte concentration. The chemical potential of the electrolyte is given by me ¼ me þ RT ln ye ce

ð10:9:5Þ

where ye is the mean activity coeﬃcient on the concentration scale. Thus, the change in electrolyte activity with organic solute concentration is zero for constant electrolyte concentration, provided the activity coeﬃcient ye is also constant. However, for the concentration range of interest, signiﬁcant changes in ye are expected (see ﬁg. 10.29). The increase in electrolyte activity for constant electrolyte concentration is known as the salting out eﬀect. It is due to the fact that the number of water molecules available to solvate the ions of the electrolyte decreases as the concentration of the organic solute increases. This eﬀect was usually ignored in early studies of organic adsorption, so that the results obtained with constant electrolyte concentration are signiﬁcantly in error. On the other hand, the task of determining the value of the electrolyte concentration needed to maintain constant electrolyte activity is formidable, so that another procedure for analyzing the thermodynamic data would help facilitate these studies. Consider now experiments carried out with a constant electrolyte concentration ce in the cell Hg j xA; ce NaF; H2 O j Hg2 F2 j Hg

ð10:9:6Þ

Fig. 10.29 Plots of the change in ionic activity coeﬃcients determined by speciﬁc ion electrodes against acetamide concentration cAM for 0.25 M NaF: Naþ ion activity coeﬃcient (!); F ion activity coeﬃcient (~); mean activity coeﬃcient for NaF(*) [53].

THE ELECTRICAL DOUBLE LAYER

573

The thermodynamic data are used to calculate the function # [53], which is deﬁned as # ¼ g þ sm E þ þ me ð10:9:7Þ If the original experimental data involve capacity measurements as a function of electrode potential E and organic solute activity aA , the data must be integrated in the usual way to obtain electrode charge density sm and interfacial tension g. In addition, the electrolyte activity must be determined as a function of aA ; this can be accomplished easily using speciﬁc ion electrodes for most electrolytes commonly used in these studies. Taking the total derivative of # , the result is d# ¼ dg þ sm dE þ E dsm þ me dþ þ þ dme

ð10:9:8Þ

Substituting this into the GAI, one obtains d# ¼ E dsm me dþ þ A dmA þ w dmw

ð10:9:9Þ

It follows that the relative surface excess of the organic adsorbate is given by @# ðwÞ ð10:9:10Þ A ¼ @mA sm ;þ If the electrolyte is not speciﬁcally adsorbed then the surface excess of the cation is constant for constant electrode charge density. It follows that introduction of the function # provides an easy route for analyzing experimental data obtained at constant electrode charge density. It also means that experimental data are not easily analyzed at constant electrode potential. Further details about the thermodynamics of these systems are given elsewhere [53, 54]. B. The Adsorption Isotherm The process of molecular adsorption at the polarizable interface involves replacement of water molecules in the inner layer which are solvating the electrode. Thus, (Aad þ rWb Ab þ rWad + ð10:9:11Þ where the subscript ‘‘ad’’ designates molecules at the interface and the subscript ‘‘b’’ molecules in the bulk. The same process can be used to describe the adsorption of molecules at the aqueous solution | air interface as described in section 8.6. The adsorption isotherm derived earlier is also applicable here, namely, ln A r lnðAm A Þ ¼ ln Bad aA þ ðhpA i rhpw iÞ

Ee RT

ð10:9:12Þ

The experimentally controlled variables which are explicit in equation (10.9.12) are the activity of the adsorbate aA and the temperature T. The experimentally observed quantity is the surface excess A. The dependence of the adsorption on the electrical state of the interface is expressed through the local eﬀective ﬁeld Ee perpendicular to the interface and the average dipole moments of the adsorbate hpAi and water molecules hpWi in the same direction. The contribution of the last term is much larger under most circumstance for adsorption at the electrode | solution interface than at the solution | air interface. As a result, further treatment of the two problems is quite diﬀerent.

574

LIQUIDS, SOLUTIONS, AND INTERFACES

The local eﬀective ﬁeld Ee cannot be determined experimentally, and must be estimated in terms of other electrical properties of the interface. This has led to considerable discussion in the literature [54] concerning the appropriate experimental electrical variable. Two choices directly available from experiment are the electrical potential E and the surface charge density sm. The choice of sm is clearly more appropriate because most experimental are measured at constant electrolyte concentration and therefore must be analyzed at constant sm as described above. Thus, the adsorption isotherm is written as ln A r lnðAm A Þ ¼ ln Bad aA

Aad A Am

ð10:9:13Þ

where ln Bad ¼

Gad þ r ln r r ln aW RT

ð10:9:14Þ

RT ln Bad is the eﬀective standard Gibbs adsorption energy, which depends on electrode charge density. This dependence can often be described by the equation RT ln Bad ¼ RT ln Bad;max bad ðsm sm;max Þ2

ð10:9:15Þ

where Bad;max and sm;max are the values of Bad and sm , respectively, at which Bad is a maximum. Aad is the interaction coeﬃcient which also can depend on the electrical state of the interface. Positive values of Aad signify repulsive interactions between the adsorbing molecules, whereas negative values indicate that the interactions are attractive. Aad is the interaction coeﬃcient which also can depend on the electrical state of the interface. The eﬀects of these interactions on adsorption for typical conditions at constant electrical state of the interface are shown in ﬁg. 10.30. When interactions are absent the value of the fractional coverage y rises sharply with bulk concentration and then levels oﬀ as coverage increases. The presence of repulsive interactions results in a signiﬁcantly slower rate of increase in y over the concentration range in which a comparison is made. On the other hand, when these interactions are attractive, y increases more rapidly. However, the presence of these interactions is not readily apparent from visual examination of y against aA data for a single system. This feature of the adsorption data can only be assessed when the value of the maximum surface excess Am is known. Some methods for determining this quantity were considered in section 8.6. Typical experimental results obtained for the adsorption of acetamide on mercury at negative charge densities [55] are shown in ﬁg. 10.31. As the electrode charge density is made more negative, the increase in surface coverage with bulk acetamide concentration is retarded. This demonstrates that the value of the interaction parameter Aad is increasing, that is, becoming more repulsive. At more positive charge densities (not shown) the plots of Am against cAm do not diﬀer signiﬁcantly from the plot for sm ¼5 mC cm–2. Further analysis of the experimental data requires determination of the maximum surface coverage Am. Various methods are available to estimate Am. In the present case, the best value in the region of maximum adsorption is 0.32 nmol cm–2 and some variation with electrode charge density is observed. The latter observation is attributed to reorientation of the polar adsorbate in the electrode’s ﬁeld.

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575

Fig. 10.30 Plots of surface coverage y ¼ A/ Am against bulk activity of the adsorbate aA with a moderately adsorbed species (RT ln Bad ¼ 0) for the cases of no interaction (Aad ¼ 0), attractive interaction (Aad ¼ 1), and repulsive interaction (Aad ¼ 1). The geometrical factor r was set equal to unity.

The experimental data for acetamide are plotted according the Frumkin isotherm (equation (10.10.13) with r ¼ 1) in ﬁg. 10.32. This plot was constructed using the acetamide concentration rather than its activity. This is equivalent to assuming that the activity coeﬃcient of the adsorbate is constant in the bulk of the solution over the concentration range considered. The slopes of the isotherm plots are negative at the most positive electrode charge densities (3 and 6 mC cm2 ) indicating that the acetamide molecules interact attractively on the adsorption plane at the interface. However, the interaction coeﬃcient decreases in magnitude

Fig. 10.31 Plots of the surface excess of acetamide Am against bulk acetamide concentration cAm for three electrode charge densities: (*) –5; (!) –10; and (^) –15 mC cm–2 [55].

576

LIQUIDS, SOLUTIONS, AND INTERFACES

Fig. 10.32 Plots of experimental data for the adsorption of acetamide at mercury according to the Frumkin isotherm (equation (10.10.13)) with r ¼ 1 and y ¼ A/Am). The activity of acetamide aA was set equal to its concentration cA. Data are shown for sm¼ 3 (*); 6 (&); 9 (~) ; 12 (!) and –15 mC cm–2 (^).

as the electrode charge density is made more negative and eventually goes to zero. The changes in the nature of the interaction with electrical state of the interface observed for acetamide are typical for polar molecules at the mercury | solution interface [55]. In early work the question of isotherm congruency was considered in the development of the theory of interfacial adsorption [54, 55]. A congruent isotherm is one for which the surface excess is speciﬁed once the values of the adsorbate activity aA and standard Gibbs energy of adsorption RT ln Bad have been speciﬁed. This theory was developed for the cases that either the electrode potential E or the electrode charge density sm is the appropriate electrical variable. More extensive examination of the experimental data has led to the conclusion that the concept of congruency is only a ﬁrst approximation. In fact, when a completely molecular approach ia applied to the problem of organic adsorption, the adsorption isotherm is clearly not congruent [56]. The most recent experimental work has involved studies of organic adsorption at the single crystal faces of polarizable solid metal electrodes [57]. These experiments provide details of the role of the metal in organic adsorption. By examining these data within the context of the new molecular descriptions of interfacial adsorption the theory of this important process will be greatly advanced.

10.10 Concluding Remarks The discussion in this chapter has focused on the metal | solution interface. The interfacial property considered in most detail was the diﬀerential capacity. It was

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577

shown that the contributions of both the metal and the solvent play an important role in determining the change in capacity with electrode potential. The adsorption of ions and solute molecules was also discussed in detail. These interfaces have been studied experimentally in great detail, both at diﬀerent metals, and for a wide variety of solvents. In addition, considerable eﬀort has been directed to developing theoretical models for the electrical double layer. Other systems which have been studied include the semiconductor | solution interface and the metal | molten salt interface. Double layers are also important in colloid chemistry. When a colloid particle is composed of an ionic crystal, it often preferentially adsorbs one of its component ions, thereby acquiring a charge. As a result the colloid particle is surrounded by a double layer. The interfacial properties are very important in determining a variety of colloidal properties, including electrophoresis and electroosmosis. It also plays a role in colloid stability and coagulation phenomena. The eﬀects of the electrical properties of the interface are well known in colloid chemistry. The description of colloid phenomena is a well-developed area of physical chemistry which is often important in industrial processes. Finally, double layers are important in a variety of biological phenomena, especially those occurring at cell walls. Although these systems are more complex in their description, the fundamental concepts applied here are also applicable in the biological systems. For example, ion transport phenomena in membranes depend on the electrical state of the membrane interfaces. Thus many concepts from the physical chemistry of polarizable interfaces and colloids are also applicable to charged interfaces in biological systems. General References G1. Grahame, D. C. Chem. Rev. 1947, 41, 441. G2. Delahay, P. Double Layer and Electrode Kinetics; Wiley-Interscience: New York, 1965. G3. Damaskin, B. B.; Petrii, O. A. An Introduction to Electrochemical Kinetics (in Russian), 2nd ed.; Vysshaya Shkola: Moscow, 1983. G4. Bockris, J. O’M.; Conway, B. E.; Yeager, E., eds. Comprehensive Treatise of Electrochemistry; Plenum Press: New York, 1980; Vol. 1. G5. Lyklema, J.; Parsons, R. Electrical Properties of Interfaces – Compilation of Data on the Electrical Double Layer, National Bureau of Standards Document NBSIR 83-2714, US Department of Commerce, 1983.

References 1. Mohilner, D. M. In Electroanalytical Chemistry; Bard, A. J., ed.; Marcel Dekker: New York, 1966; Vol. 1, Chapter 4. 2. Lipkowski, J.; Schmickler, W.; Kolb, D. M.; Parsons, R. J. Electroanal. Chem. 1998, 452, 193. 3. Heyrovsky, J. Chem. List 1922, 16, 256. 4. Lippmann, G. Compt. Rend. 1873, 76, 1407; Pogg. Ann. 1873; 149, 561; J. Phys. 1874, 43. 5. Payne, R. In Techniques of Electrochemistry; Yeager, E., Salkind, A. J., eds.; Wiley Interscience: New York, 1973; Vol. 1, Chapter 2. 6. Barradas, R. G.; Kimmerle, F. M. Can. J. Chem. 1967, 45, 109.

578 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49.

LIQUIDS, SOLUTIONS, AND INTERFACES

Gouy, G. Ann. Chim. Phys. 1903, 7, 145. Grahame, D. C. J. Electrochem. Soc. 1951, 98, 343. Pajkossy, T. J. Electroanal. Chem. 1991, 300, 1. Fawcett, W. R.; Kova´cˇova´, Z.; Motheo, A. J.; Foss, C. A. J. Electroanal. Chem. 1992, 326, 91. Lipkowski, J.; Stolberg, L. In Adsorption of Molecules at Electrodes; Lipkowski, J., Ross, P. N., eds.; VCH: New York, 1992; Chapter 4. Gale, R. J. Spectroelectrochemistry, Theory and Practice; Plenum Press: New York, 1988. Abrun˜a, H. D. Electrochemical Interfaces: Modern Techniques for In-Situ Interface Characterization; VCH: New York, 1991. Wieckowski, A. Interfacial Electrochemistry, Theory, Experiment and Applications; M. Dekker: New York, 1999. Foley, J. K.; Korzeniewski, C.; Daschbach, J. L.; Pons, S. Electroanal. Chem. 1986, 14, 309. Faguy, P. W.; Marinkovic´, N. S. Anal. Chem. 1995, 67, 2791. Marinkovic´, N. S.; Hecht, M.; Andreu, R.; Fawcett, W. R. J. Phys. Chem. 1977, 99, 6760. Marinkovic´, N. S.; Calvente, J. J.; Kova´cˇova´; Fawcett, W. R. J. Electroanal. Chem. 1997, 421, 49. Magnussen, O. M.; Ocko, B. M.; Wang, J. X.; Adzˇic´, R. R. J. Phys. Chem. 1996, 100, 5500. von Helmholtz, H. F. L. Ann. Physik 1853, 89, 211; 1879, 7, 337. Gouy, G. Compt. Rend. 1910, 149, 654. Chapman, D. L. Phil. Mag. 1913, 25, 475. Stern, O. Z. Elektrochem. 1924, 30, 508. Trasatti, S. In Trends in Interfacial Electrochemistry; Silva, A. F., ed.; Dordrecht: Boston, 1984; p. 25. Badiali, J. P.; Rosinberg, M. L.; Goodisman, J. J. Electroanal. Chem. 1983, 143, 73; 1983, 150, 25. Schmickler, W. Chem. Rev. 1996, 96, 3177. Trasatti, S.; Lust, E. Mod. Aspects Electrochem. 1999, 33, 1. Frumkin, A. N. Potentials of Zero Charge; Nauka: Moscow, 1979. Smith, J. R. Phys. Rev. 1969, 181, 522. Schmickler, W.; Henderson, D. J. J. Chem. Phys. 1984, 80, 3381. Leiva, E.; Schmickler, W. J. Electroanal. Chem. 1987, 228, 39. Boda, D.; Fawcett, W. R.; Henderson, D. J.; Sokolowski, S. J. Chem. Phys. 2002, 116, 1770. Schmikler, W.; Henderson, D. J. Prog. Surf. Sci. 1986, 22, 323. Grahame, D. C. J. Am. Chem. Soc. 1957, 79, 2093. Fawcett, W. R. Israel J. Chem. 1979, 18, 3. Watts-Tobin, R. J. Phil. Mag. 1961, 6, 133. Fawcett, W. R. J. Phys. Chem. 1978, 82, 1385. Parsons, R. F. Electroanal. Chem. 1975, 59, 229. Damaskin, B. B. J. Electroanal. Chem. 1977, 75, 359. Fawcett, W. R.; Levine, S.; de Nobriga, R. M.; McDonald, A. C. J. Electroanal. Chem. 1980, 111, 163. Levine, S.; Bell, G. M.; Smith, A. L. J. Phys. Chem. 1969, 73, 3534. Guidelli, R.; Schmickler, W. Electrochim. Acta 2000, 45, 2317. Grahame, D. C. J. Am. Chem. Soc. 1958, 80, 4201. Dutkiewicz, E; Parsons, R. J. Electroanal. Chem. 1966, 11, 100. Fawcett, W. R.; McCarrick, T. A. J. Electrochem. Soc. 1976, 123, 1325. Fawcett, W. R.; Sellan, J. B. Can. J. Chem. 1977, 55, 3871. Schmickler, W; Guidelli, R. J. Electroanal. Chem. 1987, 235, 387. Schmickler, W. J. Electroanal. Chem. 1988, 249, 25. Leiva, E. Electrochim. Acta. 1996, 41, 2185.

THE ELECTRICAL DOUBLE LAYER

579

50. Levine, S. J. Colloid Interface Sci. 1971, 37, 619. 51. Nakadomari, H.; Mohilner, D. M.; Mohilner, P. R. J. Phys. Chem. 1976, 80, 1761. 52. de Battisti, A.; Abd El Nabey, H. B.; Trasatti, S. J. Chem. Soc., Faraday Trans. 1. 1976, 72, 2076. 53. Fawcett, W. R.; Champagne, G. Y.; Komo, S. and Motheo, A. J. J. Phys. Chem. 1988, 92, 6368. 54. Damaskin, B. B., and Kazarinov, V. E. In Comprehensive Treatise of ElectroChemistry, Bockris, J. O’M., Conway, B. E., Yeager, E., eds.; Plenum: New York, 1980; Vol. 1, Chapter 8. 55. Fawcett, W. R.; Champagne, G. Y.; Motheo, A. J. J. Chem. Soc., Faraday Trans. 1990, 86, 4037. 56. Parsons, R. Chem. Rev. 1990, 90, 813. 57. Trasatti, S., Doubova, L. M. J. Chem. Soc., Faraday Trans. 1995, 91, 3311.

Problems 1. Given the following cell: Hg j K2 SO4 in H2 O j Hg2 SO4 j Hg derive an expression from the Gibbs absorption isotherm for the change in interfacial tension at the Hg | solution interface with potential drop across the cell, and chemical potential of K2 SO4 . 2. The following surface tension data were obtained for the Hg | 1.0 M CsCl interface: E/V

g/mJ cm2

E=V

g=mJ cm2

0 0:1 0:2 0:3 0:4 0:5 0:6

34.50 37.64 39.71 41.05 41.87 42.26 42.29

0:7 0:8 0:9 1:0 1:1 1:2 1:3

41.99 41.40 40.56 39.51 38.21 36.92 35.36

Determine the PZC and values of sm for 200 mV increments in E in the range 0:2 to 1:2 V. 3. The following surface tension data were obtained in LiCl solutions of varying concentration at a constant potential on the E scale (1:0 V). Determine the surface excess of the cation for all concentrations except the ﬁrst and last. y is the mean molar activity coeﬃcient. Conc/M

y

g=mJ cm2

sm =mC cm2

0.03 0.10 0.30 1.00 3.00

0.84 0.78 0.72 0.72 0.96

40.69 40.50 40.06 39.69 39.43

8:4 8:9 10:1 11:0 11:6

580

LIQUIDS, SOLUTIONS, AND INTERFACES

Use the GC model to estimate sþ and s assuming no ionic speciﬁc adsorption, and compare these estimates with those obtained from the thermodynamic analysis. 4. The following data were obtained at the PZC for the Hg | aqueous NaF interface for concentrated solutions: Conc/M

Density/g cm3

y

E/V

g=mJ cm2

0.1 0.25 0.5 1.0 4.0 6.25 8.0 10.0

1.001 1.008 1.020 1.045 1.189 1.285 1.36 1.440

0.773 0.710 0.670 0.645 0.792 1.098 1.48 2.132

0:355 0:333 0:318 0:300 0:261 0:238 0:228 0:207

42.65 42.69 42.73 42.77 43.03 43.28 43.67 44.30

Estimate the relative cationic surface excess at the PZC and plot it as a function of the mole fraction ratio xs =xw . Is there any evidence for ionic speciﬁc adsorption? If not, can an estimate of w be made from these data? Comment on the results of your analysis. 5. In his experimental study of the adsorption of iodide ion at mercury, Grahame obtained the following results at an electrode charge density of 10 mC cm2 :

Conc. KI/M

Electrode Potential E/V

Cationic Surface Excess sþ =mC cm2

0.025 0.1 1.0

0:5412 0:5896 0:6714

8.87 12.43 18.86

Calculate the adsorbed charge density due to I ion. Construct a plot of E fd against sad and determine the capacity of the outer region of the inner layer, Kad . Given that the PZC on the above potential scale in the absence of adsorption is 0:472 V, determine the integral capacity of the inner layer, and thus, the capacity of the inner region of the inner layer, Kma . 6. Using the above data construct an adsorption isotherm plot on the basis of equation (10.8.45) and determine the isotherm parameters. 7. The following surface pressure data were obtained for the cell Hg j xM DMA; 0:15 M Na2 SO4 ; H2 O j Hg2 SO4 j Hg at a constant potential E of 0:7 V at 258C, where DMA is dimethylacetamide:

THE ELECTRICAL DOUBLE LAYER

Conc DMA/M

p ¼ g g0 mJ cm2

0.04 0.063 0.1 0.16 0.25 0.4 0.63 1.0 1.60

0:575 0:645 0:848 1:135 1:135 1:753 2:101 2:506 2:947

581

Determine the relative surface excess of DMA for concentrations in the range 0.1–0.63 M, assuming the activity coeﬃcient of Na2 SO4 is independent of DMA concentration. 8. Measurements of the activity coeﬃcient of Na2 SO4 in the above solutions for a constant concentration of 0.15 M yielded the following results: Conc. DMA/M

y (0.15 M Na2 SO4 )

0.01 0.10 0.16 0.25 0.4 0.63 1.0 1.60

0.396 0.428 0.434 0.444 0.462 0.506 0.582 0.726

Correct the above results for the variation in y to obtain a better estimate of the relative surface excess of DMA. Assume that the charge on the electrode, sm , is constant and equal to 6 mC cm2 . 9. Derive the form of the Frumkin isotherm ln lnðm Þ þ

A ¼ ln bc m

which is appropriate when m . The following data were obtained for the adsorption of thiourea from ethylene carbonate solutions: Conc. Thioureau/M

Surface Excess =molec cm2

0.0095 0.058 0.26 1.06

3 1013 6 1013 9 1013 12 1013

Determine the isotherm parameters ln b and A=m .

Appendix A

Mathematical Background

In this appendix some important mathematical methods are brieﬂy outlined. These include Laplace and Fourier transformations which are often used in the solution of ordinary and partial diﬀerential equations. Some basic operations with complex numbers and functions are also outlined. Power series, which are useful in making approximations, are summarized. Vector calculus, a subject which is important in electricity and magnetism, is dealt with in appendix B. The material given here is intended to provide only a brief introduction. The interested reader is referred to the monograph by Kreyszig [1] for further details. Extensive tables relevant to these topics are available in the handbook by Abramowitz and Stegun [2].

A.1 Laplace Transforms The method of Laplace transforms is applied to the solution of ordinary and partial diﬀerential equations. It falls into the general category of operational calculus, and greatly simpliﬁes the problem to be solved. When a diﬀerential equation is solved a family of functions satisfying the equation is found. Using the boundary conditions, a speciﬁc function is selected from the family. Methods of solving diﬀerential equations is a whole subject in itself which is not discussed here. However, when the Laplace transform is applied to the diﬀerential equation, it is transformed into an algebraic equation. The algebraic equation is usually much easier to solve. 582

MATHEMATICAL BACKGROUND

583

The Laplace transform of a function fðtÞ is deﬁned as fðsÞ ¼

1 ð

est fðtÞdt

ðA:1:1Þ

0

Accordingly, a new function fðsÞ is deﬁned which depends on s as independent variable. This transformation may also be written as fðsÞ ¼ LffðtÞg

ðA:1:2Þ

where the operator L designates Laplace treansformation. Inverse Laplace transformation leads back to fðtÞ: fðtÞ ¼ L1 ffðsÞg

ðA:1:3Þ

A simple example of Laplace transformation is given now for the function eat : 1 ð

Lfe g ¼ at

at st

e e 0

1 1 eða sÞt dt ¼ as 0

ðA:1:4Þ

Substituting in the limits one obtains Lfeat g ¼

1 sa

ðA:1:5Þ

Other examples are given in section 6.5 and in table A.1. Table A.1 Laplace Transforms of Some Common Functions [1] fðtÞ a t tn t

a

eat

fðsÞ a s 1 s2 n! snþ1 ða þ 1Þ saþ1 1 sa

cos ot

s s2 þ o2

sin ot

o s2 þ o2

cosh at

s s2 a2

sinh at

a s2 a2

584

LIQUIDS, SOLUTIONS, AND INTERFACES

A.2 Fourier Transforms Fourier series are powerful tools used to describe periodic functions. Suppose that fL ðxÞ is a periodic function that is deﬁned in the interval L 0 L. A simple example is a square wave which is centered at x ¼ 0, and repeated continuously for values of x greater than L and less than L. Such a function may be represented as an inﬁnite sum of cosines and sines, so that 1 h

np

npi X fL ðxÞ ¼ a0 þ An cos þ bn sin ðA:2:1Þ L L n¼1 The technique used to generate Fourier series is easily extended to develop expressions for non-periodic functions. The Fourier series then becomes a Fourier integral. The Fourier integral expression for a function fðxÞ is 1 ð

½AðwÞ cos wx þ BðwÞ sin wxdw

fðxÞ ¼

ðA:2:2Þ

o

The coeﬃcients AðwÞ and BðwÞ are given by 1 AðwÞ ¼ 1 BðwÞ ¼

1 ð

fðvÞ cos wv dv

ðA:2:3Þ

fðvÞ sin wv dv

ðA:2:4Þ

1 1 ð

1

The Fourier integral expression is simpler in the case of odd or even functions. For example, if the function is odd, fðxÞ ¼ fðxÞ

ðA:2:5Þ

1 ð

fðxÞ ¼

BðwÞ sin wx dw

ðA:2:6Þ

o

On the other hand, if fðxÞ is an even function, fðxÞ ¼ fðxÞ

ðA:2:7Þ

1 ð

fðxÞ ¼

AðwÞ cos wx dw

ðA:2:8Þ

o

The integrals which arise can often be evaluated making use of the Laplace transforms given in table A.1. Fourier transforms are important integral transforms that can be used in the solution of diﬀerential equations. Fourier transformation is also used in modern spectroscopy to obtain spectral data of improved precision. If fðxÞ is an even function its Fourier transform is deﬁned as

585

MATHEMATICAL BACKGROUND

1=2 1 ð 2 F c ðwÞ ¼ fðxÞ cos wx dx

ðA:2:9Þ

o

Inverse Fourier transformation involves estimation of the following integral: 1=2 1 ð 2 fðxÞ ¼ F c ðwÞ cos wx dw

ðA:2:10Þ

o

In the case of odd functions a Fourier sine transform is used. Thus, if fðxÞ is an odd function, the Fourier transform is 1=2 1 ð 2 F c ðwÞ ¼ fðxÞ sin wx dx

ðA:2:11Þ

o

and the inverse Fourier transform is 1=2 1 ð 2 fðxÞ ¼ F cðwÞ sin wt dw

ðA:2:12Þ

o

Fourier transforms of some simple functions are given by Kreyszig [1]. However, usually the Fourier transform is more commonly expressed in complex form without specifying the nature of the function to be transformed. This type of Fourier transform is considered in the following section after a discussion of complex numbers and functions.

A.3 Complex Numbers and Functions Complex numbers are often used to deal with phenomena involving periodic functions. A complex number z is usually written as z ¼ x þ jy

ðA:3:1Þ

where x ispthe ﬃﬃﬃﬃﬃﬃﬃ real part of z and y is the imaginary part. The imaginary unit ‘‘j’’ is equal to 1 . The usual mathematical operations may be performed with complex numbers. Thus, addition gives z1 þ z2 ¼ x1 þ j y1 þ x2 þ j y2 ¼ ðx1 þ x2 Þ þ jðy1 þ y2 Þ

ðA:3:2Þ

Multiplication of two complex numbers gives z1 z2 ¼ ðx1 þ jy1 Þðx2 þ jy2 Þ ¼ x1 x2 þ jðx1 y2 þ x2 y1 Þ þ j2 y1 y2 ¼ ðx1 x2 y1 y2 Þ þ jðx1 y2 þ x2 y1 Þ

ðA:33Þ

The square of ‘‘j’’ is 1, so that the product y1 y2 becomes part of the real term in z1 z2 . Division of one complex number by another illustrates the use of the complex conjugate of the denominator. Thus,

586

LIQUIDS, SOLUTIONS, AND INTERFACES

z1 x1 þ j y1 x1 þ j y1 x2 j y2 x1 x2 þ y1 y2 x y x1 y 2 ¼ ¼ ¼ þ j 2 12 z2 x2 þ j y2 x2 þ j y2 x2 j y2 x22 þ y22 x2 þ y22

ðA:3:4Þ

Important complex functions are based on the Euler formula, that is, e jy ¼ cos y þ j sin y

ðA:3:5Þ

Thus, the exponential of a complex number z is given by ez ¼ ex ðcos y þ j sin yÞ

ðA:3:6Þ

e jz ¼ cos z þ j sin z

ðA:3:7Þ

This can also be written as

It follows that cos z ¼

ejz þ ejz 2

ðA:3:8Þ

sin z ¼

e jz ejz 2j

ðA:3:9Þ

The complex hyperbolic functions are e jz þ ejz ¼ cos z 2

ðA:3:10Þ

e jz ejz ¼ j sin z 2

ðA:3:11Þ

cosh j z ¼

sinh jz ¼

Now, the Fourier transform can be written in complex notation. The Fourier transform of any function fðxÞ is written as 1=2 1 ð 2 F ðwÞ ¼ fðxÞejwx dx

ðA:3:12Þ

1

The exponential in jwx can be expanded into a term in cos wx and one in j sin wx. Thus, the above formula is general and can be applied to both odd and even functions. Inverse Fourier transformation gives 1=2 1 ð 2 fðxÞ ¼ F ðwÞe jwx dw

ðA:3:13Þ

1

A.4 Power Series It is often helpful to express a function in terms of a power series. A very simple power series is

MATHEMATICAL BACKGROUND 1 X

an xn ¼ a0 þ a1 x þ a2 x2 þ

587

ðA:4:1Þ

n¼0

The coeﬃcients, ai are often related to one another in a simple way which is determined by the nature of the function. An important method of expressing functions in a power series is the Taylor and Maclaurin expansions. In a Taylor expansion the function fðxÞ is expanded about a given point x0 and the coeﬃcients are related to the values of the derivatives of the function at x ¼ x0 . Thus, the Taylor expansion of fðxÞ is ! df ðx x0 Þ2 d2 f fðxÞ ¼ fðx0 Þ þ ðx x0 Þ þ þ

dx x¼x0 2! dx2 x¼x0 ðA:4:2Þ ðx x0 Þn dn f þ þ

dxn x¼x0 n! The Maclaurin expansion is the special case in which the function is expanded about x ¼ 0. Thus, the Maclaurin expansion of fðxÞ is ! 2 df x2 d2 f fðxÞ ¼ fð0Þ þ x þ dx x¼0 2! dx

þ

þ x¼0

xn dn f þ

n! dxn x¼0

ðA:4:3Þ

A list of important functions expressed as power series follows: eax ¼ 1 þ ax þ

a2 x2 an xn þ

þ

þ 2! n!

1 ¼ 1 ax þ a2 x2 a3 x3 þ

a þ ax sin ax ¼ ax cos ax ¼ 1

a3 x3 a5 x5 a7 x7 þ þ

3! 5! 7!

a 2 x 2 a 4 x 5 a 6 x6 þ þ

2! 4! 6!

ðA:4:4Þ ðA:4:5Þ ðA:4:6Þ ðA:4:7Þ

lnð1 þ axÞ ¼ ax

a2 x2 a3 x3 a4 x4 þ þ

2 3 4

ðA:4:8Þ

sinh ax ¼ ax þ

a3 x3 a5 x5 a7 x7 þ þ þ

3! 5! 7!

ðA:4:9Þ

cosh ax ¼ 1 þ

a 2 x 2 a 4 x 4 a 6 x6 þ þ þ

2! 4! 6!

ðA:4:10Þ

These expansions are especially useful when ax 1. Under these circumstances the function can be approximated by the ﬁrst few terms in its expansion.

588

LIQUIDS, SOLUTIONS, AND INTERFACES

References 1. Kreyszig, E. Advanced Engineering Mathematics, 8th ed.; John Wiley: New York, 1999. 2. Abramowitz, M.; Stegun, I. A. Handbook of Mathematical Functions; Applied Mathematics Series 55; National Bureau of Standards, Washington, DC, 1964.

Appendix B

The Laws of Electricity and Magnetism

In this appendix, the basic laws of electricity and magnetism are reviewed. These laws are conveniently expressed in terms of Maxwell’s equations. They lead to the fundamental equations describing the propagation of electromagnetic radiation. In addition, they involve vectors and vector calculus. Thus, it is also convenient to review the basic operations of vector calculus in the following summary. In this textbook, vectors are denoted in bold type. Two examples of vectors arising in electricity and magnetism are the electrical ﬁeld E and the magnetic ﬁeld B. The magnetic ﬁeld B is usually called the magnetic induction. Electrostatics gives a description of the forces between charges distributed in a known manner in space. This description is greatly facilitated by introducing the electrical ﬁeld E which is the force per unit charge q. Both the force F and the ﬁeld E are vectors whereas the charge q is a scalar quantity. The relationship between F and E is F ¼ qE The ﬁeld due to a point charge in a vacuum is q E¼ 4pe0 r2

ðB:1Þ

ðB:2Þ

where r is the distance from the charge to the point where the ﬁeld is measured, and e0, the permittivity of free space. In the SI system of units, e0 is equal to 8.85410–12 F m–1. The force between two charges q1 and q2 separated by a distance r12 is 589

590

LIQUIDS, SOLUTIONS, AND INTERFACES

F¼

q1 q2 4 e0 r212

ðB:3Þ

This is Coulomb’s law. Gauss’ law relates the ﬁeld due to a collection of charges to the magnitude of charges. Thus, ð X qi E n da ¼ ðB:4Þ e0 i S

S is a surface which surrounds the charges, for example, a sphere. n is a vector perpendicular to the surface. The integration is carried out over the surface area a. Both Coulomb’s law and Gauss’ law have been written here for charges in a vacuum. This law may be used to estimate the ﬁeld due to a uniform distribution of charges. A well-known example is the ﬁeld due to charge on the surfaces of a parallel-plate capacitor. For this system, the ﬁeld is s E¼ ðB:5Þ e0 where s is the charge density on the plates measured in C m–2. It is important to remember that the ﬁeld points in the direction that a positive test charge would move. The vector E points from the positively charged plate to the negative one. The electrical potential is deﬁned as the work done to move a test charge from one point in a ﬁeld to another. Suppose the potential diﬀerence is estimated between a point located a distance r1 from a point charge q to a point located a distance r2. The diﬀerence in potential is q q 1 1 f f ¼ dr ¼ 4pe0 r2 r1 4pe0 r2 ðr2

2

1

ðB:6Þ

r1

The minus sign expresses the fact that work is done and the energy of the system increases when the positive test charge moves against the ﬁeld. The general expression for the potential due to a point charge q is q ðB:7Þ f¼ 4pe0 r The electrical potential is a scalar quantity. The general result relating the electrical potential to the electrical ﬁeld is E ¼ grad f ¼ rf

ðB:8Þ

where grad is a vector operator deﬁned as follows: gradf ¼

@f @f @f iþ jþ k @x @y @z

ðB:9Þ

i, j, and k are unit vectors pointing in the three directions deﬁning the Cartesian coordinate system. On the basis of equation (B.8), the ﬁeld vector points in the direction that the potential decreases, all directions in space being considered. The vector operator grad appears often in the description of systems which are not at equilibrium. A well-known example in chemistry is a concentration gradi-

THE LAWS OF ELECTRICITY AND MAGNETISM

591

ent. In this case the gradient of the concentration, grad c, deﬁnes the direction in which a concentration change occurs, and can be used to estimate the diﬀusion of the species whose concentration is changing. Another example is a temperature gradient in a system in which thermal energy is being redistributed. At this point it is convenient to introduce two other vector operations, namely, the scalar product and the vector product. The scalar product of two vectors f and g is denoted as f g and is given by f g ¼ fg cos y

ðB:10Þ

where f and g are the magnitudes of f and g, and y is the angle between f and g. The scalar product may also be written in terms of the components of f and g in the Cartesian system. Thus, f g ¼ fx gx þ fy gy þ fz gz

ðB:11Þ

where fx , fy , fz are the components of f in the Cartesian directions, and gx , gy , and gz the components of g. The vector product between f and g is denoted as f g and results in a vector h pointing in a direction perpendicular to the plane containing f and g. The value of h in terms of the Cartesian components of f and g is h ¼ ðfy gz fz gy Þi þ ðfz gx fx gz Þj þ ðfx gy fy gx Þk

ðB:12Þ

If the vector f points in the x-direction (fy ¼ fz ¼ 0) and the vector g, in the y-direction ðgx ¼ gz ¼ 0), then the vector h points in the z-direction. Now the vector operators deﬁning the derivatives of a vector are introduced. The divergence operator div simply gives the derivative of the vector in terms of its Cartesian components. Thus, div f ¼ r f ¼

@fx @fy @fz þ þ @x @y @z

ðB:13Þ

According to this equation the divergence of a vector is given by the scalar product of the vector operator r and the vector. The other operator deﬁning a vector derivative is curl. It is deﬁned as follows: @fy @fx @fx @fy @fx @fz curl f ¼ r f ¼ iþ jþ k ðB:14Þ @y @z @z @x @x @y Returning to electrostatics, a general result following from equation (B.4) is div E ¼

rz e0

ðB:15Þ

where rz is the charge density in space. Combining this with equation (B.7), one obtains div gradf ¼ r rf ¼ r2 f ¼

rz e0

ðB:16Þ

This is Poisson’s equation in vacuum. In a medium with relative permittivity es , Poisson’s equation is

592

LIQUIDS, SOLUTIONS, AND INTERFACES

r2 f ¼

rz e0 es

ðB:17Þ

The operator r2 is called the Laplacian. In Cartesian coordinates the Laplacian of f is given by r2 f ¼

@2 f @2 f @2 f þ þ @x2 @y2 @z2

ðB:18Þ

When the charge density in the medium is zero, Poisson’s equation reduces to Laplace’s equation, that is, r2 f ¼ 0

ðB:19Þ

Much has been written about the solution of Laplace’s equation in both Cartesian and spherical coordinates. Several important relationships are relevant to the properties of a dielectric. The vector P deﬁned for a dielectric is the net dipole moment per unit volume. When it is combined with the electrical ﬁeld, one obtains the deﬁnition of the electric displacement D. Thus, D ¼ e0 E þ P

ðB:20Þ

D ¼ e0 es E

ðB:21Þ

Alternatively, one may write where es is the relative permittivity of the dielectric medium; it is a dimensionless quantity which is also called the dielectric constant. Combining equations (B.20) and (B.21), it follows that P ¼ e0 Eðes 1Þ

ðB:22Þ

div D ¼ rz

ðB:23Þ

Finally, it can be shown that This is one of the fundamental Maxwell equations. The other Maxwell equation for stationary charges is curl E ¼ 0

ðB:24Þ

When charges move, one must consider the associated magnetic eﬀects. In the case that a point charge q is moving, the force on it is F ¼ qðE þ v BÞ

ðB:25Þ

where v is the velocity of the charge and B, the magnetic induction. B is the fundamental magnetic quantity just as the electrical ﬁeld E is the fundamental electrical quantity. The SI unit for B is the tesla, or weber m–2. The magnetic induction at a distance r from a charge q moving with velocity v is given by the Biot–Savart law: m vr B¼ 0q 3 ðB:26Þ 4p jr j m0 is the permeability of vacuum and is equal to 4p 107 N C–2 s–2, or 4 10–7 Tesla m A–1. This can be used to derive an expression for the magnetic induction

THE LAWS OF ELECTRICITY AND MAGNETISM

593

due to a current I in an inﬁnitely long straight wire in vacuum at a perpendicular distance r: B¼

m0 I 2pr

ðB:27Þ

Another one of Maxwell’s equations can be derived from the Biot–Savart law, namely, div B ¼ 0

ðB:28Þ

The magnetic dipole moment m can be deﬁned from a constant current I moving in a circular loop. The magnitude of m is given by m ¼ Ia

ðB:29Þ

where a is the area of the loop. The magnetic induction due to the current measured at the center of the loop in a direction perpendicular to its area is B¼

m0 I 2r

ðB:30Þ

where r is the radius of the circular loop. The energy of interaction between an external ﬁeld B and the magnetic dipole moment is u ¼ m B

ðB:31Þ

Magnetic media are classiﬁed as diamagnetic, paramagnetic, and ferromagnetic. A diamagnetic medium contains no permanent magnetic dipoles but magnetic dipoles can be induced in an external magnetic ﬁeld. A paramagnetic medium contains permanent magnetic dipoles which are randomly oriented in the absence of an external magnetic ﬁeld. A paramagnetic medium contains permanent magnetic dipoles which are randomly oriented in the absence of an external magnetic ﬁeld. In the presence of an external magnetic ﬁeld, the dipoles in a paramagnetic material align with the external ﬁeld. A ferromagnetic medium represents an extreme case of paramagnetism in which the magnetic dipoles are permanently aligned with respect to one another in domains of ﬁnite dimensions. This property of magnetic materials can only be treated by quantum mechanics. The ferromagnetic eﬀect decreases with temperature and reverts to paramagnetism at temperatures above the Curie temperature. In a magnetic medium, the vector M represents the sum of the magnetic dipole moments in unit volume. In a diamagnetic or paramagnetic medium, M is proportional to the external magnetic induction B. The proportionality constant is negative in the case of diamagnetism and positive for paramagnetism. The relationship between B and M in a magnetic material in which a current density Jel is ﬂowing is B curl M ¼ Jel ðB:32Þ m0 The quantity (B=m0 Þ M is deﬁned to be the magnetic ﬁeld H, so that B ¼ m0 ðH þ MÞ

ðB:33Þ

594

LIQUIDS, SOLUTIONS, AND INTERFACES

Thus equation (B.32) can also be written as curl H ¼ Jel

ðB:34Þ

A more general form of equation (B.34) derived by Maxwell is curl H ¼ Jel þ

@D @t

ðB:35Þ

Finally, one can deﬁne the relative permeability of a magnetic medium km by the relationship B ¼ km m0 H

ðB:36Þ

Maxwell’s equations are equations (B.23), (B.24), (B.28), and (B.35). Equation (B.21) gives the relationship between E and D, and equation (B.36), that between B and H. With these results all of the fundamental relationships in electricity and magnetism can be derived.

Appendix C

Numerical Methods of Data Analysis

The proper analysis of experimental data requires careful consideration of the numerical techniques used. Real data are subject to experimental error which can have an eﬀect on results derived from the analysis. Often, this analysis involves ﬁtting a curve to experimental data over the whole range or over part of the range in which experimental observations have been made. When thermodynamic data are involved, the relationship between the independent and dependent variable is usually not known. Then, arbitrary functions such as polynomials in the independent variable are often used in the data analysis. This type of data analysis requires consideration of the level of error in both variables, and of the eﬀects of the error on derived results. In this appendix, methods of numerical analysis used for interpolation, integration, and diﬀerentiation of numerical data are discussed. Since these procedures involve curve ﬁtting, regression analysis is also considered. Emphasis is placed on methods in which a minimum of assumptions are made regarding the functional relationship between the independent variable x and the dependent variable y.

C.1 The Principle of Least Squares Two types of error are generally recognized in discussions of experimental data. One is determinate or systematic error, that is, error which arises from the design of the experiment. The second is indeterminate or random error which arises because the experimental conditions cannot be perfectly controlled. It is assumed 595

596

LIQUIDS, SOLUTIONS, AND INTERFACES

that the latter type of error can be dealt with using statistics. It is the main focus of the discussion in this section. Determinate error usually results from experimental equipment which is faulty. Students usually ﬁrst meet this concept in the analytical chemistry laboratory in determinations of weight and volume. The quality of the equipment used is reﬂected in the accuracy of the results obtained. Accuracy is a measure of how close the experimental result is to the truth. For example, if one wishes to make up a solution of accurately known concentration in a volumetric ﬂask of 100 mL, both the ﬂask and the balance used must be carefully calibrated. The ﬂask is calibrated by weighing it empty and then ﬁlled with distilled water at a known temperature. On the basis of the weight of water, and the known density of the water, one may calculate an accurate volume for the ﬂask when it is properly ﬁlled to the mark. Calibration of the balance is based on the use of standard weights which do not corrode and which cover the range in mass used in the experiment. The accuracy of the standard weights and the quality of the volumetric ﬂask determine the accuracy of the concentration of the solution which is made. The operator’s inability to control perfectly all the variables in an experiment leads to indeterminate error. For example, errors in the concentration of the solution made up in a volumetric ﬂask can result from ﬂuctuations in the room temperature. Changes in humidity in the room can also aﬀect the weights involved in the experiment, for example, the weight of a solute which can adsorb water vapor. Indeterminate or random error aﬀects the precision of the experiment. This type of error is treated by the laws of probability. If one determines the concentration of a solution each day on ﬁve successive days, the results are the same only within the precision of the experiment. On the basis of experience, the operator averages the results, and then estimates the standard deviation in order to have an indication of the precision of the experiment. When a given quantity x is observed an inﬁnite number of times, the observed values follow a Gaussian or normal error distribution. Thus, the probability of observing a value x for the quantity being measured is given by " # 1 1 x x 2 exp PðxÞ ¼ 2 s ð2pÞ1=2 s

ðC:1:1Þ

where x is the average value of x, and s the standard deviation. The probability distribution has the well-known shape shown in ﬁg. C.1 with a maximum at x equal to x . The probability function is normalized so that the total probability found by integrating the area under the curve is unity. The height and width of the distribution curve depend on the standard deviation, becoming lower and broader, respectively, as s increases. When the observation deviates from the mean, or most probable value, by the standard deviation s the probability of its being observed drops to e-1/2 Pmax, that is, 0.607Pmax. When x x is 2s, this fraction is 0.135, and when x x is 3s, 0.011. Since the probability of observations with x greater than 3s is less than 1%, such observations are often rejected from the data pool on the basis that the observed error is not random in nature. When the same quantity is observed N times, the probability of the resulting

NUMERICAL METHODS OF DATA ANALYSIS

597

Fig. C.1 Plot of the probability of observing x, PðxÞ, against the value of x with respect to its average value x.

distribution of values of xi is given by the product of the probabilities of each observation " # N Y 1 1 X ðxi x N Þ2 PN ¼ Pðxi Þ ¼ exp ðC:1:2Þ 2 i sx ð2pÞ1=2 sx where x N is the estimate of the most probable value of x based on N observations and sx , the corresponding estimate of the standard deviation. These quantities diﬀer from the true values x and s, which are only found if x is observed an inﬁnite number of times. In order to determine x N , one must ﬁnd the maximum in the function PN . This occurs when there is a minimum in the negative argument of the exponential function. Therefore, the condition for a maximum in PN is Xxi x N 2 ¼ ¼ minimum ðC:1:3Þ sx i This is the principle of least squares applied to the determination of a single quantity x. The sum of squares is minimized, allowing x N to have the optimum value. Thus, one has the condition @ 2 X ¼ 2 ðx x N Þ ¼ 0 ðC:1:4Þ @x N sN i i Since the summation is carried out over N values of xi , the result is x N ¼

X xi i

N

ðC:1:5Þ

It follows that the best estimate of x is simply its average value, a result that is considered intuitively obvious.

598

LIQUIDS, SOLUTIONS, AND INTERFACES

Two other quantities must be deﬁned in order to complete the description of the statistical parameters for a single observed quantity. The ﬁrst is the variance vx , which is given by X vx ¼ ðxi x N Þ2 ðC:1:6Þ i

This simply is the sum of the squares of the deviation of the individual observations xi from the average value for the population x N . Expanding the expression in the summation, the variance may also be written as !2 X 2 X xi xi =N ðC:1:7Þ vx ¼ i

i

The estimate of the standard deviation is obtained from the root mean square of the variance so that

v 1=2 x sx ¼ ðC:1:8Þ N1 The factor N 1 rather than N is required in the denominator because one degree of freedom in the data set has been lost due to the estimation of the average, x N . The laws for combination of errors may be used to obtain an estimate of the error in the derived average x N . If y is a function of x, then the variance of y is given by 2 dy vy ¼ vx ðC:1:9Þ dx If y is a function of several variables, x1 ; x2 ; c3 ; . . . ; then this relationship is generalized to give X @y 2 vy ¼ v xi ðC:1:10Þ @xi i Now, applying this to the equation for x N (equation (C.1.5)), one obtains X vx v vx ¼ ðC:1:11Þ ¼ x 2 N N i The standard deviation of the average is sx ¼

sx N1=2

ðC:1:12Þ

Thus, the standard deviation of the average decreases with the square root of the number of observations. The above treatment is easily extended to data sets in which the individual observations have diﬀerent levels of error. In this case the standard deviation is designated si because it varies with index i. Under these circumstances a weighted average must be calculated using the relationship X X x N ¼ wi xi = wi ðC:1:13Þ i

where

i

NUMERICAL METHODS OF DATA ANALYSIS

wi ¼

1 1 ¼ ; vi s2i

599

ðC:1:14Þ

vi being the variance of observation i. More details about weighted least-squares procedures can be found in the monograph by Bevington and Robinson [1].

C.2 Linear Regression Curve ﬁtting is applied when a dependent variable y is measured experimentally as a function of an independent variable x. The simplest relationship between these variables is a straight line. Under some circumstances, x and y do not come directly from experiment but are calculated from the experimental data on the basis of a theory which predicts that y should be linear in x. In the present context, the independent variable x is always that with a negligible level of error, whereas any signiﬁcant experimental error is only associated with the dependent variable y. Under these circumstances the principle of least squares is applied to estimating the best straight line through the points with respect to the random error in y. The equation of the straight line is written as y ¼ a þ bx

ðC:2:1Þ

where a is the intercept on the y-axis, and b, the slope. The best values of a and b are determined in the regression analysis with the requirement that the sum of the squares of the deviations between the estimated value of y, yci , and the experimental value yi be a minimum. This sum is deﬁned by the equation X ¼ ðyi a bxi Þ2 ¼ minimum ðC:2:2Þ i

where the estimate of y is yci ¼ a þ bxi

ðC:2:3Þ

The minimization of is performed by optimizing the parameters a and b. Mathematically, this leads to two equations called the normal equations. In the present case, they are X @ ¼ 2 ðyi a bxi Þ ¼ 0 @a i

ðC:2:4Þ

X @ ¼ 2 ðyi a bxi Þxi ¼ 0 @b i

ðC:2:5Þ

and

After simpliﬁcation by placing the summation operator with each term, one obtains

600

LIQUIDS, SOLUTIONS, AND INTERFACES

Na þ X

X

bxi ¼

i

axi þ

i

X

X

yi

ðC:2:6Þ

xi yi

ðC:2:7Þ

i

bx2i ¼

X

i

i

where N is the number of values of xi and y. In order to simplify the symbols, this is written as Na þ Sx b ¼ Sy

ðC:2:8Þ

and Sx a þ S2x b ¼ Sxy ðC:2:9Þ P P where Sx equals i xi , Sy equals i yi , and so on. The equations (C.2.8) and (C.2.9) constitute two equations in the unknowns a and b. The solution of these equations is a¼

Sy S2x Sxy Sx Qx

ðC:2:10Þ

and b¼

Qxy Qx

ðC:2:11Þ

where Qx ¼ NS2x S2x

ðC:2:12Þ

Qxy ¼ NSxy Sx Sy

ðC:2:13Þ

and

The diﬀerence Qx is the variance of x multiplied by the number of points N; Qxy is the covariance of x and y also multiplied by N. Quantities like these turn up frequently in the solution of the normal equations associated with regression analysis. Another important quantity associated with the ﬁt of the best straight line to the data is the estimate of the standard deviation in y. From equations (C.2.2), (C.2.6), and (C.2.7) one obtains X ¼ ðyi a bxi Þ2 ¼ S2y aSy bSxy ðC:2:14Þ i

The estimate of the standard deviation in y is then 1=2 sy ¼ N2

ðC:2:15Þ

where the factor N 2 in the denominator indicates that two degrees of freedom were lost in the estimation of the intercept a and slope b. One may now also estimate the errors in these quantities. For the intercept a, X @a 2 2 s2y ðC:2:16Þ sa ¼ @yi i

NUMERICAL METHODS OF DATA ANALYSIS

601

On the basis of equation (C.2.10), @a S2x xi Sx ¼ @yi Qx

ðC:2:17Þ

so that s2a ¼

s2y X 2 ðSx xi Sx Þ2 Q2x i

sy2 X 2 ¼ 2 ðSx2 2xi Sx S2x þ x2i S2x Þ Qx i

ðC:2:18Þ

Simplifying, the result is s2a ¼

S2x s2y Qx

ðC:2:19Þ

In the case of the slope b, from equation (C.2.11), @b Nxi Sx ¼ @yi Qx

ðC:2:20Þ

so that s2b þ

X @b 2 i

@yi

s2y ¼

X ðNxi Sx Þ2 i

Q2x

s2y

ðC:2:21Þ

After simpliﬁcation, the result is s2b ¼

Ns2y Qx

ðC:2:22Þ

The ﬁnal quantity required in order to assess the quality of the ﬁt is the correlation coeﬃcient r. It arises from a comparison of the results of a ﬁt of y as a function of x with those from a ﬁt of x as a function of y. The deﬁnition of the correlation coeﬃcient is r¼

Qxy ðQx Qy Þ1=2

ðC:2:23Þ

where Qy ¼ NS2y S2y

ðC:2:24Þ

The value of r is zero when there is no correlation and reaches a value of 1 when there is perfect correlation. The sign of r is the same as that of the slope b and has no signiﬁcance in assessing the value of the ﬁt. Further interpretation of the correlation coeﬃcient is not straightforward without more detailed knowledge of the statistical properties of the random error in the data sample considered. In an approximate way, the value of r2 gives the percent of the observed variation in y with x that has been explained by the correlation obtained.

602

LIQUIDS, SOLUTIONS, AND INTERFACES

EXAMPLE

Using the following set of data obtain the equation of the best straight line relating the value of y to the independent variable x. Estimate the standard deviation of y, and of the slope and intercept of the resulting linear equation. Finally, estimate the correlation coeﬃcient r. x

y

1 2 3 4 5 6 7 8 9 10

19.5 9.2 4.0 2.5 0.0 14.2 25.0 26.0 28.0 40.4

First, it is necessary to calculate the sums deﬁned in the development of the normal equations and the expression for . These are Sx ¼ 55; S2x ¼ 385; Sy ¼ 98:4; S2y ¼ 4405:94; Sxy ¼ 1064:3

ðC:2:25Þ

Now, the values of Qi are calculated: Qx ¼ 10 385 ð55Þ2 ¼ 825

ðC:2:26Þ

Qy ¼ 10 4405:94 ð98:4Þ ¼ 34376:84

ðC:2:27Þ

Qxy ¼ 10 1064:3 þ 55 98:4 ¼ 5231

ðC:2:28Þ

2

It is now possible to estimate the best values of the intercept and slope: a¼

98:4 385 þ 1064:3 55 ¼ 25:0 825

ðC:2:29Þ

5231 ¼ 6:3 825

ðC:2:30Þ

and b¼

The next step in the analysis is to calculate the error estimates and the correlation coeﬃcient r. On the basis of equation (C.2.14), the value of is ¼ 4405:94 þ 25:0 98:4 6:34 1064:3 ¼ 120:91 and sy ¼

120:91 1=2 ¼ 3:9 8

ðC:2:31Þ

ðC:2:32Þ

The estimate of the error in a is sa ¼ 3:9

385 1=2 ¼ 2:7 825

ðC:2:33Þ

NUMERICAL METHODS OF DATA ANALYSIS

603

and in b,

10 sb ¼ 3:9 825

1=2 ¼ 0:4

ðC:2:34Þ

Finally, the value of r is r¼

5231 ¼ 0:982 ð825 34;377Þ1=2

ðC:2:35Þ

Thus, the ﬁtted line explains approximately 96% of the observed variation in y with x. The data are plotted in ﬁg. C.2, which also shows the best straight line from the least-squares regression. It should be noted that the calculation must be carried out using all of the signiﬁcant ﬁgures. This follows from the fact that the intermediate steps in the calculation often involve subtraction of two very large numbers to give a small result (see equations (C.2.26) – (C.2.29)). Thus, regression analysis is often carried out in computer calculations using double precision arithmetic. This is especially important in multiple linear regression which is discussed in the following section. Under some circumstances, the linear relationship between x and y involves simple proportionality so that the value of y is zero when x is zero. The equation of the straight line is then y ¼ bx

ðC:2:36Þ

so that only one parameter, namely, the slope b is determined in a regression analysis. The least-squares analysis is then considerably simpler. The condition for a minimum in the sum of squares of the deviations between the observed and calculated values of y is now

Fig. C.2 Plot of the data used in the example showing the best straight line ﬁt by least squares.

604

LIQUIDS, SOLUTIONS, AND INTERFACES

¼

X ðy1 bxi Þ2 ¼ minimum

ðC:2:37Þ

i

This minimum is obtained by optimizing the value of b. It follows that X @ ¼ 2 ðyi bxi Þxi ¼ 0 ðC:2:38Þ @b i or

X

xi yi ¼ b

X

i

x2i

ðC:2:39Þ

i

Using the notation introduced earlier, the solution giving the value of b is b¼

Sxy S2x

ðC:2:40Þ

Since there is only one parameter to be determined in the regression analysis, there is only one normal equation (C.2.39). The error in y may now be estimated using equation (C.2.37). The result is X ¼ ðyi bxi Þ2 ¼ S2y bSxy ðC:2:41Þ i

with equation (C.2.39) being used to simplify the expression after expanding the terms in the summation. The estimate of the standard deviation in y is now 1=2 sy ¼ ðC:2:42Þ N1 where N is the number of observations. Since only one parameter is estimated in this analysis, the factor in the denominator of equation (C.2.42) is N 1. The estimate of the error in the slope b is found using the equation X @b 2 s2b ¼ s2y ðC:2:43Þ @y i i On the basis of equation (C.2.40), @b x ¼ i @yi S2x

ðC:2:44Þ

It follows that s2b ¼

s2y X ðS2x Þ2

i

x2i ¼

s2y S2x

ðC:2:45Þ

Finally, the correlation coeﬃcient r is estimated as before, using equation (C.2.23). There are two points that must be remembered in applying the above analysis to experimental data. The ﬁrst is that the analysis assumes that the random error is associated only with the dependent variable y. Thus, it is important that the variables be chosen with this fact in mind. In practice, this means that the independent variable x is that quantity which has a lower level of error than that

NUMERICAL METHODS OF DATA ANALYSIS

605

estimated for the dependent variable y. The second point is that the analysis assumes that the level of error in the dependent variable y is independent of its magnitude. This is not always the case, either because the level of experimental error changes over the range that the variable y is observed, or because a quantity derived from the variable y is used in the regression analysis. Simple examples arise in the analysis of kinetic data for chemical reactions. In these experiments, one normally observes the concentration of a reactant or a product as a function of time. If the reaction is ﬁrst order, analysis of the data to obtain the rate constant requires that the logarithm of the concentration be plotted as a function of time. If the level of error in the concentration is independent of time, then the level of error in ln c changes with time. Under these circumstances, weighted least squares must be used to analyze the data properly. Details describing how to use weights in regression analysis are available elsewhere [1].

C.3 Multiple Linear Regression The analysis presented for ﬁtting a straight line to experimental data is easily extended to a curve or to a system with more than one independent variable on the basis of the principle of least squares. Consider ﬁrst a system where the observed quantity y depends on two independent variables, x1 and x2 . The resulting relationship describes the equation of a plane in three dimensions and may be written as y ¼ a þ bx1 þ cx2

ðC:3:1Þ

Three parameters, namely, a, b, and c are required to specify the relationship. The equation describing the principle of least squares is X ðyi a bx1i cx2i Þ2 ¼ minimum ðC:3:2Þ ¼ i

The sum is now minimized with respect to each of the adjustable parameters. This leads to the following three equations: X @ ¼ 2 ðyi a bx1i cx2i Þ ¼ 0 ðC:3:3Þ @a i X @ ¼ 2 ðyi a bx1i cx2i Þx1i ¼ 0 ðC:3:4Þ @b i X @ ¼ 2 ðyi a bx1i cx2i Þx2i ¼ 0 ðC:3:5Þ @c i After simpliﬁcation, and using the notation introduced earlier, the resulting normal equations are Na þ bSx1 þ cSx2 ¼ Sy

ðC:3:6Þ

aSx1 þ bSx2 þ cSx1 x2 ¼ Sx1 y

ðC:3:7Þ

aSx2 þ bSx1 x2 þ cSx2 ¼ Sx2 y

ðC:3:8Þ

1

2

606

LIQUIDS, SOLUTIONS, AND INTERFACES

This system of linear equations is usually solved using determinants, and matrix diagonalization techniques. Computer programs to do this are easily available. The important point to remember is that the calculation should be performed in double precision arithmetic because it often involves subtraction of large numbers which are close to one another. In determinant notation, the solution to these equations is 0 1 Sy S x1 Sx2 @ Sx1 y Sx2 Sx1 x2 A 1 Sx2 y Sx1 x2 Sx2 2 ðC:3:9Þ a¼ D 0 1 N Sy Sx2 @ Sx1 Sx1 y Sx1 x2 A Sx2 Sx2 y Sx2 2 b¼ ðC:3:10Þ D 0 1 N Sx1 Sy @ Sx Sx2 S x1 y A 1 1 Sx2 Sx1 x2 Sx2 y ðC:3:11Þ c¼ D where

0

N D ¼ @ Sx1 Sx2

S x1 S x2 1 S x1 x2

1 Sx2 Sx1 x2 A Sx2

ðC:3:12Þ

2

Once the coeﬃcients a, b, and c have been found, it is a simple matter to estimate using equation (C.3.2), and hence the value of the standard deviation y : 1=2 sy ¼ ðC:3:13Þ N3 where N is the number of observations. Estimates of the errors in the coeﬃcients sa ; sb , and sc can then be obtained using the same procedure outlined for linear least squares. The results are !1=2 Sx2 Sx2 S2x1 x2 1 2 ss ¼ sy ðC:3:14Þ D !1=2 NSx2 S2x2 2 ðC:3:15Þ sb ¼ sy D !1=2 NSx2 S2x1 1 sc ¼ sy ðC:3:16Þ D The same type of analysis applies when data are ﬁtted to a quadratic equation by least squares. The relationship between the dependent variable y and the independent variable x can be written as

NUMERICAL METHODS OF DATA ANALYSIS

y ¼ a þ bx þ cx2 and the least-squares condition is X ðyi a bxi cx2i Þ2 ¼ minimum ¼

607

ðC:3:17Þ ðC:3:18Þ

i

The normal equations are now Na þ bSx þ cS2x ¼ Sy

ðC:3:19Þ

aSx þ bS2x þ cS3x ¼ Sxy

ðC:3:20Þ

aS2x þ bS3x þ cS4x ¼ S2x y

ðC:3:21Þ

The solution of these equations expressed 0 Sy Sx @ Sxy Sx2 Sx2 y Sx3 a¼ D 0 N Sy @ Sx Sxy Sx2 Sx2 y b¼ D 0 N Sx @ Sx1 Sx2 Sx2 Sx3 c¼ D where

0

N D ¼ @ Sx Sx2

Sx S x2 S x3

in terms of determinants is 1 Sx2 Sx3 A Sx4 1 Sx2 Sx3 A Sx4 1 Sy Sxy A Sx2 y

1 Sy2 Sx3 A Sx4

ðC:3:22Þ

ðC:3:23Þ

ðC:3:24Þ

ðC:3:25Þ

Since the values of xi are raised to the fourth power in this calculation, a sum such as S4x can be very much smaller or very much larger than the sum Sx . As the operations of determinant diagonalization demonstrate, this means that double precision arithmetic must be used in solving the normal equations. Having obtained values of the coeﬃcients a, b, and c estimates of the errors may be made. The value of sy is again given by equation (C.3.13). The expressions for the errors in the coeﬃcients are !1=2 S2x Sx4 S2x3 ðC:3:26Þ sa ¼ sy D

sb ¼ sy

NSx4 S2x2 D

!1=2 ðC:3:27Þ

608

LIQUIDS, SOLUTIONS, AND INTERFACES

sc ¼ sy

NSx2 S2x D

!1=2 ðC:3:28Þ

It is clear that the above methods can be easily extended to equations involving more than two independent variables or to polynomials involving terms of order higher than x2. Each time a term is added to the right-hand side of the equation relating the independent and dependent variables, the order of the determinants deﬁned from the normal equations increases. Although the calculations rapidly become extremely tedious if carried out by hand, they are easily carried out using available computer routines. Unless there is a fundamental theory which gives a functional relationship between x and y, ﬁtting a polynomial over the whole range of data available should be avoided. This is especially true with thermodynamic data for which an analytical function relating x and y is usually not available. Numerical analysis of these data is best carried out using methods in which a segment of data containing a few points are ﬁt to a curve. These techniques are described in the remaining sections of this appendix.

C.4 Numerical Methods Numerical methods include those based on ﬁnite diﬀerence calculus. They are ideally suited for tabulated experimental data such as one ﬁnds in thermodynamic tables. They also include methods of solving simultaneous linear equations, curve ﬁtting, numerical solution of ordinary and partial diﬀerential equations and matrix operations. In this appendix, numerical interpolation, integration, and diﬀerentiation are considered. Information about the other topics is available in monographs by Hornbeck [2] and Lanczos [3]. The numerical procedures used to deal with experimental data are illustrated with results from an experiment in chemical kinetics recorded in table C.1. The reaction is ﬁrst order but that will be ignored in the analysis of the data. Note that the data points have been obtained for equal increments in the independent variable time. It turns out that numerical techniques are especially easy to apply when this is the case. The second feature of this data set is that the precision of the time data is much higher than that of the concentration data. The same data are plotted in ﬁg. C.3. It is apparent that there is some scatter of the results about a smooth curve ﬁtted through the experimental points. The smooth curve was obtained by ﬁtting a third-degree polynomial to the data by least squares. However, since the reaction is ﬁrst order, this is not a correct description of the relationship between c and t. The third-degree polynomial represents only an approximate description of these data in the range that they are available and illustrates that curve ﬁtting over the whole data range is often not a good idea. Serious errors in interpreting the results would be expected if this curve were extrapolated to values of time greater than 130 s. In order to avoid making assumptions about the functional relationship between the two variables involved in the experiment, it is recommended that

NUMERICAL METHODS OF DATA ANALYSIS

609

Table C.1 Values of the Fraction of Reactant Remaining as a Function of Time in a Solution Kinetics Experiment Time t=s

Fraction Unreacted c=c0

0 10 20 30 40 50 60 70 80 90 100 110 120 130

1.000 0.923 0.810 0.734 0.670 0.617 0.539 0.500 0.460 0.397 0.368 0.339 0.298 0.279

numerical procedures involve only a small segment of the data. This is illustrated by applying these techniques to a segment involving ﬁve points. A low-order polynomial, speciﬁcally, a quadratic equation, can be ﬁt to the three central points explicitly, or to all ﬁve points by least squares. Calculations such as interpolation and diﬀerentiation are limited to the region around the central point. One moves the segment of data considered through the array adding one point on one side and dropping one point from the other. For example, suppose a segment of three points is chosen. The ﬁrst calculation involves data obtained at 0, 10, and 20 s

Fig. C.3 Plot of the fraction unreacted against time using the data in table C.1.

610

LIQUIDS, SOLUTIONS, AND INTERFACES

with emphasis on results near or at 10 s. The calculation can then be carried out for the interval from 10 to 30 s with emphasis on results near or at 20 s. One moves through the data set segmentally adding one point at a higher value of time and dropping one at a lower value. If smoothing is required, the segment should contain ﬁve points with a quadratic ﬁt to these data by least squares. If results are limited to the region of the central point, then the ﬁrst calculation with smoothing is only available at or near 20 s in the present example. This segmental technique is now illustrated for several numerical procedures.

C.5 Numerical Interpolation The problem of numerical interpolation is considered ﬁrst. Suppose that an estimate of the fraction unreacted is required at 52 s. Since data are available at 50 s and 60 s the simplest way to estimate c=c0 is to draw a straight line between the two adjacent points and thereby estimate c=c0 . However, one sees clearly from the graph that the data do not fall on a straight line, so an improved estimate is obtained by using the three closest data points and ﬁtting them to a quadratic equation. In order to keep the following discussion general, time is referred to as x and c=c0 as y. Thus the equation to be ﬁt to the three points is ðC:5:1Þ

y ¼ a þ bx þ cx2

Estimation of a, b, and c requires that three equations based on the data at 40, 50, and 60 s be solved. The problem can be made simpler by redeﬁning the coordinate system for the interval in question so that the central point corresponds to (0, 0) in a new Cartesian system. Thus, new variables x and Z are deﬁned with x ¼ x x0

ðC:5:2Þ

Z ¼ y y0

ðC:5:3Þ

and

where x0 and y0 are the values of x and y in the center of the interval. This transform is illustrated for the ﬁve points centered at 50 s in table C.2. Table C.2 Transformation of Data Centered at 50 s to a New Coordinate System Position

x

y

x

Z

–2 –1 0 1 2

30 40 50 60 70

0.734 0.670 0.617 0.539 0.500

–20 –10 0 10 20

0.117 0.053 0 –0.078 –0.117

NUMERICAL METHODS OF DATA ANALYSIS

611

Since the curve goes through (0, 0) in the new coordinate system, the quadratic equation becomes Z ¼ bx þ gx2

ðC:5:4Þ

Now, only two parameters need to be obtained, namely, b and g. On the basis of the data at position –1 (40 s), one has Z1 ¼ bx1 þ gx21

ðC:5:5Þ

Z1 ¼ bx1 þ gx21

ðC:5:6Þ

and at position 1 (60 s),

These equations are now solved to obtain the values of b and g. The results are b¼

x21 Z1 x21 Z1 x1 x21 x1 x21

ðC:5:7Þ

g¼

x1 Z1 x1 Z1 x1 x21 x1 x21

ðC:5:8Þ

and

For the present example, the estimates are 100 0:053 þ 100 0:078 ¼ 6:55 103 b¼ 10 100 10 100

ðC:5:9Þ

and g¼

10 0:078 10 0:053 ¼ 1:25 104 10 100 10 100

ðC:5:10Þ

It is now a simple matter to estimate the value of y at 52 s. In the new coordinate system this corresponds to x equal to 2. Thus, from equation (C.5.4) Z ¼ 6:55 103 2 1:25 104 4 ¼ 0:0136

ðC:5:11Þ

Transforming back to the original coordinate system, y ¼ y0 þ Z ¼ 0:617 0:0136 ¼ 0:603

ðC:5:12Þ

Thus, the estimate of y at 52 s is 0.603. When the values of x and y are reported for equal increments in x, the equations deﬁning b and g are simpler. In this case, x1 ¼ x1 ¼ h where h is the value of the increment. Then the value of b is given by Z Z1 b¼ 1 2h

ðC:5:13Þ

ðC:5:14Þ

and that of g by g¼

Z1 þ Z1 2h2

ðC:5:15Þ

When the level of error in the dependent variable is high, the interpolation procedure should be carried out with smoothing. This could involve ﬁtting the ﬁve

612

LIQUIDS, SOLUTIONS, AND INTERFACES

points centered near the point of interest to a quadratic equation by least squares. Using the change of variables discussed above, this ﬁt may be realized by ﬁtting equation (C.5.4) to the data in table C.2 by least squares. The normal equations giving the values of b and g by this technique are SxZ ¼ bSx2 þ gSx3

ðC:5:16Þ

Sx2Z ¼ bSx3 þ gSx4

ðC:5:17Þ

and

where the sums involve the values of x and Z recorded in table C.2. These equations are easily solved to obtain b and g. When the values of x are equally spaced, the least-squares analysis based on the data with transposed coordinates gives particularly simple results. First of all, the sums Sx and Sx3 are zero. Thus, the normal equations immediately give the result that b¼

SxZ Sx2

ðC:5:18Þ

and g¼

Sx2 Z Sx4

ðC:5:19Þ

Expanding the sums and simplifying, the estimates of b and g are b¼

2Z2 þ Z1 Z1 2Z2 10h

ðC:5:20Þ

g¼

4Z2 þ Z1 þ Z1 þ 4Z2 34h2

ðC:5:21Þ

and

For the example being considered, the estimates are 2 0:117 0:078 0:053 2 0:117 ¼ 5:99 103 10 10

ðC:5:22Þ

4 0:117 0:0078 þ 0:053 þ 4 0:117 ¼ 7:353 106 34 100

ðC:5:23Þ

b¼ and g¼

On the basis of equation (C.5.4), the estimate of Z at x ¼ 2 is –0.0120 so that the estimated value of y is 0.605. This does not diﬀer greatly from the previous estimate. The diﬀerence reﬂects the level of error in the original data.

C.6 Numerical Integration Now, the procedure for numerical integration is examined. Since random error tends to cancel out in the additive procedure associated with numerical integration, smoothing routines are normally not used. Suppose that integration is

NUMERICAL METHODS OF DATA ANALYSIS

613

carried out in a segmental fashion involving three points per segment. On the basis of the analysis presented above, the integral involving data points i 1, i, and i þ 1 can be written as iþ1 ð

ziþ1 ¼ zi1 þ

ða þ bx þ cx2 Þdx

ðC:6:1Þ

i1

where zi is the value of the integral at point i. This integral can also be written in terms of the transformed equation (C.5.4). It then becomes iþ1 ð

ðyi þ bx þ gx2 Þdx

ziþ1 ¼ zi1 þ

ðC:6:2Þ

i1

After integration one obtains b g ziþ1 ¼ zi1 þ yi ðxiþ1 xi1 Þ þ ðx2 iþ1 x2 iþ1Þ þ ðx3 iþ1 x3 i1Þ 2 3

ðC:6:3Þ

Given the values of yi , b, and g for the interval being considered one can easily calculate the increment in the integral, ziþ1 zi1 . When the data points are equally spaced with respect to the independent variable, equation (C.6.3) can be considerably simpliﬁed. Recalling that xiþ1 is equal to h and xi–1 to h, this equation becomes 2h3 g 3

ðC:6:4Þ

1 ðyiþ1 2yi þ yi1 Þ 2h2

ðC:6:5Þ

ziþ1 ¼ zi1 þ 2hyi þ On the basis of equation (C.5.15), g is given by g¼

Substituting this expression into equation (C.6.4) and simplifying, one obtains h ziþ1 ¼ zi1 þ ðyi1 þ 4yi þ yiþ1 Þ 3

ðC:6:6Þ

This result is known as Simpson’s rule, which provides a very simple and convenient way to perform numerical integration. The above result may be generalized to a wider interval in the data set provided the number of data points is odd. The procedure involves adding up the contributions to the integral from segments of the data involving three points. After calculating the area associated with the ﬁrst three points, points 1 and 2 are dropped and points 4 and 5 added. One then adds the contribution under the new segment to that from the previous one. In this way, the total integral is evaluated. Consider an interval involving 2n þ 1 equally spaced points starting with the –nth point, and increasing through 0 to the nth point. The value of the integral is then

614

LIQUIDS, SOLUTIONS, AND INTERFACES

h zn ¼ zn þ ½yn þ 4ynþ1 þ 2ynþ2 þ 4ynþ3 þ

3 þ 2yn2 þ 4yn1 þ yn

ðC:6:7Þ

When the integration involves an equal number of equally spaced data points, a special procedure must be devised to estimate the area associated with the last segment in the data set. Suppose the last three points are designated n 2, n 1, and n, where n is an even number. Simpson’s rule may be applied in the normal way up to the (n 1)th point. In order to estimate the area associated with the last segment located between the (n 1)th and nth point, a quadratic equation is ﬁtted to the last three points. Using the above deﬁnitions of b and g, this yields the following values: b¼

Zn2 þ Zn 2

ðC:6:8Þ

g¼

Zn Zn2 2

ðC:6:9Þ

The value of the integral at the nth point is then ðn zn ¼ zn1 þ

ðyn1 þ bx þ gx2 Þdx

ðC:6:10Þ

n1

Since the value of xn1 is 0 and xn is h, the result is zn ¼ zn1 þ hyn1 þ

b h2 g h3 þ 2 3

ðC:6:11Þ

Numerical integration techniques are easily applied in the analysis of experimental data which are acquired digitally. Thus, values of the electrical current which are obtained as a function of time for a ﬁxed interval between observations are easily converted to electrical charge by numerical integration. The technique for numerical integration applied here is one of the simplest and best known. However, there are a variety of other techniques which have been used and which may be more suitable in certain applications [2–4].

C.7 Numerical Differentiation Numerical diﬀerentiation of real experimental data is a diﬃcult procedure because of the experimental error which is typically present. This means that data smoothing is usually required before the ﬁrst derivative is calculated. On the basis of the segmental technique described here, the ﬁrst derivative of y with respect to x is given by dy ¼ b þ 2cx dx

ðC:7:1Þ

where the quadratic equation ﬁt to the data in the given segment has been used. In terms of the transformed data, this result may be written

NUMERICAL METHODS OF DATA ANALYSIS

dy dZ ¼ ¼ b þ 2gx dx dx

615

ðC:7:2Þ

Calculation of the derivative should be limited to values of x close to or at the central point in the interval. Suppose one wishes to estimate dy=dx for the data in table C.1 at 50 s. In the ﬁrst calculation performed above, a quadratic equation was ﬁt to three points with the central point at 50 s and values of bð6:55 103 Þ and gð1:25 104 Þ estimated without smoothing. Since x ¼ 0 at t ¼ 50 s, the estimate of dy=dx on the basis of equation (C.7.2) is 6:55 103 s1 . This corresponds to the negative of the rate reaction in terms of the fraction unreacted. This procedure was carried out in a sequential fashion moving through the data set, making the ﬁrst calculation at 10 s, and the last at 90 s. The results are plotted in ﬁg. C.4. The level of error in the ﬁrst derivative is obviously quite high, approximately 10 times that in the observed value of y. This suggests that estimation of dy=dx should be carried out with some smoothing. When data smoothing is introduced on the basis of a quadratic ﬁt to ﬁve adjacent points, the values of b and g are estimated on the basis of equation (C.5.16) and (C.5.17). These results are also presented in ﬁg. C.4. The scatter in the estimates of the derivative is signiﬁcantly reduced when local smoothing of the data is used. When the data points are equally spaced as is the case in the present example, the ﬁrst derivative may be estimated in a much simpler way. When smoothing is not used and the derivative is estimated at or close to the central point of these values of x, then the relationships derived earlier apply. The values of b and g are estimated using equations (C.5.14) and (C.5.15) and used in equation (C.7.2) to obtain the value of dy=dx. When smoothing is applied, and a quadratic equation is ﬁtted to ﬁve data points, the derivative at or near the central point can be

Fig. C.4 Plot of the ﬁrst derivative of the data in table C.1 estimated using a segmental quadratic ﬁt without data smoothing (*) and with smoothing (*).

616

LIQUIDS, SOLUTIONS, AND INTERFACES

estimated using values of b and g obtained by applying equations (C.5.20) and (C.5.21). In summary, the techniques discussed here are designed to perform simple numerical procedures with experimental data while making the minimum of assumptions regarding the relationship between x and y. They are especially important in the analysis of thermodynamic data where procedures such as numerical diﬀerentiation are often called for. They are very easy to set up on a computer using programming languages such as BASIC and FORTRAN. Since high-order polynomials are not involved in the analysis, no special precautions are needed regarding the precision of the data at intermediate steps in the data analysis. Further information may be found in the following references. References 1. Bevington, P. R.; Robinson, D. K. Data Reduction and Error Analysis for the Physical Sciences; McGraw-Hill: New York, 1992. 2. Hornbeck, R. W. Numerical Methods; Prentice-Hall: New York, 1975. 3. Lanczos, C. Applied Analysis; Pitman: London, 1957. 4. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T. Numerical Recipes, The Art of Scientiﬁc Computing; Cambridge University Press: Cambridge, 1989.

Index

acceptor number, 191 activated complex, 324 activation energy, 323 for electron transfer, 348 activity coeﬃcients, 38, 116 electrolyte, 112, 120, 464 Henry’s law , 35 ion pair, 141 Raoult’s law, 21, 26, 31 regular solution, 26 Wilson model, 31 adiabaticity of electron transfer, 347 adsorption at interfaces of ions, 558 of molecules, 569 adsorption isotherm liquid | gas interface, 401 metal | solution interface, 566, 573 unimolecular ﬁlms, 436 atom transfer reaction, 321 Born model, 102 Bronsted equation, 375 capacitance bridge, 519 chemical exchange reaction, 358 chemical potential, 14, 78, 273

chemical shift, 219, 221, 223, 224 compensation potential diﬀerence, 408, 417, 461 computer simulations, 73 molecular dynamics, 74 Monte Carlo, 74 concentration, 4 conductivity of ions, 274 molar, 277 in non-aqueous media, 294 proton, 298 in water, 283 continuity equation, 257 coordination number, 85, 210, 244 correlation functions, 61 direct, 70 overall, 69 pair, 62 partial pair, 68 total, 64 correlation parameter, 161 Debye, P. J. W., 148 Debye equation, 157 Debye–Hu¨ckel model, 121 constants, 126 extended law, 126 617

618

INDEX

Debye–Hu¨ckel model (continued) limiting law, 127 reciprocal length, 123 Debye–Onsager model, 288 constants, 292 dielectric relaxation, 169, 180 diﬀerential capacity, 519 diﬀraction experiments, 65, 206 neutron, 66, 206, 210 surface X-ray, 528 X-ray, 65, 206 diﬀuse double layer, 531, 542 diﬀerential capacity of, 546 ionic surface excesses of, 550 potential drop across, 546 potential proﬁle of, 549 diﬀusion, 258, 264 from a wall, 266 diﬀusion coeﬃcient, 258, 264, 283 diﬀusion controlled reactions, 329 Dimroth–Reichardt parameter, 191 dipole moment, 53 of adsorbate, 563, 565 dipole potential, 533, 536 distribution functions, 62 Donnan potential diﬀerence, 484 donor number, 193, 225 elastic deformation of crystals, 514 electrical displacement, 156, 170 electrical polarization, 155 distortional, 169 orientational, 169 electrical potential, 590 discreteness-of-charge potential, 273, 567 inner potential, 272, 375 macropotential, 121, 273 micropotential, 121, 273 of point charge, 53, 104 of point dipole, 53 outer potential, 395 surface potential, 395 electroanalysis, 474 electrocapillary curve, 513 electrocapillary maximum, 518 electrochemical cell, 448 concentration cell, 468 electrochemical potential, 271 elecrode potential, 451 absolute, 460 standard, 454

electrode reaction, 449 type I, 454 types II and III, 456 electrolytes, 112 strong, 112 weak, 115 electron density in metals, 539 electron transfer reactions, 314 heteronuclear, 314 homonuclear, 314, 316 inner sphere mechanism, 315 outer sphere mechanism, 315 theory of, 346 electrophoretic eﬀect, 289 electrostatic equilibrium, 508 EMF, 449 ensemble, 48 canonical, 48 grand canonical, 50 microcanonical, 50 enthalpy, 8 activation, 325 excess mixing, 22 mixing, 18, 25 solvation, 100 Born model, 105 MSA model, 106 transfer, 188 entropy, 8, 49 activation, 325 excess mixing, 22 hard spheres, 83 mixing, 17 solvation, 100, 103 Born model, 104 MSA model, 108 enzyme catalysis, 311 equilibrium constant determination of, 466 ion association, 136 ion partition, 428 weak electrolyte dissociation, 115 femtochemistry, 338 ﬂash photolysis, 338 Flory–Huggins isotherm, 406 ﬂuid hard sphere , 79 Lennard–Jones, 59 ﬂuorescence decay time, 345 ﬂux of mass or energy, 256, 270

INDEX

Fourier transform, 584 frequency dispersion, 521 Friedman equation, 178 Frumkin, A. N., 383 Frumkin isotherm, 406 Galvani potential diﬀerence, 396, 428, 472 Gauss’ theorm, 256 Gibbs adsorption isotherm, 510 Gibbs–Duhem relationship, 10, 39, 117 Gibbs energy, 9, 50 activation, 325, 348 excess mixing, 22, 30 hard spheres, 83 mixing, 17, 26 solvation, 100, 103 Born model, 104 MSA model, 107 transfer, 185 Gibbs model for interfaces, 391 Gouy–Chapman model, 531 Guggenheim model for interfaces, 392 Hamiltonian, 50 Hammett equation, 376 heat capacity, 49 Helmholtz energy, 9, 49, 77 hard spheres, 82 Helmholtz layer, 530 Henderson, D. J., 45 Henderson equation, 480 Henry’s law, 35 Hildebrand, J. H., 3 hydrolysis reaction, 317 indicator electrode, 474 infrared spectra of solvents, 232 acetone, 237 acetonitrile, 236 dimethylformamide, 238 formamide, 235 methanol, 234 water, 233 infrared spectroscopy, 227 attenuated total reﬂection, 228 of electrolyte solutions , 242 of non-electrolyte solutions, 239 inner Helmholtz plane, 532 inner layer, 552 integral equations, 70 hypernetted chain, 71

619

mean spherical, 72 Ornstein–Zernike, 71 Percus–Yevick, 71 interface, 383 liquid | liquid, 426 metal | solution, 422 non-polarizable, 425, 431 polarizable, 425, 431, 510 solution | air, 401, 427 interfacial spectroscopy, 437, 524 interfacial tension, 385, 389, 517 intermolecular forces, 52 intermolecular potential, 63, 71 internal energy, 7, 48, 76 hard spheres, 82 ion–ion interactions, 111 Debye–Hu¨ckel model, 121 MSA model, 130 ion selective electrodes, 494 composite, 500 crystalline, 499 glass, 497 liquid membrane, 499 ion solvation, 100 in non-aqueous solvents, 184, 223 in water, 103, 219 ion–solvent interactions, 100 Born model, 202 MSA model, 106 ionic association, 135 Bjerrum model, 138 equilibrium constant, 136 Fuoss model, 140 MSA model, 140 ionic conductivity, 286, 295 ionic radii, 97 Pauling, 97 Shannon–Prewitt, 97, 99 ionic speciﬁc adsorption, 531, 558 ionic strength, 123 ionophores, 491 irreversible processes, 255 isomerization reactions, 343, 365 jellium model, 533, 539 Kalman, E., 204 Kenrick, F. B., 410 Kirkwood correlation parameter, 161, 166 Kohlrausch equation, 283 Koppel–Palm equation, 198 Krygowski–Fawcett equation, 196

620

INDEX

Lamor precessional frequency, 215 Langmuir trough, 435 Laplace transform, 267, 582 laser spectroscopy, 338 laws of electricity and magnetism, 589 Lennard–Jones interaction energy, 59 Lewis acids and bases, 191 Lewis–Sargent equation, 481 ligand exchange reaction, 319 NMR spectroscopy and, 363 limiting molar conductance, 284 linear Gibbs energy relationships, 375 linear regression, 599 liquid junction, 449, 471 liquid junction potential, 472, 477, 483 Lorentz–Lorenz equation, 157 magnetic ﬁeld, 215, 359, 592 magnetogyric ratio, 214 Marcus, R. A., 304 Marcus inverted region, 349 maximum bubble pressure experiment, 388 Maxwell’s equations, 589 medium eﬀects in solution kinetics, 366 dynamic eﬀects, 369 static eﬀects, 367 membrane potentials, 484 membranes, 484 glassy and crystalline, 485 liquid, 491, 499 mobility, 277 molar polarization, 157 molar refraction, 157 multiple linear regression, 605 Navier–Stokes equation, 261 Nernst, W. H., 447 Nernst equation, 451, 456 neutralization reaction, 317 Nikolsky equation, 489 nuclear magnetic resonance (NMR) spectroscopy, 213 and chemical exchange reactions, 358 numerical methods, 595, 608 numerical diﬀerentiation, 614 numerical integration, 612 numerical interpolation, 610 Onsager, L., 254 Onsager reciprocal relations, 257 osmotic coeﬃcient, 117

Ostwald, F. W., 95 Ostwald dilution law, 287 outer Helmholtz plane, 531 packing fraction, 73, 80, 132 Palinkas, G., 204 Parsons, R., 508 Parsons’ function, 559 partial charge transfer, 563 partition equilibria, 427 partition function, 48, 75 conﬁgurational, 52, 74 internal, 51 translational, 51 Pekar factor, 353, 373 p-functions, 502 pH, 502 phenomenological coeﬃcient, 257, 280 plastic deformation of crystals, 514 Poggendorf compensation potentiometer, 450 Poisson–Boltzmann equation, 122 potential energy, 52, 76 potential of mean force, 63, 71 potential of zero charge, 423, 521, 535, 537 power series, 586 precursor complex, 313 pre-exponential factor in rate constant 323, 355 pressure, 8, 49, 77, 80 internal, 18 vapor, 15 principle of least squares, 595 protolysis reaction, 317, proton transfer reactions, 317 NMR spectroscopy and, 359 Raoult’s law, 15 Raman spectroscopy, 229 random error, 596 rate of entropy production, 289 reaction coordinate, 324 reaction order, 306 reaction rate constant, 306 real potential, 396 reference electrode, 474 relaxation techniques, 332 ﬁeld jump, 338 pressure jump, 337 temperature jump, 336 relaxation time, 343

INDEX

621

Debye, 152, 171, 180, 262 longitudinal, 173, 181 reorganization energy, 351 inner sphere, 351 outer sphere, 353

surface excess, 391, 404 surface potential, 395 solution | gas interface, 412, 414, 416 surface pressure, 403 surface selection rule, 525

saturated calomel electrode, 476 second harmonic generation, 438 selectivity equilibrium, 488 silver | silver chloride electrode, 452, 458, 476 solvatochromic basicity, 193 solvatochromism, 245 solutions, density, 5, 119 distribution functions, 88 ideal, 15 ideally dilute, 33 non-ideal, 18 regular, 24 solvent monolayer models, 553 cluster, 557 three state, 556 two state, 554 solvent permittivity, frequency dependence, 171 solvent polarity, 198 solvent properties, 149 dielectric, 152 molecular, 153, 198 thermodynamic, 150 spin–lattice relaxation, 217, 359 spin quantum number, 214 spin–spin relaxation, 217, 359 standard deviation, 598 standard electrode potential, 454 standard hydrogen electrode, 452 static solvent permittivity, 153 Debye model, 154 electrolyte solutions, 176 Kirkwood model, 161 MSA model, 162 Onsager model, 160 stickiness parameter, 164, 166 Stokes and anti-Stokes lines, 231 Stokes–Einstein equation, 280 Stokes’ law, 260 Stokes’ radius, 280, 294 Stokes’ shift, 347 structure factor, 64, 67, 206 structure function, 68, 206

TATB assumption, 185, 427 time-of-relaxation eﬀect, 289 time resolved solvation studies, 342 transfer of ions, 184 enthalpy, 188 Gibbs energy, 185 transition state theory, 323 transport number, 276, 479 ultrasound experiments, 338 ultraviolet-visible spectroscopy, 245 underpotential deposition, 530 unimolecular ﬁlms, 433 variance, 598 vector calculus, 589 vibrational frequency shift, 239 ion induced, 243 solvent induced, 239 vibrational spectroscopy, 226 infrared, 227 interfacial, 524 Raman, 229 vibrational sum frequency spectroscopy, 440 virial adsorption isotherm, 568 viscosity, 259 Volta potential diﬀerence, 397, 461 experimental determination of, 408 metal | solution interface, 424 volume, 6 excess mixing, 22 excess molar, 13 mixing, 18 molar, 7 partial molar, 10 water structure, 84 Wertheim parameter, 107, 163, 168 work function, 398 electrons in metals, 398, 400, 535 ions in solution, 417, 420, 421 Young–Laplace equation, 386