The Thermodynamics of Soil Solution

THE THERMODYNAMICS OF SOIL SOLUTIONS THE THERMODYNAMICS OF SOIL SOLUTIONS GARRISON SPOSITO UNIVERSITY OF CALIFORNIA,...

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THE THERMODYNAMICS OF SOIL SOLUTIONS

THE THERMODYNAMICS OF SOIL SOLUTIONS

GARRISON SPOSITO UNIVERSITY OF CALIFORNIA, RIVERSIDE

OXFORD CLARENDON PRESS 1981

Oxford University Press, Walton Street, Oxford ox2 OXFORD LONDON GLASGOW NEW YORK TORONTO MELBOURNE WELLINGTON KUALA LUMPUR SINGAPORE JAKARTA HONG KONG DELHI BOMBAY CALCUlTA MADRAS KARACHI NAIROBI DAR ES SALAAM CAPE TOWN

6DP

TOKYO

© GARRISON SPOSITO 1981

Published in the United States by Oxford University Press, New York 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 British Library Cataloguing in Publication Data

Sposito, Garrison The thermodynamics of soil solutions. 1. Soil chemistry 2. Thermodynamics 3. Chemical reactions I. Title 631.4'1 S592.5 80-49694 ISBN 0-19-857568-8

Printed in the United StateR of America

To

J. WILLARD GIBBS who saw that it must be so KENNETH L. BABCOCK who made it work and

DOUGLAS, DINA, FRANK, AND JENNIFER who always took my word for it

PREFACE

Chemical thermodynamics is the theoretical structure on which the description of macroscopic assemblies of matter at equilibrium is based. This branch of physical chemistry was created 105 years ago by Josiah Willard Gibbs and was completed by the 1930s in the works of G. N. Lewis and E. A. Guggenheim. The fundamental principles of the discipline thus have long been established, and its scope as one of the five great subdivisions of physical science includes all of the chemical phenomena that material systems can exhibit in stable states. It was in the spirit of these attributes that Lewis and Randall' framed their wellknown aphorism: "The fascination of a growing science lies in the work of the pioneers at the very borderland of the unknown, but to reach this frontier one must pass over well travelled roads; of these one of the safest and surest is the broad highway of thermodynamics." Given the firm status of chemical thermodynamics, its application to describe chemical phenomena in soils would seem to be a straightforward exercise, but experience has proven different. An obvious reason for the difficulty that has been encountered is the preponderant complexity of soils. These multicomponent chemical systems comprise solid, liquid, and gaseous compounds that are continually modified by the actions of biological, hydrological, and geological agents. In particular, the labile aqueous phase in soil, the soil solution, is II dynamic, open, natural water system whose composition reflects especially the many reactions that can proceed simultaneously between an aqueous solution lind a mixture of mineral and organic solids that itself varies both temporally lind spatially. The net result of these reactions may be conceived as a dense web of chemical interrelations mediated by variable fluxes of matter and energy from the atmosphere and biosphere. It is to this very complicated milieu that chemical thermodynamics must be applied. This book is intended primarily as an introduction to the use of chemical I hermodynamics for describing reactions in the soil solution. Therefore no account is given of phenomena in the gaseous and solid portions of soil unless they impinge directly on the properties of the liquid phase. This restriction is conducive to a clarity in presentation and relevant to the interests of most soil chemists. Although the discussion in this book is self-contained, it does require 'G, N, Lewis and M. Randall. Thermodynamics and the Fret' Energy oj Chemical Substances, McGraw-Hili, New York. 1923. Used with the permission of the Mc-Graw Hill Book ( 'ompnny, uii

viii

PREFACE

exposure to thermodynamics as taught in courses on physical chemistry that employ differential and integral calculus. Since most of the examples discussed relate to soil chemistry, a background or interest m that discipime will be ot direct help in understanding the applications presented. The first two chapters of this book review the fundamental concepts of chemical thermodynamics. Care is taken to show how these concepts relate to soils and the soil solution. Much attention is given to the definitions of the Standard State and the Reference State and to the Standard State chemical potential of a substance, for these topics are seldom discussed carefully in the literature of soil chemistry. The third, fourth, and fifth chapters take up the application of chemical thermodynamics to solubility, electrochemical (including redox), and ion-exchange phenomena as they occur in soils; they contain the bulk of thermodynamics that is of concern to soil chemists. The sixth chapter digresses to consider the molecular theory of cation exchange. This topic has been included because of the widespread use of model approaches, such as diffuse double layer theory, to interpret soil exchange phenomena. A discussion of these approaches from first principles should clarify their subordinate relation to the thermodynamic theory of ion exchange and the more tenuous position they occupy as descriptions of chemical behavior. The seventh chapter presents the thermodynamic theory of water in soil from the perspective taken in soil physics (i.e., that the soil is a three-component, single-phase system). This chapter will introduce soil physicists to a thermodynamic formulation of the problem of soil water while, at the same time, bringing to soil chemists a view of soil as other than a multi component, heterogeneous system. Throughout this book there is much reference also to the limitations of chemical thermodynamics in treating natural soil solutions. These limitations refer especially to the influence of kinetics on stability, to the accuracy of thermodynamic data, and to the impossibility of deducing underlying mechanisms. The problem of mechanisms vis-a-vis thermodynamics cannot be expressed better than in the recent words of M. L. Mcfllashanr' " ... what can we learn from thermodynamic equations about the microscopic or molecular explanation of macroscopic changes? Nothing whatever. What is a 'thermodynamic theory'? (The phrase is used in the titles of many papers published in reputable chemical journals.) There is no such thing. What then is the use of thermodynamic equations to the chemist? They are indeed useful, but only by virtue of their use for the calculation of some desired quantity which has not been measured, or which is difficult to measure, from others which have been measured, or which are easier to measure." These points cannot be stated often enough. I have been helped in the development of this book by the constant encouragement and guidance of Dr Kenneth L. Babcock, who rightly may be considered the progenitor of chemical thermodynamics as applied to the soil solution. lM. L. McGlaHhan, The scope of chemical thermodynamics, Chemical Thermodynamics, Spl'C. P,riodit'al Rpl. I: I 30 (1973). Used with the permisslon of the author and the Chemical Society, London.

PREFACE

ix

To whatever extent soil physical chemists have been able to look more keenly at soil solution phenomena in the past 20 years, they must acknowledge their privileged vantage point on the firm pedestal of his seminal papers. Perhaps more often than they realize, I have benefited greatly from comments made in seminars and informal discussions by my colleagues, Dr William A. Jury, Dr Shas V. Mattigod, and Dr Albert L. Page. The lecture notes for my course on soil physical chemistry that were provided by Nancy Ball and Carlos Ramos also were of great value in organizing my thoughts, as were the suggestions from the students who read and used this book in draft form: Sabine Goldberg, Cliff Johnston, Jose Moraes, John Ojala, Marcos Pavan, Tom Quinn, Jeff Stark, Scott Strathouse, and Mohammad Yousaf. Chapter 5 was reviewed in draft by Dr Adel M. Elprince, who suggested the derivation of Eq. 5.52. Chapter 7 was reviewed by Dr William A. Jury, who exorcised a number of unclear passages and errors in the text and guided me as to the current thinking of soil physicists. The entire manuscript was read by Dr Hinrich L. Bohn and Dr James A. Kittrick, to whom I am most deeply grateful for their many suggestions and corrections. Finally, I must thank Ms Mary Campbell-Sposito for her assistance in preparing the indexes, Mr Karolyi Fogassy for his skill in drawing the figures, and Mrs Sharon Conditt for her patience in making a clear typescript from a great pile of handwritten, yellow sheets. None of these people, of course, is responsible for the errors or obscurities that may remain in this book; each only deserves gratitude for keeping the flaws to a relative minimum.

Riverside. California March 1980

G.S.

If you have built castles in the air, your work need not be lost; that is where they should be. Now put the foundations under them. DAVID HENRY THOREAU

CONTENTS

1. THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS 3 I. VARIABLES OF STATE AND THERMODYNAMIC POTENTIALS 1.1. Soils as thermodynamic systems 3 1.2. Thermodynamic processes in soils 6 1.3. Variables of state for thermodynamic soil systems 7 1.4. Thermodynamic potentials 10 1.5. Useful formal relationships 12 1.6. The Gibbs phase rule in thermodynamic soil systems 16 2. THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

20

II. CHEMICAL EQUILIBRIUM 2.1. Thermal, mechanical, and chemical equilibria in soils 20 2.2. Standard States for soil components involved in isothermal, isobaric reactions 25 2.3. Thermodynamic activity and the equilibrium constant 33 2.4. Standard State chemical potentials 41 2.5. Standard State chemical potentials and the equilibrium constant 48 2.6. Reference States and activity coefficients 53 2.7. Conditional equilibrium constants 62

3. SOLUBILITY EQUILIBRIA IN SOIL SOLUTIONS 3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7.

66 Solid phases and the activities of soil solution'species 66 Predominance diagrams 69 The Principle of Hard and Soft Acids and Bases 75 Complex formation and metal solubility: Fundamental concepts 80 Complex formation and metal solubility: Applications 83 Activity ratio and solubility diagrams 88 Coprecipitated solid phases 94

4, ELECTROCHEMICAL EQUILIBRIA IN SOILS 102 4.1. Oxidation-reduction reactions in soils 102 4.2. The electron activity 107 4.3. Soluble redox species III 4.4. pB-pH diagrams 114 4.5. Electrochemistry in soil solutions and suspensions

116

xii

CONTENTS

5. THE THERMODYNAMIC THEORY OF ION EXCHANGE 126 5.1. The ion exchange selectivity coefficient 126 5.2. Exchange isotherms 134 5.3. Mixed-exchanger systems 139 5.4. Exchanger phase activity coefficients and the exchange equilibrium constant 143 5.5. Specific adsorption 150 6. THE MOLECULAR THEORY OF CATION EXCHANGE 6.1. Fundamental principles 155 6.2. Exchange isotherms 160 6.3. The discrete site model 166 6.4. The diffuse double layer model 169 6.5. The surface complex model 178

155

7. THE THERMODYNAMIC THEORY OF WATER IN SOIL 187 7.1. Variables of state and thermodynamic potentials 187 7.2. Thermodynamic stability conditions 190 7.3. The chemical potential of soil water 193 7.4. Thermodynamic theory of the measurement of the water potential 7.5. The moisture characteristic: Hysteresis 201 7.6. Soil water in a gravitational field 205 SELECTED INDEX OF INDEX OF INDEX OF

PHYSICAL CONSTANTS 209 PRINCIPAL SYMBOLS 210 SUBJECTS 221 AUTHORS 217

198

THE THERMODYNAMICS OF SOIL SOLUTIONS

1 THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS I. Variables of State and Thermodynamic Potentials

1.1. SOILS AS THERMODYNAMIC SYSTEMS From the point of view of thermodynamics, a soil is an assembly of solid, liquid, and gaseous matter, as well as a repository of electromagnetic and gravitational fields. These characteristics, together with a surface that encloses the macroscopic region of space filled by the soil, define the thermodynamic soil system. Thus a thermodynamic soil system contains both matter and physical fields and is bounded by a surface of arbitrary shape. This bounding surface is called the thermodynamic wall surrounding the soil system. A thermodynamic soil system is studied through information about its properties which, in turn, are concepts that can be associated with numerical magnitudes obtained from experiment. The properties of a thermodynamic soil system by definition refer to a macroscopic region of space and do not depend for their measurement or interpretation on the previous history of the system. It is important to understand that the thermodynamic properties constitute only a part of all the properties of a soil. Those soil properties that describe phenomena at the molecular scale or those that relate directly to the effect of time are not included, for example. Therefore the concentration of protons and the electric potential at a point of few nanometers away from the surface of a soil colloid suspended in water are not properties of a thermodynamic soil system, nor are the length of time the soil has weathered and the number of occasions that water has percolated through it. Thermodynamics has nothing to say about these properties, although they could be studied in other disciplines of physical chemistry. I The properties of a thermodynamic soil system, hereafter to be called simply a soil, may be divided into those that are fundamental and those that are derived. This division is to some extent arbitrary. However, the fundamental properties of a soil always have the distinguishing feature that together they make up the smallest set of properties that provide a complete thermodynamic description but yet can be varied independently (i.e.• varied without any change in the values of other fundamental properties). The fundamental properties of 1

4

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

a soil are chosen on the basis of experience, as will be discussed in Section 1.3. A set of numerical values of the fundamental properties is called the state of the soil to which they refer. For this reason, fundamental properties also are termed independent variables of state. The properties of a soil, whether fundamental or not, whose numerical values depend on the quantity of matter in the soil (e.g., volume and entropy) are called extensive; the properties whose values are not dependent on the quantity of matter in the soil (e.g., pressure, bulk density, and temperature) are called intensive. The intensive properties of a soil are mathematical field variables (i.e., their values are associated with points in space that are located in the soil). Since a thermodynamic property always refers to a macroscopic region of space, it is understood that the value of an intensive property at some point in a soil applies to a macroscopically small neighborhood of that point that encloses many solid grains and interstices. If the values of its intensive properties are the same everywhere in a soil, the soil is said to be homogeneous. If anyone of its intensive properties varies (on the macroscopic scale) from point to point, the soil is said to be heterogeneous. Natural soils are invariably heterogeneous because their intensive properties vary spatially on a macroscopic scale, both from the effects of pedochemical processes and from the direct effect of the gravitational field of the earth. Thermodynamic soil systems, on the other hand, often may be treated as if they were homogeneous in respect to the analysis of experimental data. For example, a small sample of soil being studied in a pressure membrane apparatus may be regarded as homogenous if gravitational effects are ignored and if the solid and fluid portions of the soil are not differentiated chemically. This point of view, which is conventional in soil physics, will be discussed in detail in Chapter 7. Every soil consists of components, which are defined to be material substances of fixed chemical composition whose amounts can be varied independently in the soil. For example, a synthetic soil consisting of liquid water, NaCl, CaCI 2 , Na-montmorillonite, and Ca-montmorillonite contains four components. These are water plus any three of the four solid substances mentioned. There are not five components in this case because it is not possible to vary the amounts of the two chloride salts and the two forms of montmorillonite in the soil independently; a cation exchange reaction links these four compounds. The composition of a soil is specified in terms of its components. A homogeneous portion of a soil that has a variable composition is called a solution. Thus the gaseous, liquid, and solid portions of a soil each may be solutions. The interstices in the soil may contain air, a gaseous solution composed principally of nitrogen, oxygen, carbon dioxide, and water vapor, as well as the soil solution, an aqueous solution composed of liquid water and dissolved solids. The solid portion of a soil also may be a solution if its composition is mixed. Thus the montmorillonite exchanger containing both Na- and Ca-clay, mentioned above, is a solution. Solutions are special cases of phases. which are conditions of pure substances, or mixtures of pure substances, wherein the intensive properties do

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

5

not vary with position. For example, consider again the interstices of a sample of soil. If water is the only compound in the interstices and the effects of gravity are neglected, the interstices contain a homogeneous portion of the soil. That homogeneous portion can exist in three phases-gaseous, liquid, or solid. On the other hand, suppose that liquid water and dissolved NaCI were in the interstices. Again neglecting gravitational effects, the interstices now contain a homogeneous system that is in the liquid phase and that is a solution because its composition can be varied. If undissolved air, liquid water, and dissolved NaCI were in the interstices of a soil, the interstices would comprise a heterogeneous system consisting of two solutions: air and an aqueous solution. Both of these solutions, of course, are phases. The thermodynamic wall surrounding a soil is a very important part of the system. If the wall permits the free transfer of both matter and thermal energy either in or out, the soil is called an open system and the wall is said to be diathermal and permeable. If only certain types of matter may be transferred through the wall, it is said to be semipermeable. If the wall permits only the transfer of thermal energy, it is said to be diathermal and the soil is called a closed system. Finally, if the wall does not permit the transfer of either matter or thermal energy, it is said to be insulating and the soil is called an adiabatic system. Note that mechanical energy or the energy associated with gravitational and electromagnetic fields may be transferred through any of the walls that have been discussed. Thermodynamic walls are differentiated only by their behavior toward the transfer of matter and thermal energy. The intensive properties of a soil, excepting those that are simply ratios of extensive properties (e.g., the bulk density), are not strictly characteristics of the soil alone; they are determined by the thermodynamic wall and by the properties of suitable reservoirs that are separated by the wall from the system under study. (A reservoir is a large thermodynamic system whose intensive properties do not ' change in value when the transfer of matter or thermal energy into it or out of it takes place.) Consider, for example, the temperature of a soil. This intensive property is determined by the temperature of a thermal reservoir that is in contact with the soil through a diathermal wall. Under the condition that no net thermal energy transfer occurs through the wall between the reservoir and the soil, the temperature of the soil is said to be equal to the temperature of the reservoir. Similarly, the pressure on a soil is determined by a movable, diathermal wall that connects the soil to a volume reservoir (e.g., a very large cylinder of chemically inert gas). Under the condition that no net mechanical energy transfer occurs through the motion of the wall between the reservoir and the soil, the pressure on the soil is said to be equal to that of the substance in the reservoir. This basic characteristic of certain intensive properties is very important to remember when analyzing thermodynamic processes in soils. For example, needless confusion is introduced by forgetting that no internal thermodynamic pressures exist in a soil containing air and water. Only the thermodynamic pressure exists that is exerted externally on the soil.

6

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

1.2. THERMODYNAMIC PROCESSES IN SOILS A thermodynamic process takes place in a soil when its thermodynamic properties are changed in some fashion. Thus a thermodynamic process will result in a change in the state of a soil. If, during a thermodynamic process, a soil passes exclusively through states of equilibrium, that process is said to be reversible. The concept of equilibrium will be developed in Section 2.1. It suffices to recall here that equilbrium states are characterized by a relative maximum in the value of the total entropy of a system and its surroundings, including the reservoirs that control its properties. Reversible processes are limiting cases of natural processes, which are defined as the thermodynamic processes that bring a system into a state of equilibrium. In practice, it is almost always possible to arrange a natural process to be arbitrarily close in behavior to a reversible process. Consider, for example, a very wet soil slurry that is in equilibrium with a thermal reservoir at some temperature and around which is a thermodynamic wall permeable only to water vapor. A water vapor reservoir contacting the soil through the wall is maintained by a salt solution at some vapor pressure p = Peq + 1', where Peq is the equilibrium vapor pressure of water in the soil and I' < O. Under this condition, the natural process of evaporation of the soil water into the reservoir will take place. In the limit that I' t 0 ("I' goes to 0 through negative values"), the evaporation becomes a reversible process. This limit is approached as closely as one wishes by a suitable experimental adjustment of the vapor pressure in the water reservoir. There are several important special cases of thermodynamic processes in soils. If a process results only in an infinitesimal change in one or more properties, it is an infinitesimal process. Soil initially in an equilibrium state can undergo only reversible infinitesimal processes, by definition. A thermodynamic process that occurs in a soil surrounded by an insulating wall is an adiabatic process. This kind of process takes place, for example, when the heat evolved in a cation exchange reaction occurring in a clay suspension is measured in a calorimeter. It is important to note that adiabatic processes always involvea change in the temperature of a soil, since no thermal reservoir is permitted to exchange energy with the soil and control its temperature. If such a reservoir is present and the temperature is controlled during a process, the process is isothermal. If a volume reservoir is present and the pressure applied to a soil is controlled during a process, the process is isobaric. If, instead, the volume of the soil is controlled by the reservoir (by an increase or decrease in the applied pressure), the process is isochoric. Consider once again the system consisting of liquid water, NaCl, CaCI 2, Na-montrnorillonite, and Ca-montmorillonite. A movable. diathermal wall permeable to water and the two chloride salts (a wall known as a dialysis membrane) surrounds the system. As a combined thermal. volume. and matter res-

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

7

ervoir, one may employ a mixed aqueous solution of NaCI and CaClz contained in a water bath apparatus equipped with flow-through capability. The montmorillonite exchanger in the dialysis membrane is immersed in the water bath containing the mixed chloride salt solution. The amounts of NaCI and Ce.Cl, in the reservior are such that no net flow of these salts occurs across the membrane (i.e., the exchanger is in a state of equilibrium with respect to these two components). Similarly, it is arranged that there is no net thermal energy transfer across the membrane and, therefore, thermal equilibrium exists, with the temperature of the system equal to that of the reservoir. Finally, the volume of the dialysis membrane has adjusted itself until no more movement of the membrane occurs and mechanical equilibrium is achieved. Assuming that the montmorillonite structure itself does not dissolve at a significant rate, it may be concluded that the exchanger is in a state of thermodynamic equilibrium with the reservoir that controls its properties. Now suppose that a small increase in the amount of Ca Cl, and a small decrease in the amount of NaCI in the reservoir occur through an appropriate change in the composition of the solution flowing through the water bath apparatus. If these changes are quite small, the exchanger and the aqueous solution bathing it will undergo an approximately reversible process that will result in a shift in the amounts of NaCI, CaCl z, Na-clay, and Ca-clay. This shift is an isothermal cation exchange process. The process is not generally isobaric or isochoric. If, instead of altering the amounts of chloride salts in the reservoir, the temperature of the reservoir were increased by a small fraction of a degree, a nonisothermal cation exchange process would be produced in the system. This process could be arranged so as to bring the exchanger to the same final composition as did the first-mentioned adjustment of the amounts of NaCI and CeCl; These brief remarks and the description of the cation exchange process through dialysis equilibrium are perhaps familiar already. The intent here has been to stress the fundamental thermodynamic aspects of a simple laboratory exchange experiment. The soil process of interest involves changes in the macroscopic properties of the soil produced by adjustments in the reservoirs contacting the soil and controlling its state.

1.3. VARIABLES OF STATE FOR THERMODYNAMIC SOIL SYSTEMS The set of properties from which the independent variables of state for a soil may be chosen is determined by experience. This experience relates not only to experimentation with the chemical properties of soils and the subsequent development of conceptual models to interpret soil chemical phenomena but also to experience with thermodynamics itself. The following list of soil properties/ will prove adequate for the thermodynamic description of soil solution phenomena in the absence of externally applied fields. A generalization to include gravitational fields will be given in Section 7.6.

8

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

Temperature Temperature is the criterion for equilibrium with respect to thermal energy transfer. It is measured on the Kelvin scale in units of kelvins (K) and is, of course, an intensive property of a soil. Entropy Entropy is the principal criterion for thermodynamic equilibrium. It is an extensive property of a soil measured in units of joules per kelvin (J K- I). Pressure Pressure is the criterion for mechanical equilibrium and refers always to the force per unit area exerted by a volume reservoir on the thermodynamic wall enclosing a soil. Pressure is an intensive property measured in units of newtons per square meter (N m -2) or in the practical equivalent units of atmospheres (1 atm = 1.01325 X 105 N m -2) or millimeters of mercury (1 mm Hg = 1.33322 X 102 N m"). The unit N m ? is called the pascal (Pa). A pressure of 105 Pa is called a bar, a unit that is a common alternative to the atmosphere. Volume Volume is the extensive variable that describes the spatial extent of a soil and, therefore, is a means of assessing mechanical energy transfer to or from a soil. It is measured in units of cubic meters (rrr'), or in the practical units of liter (1 liter = 10- 3 rrr') or cubic centimeters (1 crrr' = 10- 6 rrr'), Chemical Potential The chemical potential is the intensive property that is the criterion for equilibrium with respect to the transfer of matter. Each component in a soil has a chemical potential that determines the relative propensity of the component to be transferred from one phase to another or to be transformed into an entirely different chemical compound in the soil. Just as thermal energy is transferred from regions of high temperature to regions of low temperature, so matter is transferred from phases or substances of high chemical potential to phases or substances of low chemical potential. Chemical potential is measured in units of joules per kilogram (J kg-I) or joules per mole (J mol"), Mass Mass is the extensive composition variable for a component of a soil. The quantity of each component is measured by its mass in units of kilograms (kg). Often the mass of a component may be replaced by a secondary variable, the mole number, n, that is simply proportional to the mass for a substance of known, fixed composition. In soils the components do not always have known or fixed compositions, however, and the use of mole numbers is not always possible. If no chemical reaction is being considered, the mole number is not necessary to a thermodynamic analysis. This set of six thermodynamic properties will be basic to the discussion in the following chapters. A number of intensive variables are omitted that can be defined simply as ratios of the extensive variables (c.g., bulk density, defined as

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

9

the mass of soil solids divided by the soil volume). These quantities are secondary variables in a thermodynamic analysis in most cases, but they will be considered further in Chapter 7. Perhaps more significant is the omission of properties relating directly to surfaces in the list of soil thermodynamic variables. Surface variables are not needed for two reasons. First, given that the thermodynamic soil system is a relatively large, macroscopic system, it is not expected that its properties will be affected greatly by any change in the area occupied by its thermodynamic wall. In more technical terms, a change in total energy because of a change in the surface area of the thermodynamic wall can be assumed negligible for a soil sample of typical dimensions. Thus the surface area of the wall can be omitted from the list of variables of state. Second, since thermodynamic properties must be macroscopic properties, the surface area of the many interfaces in a soil also is not a thermodynamic property. As indicated in Section 1.1, a thermodynamic soil property refers to a region of space that includes many solid grains and interstices. The individual grains and interstices, as well as the interfaces between them, are not accounted for except as they contribute in large numbers to a macroscopic soil property. For this reason, neither the diameter of a solid grain in the soil nor its interfacial area is a thermodynamic soil property. On the other hand, instead of a whole soil one could consider, for example, a single solid grain of soil material in contact with an aqueous solution. In this case the interfacial area can be controlled independently and surface variables would enter the thermodynamic analysis of this system. When soil samples instead of individual small grains of solid soil material are considered, however, surface variables are not necessary. The six thermodynamic variables for a soil in the absence of external fields are summarized in Table 1.1. It will be shown in Section 1.4 that T and S, P and V, and ILl and m, each are conjugate pairs of thermodynamic properties in that one variable in the pair always may be derived from the other by differentiation of a thermodynamic potential. Therefore only one variable from each conjugate pair may be chosen as an independent variable of state with which to describe a soil. For example, a soil may be described nonredundantly by the variables T and V and the set of m.; but not T, S, V, and the set of mi. nor T, V, the ILl, and the mi.

TABLE 1.1. Summary of thermodynamic variables for a soil Variable Temperature Entropy Pressure Volume Chemical potential of Ith component Mass of Ith component

Symbol

Extensive

T

S

X X X X

ILl

m/

X

Units K J K- 1

X

P

V

Intensive

N m? m3 J kg-I kg

10

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

1.4 THERMODYNAMIC POTENTIALS

The concepts of internal energy and entropy as discussed in standard reference works on thermodynamics may be applied to soils without modifications. In particular, according to the First and Second Laws of Thermodynamics, the internal energy of a soil is expressed as a function of the entropy, volume, and composition: (1.1)

where U is the internal energy, measured in units of joules, {m ia } is a set of component masses, i indexes a component, and a indexes a phase in the soil. Equation 1.1 is known as a fundamental relation for a soil because the equilibrium states of the system are characterized completely once the dependence of U on the independent variables of state S, V, and {m ia } has been established by experiment. Fundamental relations often may be developed by investigating infinitesimal processes, whose complete description is contained in the differential form of Eq. 1.1: (1.2) In Eq. 1.2 it is assumed that all phases in the soil are at the same temperature and under the same applied pressure. If this assumption is not correct then, for example, the entropy term in Eq. 1.2 would be replaced by 2:a T; dSa , and so forth. The mathematical interpretation of Eq. 1.2 in terms of the theory of exact differentials produces two useful pieces of information about the coefficients T, P, and Ilia' First, each of these intensive variables may be written as a partial derivative:

where it is understood that mj~ =1= m ia • Equations 1.3 demonstrate the general fact that intensive properties are partial derivatives of one extensive property with respect to another. Second, each of the coefficients T, P, and Ilia are functions of all of the independent variables of state: T = T(S, V,{m ia })

Equations 1.4 are called equations of state. If, by experimentation, all of the equations of state can be determined, Eq. 1.2 can be integrated for any set of states and the fundamental relation, Eq. 1.1, can be established. Thus, as will become clear in Section 1.5, a knowledge of all of the equations of state for a soil is the same as a complete thermodynamic characterization of a soil. Another very important mathematical property of Eq. 1.1 is that an operation known as a partial Legendre transformation may be performed on

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

11

U(S, V,{m j ,, }) to create systematically a set of functions called thermodynamic potentials. The mathematical aspects of the partial Legendre transformation will not be discussed here,' but the fundamental characteristics of the thermodynamic potentials may be described as follows.

1. A thermodynamic potential is an extensive quantity that is equivalent in all respects to the internal energy. A knowledge of a thermodynamic potential is the same as a complete thermodynamic description of a soil. 2. Each thermodynamic potential is uniquely determined by the independent variables of state chosen to describe a soil. Once these variables are picked, the differential form of the thermodynamic potential follows immediately and unambiguously. 3. Each thermodynamic potential has the property that the equilibrium state of a soil, in contact with reservoirs that control the intensive variables on which the potential depends, always is such as to produce a relative minimum in the potential. Moreover, any decrease in the thermodynamic potential under these conditions is equal to the reversible work delivered by the soil in the process. By choosing different sets of independent variables of state, many different thermodynamic potentials can be defined by means of the partial Legendre transformation. The three most important of these will be considered now. If S, P, and {m j,, } are chosen as independent variables of state, the resulting thermodynamic potential is called the enthalpy, H. In differential form this potential is expressed as Enthalpy

(1.5)

and the corresponding equations of state are

v=

V(S,P,{m j ,, })

A decrease of enthalpy in a soil contacting a volume reservoir that controls the applied pressure, P, is equal to the reversible work delivered by the soil. If T, V, and {mj,,} are the chosen independent variables of state, the thermodynamic potential is called the Helmholtz energy. A. In differential form this potential is Helmholtz Energy

(1.7)

The equivalent equations of state are S = S( T, V,{ mj,,})

P = P( T, V,{m j ,, })

A decrease of Helmholtz energy in a soil for which T is controlled by a thermal reservoir is equal to the reversible work delivered by the soil.

12

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

Gibbs Energy If T, P, and {m ia} are the chosen independent variables, the thermodynamic potential is called the Gibbs energy, G. In differential form this potential is

(1.9) The equivalent equations of state are S

=

S( T,P,{mia})

v=

V( T,P,{m ia})

A decrease of Gibbs energy in a soil for which T and P are controlled by thermal and volume reservoirs is equal to the reversible work delivered by the soil. The properties of the principal thermodynamic potentials are summarized in Table 1.2. The units of each potential are joules, the same as for the internal energy. In addition, each coefficient before a differential on the right sides of Eqs. 1.5, 1.7, and 1.9 is also a partial derivative. For example,

ILia

= (aH/amia)S,p,{mj~) = (aA/amia>r.V,{mj~) = (aG/amia>r.p.{mj~)

(1.1 1)

where mjfJ "1= mi; Thus the chemical potential may be expressed in a variety of equivalent ways through the thermodynamic potentials. The choice of which thermodynamic potential to employ in describing processes in soils depends entirely on the choice of independent variables of state. Often it is convenient to control the temperature, applied pressure, and composition of a soil, in which case the thermodynamic potential that must be used is the Gibbs energy. Clearly, then, the very design of the experiment on a soil dictates the proper thermodynamic function with which to describe the experiment. In any case, the thermodynamic potentials are all equivalent to one another and to the internal energy. The use of one in preference to another is only a matter of convenience in applications.

1,5. USEFUL FORMAL RELATIONSHIPS

Three purely mathematical properties of the fundamental relation and its equivalents (Eqs. 1.2, 1.5, 1.7, and 1.9) have very important applications in the ther-

TABLE 1.2. The principal thermodynamic potentials Independent Variables of State

S,V,{m/,,} S,P.{m'a} T,V,{m,,,} T,P.{m,,,}

Thermodynamic Potential internal energy enthalpy

Helmholtz energy Gibbs energy

Symbol

U(S,V,{m'a}) H(S,P,{m ia}) A( T, V,{m/ a }) G( T.P,{m,,,})

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

13

modynamic analysis of soil solutions. These properties are derived very simply from the facts that the total differentials of U and the thermodynamic potentials are exact and that U, H, A, and G are extensive quantities. The Maxwell Relations lows, for example, that

Consider Eq. 1.9 as an exact differential. Then it fol-

(1.12) and so on for all of the cross partial derivatives. However, 2G

a apaT

= -

(as) ap T.{mla)

(1.13)

according to the form of Eq. 1.9. From Eqs. 1.12 and 1.13 the equality

- (as/aph.{m,

a}

=

(a v/anp.{m,

(1.14)

a}

may be derived. Equation 1.14 is an example of a Maxwell relation (i.e., an equality between two cross partial derivatives of a thermodynamic potential). In an exactly similar fashion, the Maxwell relations

-

(as/am'ahP.{m}~) = (av/amiahp,{m}~) =

(alJ-ia/anp,{m, (alJ-ia/aph{m,

a) a)

(1.15) (1.16)

may be derived from Eq. 1.9. The set of Maxwell relations that can be calculated from Eqs. 1.2, 1.5, 1.7, and 1.9 are listed in Table 1.3. The principal utility of the Maxwell relations is that they permit experimental alternatives in methods for measuring changes in thermodynamic quantities. For example, Eq. 1.16 shows that if one wishes to study the pressure dependence of the chemical potential of a component at constant temperature and composition, one need only measure the change in total volume with respect to a change in the mass of the component (a partial specific volume), while T, P, and the remaining composition variables are held fixed. The Euler Equation An Euler equation can be derived for the internal energy and for each thermodynamic potential based only on the fact that these quantities are extensive. Consider, for example, the Gibbs energy. Since G is extensive, it has the mathematical property (1.17) where G; is the Gibbs energy of the ath phase in a soil and A > 0 is a scale factor of arbitrary magnitude. Now, by the chain rule for partial differentiation, ( 1.18)

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

14

TABLE 1.3. Some of the Maxwell relations Independent Variables of State

Maxwell Relations

S,V,{m;a}

(aT/aV)s.. = -(ap/aS)v..

S,P,{m;a}

(iJT/ap)

= (iJV/iJS)

s~

(iJT/am;a) s.v.. = (a/l;a/iJS) v..

(iJT/am;a) S,P.-

P..

= (iJ/l;a/iJS)p..

(iJV/iJm'a)S,P,_ T, V,{m ..}

= (iJP/iJn v..

(iJs/aV) r.. -(iJP/iJm;a) (as/iJP)

T..

T,V,.

= (a/lla/aV)

= (av/an

(iJV/iJm;a) T,P,_

= (iJ/l;a/an v..

- (as/am;a)

= (iJ/l'a/iJnp,.

T••

p ..

= (iJ/l;a/iJP)

- (iJS/iJm'a) T,V..

T,P.-

T.-

Note, The symbol m refers to the set of composition variables {m;a}' If the partial derivative contains m;a' it is understood that m includes only the composition variables other than one in the derivative, Each set of Maxwell relations in this table can be increased by including cross partial derivatives involving only the /l'a and the m'a, for example, (iJ/l;a/iJmj#h,p.. = (iJ/lj#/am'ah,p,., where i "!= j and/or ex "!= {3,

where

Xi"

=

Ami'" Also, considering only the right side of Eq. 1.17,

eo, aA =

{}

(1.19)

G" T,P, m.; )

From Eqs. 1.18 and 1.19 it follows that G,,( T,P,{mi,, }) =

z, aaG" m.: X;a

or, after multiplying through by A and referring to Eq. 1.17,

AG,,( T,P,{mi,,}) = G,,( TP,{x i ,, }) =

z, aG" Xi" axi"

( 1.20)

Finally, A is set equal to I in Eq. 1.20 in order to derive the Euler equation G,,( T,P,{m/,,}) • 2:1aaG" min • 2:,Iot'n mln

min

(1.21)

15

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

The Euler equations listed in Table 1.4 are derived in precisely the same manner as was Eq. 1.21, with the factor A inserted before each extensive variable on which the given thermodynamic potential depends, when the analog of Eq. 1.17 is written down. The Euler equations show that U and all of the thermodynamic potentials can be expressed as sums of the extensive variables on which they depend, with the coefficient of each extensive variable being just its conjugate intensive variable. One variable in each term of the Euler equation is an independent variable of state; the other variable can be expressed as a function of all the independent variables of state through the equations of state (Eq. lA, 1.6, 1.8, or 1.10). Therefore, because of the Euler equation, a knowledge of the equations of state is the same as knowing either U or the thermodynamic potential that corresponds to those equations of state. Consider once again the Gibbs energy of a single phase a in a soil. According to Eq. 1.9, the total differential of this thermodynamic potential is The Gibbs-Duhem Equation

(1.22) However, the total differential of G; given in the Euler equation (Eq. 1.21) is dG a = ~;(/liadmia

+

miad/l ia)

(1.23)

Equations 1.22 and 1.23 are compatible only if the relation SadT -

VadP

+ ~imiad/lia

= 0

( 1.24)

holds identically. Equation 1.24 is the Gibbs-Duhem equation. This expression is an identity relationship among the differentials of the intensive variables that describe a soil. It has nothing directly to do with whatever independent variables of state and thermodynamic potential actually are employed to account for the behavior of a particular soil. Equation 1.24 could pave been derived just as well by choosing, for example, the Euler equation for U from Table 1.4 and comparing its total differential with Eq. 1.2. Thus the Gibbs-Duhem relation makes the completely general statement that, for a phase containing C components, the C + 2 intensive properties of the phase cannot be varied independently; only C + 1 of them can be so varied.

TABLE 1.4. The Euler equations Independent Variables of State S,V,{m l ,, } S,P,{m l ,, } T,V,{m,,,} T,P,{m,,,}

Euler Equation U = TS - PV + "J:.""J:.i!J.i,,m 1a H = TS + "J:.a"J:.i!J.iamla A = - PV + "J:.,,'J:.i!J.lamia

G - 'J:."'J:.i!J.l"ml,,

16

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

One of the most useful applications of Eq. 1.24 occurs when T and Pare among the controlled variables of state. In that case isothermal, isobaric processes may be studied, and Eq. 1.24 reduces to (T, P fixed)

( 1.25)

for a given phase. As a simple but important example of the use of Eq. 1.25, an ~queous solution phase containing only water and NaCI may be considered. In tryat case the Gibbs-Duhem equation becomes

midi»;

+

mNaC1d}.tNaCI

= 0

or, if mole numbers are used instead of masses,

ncdu;

+

nNaCld}.tNaCI

= 0

( 1.26)

where }.t is now expressed in joules per mole. The dependence of the chemical potential of water in the solution on its content of NaCI can be determined readily by means of vapor pressure measurements or through measurements of any of the other well-known colligative properties of aqueous solutions. On the basis of these kinds of measurements, Eq. 1.26 can be integrated to obtain the chemical potential of NaCI in the solution at any desired content of the salt. In this way the Gibbs-Duhem equation has reduced the problem of determining }.tNaCI to the simpler experimental task of determining }.tw.

1.6 THE GIBBS PHASE RULE IN THERMODYNAMIC SOIL SYSTEMS

Consider a particular phase in a soil in which no chemical reactions are presumed to occur. The intensive properties of this phase will include T, P, and the C chemical potentials of its C components. Thus there are C + 2 intensive properties in all. But in Section 1.5 it was shown that only C + 1 of these properties could be varied independently because of the Gibbs-Duhem equation (Eq. 1.24). Therefore, in the case of a single phase, the number of independently variable intensive properties is C + 2 - 1 = C + 1. If P phases are considered now, there will be one Gibbs-Duhem equation for each phase, and the number of independently variable intensive properties will be reduced by P instead of by 1. Thus

!=C+2-P

( 1.27)

gives the number, f, of intensive properties that can be varied independently for any soil. Equation 1.27 is the Gibbs phase rule for a system in which no chemical reactions take place. As an example of the application of the phase rule, consider a water-saturated soil sample taken from a calcic horizon that happens to be completely frozen, In the simplest possible situation, this sample consists of two components, H10(s) and CaCO,(s), and two solid phases, ice and calcite. According to Eq. 1.27, the number of independent intensive variables is! - 2 + 2 - 2 - 2. The

THE

CHEMICAL~RMODYNAMICS OF SOIL SOLUTIONS

17

total number of intensive properties is four: T, P, and the chemical potentials of H 20(s) and CaCOis). However, there are two Gibbs-Duhem equations relating these properties, one for each solid phase and, therefore, only two of the properties can be varied independently. These two could be T and P, or T and J.L(HP) or J.L(H 20) and J.L(CaCOJ); the particular choice depends on what experiment is to be interpreted. If chemical reactions take place, the value of f must be reduced by I for each independent reaction. This reduction is necessary because a chemical reaction produces an equation relating the chemical potentials of the reactants and products to one another, as will be shown in Section 2.1. Therefore, if N independent chemical reactions occur in a soil containing P phases, there are P Gibbs-Duhem equations and N chemical equilibrium conditions relating the intensive properties of the soil. The Gibbs phase rule then takes the form

f= Co + 2 - P - N

(1.28)

where Co is the number of components the soil would have if no chemical reactions were occurring; that is, Co is the number of chemical compounds in the soil. Alternatively, the number of actual components in the soil, taking into account chemical reactions, is C = Co - N and the phase rule may be written

f=C+2-P

(1.27)

Obviously, Eqs. 1.27 and 1.28 express the phase rule in completely equivalent ways. The preference of one expression over the other is only a matter of taste. To illustrate the more general form of the phase rule, consider again the soil sample from a calcic horizon. This time let the water be in the liquid phase. Then the soil, in the simplest case, will consist of three compounds (Co = 3)H 20(l), CaCOJ(aq), and CaCOJ(s)-and will contain two phases, an aqueous solution phase and a solid phase. There is one chemical reaction involved, the precipitation-dissolution reaction: CaCOiaq) = CaCOis). According to Eq. 1.28, the number of independent intensive variables is f = 3 + 2 - 2 - I = 2. If, for example, T and P are chosen as the two independent intensive variables and then are controlled at fixed values, the chemical potentials of H 20 and CaCO J in the soil are completely determined. Conversely, if J.L(H 20) and J.L(CaCOiaq» are chosen and controlled, only one set of values of T, P, and J.L(CaCOis» can be found that corresponds to thermodynamic equilibrium in the soil. As another, somewhat more complicated example of the use of the phase rule, consider a model soil that is water saturated and contains H 20( l), Na2CO(aq), CaCOiaq), CaCOJ(s), NaX(s), and CaXls), where X refers to one equivalent of montmorillonite clay. This model soil consists of six compounds (Co = 6) and three phases, an aqueous solution phase, a solid solution exchanger phase made up from NaX and CaX 2, and a solid CaCOJ phase. There are two independent chemical reactions Na 2C0 1(aq)

CaCOJ(aq) - CaCOJ(s) + CaX 2(s) - CaCOJ(aq)

+ 2 NaX(s)

18

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

that account for CaC0 3 precipitation and Na-Ca exchange, respectively. Therefore, according to Eq. 1.28, the number of intensive properties that can be varied independently is f = 6 + 2 - 3 - 2 = 3. It follows that if one chooses to control T,P, and anyone chemical potential [e.g., JL(H 20)], all of the other chemical potentials are completely determined at equilibrium. It will be shown in Section 2.6 that the chemical potential of a substance in a solution can be related to its concentration in that solution. Accordingly, the statement that chemical potentials of solution components are determined completely after controlling, for example, T and P, or T and P plus one chemical potential, means that the compositions of the solution phases are completely determined at equilibrium. For example, the control of T, P, and JL(H 20) in the model soil just discussed is tantamount to the complete specification of the equilibrium solution concentrations of Na 2C03 and CaC0 3 in the aqueous solution phase as well as of NaX and CaX 2 in the exchanger phase. This perhaps surprising result is just one illustration of the great utility of the Gibbs phase rule in the thermodynamic analysis of soils.

NOTES 1. The properties of a thermodynamic system are discussed in detail by K. L. Babcock, Theory of the chemical properties of soil colloidal systems at equilibrium, Hilgardia 34:417-542 (1963), and G. N. Hatsopoulos and J. H. Keenan, Principles of General Thermodynamics. John Wiley, New York, 1965. 2. These basic thermodynamic properties are discussed briefly, with examples, by A. M. James, A Dictionary of Thermodynamics. John Wiley, New York, 1976. 3. For introductory discussions of the Legendre transformation, see Chapter 5 in H. B. Callen, Thermodynamics. John Wiley, New York, 1960, and Section 9.5 in G. Sposito, An Introduction to Classical Dynamics. John Wiley, New York, 1976.

FOR FURTHER READING H. B. Callen, Thermodynamics. John Wiley, New York, 1960. The first seven chapters of this book provide an outstanding discussion of the formal structure of thermodynamics. In particular, Chapter 1 discusses entropy and Chapters 5 to 7 thoroughly discuss the thermodynamic potentials and the Maxwell relations. J. W. Gibbs, The Scientific Papers of J. Willard Gibbs. Vol. I, Thermodynamics. Dover Publications, New York, 1961. These classic papers by the founder of chemical thermodynamics should be read by anyone with a serious interest in the subject. The principal paper, "On the Equilibrium of Heterogeneous Substances," is not simple, but it is very rewarding for the patient reader. E. A. GUKgenhelm, Thermodynamics. North-Holland, Amsterdam, 1967. Chapter I of this seminal reference work is a solid introduction to chemical thermodynamics. Guggenheim's book should be consulted frequently as a companion to the present book.

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

19

J. Kestin, The Second Law of Thermodynamics. Dowden, Hutchinson and Ross, Stroudsburg, Pa., 1976. This is a fine collection of founding papers on thermodynamics by Camet, Clausius, Gibbs, Kelvin, and others. The editor has provided an introduction to each set of papers, which are grouped chronologically. G. N. Lewis and M. Randall, Thermodynamics. rev. by K. S. Pitzer and L. Brewer, McGraw-Hill, New York, 1961. If Guggenheim's text is the classic theoretical introduction to chemical thermodynamics, this book is rightly the classic experimentalist's introduction. Chapters 1 to 3 cover the foundational aspects of thermodynamics.

2 THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS II. Chemical Equilibrium

2.1 THERMAL, MECHANICAL, AND CHEMICAL EQUILIBRIA IN SOILS

In Section 1.3 it was pointed out that the intensive soil properties, temperature, applied pressure and the chemical potentials, are the criteria for thermal, mechanical, and chemical equilibria, respectively. This important conclusion depends only on the First and Second Laws of Thermodynamics and on the fact that all of the intensive properties mentioned exist for the soil because of its contact with thermal, volume, and matter reservoirs. At this juncture, it is useful to demonstrate these criteria for equilibrium in a rigorous and general manner. Thermal Equilibrium Consider an adiabatic system consisting of a sample of soil in contact with a thermal reservoir through a rigid, diathermal wall. The soil and the reservoir themselves are surrounded by an insulating wall. A simple example of this kind of composite system would be a small amount of soil placedin a metal beaker fitted tightly with an insulating plastic top and immersed in; a water bath (the thermal reservoir) that has insulating double walls. A funda-l mental relation in differential form can be written down for the soil/thermal! reservoir system to describe any infinitesimal p r o c e s s : / i dU = dUs + dUR = TsdSs - PsdVs

i

+ ~i~a J,l.iaSdmiaS +

i

TRdS R -

PRdVR

+ ~i~a J,l.iaRdmJ

(2.1 ») where the subscripts Sand R refer to "soil" and "reservoir," respectively. Thej general condition for equilibrium in the composite system is that the value of ,1 I the total internal energy be a relative minimum under whatever constraints may i, be imposed. This condition follows from the Second Law, which stipulates thata relative maximum must occur in the value of the total entropy as the general ' criterion for equilibrium. Indeed, if an equilibrium state corresponded to a maximum in S, but not a minimum in U at the same time, it would be possible to lower U further by withdrawing mechanical energy at fixed S. Then this mechanical energy could be converted entirely to thermal energy and transferred back into the system to restore U to its former value. But, with such an influx

20

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

21

of heat, S would have to increase, thus violating the initial hypothesis that S was a maximum. This contradiction means that U must have been a minimum in the first place when S was a maximum and that the imagined process to lower U is not possible. Suppose that an infinitesimal transfer of thermal energy takes place between the soil and the reservoir. Because of the nature of the wall separating the two systems, dVs = dVR = dm/"s = dm/"R = for any such process, and Eq. 2.1 reduces to

°

Because the composite system is an adiabatic system, it operates under the constraint dS s = - dS R (i.e., no thermal energy can be transferred in or out to alter the total entropy). Therefore (2.2)

for an arbitrary infinitesimal change in Ss. If the composite system was initially in equilibrium, the infinitesimal process considered here is a virtual process subject to the condition dU = 0, which is one of the requirements for U to be a minimum. Since dS s is an arbitrary virtual change, it may be concluded that, at equilibrium, the coefficient of dSs in Eq. 2.2 vanishes and (2.3) The criterion for thermal equilibrium between the soil and the thermal reservoir is that their absolute temperatures be equal. If the composite system was not initially in equilibrium, the infinitesimal process simulates an actual process subject to the condition dU < 0. Therefore Eq. 2.2 becomes the inequality

(2.4) Now, if dS s > 0, thermal energy must be moving from the reservoir into the soil. The inequality in Eq. 2.4 then indicates that T s < T R• On the other hand, if dS s < 0, thermal energy is lost from the soil and the inequality requires Ts > T R. It follows that thermal energy always is transferred from the system at higher T to the system at lower T. At equilibrium, the two systems have the same value of T. Consider now an adiabatic composite system that contains a soil in contact with thermal and volume reservoirs through a movable, diathermal wall. The soil and the reservoirs are surrounded by an adiabatic wall. An example of this type of system would be an unsaturated, swelling soil placed inside a cylindrical metal cell adjacent to a piston in the cell that can be actuated by N 2 gas under pressure, which fills the rest of the cell. The entire assembly is placed in an air bath that is surrounded by insulating walls. The air bath is a thermal reservoir. The piston and N 2 gas together act as a volume reservoir for Mechanical Equilibrium

22

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTION:

the control of the pressure applied to the soil. During an infinitesimal transfe of thermal and mechanical energy between the soil and the reservoirs, the tota internal energy change will be (2.5; according to Eq. 2.1. In this case, because the thermodynamic wall is impermeable, dmias = dm;« = O. Equation 2.5 is subject to the conditions dS s = - dS R and dVs = - dVR (i.e., no changes in the total entropy and volume 01 the system). Therefore, in any infinitesimal process, (2.6) gives the change in internal energy. After thermal equilibrium is established, - T R by Eq. 2.3, and Eq. 2.6 reduces to

T~

(2.7)

If the system also is in a state of mechanical equilibrium, Eq. 2.7 describes a virtual process subject to dU = O. Since dVs is an arbitrary, infinitesimal volume change, it follows that, at equilibrium, the coefficient of dVs in Eq. 2.7 vanishes and

(2.8) The criterion for mechanical equilibrium between the soil and the volume reservoir is that their pressures be equal. If the composite system is not initially in equilibrium, Eq. 2.7 is subject to dU < 0, and the inequality -cps - PR)dVs

<

0

(2.9)

applies. If mechanical energy is transferred to the soil by the reservoir, the soil will shrink. Then dVs < 0 and Ps < P R , according to the inequality in Eq. 2.9. If mechanical energy is transferred out of the soil to the reservoir, the soil swells, and dVs > 0, which results in P« > PR according to Eq. 2.9. Therefore mechanical energy always is transferred from the system at higher P to the system at lower P. At equilibrium, the two systems have the same value of P.

Phase Equilibrium For a composite system consisting of a single chemical component in a single phase that contacts thermal, volume, and matter reservoirs through a movable, permeable, diathermal wall, infinitesimal processes may be described by the equation dU = TdS - PdV

+ udm +

TRdS R - PRdVR - J.LRdmR

(2.10)

Equation 2.10 is subject to the constraints dS = -dS R, dV = -dVR, and dm = - dnu; assuming as usual that the composite, soil-reservoir system is closed. Under these constraints and the conditions of thermal and mechanical equilibrium (Eqs. 2.3 and 2.8), Eq. 2.10 becomes

dU - (J.L - J.L1t)dm

(2.11)

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

23

Arguments exactly similar to those employed in connection with Eqs. 2.2 and 2.7 now may be made to show that the condition for matter flow equilibrium between the system and the matter reservoir is (2.12) and that matter always is transferred from a system at higher 11 to one at lower 11. In a soil there are several components and phases. The behavior of anyone component in anyone phase (e.g., water in the vapor phase) may be described in terms of Eq. 2.10 if the soil is an open system with respect to that component and phase. In some experiments, however, the soil may be closed with respect to a component of interest that exists in more than one phase. For example, the precipitation and dissolution of CaC03 might be studied in a soil whose thermodynamic wall is not permeable to that compound. In that case, the relevant chemical equilibrium is a phase equilibrium for a single component. Consider an adiabatic system comprising a water-saturated soil in contact with thermal and volume reservoirs. These reservoirs are used to maintain thermal equilibrium in the soil at a temperature T s = T R and mechanical equilibrium in the soil under a pressure P s = PRo Under these conditions, infinitesimal processes in the soil may be described by the equation (2.13) Suppose that ex = A corresponds to the liquid phase in the soil and ex = (J corresponds to a solid phase in the soil. If a process takes place involving only component k, which is assumed to exist in both phases (i.e., a precipitable component), Eq. 2.13 becomes

Because the soil is a closed system in this example, it is subject to the constraint dtllk~s = - dmkus, Therefore (2.14) gives the change in internal energy of the soil for any infinitesimal process under I he assumed conditions. If the initial state of the soil is an equilibrium state, dUs - 0, and the condition of phase equilibrium for component k is (2.15) This result applies to any component that can be partitioned between two phases III II soil. For example, returning to the case of CaC0 3, one may write Eq. 2.15 III the form

where (aq) refers to the aqueous liquid phase and (s) to the solid phase. At oquilibrum, the chemical potential of dissolved CaCO J must be the same as that 01 t he precipitated solid CaCO,.

24

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

The transfer of a compound from one phase to another is a simple example of a chemical reaction. In general, of course, compounds in soils transform into other compounds as well as between phases. A typical example of a change in compounds is the reaction Chemical Reactions

(2.16)

with reactants always understood to be on the left side and products on the right side. If an infinitesimal mass transfer is imagined in connection with Eq. 2.16, one may define an extent of reaction parameter, d~, such that d~/ dt is the rate of the reaction and the stoichiometric relations dn(CaCI 2) = -d~ = dn(Na 2C03) dn(NaCl) = 2 d~ dn(CaC0 3) = d~

(2.17)

hold, where n(CaCI 2) is the number of moles of CaCliaq), and so on. The numbers appearing in front of d~ in Eq. 2.17 whether positive or negative, are the stoichiometric coefficients of the reaction in Eq. 2.16. By convention, these coefficients are positive for products and negative for reactants. Consider now a closed thermodynamic soil system that is in thermal and mechanical equilibrium and in which the reaction in Eq. 2.16 is occurring as an infinitesimal process. Assuming that no other chemical reactions are taking place, the internal energy change in the soil is given by dUs =

~(CaCI2)dn(CaCI2)

+ ~(Na2C03)dn(Na2C03) + ~(CaC03)dn(CaC03) + ~(NaCl) dn(NaCI)

where it is understood now that the chemical potentials are measured in units of joules per mole. This expression can be written in the more compact form

after Eqs. 2.17 have been incorporated. If the initial state of the soil was an equilibrium state, the mass transfer d~ is a virtual process, and Eq. 2.18 is subject to dU = O. Since d~ is an arbitrary, infinitesimal parameter, the condition for chemical equilibrium becomes

Equation 2.19 is a constraint placed on the chemical potentials by the reaction in Eq. 2.16. The discussion given here can be made completely general by writing in place of Eq. 2.16 the arbitrary expression for a chemical reaction ~)

II)A)

= 0

where A) is the symbol for a chemical species and ficient. The convention II)

> 0 for a product

II}

(2.20) II)

is its stoichiometric coef-

< 0 for a reactant

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

25

is employed in Eq. 2.20. The generalization of Eq. 2.18 is then dUs

=

}";j vjl.L(A)d~

and the condition for chemical equilibrium becomes (T, P fixed)

(2.21)

Equation 2.21 is a direct result of the First and Second Laws of Thermodynamics applied to a chemical reaction. It and Eq. 1.24 are perhaps the most important theoretical expressions in the thermodynamic analysis of soil solutions. To give Eq. 2.21 a full experimental interpretation obviously requires a discussion of the measurement of the chemical potential, a topic to be considered at length in the remainder of this chapter.

2.2. STANDARD STATES FOR SOIL COMPONENTS INVOLVED IN ISOTHERMAL, ISOBARIC REACTIONS

Most experiments designed to study soil chemical processes employ thermal and volume reservoirs to control the absolute temperature and the pressure. If these two intensive properties are fixed by the reservoirs during a process of interest, the process is, by definition, isothermal and isobaric. Virtually all of the chemical reactions in soils are studied as isothermal, isobaric processes. It is for this reason that the measurement of the chemical potentials of soil components involves the prior designation of a set of Standard States that are characterized by selected values of T and P. Unlike the situation for T and P, however, there is no thermodynamic method for determining absolute values of the chemical potential of a substance. The reason for this is that /-L represents an intrinsic chemical property that, by its very conception, cannot be identified with a universal scale, such as the Kelvin scale for T, which exists regardless of the chemical nature of a substance having the property. Moreover, /-L cannot usefully be accorded a reference value of zero in the complete absence of a substance, as is the pressure, because there is no thermodynamic method for measuring /-L by virtue of the creation of matter. Therefore it is necessary to adopt a conventional definition of a state of a substance in which the chemical potential of that substance vanishes. This definition will require a statement about T and P (solely because these properties are convenient to maintain under control) as well as a specification of the phase in which. II substance occurs. The convention agreed on in thermodynamics is expressed: The chemical potential of any chemical element in its most stable phase under Standard State conditions is by convention equal to the value O. This definition of the chemical potential of an element in the Standard State npplies to every entry in the Periodic Table. For a chosen chemical element, all that one must do is establish what phase is the most stable one "under Standard

26

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

State conditions." These conditions will be discussed at length in this section. It should be noted that the convention adopted above concerning J.L does not permit a comparison of chemical potentials among the elements, nor is this kind of comparison necessary in thermodynamics. On the other hand, the chemical potentials of a given element in states other than the Standard State can be compared, as can the chemical potentials of compounds formed from elements.

Gases The Standard State for a substance in the gas phase is taken to be that of an ideal gas composed of the pure substance, with a fugacity equal to 101.325 kPa 0 atm), at a designated temperature. Usually this temperature is 298.15 K. The fugacity,f, of a gas can be defined by P

In f=lim[lno+ hlO

J

( v / nR 1) dr ]

(2.22)

h

where 0 ~ 0 means "0 goes to 0 through positive values," and R - 8.3144 J mol-1K- 1 is the molar gas constant. According to Eq. 2.22, the value offat the designated temperature, T, and a chosen value of P will depend on the particular equation of state, V( T,P,n), that describes the gas. If the ideal gas expression

Vd (T,P,n) = nRT/P is employed in Eq. 2.22, the fugacity is found to be equal to the pressure: Injid = lim [In 0 + hlO

J

P

0/ pl)dr]

= In P

h

On the other hand, if V( T,P,n) was given by the expression V( T,P,n) = (nRT/P)

+

nBi 1)

where Bi 1) is the second virial coefficient, the fugacity would be given by the equation

Inf =

l:~ { In 0 + J

= In P

+

P

d In pI + [ B 2( 1)/ RT] {P dr h Bi 1)P/RT

In summary, the experimentally determined equation of state for V leads to a well-defined fugacity for every T and P. If a gas shows ideal behavior to a good approximation (as many gases do for P < 10 atm), the fugacity of the gas is equal to its pressure. This fact, in turn, means that the Standard State pressure of a gas is equal to 1 atm, since the Standard State fugacity is 1 atm. The fugacity of a pure liquid or solid can be defined by applying Eq. 2.22 to the vapor in equilibrium with the substance in either condensed phase. Usually the volume of the vapor will follow the ideal gas equation of state very closely, and the fugacity of the vapor may be set equal to the equilibrium vapor pressure. The thermodynamical basis for associating the fugacity of a condensed

27

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

substance with that of its equilibrium vapor may be seen by combining Eq. 2.22 with the Gibbs-Duhem equation (Eq. 1.24) applied to the vapor under the condition of fixed T: Infvap = lim 610

[

In 0

+ (1/RT)

~(P)

J

]

du

,.(6)

where the form of the integral term comes from replacing VdP by ndu. It follows from this expression that Infvap

=

I-/-(P)/ RT

+

C( n

(2.23)

where

C(n == lim [In 0 -

1-/-(0)/ RTJ

(2.24)

610

Equation 2.23 demonstrates that hap is related directly to the chemical potential of the equilibrium vapor. But this latter quantity, in turn, equals the chemical potential of the substance in the condensed phase, according to Eq. 2.15. Therefore the fugacity of a condensed phase may be defined by the expression (2.25)

n

is defined in Eq. 2.24 and I-/- is the chemical potential of the subwhere C( stance in the condensed phase. Alternatively, Eq. 2.22 may be used to define the fugacity of a substance in any phase since, as the pressure oiO, a condensed phase will vaporize to become a gas and in Eq. 2.24 will have the same numerical value regardless of what phase actually exists when the applied pressure equals P. It follows that Eqs. 2.22 and 2.25 are completely equivalent. Moreover, an important corollary to Eq. 2.15 is that the fugacities ofa substance coexisting in two phases that are in equilibrium are the same. Equation 2.25 may be employed, together with the definition of the Standard State, to produce a formal definition of the Standard State chemical potential of a gas. Applied to the Standard State, Eq. 2.25 can be written

C( n

O=I-/-°/RT+C(n

where 1-/-0 designates the value of I-/- in the Standard State. It follows that in Eq. 2.25

Inf = 1-/-/ RT - 1-/-0/ RT und, according to Eq. 2.22, that

1:~~ll

Ino+ {P(v/nRndPI] =I-/-/RT-I-/-°/RT =

I!~ r

-c

(1/P')dP'

+ In P +

{P (V/nRndP'

J

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

28

where

the

second

equality

comes

from

the

identity

1n 0 = 1n P

- J P(I I P')dP'. Upon solving this expression for JLo, one finds that s JL o = JL -

RT In P - lim ~IO

J

P [(

VI nRT)

- (II P')] dP'

(2.26)

~

Equation 2.26 shows clearly how JL for a gas shifts from its Standard State value JL o, both because of a change in pressure from P = 1 atm and because of nonideality in the gas as expressed in the difference between VI nRT and 1IP. Liquids, Solids, and Solvents The Standard State for a substance in the liquid or solid phase is that of the pure substance at a designated temperature (usually T = 298.15 K) and under an applied pressure of 101.325 kPa (1 atm). The fugacity in the Standard State is not specified, since it is determined by the value of the equilibrium vapor pressure of the solid or liquid under Standard State conditions. The component of largest content in a solution is, by convention, called the solvent. This arbitrary designation is most convenient when aqueous solutions are under consideration because liquid water is invariably the component with the largest mole number. In the cases of solid and gaseous solutions (with the exception of air), the naming of one component as the solvent usually is of small utility, since broad ranges of composition often are involved. At any rate, the Standard State for a liquid or solid solvent is the same as that for a liquid or solid: the pure substance at a designated value of T and under 1 atm pressure. For a gaseous solvent, the Standard State is the same as that for a gas: the pure substance treated as an ideal gas at a designated value of T and at unit fugacity. Solutes By convention, the solutes in a solution are all of the components other than the solvent. In the case of a solute in a gaseous, solid, or nonaqueous liquid solution, the Standard State is that of the substance at unit mole fraction with a fugacity equal to its Henry's Law constant in the solution at some designated temperature (usually 298.15 K) and under a pressure of 101.325 kPa (1 atm). Henry's Law for the solute may be expressed mathematically as (N ~ 0)

(2.27)

where fis the fugacity of the solute, k H is its Henry's Law constant, and N is its mole fraction. For a component A, the mole fraction is defined by the equation NA

n =-----'.:..-_-n + n + nc + ... A

A

(2.28)

B

where the denominator equals the sum of mole numbers for all of the chemical constituents in the solution. When the solute is at unit mole fraction, it is the only component in the solution, Its actual fugacity will be equal approximately to its vapor pressure, which, in turn, is determined by the values of T and P. In

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

29

general, this fugacity will not be equal to k H because Eq. 2.27 usually applies only in the limit as N ~ O. Therefore the Standard State of a solute is usually a hypothetical state. The fugacity of a solute in the Standard State may be determined experimentally, however, according to the construction shown in Fig. 2.1. The measured values offare plotted against the mole fraction of the solute and a line tangent to the graph at the origin is drawn. The extrapolation of this line, to the point where N = I, permits the Standard State fugacity to be read

FIGURE 2.1. Determination of the Standard State fugacity.j", of a solute by extrapolation from the Henry's Law region. 0.5~-----------------------.

0.4

0.3 ,-...

.s E

It...,

0.2

0.1

0.5 N

30

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

from the graph. Alternatively, the Standard State fugacity is simply equal to the slope of the limiting tangent line, k H , as indicated in Eq. 2.27. In the case of a solute in an aqueous solution or of an electrolyte in any kind of solution, a concentration variable usually is employed to denote the content of the solute in the solution. However, the mole fraction concentration scale is seldom used. Instead, either the molal or the molar scale is used (the latter only if the temperature is maintained at a single value for all experiments of interest):

m =

nsolute/ m,olvent

M =

nsol ute/ V,olution

(2.29)

The molality, m, is measured in mol kg- 1 and the molarity, M, is measured in mol dm- 3• With the molal concentration scale introduced, Henry's Law for a solute in an aqueous solution may be expressed mathematically:

f=

(m ~ 0)

k~mv

(2.30a)

where v -

-

r 1 for nonelectrolytes l VI + V2 for the electrolyte C

(2.30b) v,A v2

Since both Eqs. 2.27 and 2.30a usually apply only in the limit of infinite dilution of a solute (i.e., as N ~ 0 and m ~ 0), there is no inconsistency between the two equations when V = I: as an aqueous solution becomes more dilute, the molality of a solute becomes proportional to its mole fraction. However, it is clear that, even in this case, k~ is not directly comparable with k H because the units of the former are N m ? kg' mol>, while those of the latter are pascals or an equivalent unit. The Standard State of a solute in an aqueous solution is that of the solute at unit molality, with a fugacity equal to its Henry's Law constant in the solution, at a designated temperature and under a pressure of 101.325 kPa (1 atm). For an electrolyte, it is strictly the mean ionic molality, defined in Eq. 2.88, that has unit value in the Standard State. The relation between the mean ionic molality of an electrolyte and the molality of the electrolyte is discussed in Section 2.6. The method for obtaining the fugacity in the Standard State is quite analogous to what is done for nonaqueous solutions and is illustrated in Fig. 2.2. In this instance the fugacity is plotted against the variable m'. Equations 2.25 and 2.27 may be combined to derive an explicit formal expression for the Standard State chemical potential of a solute in an aqueous solution. Applied to the Standard State, Eq. 2.25 has the form In k~ = INRT

+

C(7)

from which it follows that, in general, the fugacity of the solute is

Inf= J.l/RT

+

In k~ - J.l°/RT

Therefore J.l 0

-

J.l - NT I n(f/ k{,)

(2.31 )

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

31

0'05r---------------------...,

0.04

0.03

0.02

0.01

0.5 m 2 (moI2 kg- 2 )

1.0

FIGURE 2.2. Determination of the Standard State fugacity,f°, of an aqueous electrolyle. CA(aq), by extrapolation from the Henry's Law region.

With the information provided by Eq. 2.30a, Eq. 2.31 may be written as a limit expression: J.L0 =

lim (J.L - RT 1n m')

(2.32)

mlO

1\'1uation 2.32 demonstrates clearly the close relationship between the behavior of II dilute aqueous solution and the Standard State chemical potential of a solute.

32

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTION~

As is well known, electrolyte solutions pose a special problem in thermodynamics. Equations 2.30 show that electrolytes do not follow the classical form of Henry's Law, for example. This fact is related to the tendency of electrolytes to dissociate in water into ionic species. It proves to be less cumbersome at times to describe an electrolyte solution in thermodynamic-like terms if dissociation into ions is taken into account explicitly. Now, the properties of ionic species in an aqueous solution cannot be thermodynamic properties, according to the discussion in Section 1.1, because ionic species are strictly molecular concepts. Therefore the introduction of ionic components into the description of a solution is an extrathermodynamic innovation that must be treated with care to avoid errors and inconsistencies in formal manipulations. This point will become clear in the sections to follow. It suffices to remark here that ionic solutes are introduced for convenience only. They are not necessary and, in fact, are inimical to a strict thermodynamic description of an aqueous solution. The Standard State of an ionic solute is that of the solute at unit molality in a solution (at a designated temperature and under a pressure of 1 atm) in which no interionic forces are operative. This convention differs from that for a neutral solute in aqueous solution by its substitution of a condition on the ionic forces for a condition on the fugacity. In Section 2.6 it will be demonstrated how these two conditions are related. Besides the designation of a Standard State, a convention must be adopted for the Standard State chemical potentials of ionic solutes. This requirement derives from the fact that a change in U (or in anyone of the thermodynamic potentials) with respect to the mole number of an ionic solute, carried out while all other composition variables remain fixed, is impossible to measure. This particular type of change would define the chemical potential of an ionic solute, but it is impossible because of the requirement of electroneutrality, which stipulates that, for example, a shift in the number of moles of a cation in a solution must always be accompanied by a balancing shift in the number of moles of the anions. Therefore it is not possible to determine the chemical potential of an ionic solute experimentally, even given the convention already provided for chemical potentials of the elements, without specifying arbitrarily the Standard State chemical potential for one ionic solute as a reference. This specification is made for the proton, in the case of acid-base reactions, and for the electron, in the case of oxidation-reduction reactions. The chemical potential of the proton and of the electron in aqueous solution in the Standard State is, by convention, equal to the value O.

The Standard States discussed in this section are summarized for easy reference in Table 2.1. It is important to note that the Standard States for gases and for solutes are hypothetical, ideal states and not actual states. For gases, this choice of Standard State is useful because the ideal gas represents a good limiting approximation to the real behavior of gases and possesses equations of state that are mathematically tractable in applications. For solutes, the choice of a hypothetical Standard State is of value because the alternative choice of a

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

33

TABLE 2.1. Standard States employed in thermodynamics" Substance Gas Liquid Solid Solvent (I, s) Solvent (g) Solute (g, s; neutral) Solute (aq; neutral) Solute (aq; ionic)

TO

pO

298.15 K 298.15 K 298.15 K 298.15 K 298.15 K 298.15 K

1 atm 1 atm 1 atm 1 atm 1 atm 1 atm

pure substance pure substance pure substance t" = 1 atm i" = k H ; N° = 1

298.15 K

1 atm

t" = k~;

298.15 K

1 atm

mO = 1 mol kg"; no ionic interactions

Special Conditions

I" = 1 atm

m~

= 1 mol

kg- I

* TO = Standard State temperature; pO = Standard State pressure. t" = Standard State fugacity; N° = Standard State mole fraction; m~ = Standard State mean ionic molality; and mO = Standard State molality.

Standard State, consisting simply of the pure solute at unit mole fraction, is not very relevant to a solution component whose content in a solution must always remain small. Moreover, by making the Standard State have the property of no interactions among the solute molecules or ions, it is possible to define a useful thermodynamic parameter that accounts for differences among solutes in their solution behavior, as will be seen in Section 2.6. A hypothetical Standard State also is necessary to a self-consistent description of ionic solutes in thermodynamic language. This fact will be demonstrated in Sections 2.3 and 2.6.

2.3. THERMODYNAMIC ACTIVITY AND THE EQUILIBRIUM CONSTANT

With the establishment of conventions for the Standard State and for the reference zero value of the chemical potential, it is possible to give a full development of the thermodynamic description of chemical reactions. This development begins with the application of Eq. 2.25 to both the equilibrium state of a substance and to its Standard State. If Eq. 2.25 is applied separately to both states and the two resulting equations are subtracted, one obtains the expression

lnf - ln j" =

J.tl RT - J.t°l RT

or (2.33) where the superscript ° is a conventional designation for the Standard State. Equation 2.33 applies to any substance, in any phase, in any kind of mixture at equilibrium. It expresses the idea that the chemical potential of a substance nlways may be written as equal to the Standard State chemical potential plus a

34

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

logarithmic term in the ratio of equilibrium to Standard State fugacities. For convenience, this last ratio is given the symbol a and defined to be the relative activity or, more commonly, the thermodynamic activity of the substance: (2.34) The activity, therefore, is a dimensionless quantity that serves as a measure of the deviation of the chemical potential from its value in the Standard State. By definition, the activity of any substance in its Standard State is equal to 1. Thus, for example, the activity of pure CaC0ls) at 298.15 K and under a pressure of 1 atm is 1.0, as is the activity of pure liquid water in a beaker under the same conditions. Although the activity and the fugacity are closely related, they have quite different characteristics in regards to phase equilibria. Consider, for example, the equilibrium between liquid water and water vapor in the interstices of an unsaturated soil. At a given temperature and pressure, the principles of thermodynamic equilibrium demand that the chemical potentials and fugacities of water in the two phases be equal (Eqs. 2.15 and 2.25). However, the activity of water in the two phases will not be the same because the Standard State for the two phases is not the same. Indeed.j" = 1 atm for the water vapor, so its activity is numerically equal to its own vapor pressure (assuming ideal gas behavior). In the case of the liquid soil water, f O is not equal to 1 atm but, instead, is equal (approximately) to the much smaller equilibrium vapor pressure over pure liquid water at T = 298.15 K and P = 1 atm. Therefore the activity of the liquid water is (approximately) equal to its relative humidity divided by 100. The general conclusion to be drawn here is that the activity of a substance coexisting in two phases at equilibrium cannot be the same in both phases unless the Standard State for both is the same. On the other hand, the chemical potential and fugacity are always the same in the two phases at equilibrium. The combination of Eqs. 2.33 and 2.34 produces the expression f.L

= f.L 0 + RT In a

(2.35)

for the chemical potential of any substance at equilibrium. This expression may be inserted into Eq. 2.21 to derive the identity AGO

+ RT };jvjln aj

=

°

(2.36)

where (2.37) is called the standard free energy change of the reaction. The second term on the left side of Eq. 2.36 may be collected in part into the parameter (2.38) which is called the thermodynamic equilibrium constant for the reaction. Thus Eq. 2.36 can be written in the form /iao •

- NT InK

(2.39)

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

35

upon noting that, according to the mathematics of logarithms,

It should be stressed that the chemical significance of Eq. 2.39 is quite the same as that of Eq. 2.21. The definitions in Eqs. 2.37 and 2.38 are made solely for convenience in applications. Also, in a strict sense, K should be written [(0 to denote the Standard State T and P. The principal utility of ti.Go is that it may be employed to determine the stability of products relative to reactants when all of these compounds are in their Standard States. The criteria for stability are: Products are more stable than reactants if ti.Go than products if ti.Go > O.

<

O. Reactants are more stable

As an example, consider the weathering of kaolinite in a soil to produce gibbsite. This process may be described by the hydrolysis reaction AI2Si20s(OH)4(s)

+ 5 H 20(1)

= 2 H 4Si04(aq)

+ AI(OH)3(s)

(2.40)

sa:

where 0) refers to a liquid phase. In order to compute the values of f.L 0 for the four compounds involved must be known. The methods for determining these values will be discussed in Section 2.4. It suffices for now simply to quote the values: f.L 0 (kaolinite) = -3799.4 kJ mol"; f.L 0 (H 20(l)) = -237.1 kJ mol"; f.L 0 (H 4SiOiaq)) = -1308.0 kJ mol"; and f.L 0 (gibbsite) = -1154.9 kJ mol-I. These data all refer to T = 298.15 K and P = 1 atm. The value of ti.Go for the kaolinite weathering reaction is, accordingly, ti.Go = 2(-1308.0) + 1(-1154.9) - 1(-3799.4) - 5(-237.1) = 1214.0 kJ mol" Therefore, since ti.Go > 0, kaolinite and water are more stable than silicic acid and gibbsite, relative to the Standard State. As another example, take the reaction that transforms aragonite into its polymorph, calcite: CaC0 3(s, aragonite) = CaC0 3(s, calcite) The values of f.L 0 are, at 298.15 K and 1 atm pressure, f.L°(aragonite) = -1127.8 k.l mol" and f.L°(calcite) = -1128.8 kJ mol", from which it follows immediately that ti.Go = -1.0 kJ mol-I. In this example, ti.Go < 0 and the product, calcite, is more stable than aragonite, relative to the Standard State. If the Standard State temperatures were to be altered to some other value, it is possible that ti.Go for the transition, being relatively small in absolute magnitude, could change sign. Then aragonite would be the more stable polymorph, relative to the new Standard State. Clearly, then, the sign of ti.OO is only a relative indicator of product versus reactant stability in a chemical reaction. The value of ti.Go, according to Eq. 2.39, also leads directly to the value of the equilibrium constant at the Standard State temperature and pressure. For

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

36

T

=

298.15 K, Eq. 2.39 may be written in the practical form: ,6.Go = -5.708 log K

(2.42)

where log refers to the common logarithm and ,6.Go is measured in kilojoules per mole. If the values of the Standard State chemical potentials for the reactants and products are available, the equilibrium constant for any reaction may be found (at T = 298.15 K and P = 1 atm) from Eqs. 2.37 and 2.42. This method for obtaining equilibrium constants will be discussed in Section 2.5. The equilibrium constant is a dimensionless quantity equal to the weighted > 0) to those of the reactants (v j < 0) ratio of the activities of the products in a chemical reaction. Often this important thermodynamic parameter may be determined by direct or indirect measurements of the activities themselves. In the thermodynamic analysis of soil solutions, two kinds of equilibrium constant often are employed. These are the formation constant, K; which describes the precipitation of a solid from aqueous solution, and the ion exchange equilibrium constant, K m which describes an ion exchange reaction. For example, consider the precipitation of CaCO], described by the reaction

sr.

CaCOlaq) = CaCOls) The formation constant is

«xco.e»

K, = -'--_'::"":-:":-

(2.43a)

(CaCObq»

where the commonly used symbol ( ) refers to the activity of a compound. Parentheses henceforth will be employed to denote activities whenever the use of the symbol a would be cumbersome. Now, if the reaction takes place at T = 298.15 K and P = 1 atm and if the precipitate formed is pure CaCO], then Eq. 2.43a simplifies to the expression K, = 1/(CaCOlaq»

since the solid phase would be in its Standard State with (CaCO](s» = 1. In this case, it is convenient to work with the solubility product constant, K so' instead of K; (2.43b) A measurement of the activity of CaCO] in aqueous solution suffices to determine Kso- However, if the CaCO] had coprecipitated with, for example, MgCO], the activity of CaCO](s) would not, in general, be equal to 1 and Eq. 2.43a could not be simplified by the use of Eq. 2.43b. Indeed, in that case, the activity of CaCOlaq) would not be equal to an equilibrium constant but, instead, would be equal to the product K,o (CaC01(s». The value of (CaCOls)) would depend entirely on the mode of solid solution between CaeO, and MgCO,. As an example or the lise or K w consider the calion exchange reaction Na ..( 'O,(aq)

I ('aX ..(s) = ('a( 'O,(aq)

I 2 NaX(s)

(2.44 )

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

37

where X refers to one equivalent of the anionic part of the exchanger. The equilibrium constant for this reaction is (2.45) Equation 2.45 cannot be simplified further because NaX and CaX 2 form a solid solution and, therefore, are not in the Standard State with unit value of the activity. The concepts of t:.Go and K can be extended to ionic solutes in aqueous solutions by making the formal definition (2.46a) where cm+ is the cation of valence + m and A n- is the anion of valence - n in the electrolyte C"A". It is assumed in writing Eq. 2.46a that a method exists for obtaining the chemical potential of an ion in aqueous solution in its Standard State, with the result that (2.46b) holds for any neutral electrolyte, cation, cm+, and anion, A n-. Since the Standard State for an electrolyte is not defined in quite the same way as that for an ionic solute (see Table 2.I), the verification of Eq. 2.46b is not trivial. The consistency of the Standard State conventions for solutes with Eq. 2.46a will be examined in Section 2.6. Given the validity of Eq. 2.46, it is possible to define the activity of an ionic solute formally through Eq. 2.35: (2.47) where IJ,c = IJ,(cm+), and so on. Equations 2.47 and the corresponding expression for IJ,( C"A,,) may be introduced into Eq. 2.46a to obtain In (C"A,,)

=

VI

In (C?")

+

V2

In (An-)

(2.48a) The activity of the electrolyte is equal to the ion activity product (lAP), which is the weighted product of individual ionic activities. If the additional definition (2.48b) is made, where

V

=

VI

+

V2,

Eq. 2.48b becomes (2.48c)

The quantity defined in 1~4. 2.4Sb is called thc mean ionic activity. It is the geometric mcan value of thc individual ionic activitics. Note thatthc mean ionic

38

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

activity is a strictly thermodynamic quantity, even though the individual ionic activities are not. As examples of the use of individual ionic activities, consider the reactions in Eqs. 2.41 and 2.44. The precipitation reaction for CaC0 3 may be written in terms of ions in the form (2.49) with the equilibrium constant K r = (CaC0 3(s))/(Ca H)(COi-)

(2.50a)

If CaCOls) is in the Standard State, Eq. 2.50a may be replaced by

K;

=

(CaH)(COi-)

(2.50b)

The equilibrium constant K, in Eq. 2.50a is the same as K, in Eq. 2.43a, and K; in Eq. 2.50b is the same as K; in Eq. 2.43b, because of Eq. 2.48a. In the case of the cation exchange reaction, the ionic form is (2.51) with the equilibrium constant

s; = (NaXnCaH)/(CaX2)(Na +)2

(2.52)

Now

according to Eq. 2.48a. It follows that Kex has the same numerical value in Eqs. 2.52 and 2.45. These two examples emphasize that equilibrium constants and other strictly thermodynamic parameters always will involve thermodynamically meaningful combinations of ionic activities. A particularly interesting chemical reaction in the aqueous solution phase of a soil is soluble complex formation. This process may be described by the equation Pc

C?"

+ VA An-

-- C lICAq"A

(2.53)

where q = Vern - vAn is the valence of the soluble complex, C'CA;A' The equilibrium constant for this reaction is called a stability constant: (2.54) The inverse of K, is known as an instability constant or, especially if C?" is the proton, H+, as a dissocation constant. It is important to note that the complex C'CA;A represents a molecular concept and, therefore, is not a strictly thermodynamic entity. The notation CveA~A never refers to the neutral electrolyte, C.,A." even when q = O. For example, the reaction CaCO~(aq) - CaC01(s)

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

39

describes the disappearance of the neutral ion-pair CaCO~ from aqueous solution to form solid CaC0 3. The equilibrium constant for the reaction is K = (CaC03(s))/(CaCO~) = K f / K,

where

s, = (CaCO~)/(Ca H)(COJ) is a stability constant and K, is given in Eq. 2.50a. In connection with the aragonite-calcite transition (Eq. 2.41), it was emphasized that the value of toGo depends in part on the values of T and P with which one defines the Standard State. It is of interest, therefore, to develop expressions that show how toGo changes when the Standard State T and Pare shifted. Consider first a change in T. The response coefficient of interest is

(2.55) where the Maxwell relation in Eq. 1.15 has been employed and toSO, measured in J mol-IK- I, is called the standard entropy change of the reaction under consideration. The computation of toSO requires experimental data on the partial molal entropy, S~ = (aSO/an)T,p, for each chemical species involved in the reaction. For the aragonite-calcite transformation, at T = 298.15 K and P I atm, SO(aragonite) = 88.0 J mol-IK- I and SO(calcite) = 91.7 J mol-IK- I, It follows that O

G ( ato aT

_

)

-(91.7 - 88.0) = -3.7 J molv'K:"

P

Accordingly, a shift in the Standard State temperature at atmospheric pressure will not change toGo very much from its value of -1.0 kJ mol", In the example of the hydrolysis reaction given in Eq. 2.40, the temperature coefficient of toGo IS

ato GO )

( aT

_

= SO(kaolinite)

+ 5 _SO(H 20 (l ))

P

- 2 SO(H 4SiOi l )) - SO(gibbsite) = 203.1 + 5(69.95) - 2(180.0) - 68.44 = 124 J mol-IK- I I n this case, also, the shift in toGo with a unit temperature change is small relative to toGo = 1214 kJ mol-I. The corresponding temperature coefficient for In K may be calculated quite

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

40

easily with the help of Eqs. 2.39 and 2.55: O

_ (at::.G aT

)

p

=

+ RT (a In K) er p

R In K + RT (a In K) _ _ t::.Go aT p T

or (

a In K)

er

p

= t::.Go __1_ (at::.G RT z RT

er

_

-

O )

p

= _1_ (t::.Go RT z

+

Tt::.S O)

1 t::.J-!O + J}°. _ l ) = -Z} : · v · n , = -z RT } } } RT

1("'() }} }

TrO _

-}:.v.(T~; z

RT

(2.56)

where t::.J-!O, measured in J mol", is the standard enthalpy change of the reaction under consideration. For the kaolinite hydrolysis reaction in Eq. 2.40, at T = 298.15 K, (a In K/anp

=

-(1/739.10)[ 1:fO(kaolinite) + 5J-!O(H z0(1)) -2J-!O(H 4SiOiaq)) - J-!O(gibbsite)] = -(1/739.10)[-4120.1 + 5(-285.83) - 2(-1460.0) - (-1293.1)] = + 1.81K- '

This value may be compared with In K = -489.7. The temperature coefficient represents about 0.4% change per degree in the In K value. The response coefficient for the effect of pressure on t::.Go is O

O

at::. G ) ( aPT

=}:.

v, (aJ.l

}}

)

aP

= }:j v, (a

T,'j

YO)

anj

=

t::. yo

(2.57)

T.P

where the Maxwell relation in Eq. 1.16 has been employed and t::. YO, measured in rrr' mol", is the standard volume change of the reaction under consideration. The computation of t::. VO requires data on the partial molal volume, VJ = (a VOl an) T,P for each chemical species involved. Taking the aragonite-calcite transition in Eq. 2.41 as an example, O

(

at::. G

ap

) T

= (36.93 _ 34.15) X 10- 6 rrr' mol- '

=

+2.78 X 10- 6 m! mol"

= 0.282 J

mol r'atm"

where VO(calcite) = 3.693 X 10- 5 rrr' mor ' and VO(aragonite) = 3.415 X 10- 5 m' mor ' have been employed along with the conversion factor I rrr' = 1 J m' N- 1 = 1.01325 X 105 J arm ". The pressure coefficient for In K follows from Eqs. 2.39 and 2.57:

(

K)' _ oP

- t::. VO

8 In

T

RT

(2.58)

The calculation of t::.Go and its temperature and pressure response coefficients is facilitated by the availnbllity of reliable compilations of datu on J.l 0 ,

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

41

SO, and VO for chemical compounds of interest in soils. Good compilations for many compounds are available, and statistical correlation techniques exist for obtaining at least 11° values for a variety of solids and complexes not yet fully characterized thermodynamically by experiment. These sources of useful data will be described in the next section.

2.4. STANDARD STATE CHEMICAL POTENTIALS

The Standard State chemical potentials of substances in the gas, liquid, and solid phases, as well as of solutes in aqueous solution, can be determined by a variety of experimental methods, among them spectroscopic, calorimetric, solubility, colligative-property, and electrochemical techniques. I The accepted values of these fundamental thermodynamic properties are and should be undergoing constant revision under the critical eyes of specialists. It is not the purpose here to discuss the practice of determining values of 11° for the compounds of interest in soils;' this is best left to special works on experimental thermodynamics. Suffice it to say that Standard State chemical potentials must be selected from the published literature on the basis of precision of measurement, internal experimental consistency, and consistency with other, currently accepted 11° values. Because of the continual revision in 11° values, no attempt will be made here to present a list of critically compiled data, even for the compounds of principal interest in soil solutions. In this and subsequent chapters, Standard State chemical potentials for gases, liquids, solids, and solutes will be taken directly from or calculated based on data in the following critical compilations.'

1. R. A. Robie, B. S. Hemingway, and J. R. Fisher, Thermodynamic properties of minerals and related substances at 298.15 K and 1 bar (l0 5 Pa) pressure and at higher temperatures, Geol. Survey Bull. 1452, U.S. Government Printing Office, Washington, D.C., 1978. 2. M. Sadiq and W. L. Lindsay, Selection of standard free energies of formation for use in soil chemistry, Colorado State Univ. Tech. Bull. 134, Colorado State Univ. Experiment Station, Fort Collins, Col., 1979. 3. A. E. Martell and R. M. Smith, Critical Stability Constants, 4 vols., Plenum Press, New York, 1974-1977. 4. G. Milazzo and S. Caroli, Tables ofStandard Electrode Potentials, John Wiley, New York, 1978. 5. C. F. Baes and R. E. Mesmer, The Hydrolysis of Cations, John Wiley, New York, 1976. The compilations by Robie et al. and by Sadiq and Lindsay are quite extensive, including in their entries .nany solids as well as ionic solutes in aqueous solution. Since a compound may be written as the product of a chemical reaction I hat involves only chemical elements as reactants, and since 11° for an element is equal to zero, 11° for a compound may be considered to be a special example of !i(jCl for a reaction that forms the compound from its constituent elements. Thus l1Cl values also are termed standard free energies offormation and are given the

42

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

symbol t1G~. This nomenclature is particularly preferred by thermodynamists who work with substances in the solid state. The bulletins authored by Robie et al. and by Sadiq and Lindsay follow this convention. In addition to 11° (or t1G~) values, Robie et al. list VO, SO, and flO for many substances. The compilation by Martell and Smith applies to soluble complexes in aqueous solution and presents equilibrium constants for the reaction in Eq. 2.53. These constants are related to 11° through Eqs. 2.46 and 2.47. The compilation by Milazzo and Caroli is useful for species related to one another by oxidation-reduction reactions, as will be demonstrated in Chapter 4. A wide variety of published, critical compilations of thermodynamic data that have been organized for many different applications has been described in a review prepared for the Specialist Periodical Report series in chemical thermodynamics." As extensive as the recent compilations of critically selected 11° values are, it still would be naive to presume that thermodynamic data for all of the compounds likely to form in soils will be readily available for all applications of interest. Indeed, in a typical soil solution, it would be normal to consider 200 to 300 soluble complexes and 20 to 30 solid phases as possible compounds under normal conditions of temperature and pressure. In order to characterize such a system thermodynamically, it would be necessary to have at hand 250 to 350 equilibrium constants. But there is no certainty that these equilibrium constants actually will have been measured experimentally, nor is there certainty that the 11° values requisite to their calculation will have been determined and published. This problem of a lack of thermodynamic data has been appreciated widely by natural water chemists and has led to attempts to predict 11° values by a method known as linear free energy relations (LFER). The basic premise of LFER is that a small set of chemically significant parameters can be found that characterizes a class of compounds of interest and permits a linear relationship to be established between a suitable function of the parameters and the Standard State chemical potentials of the compounds. In some instances, the set of parameters will include basic properties of the principal components in the compounds, while in others the set will include properties of the principal components as well as 11° values for a set of related compounds. Whatever the chosen set of parameters may comprise, the objective always is to employ it in a statistical correlation technique to predict 11° values of compounds for which reliable, experimentally determined Standard State chemical potentials do not exist. The LFER approach is best understood by a detailed consideration of three typical applications. METAL OXIDES AND HYDROXIDES. When a metal cation in aqueous solution hydrolyzes and precipitates to form an oxide or hydroxide solid phase, its local bonding environment changes from that of the cation coordinated principally through ion-dipole interactions to water molecules to that of the cation coordinated principally through ionic bonds to oxide or hydroxide anions. Actually, the chemical bonds in oxide or hydroxide minerals are known to possess both covalent and ionic characteristics. The detailed properties of these

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

43

bonds are complicated when considered from the point of view of quantum chemistry, but progress in understanding the stability of the oxides and hydroxides nonetheless has been made through simple physical models based primarily in classical electrostatics. The reason for the success of the electrostatic model lies in the observation that the short-range coordination environment of a cation that is found in, for example, hydroxide minerals, varies little from one hydroxide mineral to another. Since covalency plays its most important role in nearestneighbor interactions, it follows that its contribution to the structural energy of a mineral will be about the same among either oxides or hydroxides and, therefore, that the relative stabilities of the minerals in either group largely will be determined by the longer-ranged ionic interactions. These considerations suggest that the J.,L0 values of metal oxides and hydroxides may depend primarily on the nature of the metal cations and, therefore, that the J.,L0 values of the metal oxides may be correlated strongly with those for the hydroxides. Such a correlation may be useful as a predictive tool if it is linear and reasonably straightforward to establish. One approach that turns out to be successful is based on the concept of J.,L0 for a metal cation in a mineral. 5 In the cases of hydroxides and oxides, one may define J.,L°(Mm+(hy» J.,L°(Mm+(ox»

= =

J.,L°(M(OH) m(s» - mJ.,L°(H 20 (s» J.,L°(MO m/2(s» - (mI2)J.,L°(H 20(s»

(2.59a) (2.59b)

The quantities on the left sides of Eq. 2.59 are Standard State chemical potentials of the metal cation M m + in an hydroxide or an oxide mineral, by definition. H 20(s) refers to ice Ih in these equations, and MO m/ 2 is the chemical formula for one equivalent of a metal oxide. The chemical potential J.,L°(MO mils»~ is related to the more common chemical potential J.,L°(MpOq(s», which refers to 1 mol of the oxide M pOq' by the expression:

J.,L°(MOm/ls» =

(11 p)J.,L°(MpOq(s»

Equations 2.52 indicate that J.,L0 for a metal cation in an oxide or an hydroxide mineral equals J.,L0 for the mineral less J.,L0 for ice (i.e., hydrogen oxide) times the stoichiometric coefficient of 0 or OH in the oxide or hydroxide. This "singleion" chemical potential is, therefore, well defined thermodynamically. Now consider the dissolution reactions Mm+(hy) = Mm+(aq) Mm+(ox) = Mm+(aq)

for which the standard free energy changes are, respectively, LlGgy = J.,L°(Mm+(aq» - J.,L°(Mm+(hy» LlG~. = J.,L°(Mm+(aq» - J.,L°(Mm+(ox»

(2.60a) (2.60b)

The LlGo in Eq. 2.60 represent the change in J.,L0 when a metal cation undergoes II change in phase from the solid mineral to an aqueous solution. On the basis of the discussion given above concerning what kind of change in bonding this

44

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

change of phase implies for the metal cation and its coordinating ligands, it is reasonable to suppose that both ~G~y and ~G~x are determined principally by the nature of the metal cation, M?" and, therefore, that a linear relationship exists having the form (2.61) where a is an empirical parameter to be determined through statistical correlation. A study of a large number of metal oxides and hydroxides' demonstrated that a highly significant linear correlation of the type in Eq. 2.61 does indeed exist with a = 0.58. With this correlation established, it is possible to estimate, for example IlO(MOm/z(s» for a given metal oxide if IlO(M(OH) m(s» and IlO(Mm+(aq» are known and Eq. 2.59 and 2.60 are employed. A value of a < 1 is expected, incidentally, since it implies that Ilo of a metal cation in a hydroxide is nearer in value to IlO(M m+ (aqj) than is Ilo of the metal cation in an oxide. The hydroxyl anion is likely to provide a bonding environment more similar to that in aqueous solution than is the oxide anion, and this situation should be reflected by a smaller absolute value for ~G~y than ~~x' A specific numerical example of the use of Eq. 2.61 to estimate a Ilo value will illustrate the expected accuracy of the LFER approach in this application. The Standard State chemical potential of Al(OH)ls), gibbsite, is -1154.9 kJ mol", while that of AI3+(aq) is -489.4 kJ mol" and that of ice Ih is -223.6 kJ mol", Therefore, by Eqs. 2.59a and 2.60a, IlO(AI3+(hy»

= -1154.9 - 3( -223.6) = -484.1 kJ mol " ~G~y = -489.4 - (-484.1) = -5.3 kJ mol "

Equation 2.61 and the value a = 0.58 now lead to the estimate ~G~x

=

-9.14 kJ mol"

from which it follows that IlO(AI3+(ox» = -480.3 kJ mol", Equation 2.59b then leads to the estimate IlO(AI0 3/z(s» = -815.7 kJ mol', or IlO(AI 203(s» = -1631.3 kJ mol:", after conversion from one equivalent of corundum to 1 mol as a basis for expressing Ilo. In this case a measured value of Ilo is available, IlO(AI 203(s» = -1582.2 ± 1.3 kJ mor". This value differs from the estimate by about 3%. SMECTITES. The smectites are a class of widely occurring layer silicates whose chief importance in soils derives from their relatively high cation exchange capacity. A distinguishing characteristic of the smectites is the variability in their chemical composition. Although Ilo values have been determined experimentally for several varieties of smectites, including the most important type, montmorillonite, it is unlikely that these data will bcome available for every variant found in soils. Thus a method for estimating the Standard State chemical potentials of smectites based only on their chemical composition should prove to be quite useful. One reasonably accurate method] that is based on the

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

45

general picture of ionic bonding discussed above involves the consideration of the reaction

+ n3Ca(OH)2 + n4Mg(OH)2 + + n6Fe(OH)2 + n7AI(OH)3 + nsSi(OH)4

nlNaOH

+

n2KOH

.o

= Na'IK'2Ca'3Mg•• Fe~;Fe~: AI.7Si •

+ (~jnjZj

lO

nsFe(OH)3

(OH) l s)

- 12)H 20(l)

(2.62)

in which a smectite and liquid water are formed from a set of hydroxide components. In Eq. 2.62, all the reactants are solids, n, is the stoichiometric coefficient for the ith reacting hydroxide, and Z, is the valence of the principal cation in the ith hydroxide. The sum on the right side of the equation is over all eight reactants. The Standard State chemical potential of the smectite formed in Eq. 2.62 may be expressed by rearranging Eq. 2.37: /-L°(smectite) =

~jnj/-L°(r;)

- (~jnjZj - 12)/-L°(H 20(l»

+ I1G~

(2.63)

where /-L0( rj) is the Standard State chemical potential of the ith reactant hydroxide (denoted rj ) and I1G~ is the standard free energy change for the reaction in Eq. 2.62. Since the /-L°(rj) and /-L°(H 20(l» generally are available, /-L°(smectite) can be estimated with the help of Eq. 2.63 if 11~ can be estimated. Granting the applicability of the ionic bonding picture to the estimation of /-L0 for layer silicates, it follows that I1G~ should be regarded principally as a measure of the changes in ionic potential that occur because of changes in the coordination environment of the cations in the smectite minerals relative to the component hydroxides. An important change in coordination environment that occurs in transferring a cation from an hydroxide to a constant-charge clay mineral, such as montmorillonite, takes place for those cations found in the interlayer space of the clay mineral (e.g., Na ", K+, and Ca2+). These exchangeable cations must be taken from relatively high-potential sites in hydroxides and be placed in relatively low-potential, interlayer sites in the clay mineral. If the number of these sites is large, 111~ I also should be large. With respect to 11~ for different homoionic clay minerals, the ionic bonding picture would suggest that either decreasing the ionic radius or increasing the ionic charge of the exchangeable cation will decrease the values of 111~ I. A small ionic radius or a large ionic charge enhances the role played in ionic bonding by the short-ranged repulsive interaction at the expense of the long-ranged, attractive Coulomb interactions. Since the principal characteristic of an exchangeable cation in a constantcharge clay mineral is its relatively large separation from the site of the negative charge to which it is bound, the cations for which repulsive interactions are most important will have the least perturbation in ionic energy in transferring from the hydroxide or oxide to the clay mineral. These suggested trends in 11~ can be substantiated by multiple regression of known I1G~ values for homoionic montmorillonites on the cation exchange capacity due to isomorphous substitution (expressed as the number of equiva-

46

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

lents per formula weight, eq/fw), the Pauling radius of the exchangeable cation, and the valence of the exchangeable cation. The result is the linear expression' In 1.::l~1 = 1.9283C

+ 3.501R

- 0.28192

+ 3.5427

(2.64)

where C is the cation exchange capacity (eq/fw) due to isomorphous substitution, R is the Pauling ionic radius (nm), and 2 is the valence. Equations 2.63 and 2.64 have been used successfully to estimate J.l,0 within about 13 kJ mol:" for a number of smectites that were not considered in the development of Eq. 2.64 but for which J.l,0 values have been determined experimentally. As a numerical example of this estimation procedure, consider the magnesium-saturated Aberdeen montmorillonite with the composition Mg02075( Alo.18Si382)(AII.29Fe~.t35Mgo.445)O JO( 0 H) 2 According to Eq. 2.63: J.l,°(mont) = 0.6525J.1,°(Mg(OH) 2) + 0.335J.1,°(Fe(OH) 3) + 1.47J.1,°(Al(OH)3) + 3.82J.1,°(Si(OH)4) - 10J.l,°(H 20(l» - exp (In I.::lG~ I)

= 0.415 eq/fw, R = 0.065 nm, and 2 = 2, = 1.9283(0.415) + 3.501(0.065) - 0.2819(2) +

By Eq. 2.64, with C In I.::lG~1

=

3.5427

4.007

It follows that

J.l,°(mont)

0.6525( -833.58) + 0.335( -696.64) + 1.47( -1154.90) + 3.82( -1322.9) - lO( -237.18) - 54.97 = -5211.6 kJ rnol"

=

This estimate may be compared with the measured value of -5218.7 kJ mor '. METAL-CHLORIDE COMPLEXES. Tabulated J.l,0 values for soluble complexes in aqueous solution are not common. Instead, thermodynamic data for metal complexes are presented as common logarithms of stability constants (see Eq. 2.54). Consider, for example, the formation of the metal-chloride complex MCI(m-I): M'"'

+ Cl" =

MCI(m-l)

in aqueous solution, for which the stability constant is K, = (MCI(m-I»)/(Mm+)(CI-) The Standard State chemical potential of MCI(m-')(aq) follows from this expression along with Eqs. 2.37 and 2.42 (all terms in units of kJ mol:"): J.l,°(Mcpm-I)(aq» .. - 5.708 log K.

+ J.l,°(M

mt

(aql)

+ J.l,°(CI

(aql)

(2.65)

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

47

Given the value of the stability constant, K" and a set of J-L 0 values for ionic solutes in aqueous solution, J-L 0 is calculated readily for any metal-chloride complex, MCl(m-I)(aq). A number of schemes has been developed for predicting log K, values of soluble metal-ligand complexes. One relatively successful method" employs a linear correlation of log K" for a chosen ligand and a set of metal cations, with a function, Q, that depends on the valence and the electronegativity of a metal cation: (2.66) where X M is the Pauling electronegativity and a and (3 are empirical parameters characteristic of the ligand." The values of Q calculated for several metal cations are listed in Table 2.2. The application of LFER in this case consists of establishing a linear correlation relationship between measured values of log K, and the stability parameter, Q. With the log K, values listed in Table 2.2, the statistical relationship" log K, = 2.64( Q - 3.30)

(2.67)

can be established by conventional least-squares techniques. With this expression included, Eq. 2.65 becomes the working equation (all terms in kilojoules per mole):

+ J-L°(M m+(aq)) (2.68) for a complex between cr and any

J-L°(MCI "(aqj) = 15.069(3.30 - Q) - 131.27

This expression can be used to compute J-L 0 of the metal cations listed in Table 2.2. Equations of the same mathematical form as Eq. 2.67 have been developed to predict log K, values for higher-order complexes of transition metals with cr.7

TABLE 2.2. Values of the stability parameter, Q, and the common logarithm of the stability constant for MCI complexes of several metal ions* Metal Cation

Q

log K,

TIJ+ Pdl+ InJ+ Cdl+ Cul+ Felt Pb l +

6.12 4.84 4.26 4.23 4.00 3.84 3.74

7.72 4.34 (6.1) 2.04 2.00 (1.98) 0.95 (0.40) 1.48 1.60 (1.59)

-- ..-,.-

.. ,-

..•...----

_.__.•._--.-

Metal Cation Nj2+

Sn"' Zn H GaJ+ Bel+ Mnl+

Q

3.65 3.60 3.38 3.28 3.14 3.10

log K, 0.89 (-0.43) 1.51 (1.16) -0.19 0.00 -0.50 0.00 (-0.14)

·See note 2 in the article by Nieboer and McBryde, cited in note 6. More accurate values of log K. than used by Nieboer and McBryde are available now in some cases. These are shown in parentheses.

48

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

2.5. STANDARD STATE CHEMICAL POTENTIALS AND THE EQUILIBRIUM CONSTANT

The relationship between the thermodynamic equilibrium constant and the Standard State chemical potentials of the reactants and products in a chemical reaction is specified by Eqs. 2.37 and 2.39 (or 2.42). This relationship makes it evident that, given a table of experimentally determined f.L 0 values for a variety of compounds and ionic solutes, the equilibrium constant for any conceivable chemical reaction among some species in a set chosen from the table can be calculated at once. This most useful possibility is tempered, however, by the requirement that the f.L 0 data be of high precision, since the percentage error in the computed K, according to elementary calculus, will be approximately equal to the error in I!1Go I expressed as a percentage of RT. For example, a reasonable error of + I kJ mol" in !1Go is about 40% of RT at 298.15 K, and this is the estimated percentage error in the corresponding K value. On the other hand, if log K values were desired instead of K values, the percentage error would be equal to the percentage error in I!1Go I. For example, if there is an error of + 10 kJ mol- 1 in a !1Go value of 1000 kJ mol", this amounts to a 1% error in the corresponding log K value. Besides the important requirement for good precision, several other conditions must be imposed on tabulated f.L 0 values if these data are to be employed successfully to characterize chemical reactions in soil solutions. These conditions can be brought out through a consideration of the relation between f.L 0 values and K values for three commonly studied soil chemical phenomena. PRECIPITATION-DISSOLUTION REACTIONS. Consider the precipitation of the minerals gypsum and calcite from aqueous solution in an arid-zone soil. The relevant chemical reactions may be written: CaH(aq)

+ SO~-(aq) + 2H = CaH(aq) + COi-(aq) = 20(l)

CaS0 4 • 2H 20(s) CaCOls)

The Standard State chemical potentials that contribute to !1Go for these reactions are, together with their reported uncertainties, f.L°(Ca H(aq)) - - 553.5 ± 1.2 kJ mor ' f.L°(COi-(aq)) - -527.9 ± 0.1 kJ mol- 1 f.L°(SO~-(aq)) - -744.6 ± 0.1 kJ mol" f.L°(gypsum) - -1797.2 ± 4.6 kJ mol" f.L°(H 20 (l )) - -237.1 ± 0.1 kJ rnol" f.L°(calcite) - -1128.8 ± 1.4 kJ mol " It follows that, for the formation of gypsum, !1G~p •

- 24.9

± 4.8 kJ mol

I

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

49

and, for the formation of calcite, LlG~1

=

-47.4

+

1.8 kJ mol"

where the uncertainties in LlGo have been computed with the help of the equation (2.69) where Vj is the stoichiometric coefficient and (Jj is the uncertainty in the ILo value for the jth reactant or product in the reaction described by Ll(JJ. As is commonly observed, the uncertainty in LlGo is dominated by that in ILo for the solid phase formed. The corresponding formation constants are, at 298.15 K, log Kr
=

4.36

± 0.84

log K, (cal) = 8.30

+

0.32

where it is understood that, for example, log K, (gyp) may be in error by as much as 20%, primarily because of the uncertainty in ILO(gypsum). The results of these two calculations can be converted to solubility product constants if certain chemical criteria are met. These criteria are related to the step of setting the activity of the solid phase equal to 1.0 in the expressions

K,o(gyp) = (CaS04' 2H 20(s»/ s, (gyp) K,o(cal) = (CaC0 3(s»/ K, (cal) The solid phase activities are of unit value, and log K, = -log K,o == pK,o if, in the soil under study, it can be ascertained that the solid phases are indeed in their Standard States. This requirement means that T = 298.15 K, P = 1 atm, and the solid phase is a pure, macroscopic substance. If T and P differ from their standard values, an adjustment in K, can be made according to Eqs. 2.56 and 2.58 with no change in the relation between K, and Kso- It is more important to determine whether the solid for which ILo was determined is, in fact, the one that actually has formed in the soil. For example, the solid phase in the soil may be X-ray amorphous or, otherwise, may be of significantly smaller particle size than the one to which ILo refers. In that case, the Standard State of a pure, crystalline solid is not achieved by the solid precipitated in the soil, and the activity of the solid will not be equal to 1.0. Alternatively, the soil solid phase may contain more defects or structural irregularities than the one to which ILo refers because the former solid is precipitated onto the surfaces of previously existing soil minerals instead of onto the bottom of a laboratory flask. In this case, also, the activity of the solid in the soil may not be 1.0. Finally, it is possible that substitution for one or more of the components of the soil solid (e.g., Mg2+ for CaB in calcite) has occurred when the solid precipitated from the mixed electrolyte soil solution. In this case, the solid is actually a solid solution and the activity of anyone of its components (e.g., CaC0 3 in a magnesian calcite) cannot be 1,0 except fortuitously. Thus, for solid phases formed in soils, a variety of Xray diffraction, chemical analysis, spectroscopic, and thermal experiments may be needed in order to specify whether Standard State conditions actually obtain lind K, values calculated from tabulated ILo data are applicable.

50

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

As a simple example of the problems just discussed, consider the precipitation of the mineral variscite, AIP0 4 • 2 H 20, in an acid soil. The relevant chemical reaction is AI3+(aq)

+ PO~-(aq) + 2H 20(l)

= AIP0 4 • 2H 20(s)

The standard free energy change for this reaction is LlGO(var) = -174.1 kJ mol", which follows from Eqs. 2.37 and 2.69 along with the data:

± 2.4

J.l°(AI3+(aq» - -489.4 ± 1.4 kJ mol" J.l°(H 20(l» - -237.1 ± 0.1 kJ mol- t J.l°(PO~-(aq» - -1019.0 ± 0.9 kJ mol" J.l°(variscite) - -2156.7 + 1.7 kJ mor ' These data also lead to log K, (var) = 30.5 ± 0.5. However, it has been observed that crystalline AIP0 4 • 2H 20 seldom precipitates in soils. Instead, an X-ray amorphous compound forms that can remain in a metastable state for significant lengths of time. For this compound, which may be formulated as Al(OH)2H2POiam), measurements made on allophanic soils" indicate that, at T = 298.15 K, log K, (var, am) = 28.1 ± 0.1. This experimental result may be used to calculate a value of J.l 0 for Al(OH)2H2POiam): J.l°(variscite, am) = -5.708(28.1) + (-489.4) = -2143.0 + 1.8 kJ mol- t

+ (-1019.0) + 2( -237.1)

Two principal conclusions may be drawn from this example. First, since LlGo <0 for the reaction

there is no doubt that, at T = 298.15 K and P = 1 atm, variscite is more stable than its amorphous analog. On the other hand, the amorphous compound is favored kinetically and persists in soils. Second, if J.l0 values for AP+(aq), PO~ (aq), H 20(l), and Al(OH)2H2POis, am) are employed to calculate log K r, then pK so for the soil phosphate is equal to log K, because Al(OH)2H2POis, am) is in the Standard State and, therefore, has unit activity. However, if J.l0 values for the ionic solutes and liquid water are combined with J.l0 for AIP04 • 2H 20(s) to calculate log Kr, then pKso for the soil phosphate is equal to 28.1 = - 2.4 + log K, because the activity of the soil phosphate in this case is not 1.0 but, instead, is 251.19. This much larger activity reflects the more surface-active, X-ray amorphous state of the precipitated compound. In solubility studies, it is much more common to use a J.l 0 value for the solid that actually forms, when calculating pKso' than it is to use a J.l0 value for a solid phase that does not form and then adjust the value of its activity to account for the actual state of the solid in the soil. CHEMICAL WEATHERING REACTIONS. Consider, as a first example, the hydrolysis of a soil montmorillonite to form kaolinite:

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

51

Mgo225(Alo.30Si3.7o)(AI1345Fe6!osMgo27)OIO(OH)ls) + 3.6225 H 20(l) + 2.265 H+(aq) = 0.8225 AI2Si20s(OH)ls) + 2.055 H4SiO~(aq) + 0.405 FeJ+(aq) + 0.495 Mg2+(aq) The standard free energy change for this weathering reaction may be calculated with the help of Eq. 2.37 and the following J,l.0 values: J,l.°(smectite) J,l.°(H 4SiOlaq» J,l.°(H 2O(l» J,l.°(FeJ+(aq» J,l.°(kaolinite) J,l.°(Mg2+(aq»

-

-

-

-5214.7 + 1.7 kJ mol' -1308.0 + 1.7 kJ mor ' -237.1 ± 0.1 kJ mol" -4.6 ± 1.0 kJ mol" -3799.4 + 4.0 kJ mol' -454.8 ± 1.7 kJ mol"

Accordingly, f::J.Go = 33.7 + 5.2 kJ mol" for the reaction and log K = -5.9 ± 0.9 at 298.15 K. Given the assumptions that (1) the two soil clay minerals are described properly by the J,l.0 values available, and (2) each remains in its Standard State, the activities of the species participating in the reaction are related by the expression

+

- 5.90 = 2.055 log (H4SiO~(aq» + 0.405 log (FeJ+(aq» 0.495 log (Mgj'{aqj] + 2.265 pH - 3.6225 log (H 20(l»

(2.70b)

where pH = -log (H+(aq» as usual. Equation 2.70b is just an expression for log K written out according to Eq. 2.38 with due attention to the sign convention for the "r Since f::J.Go > 0 in this example, the reactants, montmorillonite, liquid water, and the proton, are stable relative to the products in the Standard State. This result may appear surprising because of the known, normal progression from montmorillonite to kaolinite in the soil clay mineral weathering sequence. The apparent paradox brought on by the sign of f::J.CO will be resolved in Section 3.5 when activity-ratio diagrams are discussed. Suffice it to point out here that the transformation of one clay mineral into another through a dissolution-precipitation process in the soil solution does not depend only on the Standard State chemical potentials of the reactants and products in that process. The transformation of matter between two states depends on the actual chemical potentials of all the participants (i.e., on the Standard State chemical potentials and the activities of the reactants and products). Both J,l.0 and the activity of a substance determine its actual stability relative to some other substance. The importance of ascertaining the appropriateness of a calculated f::J.CO value for describing an actual chemical weathering process may be illustrated in another way by a consideration of the reaction between kaolinite and o-phospilate ions to form an aluminum phosphate in an acid soil: AI1Si 20s(OH)4(s)

+ 2PO~-(aq) + 3H 20(1) + 6H+(aq) - 2AI(OH)2H2P04(am)

+ 2H4SiO~(aq)

52

THE CHEMICAL THERMODYNAMICS OF SOIL

SOLUTION~

With the help of the Jlo values already presented in this section, the standard free energy change for this phosphate-fixation reaction is found to be ti.Go = -353.3 + 6.6 kJ mol " and the equilibrium constant is log K = 61.9 + 1.2. If the product phosphate had been written as AlP0 4· 2H 20(s) and the Jlo value for variscite had been employed in the calculation of log K, the result would have been 7% larger. This increase, in turn, would have produced serious errors in the prediction of PO~- activity in the soil solution based on the value of K. Thus it is important to establish as well as possible the condition of the solid phases in a soil before attempting to describe their weathering reactions with Jlo data. CATION EXCHANGE REACTIONS. The cation exchange reaction between the Na-saturated and Ca-saturated forms of Camp Berteau montmorillonite: may be expressed:

+ 0.1675Ca2+(aq) Cao.I675Si4(AlI.46Fe~.to5Fe~tI5Mgo.32)OlO(OH)2(s) + 0.335Na "(aq)

Nao335Si4(AlI.46Fe~.to5Fe~.tI5Mgo.32)OlO(OH)2(s)

=

(2.7Ia if

The standard free energy change for this reaction can be calculated with t .•' help of the Jlo v a l u e s : ) , JlO(Na-mont) JlO(Ca-mont) JlO(Ca2+(aq)) JlO(Na+(aq))

-

-5221.0 ± 2 kJ m o l - I ' , -5225.9 + I kJ mol" -553.5 + 1.2 kJ mol- 1 -261.9 + 0.1 kJ mol "

However, the thermodynamic equilibrium constant corresponding to ti.Go so ca culated cannot be compared directly with the cation exchange equilibrium stant that is discussed commonly in the soil chemical literature. This is becau cation exchange reactions usually are not expressed in the format of Eq. 2.71" Instead, the format leading to an exchange equilibrium constant of the ty given in Eq. 2.52 is employed, wherein the relatively insoluble, aluminosilicai portion of the clay mineral is considered as a single anion and is specifiedii units of one equivalent. This can be done in the present example by dividi' each stoichiometric coefficient in the chemical formula of Camp Berteau moo morillonite by 0.335 eqjfw, the cation exchange capacity of the clay mine! due to isomorphous substitutions. If this is done, the aluminoscilicate part of th mineral may be given the designation X-I, where X -I

=

[Si 1I.9iAI4358F e~.tI2Fe~.t45Mgo955)029.85( 0 H) 597] -I

The chemical formula within the brackets is just that in the first term on the Ie ' side of Eq. 2.71a after division of each subscript coefficient by 0.335 and deletlon of Na. Equation 2.71a now may be divided on both sides in all terms by 0.33 . and then multiplied by 2 in order to clear the fraction before the term in Ca2+(aq):

2NuX(s)

+ Ca 2

t

(uq) - CaX 2(s)

+

2Na 2 I (aq)

(2.7Ib)

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

53

where CaX is)

= 2Cao.5Si11.9iA1 358Fe~~12Fe~"~5Mg0955)0 29.85(0 H) 5.9'(s) 4.

The standard free energy change for the reaction in Eq. 2.71b can be calculated immediately after the 1-/-0 values for the Na- and Ca-montrnorillonites are divided by 0.335 and multiplied by the valence of the exchangeable cation to place them on an equivalent basis with respect to the aluminosilicate part of the clay mineral: 1-/-°(NaX(s» = (ljO.335)1-/-°(Na-mont) - -15585.1 1-/-°(CaX 2(s» = (2jO.335)1-/-°(Ca-mont) - -31199.4

+ 6.0 kJ mol- 1 + 6.0 kJ mol "

Accordingly, !:i.Go = -31199.4 + 2( -261.9) - (-535.5) - 2( -15585.1) - -17.5 + 13.5 kJ mol- 1

and the cation exchange equilibrium constant for the reaction in Eq. 2.71b is log Kex = 3.07 + 2.37 at 298.15 K. The effect of the precision of the 1-/-0 values is especially notable in this example. The equilibrium constant may be related to the activities of the participating species in Eq. 2.71b just as was Kex in Eq. 2.52. At this point, a recent discussion? in which attention is called to some important problems in the interpretation of 1-/-0 values for layer silicates should be mentioned. It has been suggested that smectites, for example, may not be stable minerals in the strict thermodynamic sense because of their extensive, random isomorphous substitution in the octahedral and tetrahedral sheets. (Note the fractional stoichiometric coefficients in Eq. 2.71a.) These substitutions, it can be argued, are not normal for stable minerals formed at low temperatures under sedimentary conditions, and their existence leads to the idea that phyllosilicates such as the smectites should be considered to be solid solutions of truly stable minerals such as pyrophyllite, muscovite, and margarite, which do not exhibit random isomorphous substitution. If, indeed, the smectites and the vermiculites were completely unstable solid solutions, the thermodynamic significance of 1-/-0 for these minerals would be open to serious question. On the other hand, if these minerals are metastable on a time scale that is long in the context of the weathering environments of soils and sediments, and if a practicable Standard State for these minerals can be defined for a chosen temperature, pressure, and set of solid composition variables, measured values of 1-/-0 will continue to be useful in studies of cation exchange phenomena and of mineral formation and dissolution in soil solutions and other interstitial waters.

2.6. REFERENCE STATES AND ACTIVITY COEFFICIENTS

The Standard State of a substance is characterized by designated values of T and P and by an activity equal to 1.0. The Reference State of a substance. on

54

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

the other hand, is characterized by a designated value of T and by an activity coefficient equal numerically to 1.0. In general. the Standard State and the Reference State of a substance are not the same. The properties of Reference States may be understood by a consideration of solution phases paralleling the discussion in Section 2.2 for Standard States. Gaseous Solutions According to Eq. 2.22, the fugacity of a single-component gas phase is given by the equation

+ lim

Inf= In P

alo

I

P

[(VjnRn - (ljP')]dP'

(2.72)

a

where the identity employed to derive Eq. 2.26 has been incorporated. The gas activity coefficient, x, is defined by the expression:

x=f/P

(2.73)

[(VjnRn - (ljP')]dP'

(2.74)

Therefore, by Eq. 2.72, InX = lim alo

I

P

a

The activity coefficient, x, is a dimensionless quantity that does not depend on the unit of pressure chosen. Another important consideration is that Eq. 2.74, shows X to be a measure of the departure of a gas from ideal behavior. If a gas' obeys the ideal gas expression PV = nRT, the right side of Eq. 2.74 vanishes identically, and X = 1.0. In general, ideal behavior in a real gas is its limiting behavior as the pressure tends to zero. Thus, for any gas, it is expected, and 1 indeed required by Eq. 2.74, that the value of X tends to 1.0 as P becomes van-] ishingly small. This result leads directly to the definition of the Reference Statcl for a gas: the Reference State for a substance in the gas phase is that of the pun~i substance at a designated temperature (usually 298.15 K) in the limit as P ! 0': In this state the gas activity coefficient is equal to 1.0. Note that the Referen " State of a gas is not the same as its Standard State, although in both states the: gas will show ideal behavior. In the Reference State the pressure of the gas is vanishingly small, whereas in the Standard State the pressure of the gas equal' 1.0 atm. For a gas, the thermodynamic activity is equal numerically to the fugacitj because the fugacity in the Standard State equals 1.0 atm (see Eq. 2.34). Theres fore, a = f = xP, or (numerically)

X = alP

(2.75)

provides an alternate way to define the gas activity coefficient. In a gaseous solution, the definition of the fugacity becomes Inf- In NP r

+ lim '10

/' P1[( Vj Rn - (lj pmdP~ '.

(2.76)

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

55

for a component of the solution whose mole fraction is N and whose partial molal volume is V (a VT / anh,p,{n,), where VT is the total volume of the mixture. The pressure P T in Eq. 2.76 is the total pressure of the gaseous solution. The definition of the gas activity coefficient is now

=

X

=f/NPT

(2.77)

from which it follows that Eq. 2.74 still applies, with V/ n replaced by V and P replaced by PT' The Reference State for the mixture is that which is attained as P T ~ O. In the case of a mixture of nonaqueous liquids or of solids, the definition of the activity coefficient, fA' of a component A is Nonaqueous Solutions

(2.78) This activity coefficient is called a rational activity coefficient. It is dimensionless and is related to the chemical potential of the component A through the expression (2.79) which follows from the combination of Eqs. 2.35 and 2.78. The Reference State of a component A in a nonaqueous liquid solution or in a solid solution is that of the component at T = 298.15 K and P = 1 atm as N A approaches 1.0. This Reference State is to be used when no component can be designated conveniently us either the solute or the solvent. According to Eq. 2.79, the chemical potential of the solution component A is equal to J.L~ in the Reference State, since fA = 1 lind N A = 1 in that state by definition. Therefore, in this case, the Standard State and the Reference State are the same. If the rational activity coefficient of each component of a nonaqueous liquid solution or of a solid solution is equal to 1.0, regardless of the composition of the mixture, the solution is said to be ideal. For the ideal solution, Eq. 2.79 becomes (2.80)

for each component A regardless of the value of N A' This expression provides a reason for calling the mixture to which it applies ideal. In the case of an ideal tlllS

mixture, Eq. 2.76 takes the form Inf~ = In (NAP T )

(2.81)

Nincc PTVA = RT in the mixture for any gaseous component A. Equation 2.81 mny be introduced into the appropriate form of Eq. 2.26 for ideal gas mixtures 10 obtain the result (2.82) limier the condition that PT = I atrn, Eq. 2.82 reduces to Eq. 2.80. Thus the chemical potential of a component of an ideal, nonaqueous liquid or solid solu-

56

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

tion has the same dependence on the mole fraction as does the chemical potential of a component of an ideal gas mixture whose total pressure is equal to one atmosphere. If it is convenient to call one component of a solution the solvent, the Reference State for that component may be chosen exactly as described for any component in the discussion following Eq. 2.79. In this case the Standard State and the Reference State also are the same. For a solute, A, the Reference State is defined by T = 298.15 K, P = 1 atm, and N A = 1.0. This state is not the same as the Standard State of the solute. In the Reference State, the fugacity of the solute is equal to its actual fugacity as a pure substance at T = 298.15 K and P = 1 atm (i.e., fA is approximately equal to the vapor pressure of the pure solute A). In the Standard State, . on the other hand, the fugacity of the solute is by definition equal to the Henry's Law constant, according to Table 2.1. Thus the Reference State of a solute is an actual state, whereas the Standard State is a hypothetical state. In the case of a very dilute solution, Eq. 2.27 applies to the solute I

(2.83) Since the activity of the solute is equal to fA/f~ = fA/ k H, it follows that the activity coefficient of the solute is equal to 1.0 when Henry's Law applies:

which follows from the combination of Eqs. 2.78 and 2.83. For a solute in a nonaqueous liquid solution or in a solid solution, the rational activity coefficient . approaches 1.0 as N approaches both zero and one. Solutes In Aqueous Solutions There are two principal definitions of the Ref- . erence State for a solute in an aqueous solution. In the first, the Infinite Dilution Reference State, the activity coefficient of a solute is defined to approach unit i value as the concentration approaches zero for each solid component of an aqueous solution at T = 298.15 K and P = 1 atm. In the second, the Constant' Ionic Medium Reference State, the activity coefficient of a solute approaches unit value as the concentration of only that solute approaches zero, while the concentrations of all the other solid components of the aqueous solution (the' "background ionic medium") remain fixed. Both definitions are valid thermo- , dynamically, and each has advantages and disadvantages. For example, in the . case of the proton, the use of the Constant Ionic Medium Reference State means that the activity coefficient of H+ in most soil solutions will have very nearly unit value and, therefore, that a glass electrode will measure directly the proton concentration in these solutions. There is no need to calibrate the electrode against a set of standard buffer solutions either, since one may, in principle, simply make known additions of protons to a soil solution and read the corresponding EMF values of the electrode in order to calibrate it. On the other hand, this kind of calibration would have to be done for every soil solution of interest instead of for I,

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

57

just one set of standard buffer solutions (assuming that liquid junction potentials in the buffer solutions are negligibly different from those in the soil solutions). The Infinite Dilution Reference State will be employed in this book. However, note that many published thermodynamic properties of pure electrolyte solutions are based on the Constant Ionic Medium Reference State (usually with NaCI04 providing the background ionic medium), and that this choice of Reference State is becoming increasingly popular among those who study seawater and other saline natural waters. The activity coefficient, 'Y, of a neutral solute in an aqueous solution, may be defined by the expression 'Y

= aim

(2.84)

where a is the activity of the solute and m is its molal concentration. Alternatively, the molar concentration scale may be used and a molar activity coefficient, y, may be defined by analogy with Eq. 2.84, where y = 'Y I Pwater

(2.85)

and Pwater is the mass density of water in the solution (kg dm ") when the molality equals m. It is evident from an examination of Eqs. 2.84 and 2.85 that 'Y has the units kg mol " while y has the units drrr' mol ", Because the mass density of water is very nearly equal to 1 kg dm -3 at T = 298.15 K, the numerical values of 'Y and y will be almost exactly the same at that temperature. Equation 2.84 also may be applied to a neutral electrolyte, but it is more common to use another definition of the activity coefficient. Consider again Eq. 2.48a for the electrolyte C.,A P2: (2.48a) The MacInnes activity coefficients, 'Ys+ and 'Ys-, now may be defined according to the expressions (2.86) where m is the molality of the electrolyte. The combination of Eqs. 2.48a and 2.K6 leads to the expression (C "'I A112 ) = (""" 1 s+ ",," I s- )(v"v") 1 2 m' where, as before, v = VI + V2' The right side of this equation may be collected Into two factors, the stoichiometric mean ionic activity coefficient, (2.87) lind the mean ionic molality,

(2.88)

58

THE CHEMICAL THERMODYNAMICS OF SOIL

SOLUTION~

Therefore (2.89) and, according to Eq. 2.48c, (2.90) Equations 2.88 and 2.89 make it clear that "Y± is a well-defined thermodynamic quantity. Equation 2.89, in fact, may be taken as the definition of the stoichiometric mean ionic activity coefficient for an electrolyte. The MacInnes activity coefficients that are contained in "Y + have no welldefined thermodynamic meaning in themselves. They can be evaluated, however, if an arbitrary convention is established for the numerical estimation of any pair of them describing a neutral electrolyte. In the MacInnes convention, "Ys+ for K+(aq) and "Ys- for Cl Taq) each are set equal to "Y± for KCl (aq). Then, for example, the value of "Ys+ for CaH(aq) in a solution of CaCl 2 can be estimated according to the expression b+(CaCI 2}P b±(KCOj2

= "Ys+(CaH)')';_(CI-) = "Ys+(K+)')'s-(CI-)

"Ys+(CaH)')';_(Cl-) "Y;_(CI-)

=

(Ca H) "Ys+

where both values of "Y ± employed are measured at the same temperature, pressure, and ionic strength. The MacInnes convention usually is not a good approximation for mixed electrolyte solutions, but it has been employed often in geochemical studies. Given Eq. 2.89, the Reference State for an electrolyte can be defined as that attained when m± approaches zero along with the molal concentrations of all other solutes. Typical values of m ±' expressed in terms of the total molality, m, are listed, with examples, in Table 2.3. As indicated in Section 2.2, the molality that appears in Henry's Law for electrolytes (Eq. 2.30) is m± and not m alone. This specification is made in order that the Standard State and the Reference State for an electrolyte be mutually consistent. Thus, in Eq. 2.30a, f TABLE 2.3. The mean ionic molality for several types of electrolyte, C"A", Type

1-1 1-2 1-3

2-1 2-2 2-3 3-1 3-2

Example NaCI Na2S04 Na 3P04 CaCI 2 CaS0 4 Ca 3(P04h· AICI 3 AI1(S04)1

m

41/ 3 m 271/ 4 m 4 1/ 3m m 108 1/' m 27'/4 m I081/~m

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

59

= k~ when m± is 1.0 mol kg-I and the electrolyte is in its Standard State.

Indeed, when Henry's Law applies, the activity of the electrolyte is equal to m ± , according to Eqs. 2.30a and 2.34. It follows from Eq. 2.89 that "y± = 1.0 kg mol" under this condition also, as would be expected. For both single and mixed electrolyte solutions, the values of "y ± have been measured for a large variety of electrolytes. This activity coefficient also can be evaluated numerically for a chosen electrolyte with the help of a very accurate (but complicated) regression equation whose mathematical form was suggested by statistical mechanics. 10 It is possible to define the single-ion activity coefficients, "Y + and "Y_, in a formally correct manner through the expressions V

(2.91) where m+ and m.. are the actual molal concentrations of the ions C m+ and A'in a solution containing the electrolyte C at a molality m. Unless the electrolyte is completely dissociated in the solution, m+ =1= v-m and m : =1= vzm. However, according to Eqs. 2.48a and 2.91, V,AV2

(2.92) and, therefore, the right sides of Eqs. 2.89 and 2.92 are equal even though, in general, (2.93) The definitions in Eq. 2.91 suggest that the Reference State for an ionic solute is that attained (at T = 298.15 K and P = 1 atm) as either m+ or m.. and all other solute concentrations tend to zero. In this limiting state, the values of "Y + and "Y_ approach 1.0 kg mol" and single-ion activities become numerically equal to molal concentrations. There is a long history of attempts to find simple expressions for "Y + and "Ythat can be employed in multicomponent electrolyte solutions. One such expression that has found much use in the study of soil solutions is - AZzP/ z 10g"Y = 1 + /I/Z + B/ (2.94) in which "Y may be "Y +, "Y_, or "Y for a neutral solute (Eq. 2.84), depending on the values of the empirical coefficients A and B. In Eq. 2.94, Z is the valence of the solute and / is the ionic strength of the electrolyte solution (2.95) where Z/ is the valence and m, is the actual molality of the ith charged species in the solution. The sum in Eq. 2.95 is over every charged species, including charged complexes. For the calculation of "Y + or "Y_. A is the well-known Debyel luckel Limiting Law constant and B = 0.3AZz, Values of A are listed in Table 2.4 for several temperatures and in the units kgl/zmol- I/z or dm 3/ zmol -I/Z, The

60

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS TABLE 2.4. Values of the Debye-Huckel Limiting Law constant for aqueous solutions* A

rc o 5 10 15 18 20 25 30 35 38 40 45 50

55 60 65 70 75 80 85 90 95 100

Mass .basis, kg'/2mol-I/2

Volume basis, dm J / 2mol- I / 2

0.4918 0.4953 0.4989 0.5027 0.5050 0.5067 0.5108 0.5151 0.5196 0.5224 0.5243 0.5292 0.5342 0.5395 0.5449 0.5505 0.5563 0.5623 0.5685 0.5750 0.5817 0.5886 0.5959

0.4918 0.4953 0.4990 0.5029 0.5054 0.5072 0.5116 0.5162 0.5212 0.5242 0.5263 0.5318 0.5374 0.5434 0.5495 0.5559 0.5625 0.5695 0.5767 0.5843 0.5921 0.6001 0.6087

*Adapted from a table in the NBS publication by Harner."

former set of units applies to I in mol kg" whereas the latter set applies to I in . mol dm ", With these values for A and B, Eq. 2.94 becomes the Davies : equation: II log l' = - Az2

(I :1/;1/2 - 0.31)

(2.96)

Equation 2.96 may be used in mixed electrolyte solutions for I < 0.5 mol kg", For the calculation of l' of a neutral solute (e.g., a sugar or a neutral complex), B is a small number known as a salting coefficient. Equation 2.94 then becomes the Setchenow-Harned-Owen equation: (2.97) where k m == B is the salting cocflicient. Compiled values of k m for a variety of compounds at T • 298.15 K arc nvailablc." It is not an unreasonable approx-

I

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

61

imation to set k m = 0.1 mol kg-I, or even k m = 0.0 mol kg-I, in many soil solutions where I < 0.1 mol kg-I. Before leaving the subject of single-ion activity coefficients, it is important to consider the relation between the Reference State and the Standard State for an ionic solute and to check the mutual compatibilities of Eqs. 2.46, 2.47, 2.89, and 2.92. First, consider the Standard State and, to make the discussion concrete, a solution containing the electrolyte CaCI 2• Now, in general, ~(CaCllaq)) = ~O(CaCllaq))

+ RT In (CaCI 2)

In the Standard State (CaCI 2) = 1.0 and the salt is at unit mean ionic molality. According to Eq. 2.89, these conditions imply that 'Y± = 1.0 kg mor ' in the Standard State. Thus the Standard State exhibits the same "ideality", or dilutesolution property, as does the Reference State, except that m~ = 1.0 mol kg- 1 in the Standard State and m± ~ 0 mol kg-I in the Reference State. It is this important difference that makes the Standard State a hypothetical one while the Reference State remains an actual (limiting) one. In the Standard State for the Ca H and cr ions in the solution of CaCI 2, the single-ion activities and molalities each have unit numerical value. According to Eq. 2.91, these conditions imply that 'Y+ = 'Y- = 1 kg mor ' in the Standard State. Thus, in Table 2.1, the stipulation of "no ionic interactions" in the Standard State of an ionic solute may be taken to mean unit numerical value for the single-ion activity coefficients. In this sense, the Reference State and the Standard State for ionic solutes also share the dilute-solution property. But, once again, they are not the same state, because m+ and m.. tend to zero in one and have unit value in the other. Now consider Eq. 2.46b as applied to CaCllaq): (2.98) In the Standard State, m~ = I mol kg-I, which means that the molality of CaCl 2 is 4- 1/ 3 mol kg- I (see Table 2.3). It follows from this fact and the stipulated complete ionic dissociation in the Standard State that the molality of Ca H is also 4- 1/ 3 mol kg- I and that of Cl- is 2 1/ 3 mol kg-I. Thus the individual ions are not at Standard State molality, and it would seem at first that Eq. 2.98 cannot be correct. However, imagine the molality of Ca H to increase from 4 1/3 mol kg-I to 1 mol kg", and the molality of Cl- to decrease from 2 1/ 3 mol kg I to I mol kg-I, while the activity coefficientsof these two ions maintain their Standard State values of 1.0 kg mol'. For this process ~(CaH) will increase by RT In 4 1/ 3 and ~(Cl-) will decrease by - RT In 21/ 3• The net change in the chemical potential of CaCl 2 to bring Ca H and Cl" into the Standard State is then RT In 4 1/ 3 - 2RT In 2 1/ 3 = O. Therefore Eq. 2.98 is indeed perfectly compatible with the defined Standard States of neutral electrolytes and ionic solutes. This compatibility depends critically on the hypothetical nature of the Standard State (i.e.. the property of maintaining unit activity coefficients regardless of molality).

62

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

TABLE 2.5. Reference States employed in thermodynamics for mixtures Substance

Reference T

Reference P

Gas Liquid Solid Solute (nonaqueous solution) Solute (aq, neutral)

298.15 K 298.15 K 298.15 K 298.15 K

r, ~ 0

X = f/NP r-+ 1.0

1 atm 1 atm 1 atm

N N N

298.15 K

1 atm

Solute (aq, electrolyte)

298.15 K

1 atm

Solute (aq, ionic)

298.15 K

1 atm

0 mol kg- 1 1-+ 0 mol kg- 1 m s -+ 0 mol kg- 1 1-+ 0 mol kg-I m+, m : -+ 0 mol kg- 1 1-+ 0 mol kg"

• N = mole fraction; m = total molality; m± molalities; and I = ionic strength.

=

Special Conditions"

m

1.0 1.0 1.0

-+ -+ -+

-+

mean ionic molality; m

i

,

m:

= actual ionic

The Reference States discussed in this section are summarized for convenience in Table 2.5.

2.7. CONDITIONAL EQUILIBRIUM CONSTANTS

A conditional equilibrium constant is obtained from a thermodynamic equilibrium constant by substituting a pressure or a concentration variable for one or more of the activities. For example, in the case of the solubility product constant for calcite, given in Eq. 2.50b, the conditional solubility product constant would be: (2.99) where [ ] refers to a molar concentration. According to Eqs. 2.50b, 2.91, and} 2.96, . log c s; = log

s; + 8A

(1 :/~1/2 - 0.3/)

which shows that °Kso increases with ionic strength (for 1 < 0.5M). (A is evaluated in dm3/2mol-1/2.) Thus the conditional solubility product constant depends .•~. on T, P, and the aqueous solution composition as reflected in I. ' In the case of the stability constant for the soluble complex, CVCA~A' given, in Eq. 2.54, the conditional stability constant would be

-«,

=

[CVCA~AJI[cm+]v'[A'-]'A

(2.100)

where, by Eqs. 2.91 and 2.94,

log "K, - log K, - (vem2

+ vAn2 -

APl2

q2)

1+ /'/ 2 + (v(' +

VA -

I)B/ (2.101)

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

63

and A, E, and 1 are evaluated on the molar concentration basis. Whether K, increases or decreases with ionic strength depends on the magnitudes of m, n, and q. For example, if the complex formed is neutral, q = 0, vAm = Ven and Eq. 2.101 reduces to C

log "K, = log K, - vmnA

[1/2

(

1+

/

12 -

0.31

)

- kml

where v = Ve + VA' In this instance, K, will decrease with an increase in ionic strength. For the Na-Ca cation exchange reaction described by Eq. 2.71 b, the conditional exchange equilibrium constant may be expressed C

CKex = N Ca(Na +)2/MNa (Ca H

(2.102)

)

where N ea is the mole fraction of CaXls) in the exchanger phase and NNa is that of NaX(s). In this example, only the activities in the exchanger phase have been replaced by concentration variables (those appropriate for a solid solution, according to Eq. 2.78). By Eqs. 2.71b and 2.78, log Kex = log Kex C

+ 2 log fNa

(2.103)

- log fea

The dependence of Kex on the exchanger phase composition requires the evaluation of the two activity coefficients, fNa and fea. This calculation will be described in Section 5.4. The conditional equilibrium constant Kex often is denoted by the symbol K; and is called the Vanselow selectivity coefficient. Conditional equilibrium constants may be developed in a variety of ways, even for the same chemical reaction. The only criteria for defining them are that at least one activity in the thermodynamic equilibrium constant be replaced by a concentration variable and that the conditional constant be related to the equilibrium constant through activity coefficientsdefined on the mole fraction, molal, or molar concentration scales. C

C

NOTES I. Methods for measuring Standard State chemical potentials are surveyed by M. L. McGlashan, The scope of chemical thermodynamics, Chemical Thermodynamics, Spec. Periodical Rpt. 1:1.'0, The Chemical Society, London, 1973. The volumes of Specialist Periodical Report in the series on chemical thermodynamics provide excellent discussions of the methods employed to determine /10 for gases, liquids, and nonelectrolyte solutions. A discussion of methods for solids is given by W. E. Brown, Solubilities of phosphates and other sparingly soluble compounds, pp. 2032.W in Environmental Phosphorous Handbook, ed. by E. J. Griffith et aI., John Wiley, New York, 1973. Chapter 2 in C. F. Baes and R. E. Mesmer, The Hydrolysis of Cations, John Wiley, 1976, ulso contains information and references relating to methods for solids. For methods pertaining to electrolytes and individual ionic species in aqueous solution, Chapter 8 in R. A. Robinson and R. II. Stokes, Electrolyte Solutions, Butterworths, London, 1970, may be consulted. See as well the chapters in G. N. Lewis and M. Randall (op. cit.). } lor II good introductory discussion of some of the problems involved, sec J. A. Kittrick, Mineral equilibrin and the soil system. pp. I 2~ in Mlnl'ral.f In Soll Environments, cd. by .I. B. Dixon et III., Soil Science Society of Amerlc«, MUdison, Wis., 1977. Especially good working discussions

64

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

are to be found in B. S. Hemingway and R. A. Robie, The entropy and Gibbs free energy of formation of the aluminum ion, Geochim. et Cosmochim. Acta 41:1402-1404 (1977) and in J. A. Kittrick, Solubility product of Belle Fourche and Colony montmorillonites in acid aqueous solutions, Soil Sci. Soc. Amer. J. 42:524-528 (1978). 3. A compilation of IJ.0 values for smectite minerals may be found in S. V. Mattigod and G. Sposito, Improved method for estimating the standard free energies of formation (IlGJ.'98.15) of smectites, Geochim. et Cosmochim. Acta 42:1753-1762 (1978). Data on vermiculites are in J. O. Nriagu, Thermochemical approximations for clay minerals, Am. Mineralogist 60:834-839 (1975). 4. E. F. G. Herington, Thermodynamic quantities, thermodynamic data, and their uses, Chemical Thermodynamics, Spec. Periodical Rpt, 1:31-94, The Chemical Society, London, 1973. 5. Y. Tardy and R. M. Garrels, Prediction of Gibbs energies of formation. I. Relationships among Gibbs energies of formation of hydroxides, oxides and aqueous ions, Geochim. et Cosmochim. Acta 40: I051-1 056 (1976). The empirical parameter m used in the paper by Tardy and Garrels (p. 1054) is equal to (a - 1)/2, where a is defined in Eq. 2.61 in the present section. 6. E. Nieboer and W. A. McBryde, Free-energy relationships in coordination chemistry. III. A comprehensive index to complex stability, Can. J. Chem. 51:2511-2524 (1973). 7. S. V. Mattigod and G. Sposito, Estimated association constants for some complexes of trace metals with inorganic ligands, Soil Sci. Soc. Amer. J. 41:1092-1097 (1977). 8. J. Veith and G. Sposito, Reactions of aluminosilicates, aluminum hydrous oxides, and aluminum oxide with o-phosphate: The formation of X-ray amorphous analogs of variscite and montebrasite, Soil Sci. Soc. Amer. J. 41:870-876 (1977). 9. F. Lippmann, The solubility products of complex minerals, mixed crystals, and three-layer clay minerals, N. Jb. Miner. Abh. 3:243-263 (1977). 10. For discussions and applications of this equation, see K. S. Pitzer, Electrolyte theoryImprovements since Debye and Huckel, Acc. Chem. Research 10:371-377 (1977); K. S. Pitzer, Theory: Ion interaction approach, in Activity Coefficients in Electrolyte Solutions. ed. by R. M. Pytkowicz, Vol. I, pp. 157-208, CRC Press, Boca Raton, Fla., 1979; and C. E. Harvie and J. H. Weare, The prediction of mineral solubilities in natural waters: The Na-K-Mg-Ca-CI-SO.-H,O system from zero to high concentration at 25°C, Geochim. et Cosmochim. Acta 44:981-997 (1980).

I

II. C. W. Davies, Ion Association. Butterworths, London, 1962. A discussion of alternatives to Eq. 2.96 is in W. J. Hamer, Theoretical Mean Activity Coefficients of Strong Electrolytes in Aqueous Solutions from 0 to 100"C, NSRDS-NBS 24, U.S. Government Printing Office, Washington, D.C., 1968. 12. H. S. Harned and B. B. Owen, The Physical Chemistry of Electrolytic Solutions, Reinhold Pub!. Corp., New York, 1958, pp. 736-737.

FOR FURTHER READING R. M. Garrels and C. L. Christ, Solutions, Minerals, and Equilibria, Freeman, Cooper and Co., San Francisco, 1965. Chapter 2 of this excellent geochemistry textbook covers the definitions of Standard States and Reference States with numerous examples. E. A. Guggenheim and R. H. Stokes, Equilibrium Properties of Aqueous Solutions of Single Strong Electrolytes, Pergamon Press, Oxford, England, 1969. This small book provides a fine introduction to the thermodynamic theory of electrolyte solutions. J. A. Kittrick, "Mineral Equilibria and the Soil System," pp. 1··25 in Mlnerals in Soil

I

THE CHEMICAL THERMODYNAMICS OF SOIL SOLUTIONS

65

Environments, ed. by J. B. Dixon and S. B. Weed, Soil Science of America, Madison, Wis., 1977. This article presents a good introduction to the thermodynamic theory of solid phases as they form in soils. G. N. Lewis and M. Randall, Thermodynamics, rev. by K. S. Pitzer and L. Brewer, McGraw-Hill, New York, 1961. Chapters 20 to 22 of this classic text present an extensive discussion of the methods by which activities are measured. J. E. Prue, Ionic Equilibria, Pergamon Press, Oxford, England, 1966. Another excellent little book that tells all about the experimental techniques for measuring activities and equilibrium constants in aqueous electrolyte solutions. R. M. Pytkowicz, Activity Coefficients in Electrolyte Solutions, CRC Press, Boca Raton, Fla., 1979. This two-volume book contains reviews of all aspects of the physical chemistry of electrolyte solutions at equilibrium. Each chapter is an advanced summary written by a recognized expert. R. A. Robinson and R. H. Stokes, Electrolyte Solutions, 2nd ed., rev., Butterworths, London, 1965. This book is accepted as the modern standard introduction to the physical chemistry of ionic solutions. Carefully selected data relating to the activity coefficients of aqueous electrolytes are given in about 100 pages of appendices.

3 SOLUBILITY EQUILIBRIA IN SOIL SOLUTIONS

3.1. SOLID PHASES AND THE ACTIVITIES OF SOIL SOLUTION SPECIES

The general precipitation-dissolution reaction for a two-component, dissociable solid, in contact with an aqueous solution, may be expressed (3.1) where the ionic valences and stoichiometric coefficients are subject to the electroneutrality condition, vIm = V2n. The equilibrium constant for this reaction is (3.2) which is the inverse of Kr. defined for the special case of CaCOls) in Eq. 2.50a. The ion activity product, lAP, defined by Eq. 2.48a, is related to K dis through the expression lAP = Kdis(Cv,Av2(s»

(3.3)

When the activity of the solid phase has unit value, lAP = K so, the solubility product constant, which was defined in Eq. 2.50b for CaCOls). If the solid ' phase is not in its Standard State, lAP will be a function of all the thermodynamic variables that affect the value of the activity of the solid (the mole numbers of coprecipitated solids, etc.). According to Eqs. 2.37 and 2.42, the value of log K dis may be calculated with the help of the equation log Kdis = (l/5.708)[JLO( C ,A (s» - V,JLO( C m+) - v2JLO(A'-)] V

V2

(3.4)'

at T = 298.15 K. The application of Eq. 3.4 requires precise data on the Standard State chemical potentials involved. Alternatively, by measurements of the activities of the aqueous solution species in a solubility experiment, the value of lAP may be determined directly. If it can be ascertained with certainty that a solid substance has produced the observed activities in the aqueous solution phase and that the solid was in its Standard State, the measured lAP = KIO• These considerations can be extended quite readily to solids that comprise three 66

SOLUBILITY EQUILIBRIA IN SOIL SOLUTIONS

67

or more components. This extension was illustrated already in Section 2.5 with the examples of gypsum and variscite. One of the central problems in the thermodynamic analysis of soil solutions is the identification of the solid phase which controls the observed aqueous solution concentration of a cation or anion of interest. The development of methods for solving this problem is the principal theme of this chapter. To begin the discussion, consider the following simple question. Under given conditions of T, P, pH, and aqueous solution concentrations of the relevant ions, how does one predict which of a variety of possible solid phases will control the soil solution concentration of a chosen ion? This question can be answered immediately by saying that the controlling solid phase will be the one that results in the smallest value of the aqueous solution activity of the ion. This is true because the chemical potential of the ion in an aqueous phase will be smallest wherever the activity of the ion is least. The tendency of the ion, in case several solids containing it are present initially, will be to diffuse ultimately to the region of the aqueous phase where its chemical potential will be least. Therefore the solid phase capable of producing the smallest soil solution activity of an ion also will be the most stable repository of that ion in a precipitate. As an example of these ideas, suppose that Cd has been added to a soil as part of some contaminating waste material and that one wishes to deduce whether Cd(OH)z(s) or CdCOls) will control the activity of Cd H in the soil solution. The available data indicate that the pH value of the soil solution is 7.6 and that the bicarbonate activity is 10- 3• For the reaction Cd(OH)2(s)

=

Cd H

+ 20H= 1 atm. The reverse of the ioni-

log K so = -14.39 at T = 298.15 K and P zation reaction for liquid water may be added twice to this reaction to produce the results: Cd(OH)is) = Cd H + 20H= -14.39 log K so 2H+ + 20H- = 2H 20(1) -2 log K w = 28.0 log *K so = 13.61 (The notation *Kso commonly is used to denote the equilibrium constant for the dissolution reaction of an hydroxide solid as obtained through adding algebraically the ionization reaction of water.) Under the assumption that both Cd(OH)z(s) and H 20(1) are in their Standard States (the latter because the soil solution is dilute), the activity of Cd H is predicted to be log (Cd H

)

= log

«s; +

2 log (H+) = log *Kso

-

2pH

(3.5a)

since *Kso = (Cd H)/(H+)2. Under the given condition of pH, log (Cd H . 1.59, if Cd(OH)z(s) is the controlling solid phase. For the carbonate solid phase, the dissolution reaction is: CdCO (s) - Cd!" + C0 2log K = -11.2 3

J

oo

)

-

SOLUBILITY EQUILIBRIA IN SOIL SOLUTIONS

68

The formation reaction for the ion-pair HC0 3(aq) must be added if the data given are to be employed: CdCOls) = Cd H + H+ + CO~- = HC0 3

CO~

log K", = -11.2 log K 2 = 10.33 -0.87 /ogK =

In this case, log (Cd H

)

= log K

- pH - log (HCOJ)

(3.6a)

and log (Cd H ) = -5.47, for the given pH value and bicarbonate activity, if :i CdC0ls) is the controlling solid phase. It is evident at once from this calculation ~,~ that, in fact, CdCOls) would be the controlling solid phase, since the Cd H il activity in the soil solution is about four orders of magnitude smaller for the carbonate than for the hydroxide. Of course, the conclusion just drawn rests on the very important assumption that the two solids and liquid water in the soil are in their Standard States. If there is good reason to question this assumption, Eqs. 3.5a and 3.6a must be generalized, respectively, to the expressions: log (Cd H log (Cd H

)

-

)

-

13.61 - 2pH -0.87 + pH

+ log (H 20(l)) + log (Cd(OH)2(s)) + log (HCOJ) + log (CdC0 3(s))

(3.5b) (3.6b)

Often the kind of calculation just illustrated is reversed through an experimental measurement of an ion activity in a soil solution and a comparison of this empirical result, combined with relevant data on the activities of other aqueous ions, to a chosen value of KSO" For example, one could determine (Cd H ) in a soil solution with a cadmium-selective electrode, then form the lAP with a calculated value of (CO~-), and compare the resulting product with 10- 11.2 = K so• If the lAP =/= 10- 11.2 and the activity data are accurate, the conclusion to be drawn is one or more of the following statements. 1. Equilibrium with a solid phase does not exist. 2. The solid phase controlling the ion activity is not the one suspected. 3. The solid phase controlling the ion activity is the one suspected, but ( the solid is not in its Standard State. Just which conclusion is correct must be decided on the basis of additional experimentation. As an example of data related to conclusion (1), Fig. 3.1 shows the time variation of the lAP for the dissolution of gibbsite:' Al(OH)ls) = Al H

+ 30H-

with log K", = - 33.90 ± 0.33 at 298.15 K. The data in Fig. 3.1 refer to the common logarithm of the lAP = (Al H)(OH-)3 determined in a leachate from an Oxisol (Molokai soil) once each day after an elution experiment was begun. The gradual approach of the lAP to K is apparent. Evidently, after 40 days of tI ,

SOLUBILITY EQUILIBRIA IN SOIL SOLUTIONS

69

log K so=-34

-o -----

1'0

I

- --- - ----- ----

---.,..~...---=--.

----0' o

-30 '--_ _----''--_ _----''-o 240

L-

-'--

-'--_

1200

960

elution, the soil solution was in equilibrium with gibbsite in the soil. (The presence of gibbsite was established independently by X-ray diffraction analysis.) An example of data related to conclusion (3) may be found in a set of extensive measurements' of the lAP for the dissolution of CaC0 3(s); that is, lAP = (Ca2+)(COi-). In this case, the value of log lAP for 63 water samples taken from wells pumping aquifers in Arizona and Colorado that contained CaCOls) was found to be -7.96 + 0.01 at 298.15 K. This result is larger than log Kso = -8.30 + 0.32, which was calculated in Section 2.5. Since the solid phase CaC03 in the aquifers was shown to be stable and coprecipitation phenomena could be ruled out, it was possible to suggest that the controlling solid probably has a much smaller particle size than does thermodynamically stable calcite. Indeed, according to Eq. 3.3, (CaCOb)) = IAP/ Kdis =

1O-7.96/l0-8.3

=

10°·34

= 2.2

This enhanced solid-phase activity may be brought on by microcrystallinity in the precipitated soil carbonate.

3.2. PREDOMINANCE DIAGRAMS The kind of solubility information epitomized in Eqs. 3.5 and 3.6 may be represented in a convenient fashion through a predominance diagram. A predominance diagram is a two-dimensional field consisting of well-defined regions whose coordinate points are specified by the pH value and the common loga-

'~ )

70

SOLUBILITY EQUILIBRIA IN SOIL SOLUTIONS

rithm of a second relevant activity variable. The boundary lines that enclose regions in the diagram are specified by equations based on equilibrium constants, such as Eqs. 3.5 and 3.6. In each region of a predominance diagram, either a particular solid phase containing an ion of interest, or the free ion itself in the aqueous solution phase contacting the solid, will be predominant. Thus a predominance diagram gives information about changing relative stabilities, at equilibrium, among the solid phases formed by an ion as the pH value and one other controlling activity variable are altered in a soil solution. The construction of predominance diagrams is straightforward if the relevant thermodynamic data are available. Consider, for example, the system ZnCO 2-H 20, which has a bearing on the solubility of zinc in soils. The chemical species of interest in this system are Zn2+(aq) and the solid phases Zn(OH)2' ZnO, ZnC0 3, and Zn s(OHMC0 3h In the case of Zn(OH)2' a number of polymorphs has been identified, the most stable one thermodynamically being fZn(OH)2' However, the dehydration reaction f-Zn(OH)is)

=

ZnO(s)

+ H 20(l)

has a standard free energy change of -2.05 kJ mol", which means that all forms of Zn(OH)is) are unstable relative to ZnO(s), as long as the solid phases and liquid water remain in their Standard States. Therefore Zn(OH)2 may be excluded formally from the list of solids to be considered. This exclusion would have to be reconsidered if data for a soil under study indicated, for example, that X-ray amorphous or crystalline Zn(OH)2 were actually present as a metastable phase. In the present example, the second activity variable that, along with the pH value, defines the regions of the predominance diagram could be log (COi-). But an equivalent and more commonly employed variable is log Pco, the common logarithm of the partial pressure of COig) in the system. This variable is \ especially convenient because natural soils exhibit a range of CO 2 pressures. In terms of COig), the dissolution reaction of ZnCOls) may be represented by the equation ZnC0 3(s) + 2H+ = Zn2+ + COig) + H 20(l) for which log K = 7.95 at 298.15 K. The analog of Eq. 3.6b for this reaction is log (Zn2+) = 7.95 - 2pH - log Pco, - log (H 20(l)) + log (ZnCOb))

(3.7)

where it has been assumed that the activity of COig) is equal numerically to its partial pressure. In the predominance diagram, only log Pco, and pH are variables. Therefore all the other activity terms in Eq. 3.7 must be specified and held constant. It will be assumed that H 20(l) and ZnC0 3(s) are in their Standard States and that (Zn2+(aq» = 10- 6 , a representative value for a soil solution. Then Eq. 3.7 simplifies to

log I'('(), • 13.95 - 2pH

(3.8)

SOLUBILITY EQUILIBRIA IN SOIL SOLUTIONS

71

O~-----------.----------,

-I

Soil air

ZHC

Atmosphere

Zn 2+(aq)

ZnO

-5

_6l--

....l..-

4

5

...l...-

----L

7

6

- - ' -_ _...l...------J

8

9

pH FIGURE 3.2. Predominance diagram for the system Zn-C0 2-H20 under the condition (Zn H ) = 10- 6 •

Equation 3.8 specifies the boundary between the region of stability of Zn2+(aq) (at the chosen, fixed activity) and ZnCOls). This boundary is shown in the upper part of Fig. 3.2. Note that the boundary would have been shifted to the left (right) if a larger (smaller) Zn2+(aq) activity had been chosen. The dissolution reaction of Zn s(OHMC0 3Ms) (which is abbreviated as ZHC) may be represented by the reaction Zn(OH) 12(COl )04(S)

+

2H I - Zn!"

+ 0.4 CO2(g) +

1.6H 20(1)

SOLUBILITY EQUILIBRIA IN SOIL SOLUTIONS

72

for which log K = 9.80 at 298.15 K. The corresponding relation between log Peo, and pH is (3.9)

log peo, = 39.50 - 5pH

where all the condensed phases are assumed to be in their Standard States and the activity of ZnH(aq) is set at 10- 6 • Equation 3.9 defines the boundary between the region of stability of ZnH(aq) and ZHC. The boundary between the two carbonate solid phases, ZnC0 3 and ZHC, is found by considering the reaction ZnCOls)

+ 0.6H 20(l) =

Zn(OH) u(C0 3)04(S)

+ 0.6C0 2(g)

for which log K = 7.95 - 9.80 = -1.85 at 298.15 K. In this case, H+ does not appear in the reaction and log K = 0.6 log Pco, if all the condensed phases are in their Standard States. Therefore the boundary line between the two solid phases is the horizontal line in Fig. 3.2 along which log Pco, = -1.85/0.6 = -3.08. The dissolution reaction of ZnO may be represented by ZnO(s) + 2H+ = Zn H + H 20(l) for which log *Kso = 11.14 at 298.15 K. Here COlg) does not enter the reaction and, therefore, the boundary line between the region of stability of ZnH(aq) and ZnO(s) is given by the vertical line in Fig. 3.2, corresponding to pH = 0.5[log *Kso - log (Zn H)] = 8.57. The boundary line between the regions of stability for ZHC and ZnO(s) may be calculated after considering the reaction Zn(OH) dC0 3)04(S) = ZnO(s)

, "

+ 0.4 COlg) + 0.6 H 20(l)

for which log K = 9.80 - 11.14 = -1.34 at 298.15 K. In this case, the boundary line is log Pco, = 2.5 log K = - 3.35, which appears on the lower right side in Fig. 3.2. The predominance diagram for the Zn-C02-H 20 system at 298.15 K indicates that, when the activity of ZnH(aq) is fixed at 10- 6, thermodynamically stable, solid oxide or carbonate phases will not form until the soil solution pH value increases to about 8.0. If the soil solution is saturated with ordinary atmospheric air (log Pco, = - 3.52), the thermodynamically stable solid phase is ZnO. This particular conclusion would be different if, for example, amorphous Zn(OH)2 [log K = 12.45 for Zn(OH)z(am) + 2H+ = Zn H + 2H 20(l)] had been chosen as a kinetically favored, metastable solid phase instead of the more stable ZnO. In that case the lower boundary of stability for ZHC would have been the horizontal line log Peo, = -6.7, and ZHC would have been the stable phase in equilibrium with CO 2(g) in the earth's atmosphere. For soil air, perhaps a representative value of log Pco, is - 2.5, although values at least one logarithmic unit higher have been observed in soils at depth. In this case, ZnC0 3 is predicted to be the stable solid phase at equilibrium above pH - 8.2. If this

I

I

SOLUBILITY EQUILIBRIA IN SOIL SOLUTIONS

73

solid phase actually formed, the relation between the activity of ZnH(aq) and that of the proton would be (ZnH) = 107.9S(ZnCOb)) (H+)2 Pco2(H 20(l))

(3.10)

according to Eq. 3.7. If all the condensed phases were in their Standard States and log PC0 2 = -2.5, the coefficient of (H+)2 in Eq. 3.10 would equal 1010 4s. Unfortunately, an observed linear relationship between (Zn H) and (H+)2 cannot be used to determine which solid phase in the Zn-C0 2-H20 system could control the activity of ZnH(aq) because such a linear relationship obtains for all three solid phases considered in the present example. Throughout this discussion, it has been assumed that the activity of liquid water and that of all solid phases has unit value. For natural soils, this assumption certainly can be questioned, since both high solute concentrations in the soil solution and coprecipitation phenomena are common. Coprecipitation phenomena will be discussed in Section 3.7. In the remainder of the present section, the effect of the water activity on predominance diagrams will be considered. The solids in the system AI-K-Si-H 20 offer a good example of the point in question. The relevant solid phases may be chosen to be gibbsite, Al(OH)3' kaolinite, AI2Si20s(OH)4, and muscovite, KAI2(Si3AIO IO)(OHh The chemical reactions that relate these solid phases to one another are: AI2Si20s(OH)4(s) + 5H 20(l) = 2Al(OH)b) + 2Si(OH)~(aq) (3.lla) KAllSi 3AIO IO)(OH)ls) + 9H 20(l) + H+ = 3Al(OH)3(s) + 3Si(OH)~(aq) + K+ (3.llb) 2KAllSi 3AIO IO)(OH)is) + 3H 20(l) + 2H+ = 3AI2Si20s(OH)4(s) + 2K+ (3.1lc) An inspection of these three equations indicates that two possible activity variables-(Si(OH)~) and (K+)-can be employed along with the pH value to construct a predominance diagram. In this example, log (Si(OH)~) will be chosen as the additional variable, with (K+) = 10- 3, a representative value for soil solutions. The common logarithms of the equilibrium constants for the reactions in Eq. 3.11 are equal to -10.39, -11.17, and 8.81, respectively, at 298.15 K. The expressions for the boundary lines in the predominance diagram then may be computed using the three different equations for log K and the single, imposed K+(aq) activity. The results of the computation are: log (Si(OH)~) = -5.20 + 2.5 log (H 20)(l)) log (Si(OH)~) = -2.72 - jpH + 3 log (H 20(l)) 7.40 = pH - 1.5 log (H 20(l))

(kaolinite-gibbsite) (muscovite-gibbsite) (muscovite-kaolinite)

where each solid phase is taken to be in its Standard State. These boundary lines arc plotted in Fig. 3.3a for the case when (H 20(l)) = 1.0 and in Fig. 3.3b for the case when (H 20(1)) - 0.5 (i.e., when the relative humidity over the soil

SOLUBILITY EQUILIBRIA IN SOIL SOLUTIONS

74

-4r--------------,----------,

-5 -

Muscovite

Kaolinite

~

I

Q

I

~ - 6 f : : - - - - - - - - - - - -................ ~

'I l

1

d

:1 Gibbsite

-7 -

(a) I

I

I

I

5

6

7

8

9

pH

-4r----------------,----------,

Kaolinite

Muscovite

-5 ~

0<;1"

I

Q i:7i -6 r0' o Gibbsite

-7r-

(b)

-8 '--4

-

I

--'--

5

-

-

I

----'--

-

-

I --1.-

7

6

-

-

I --'-

8

-

-

---'

9

pH

FIGURE 3.3. Predominance diagram for the system AI-K-Si-HlO under the condition (K+) - 10- 3• (a) (HlO(l» - 1.0. (b) (HlO(I» - 0.5.

," ,!

SOLUBILITY EQUILIBRIA IN SOIL SOLUTIONS

75

solution is 50%). The decrease in the size of the stability field for gibbsite and the corresponding increase in those for kaolinite and muscovite as the activity of water decreases are quite apparent in the figure. In particular, it seems that alkaline soils with a relatively low water activity and (Si(OH)~) :::::: 10- 6 generally would contain muscovite as the thermodynamically stable solid phase instead of gibbsite.'

3.3. THE PRINCIPLE OF HARD AND SOFT ACIDS AND BASES The thermodynamic equilibrium constant for the dissolution of a solid substance in water conventionally is expressed in terms of the activities of ions, as in Eq. 3.2. For certain ions, conventions exist that make it possible to measure the activity directly by means of an ion-selective electrode that has been standardized through an accepted calibration technique. But, for most elements that can become ionic solutes, the activity must be calculated after correcting the total concentration of the element for soluble complex formation. In a typical soil solution, which may contain about 30 different metals and ligands at significant concentrations, approximately 300 soluble complexes can be present. Thus the estimation of the degee of complexation of an element can be a complicated matter, and some kind of general overview concerning complex formation reactions is useful. One approach that is simple and directly applicable to soil solutions is the Principle of Hard and Soft Lewis Acids and Bases (HSAB).4 Before discussing the HSAB Principle, it is important to consider the nature of soluble complexes themselves. These chemical compounds may be pictured as adducts (addition products) of Lewis acids and bases. A Lewis acid is any chemical species that employs an empty electronic orbital in initiating a reaction, while a Lewis base is any chemical species that employs a doubly occupied electronic orbital in initiating a reaction. Lewis acids and bases can be neutral molecules, simple or complex ions, and neutral or charged macromolecules. The electronic orbitals referred to can be molecular orbitals, bands, or atomic orbitals, depending on the chemical structure of the Lewis acid or base. The passage "in initiating a reaction" emphasizes the point that complex formation is brought on by, but may not involveonly, the empty and doubly occupied orbitals that differentiate Lewis acids from Lewis bases. The proton and all of the metal cations of interest in soil solutions are Lewis acids. The Lewis bases include H 20 , oxyanions such as OH-, COO-, CO~-, SO~-, and PO", and organic N, S, and P electron donors. Some common examples of Lewis acid-base reactions are the hydration of a proton or a metal cation, hydrolysis reactions, the protonation of an oxyanion, and the coordination of a transition metal cation with an organic functional group containing nitrogen. In a thermodynamic context, the formation of a complex in aqueous solution is described fully by Eq. 2.54. However, it is helpful to add to this description a classification of complexes according to their molecular structural features. If a complex forms between a Lewis acid and base, such that one or more

76

SOLUBILITY EQUILIBRIA IN SOIL SOLUTIONS

H 20 molecules is interposed to separate the acid unit from the base unit, the adduct is called an outer-sphere complex. Outer-sphere complexes often are called "ion-pairs" because the principal bonding mechanism in this case necessarily must involve electrostatic interactions acting over 0.1 nm or so if the Lewis acid and base involved are charged species. However, the term "ion-pair" also is employed to denote any complex that comprises two ions, with "ion-triple" applied to three-ion complexes, and so forth, without regard to the structural role of water molecules. For this reason, the term outer-sphere complex is to be preferred over ion-pair. If a complex containing a Lewis acid and a Lewis base forms without the interposition of water molecules between the coordinated units, it is called an inner-sphere complex. Generally, inner-sphere complexes involve a greater degree of covalency in the acid-base bonding mechanism and exhibit larger values of the stability constant than do outer-sphere complexes. A hard Lewis acid is a molecular unit of relatively small size, high oxidation state, high electronegativity, and low polarizability. (The polarizability is a measure of the ease with which the electronic orbitals deform in the presence of an electric field.) It tends also to possess outer electrons that are relatively difficult to excite to higher energies. A soft Lewis acid on the other hand, is of relatively large size, low oxidation state, low electronegativity, and high polarizability. It " tends to possess easily excitable outer electrons that often are d-orbital electrons. :1 Examples of hard Lewis acids are the proton and the cations of the metals Na, K, Mg, and Ca. Examples of soft Lewis acids are Cd H , Cu ", Hg ", and pi-ij acceptors such as quinones. A list of hard, borderline, and soft Lewis acids that' are important in soil solutions is given in Table 3.1. The borderline Lewis acids include most of the significant bivalent trace metal cations. Their existence is an I:

TABLE 3.1. A catalog of hard and soft Lewis acids and bases in soil solutions

LEWIS ACIDS Hard Acids H +, u: , Na+ " K+ (Rb+ , Cs+) , Mg H , Ca H , Sr H , (BaH) , Ti H , ZrH , CrH , Cr6+, MoOJ+, Mn H, Mn7+, Fe H , Co H , AlJ+, Si H , CO 2 Borderline Acids Fe H, Co H, NiH, Cu H, Zn H, (Pb H) Soft Acids Cu", Ag ", Au ", Cd H, Hg ' , HgH, ChJHg+; pi-acceptors such as qui nones; bulk metals LEWIS BASES Hard Bases NH J, RNH 2, H 20, OH-, 0 2 - , ROH, CHJCOO-, CO;-, NO;, PO~-, SO~-, FBorderline Bases C 6H sNH 2, CsHsN, N 2, NO;, SO;-, Br-, (CI-) Soft Bases C 2H., C 6H 6, RJP. (RO)JP, RJAs, R 2S. RSH. SIOl , Sl , I R • organic molecular unit. ( ) indicates II tendency to softncNN.

SOLUBILITY EQUILIBRIA IN SOIL SOLUTIONS

77

TABLE 3.2. Misono softness parameters for metal cations in aqueous solutions Valence +1

+2

+3

Metal ion

Softness

Metal ion

Softness

Metal ion

Softness

Li K Na Rb Cs Cu Ag

0.36 0.92 0.93 2.27 2.73 3.45 3.99

Mg Ca Sr Zn Ba Ni Cu Co Mn Cd Fe Sn Pb Hg

0.87 1.62 2.08 2.34 2.62 2.82 2.89 2.96 3.03 3.04 3.09 3.17 3.58 4.25

Al Fe La Co Cr Ti

0.70 2.37 2.45 2.56 2.70 3.23

indicator of the fact that no sharp dividing line separates hard acids from soft acids. Even within the set of hard acids, for example, there is a spectrum of hardness instead of a single degree of hardness applicable to all members. This point is illustrated in Table 3.2, which lists the Misono softness parameter' for a number of metal cations. The degree of softness of a metal cation is proportional to this parameter. For a given oxidation state, as the cationic radius increases, the polarizability increases and the degree of softness increases. Thus, although Li" and K+ are both classified as hard Lewis acids, K+ is much softer than Li +, as shown by the increasing value of the Misono parameter. Generally, the hard Lewis acids have Misono softness parameters less than 2.8, the borderline Lewis acids between 2.8 and 3.2, and the soft Lewis acids greater than 3.2. A hard Lewis base is a molecular unit of high electronegativity and low polarizability. It tends to be difficult to oxidize and does not possess low-energy, empty electronic orbitals. A soft Lewis base, on the other hand, shows a low electronegativity, a high polarizability, and a proclivity toward oxidation. Examples of hard Lewis bases are, in order of decreasing hardness, H 20, OH-, F-, and NH 3• Examples of soft Lewis bases are R 3P, R 2S, RSH, and 1-, where R denotes an organic molecular unit. A list of hard, borderline, and soft Lewis bases that are of significance in soil solutions is given also in Table 3.1. Note that almost every inorganic anion of concern in soil solutions is a hard Lewis base.

78

SOLUBILITY EQUILIBRIA IN SOIL SOLUTIONS

With many of the important soil solution constituents classified according to their Lewis acidity or basicity, it is possible to state a general, empirical rule regarding their reactions to form complexes. Hard bases prefer to complex hard acids and soft bases prefer to complex soft acids, under comparable conditions of acid-base strength. This rule is known as the HSAB Principle. It can be illustrated in terms of general chemical reactions as follows: A H + AsBH~ AHB H + As B, + AsBH~ AsBs + BH A H + BH~ AHB H As + Bs~ AsBs AHB s + AsBH~ AHB H + AsBs

(3.l2a) (3.l2b) (3.l2c) (3.l2d)

where A H is a hard acid, As is a soft acid, BH is a hard base, B, is a soft base, and the longer arrow denotes the direction in which the reaction is more favorable. The phrase "under comparable conditions of acid-base strength," which appears in the HSAB Principle, may be elucidated by considering two complexation reactions involving the methyl mercury cation, which is a soft Lewis acid. Let As = CH 3Hg+, A H = H+, BH = OH-, and Bs = SO~- in Eqs. 3.12b and 3.12d, respectively: SO~-

HSO;

+ CH + CH 3HgOH

3HgOH

= CH 3HgSO; = HOH

+ OH-

+ CH 3HgSO;

log K log K

= ~

-1 7

In the first reaction, log K is small, in contradiction to Eq. 3.l2b, because OHis so much stronger a base (as measured by its protonation constant) than SO~- that it remains quite stable in a complex with methyl mercury, even though that complex is a soft-acid-hard-base adduct. On the other hand, in the second reaction, the preferred direction in Eq. 3.12d is reflected clearly in the value of log K because the proton and the hydroxide ion are of comparable acid and base strength, respectively. This example emphasizes that both acid-base" strength and hardness-softness determine the magnitude of the equilibrium constant for a Lewis acid-Lewis base reaction. The HSAB Principle can be masked, in a given complexation reaction, by acid-base strength considerations. According to Table 3.1, the metal cations Na ", K+, Mg2+, and Ca2+, which usually are of greatest concentration in neutral and alkaline soil solutions, and the metal cations Fe H and AlH , which are signficant in acidic soil solutions, are all classified as hard Lewis acids. The application of the HSAB Principle then leads to the conclusion that these cations preferentially will form complexes with ligands that contain atoms. In particular, H 20, OH- and the other inorganic oxyanions, and RCOO- should be much preferred relative to CI- , S2-, and most organic electron donors. Because ionic size and charge are the dominant properties of hard Lewis acids and bases, it is expected that their complexes will involve ionic chemical bonds and, therefore, will tend to form outer-sphere com-

°

SOLUBILITY EQUILIBRIA IN SOIL SOLUTIONS

79

plexes. Indeed, since H 20 is one of the hardest Lewis bases, the existence of an outer-sphere complex comprising a hard Lewis acid means simply that, according to the HSAB Principle, H 20 is retained in preference to the other ligand in the complex. On the other hand, if the other ligand is a sufficiently hard Lewis base, the complex formed could be inner-sphere. In that case, the hydration shells that surround the metal cation and the ligand before they react must be disrupted in order for the complex to form. Since the metal-ligand bond is primarily ionic in nature, the energy released when it forms is about the same, in absolute value, as the energy required to disrupt the hydration shells. Thus !:i.IfJ :::::: 0 for inner-sphere, hard acid-hard base complexes. However, !:i.SO will be large and positive because of the hydration shell disruption, with the result that !:i.Go :::::: - T!:i.SO will be strongly negative (i.e., a large stability constant). The bivalent transition metal cations and the cations formed by the metals in Groups IB and lIB of the periodic table tend to be either borderline or soft Lewis acids. One exception is Mn H , which is a hard Lewis acid principally because it possesses no unpaired d electrons. Aside from this exception, the HSAB Principle leads to the conclusion that the bivalent and univalent trace metal cations will tend to form complexes most readily with organic N, P, and S electron donors, cr, and S2-. Because a high polarizability is their dominant property as Lewis acids, these cations are expected to engage in more covalent bonding to ligands and, therefore, to form inner-sphere complexes to a greater extent than the macrocations in the soil solution. In this case, the hydration shells surrounding the metal cation and ligand will be easy to disrupt (because H 20 is a hard species), and !:i.IfJ will be strongly negative because of the energy release upon covalent bond formation in the complex. The value of !:i.SO will not be large, since the hydration shells are more weakly ordered. Thus !:i.Go :::::: !:i.IfJ < 0 for strong complexation reactions between soft acids and bases. These characteristics are illustrated in Fig. 3.4 through two graphs of log K, versus the Misono softness parameter for a set of bivalent metal cations comprising hard, borderline, and soft Lewis acids. In the case of the SO~- ligand, no particular trend with softness is seen. This result comes directly from the proclivity of the hard base SO~- to form ionic bonds in complexes and from the relatively small variation in the ionic potential (valence/ionic radius) among the metal cations. Since the ionic potential does not undergo large changes, the ionic bonds in the sulfate complexes are relatively similar in nature. In the case of the EDTA ligand, the values of log "K, are positively correlated with softness because of the proclivity of EDTA to form more covalent bonds with the metal cations. The HSAB Principle, although empirical, does make some broad statements possible concerning the degree of complexation to be expected for a given metal cation in a soil solution of a given composition. For example, it is clear that the speciation of the trace metals will depend more sensitively on the content and type of organic material present in a soil solution than will the speciation of, for example, Na, K, Ca, or Mg. Moreover, since CI- and S2- tend to behave as relatively soft Lewis bases, an increase in their concentrations will

80

SOLUBILITY EQUILIBRIA IN SOIL SOLUTIONS 25r--------------------,

5-

• Mg I

I

•

Sr

c:u_C.d • •80•t1f4

Co I

2

Zn Ni 1 Fe Co M,n

3

I

4

5

Misono softness parameter

FIGURE 3.4. Plots of log "K, for bivalent metal-EDTA and bivalent metal-sulfate complexes versus the Misono softness parameter for the metal cations.

tend to affect the trace metals more than the alkali metals or the alkaline earth metals. On the other hand, changes in the concentrations of the relatively hard , Lewis bases, CO~-, SO~-, and PO~-, should result in important changes in the speciation of the Group IA and IIA metals. These kinds of consideration can be a helpful guide in the application of strictly theoretical concepts regarding the effect of complex formation on metal solubilities in soil solutions.

3.4. COMPLEX FORMATION AND METAL SOLUBILITY: FUNDAMENTAL CONCEPTS

The basic principles that lead to a calculation of the ion activity product for a solid phase whose components ionize and form complexes in aqueous solution perhaps are understood best by considering a series of concrete examples. Before ' a discussion of these examples is undertaken, it is worth noting that the concept of ion association, although central to the approach taken here, is not essential to the thermodynamic analysis of solubility phenomena. The ion activity product for the solid C.,A., is, according to Eqs. 3.3 and 2.89, (3.13)

SOLUBILITY EQUILIBRIA IN SOIL SOLUTIONS

81

where 'Y ± and m± are given in Eqs. 2.87 and 2.88. The mean ionic molality can be computed directly from the measured molality of C.,A.,(aq), and the mean ionic activity coefficient can be determined by purely thermodynamic methods. Therefore no appeal to an ion-association model is necessary in order to calculate an lAP if a value of 'Y ± is available for the aqueous solution ofinterest. The ion-association model is simply a conceptual framework, with its own set of conventions, that is useful for interpreting the web of interactions among the solutes in a soil solution. Consider now an aqueous solution of Cu(Cl0 4)2 at a concentration of 10- 7 M. The species of copper that occur in this solution have been determined experimentally to be Cu H, CuOH+, Cu(OH)g, CU(OH)3-' Cu(OH)~-, and CU2(OH)~ +. No perchlorate complexes have been identified. With these hydrolytic complexes in mind, the equation of mole balance (i.e., mass conservation) may be written down for copper in solution: CUTS = [Cu H] + [CuOH+] + [Cu(OH)g] + [Cu(OH))] (3.14)

+

[Cu(OH)~-]

+ 2[CU2(OH)~+]

where CUTS is the total molar concentration of copper in solution (i.e., CUTS = 10- 7 M). At this total concentration, if the pH value remains below 8.0, the species Cu(OH)g, CU(OH)3-' and Cu(OH)~- will make a negligible contribution to CUTS' Therefore Eq. 3.14 may be approximated by the expression: (pH < 8.0)

(3.15)

Thermodynamic data now are necessary in order to proceed. The relevant chemical reactions are Cu H + H 20 = CuOH+ + H+ log * K, = -7.9 H log *fJ22 = -10.36 2 Cu + 2H 20 = CU2(OH)~+ + 2H+ where (CuOH+)(H +)j(Cu H)(H 20) *fJ22 = (Cu2(OH)~+)(H +)2j(CU H)2(H 20)2

*K] =

at 298.15 K and 1 atm applied pressure. Because CUTS is low, the conditional stability constants c* K 1 and c*fJ22' which correspond to the two complexation reactions in the sense of Eq. 2.100, may be approximated by 10- 7.9 and by 10- 1036 mol drn", respectively. Then Eq. 3.15 may be expressed

{I

C UTS = [C U2+] = [Cu H]{1

[CuOH+]

2 [Cu 2(OH)i+] }

+ [CuH] + [Cu H] + c*K./[H+] + 2 c*fJdCuH]j[H+]}

(3.16)

The quantity of interest here is the distribution coefficient, alO lEI

[Cu H ] CUTS

(3.l7a)

.•.~

SOLUBILITY EQUILIBRIA IN SOIL SOLUTIONS

82

which, when it is introduced into Eq. 3.16, produces the quadratic equation 1 = aIO{1

+ c*K1/[H+] + 2 c*J322aIOCUTS/[H+P}

or 2 aIO

+

[H+P 1 + c* K1/[H+] 2 c*J322 CUTSj[H+P aIO - 2 c*J322 CUTS = 0

(3.18)

In the present case, the numerical form of Eq. 3.18 is 1 + 10pH- 7.9 I7.06-2pH - 0 2 aIO + 102pH-17.06 aIO - 10 where the approximation [H+] :::::: (H+) (numerically) has been employed also. This expression can be solved readily to obtain a IO and, therefore, to calculate [Cu H] at any desired pH value less than 8.0. The ion activity product, (Cu H)(OH-)2, then could be estimated with a single-ion activity coefficient for CuH(aq), based on the Davies equation, and the activity of OH-(aq), computed from the chosen pH value and K; For completeness, the additional distribution coefficients [CuOH+] [CuOH+] [Cu2+] all = C UTS = [C U2+] C uTS _ 2[Cu 20Hg+] _ 2 c*J322CUTS 2 a22 = CUTS [H+P aIO

(3.17b) (3.17c)

may be defined and related directly to the distribution coefficient aIO' Each distribution coefficient equals the fraction of the total copper concentration that may be attributed to a given complex. This fraction, in the present example, depends on pH and CUTS alone. For example, at pH 5.0, with CUTS = 10- 7 M, Eqs.3.18,3.17b,and3.17cleadtoaIO = 0.9987, all = 1.257 X 1O-3,anda22 = 8.708 X 10- 8• At pH 7.5, the distribution of copper species shifts to aIO = 0.712, all = 0.283, and a22 = 4.43 X 10- 3. Now a solution containing both 10- 7 M Cu(CI0 4)2 and 10- 3 M NaCI will be considered. The species that occur in this solution are all of the copper- . hydroxy species already enumerated plus Cu'Cl", NaOHo, and Na Cl", The complexes containing Na" may be neglected in this example, as may be the copper hydroxy species other than those enumerated in Eq. 3.15. Therefore the only additional chemical reaction to consider is Cu H + Cl- = CuCl+ log K = 0.4 II

The mole balance equations for copper and chlorine in solution are:

+

CUTS = [Cu2+] + [CuOH+] 2[CU20H)~+] + [CuCI+] = [Cu2+]{1 + c* K,j[H+] + 2 c*J322[Cu H]/[H+P + CKI[CI oJ} CI - rCI ] + rCuCI f J - rCI J(I + cK d Cu211} TS

l

(3.19a) (3.20a)

83

SOLUBILITY EQUILIBRIA IN SOIL SOLUTIONS

In this case there are two equations into which the distribution coefficients 0'10 and

must be introduced. The resulting expressions are: I = alO{1 I = alO{1

+ c* Kt/[H+] + 2 c*1322aIOCuTsj[H+P + cKllaotChs} + cKllalOCuTS}

(3.19b) (3.20b)

These equations can be combined to yield a single, third-degree equation for 0'10 in terms of the three stability constants and the concentration variables, [H+], CUTS' and Chs. Once this equation is solved numerically, 0'01 can be calculated directly from Eq. 3.20b, and all and 0'22 can be obtained from Eqs. 3.17b and 3.17c, respectively. The lAP can be calculated with the value for 0'10' as described before. The general procedure in the case of a multielectrolyte solution containing copper and other metals of interest may be developed by induction from the example just considered. 1. Mole balance equations are written down for each metal and each ligand in the solution in terms of the concentrations of all species known from experiment to exist in the solution. 2. Chemical reactions are written for the formation of each complex of interest, and the corresponding thermodynamic stability constants are compiled. 3. The mole balance equations then are rewritten in terms of conditional stability constants, total concentrations of metals and ligands, and free ionic concentrations of metals (including H+) and ligands to provide a set of coupled, nonlinear algebraic equations. 4. The set of coupled equations is solved numerically to obtain the free ionic concentrations or, equivalently, the distribution coefficients for the free ionic species. The free ionic concentrations so calculated may be used in turn to compute ion activity products. This procedure is straightforward in concept but may be quite involved numerically. Usually the set of coupled algebraic equations is too complicated to solve analytically, and approximation procedures carried out with the help of a digital computer are required.

3.5. COMPLEX FORMATION AND METAL SOLUBILITY: APPLICATIONS

Several different computer programs are available that can calculate the free ionic concentrations of metals and ligands in a multicomponent aqueous solution where solids can precipitate." One of them' will be employed to illustrate the

Constituent metal and ligands set of solids

Mr,A r

Mole balances

Mr=[ Mm+] + ~Vc[MvcHyAva(aql] + ~vl [MVIH8Av2(Sl] Ar

=[An -

]

+ ~va [MvcHyAva(aql] + ~V2[Mvl H8Av2(Sl]

Elimination of solids cKso

Elimination of complexes

CK5

Newton - Raphson- Algorithm

Concentration of free ionic species, complexes, and solids

No I

i 'l

Yes

Precipitation dissolution of solids

FIGURE 3.5. Flow diagram for the numerical. iterative solution of thc mole balance cquutions in a soil system.

SOLUBILITY EQUILIBRIA IN SOIL SOLUTIONS

85

computation of solubility equilibria in soil solutions. In calculating the extent of complexation of a metal or ligand in the soil solution, the computer program operates in close accordance with the procedure outlined at the end of Section 3.4 and illustrated in Fig. 3.5. The data input to the program are the total molar concentrations of all metals and ligands, as well as the pH value of the solution. (If desired, the soil solution may be treated as an open system, with the partial pressures of CO 2, O 2, and/or N 2 specified in addition to the total concentrations.) A mole balance equation is set up for each metal and ligand in terms of the molar free ionic concentrations of the proton and all metals and ligands in the solution through the use of conditional stability constants at 298.15 K, as was done, for example, in Eq. 3.16 or 3.19a. The values of the conditional stability constants are estimated with the help of the thermodynamic stability constants stored in the computer program, and single-species activity coefficients are calculated according to Eq. 2.94 with an input guess of the ionic strength. The set of coupled, nonlinear algebraic equations represented by the mole balance equations is solved numerically, using the Newton-Raphson method, to find the free ionic concentrations and to compute the molar ionic strength. The results are substituted into the mole balance equations, to examine them for self-consistency, with the conditional stability constants used now being those estimated at the newly computed ionic strength. The calculation is repeated until self-consistency is achieved within a prearranged tolerance. Then the concentrations of all the complexes and free ionic species are tabulated. As a first example of complexation phenomena in soil solutions, the lAP of CaC0 3 in a saturation extract obtained from Holtville soil [clayey over loamy, montmorillonitic (calcareous), hyperthermic, Typic Torrifluvent] will be calculated. The total molar concentrations of the metals and ligands in the soil solution are listed in Table 3.3. In this case, there are 13 metals (including H+) and 16 ligands (including OH-) that can form 397 known complexes. The 16 ligands include eight organic acids that together constitute a model for the functional groups in the water-soluble organic fraction of the soil. The simultaneous numerical solution of the 27 mole balance equations produces, among other data, the calculated true ionic strength of 0.035M and the molar concentrations of the 20 calcium and 35 carbonate complexes that are shown in Table 3.4. The lAP, (Ca H)( CO~-), in the Holtville soil solution now can be calculated, given the values of the molar concentrations of Ca H and CO~- in Table 3.4 and the singleion activity coefficients, computed at I = 0.035M with the Davies equation. Therefore lAP = Yc.[CaH]yco,[CO~-] = (0.499)(3.96 X 10- 3) (0.499)(2.82 X 10- 5) = 2.78 X 10- 8 or log lAP = -7.6. This result may be compared with log Kso = - 8.30 ± 0.32 for calcite, which was computed in Section 2.5. The conclusion to be drawn from this comparison is the same as that discussed at the close of Section 3.1. The soil solution is not in equilibrium with calcite but could be in equilibrium with a microcrystalline form of CaC03(s). The example just discussed is typical of any calculation of an lAP value based on an analysis of the speciation of a metal or ligand in a soil solution. Besides the computation of the ion activity product. there is another common

86

SOLUBILITY EQUILIBRIA IN SOIL SOLUTIONS TABLE 3.3. Analytical data (expressed as pC values) for a saturation extract of a Holtville soil Component

pC*

Component

pC

Ca Mg Sr K Na Fe Mn Cu Zn Ni Ph Cd C0 3 S04

2.28 2.64 4.20 2.89 2.00 5.75 4.66 5.33 4.97 5.99 7.02 7.57 2.40 2.12

CI F P04 Citrate Salicylate Phthalate Arginine Ornithine Lysine Valine B N0 3 Maleate pH = 7.90

2.03 4.15 4.19 5.27 5.57 5.57 5.79 5.66 5.66 5.66 4.42 3.56 5.44

*pC

=

-log molar concentration.

use of speciation analysis that is important to soil chemistry. This second use' becomes necessary when the effect of a solid or liquid mixture (e.g., a chemical.l fertilizer, a manure, or a wastewater) added to a soil must be evaluated. The!: metals added in this way may react with ligand constituents already present in i'l a soil solution to produce new solid phases or to add to existing ones. On the I other hand, the ligands in the compounds applied may react with metals to pre- j cipitate them or to complex them and increase their solubility. These kinds of phenomena may be described in a straightforward manner by extending the, mole balance equation approach to include solid phases. The general expression of mass conservation for a metal added to a soiL solution to produce a total molar concentration, M T , may be expressed in tho form M T = [Mm+]

+ ~ vccK'C'Y'A[Mm+Yc[H+p[An-]'A + ~ vl[M"H~A'2(s)1

(3.21

where the first sum on the right side is over each complex, M'cH'YA'A' that is formed with the metal, M?", and the second sum is over each solid, M"H~A'2t' formed by the metal. The first two terms on the right side of Eq. 3.21 are the same collectively as M TS, the total molar concentration of the metal in the soil solution (see Eq. 3.16 for CUTS)' An equation having the same mathematica]' form as Eq. 3.21 can be written down for each metal and each ligand in a soil solution of interest. By choosing an appropriate linear combination of these equations, it is always possible to eliminate the concentrations of the solid phases

SOLUBILITY EQUILIBRIA IN SOIL SOLUTIONS

87

completely from the set of equations to be solved numerically. Moreover, some of the free ionic concentrations of the metals and ligands also can be eliminated from the equations because of the relation

-«; =

[Mm+]"I[H+]6[An-]",

which holds for each solid phase formed. The final set of nonlinear algebraic equations obtained from this elimination process will involve only independent free ionic concentrations as well as conditional stability and solubility product constants. The numerical solution of these equations then proceeds much like the iteration scheme outlined for the case where only complexation reactions were considered, with the exception of an added requirement of self-consistency, that the calculated concentration of the amount of each solid formed be a positive number and that lAP values not be greater than K so values (see Fig. 3.5). Consider, as a simple illustration of these ideas, the problem of predicting the fate of the salt CdS04 when it is added to a soil solution to produce a total

TABLE 3.4. Molar concentrations of Ca H and saturation extract described in Table 3.3 Species CaH CaCO~

CaHCOj CaSO~

csciCaF+

CaHPO~

CaH,PO: CaPO.. CaCitCaHCitO CaH 2Cit+ CaHSal+ CaPhthO CaArg" CaHOrnH CaH 2BOj CaNOj Ca(NOl)~

CaMalo CaOH+

- ---._------_...

pC· 2.40 4.39 4.04 2.97 4.16 5.90 4.81 6.77 5.40 11.43 14.64 19.70 7.88 6.30 7.25 6.82 6.58 5.59 9.40 6.18 7.27

__.

Species CO~-

MgCO~

MgHCOt SrCO~

SrHCOj KCO) KHCO~ K,CO~

NaCO) NaHCO~

Na2CO~ FeCO~ Fe(COl)~-

FeHCOj MnCO~

Mn(COlnMnHCOj CuCO~ CU(COl)~-

CuHCOj ZnCO~

*pC - -log molar concentration.

CO~-

and their complexes in the

pC

Species

pC

4.56 5.03 4.48 7.02 6.47 6.88 5.59 10.13 6.30 4.41 9.74 6.18 9.03 6.33 5.60 9.25 5.65 5.69 7.14 7.94 5.88

Zn(COlW ZnHCOj

9.33 5.43 6.31 8.76 6.56 7.12 8.27 7.97 9.01 13.06 8.56 2.44 4.05

NiCO~ Ni(COl)~-

NiHCOj PbCO~

Pb(COlHPbHCOj CdCO~

Cd(CO l)2CdHCOj HCO) H2CO~

SOLUBILITY EQUILIBRIA IN SOIL SOLUTIONS

88

initial concentration of 10- 6 M. In Table 3.5 analytical data are listed for four saturation extracts obtained from soils exhibiting wide differences in chemical and physical properties. The system of six metals and four ligands (including H+ and OH-) representing the saturation extracts can form 51 complexes and up to nine solids. The results of a speciation calculation pertaining to the added cadmium are given in Table 3.6. Only in the Arizo soil is cadmium predicted to precipitate as a carbonate solid. Most of the soluble cadmium is in the free ionic form in each soil solution. Since Cd H is a soft Lewis acid and since none of the complexing ligands listed in Table 3.5 is a soft Lewis base, extensive complex formation involving cadmium is not expected, according to the HSAB Principle.

3.6. ACTIVITY RATIO AND SOLUBILITY DIAGRAMS

A simpler method than computer analysis for predicting the fate of a metal or ligand added to a soil solution often is of value in practical work. In particular, one might wish to know only whether the added element will precipitate under given soil conditions and, if it does, what its aqueous phase activity as a free ionic species will be. These kinds of questions can be answered quickly with the help of an activity ratio diagram. This diagram is a graph of the variable log [(solid phase)j(free metal or ligand)] versus some important soil solution parameter such as pH, where ( ) refers to a thermodynamic activity. The graph includes plots of the logarithm of the activity ratio for each possible solid phase involving the metal or ligand of interest. For a given value of the soil solution parameter being treated as the independent variable and under the assumption that all solid phases are in their Standard States, the solid that produces the largest value of the logarithm of the activity ratio is the one that is most stable:: TABLE 3.5. Analytical data on the composition* of four saturation extracts to which ;~ a hypothetical addition of CdS0 4 has been m a d e N - - - - - - - - - - - - - - - - - - - - - - - - - - - ill

Soil

pH

Ca

Mg

Na

Altamont clay loam Arizo sandy loam Domino silt loam Redding f. sandy loam

6.9 8.2 7.6 5.6

2.60 2.68 2.23 3.15

3.10 3.40 2.87 2.82

3.00 2.96 2.72 2.55

Soil

pH

Cd

CO)

S04

Altamont clay loam Arizo sandy loam Domino silt loam Redding f. sandy loam

6.9 8.2 7.6 5.6

6.00 6.00 6.00 6.00

2.70 3.00 2.52 2.68

2.60 2.54 2.82 2.17

·Conccntrations expressed as pC values: data courtesy of Dr. R. J. Mahlcr.

K

,~

..

3.22 3.22 2.89 'j;1 )l:' 3.40 ,', Cl 3.00 2.68 3.22 2.92

- -""---

:.'J I:':~

·1 ','Ii

89

SOLUBILITY EQUILIBRIA IN SOIL SOLUTIONS TABLE 3.6. Calculated speciation at 298.15 K for cadmium in the saturation extracts described in Table

3.5 Percentage Speciation" Soil

Cd H

CO)

S04

CI

OH

caco.oi

Altamont Ariza Domino Redding

78.9 13.0 72.8 78.2

4.0 1.2 2.5 0.4

12.5 1.6 20.5 11.9

4.6 0.5 4.0 9.5

0.0 0.2 0.2 0.0

0.0 83.5 0.0 0.0

*Columns headed by ligands refer to soluble complexes between Cd and those ligands.

and, therefore, the only one that will form at equilibrium. This conclusion follows directly from the discussion in Section 3.1, where it was shown that the solid phase that produces the smallest soil solution activity of a free ionic species will control the solubility of that species. Consider, as a first example of an activity ratio diagram, the problem of deciding which solid phases of copper will form in the Redding soil, described in Table 3.5, as the pH value of its aqueous phase is increased through liming. The relevant crystalline solid phases in this case are tenorite, CuO(s), malachite, CUz(OH)2COls), and azurite, CulOH)lC03)ls). The dissolution reactions of these three solids at 298.15 K are:

+ 2H+ = Cu2+ + H 20(l) CuOH(C0 3)ds) + 2H+ = Cu2+ + 1.5H 20(l) + 0.5C02(g) Cu(OH)o.dC0 3)067(S) + 2H+ = Cu2+ + 1.33H 20(l) + 0.67COz(g) CuO(s)

log K = 7.65 log K = 7.08 log K = 7.08

The relations between the thermodynamic equilibrium constants and the activities of the reactants and products lead to the three equations: log [(CuO(s))/(Cu 2+)] = -7.65 + 2pH + log (H 20(l)) (3.22a) log [(CuOH(C0 3)o.s(s))/(Cu2+)] = -7.08 + 2pH + 0.5 log Pea, + 1.5 log (H 20(l)) (3.22b) log [(Cu(OH)o.dC0 3)o.67(S))/(Cu2+)] = -7.08 + 2pH + 0.67 log Pea, + 1.33 log (H 20(l)) (3.22c) In order to plot an activity ratio diagram, one of the soil solution parameters, pH, log Pco" or log (H 20(l)), must be chosen as an independent variable while the other two are held fixed. Since the pH value is of concern in the present example, it will he chosen as the independent variable. The values of the other

10.------------------------,

--'log P

=-3.52

---log P

=-1.0

cO 2

CO 2

8

~

,--..

+

'"U

6

::l

:s ""'0

o

(/) ......... L--.J

O'l

o - 4

2

7

6

8

pH FIGURE 3.6. Activity ratio diagram for solid phases containing copper in a Redding soil. Note that malachite becomes the most stable solid phase as Pc», increases, if (HIO(l» • 1.0.

SOLUBILITY EQUILIBRIA IN SOIL SOLUTIONS

91

two parameters will be set at log Peo, = - 3.52 and log (H 20(l» = 0.0. The resulting activity ratio diagram is shown in Fig. 3.6. Under the given conditions, the most stable solid phase formed is tenorite at all pH values. Therefore the activity of Cu H in the soil solution may be calculated from Eq. 3.22a with (H 20(l» = 1.0. The conclusion that tenorite is the most stable solid will not be affected by a decrease in the activity of water (at the fixed Peo,) because the coefficient of log (H 20(l» in Eq. 3.22a is smaller than those in Eqs. 3.22b and 3.22c. On the other hand, if (H 20(l» = 1.0 and Peo, rises above 10- 1 atm, as it can in flooded soils, the most stable solid phase would be malachite, since (-7.08 + 0.5 log Peo,) > -7.65 in this case and, therefore, the logarithm of the activity ratio for malachite always will be greater than that for either tenorite or azurite. This possibility also is shown in Fig. 3.6. Finally, it may be noted that, with the proper combination of a high Peo, and a water activity below 1.0, the most stable solid phase could be azurite since, in Eq. 3.22c, the coefficient of log Peo, is larger than it is in Eq. 3.22b, whereas for the coefficient of log (H 20(l» the opposite relationship holds true. The calculation illustrated in connection with Fig. 3.6 can be carried out equally well for a ligand. For example, if one is interested in aluminum phosphates at low pH values, a plot of log [(AI-phosphate)/(P0 34-)] versus pH can be made at fixed (Al H) for each phosphate solid phase that is possible and the most stable of these solid phases can be identified. Equations analogous to Eq. 3.22 can be written down corresponding to the dissolution of the phosphates into AlH(aq) and PO~-(aq). These expressions then become working equations for the activity ratio diagram, once all the activities but the one to be varied are specified. Some additional examples of activity ratio diagrams are shown in Figs. 3.7 and 3.8. In Fig. 3.7, log [(Cd-solid)/(Cd H ) ] is plotted against pH for a Hanford sandy loam soil solution." In this application, Peo, = 10- 3 atm and [PO~-] = 10- 6 .6 M. The activity coefficient of PO~- was estimated at each pH value in order to compute (PO~-). The diagram which, as is common practice, has the logarithm of the activity ratio increasing downward on the j-axis, shows that CdlP04)2 is the solid phase expected to control the solubility of cadmium around pH = 6.6, the natural value in the Hanford soil solution. The data points in the diagram are values of (Cd H ) based on measurements of Cd-, and subsequent calculations with an ion association model under the assumption that (CdlP0 4Ms» = 1.0. In Fig. 3.8, log [(Al-solid)/(AP+)] is plotted against -log (H4SiO~) for three soil minerals having a layer structure." In this example, the pH value was fixed at 6.0, (Mg H ) = (Na") = (K+) = 10- 3, and (Fe H) = 10- 15.5, while (H4SiO~) was chosen as the independent variable. The diagram shows that, as the activity of H4SiO~ decreases in the soil solution, the most stable solid phase under the given fixed conditions shifts from montmorillonite to kaolinite to gibbsite. This progression agrees with what is observed in natural soils with increased leaching and removal of silica from the soil profile. In Section 2.5 it was noted that t:J.Go for the transformation of kaolinite to montmorillonite plus dissolved

92

SOLUBILITY EQUILIBRIA IN SOIL SOLUTIONS Or------~o;;::----------------_,

O'l

o

Hanford soil

10 '----_ _- l - -_ _--l....-_ _--L.._ _--lL.--_ _..L-_ _--l 6.0 6.5 7.0 7.5 8.5 8.0 9.0

pH FIGURE 3.7. Activity ratio diagram for solid phases containing cadmium in a Hanford soil.

silica is positive, which indicates that montmorillonite and silica are more stable than kaolinite in the Standard State. However, when the activity of silica drops from its Standard State value of 1.0 to around 10- 4 and acidic conditions prevail, kaolinite becomes the more stable solid phase, as is demonstrated in Fig. 3.8. Once the controlling solid phases are known (often a difficult task to accomplish for a soil solution), the activities of the solute species in equilibrium with those solid phases can be calculated, if the latter are assumed to be in their Standard States. These activities, in turn, can be related by mole balance to total solution concentrations if the relevant single-ion activity coefficients and distribution coefficients can be estimated. In a full speciation calculation such as was illustrated in Table 3.6, the activity coefficients and the distribution coefficients are computed self-consistently, and the total solution concentration of each element may be found from the computer output. In some instances it is useful to calculate the speciation of an element at several pH values. The total molar solution concentration of a metal or ligand of interest then may be computed at each pH value at which a solid phase containing the metal or ligand is predicted to precipitate. A graph of log MTS for a metal or log A TS for a ligand versus pH is called a solubility diagram when a solid phase is present. These diagrams are useful if the total level of an element in the soil solution is more important than the concentrations of individual species containing the element. In Table 3.7,

SOLUBILITY EQUILIBRIA IN SOIL SOLUTIONS

93

7

01

o

5.5

2.5

FIGURE 3.8. Activity ratio diagram for three solid phases containing aluminum, at pH 6.0 under the conditions (Mg H ) = (Na ") = (K+) = 10- 3, and (Fe H ) = 10- 15.5.

TABLE 3.7. Analytical composition data for a saturation extract obtained from an arid-zone soil Element

pC

Ca Mg Na

2.27 2.34

K

2.92 3.05

Element

pC

6.00 2.59 2.38 3.05

SOLUBILITY EQUILIBRIA IN SOIL SOLUTIONS

94

analytical data are listed for a saturation extract obtained from an arid-zone soil to which CdS0 4 has been added to produce 10- 6 M Cd in the soil extract. The results of a speciation calculation, carried out at pH 6.5, 7.0, and 8.0, are shown in the solubility diagram illustrated in Fig. 3.9. These data could be useful for predicting the value of Cd TS in the soil, although adsorption phenomena could make important differences in the results were they to be included in the calculation as possible mechanisms for controlling cadmium solubility.

3.7. COPRECIPITATED SOLID PHASES

Throughout all of the applications considered in this chapter, it has been assumed that the solid phases formed were in their Standard States. In natural water systems, such as soil solutions, it is well known that this assumption often I

,j

I

FIGURE 3.9. Solubility diagram for cadmium in an arid-zone soil. -6r--------~-------------......,

............ (/) Q)

'u -7 Q) a. (/)

'----'

0" 0 >-

0

,

complexes

.......

(J')

I-

-,

"'0

u

0" 0

-8

Cd-CI complexes

.......\

\.

\.

\\

-9 L-.. 5

\.

"""", "-" ........ .........

'-

-

.....J-_ _--:::..-_-----J

........I.

7

6

pH

"'-

8

SOLUBILITY EQUILIBRIA IN SOIL SOLUTIONS

95

is invalid and that the solid phases formed are mixtures because of coprecipitation phenomena. The term coprecipitation refers to the simultaneous precipitation of a compound in conjunction with other compounds by any mechanism and at any rate. Three broad categories of coprecipitation phenomena have been identified in soils: mixed solid formation, adsorption, and inclusion. Mixed solid formation is common, for example, when secondary minerals, such as montmorillonite, precipitate from the soil solution (see Section 2.4). These solids are characterized by a wide range of isomorphous substitutions, in which both cations and anions in the structure may be replaced by ions of the same charge sign and comparable size. Examples of this solid solution phenomenon occur during the formation of clay minerals if metal cations replace either SiH in the tetrahedral sheet or AIH in the octahedral sheet; during the formation of CaCOls), if Mg H , Sr H , Fe H , Mn H , or Na + replaces Ca H , and during the formation of hydroxyapatite, Ca sOH(P0 4)ls), if Ca H is replaced by SrH , or other cations, or if OH- is replaced by F-, or other anions. In every case of coprecipitation through the formation of a solid solution, the solid phase is a homogeneous mass with its minor substituents distributed uniformly. Thus two requirements for this type of coprecipitation are the free diffusion and relatively high structural compatibility of the minor substituents within the precipitate as it is forming. If free diffusion within the precipitate is not possible for a given cation or anion, that ion may still coprecipitate by an adsorption process. For example, if an iron or aluminum hydrous oxide were precipitating in a soil solution to form colloidal material with a relatively large surface area, and if the pH value were in the alkaline range, metal cations could adsorb on the surface of the precipitate. These coprecipitated metals might not be able to diffuse freely into the bulk solid and, therefore, the development of a true solid solution would be prevented. The same type of phenomenon could occur in the acidic pH range for an oxyanion in the soil solution, such as PO~- or SO~-. Generally, it can be expected that coprecipitation through adsorption will be much more dependent on the kinetics of precipitation and on the composition of the soil solution than is solid solution formation. The conditions that favor adsorption would be rapid precipitation initiated under pronounced supersaturation conditions and a relatively high degree of incompatibility between the adsorbing species and the bulk structure of the precipitate. If the coprecipitating elements naturally would tend to form solids with very different structures, it is likely that a mixed solid comprising the elements will be a heterogeneous system instead of a solution. This type of coprecipitation is illustrated by the inclusion of separate Ti0 2 and o-phosphate solid phases within secondary clay minerals. Insofar as the soil solution is concerned, the principal effect of coprecipitation is on the solubility of the elements to be found in the solid mixture. If the soil solution is in equilibrium with the mixed solid phase, the activity in the aqueous phase of an ion that is a minor or trace element of the solid may be significantly smaller than what it would be in the presence of a pure solid phase

SOLUBILITY EQUILIBRIA IN SOIL SOLUTIONS

96

comprising that element. For example, in Eq. 3.10 an expression was developed that relates the activity of ZnH(aq) to the activities of H+(aq), H 20(l), CO 2(g), and a ZnCOJCs) phase presumed to control the solubility of zinc in the soil solution under certain prescribed conditions. If, under those conditions, the solid carbonate phase of zinc has unit activity, Eq. 3.10 shows that, at 298.15 K, the relation between the aqueous solution activity of Zn H and that of H+ is (Zn H) = 10104 5 (H+)2. However, if ZnCOJCs) were coprecipitated as, for example, 1 mole percent of a bulk calcite phase, and if (ZnC0 3(s)) ~ 0.01, then (Zn H) = 1085 (H+)2 according to Eq. 3.10. Clearly, the extent to which coprecipitation phenomena occur in a soil can be very important to know in ascertaining quantitatively how the composition of the aqueous phase is controlled by the solid phase. In many instances the formation of mixed solid phases in a soil will be dictated primarily by complicated kinetic considerations, and the prediction of the composition of the soil solution as influenced by those solid phases will be quite difficult. On the other hand, if chemical equilibrium actually exists between the solid and aqueous phases, or even if it is desired only to have a general understanding of the reaction pathways and ultimate end products in a precipitationdissolution phenomenon of interest, a thermodynamic description of coprecipitation can be valuable. Consider the coprecipitation of two metal cations, A a+ and Bb+, with a ligand U- to form a solid solution consisting of the two components AmL.(s) and BpLq(s). These two components react according to the equation (3.23a) Equation 3.23a describes the replacement of A a+ in the mixed solid by B b+. It is subject to the electroneutrality conditions, am = In and pb = Iq, which guarantee that the mixed solid phase has zero electrical charge. If the two electroneutrality conditions are introduced into Eq. 3.23a and the compounds in the aqueous phase are separated into cations and anions, the replacement simplifies to the expression

+

(bjm)AmL.(s)

aBb+(aq)

=

bN+(aq)

+ (ajp)BpLq(s)

(3.23b)

The equilibrium constant for the reaction in Eq. 3.23b is K = (BpLq(s)) a/peA a+)b j(AmL.(s)) b/m(Bb+)a

(3.24)

The conditional equilibrium constant that corresponds to K is C

K = Nr/P(Aa+)b j N"lm(Bb+)a

(3.25)

where N B is the mole fracion of BpL q in the mixed solid phase and N A is the mole fraction of AmL. in the mixed solid phase. If it so happens that a = band m = p = I, then Eq. 3.25 reduces to "K = D", where (3.26) is called the separation factor. This special case is of widespread interest because

f

,I

I,

i I "

I

,j

SOLUBILITY EQUILIBRIA IN SOIL SOLUTIONS

97

mixed solid formation in precipitates is often most expected when ions of like charge and size replace one another in one-to-one solids. For each component of the solid phase, an lAP can be written that has the same mathematical form as Eq. 3.3:

(Bb+)P(U-)q = KB(BpLq(s» (A a+)m(U-)n = KA(AmLn(s»

(3.27a) (3.27b)

where K B and K A are thermodynamic equilibrium constants for the dissolution of the two solid phases. The introduction of Eq. 3.27 into Eq. 3.24 reduces that expression to (3.28) The thermodynamic equilibrium constant K, therefore, is related directly to the dissolution equilibrium constants, K A and K B. If a = band m = p = 1, K =

(KAI KB)a. The relation between K and K involves the solid-phase activity coefficients, fA and fB' as described in Eqs. 2.78 and 2.103. In particular, C

In K = In CK

+ (alp)

InfB - (blm) In fA

(3.29a)

from which it follows that

d In K = (b 1m) d In fA - (a I p) d In fB C

(3.29b)

for any infinitesimal change in CK as a result of a change in the composition of the mixed solid, since din K = 0 for such changes. Equation 3.29b is a condition on the activity coefficients expressed in terms of C K. Another condition can be derived from the Gibbs-Duhem equation (Eq. 1.25) applied to the mixed solid at fixed T and P:

NAdlJ.A

+

NBdlJ.B = 0

In this expression, the mole number before each dlJ. has been divided by the total number of moles in the solid to yield a mole fraction. If each chemical potential is written in terms of an activity coefficient and a mole fraction, the result is

NAd In (fANA)

+ NBd In (fBNB) = dNA + NAd In fA + dNB + NBd In fB = NAd In I; + NBd In fB = 0

(3.30)

where the mass balance condition dNA = -dNB has been employed to obtain the final form of the equation. Equations 3.29b and 3.30 are a pair of differential equations that may be solved and integrated in order to calculate the activity coefficients. For example, to obtain fA one writes: din CK = (b] m)d In fA + (a] p)(NAI NB)d In fA = (b] m)[ 1 + (maNAI pbNB)] d In fA or

(hlm)dlnfA - E.dln"K

(3.31)

98

SOLUBILITY EQUILIBRIA IN SOIL SOLUTIONS

where (3.32) is the equivalent fraction of Bb+ in the solid phase. The value of fA when the equivalent fraction of Bb+ is equal to E B may be calculated by integrating Eq. 3.31 between E B = 0, when fA = 1.0 (see Table 2.5), and the desired equivalent fraction: (him) In fA =

I

Eo

o

E~ dIn CK

= E B In CK

I

-

=

I

Eo

[d(E~ In cK) -

In CK

dE~]

0

Eo

dE~

In CK

(3.33)

o

where the second step is an integration by parts. Equation 3.33 shows that fA may be computed if CK has been measured as a function of the composition of the solid phase. The corresponding expression for fB is (noting that fB = 1.0 when E B = 1.0): (alp) InfB = -(1 -

I

+

E B) In CK

I

In CK dE~

(3.34)

Eo

Equations 3.33 and 3.34 combine to give In K

=

I

1

In K dEB

(3.35)

C

o

after they are substituted into Eq. 3.29a. When a = hand m = p = 1, Eq. 3.35 may be written as a relationship between the separation factor, D, and the dissolution equilibrium constants, K A and K B

I

I

In D dNB = In K A

In K B

-

(3.36)

o

where the fact that E B = N B in this case has been employed. One application of the expressions derived here is to the precipitation of magnesian calcites, Ca 1 - xMg xC0 3(s), where x is the mole fraction of MgC0 3 in the solid. For this example, with A a+ = Ca H and Bb+ = Mg H , Eqs. 3.33, 3.34, and 3.36 become, respectively: In fea = x In D -

I

x In D dNMg

o

In fMa = -(1 - x) In D ,

,

r

I

In D dN Ma

...

+

II

In D dN Mg

x

In KIO(calcilc) - In KIO(magncsile)

()

- -19.1218

+

18.8709 - 0.2509

99

SOLUBILITY EQUILIBRIA IN SOIL SOLUTIONS

where D = [NMg/(l - NMg)]/[(MgH)/(Ca H)]

If the magnesian calcite were an ideal solution, then fe. = fMg = 1.0 by definition, the separation factor would be constant and equal to exp(0.2509) = 1.29, and the relation between the cation activity ratio, (MgH)/(Ca H), and the mole fraction of MgC0 3 in the solid would be, in logarithmic form, (3.37) log [(MgH)/(Ca H)] = -0.1106 + log [x/(l - x)] This expression is plotted in Fig. 3.10, along with some recent measured values of the activity ratio, for x < 0.05. 10 It is apparent from an examination of the

FIGURE 3.10. Measured values of (MgH)/(Ca H) in aqueous solutions contacting magnesian calcites, plotted against the mole fraction of MgC0 3(s) in the calcites, x. The solid line is a graph of Eq. 3.37.

+5r-------------------------, ~

• •

Winland (1969) McCauley and Roy(1974) Thorstenson and Plummer(l977)

+3-

, .--.. ' + o

N

U

-S +

+1 -

•

N CJl

~ '---' ~

CJl

o

-1-

-3L...--

o

• •

...L-I

0.01

---L.. I

0.02

s

•

Eq. 3.37~

.1..-I

0.03

.....

0.04

100

SOLUBILITY EQUILIBRIA IN SOIL SOLUTIONS

figure that either the measured values of [(MgH)/(Ca H)] do not reflect true equilibrium or the solid solution formed by magnesian calcite is not ideal. The data in Fig. 3.10 suggest that, when log [(MgH)/(Ca H)] ;S 0, which is typical of soil solutions, the value of x will lie between 0 and 0.02. This result appears to agree with those of X-ray diffraction studies on soil carbonates.

NOTES J. G. M. Marion, D. M. Hendricks, G. R. Dutt, and W. H. Fuller, Aluminum and silica solubility in soils, Soil Sci. 121:76-85 (1975).

2. D. L. Suarez, Ion activity products of calcium carbonate in waters below the root zone, Soil Sci. Soc. Amer. J. 41:310-315 (1977). 3. This example is considered in fuller detail in S. V. Mattigod and J. A. Kittrick, Temperature and water activity as variables in soil mineral activity diagrams, Soil Sci. Soc. Amer. J. 44:149154 (1980). 4. R. G. Pearson, Hard and soft acids and bases, Part I, J. Chern. Educ. 45:581-587 (1968), Part II, J. Chern. Educ. 45:643-648 (1968). A complete introductory discussion of the Lewis acid-base concept is given in W. B. Jensen, The Lewis acid-base definitions: A status report, Chern. Rev. 78: 1-22 (1978), and in W. B. Jensen, The Lewis Acid-Base Concepts: An Overview, John Wiley, New York, 1980. 5. M. Misono, E. Ochiai, Y. Saito, and Y. Yoneda, A new dual parameter scale for the strength of Lewis acids and bases with the evaluation of their softness, J. Inorg. Nuc/. Chern. 29:26852691 (1967). 6. A review of 14 such computer programs is given in D. K. Nordstrum et aI., A comparison of computerized chemical models for equilibrium calculations in aqueous systems, in Chemical Modeling in Aqueous Systems. Speciation, Sorption, Solubility, and Kinetics (ed. E. A. Jenne), pp. 857-892. ACS Symposium Series No. 93, American Chemical Society, Washington, D.C., 1979. A discussion of the numerical methods employed in some of these programs is in D. J. Leggett, Machine computation of equilibrium concentrations-some practical considerations, Talanta 24:535-542 (1977). 7. G. Sposito and S. V. Mattigod, GEOCHEM: A Computer Program for the Calculation of Chemical Equilibria in Soil Solutions and Other Natural Water Systems, Kearney Foundation of Soil Science, University of California, Riverside, 1980. 8. J. Santillan-Medrano and J. J. Jurinak, The chemistry of lead and cadmium in soil: Solid phase formation, Soil Sci. Soc. Amer. Proc. 39:850-856 (1975). 9. D. Rai and W. L. Lindsay, A thermodynamic model for predicting the formation, stability, and weathering of common soil minerals, Soil Sci. Soc. Amer. Proc. 39:991-996 (1975). See also J. A. Kittrick, Soil minerals in the AI20,-Si02-H 20 system and a theory of their formation, Clays and Clay Min. 17:157-167 (1969). 10. D. C. Thorstenson and L. N. Plummer, Equilibrium criteria for two-component solids reacting with fixed composition in an aqueous phase-Example: The magnesian calcites, Am. J. Sci. 277:1203-1223 (1977); 278:1478-1488 (1978) [see also the discussion of this article by M. G. Lafon, Am. J. Sci. 278:1455-1468 (1978)J; J. W. McCauley and R. Roy, Controlled nucleation and crystal growth of various CaCO, phases by the silica gel technique, Am. Mineralogist 59:947963 (1974); and H. D. Winland, Stability of calcium carbonate polymorphs in warm, shallow sea water, J. Sed. Petrol. 39: I S79 I Sll7 (1969).

SOLUBILITY EQUILIBRIA IN SOIL SOLUTIONS

101

FOR FURTHER READING J. B. Dixon and S. B. Weed, Minerals in Soil Environments, Soil Science Society of America, Madison, Wis., 1977. The first 18 chapters of this comprehensive book survey the properties of the solid phases found in soils. E. A. Jenne, Chemical Modeling in Aqueous Systems, ACS Symposium Series No. 93, American Chemical Society, Washington, D.C., 1979. This book is a veritable encyclopedia of natural water chemistry, with chapters covering precipitation-dissolution, complexation, oxidation-reduction, and adsorption phenomena. Of special interest to solubility studies are: Chapter 18. Techniques of estimating thermodynamic properties for some aqueous complexes of geochemical interest by D. Langmuir; Chapter 19. Critical review of the equilibrium constants for kaolinite and sepiolite by R. L. Basset, Y. K. Kharaka, and D. Langmuir; Chapter 23. Calcium phosphates-Speciation, solubility, and kinetic considerations by G. H. Nancollas, Z. Amjad, and P. Koutsoukos; and Chapter 25. Critical review of the kinetics of calcite dissolution and precipitation by L. N. Plummer, T. M. L. Wigley, and D. L. Parkhurst. W. L. Lindsay, Chemical Equilibria in Soils, John Wiley, New York, 1979. This excellent monograph presents an exhaustive discussion of equilibria between soil solutions and soil minerals. Its 23 chapters discuss all of the soil solution components of general interest and make abundant use of activity ratio diagrams. F. Morel, R. E. McDuff, and J. J. Morgan, Interactions and chemostasis in aquatic chemical systems: Role of pH, pE, solubility, and complexation, in Trace Metals and Metal-Organic Interactions in Natural Waters, ed. by P. C. Singer, pp. 157200, Ann Arbor Science, Ann Arbor, Mich., 1973. This chapter is an outstanding introduction to natural waters considered as multicomponent systems and modeled by chemical equilibrium computer programs. W. Stumm and P. A. Brauner, Chemical speciation, in Chemical Oceanography, ed. by J. P. Riley and G. Skirrow, 2nd ed., Vol. I, pp. 173-239, Academic Press, London, 1975. The first 30 pages of this review chapter comprise an excellent introduction to complex formation by metals in any natural water system. W. Stumm and J. J. Morgan, Aquatic Chemistry, John Wiley, New York, 1970. Chapters 5 and 6 of this standard textbook provide a discussion of all of the topics considered in the present chapter. A. G. Walton, The Formation and Properties of Precipitates, John Wiley, New York, 1967. Chapter 3 in this book is a fine introduction to coprecipitation phenomena from the experimental point of view.

4 ELECTROCHEMICAL EQUILIBRIA IN SOILS

4.1. OXIDATION-REDUCTION REACTIONS IN SOILS

An oxidation-reduction reaction, or redox reaction, is a chemical reaction in which one or more electrons are transferred completely from one species to another. Every redox reaction may be separated into a reduction half-reaction and an oxidation half-reaction. A reduction half-reaction, in which a chemical species accepts electrons, usually may be written in the form (4.1) where A represents a chemical species in any phase and "ox" and "red" denote its oxidized and reduced states, respectively. The parameters m, n, p, and q are stoichiometric coefficients, whereas H+ and e refer to the proton and the electron in aqueous solution. A complete redox reaction is a combination of Eq. 4.1 and an oxidation half-reaction, in which a chemical species gives up electrons. This combination yields a chemical equation that does not exhibit the electron as a reacting species. The designations "oxidized" and "reduced" for a given chemical species are provided with quantitative significance through the concept of the oxidation number, the hypothetical valence that may be assigned to each atom in a compound. The oxidation number is denoted conventionally by a positive or negative Roman numeral that is computed according to a set of rules. 1. For a monoatomic species, the oxidation number is the same as the valence. 2. For a molecule, the sum of oxidation numbers of the constituent atoms must equal the net charge on the molecule expressed in terms of the protonic charge. 3. For a chemical bond in a molecule, the shareable, bonding electrons are assigned entirely to the more electronegative atom participating in the bond. If there is no difference in electronegativity between the two bonding atoms, the electrons are assigned equally. These rules may be illustrated by working out the oxidation numbers for the atoms in NO;-, HCO;-, N 2, and CH•. In the case of NO;-, oxygen is the bond. (See almost any chemistry more electronegative atom in each N handbook or inorganic chemstry textbook for a Jist of elcctronegativities.) There-

°

102

,~

, !

ELECTROCHEMICAL EQUILIBRIA IN SOILS

103

°

fore two sets of bonding electrons are assigned to each in NO;- and this atom is denoted O( - II). By rule (2), the N atom must be denoted N(V) because 5 + 3( - 2) = -1, the net charge of NO;-. In the case of HCO;-, oxygen is more electronegative than either carbon or hydrogen and is once again denoted by O( - II). The hydrogen atom in O-H is denoted H(I) while the carbon atom is qIV), since 4 + 3( - 2) + 1 = - 1, the net charge of HCO;-. In the case of N 2, both atoms have the same electronegativity and, therefore, neither receives an additional bonding electron. Since the molecule is neutral, each atom must be denoted N(O). In the case of CH 4 , the more electronegative atom in each CH bond is carbon. Thus carbon is assigned one additional electron from each shared electron pair and is denoted q - IV). The hydrogen atoms then must be denoted H(I) according to rule (2). In soil solutions the most important chemical elements that undergo redox reactions are C, Fe, Mn, N, 0, and S. For contaminated soils, the elements As, Cr, Cu, and Pb could be added to this list. Table 4.1 lists some reduction halfreactions and their equilibrium constants at 298.15 K and under 1 atm pressure

TABLE 4.1. Some reduction reactions of interest in soil solutions Reaction*

+ H+ + e = ! H 20(l) ! NO) + ~ H+ + e = fo N 2(g) + ~ H 20(l) t NO; + ~ H+ + e = ~ Nlg) + i H 20(l) i N 2(g) + ~ H+ + e = t NHt ! MnOb) + 2H+ + e = ! Mn H + H 20(l) MnOOH(s) + 3H+ + e = Mn H + 2H 20(l) ! Mn20J(S) + 3H+ + e = Mn + ~ H 20(l) ! Mn 30b) + 4H+ + e = ~ Mn H + 2H 20(l) ! Mn30b) + 4H+ + ~ COi- + e = ~ MnCOls) + Fe(OHMs) + 3H+ + e = Fe H + 3H 20(l) ! Fe304(s) + 4H+ + e = ~ Fe H + Hp(l) ! HCOO- + ~ H+ + e = ! HCOH + ! H 20(l) i CO2(g) + H+ + e = i CH 4(g) + ~ HP(1) ! HCOH + H+ + e = ! CH 30H ! CH 30H +' H+ + e = ! CHig) + ! H 20(l) t sot + H+ + e = t S2- + ! H 20(l) t SOl- + t H+ + e = t HS- + ! H 20(l) i SOl- + i H+ + e = i H 2S(g) + ! H 20(l) i HSOi + t H+ + e = i H 2S(aq) + ! H 20(l) ! S(s) + e - ! S2 ~ 02(g)

20.78 21.06 25.70 4.65 20.69 22.86

H

.Chargcd specie» are in "'IUCOUK solution; log K.

VII)UCK

24.39 2 H 20(1)

30.83 46.59 15.87 20.79 2.82 2.86 3.92 9.94 2.52 4.26 5.26 4.89 -8.0512

refer 10 298.15 K lind I 111m.

ELECTROCHEMICAL EQUILIBRIA IN SOILS

104

for the six principal elements involved in soil redox phenomena. Although the reactions listed in the table are not full redox reactions, their equilibrium constants have thermodynamic significance and may be calculated with the help of Standard State chemical potentials in the manner described in Section 2.5. Two important points are worth mentioning in this connection. First, the fact that the reactions in Table 4.1 describe electrochemical equilibria has no direct bearing on the method by which the equilibrium constants were determined. The Standard State chemical potentials employed to compute a value of log K R may be the results of solubility, thermochemical, spectroscopic, or electrochemical experiments. No particular experimental method is implied just because the reaction of interest is a redox reaction. Second, the convention that IJ-0 0 for H+(aq), e(aq), and any element in its most stable phase (see Section 2.2) often simplifies the calculation of log K R for a reduction or oxidation reaction. For example, log K R for the reduction of Oz(g) at 298.15 K and 1 atm pressure is given by

==

log K R = -

=

B1J-°(H zO(1))

-

! IJ-°(Olg))

- IJ-°(H+) - IJ-°(e)] /5.708

-1J-°(Hz0(1))/11.416 kJ mol"

=

20.78

Because of the convention for IJ-0 , the value of log K for the oxidation of Hlg) is equal to zero: = H+ + e IJ-°(e) -! 1J-°(H 2(g))] /5.708

! Hlg) log K = -[IJ-°(H+)

+

=

0

It follows that log K R for any reduction half-reaction may be considered equally as log K for a complete redox reaction in which electrons are transferred from Hlg) to the reduced species in the reduction reaction. In the case of Eq. 4.1, for example, log K R for the reduction half-reaction is the same as log K for the redox reaction:

i. j

'!

Taking the reduction of N01 as a concrete example from Table 4.1, one has: log K = 21.06 "

The firm thermodynamic status of log K R values for reduction half-reactions permits the use of these parameters in the normal way (see Section 2.3) to evaluate equilibrium activities of oxidized and reduced species and to compare the stabilities of reactants and products in redox reactions. As an example of a stability comparison, consider the possible reduction of S(VI) to S( - II) through the oxidation of C(II) to C(IV) in a soil solution. The reduction half-reaction for sulfur may be written in the form: log KR

-

2.52

ELECTROCHEMICAL EQUILIBRIA IN SOILS

105

The oxidation of carbon can be described by the half-reaction that results from the following combination of reactions in Table 4.1:

! HCOO- + i H+ + e = ! HCOH + ! H 20(l) ! HCOH + H+ + e =! CH 30H ! CH 30H + H+ + e =! CH 4(g) + ! H 20 ! CHig) + H 20(1) = ! COig) + 4H+ + 4e ! HCOO = ! COig) + ! H+ + e

log K R = log KR = log K R = -4 log KR = log K =

2.82 3.92 9.94 -11.44 5.24

Finally, the complete redox reaction is:

l SO~- + H+ + e ! HCOO! SO~ +! HCOO

= =

! S2- + ! H 20(l) ! COig) + ! H + +

log K R = 2.52 log K = 5.24 log K = 7.76

e

Since log K for this reaction is positive, ti.Go is negative, and the products are more stable than the reactants in the Standard State. The situation in a state other than the Standard State may be evaluated, as usual, by a consideration of the equilibrium constant itself. In the present case (assuming unit water activity):

=

= llog [(S2-)/(SO~-)] +! log PeOt + ! pH - ! log (HCOO-) soil solution is in equilibrium with Pco; = 10- 3.52 atm at pH log K

7.76

5.6 and has (HCOO-) = 10- , the above expression for log K leads to log [(S2-)/ (SO~-)] = 29.76, which shows that sulfate reduction is highly favored thermodynamically. As another illustration of a stability calculation, consider the possible oxidation of Cr(III) to Cr(VI) through the reduction of Mn(IV) to Mn(II). This redox reaction could be important in soils high in manganese that are subject to contamination by chromium disposed of in landfill sites. I The complete redox reaction may be expressed as the combination: If a

6

! MnOb) + 2H+ + e =! Mn H + H 20(l) t Cr(OH)3(S) + i H 20(l) = i Cr20~- + j H+ +

e

! MnOis) + t Cr(OH)ls) + j H+ = ! Mn H + i Cr 20?- + ~ H 20(l)

log K R = 20.69 -log K R = -21.00

log K = -0.31

In this case the reactants are slightly more stable than the products, relative to the Standard State. The equilibrium constant expressed in terms of activities once again may be employed to estimate the possible extent of chromium oxidation in a soil solution. In a soil solution at pH 4 contacting a coprecipitate' of Mn02 and Cr(OH)3' with (MnOz(s» ~ 1.0 and (Cr(OHMs» :::::: 10- 3, and

106

having (Mn H activity):

ELECTROCHEMICAL EQUILIBRIA IN SOILS )

-

10-6, the activity of

log (Cr20~-) - 6 log K

Cr20~-

would be (assuming unit water

+ 2 log (Cr(OH) ls» - 4pH - 3 log (Mn H

)

=

-5.86

This result suggests that chromium oxidation would be significant under the given conditions. It is important to understand clearly that the two examples just considered indicate that certain redox reactions can occur in soils, but not that they will occur: a chemical reaction that is favored thermodynamically is not necessarily favored kinetically. This fact is especially applicable to oxidation-reduction reactions because they often are extremely slow and because reduction and oxidation half-reactions often do not couple well to one another. For example, the coupling of the half-reaction for Oig) reduction with that for HCOH oxidation leads to the redox reaction: log K

=

18.36

For a soil solution that is in equilibrium with the earth's atmosphere, the above value of log K predicts complete oxidation of carbon from C(O) to C(IV) at any pH value. But this prediction is contradicted quite plainly by the persistence of dissolved organic matter in soil solutions under surface terrestrial conditions. The same kind of contradiction would occur if Nlg) were considered instead of CHOH; indeed, the N 2 of the atmosphere should be oxidized to NO;- almost everywhere, if thermodynamic equilibrium prevailed. The lack of effective coupling and the slowness of redox reactions mean that catalysis is required if equilibrium is ever to come about. In soil solutions and other natural water systems, the catalysis of redox reactions is mediated by microbial organisms. In the presence of the appropriate microbial species, a redox reaction can proceed quickly enough in a soil to produce activity values of the reactants and products that agree with thermodynamic predictions. Of course, this possibility is dependent entirely on the growth and ecological behavior of the soil microbial population and the degree to which the products of the attendant biochemical reactions can diffuse and mix in the soil solution. In some cases, redox reactions will be controlled by the highly variable dynamics of an open biological system, with the result that redox products at best will correspond to local conditions of partial equilibrium in a soil. In other cases, including often the very important one of the flooded soil, redox reactions will be controlled by the behavior of a closed, isobaric, isothermal chemical system that is catalyzed effectively by bacteria and for which a thermodynamic description is especially apt. Regardless of which of these two extremes is the more appropriate to characterize the redox reactions in a given soil, the role of organisms deals only with the kinetics aspect of redox. Soil organisms affect the rate of a redox reaction, not its standard free energy change. If a redox reaction is not favored thermodynamically, microbial intervention cannot change that fact.

1,I 1

'(

.'

ELECTROCHEMICAL EQUILIBRIA IN SOILS

107

4.2. THE ELECTRON ACTIVITY

In aqueous solutions, Brensted acidity is measured by the negative common logarithm of the free proton activity (i.e., the pH value). As is well known, large positive values of pH strongly favor the existence of Brensted bases (free proton acceptors), while small values of pH strongly favor Brensted acids (free proton donors). This role of the pH value in Brensted acid-base reactions (proton transfer reactions) may be understood directly in terms of the fact that, at a given temperature, pH is proportional to the chemical potential of H+(aq). The larger the pH value, the smaller is the value of ~(H+) in solution and the greater is the tendency of a Brensted acid to lose its transferable protons. Statements exactly analogous with these can be made about the free electron in aqueous solutions. The Jorgensen oxidizability is measured by the negative common logarithm of the free electron activity, the pE value. Large positive values of pE strongly favor the existence of oxidized species (free electron acceptors), while small values of pE strongly favor reduced species (free electron donors). The role of the pE value in redox reactions (electron transfer reactions) may be understood in terms of the direct proportionality of the chemical potential of e(aq) to pE at a given temperature. The larger the pE value, the smaller is the value of ~(e(aq» and the greater is the tendency of a reduced species to lose its transferable electrons.' The range of pE values possible in soil solutions can be estimated through a consideration of the decomposition of liquid water. The upper extreme of pE will occur when H 20 (l ) decomposes to form protons and Olg) according to the first half-reaction in Table 4.1: -log K R

=

-20.78

Assuming that water is in its Standard State, the pE value at equilibrium is given by the expression pE

== -log(e) = 20.61 -

= log K R pH

-

pH

+ ~ log Po,

where Po, = 0.21 atm, the atmospheric partial pressure of Olg), has been introduced in the last step along with the value of log K R • The pE value will be greatest when the pH value is least. With pH = 4 taken as a typical lower bound for soil solutions, the estimated upper extreme value of pE is + 16.6. The lower extreme of pE will occur when H 20 (1) decomposes to form hydroxide ions and H 2(g) according to the reaction log K = -14 When the water ionization reaction, H 20 (l ) = H+ + OH-, is subtracted from this reaction once, in order to introduce H t into the products in place of OH . ~,

ELECTROCHEMICAL EQUILIBRIA IN SOILS

108 the half reaction

log K R = 0 is the result. In this case, the pE value is given by the expression pE

= -log(e) =

log K R = -pH

-

pH +

! log PH,

if PH, = I atm is employed to simplify the calculation. With pH = 9 taken as a typical upper bound for soil solutions, the estimated lower extreme value of pE is -9.0, although a still lower value would result if a less extreme value of PH, were used instead of 1.0 atm. Thus the range of pE values exprected in soil solutions is -9.0 < pE <

+ 16.6

which corresponds to a change in electron activity by 26 orders of magnitude. In actuality, the pE values in soil solutions tend to remain between -6.8 and + 13.5, regardless of the pH value. At pH 7.0, oxidized soils usually have + 7 < pE < + 13.5; moderately reduced soils have + 2 < pE < + 7; reduced soils have -2 < pE < +2; and highly reduced soils have -6.8 < pE < -2.0. The chemical significance of these ranges of pE value perhaps can be appreciated most rapidly through a brief survey of the reduction reactions of 0, N, Mn, Fe, and S in soils. For the general reduction half-reaction given by Eq. 4.1, the pE value is expressed in the form pE = log K R + m 10g(Aox) - P 10g(Ared) -

n pH

(4.3)

where, as usual, liquid water is assumed to be at unit activity. The "critical value" of pE at pH 7 for the situation when either (Aax) is "undetectable" or (Ared) is "just detectable" is given by a special case of Eq. 4.3: (4.4)

Estimates of pEecit for typical reduction half-reactions involving 0, N, Mn, Fe, and S are presented in Table 4.2. In making these estimates, it was assumed that the values of (Aox)uodet or (Ared)det were Po, 10- 4 atm; (NOJ)j(NHn = (SO~-)j(S2-) = 10- 6, and (MnH ) = (Fe H ) = 10-7 . All solid phases and liquid water were assumed to be in their Standard States. The data in the second column of Table 4.2 show that, as the pE value in a soil solution drops, first oxygen is reduced, followed by nitrate, then manganese oxides, then iron oxides and, finally, sulfate. This predicted sequence is in agreement with the results of experiments on flooded soils, although a comparison between the second and third columns in Table 4.2 demonstrates that quantitative agreement between thermodynamic estimates of pEeril and measured values does not always obtain. J The particularly large discrepancy in the case of

109

ELECTROCHEMICAL EQUILIBRIA IN SOILS

TABLE 4.2. Sequential reduction of 0, N, Mn, Fe, and S in soil solutions at pH 7.0 pEeri! Reduction Reaction

Calculated

+ H+ + e = ! H 20(l) i NO) + ~ H+ + e = l NHt + i H 20(l) ! Mn02(S) + 2H+ + e = ! Mn2+ + H 20(l) Fe(OHMs) + 3H+ + e = Fe2+ + 3H 20(l) i SOJ- + H+ + e = i S2- + ! H 20(l) ;\ 02(g)

Observed

5.4

5.6 3.7

3.2 1.8

3.4 2.0

-5.2

-2.5

12.8

Olg) reduction may be partly due to experimental difficulties in the determination of oxygen by electrochemical methods. The pE value can be an especially useful guide to redox chemistry in a soil solution if flooded conditions prevail, which largely eliminates spatial variability in the reducing environment, and if a single reduction half-reaction dominates. This situation may be expected to occur in a waterlogged soil that has a high content of iron oxides and a low content of manganese oxides, sulfate, and nitrate. During the initial phase of reduction, which takes place within a few months after flooding, the half-reaction (4.5) may be controlling the pH value and Fe H activity in the soil solution. The relation between pE and these two quantities is pE = 15.87 - 3pH - log (Fe H

(4.6)

)

assuming that both liquid water and Fe(OH)ls) are in their Standard States. Under oxidizing conditions, ferric hydroxide is, in fact, only a metastable solid phase since, for example, I::1Go = -25.98 kJ mol" for the reaction Fe(OH)ls) = FeO(OH)(s)

+ H 20 (l)

However, in soils that undergo alternate flooding and drying, it seems that ferric hydroxide is favored kinetically over the more stable goethite, FeO(OH)(s). In that case, Eq. 4.5 could describe the control of pH by Fe(III) reduction. This expectation was confirmed reasonably well in measurements of pE, pH, and (Fe H ) made on soil solutions extracted from 32 flooded soils found in the tropics.' The experimental data produced the statistical correlation expression pE

+ 3pH

= 17.90

± 0.24

- 1.02

± 0.06 log (Fe H

)

The first term on the right side of this expression differs by about 11 % from the constant term on the right side of Eq. 4.6, but this difference reflects only a very small increase in ,.,.0 for Fe(OH)J(s) in the soil relative to ,.,.0 for Fe(OHMs) as

ELECTROCHEMICAL EQUILIBRIA IN SOILS

110

calculated with the help of log K R given in Table 4.1. If log K R = 17.90 then, by Eq. 4.5, JLO(Fe(OHUs»

JLO(Fe2+) + 3JLO(H zO(l» + 5.708 (17.90) = -78.87 + 3( -237.141) + 102.17 = -688.12 kJ mor '

=

whereas if log K R = 15.87, JLO(Fe(OH)ls» = -699.71 kJ mor ' by the same method of calculation. This difference of only 1.7% in the value of the Standard State chemical potential of ferric hydroxide results in a difference of II % in the value of log K R , as noted above. The chemical potential of the electron in aqueous solution is related to the pE value by the equation

JL(e) = - RT(ln 10) pE

(4.7)

since JLO( e) = O. It is always possible to express JL( e) in units of energy per unit charge, instead of energy per mole, by dividing both sides of Eq. 4.7 by the Faraday constant, F = 96484.56 C mol", If this is done, the chemical potential of the electron will have the units of joules per coulomb, or volt, and Eq. 4.7 may be written in the form (at T = 298.15 K) EH =

RT In 10 F pE = 0.059155 pE

(4.8)

where E H = -JL(e)/ F is called the electrode potential. (The second step in Eq. 4.8 results from substituting 5708 J mor ' for RT In 10.) The electrode potential, with the convention that its sign be the same as that of pE, is just an alternate form of the chemical potential of the electron. Its use in the thermodynamic description of redox reactions is convenient if JL( e) can be determined for the system of interest with the help of an electrochemical cell. Often, in soil solutions and other natural waters, it is not possible to make an accurate electrochemical measurement of the electrode potential. When this situation prevails, there is no particular advantage in employing E H instead of the more fundamental pE value. On the other hand, if the standard electrodepotential, E~, has been determined for a reduction half-reaction, the value of log K R for the reaction may be computed readily with the equation (at T = 298.15 K): log K R

_ -

ne ° 0.059155 E H

(E~

in volts)

(4.9)

where n, is the number of electrons transferred in the reaction. Equation 4.9 can be derived for the case n, = I by writing down Eq. 4.3 and employing Eq. 4.8 under the condition that all species in the reaction, except the electron, are in their Standard States. The case n, "f- I can be handled after deriving Eq. 4.3 for the more general form of Eq. 4.1, wherein the stoichiometric coefficient of e is n, instead of 1.0.

ELECTROCHEMICAL EQUILIBRIA IN SOILS

111

4.3. SOLUBLE REDOX SPECIES When two or more water soluble chemical species contain the same element but contain it in different oxidation states, their concentrations in a soil solution can be predicted as a function of pE and pH in precisely the same way as the concentrations of hydrolytic species were predicted as a function of pH alone in Section 3.4. As a typical example of the calculation of the concentrations of soluble redox species, consider the speciation of nitrogen in the absence of and denitrification. The soluble species to be considered are NO;, NO NHt. These three ions are linked through the nitrate reduction reactions

z,

NO; + 2H+ + 2e = NO NO; + 10H+ + 8e = NHt

z + H 20 (l) +

3H 20 (l)

log K , = 28.2 log K 2 = 119.25

These two reactions, in turn, are obtained by combining the second, third, and fourth reduction half-reactions in Table 4.1 in the appropriate manner and then clearing fractional stoichiometric coefficients. The equation of mole balance in this example is: N TS = [NO;]

+

[NOn

+

[NH:J

(4.10)

This expression can be solved for the distribution coefficient aNO,

_[NO;] N

=

(4.11a)

TS

in terms of pE, pH, and the equilibrium constants K, and K 2• After factoring the nitrate concentration from the left side of Eq. 4.10 one obtains N TS = [NO;]{l - [NO;]{l _ [NO;]{l

+ ([NOn/[NO;]) + ([NH:J/[NO;])} +K + KiH+)IO(e)8} + 102(14.I-pH-pE) + 108(14.91-1.25 PH-PEl} 1(H+)2(e)2

(4.12)

where it has been assumed that the Davies equation (Eq. 2.96) may be used to express the activity coefficient of each ion, thereby making all concentration ratios equal to activity ratios. It follows from Eqs. 4.11a and 4.12 that

+

aNO, = {1

102('4.,-pH-pEl

+

108(14.91-1.25 PH-PElt l

(4.11b)

The other two distribution coefficients of interest now may be written out immediately: aN02

a

Nil,

z]

z]

_

=

[NO N

-

[NH:J = [NHt] a 9 . - 108('4.'-1.25pH-pl:)a N [NO!] NO) NO,

=

-

[NO 102(I41-pH-pEl [NO;] aNO, aNO,

(4.11c)

TS

TS

(4.1Id)

.....

1.00

1.00 1.00

a N0 2

0.75

'I> 'I> 'I>

oJ

~

FIGURE 4.1. (a) Graph of the distribution coefficient aNO, in Eq. 4.11b. (b) Graph of the distribution coefficient aNo, in Eq. 4.11c. (c) Graph of the distribution coefficient aNH. in Eq. 4.11d. • - -_c

113

ELECTROCHEMICAL EQUILIBRIA IN SOILS

Equations 4.11b, 4.11c, and 4.11d provide the complete solution to the problem in this example. For a given pH value of the soil solution, the values of the distribution coefficientscan be calculated as functions of the pE value, or vice versa. Figure 4.1 shows graphs of the three distribution coefficients for 4 -< pH -< 8 and 3 -< pE -< 12. The most important features of the graphs are the increasing range of pE over which nitrate predominates as the pH value is increased and the very narrow pE range over which predominance shifts from nitrate to nitrite to ammonium at any fixed pH value. This latter property suggests that the three nitrogen species will exhibit a great deal of variability in their concentrations in soils whose degree of aeration fluctuates significantly. In an actual solution, of course, redox reactions do not occur in isolation but are coupled through complexation and precipitation reactions to other species in the system. For example, Eq. 4.10 would include terms for nitrate and amine complexes, in addition to those for free nitrate, nitrite, and ammonium ions, if a typical soil solution were under consideration. The calculation of nitrogen speciation then would proceed just as described in Sections 3.4 and 3.5. Indeed, redox reactions introduce no new mathematical elements into a speciation computation, any more than would the consideration of, for example, CO 2(g) reactions. The only new item brought in is an additional variable, the pE value, which, like the partial pressure of CO 2(g), must be specified in order to solve the mole balance equations. A more complicated example of speciation with redox reactions may be illustrated by a calculation using the soil solution composition data in Table 4.3. These data represent the concentrations of inorganic components in a saturation

TABLE 4.3. Composition of a saturation extract obtained from an Altamont soil (Typic Chromoxerert) Component

pC·

Component

pC

Component

pC

Ca

2.07 3.70 3.00 4.75 4.70 5.72 5.85 5.13

C0 3 S04 CI P04 CIT SAL PHTH ARG

2.70 2.70 2.28 4.00 4.14 4.27 3.97 4.49

ORN LYS VAL N0 3 MAL BES pH

4.36 4.36 4.36 2.17 3.97 4.27 6.30

K Na Fe Mn Cu Cd Zn

*pC = -log total molar concentration. CIT = citrate SAL = salycylate PHTH = phthalate ARG = arginine ORN - ornithine LYS - lysine VAL - valine MAL - maleate BES - benzylsulfonote

ELECTROCHEMICAL EQUILIBRIA IN SOILS

114

extract, plus the concentrations of organic ligands in the extract at values expected after the addition of sewage sludge to the soil. The problem in this example is to predict the concentrations of the metals Cd, Cu, Fe, and Zn in solution after the pE value is lowered to - 2.00, which represents a highly reduced soil, with the pH value remaining at 6.30. The results of the speciation calculation for the four metals are shown in Table 4.4. The principal conclusions to be drawn from the data in the table are that the reduction in pE value induces a dramatic change in the concentration of each metal cation in solution through sulfide precipitation and that either organic complexes or the free ionic form are the predominant soluble species.

4.4. pE-pH DIAGRAMS A pE-pH diagram is a predominance diagram in which the electron activity is the particular activity variable chosen to plot against pH. Thus the pE value plays the same role as did, for example, the value of log Pea, in Fig. 3.2. The construction of a pE-pH diagram is, accordingly, just another example of the construction of a predominance diagram, as described in Section 3.2. The difference lies only with the interpretation of the diagram, which is in terms of redox species instead of, say, carbonate and hydroxide solids. To illustrate the development of a pE-pH diagram, consider the system FeH 20 . The species to be accounted for are FeH , FeH , Fe(OHMs), FelOH)g(s), and Fe(OHMs). Other solid phases [e.g., FeO(OH)(s)] are more stable thermodynamically than the three just listed, but kinetic data suggest the metasta-

TABLE 4.4. Distribution" of Cd, Cu, Fe, and Zn in the soil solution whose composition is described in Table 4.3 Species Free ionic Sulfide solid Carbonate complexes Chloride complexes Hydroxide complexes Nitrate complexes Phosphate complexes Sulfate complexes Sulfide complexes Organic complexes

Cd 8.82 99.8% 10.49 9.38 12.77 10.78 13.09 9.93 11.00 9.8 _.._-------_.-..

Cu

Fe

Zn

17.22 100.0% 16.19 19.39 19.07 19.28 20.25 18.23 21.64 14.6

5.42 42.1% 5.99 8.06 8.77

6.82 97.0% 7.30 9.54 9.66 8.88 10.46 7.83 13.26 8.3

8.52 6.63 5.2

_-

-_..__."-------._-'-' _.. ·Free ionic species are reported as -log molar concentration. Soluble complexes are reported as -log molar concentration of all complexes formed. Solids are reported as percentages of the total concentrations in Table 4.3 thai precipitated al pI:' - - 2.00. _-~--~.-.

__ _.._,-_. __ .---.-.__.. ...

"

ELECTROCHEMICAL EQUILIBRIA IN SOILS

115

bility of the hydroxide phases in soil redox systems.' The reduction reactions that define the boundary lines in the pE-pH diagram are: log K R = 13.01 Fe H + e = Fe H (4.13) pE = 13.01 - log {(FeH)j(Fe H)} H log K R = 15.87 Fe(OH)ls) + 3H+ + e = Fe + 3H 20(l) H) (4.14) pE = 15.87 - 10g(Fe - 3pH log K = 5.27 3 Fe(OH)ls) + H+ + e = FelOHMs) + H 20(l) (4.15) pE = 5.27 - pH H log K = 42.34 Fe3(OHMs) + 8H+ + 2e = 3Fe + 8H 20(l) H) (4.16) pE = 21.17 - 1.5 log(Fe - 4pH log K = 3.79 FelOHMs) + 2H+ + 2e = 3Fe(OH)z(s) + 2H 20(l) pE = 1.89 - pH (4.17) H Fe(OH)2(S) + 2H+ = Fe + 2H 20(l) log "K; = 12.85 H) o = 6.425 - 0.5 log(Fe - pH (4.18) For each chemical reaction, the expression relating pE to pH was obtained by writing down the equilibrium constant in terms of activities [with all solids and H 20(l) assumed to be in their Standard States], taking the common logarithm of both sides of the resulting equation, and rearranging terms to solve for pE. For example, in order to derive Eq. 4.16, one writes: K = (Fe H)3 j(H +)8( e)2 = 1042.34 3 log(Fe H) + 8 pH + 2 pE = log K = 42.34 and Eq. 4.16 follows after moving all terms but 2pE to the right side, then dividing by 2. Just as was required in order to compute the predominance diagram for Zn-H 20-C02 in Fig. 3.2, the activity of the principal metal cation must be specified to compute a pE-pH diagram with the help of Eqs. 4.13 to 4.18. In this case, (Fe H) = 10- 5 is taken as a reasonable estimate for soil solutions. The resulting pE-pH diagram is shown in Fig. 4.2. The uppermost and lowermost boundary lines in the figure are, respectively, plots of pE = 20.61 - pH, which describes the oxidation of H 20(l) to 02(g) at Po, = 0.21 atm, and of pE = - pH, which describes the reduction of H 20(l) to Hig) at PH, = 1.0 atm. These two equations prescribe the limits of stability of liquid water and, therefore, set the boundaries for redox species in any soil solution. Because FeH(aq) does not predominate until pE > 13.0 and pH < 2.5, according to Eqs. 4.13 and 4.14 when (Fe H) = 10- 5, it does not appear in Fig. 4.2. The stability fields in the normal range of soil pH are filled largely by FeH(aq) and Fe(OH)ls), with the solid phases FelOH)g(s) and Fe(OHMs) entering as dominant species only for high pH and low pE values. Thus, under typical oxidizing conditions, with pE > + 7.0, iron hydroxide is the expected iron species, whereas under typical reducing conditions, with pE < + 7.0, Fe2+(aq) is expected, given the imposed condition, (Fe H ) - 10. 5• If the Fe2+(aq) activity were larger, all of the boundary lines in Fig. 4.2 would be shifted left.

ELECTROCHEMICAL EQUILIBRIA IN SOILS

116

+15.--------;:;:::-------:-----------,

, i

,"I

'I

J )

"

1

o

Fe 2+{oq)

-5

-IOL4

L-

L-

L-

5

6

7

L.....-_ _-----I

8

9

pH FIGURE 4.2. A pE-pH diagram for the system Fe-H 20 under the condition (FeH = 10- 5.

)

4.5. ELECTROCHEMISTRY IN SOIL SOLUTIONS AND SUSPENSIONS It was pointed out in Section 2.2 that the introduction of ionic species into the framework of a thermodynamic analysis of chemical phenomena in a soil solution is principally a convenient mathematical device, since an ionic species strictly cannot be described in thermodynamic terms. For example, in Section 2.3 the chemical potential of an ionic species was accorded a formal thermodynamic significance and manipulated in the same way as the chemical potential of a macroscopic quantity of matter, but with care taken always to meet the condition of overall electroneutrality. In this fashion, the chemical potentials of ionic species were incorporated usefully into discussions of complexation, precipitation-dissolution, and redox phenomena to derive results of thermodynamic

ELECTROCHEMICAL EQUILIBRIA IN SOILS

117

importance in Chapters 2 and 3 as well as in the present chapter. However, in none of these applications was it necessary to consider the process of charge transfer in which, by definition, a single ionic species passes alone from one phase to another. This kind of process occurs at the interface between an electrode and a soil solution or suspension. Therefore charge transfer processes are of paramount concern in the study of the electrochemistry of these aqueous media. By its very definition, a charge transfer process is not subject to the direct imposition of the electroneutrality condition. This fact necessitates a more careful examination of the concept of the chemical potential of an ionic species than was carried out in Chapter 2. To begin this examination, consider two teflon beakers, each of which contains an aqueous solution of 0.01 M Cu(N03)2 along with a wire made of pure copper that is immersed in the solution. Each of the two identical copper wires, in turn, is connected to one pole of a dry cell, such that an electrical potential difference V exists between the wires. Within either wire, the reaction CuH

+ 2e

= Cu

(or its reverse) takes place. According to Eq. 2.21, this reaction can be described by the expression J.L(Cu) = J.L(Cu H

)

+ 2J.L(e)

at equilibrium. Because J.L(Cu) = J.L°(Cu) in this case and the Standard State chemical potential of copper metal in the wire is zero by definition (see Section 2.2), this expression reduces to the equation (4.19) between the chemical potentials of Cu H and the electron in each wire. Now, if N denotes the wire connected to the negative pole of the dry cell and P denotes the wire connected to the positive pole, (4.20) relates the electrical potential difference to the chemical potential difference between the electrons in the two identical wires (F is the Faraday constant). The combination of Eqs. 4.19 and 4.20 produces the equation: J.LN(CU H

)

-

J.LP(Cu H

)

=

2 FV

(4.21)

Moreover, since equilibrium exists with respect to the transfer of Cu H between each copper wire and the solution of Cu(N03)2 bathing it, Eq. 4.21 can be written in the alternate form (4.22) where Nand P now refer to the aqueous solutions in which the copper wires labeled Nand P, respectively, are immersed. Suppose now that the copper wires arc removed very carefully from the two solutions without otherwise perturbing them. In that case, Eq. 4.22 still must

ELECTROCHEMICAL EQUILIBRIA IN SOILS

118

relate the chemical potentials of Cu H in the two solutions of 0.01 M Cu(N03b This fact makes it quite evident that, in general, the value of ~(CuH(aq» will depend not only on the chemical properties of the solution of CU(N03)2 (which in this example are exactly the same in each beaker) but also on the electrical state of the solution. This dependence on the electrical state does not exist, of course, for neutral species (nor in fact must it be taken into account explicitly if the condition of overall electroneutrality is imposed on a reaction involving ionic species). For this reason, it is more precise to call jL(CuH(aq» the electrochemical potential of Cu H in an aqueous solution and to denote it with a tilde, while reserving the name chemical potential for ~(Cu(N03Maq» and generally for u, which refers to any neutral species. The possibility that the electrochemical potential of an ionic species can differ between two chemically identical aqueous solutions, by virtue of a difference in electrical potential, suggests that the formal definition jL

= ZF¢ + A + RT In a O

(4.23)

should be examined as to its general applicability to a species of valence Z that is subjected to an electrical potential ¢. The quantity A0 is defined to be a function only of the Standard State temperature and pressure, as well as of the "purely chemical" nature of the substance whose activity is a. For a neutral species, Z = 0 and Eq. 4.23 reduces unambiguously to Eq. 2.35, with AO = ~o. In the case of the two solutions of CU(N0 3)2 just discussed, jLN(CuH(aq» = 2F¢N + A~ + RT In (CuHh V(CuH(aq» = 2F¢P + A~ + RT In (Cu'"); A~ = A~ (CUH)N = (Cu?"); and (4.24) which is consistent with Eq. 4.22. Here again there is no ambiguity in the separation of jL into "purely electrical" and "purely chemical" parts. If, instead of two 0.01 M CU(N03)2 solutions, Eq. 4.23 is applied in turn to CuH(aq) in solutions of 0.1 M CU(N03)2 and 0.001 M CU(N03)2, the difference in electrochemical potential of CuH(aq) will be given by the equation jL"(CuH(aq» - jL'(CuH(aq» = 2 F (¢" - ¢') + RT In{[Cu H]" j[CuH]'}

+ RT In ('Y~uhtu)

(4.25)

where" refers to the 0.1 M CU(N0 3)2 solution and' refers to the 0.001 M CU(N0 3)2 solution. The left side of Eq. 4.25 is well defined because it is measurable, for example, by means of a voltmeter and a pair of suitable electrodes. The molar concentrations on the right side of Eq. 4.25 also are measurable. Therefore the extent to which the electrical potential difference, (¢" - ¢'), which cannot be measured directly, has chemical meaning depends entirely on whether the ratio of single-ion activity coefficients, 'Y~uhtu, in Eq. 4.25 can be evaluated unambiguously. If, for example, this ratio can be calculated accurately

I

1

I

119

ELECTROCHEMICAL EQUILIBRIA IN SOILS

with the Davies equation, the electrical potential difference becomes well defined. If, on the other hand, the ratio of activity coefficients could not be represented accurately with the help of some model expression, or, otherwise, if the activity ratio for Cu H in the two solutions could not be measured, there would be no chemical significance to the separation of the electrochemical potential into "purely electrical" and "purely chemical" parts, as in Eq. 4.23. This conclusion applies, in particular, to the problem of defining the electrical potential difference between an aqueous solution in the pore space of a soil and a supernatant solution contacting the soil, that is, a so-called Donnan potential. Unless the ratio of activity coefficients on the right side of Eq. 4.25 is well defined, the Donnan potential, cJ>" - cJ>', is without empirical significance. This important point can be made even more forcefully through a consideration of the equilibrium between Cu H in an aqueous solution and in a solid phase [e.g., a solid cupric electrode, or CulOH)2C03(S), or Cu-montmorillonite]. In this case, even though the electrochemical potential difference ji(CuH(s» - ji(CuH(aq» may be well defined, there is no known experimental means by which the terms in cJ> and ")...0 can be distinguished from one another unambiguously. Therefore the formal definition in Eq. 4.23 cannot be applied usefully to this example, and only the electrochemical potential itself may be employed to describe thermodynamic equilibrium. The principal conclusion to be drawn from these examples is that an electrical potential difference between two phases of different chemical composition cannot be defined thermodynamically. The electric potential is a concept that has meaning only when identical phases (e.g., two pieces of copper wire) are under consideration, or when it is possible to determine unambiguously the ionic activity coefficient ratios in two aqueous solutions that contain the same components. In all other situations and, in particular, when ionic equilibria between aqueous solutions and solid phases are investigated, the division of the electrochemical potential into a term in cJ> and a term in ")...0 is entirely arbitrary. With the concept of the electrochemical potential established, it is possible to describe a number of important electrode phenomena in soil solutions and suspensions. Consider first a pair of silver-silver chloride electrodes immersed in a composite clay suspension-aqueous solution system, as diagramed here:

L Ag; AgCl

R

W NaX(s), NaCl(aq)

NaCl(aq)

AgCl; Ag

(4.26)

where the single vertical line refers to an electrode-solution interface and the double vertical line marked W refers to a thermodynamic wall that is impermeable to the colloidal anion, X-I, but is permeable to Na ", Cl-, and water (e.g., a dialysis membrane). At the left electrode (L), the oxidation reaction Ag(s)

+ CI

(I., aq) - AgCI(s)

+e

ELECTROCHEMICAL EQUILIBRIA IN SOILS

120

occurs, where L refers to a point inside the electrode assembly, and jiL(e) = ~(Ag(s»

+ jiL(CI-)

- ~(AgCl(s»

(4.27)

at equilibrium. At the right electrode (R), the reduction reaction AgCl(s)

+ e = Ag(s) + Cl-(R, aq)

occurs and (4.28) at equilibrium. The EMF across the electrode pair is given by the equation:" (4.29) where E is the EMF in units of volts. The introduction of Eqs. 4.27 and 4.28 into Eq. 4.29 produces the expression (4.30)

If equilibrium with respect to Cl-(aq) exists in the composite electrode-clay suspension-aqueous solution system, then jiR(CI-) = jisU(CI-) = jiSo(CI-) = jiL(Cl-), where "Su" refers to the suspension and "So" refers to the aqueous solution of NaCl. Under this condition, it follows that E = 0 in Eq. 4.30. Therefore the absence of an EMF for an electrode pair as portrayed in Eq. 4.26 can be employed as an indicator of chloride ion equilibrium between a soil suspension and an aqueous solution. This conclusion does not depend on the type or concentration of the suspended colloid, or on the type or concentration of the cation balancing cr in the aqueous solution phase. Next, consider a soil suspension into which is immersed an electrode pair consisting of a glass electrode reversible to Na + and a silver-silver chloride electrode: R L Na(gl)

NaX(s), NaCl(aq)

AgCl; Ag

(4.31 )

Other colloidal solids than NaX(s) and other electrolytes than NaCI may be present in the soil suspension, but they are assumed not to interfere with the performance of the electrode pair in Eq. 4.31. The development of electrical potential at the sodium electrode is a somewhat complicated process that includes the creation of a difference in ji(Na+) across the glass membrane separating the soil suspension from the inner solution of the electrode, the slow diffusion of Na + inside the glass membrane, and the oxidation reaction at the inner electrode.' Since a detailed account of these processes is not required here, it suffices to describe the electrode reactions in Eq. 4.31 formally by the expression Na(gl)

+

AgCI(s) - Na+(L, aq)

+ CI-(R, aq) + Ag(s)

(4.32)

ELECTROCHEMICAL EQUILIBRIA IN SOILS

121

TABLE 4.5. Conventions in the assignment of EMF values to galvanic cells 1. The cell reaction is written as if oxidation occurs spontaneously at the left electrode and reduction at the right electrode. 2. The cell EMF may be defined by the relation jiR(e) -

jiL(e)

== - FE

where ji(e) is the electrochemical potential of the electron; Rand L refer to the right and left electrodes, respectively; F is the Faraday constant; and E is the cell EMF, in volts. 3. If the overall cell reaction is aA

+ bB + ... = xX + yY +

the cell EMF (in volts) may be defined by the relation XIL(X)

+ YIL( Y) + ... -

alL(A) -

blL(B) -

...

== -

FE

4. In these conventions, it is assumed that all reduction or oxidation half-reactions are written in terms of the transfer of 1 mol of electrons.

where Land R refer to points just outside the sodium electrode and inside the silver-silver chloride electrode assembly, respectively. The EMF created by the reaction in Eq. 4.32 is given by the expression - FE

=

ji,L(Na+)

+ ji,R(CI-) + }L(Ag)

- ji,(Na(gl)) - }L(AgCI(s))

(4.33)

The conventions that lead from Eq. 4.32 to Eq. 4.33 are outlined in Table 4.5. (Note that the two electrode reactions in 4.26 could have been added and treated like Eq. 4.32 to obtain Eq. 4.30.) If equilibrium exists with respect to Na+(aq) and Cl Taq), then ji,L(Na +) = ji,sU(Na+)

ji,R(CI-) = ji,SU( CI-)

and Eq. 4.33 may be written in the form - FE = [}L(Ag) - ji,(Na(gl)) - }L(AgCI(s))]

+ ji,sU(Na+) + ji,sU(CI-)

But

according to Eqs. 2.46a and 2.35, where ( )su refers to an activity in the suspension. Therefore

RT

E = A - Fin (NaCl)su

where

A

== (1/ F)[}L(AgCI(s)) + ji,(Na(gl))

- }L(Ag) - }L°(NaCI)]

(4.34)

122

ELECTROCHEMICAL EQUILIBRIA IN SOILS

Equation 4.34 demonstrates that the EMF of the electrode pair in 4.31 can be employed to measure the activity of NaCI in a soil suspension." Equation 4.34 is strictly thermodynamic and, therefore, cannot provide direct information about the electrical nature of colloid-electrolyte interactions in soil suspensions. Whatever contributions electrical potentials may make to ~SU(Na +) and ~SU(CI-) must cancel completely when these two electrochemical potentials are added. Suppose now that the silver-silver chloride electrode in Eq. 4.31 is replaced by a saturated calomel electrode: L

R

I I

Na(gl)

I

NaX(s), NaCI(aq) : KCI(satd.)

Hg 2Clb); Hg(l)

(4.35)

I

I

The dashed vertical line in 4.35 refers to a liquid-liquid junction between the suspension and the saturated KCI salt bridge (as well as to a thermodynamic wall through which colloidal material cannot pass). The overall reaction in Eq. 4.35 can be written out formally in a manner similar to Eq. 4.32: Na(gl)

+ ! Hg 2CI2(s) = Na+(L, aq) + Cl-(R, aq) + Hg(l)

(4.36)

I

:l

The EMF developed in this case differs somewhat from that in Eq. 4.33 because of the electrical potential difference, E J , which arises across the liquid-liquid junction: - F(E - E J ) = ~L(Na+)

+ ~R(CI-) + ~(Hg(l»

- ~(Na(gl»

-! ~(Hg2Cl2(S»

(4.37)

The liquid junction potential comes about because of the variation in chemical composition in passing from the suspension into the KCI salt bridge. This variation in composition induces mixing processes across the liquid junction. However, because of the construction of the saturated calomel electrode, these mixing processes are unable to produce equilibrium with respect to Cl-(aq) transfer. The steady-state transport process that ensues does not permit the equality ~R(CI-) = ~SU(CI-) to be applied in this case. Instead, the permanent differences in the electrochemical potentials of Cl-(aq) and other charged species in the system across the junction combine to produce the electrical potential difference, E J • The calculation of E}> accordingly, cannot be done by purely thermodynamic methods but requires information relating to steady-state ionic transport processes. Since the only variable quantity on the right side of Eq. 4.37 is the first term, and since ~L(Na+) = ~SU(Na+), the EMF developed in Eq. 4.35 may be expressed (4.38) where B equals ( - I) times the sum of the constant electrochemical and chern-

~

ELECTROCHEMICAL EQUILIBRIA IN SOILS

123

ical potentials in Eq. 4.37 divided by the Faraday constant. Now suppose that the electrochemical potential of Na" in the suspension is separated arbitrarily into the two terms (4.39) where ~ is a parameter that incorporates both the electrical state of the suspension and the Standard State properties of Na "(aq). The combination of Eqs. 4.38 and 4.39 then produces the formal relationship F

-In (Na+)gu = RT (E - E J

-

B')

(4.40)

where

B' = B - (Rn~~/ F) Equation 4.40 could be employed to calculate the activity of Na + in the suspension, provided that E J and B' can be determined experimentally or eliminated by some method of calibration. If the soil suspension in Eq. 4.35 were an aqueous solution instead, a scale of activity values for Na" could be defined in terms of EMF data obtained for standard reference solutions of prescribed (Na+) in exactly the same way as the scale of (H+) values (the pH scale) is defined, arbitrarily, in terms of EMF data for standard buffer solutions. However, the success of this extrathermodynamic calibration technique depends entirely on the extent to which E J and B' in the standard reference solutions are the same as E J and B' in the solution of interest. For the case of a soil suspension, the presence of colloidal material may cause these two parameters to differ very much from what they would be in a reference aqueous solution. If the difference is indeed large, the value of (Na"), (H+), or any other ionic activity estimated with the help of standard solutions and an equation like Eq. 4.40 would be of no chemical significance. This point can be illustrated in a concrete manner by a consideration of the double electrode-pair system: w

L

H,(I); H"CI,(s)

KCI(sald.)

NaX(s), NaCI(aq)

Na(gl)

R

,I Na(gl)

NaCI(aq); KCI(satd.)

H"CI,(s); HI(I)

(4.41)

,

The electrode system in 4.41 is essentially a double version of 4.35, except that one of the aqueous media is just a solution of NaCl (and other, noninterfering salts). As in 4.26, the thermodynamic wall in 4.41, marked W, does not permit the transfer of colloidal material. The EMF developed by this system is the algebraic sum of those developed by the two calomel-sodium electrode pairs: EI. -

f.:R -

- E~ -

B

f.:' + B -

+ (1/ F),:lsU(Na +) (J / F),:lSo(Na I)

ELECTROCHEMICAL EQUILIBRIA IN SOILS

124

according to Eq. 4.38, and E = ER

+

EL =

(1/ F)[ji,sU(Na+)

- ji,So(Na+)]

+ E~

- E~

(4.42)

If equilibrium exists with respect to Na+(aq) between the suspension and the solution, then ji,sU(Na+) = ji,So(Na +), and Eq. 4.42 reduces to the simple expression E=E~-E~

(4.43)

Thus, when equilibrium conditions obtain, the EMF of the double electrode-pair system is determined solely by the electrical potential differences developed at the two liquid junctions that involve KCI salt bridges. The fact that this EMF develops is known as the suspension effect? The significance of the suspension effect for ionic interactions within soil suspensions was at one time the subject of much debate. However, Eq. 4.43 makes it clear that only ionic transport processes across the liquid junctions need to be taken into account in order to evaluate E. Therefore ionic transport processes across the semipermeable membrane between the suspension and the solution are not germane. Moreover, since neither E~ nor E~ can be calculated by strictly thermodynamic methods, the interpretation of E must be made in terms of specific models of ionic transport across salt bridges contacting suspensions and solutions. Thus the relation between E and the behavior of ions in soil suspensions is not direct. Finally, the equality ji,sU(Na+) = ji,So(Na+), provided by thermodynamics, is only a statement of electrochemical equilibrium, not a conclusion relating to ionic interactions within soil suspensions. This equality says nothing, for example, about the electrical potentials experienced by Na "(aq) in either the suspension or the solution, nor can it do so, because the division of each electrochemical potential into electrical and chemical parts according to Eq. 4.23 would be, in this case, a completely arbitrary step.

NOTES I. See R. Bartlett and B. James, Behavior of chromium in soils: III. Oxidation, J. Environ. Qual. 8:31-35 (1979). 2. The analogy between pH and pE is developed in great detail on pp. 303-315 in W. Stumm and J. J. Morgan, Aquatic Chemistry, John Wiley, New York, 1970. The concept of pE appeared first in H. Jargensen, Redox-malinger, Gjellerup, Copenhagen, 1945. 3. The measured pE cri, values were calculated with the help of Eq. 4.8 from data presented in F. T. Turner and W. H. Patrick, Chemical changes in waterlogged soils as a result of oxygen depletion, Trans. Int. Congr. Soil Sci., 9th 4:53-65 (1968), and in W. E. Connell and W. H. Patrick, Sulfate reduction in soil: Effects of redox potential and pH, Science 159:86-87 (I 968). 4. F. N. Ponnamperuma, E. M. Tianco, and T. Loy, Redox equilibria in flooded soils: I. The iron hydroxide systems, Soil Sci. 103:374-382 (1967). 5. See, for example, F. N. Ponnamperuma, E. M. Tianco, and T, Loy, cited in note 4, and U.

"

ELECTROCHEMICAL EQUILIBRIA IN SOILS

125

Schwertmann and R. M. Taylor, Iron oxides, pp. 145-180 in Minerals in Soil Environments (ed. J. B. Dixon et al.), Soil Science Society of America, Madison, Wis., 1977. 6. The conventions employed in assigning EMF values to electrode pairs are discussed in R. Parsons, Manual of symbols and terminology for physicochemical quantities and units. Appendix III. Electrochemical nomenclature, Pure and Appl. Chern. 37:500-516 (1974). 7. For a complete review, see Chapter 1 in R. A. Durst (ed.), Ion-Selective Electrodes, NBS Spec. Pub!. 314, U.S. Government Printing Office, Washington, D.C., 1969. 8. See I. Shainberg and A. Caiserman, Electrochemical potential of NaCI in Na-montmorillonite suspensions, Soil Sci. 104:410-415 (1967). 9. This result was derived independently by K. L. Babcock and R. Overstreet, On the use of calomel half cells to measure Donnan potentials, Science 117:686-687 (1953), and by J. Th. G. Overbeek, Donnan-E.M.F. and suspension effect, J. Colloid Sci. 8:593-605 (1953). See also J. Th. G. Overbeek, The Donnan equilibrium, Progr. in Biophysics 6:58-94 (1956).

FOR FURTHER READING R. G. Bates, Determination of pH, 2nd ed., John Wiley, New York, 1973. This book contains fine discussions of galvanic cells, liquid junctions, and methods for the measurement of proton activity in aqueous solutions and suspensions. Chapters 1 to 3 and 9 are especially relevant. H. L. Bohn, Redox potentials, Soil Sci. 112:39-45 (1971). This review article presents a careful survey of the problems associated with the electrical measurement of pE in soil solutions. C. W. Davies and A. M. James, A Dictionary of Electrochemistry, John Wiley, New York, 1976. This is a fine ready-reference on all aspects of electrochemistry in aqueous solutions. A brief discussion is given for every important experimental device or theoretical concept, with each entry arranged alphabetically. E. A. Guggenheim, Thermodynamics, 5th ed., North-Holland, Amsterdam, 1967. Chapter 8 is an essential prerequisite to the reading of Section 4.5. D. A. Macinnes, The Principles of Electrochemistry, Dover Publications, New York, 1961. Chapter 13 in this classic monograph gives a complete introduction to the theory of liquid junction potentials. F. N. Ponnamperuma, The chemistry of submerged soils, Advan. in Agronomy 24:2996 (1972). This article presents a thorough review of redox reactions in flooded soils. W. Stumm and J. J. Morgan, Aquatic Chemistry, John Wiley, New York, 1970. Chapter 7 in this stanard textbook gives a thorough introduction to redox phenomena in natural waters, including both kinetics and equilibria.

5 THE THERMODYNAMIC THEORY OF ION EXCHANGE

5.1. THE ION EXCHANGE SELECTIVITY COEFFICIENT

The cation exchange reaction between Na+ and Ca H in Eq. 2.44 is a special case of the binary, stoichiometric exchange reaction vAXu(s)

+ uBClv(aq)

= uBXis)

+ vACl.{aq)

(5.1)

where A u+ and B v+ are the two cations exchanging on X-I, which represents one equivalent of an exchange complex. The anion balancing the cationic charge in the aqueous solution phase, Cl-, is a common one in expetiments, but the fact that it is Cl" and not, for example, CO~-, is not essential: the equilibrium constant that describes the reaction in Eq. 5.1 is (5.2) which is a generalization of Eq. 2.45. According to the discussion in Section 2.7, the conditional equilibrium constant that corresponds to Kex in Eq. 5.2 is

s, =

N~(ACl.)v/ N;.(BClv)u

(5.3)

where N B is the mole fraction of BXv(s) in the exchanger phase and N A is that ofAXu(s). Eq. 5.3 is a generalization of Eq. 2.102. The parameter K; will not be equal to Kex and, therefore, cannot be used to calculate !100 for the reaction in Eq. 5.1, unless the exchanger phase happens to be an ideal solid solution. On the other hand, K y is related to the conditional separation/actor, which indicates quantitatively the preference of the exchanger for BV+ relative to A "": (5.4) where m denotes a molality. If the cation BV+ is preferred, then "D > 1; if the cation A u+ is preferred, CD < 1. According to Eqs. 2.87, 2.88, 5.3, and 5.4, the relation between K; and "D is

s;

=

[CD]UN~-vm~-u[-y±(AClu)]('+U)V(u/v)Uv/[-y±(BCl,)](I+V)U

(5.5)

This complicated expression reduces to Ky 126

-

[CD]U

(5.6)

THE THERMODYNAMIC THEORY OF ION EXCHANGE

127

for homovalent exchange (i.e., u = v in Eq. 5.1) under the condition that the mean ionic activity coefficients are functions only of the ionic strength. In this case, K; > 1 indicates a selectivity for BV+ in the exchanger, whereas K; < 1 indicates that AU+ is preferred instead. Because of this fact, K; is known as a selectivity coefficient. In particular, as mentioned already in Section 2.7, K; is the Vanselow selectivity coefficient. The concepts just described may be applied equally to the anion exchange reaction

where CX- and DW- are the two anions exchanging on Y+, which denotes one equivalent of the anion exchange complex, and the cation balancing the anionic charge in aqueous solution has been chosen to be Na". The equilibrium constant that describes the reaction in Eq. 5.7 is (5.8) and a selectivity coefficient for the reaction may be defined analogously to K; in Eq.5.3. The equilibrium constant Kex is the appropriate quantity with which to characterize ion exchange reactions as long as they do not exhibit hysteresis effects. When an exchange reaction is reversible, it is useful to measure K; as a function of the exchanger composition, since any variation in K; is related directly to that of the activity coefficients of the components of the exchanger phase. For example, in the case of the cation exchange reaction in Eq. 5.1, (5.9) where, in accordance with Eq. 2.78, (5.10) Thus the dependence of K; on the composition of the exchanger is the same as that of the ratio of activity coefficients,fl.I,ro. In Section 2.6, the Reference State in which the rational activity coefficient of a solid solution component has unit value was chosen to be that of the pure component at T = 298.15 K and P = 1 atm (see Table 2.5). This specification of the Reference State is not wholly adequate when the solid solution component is part of an exchanger phase, however, because ion exchange reactions invariably are carried out in aqueous media. The question arises, therefore, as to what relationship a homoionic exchanger phase at T = 298.15 K and P = 1 atm should have to the aqueous solution that must be brought into contact with it in order to initiate an exchange reaction. If the prescription in Table 2.5 is followed exactly, the homoionic exchanger should be entirely unhydrated when it is in the Reference State. This condition, however, is difficult to achieve experimentally and bears little resemblance chemically to that of an exchanger bathed by a soil solution. Moreover, the chemical potential of a homoionic exchanger in the Stan-

128

THE THERMODYNAMIC THEORY OF ION EXCHANGE

dard State (which is the same as the Reference State) no longer could be measured directly in a solubility experiment. For example, the measured values of /.L0 for homoionic montmorillonites, such as those employed in Section 2.5 to calculate Kex for Na+-Ca H exchange, would be invalid generally because they are determined routinely with the help of solubility data. For these kinds of reasons, it is necessary to include a statement concerning an aqueous solution phase when defining the Reference State and the Standard State of an exchanger phase component. At first it may seem enough to define the Reference State of a component of an exchanger phase as the homoionic exchanger, at 298.15 K and 1 atm pressure, in contact with an aqueous solution containing a salt of the exchangeable cation. That this statement will not suffice may be seen as follows. Consider a soil exchanger that is saturated completely with Na" and is in equilibrium with a 0.1 M solution of NaCI at 298.15 K and under a pressure of 1 atm. In this situation, ji,(Na+, ex) = ji,(Na +, aq), according to the discussion in Section 4;5, where ji, is an electrochemical potential and ex refers to the exchanger phase. This equality could provide a value for the Reference State electrochemical potential of Na" in the exchanger phase, since ji,(Na+, aq) can be measured. Now suppose that the same soil exchanger is put into contact with a solution of 0.05 M NaCI at the same temperature and pressure. The exchanger is still in its Reference State by hypothesis. At equilibrium, the electrochemical potentials of Na + in the exchanger and aqueous solution phases again must be equal, but this time they cannot have the same values as when the concentration of NaCI was 0.1 M. In both cases, then, the exchanger supposedly was in its Reference State with unit activity coefficient, yet the value of ji,(Na+, ex) is different in each case. This paradox can be avoided only by acknowledging that something specific must be said about the composition of the aqueous solution phase when the Reference State of an exchanger is defined.' The prescription that is most widely accepted at present (in principle, if not always in practice) is that given by Gaines and Thomas.' It specifies that an exchanger phase component is in the Reference State if it is at unit mole fraction, at 298.15 K and under 1 atm pressure, and is in equilibrium with an infinitely dilute solution containing a salt of the exchangeable ion. For ion exchange experiments, the Gaines-Thomas definition of the Reference State may be incorporated by measuring K y for a fixed exchanger composition at several ionic strengths and extrapolating to zero ionic strength to obtain a value of K; based on the Infinite Dilution Reference State.' This procedure, or one equivalent to it, very seldom is followed in actual ion exchange experiments. Fortunately, it seems that K; for most exchange reactions of interest in soils exhibits only a slight dependence on ionic strength. Thus the common neglect of the extrapolation procedure may not be too serious an error. This definition applies to the Standard State as well, once it is specified additionally that the infinitely dilute solution must contain salts of the ions in the exchanger structure. With respect to the determination of /.L0 for exchangers

THE THERMODYNAMIC THEORY OF ION EXCHANGE

129

by solubility measurements, the Gaines-Thomas definition may be taken into account by determining the lAP for the exchanger at several ionic strengths, then extrapolating the data to obtain the lAP at zero ionic strength. In this limit, the lAP equals the solubility product constant for the exchanger, since the activity of the exchanger now has unit value by definition. Equation 5.9 makes it clear that the dependence of K; on the exchanger phase composition is related, in turn, to that of the rational activity coefficients, fA and fB' One approach to obtaining a systematic view of how K; depends on exchanger composition, therefore, has been the use of model expressions for the rational activity coefficients as functions of the mole fraction of one of the exchanger components. For example, suppose that In fA and In fB' pertaining to the univalent exchangeable cations, A + and B+, respectively, are expressed as the power series: In fA = alNB + a2M In fB = s,», + b2~

+ a3N~ + a4M + b3Ni + b4N'l

(5.11a) (5.11b)

where the a, and b, are empirical coefficients. Not all of these coefficients are independent, however, because the Gibbs-Duhem equation (Eq. 1.25), applied to the exchanger phase at fixed T and P, imposes a constraint on the variation with composition of the two activity coefficients. With the help of an argument exactly like that employed to derive Eq. 3.30, one can show that, for uni-univalent exchange (u = v = 1 in Eq. 5.l): (5.12) The combination of Eq. 5.11 with Eq. 5.12 then yields the expression: (aINA + 2a 2NANB + 3a3NAM + 4a4NA~)dNB + (bINB + 2b 2NANB + 3b3NB~ + 4b 4NBNDdNA = (aINA + 2a 2NANB + 3a 3NAM + 4a4NAN~ - blNB - 2b 2NANB - 3b3NB~ -4b4NBNi)dNB = [(al + 2a 2 + 3a 3 + 4a 4 + b, - 2b 2)NA + (-2a 2 - 6a 3 - 12a4 + 2b 2 - 3b3)~ + (3a3 + 12a4 + 3b 3 - 4b 4)Ni + (-4a 4 + 4b 4)N'l - b l ] dNB = 0 (5.13)

where the second step comes from invoking the condition dNA = -dNBand the third step from the condition N B = 1 - N A , both conditions being statements of the law of mass conservation in the exchanger phase. Equation 5.13 is true in general only if each coefficient of a power of N A vanishes identically. Therefore

a l + 2a 2 + 3a 3 + 4a 4 + b, - 2b 2 = 0 2a2 + 6a] + 12a4 - 2b 2 + 3b 3 = 0 3a] + 1204 + 3b] - 4b 4 = 0 0 4 - b, - 0 hi - 0

(5.14a) (5.14b) (5.14c) (5.14d) (5.14e)

THE THERMODYNAMIC THEORY OF ION EXCHANGE

130

The incorporation of Eq. 5.l4c into Eq. 5.l4b produces the expression

2a 2 + 3a 3

+ 4a 4 -

2b 2 = 0

after a glance at Eq. 5.l4d. This last expression is compatible with Eqs. 5.l4a and 5.14e only if a l = o. Thus

al = b2 = b, = b, =

b, = 0 a2 + (3/2)a 3 + 2a4 -a 3 - (8/3)a 4 a4

Accordingly, Eqs. 5.lla and 5.llb are reduced to the following expressions:

+ a3~ + a4M + 2a 4)M, - (a 3 + ~ a4)N i +

In IA = a2M

In/B

= (a 2 + ~ a,

a4N';,

(5.llc) (5.lld)

These equations are known as Margules expansions. If they are truncated by setting a3 = a4 = 0, they are known as the regular solution model. Whether truncated or not, Eqs. 5.11c and 5.1ld may be introduced into Eq. 5.9 (with u = v = 1) to provide a series expansion for In K; in terms of powers of either N B or N A • A useful and simple set of model expressions for In IA and In IB that may be applied to produce a closed-form equation for In K; are the van Laar

equations: In/A = al/[l In/B = a2/[1

+ (a N + (a 2N I

N BW B/a I N A )p A/a2

(5.l5a) (5.l5b)

The values of the empirical parameters a l and a2 in Eq. 5.15 are given by the equations: al

=

lim In/A

(5.l6a)

NAIO

a2 = lim In/B

(5.l6b)

Nal O

Therefore measurements of IA and IB as functions of exchanger composition could be extrapolated to obtain a l and a-. Alternatively, Eq. 5.15 could be solved uniquely for a l and (12 in terms of In lA' In IB' N A, and N B, and the two pararneters could be calculated from data on the activity coefficients at a single exchanger composition. For an arbitrary cation exchange reaction, the combination of Eq. 5.15 with Eq. 5.9 gives the simple, closed-form equation In K, = In K ex

+

va2M - ualN;.] ala2 [ ( N )2 . a2 B + alNA

In Fig. 5.1, In K; for the cation exchange reaction

MgX 2(s)

+ 2 J
- 2 J<X(s)

+ MgCI 2(aq)

(5.17)

THE THERMODYNAMIC THEORY OF ION EXCHANGE

131

5.0.__---------------------,

4.0

>

3.0

< c

2.0

• -

1.0

0.1

0.2

EXPERIMENT Eq. 5.17

0.3

0.4

05

0.6

0.7

0.8

09

1.0

NK FIGURE 5.1. Graph of In K; versus the mole fraction of KX(s), N K , for Mg H -K+ exchange on Yolo loam at room temperature. The solid line through the data points is a fit of Eq. 5.17.

on Yolo loam is shown fit to Eq. 5.17. It is evident that the van Laar equation, with its two empirical parameters, provides a fair description of the composition dependence of Kc, which is quite pronounced in this example. Often, instead of the rational activity coefficients, the activities of exchanger components themselves are modeled mathematically. These models usually are based on some empirical, chemical picture of the ion exchange reaction and are constructed for the specific purpose of developing a quantity that will be equal numerically to K.. in Eq. 5.2 or 5.8, regardless of changes in exchanger composition. Thus the principal objective is to find an equation for K. x written in terms of exchanger composition variables. Generally. these empirically based equations do not approximate K.. well over the entire range of exchanger composition and. therefore. they display some kind of composition dependence. just

132

THE THERMODYNAMIC THEORY OF ION EXCHANGE

as does K; in Eq. 5.3. For this reason, the empirically based model equations for Kex also are called selectivity coefficients in the literature of ion exchange.' As an example of the construction of an empirical selectivity coefficient, consider the uni-bivalent cation exchange reaction 2 AX(s)

+ BCI

2(aq)

= BXls)

+ 2 ACI(aq)

(5.l8a)

This reaction, which is a special case of Eq. 5.1, employs the so-called Vanselow convention by showing the exchangeable cations to react in moles with X-I. It is equally correct thermodynamically, however, to rewrite Eq. 5.l8a with the exchangeable cations reacting in equivalents with X-I: 2 AX(s)

+ BCllaq)

= 2 B 1/ 2X(S)

+ 2 ACI(aq)

(5.l8b)

This form of the exchange reaction represents the so-called Gapon convention. The equilibrium constant corresponding to Eq. 5.l8b is: (5.19) and, of course, has the same numerical value as K ex for the reaction in Eq. 5.l8a. In the Gapon model for K,.. it is assumed that (AX) = E A

(5.20)

where (5.21) are equivalent fractions of BXls) and AX(s), respectively, in the exchanger phase. The Gapon model expression for Kex is obtained by substituting Eq. 5.20 into Eq. 5.19 and taking the square root of both sides of the resulting expression: (5.22)

I

If the Gapon model is correct, Kla should have the same value as Kex for the exchange reaction in Eq. 5.18, regardless of the values of E B and EA' It may be ' noted, by comparison of Eq. 5.2 (for the case u = I, v = 2) with Eqs. 5.19 and 5.20, that the Gapon model can be derived from the specification: = E A / N A = (NA + 2NB) - 1 = E~/ N B = 4NB(NA + 2NB) -

fA

. fB

2

(5.23)i

Equations 5.23 provide for a connection between the approach of modeling the rational activity coefficients and that of modeling the exchanger component activities. This connection is illustrated in Table 5.1 for several empirically based models of Kex that have been proposed to describe the cation exchange reaction in either Eq. 5.18a or 5.18b. In each case, the model expression for Kex can be derived by writing

i

TABLE 5.1. Four model expressions for K.. that describe the uni-bivalent cation exchange reaction in Eq. 5.18

Vanselow Gaines and Thomas

N B(ACI)2 = N;. (BCI 2) 2(2 - N A)2NB(ACW GT K = (1 + N B)N;.(BCI2)

K RK

_ [2(2 - NA)NBJ2/~ (ACI)2 (1 + NB)NA (BCI 2)

Kb =

= NA =N

B

(AX) = E A (BX 2 ) = E B

-

[2(2 - N A)NBJ2 (ACI)2 (1 + NB)N A (BCI 2)

*The parameter fJ is a positive-valued, empirical constant. The case fJ

........

(AX) (BX 2)

x;

Rothmund and Kornfeld" Gapon

Activity Relation Assumed

Selectivity Coefficient

Authors

=

Activity Coefficients

I, = I

fB = I fA = (2 - NA)-l fB = 2(1 - NB)-l

(AX) = EIj~ (B 1/ 2X) = E'r/~

I,

(AX) = E A (B 1/ 2X) = E B

fA = (2 - NA)-I fB = 4NB(l + N B) -

1 leads to KRK

= /G; .

= ~./.~-l (2 -

fB =

41/~ Nlrf~-1

(1

NA)-I/~

+

2

NB)-2/~

134

THE THERMODYNAMIC THEORY OF ION EXCHANGE

and then substituting into this expression the empirical equations given in the table for the rational activity coefficients, fA and fB'

5.2. EXCHANGE ISOTHERMS

An exchange isotherm is a graph, at a fixed temperature, of the equivalent fraction of an exchangeable ion in the exchanger phase versus its equivalent fraction in the equilibrium bathing solution. The points along an exchange isotherm may be used either to calculate selectivity coefficients or to ascertain the general features of an ion exchange process. To aid in this latter objective, exchange iso- 1 ,j therms may be classified according to their behavior at low values of the ordinate and abscissa. The four commonly encountered classes of exchange isotherm are illustrated in Fig. 5.2. The S-curve isotherm is indicative of an exchangeable ion whose relative affinity for the exchanger is not large. This type of isotherm is possible when the exchangeable ion meets strong competition from the ion it replaces on the exchanger, or when it tends to cluster with like ions instead of mixing randomly. The I-curve isotherm indicates that the ion has a reasonably high relative affinity for the exchanger. This type of isotherm occurs when steric factors (e.g., the number of exchange sites available) are as important as direct competition effects. The H-curve isotherm is an extreme case of the L-curve isotherm. It reflects a very high relative affinity for the exchanger on the part of the ion. The C-curve isotherm is strictly linear. If an exchange isotherm is a C-curve type, it is possible that the exchangeable ion undergoes very little competition from the ion it replaces on the exchanger. This characteristic may result from equal affinities of the two ions for the exchanger." It is important to understand that the shape of an exchange isotherm and, therefore, its classification, depend on what takes place in the aqueous solution phase as much as in the exchanger phase. For example, dilution effects in the " aqueous solution phase can, in the case of heterovalent exchange, by themselves transform an L-curve isotherm into an S-curve isotherm or vice versa. To see this point in detail, consider again the cation exchange reaction in Eq. 5.1 and its equilibrium constant: (5.24) The introduction of the equivalent fractions, EA

_ -

UNA uNA + vNB

vN E B = - - - - -B" - uNA + vNB

(5.25a)

and (5.25b) which refer to the exchanger and aqueous solution phases, respectively, permits

THE THERMODYNAMIC THEORY OF ION EXCHANGE

S-curve

135

L-curve

H-curve

FIGURE 5.2. The four classes of exchange isotherm. In each graph the ordinate is the equivalent fraction of a cation in the exchanger phase and the abscissa is the equivalent fraction of the cation in the aqueous solution phase.

Eq. 5.24 to be rewritten in the form: .fBE~

s; = fl.E~

(vE A

+

uEBr- u Q

rE~

(5.26)

E~

where

r

= h'±(ACl u)](I+U)v/h'±(BC\J](I+v)U

Q = (TN/uv)U-V

TN = um.;

+

vmB

The parameter TN is, to a very good degree of approximation, the total normality in the aqueous solution phase, since m is a molality and the mass density or water is very nearly I kg dm 1. When the exchanger phase composition

136

THE THERMODYNAMIC THEORY OF ION EXCHANGE

s

remains fixed, a variation in TN must leave the ratio E~r I E Q constant because the exchange equilibrium constant, the rational activity coefficients, and the exchanger phase equivalent fractions in Eq. 5.26 all will not change. (Any variation in the value of fllfl with TN may be neglected.) The parameter r will depend weakly on TN, whereas Q varies as TN"-V. If the exchange is homovalent, U = v and Q = 1. Then any shift in TN will alter only r and the ratio EAI Eo will adjust slightly to compensate, in order that (EAI Ear r remains fixed in value. If the exchange is heterovalent, a shift in TN may cause a significant change in Q and, therefore, a large, compensating one in E~I E Suppose, for example, that A = Na and B = Ca in Eq. 5.26. Then

s.

,

K ex

-2

fcaEca (2E Na + E ca) rENa = 11. 2 --iNaENa Q E ca

(5.27)

+ 2meaCl2 will leave the ratio E~ar I e; Q = (1 - EC.)2r( TN12Ec.)

Any change in TN = mNaCI

unchanged at a fixed value of E ca. For example, a tenfold reduction in TN will require approximately a tenfold increase in (1 - ECa)21 Eca, assuming thatI' is not altered much in value. Therefore, at a chosen, fixed value of E ea , a reduction in TN will cause Eca to decrease also. A smaller Eca now corresponds to the chosen value of E cathan before dilution. Figure 5.3 illustrates this dilution effect through a set of computer-calculated isotherms pertaining to Na + replacement by Ca H on an exchange resin that displays a relatively small value of K ex • 7 At E ca = 0.4, for example, Eca decreases from 0.8 to 0.16 as TN decreases from 0.1N to 0.001 N. Indeed, the exchange isotherm for TN = 0.1 N is an S-curve, whereas that for TN = 0.001 N is an L-curve. The common procedure of determining exchange isotherms at a single, fixed, total normality could, therefore, lead to erroneous conclusions if the shape of the isotherm were interpreted solely in terms of exchangeable ion-exchanger interactions. Perhaps the most dramatic example of the effect of dilution on the exchange isotherm occurs in the case of the nonpreference isotherm. An exchange complex is said to show no preference toward either ion in a binary exchange reaction if: (1) K ex = 1 for the reaction, and (2) the exchanger phase is an ideal solid solution. The corresponding exchange isotherms then are called nonpreference isotherms. For cation exchange, the conditions for nonpreference reduce Eq. 5.26 to the form

(5.28) The nonpreference isotherm for either A u+ or BV+ can be calculated through an appropriate rearrangement of Eq. 5.28. For example, if A u+ is of interest, the nonpreference isotherm is the solution of the equation E; [u A

+ (v

- u)EA]U-V _ r E~ (I - E'A)" Q (I A)"

E

(5.29)

, "

THE THERMODYNAMIC THEORY OF ION EXCHANGE

137

where the charge balance conditions, E B = I - E A and EB = 1 - EA , have been employed. If the exchange is homovalent, u = v, Q = 1 and, to a good approximation, r has unit value. In this case, it is easy to show that, according to Eq. 5.29, the nonpreference isotherm, a plot of E A versus EA> will be a straight line that makes an angle of 45° with both axes (i.e., a C-curve isotherm). If the exchange is heterovalent, a plot of E A versus EA will not generally be a straight line. Consider again the case A = Na and B = Ca, for which Eq. 5.29 reduces to the expression E~a (l - E~a) -

r TN 2

E~a (l -

EN a )

FIGURE 5.3. Exchange isotherms for Na"-Ca H exchange on a resin. The total normality in the aqueous solution phase is a fixed parameter for each curve.

0.8

0.6

0.4

0.2

0.2

0.4

0.6

0.8

THE THERMODYNAMIC THEORY OF ION EXCHANGE

138

Then the nonpreference isotherm is

a [1 + r ~N (l~a - iNa) EN =

r

1 2

(5.30a)

/

Equation 5.30a is plotted in Fig. 5.4 for values of TN equal to 1.0N, 0.1N, and 0.01N. (The parameter I' was calculated with the help of Eq. 2.96 under the assumption that NaCI and CaCl2 are completely dissociated.) It is evident that the nonpreference isotherm for Na-Ca exchange will approach a straight line only at large values of TN. The isotherms at lower values of TN are decidedly S-curve isotherms and, therefore, would give the impression that the exchanger is selective for Ca H . However, in this case, Na" and Ca H are preferred equally by the exchanger, because of the imposed conditions that Kex = I and that the

FIGURE 5.4. Nonpreference exchange isotherms for Na+-Ca H exchange. The total normality is a fixed parameter for each curve.

1'Or--------------------"":::iI

0.8

0.6 o

z

It..J

0.4

0.2

02

0.4

06

08

1.0

THE THERMODYNAMIC THEORY OF ION EXCHANGE

139

exchanger phase form an ideal solid solution. Therefore no selectivity exists for Ca H over Na +, insofar as selectivity is determined by the values of Kex and the rational activity coefficients. This result shows clearly that the magnitude ofK ex cannot be inferred simply from the appearance of an exchange isotherm. On the other hand, the isotherm is a reliable indicator of selectivity as epitomized in the conditional separation factor, defined in Eq. 5.4. In the present example,

co =

N ca mNa/ NNamca

= EcaENa/ ENaEca =

(1 - ENa)E Na/ E Na(1 - E Na)

and it is apparent from Fig. 5.4 that CD, for a chosen, fixed value of EN aincreases considerably as TN decreases from 1.0N to O.OIN. These same conclusions may be drawn from an investigation of Na-Ca exchange at constant ionic strength instead of constant total normality. In that case, the nonpreference isotherm takes the form

E Na = [ 1 +:1

(

E~a -

- 1/ 2

lNa +

1)]

(5.30b)

where I is the ionic strength in mol dm ", Equation 5.30b is qualitatively very similar to Eq. 5.30a and produces graphs much like those in Fig. 5.4.8

5.3. MIXED-EXCHANGER SYSTEMS The ion exchange complexes found in soils almost always are mixtures of inorganic and organic substances whose individual exchange reactions are described by different equilibrium constants. For example, in an agricultural soil, the exchange complex might consist of the clay minerals montmorillonite and kaolinite, oxides of aluminum and iron, and decomposed plant material, with each substance reacting differently toward Na +-Ca H exchange. Montmorillonite and soil organic matter themselves possess more than one class of metal-eomplexing functional group and, therefore, more than one exchange equilibrium constant is needed for these exchangers in order to account completely for a given exchange reaction. Thus soil ion exchangers are polyfunctional ion exchangers. The description of this polyfunctionality in thermodynamic terms requires an extension of the concepts developed in Sections 5.1 and 5.2. In order to be concrete, the discussion here will focus on the cation exchangers participating in the reaction in Eq. 5.1. However, all of the results to be derived apply just as well to anion exchangers participating in the reaction in Eq. 5.7, after the correspondence between Eqs. 5.2 and 5.8 is noted. The overall equilibrium constant for a soil cation exchange complex is given in Eqs. 5.2 and 5.24. If there are n classes of exchanger within this complex, each class undergoes a reaction like that in Eq. 5.1 and each may be described by the equilibrium constant (5.31 )

THE THERMODYNAMIC THEORY OF ION EXCHANGE

140

where i - I , . . . , n refers to the ith class of exchanger. It is important to understand that each distinct Ke• i defines a class of exchanger in the mixture without regard to its chemical structure or its chemical behavior other than the reaction in Eq. 5.1. Therefore a given class of exchanger may itself be a mixture of exchangers (which happen to have the same value of K e. ) , and the partitioning of several soil exchangers into classes may change as the cations A u+ and B Y+, or the conditions under which the exchange reaction in Eq. 5.1 occurs, are varied. A comparison of Eq. 5.2 with Eq. 5.31 indicates that (BCIy)U (ACluY

-

(BX.)U Ke.(AXu)Y

(BXnv)U Ke.n(AXnv)V

(BX 1y)"

- K• e 1(AX 1u)Y

(5.32)

is an equation of constraint for the soil exchange complex and for each class of exchanger in it. Let Pi = CEC,.jCEC n where CEC i is the cation exchange capacity of the ith class of exchanger and CEC T is that of the entire exchange complex. According to Eq. 5.32, n

II

(BX)U n [(BX.)uKe.i/(AXuYlP; = (AXv)V K~ii

II

u

,.=1

,-I

n

=

II [(BXi.)uKe./(AXi.)v)p; i=1

n

= x: II

[(BXiV)u/(AXi.)")p;

(5.33)

where the first and third steps are based on the fact that 'I

Equation 5.33 shows that K e• is a kind of weighted geometric mean of the K e. / For example, for n = 2, Eq. 5.33 reduces to

In this case, Ke• is a product of the two Ke• i , with each factor weighted according to the Pi' and of the exchanger phase activities for the soil exchange complex and each class of exchanger it contains. The overall exchange equilibrium constant can be written (see Eq. 5.26)

K•• =

i:..fB Vu" E~E"

(vE A

+

(ACI )Y uEB)V-"

(BCl~)U

(5.34)

where the equivalent fractions E A and E B are given in Eq. 5.25a, in terms of mole fractions, and are related to the equivalent fractions pertaining to each

THE THERMODYNAMIC THEORY OF ION EXCHANGE

141

class of exchanger through the equations n

L

EB = i= I

(5.25c)

PiEBi

i= I

Each of the E Ai and E Bi is defined by expressions analogous to Eq. 5.25a. Moreover, E A ; + E B ; = 1 (i = 1, ... , n). The Vanselow selectivity coefficient that corresponds to Kex in Eq. 5.34 may be written in terms of the E Ai and E Bi: uU[~;PiEB;] v-u (ACl u)" «; = vv [~.,P,E A,.] v [v~iPiEAi + u~iPiEBi] (BCL v)U (5.35) U

This equation, in turn, may be expressed in terms of only the E Bi and the individual Vanselow selectivity coefficients, K y i , where K Yi

U

U

E;;

= vVE~i

(vE Ai

+

uEB;)

v-u (ACl.)v (BCl.)u

(5.36)

The result is the equation

Ky =

[~iPiEBJU[v

+ (u

-

. E;{v [v + (u _ { ~,.p'KI/v

v)~iPiEBi]V-U v

(5.37)

v)E .](V-Ul/V} B,

Vi

Because the K Vi (i = 1, ... , n) are different from one another by hypothesis, Eq. 5.37 leads to the important conclusion, that K, will be a function of the overallexchanger composition even if each ofthe K Yi is a constant. For example, in the case of uni-univalent exchange (u = v = 1), Eq. 5.37 reduces to (5.38a) or (5.38b) Even in this simple case, K;I is a weighted average of the K;;I wherein the weights, PiEBi, will vary with E B. Therefore the soil exchange complex will behave as a nonideal exchanger phase, regardless of whether the individual classes of exchangers it contains are ideal mixtures ofAXiu(s) and BXiv(s). Equation 5.38a can be developed further to show the explicit dependence of K; on E B in the special case n = 2. First, Eq. 5.35 is written entirely in terms of the E Ai and the K Vi with the help of Eq. 5.36:

K = (PIERI y

(PIEAI

...

+ P2 ER2)

+

(ACl) P2Ed (BCI)

(p,KVIE A1 + P2 K V2E A2) (p,E", + P2 EA2)

PIEA1K vt

+ P2 EA2 K V2

..;.:....;'--'-'--.:.:.:...._..:....:......:.;=----'~=:....:.......:.:.:----'-'-:::--'"-=-..:.:.::........:..::

Ell.

(5.39)

THE THERMODYNAMIC THEORY OF ION EXCHANGE

142

This equation is the analog of Eq. 5.38b and shows once again that K; is a weighted average of the K Vi with composition-dependent weights. Since E A i = 1 - E B i , Eq. 5.39 may be rewritten in the form K

E BI ) + P2 K VD - E B2) 1 - EB P2 Kn - PtKVlE BI - P2 KnEB2 1 - EB

_ P IKVl (1 y

-

.

-

PIK Vl

+

(5.40)

Now Eq. 5.40 is multiplied on both sides by K; and the identity PIKvIEBI

+ P2KnEB2 =

EB(K vI

+

K n) - (PIEBIK n

+ P2 EB2 KVl)

is noted. The result of this manipulation is

K2v

+

+

[PIK Vl P2Kn _ - _ EB(K K e.:::.-,,----,Vl n)] K; = ~:-.....:,-,------,,-,,---'-=. =-:...--:...:._ _

1 - EB

+

(PIEB1K n + P2 EB2 KVl)Ky 1 - EB

(5.41)

According to Eq. 5.38a,

s;

+ P2(E Bd K n)] KVIKnEB/[PIEBIKn + P2 EB2 KVl]

= EB/[PI(E B1/ K Vl) =

The combination of Eqs. 5.41 and 5.42 produces the quadratic equation K!; _ [PIK v1 + P2 Kn - EB(K Vl v 1 - EB

+ K n)] K y

_

KVlKnE B _ 1 - EB - 0

(5.42) 'j

l

(5.43)'1 i

Ii

~

The solution of Eq. 5.43 for s; as a function of E B , s.; and KY2 is straightfor-,! ward. If the exchangers of classes 1 and 2 are ideal mixtures ofAX;(s) and BX;(s) (i = 1,2), then K Y 1 and K Y 2 are constant parameters and the composition dependence of K; may be calculated numerically without difficulty. Sample calculations of this type'? show that K y varies with E B in a more pronounced fashion, the greater the difference in value between K Y 1 and K y 2• In addition, if KY 1 > Ky 2, then the variation of K; with E B occurs primarily at low equivalent fractions of BX(s), if PI is small, and primarily at high equivalent fractions of BX(s), if PI is large. The general trend in Ky(E B) is much like that observed commonly for soil exchangers. For the general case expressed in Eq. 5.37, the analysis of K; pertaining to a mixed-exchanger system can be made somewhat easier to carry out if the K y, can be developed in either power series or closed-form equations in the E Bi along the lines of what was described in Section 5.1. Even with relatively simple equations for the Ky i , it is evident that the composition dependence of K; can be complicated. In particular, K y may display maxima or minima as E B is varied. I I As a general rule, a lack of monotonicity in the composition dependence of Ky may be taken as a sign of polyfunctionality in a soil exchange complex.

1

THE THERMODYNAMIC THEORY OF ION EXCHANGE

143

5.4. EXCHANGER PHASE ACTIVITY COEFFICIENTS AND THE EXCHANGE EQUILIBRIUM CONSTANT

It was demonstrated in Section 2.5, for the example of Na" -Ca H exchange on

Camp Berteau montmorillonite, how the equilibrium constants in Eqs. 5.2 and 5.8 can be calculated in terms of the Standard State chemical potentials of homoionic exchangers and ions in aqueous solution. This method of obtaining Kex is practicable if the exchanger is a unique mineral with a known chemical composition. Soil exchange complexes, however, seldom are single minerals and usually are not analyzed for their total elemental composition. For these exchangers, Kex cannot be determined through measurements of ,.,,0 of their homoionic forms in a convenient manner. The alternative to applying Eqs. 2.37 and 2.39 for the calculation of Kex is, of course, an application of Eq. 2.38, which expresses the equilibrium constant directly in factors of the activities of products and reactants. Taking as an example Eq. 5.2, it is evident that a determination of the activities of BXv(s) and AX.(s) would suffice to compute Kex for a cation exchange reaction, since the activities of ACl.(aq) and BClv(aq) can be obtained readily according to the methods outlined in Chapter 2. The determination of the activities of the exchanger components, in turn, would require only measurements of their activity coefficients, so the problem of computing Kex in Eq. 5.2 reduces to that of computing j', and!B in Eq. 5.24. For a binary ion exchange reaction, given that K; can be determined experimentally without difficulty, it turns out that the exchanger phase activity coefficients can be calculated exactly, without the need for model expressions such as those described in Section 5.1. The reason for this important result is that there are always two equations of constraint on binary exchanger activity coefficients. Since there are also just two activity coefficients, they are determined uniquely, given a workable definition of the Reference State of an exchanger phase. The two equations of constraint derive from two conditions: (1) the composition dependence of K y is the result of the composition dependence of the activity coefficients; and (2) the composition dependence of the activity of one exchanger component is compensated for by that of the other in such a way that mass is conserved in the exchange complex. The first of these two conditions can be expressed mathematically in a convenient manner by taking the natural logarithm of both sides of Eq. 5.9, then forming the differential of In K; (see Eqs. 2.103 and 3.29b): din Ky = v dui f; - u din!B

(5.44)

It is understood in Eq. 5.44 that the differentials refer to infinitesimal changes brought on by variation of Nil or N B• The second condition just mentioned is merely an application of the Gibbs-Duhem equation (Eq. 1.25) to an exchanger phase at constant temperature and pressure. In making this application, one

.'

144

THE THERMODYNAMIC THEORY OF ION EXCHANGE

,

;.1.·······' .

should allow for any change in the activity of the water adsorbed by the exchanger as the exchanger composition is varied. Thus an exchanger phase should be regarded as a three-component system whose intensive properties vary according to the equation (5.45a) at constant T and P, where n refers to a mole number in the exchanger phase. The division of both sides of Eq. 5.45a by (n A + nB ) and the introduction of Eqs. 2.28, 2.35, and 5.10 into the resulting expression yields (5.4Sb) where n~ is the number of moles of water in the exchanger divided by the total moles of exchangeable cations and a; is the activity of water in the exchanger. Since the mole fractions N A and N B are defined in terms of n A and n B only [e.g., N A = nA/(n A + nB) ], they are still subject to the constraint dNA = -dNB regardless of the variation in n; or a.; Therefore Eq. 5.45b may be reduced to the differential expression (see Eq. 3.30) (5.45c) Equations 5.44 and 5.45c form a set of two differential equations that can be solved uniquely for the activity coefficients, fA and fB' The solution of Eqs. 5.44 and 5.45c for In fA is found by expressing dIn fB in Eq. 5.45c in terms of In fA and In a.; then substituting the result into Eq. 5.44:

and (5.46a) where E B is the equivalent fraction of BXv(s) in the exchanger phase (see Eq. 5.25a) and (5.47) is the number of moles of water in the exchanger per equivalent of cation exchange capacity. The corresponding solutions for In fB are dIn K; = - v

(Z:

dIn fa

+

~: dIn a

w)

-

u dIn fa

and (S.46b)

d,

1

:1

THE THERMODYNAMIC THEORY OF ION EXCHANGE

145

Equations 5.46 can be integrated on both sides, going from the appropriate Reference State to a state of arbitrary exchanger composition and water activity, in order to calculate the rational activity coefficients. Since the path of this integration may be chosen at will, the following convenient sequence will be employed: (1) integration from the Reference State to a state characterized by unit mole fraction of the exchanger component of interest and by a water activity that corresponds to the total normality (or ionic strength) maintained in the exchange experiments; and (2) integration from the final state in (1) to a state characterized by a chosen exchanger composition and the same total normality (or ionic strength) of AClu(aq) and BCUaq) as found in the final state in (1). The Reference State to be employed is that specified in Section 5.1, the GainesThomas Reference State. The equations that result from applying (1) and (2) to Eq. 5.46a are: In fA( I)

V

Jo

E.

-

=

Jo

J

InQw(A)

dIn fA = v In fA(1) = - uv

n~ dIn

a;

0

[d(E~

In K y) -

l~

E.

E BIn K y

-

In K;

Jo

K; dE~] - uv dE~

- uv

J

J

In Qw(AB)

n~BdIn

a;

In Qw(A) In Qw(AB)

n~BdIn

a;

In Qw(A)

with the final result:

v In I, = E BIn

J

s; -

E.

In

o In Qw(AB)

+ J

s, dE~ -

nABdIn aw w

In Qw(A)

UV

[J 0

n~ dIn a;

]

(5.48a)

In Qw(A)

Equation 5.48a is the sum of the two equations it follows. In obtaining the equation, the result d(E B In K y) = In KydE B + EBdInK y was used. The quantity n~ is n; in the homoionic exchanger, while n~B is n; in the exchanger phase when N A < 1.0. A similar designation applies to aw(A) and aw(AB). The equation corresponding to Eq. 5.48a for the activity coefficientfB is

u In fB = -(1 - E B) In K y -

J

E.

In

I

- uv

[r

. 0

n:

In Qw(B)

n: din a;

+

s; as;

r

In Qw(AB)

n~Bd In

]

a;

(5048b)

. In Qw(B)

with and aw(B) indicating the number or moles or water and the water activity in the homoionic exchanger, BX .(s), respectively. Equations 5.48 are exact

THE THERMODYNAMIC THEORY OF ION EXCHANGE

146

thermodynamic results for the rational activity coefficients. The exchange equilibrium constant, K... now may be calculated by combining Eq. 5.9 (in logarithmic form) with Eq. 5.48:

f

In K ex =

0

-

1

In KydE B

f

in Q,.(A)

[

+

UV

n~ dIn

a;

f

0

+

In Qw(B)

n: dIn a;

f

o

In Qw(B)

n~B dIn

]

a;

(5.49)

In Qw(A)

According to Eqs. 5.48 and 5.49, measurements of K y , nw , and a; as functions of exchanger composition are necessary in order to evaluate fA' fB' and K ex for a given cation exchange reaction. (Precisely the same considerations and the exact analogs of Eqs. 5.48 and 5.49 would apply to anion exchange.) The integrals in Eqs. 5.48 and 5.49 that involve n; and In a; have been discussed extensively and studied experimentally a number of times." The available data indicate that the contribution of the last three integrals on the right side of Eq. 5.49 is usually quite negligible when compared with the integral over In Ky. The contributions of the last two integrals on the right sides of Eq. 5.48 usually are small, as pointed out already in Section 5.1, but are not necessarily so when compared with the terms in In K y in the two equations. However, the common practice, partly because of the difficulty in measuring nw , has been to assume that aw(A) = aw(B) = aw(AB) and that n~ = n: in Eqs. 5.48 and 5.49. The effect of these assumptions is to produce the simplified equations

v In fA = E B In U

s, - fEB In s, as; o

InfB = -(1 - E B) In In K ex =

f

«, +

f

I

In

(5.50a)

s, dE'

(5.50b)

EB

1

In

s, dEB

(5.50c)

o

An example of the application of Eq. 5.50 to the cation exchange reaction MgX 2(s)

+ 2 KCI(aq)

= 2 KX(s)

+ MgCI 2(aq)

on Yolo loam" is shown in Table 5.2. The rational activity coefficients listed in the third and fourth columns of the table were calculated with the help of the data in the first and second columns as well as Eqs. 5.50a (with v = 1, A ... Mg) and 5.50b (with U = 2, B = K). The generally small effect on fM. and fK of a tenfold change in ionic strength is evident. The values of K ex calculated according to Eq. 5.50c at the two ionic strengths were 22.85 (I = 0.01 M) and 22.93 (I = 0.001 M). The fifth column of Table 5.2 lists values of K.. calculated at the two ionic strengths and at each value of E Kaccording to the equation

K.. • J1KY lf M .

TheIc values compare well with those derived from Eq. S.SOC. as they should if Eq•. .5..50 are mutually consistent.

147

THE THERMODYNAMIC THEORY OF ION EXCHANGE TABLE 5.2. Values of the Vanselow selectivity coefficient, exchanger phase activity coefficients, and the exchange equilibrium constant for Mg H - K + exchange on Yolo loam" E K*

fMg

Kv

«;

!K

f = 0.Ql M

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

142.0 107.6 51.8 28.3 20.4 15.7 13.1 11.9 11.3 11.4 11.2

1.000 0.985 0.886 0.760 0.677 0.603 0.543 0.512 0.496 0.496 0.493 f = 0.001 149.0 1.000 113.2 0.986 54.6 0.884 31.2 0.769 20.8 0.666 15.6 0.584 13.4 0.538 11.3 0.482 10.3 0.450 10.7 0.465 10.3 0.448

0.400 0.460 0.624 0.785 0.873 0.938 0.979 0.995 0.999 0.999 1.000 M 0.393 0.446 0.609 0.752 0.859 0.931 0.962 0.992 1.000 0.999 1.000

22.72 23.11 22.76 22.95 22.97 22.91 23.12 23.01 22.74 22.94 22.72

)

23.01 22.84 22.91 22.94 23.04 23.15 23.05 23.07 22.89 22.96 22.99

*E K = equivalent fraction of potassium in the exchanger.

In some cases, the computation of lA' IB' and In K e• can be made simpler, numerically, by fitting a power series expression to the measured values of In K v . For example, the Margules expansion In K; = 3.628 - 2.29Ec•

+ 6.03E2. -

4.02E~.

has been shown to describe the composition dependence of K; for the cation exchange reaction

- CsX(s) + NaCI(aq) on Camp Berteau montmorillonite at 25.5C. 'J The introduction of this leastsquares equation into Eq, 5.50 and term-by-term Integration of In Kv result in NaX(s)

+ CsCI(aq)

THE THERMODYNAMIC THEORY OF ION EXCHANGE

148

the following expressions: InfNa = -1.145E~s + 4.02E~s - 3.015Ets lnfes = -0.14 + 2.29Ees - 7.175E~s + 8.04E~s - 3.015Ets In K; = 3.488 These equations may give more precise information than would the results of ordinary numerical integration of In K; using only tabulated experimental values. The consistency of Eq. 5.50 with the Margules expansion of In K v can be checked in the present example by examining the equations for In fNa and In fes when E es = 0 and E es = 1, respectively. With E es = 0, the equation for fNa predicts fNa = 1.0, as it should. With E es = I, the equation for fes predicts lnfes = -0.14 + 2.29 - 7.175 + 8.04 - 3.015 = 0, which also is correct. Another consistency check of any set of measured values of the rational activity coefficients can be made on the basis of an expression derivable from Eq. 5.50. According to Eq. 5.50, the rational activity coefficients must satisfy the relation

In(fl/~) = In s; - J

I

In

x, dEB = In s; -

In

«;

(5.51)

o

Upon integrating both sides of Eq. 5.51 with respect to E B between the limits of o and 1, one finds directly the result

J

I

o

In

(fl/~)dEB

= 0

(5.52)

Equation 5.52 may be used to examine the self-consistency of either measured values of fA and fB or a set of model equations for these parameters, such as the equations listed in Table 5.1. In the particular case of the Na-Ca exchange experiment discussed in the foregoing paragraph, In(fNa/fes) = 0.14 - 2.29Ees + 6.03E~s - 4.02E~s and

J

I

In(fNa/fes)dEes = 0.14 - 1.145 + 2.03 - 1.005 = +0.02

o

which equals zero within the precision of measurement of the activity coefficients. Yet another kind of consistency check of Eq. 5.50 can be made if Kex is determined for three or more binary exchange reactions. Consider, for example, the three cation exchange reactions: vAX.(s)

wBX.(s) wAX.(s)

+ +

+

uBCI.(aq) = uBX.(s) + vACI.(aq) vEClw(aq) - vEXw(s) + wBCJ.(aq) uEClw(aq) • uEXw(s) + wACI.(aq)

(5.53a) (5.53b) (5.53c)

j

I

149

THE THERMODYNAMIC THEORY OF ION EXCHANGE

where A U+, B v+, and E w+ are the three exchanging cations. An expression such as Eq. 5.2 can be written for the equilibrium constant pertaining to each of these' reactions. Equation 5.2 describes the reaction in Eq. 5.53a, whereas I0.~) =

(EX w) V(BCl v) w/ (BXvY(EClwY I0.~) = (EXw)U(AClu)W /(AXuY(EClw)U

(5.54) (5.55)

respectively describe the reactions in Eqs. 5.53b and 5.53c at equilibrium. It is evident from an inspection of Eqs. 5.2, 5.54, and 5.55 that K;[I0.~]U = [I0.~)]V

(5.56)

or, in terms of /lGo for the reactions, w/lG~ + u/l~(a) = v/lG~(b)

(5.57)

according to Eq. 2.39. Equation 5.57 provides for a test of both experimental precision and self-consistency when the /lGo values are calculated using values of In Kex determined with the help of Eq. 5.50. Table 5.3 shows some typical examples of this test applied to cation exchanges on three montmorillonites." The values of v/l~(b) expected according to Eq. 5.57 are +0.77, -4.51, - 2.62, and -0.67 kJ mor ', respectively, for the four entries in the seventh column of the table. In each case the measured value of v/lG~(b) is in fair agreement with the expected value based on the requirement of self-consistency, the mean deviation between the two values being ±0.62 kJ mol", Before this discussion of exchanger phase activity coefficients and the exchange equilibrium constant is ended, it must be pointed out that, in many investigations of cation exchange, the Gaines-Thomas selectivity coefficient, KGT

TABLE 5.3. Consistency tests according to Eq. 5.57 for cation exchange reactions on three montmorillonites" kJ mol:"

Montmorillonite Wyoming

Chambers (25°C)

Chambers (30°C)

Camp Berteau

AU+

Na+ Na+ Na+ Na+ Na+ Na+ Na+ Na'

BV+ K+ K+ Cs+ Cs+ Cs+ Cs+ NH.+ NH:

EW+

w~~x

u~~x(a)

v~~x(b)

-1.28 Li+ Li+

+2.05 +0.20 -7.89

Rb+ Rb+

+3.38 -5.64 -18.00

BaH BaH

+ 15.38 -2.08 -8.54

BaH

BaH

+7.87 -0.42

150

I

THE THERMODYNAMIC THEORY OF ION EXCHANGE

(Table 5.1), is employed instead of K; in order to derive Eqs. 5.49 and 5.50c. Since, for the exchange reaction in Eq. 5.1, EB(ACluY K

GT

=

E~(BCUU

'J.

(5.58)

where E B and E A are equivalent fractions (Eq. 5.25a), the transcription of the equations for fA' fB' and Kex into expressions containing KGT can be made readily with the help of the relationship (see Eq. 5.34) uU Kv = - V (v V

+ (u

-

v)EBy-uKaT

(5.59)

which follows from a comparison of Eqs. 5.2, 5.25a, and 5.58. Thus there is no difficulty, in principle, with employing KaT in place of K v in the thermodynamic description of cation exchange, and to do so will be convenient if experiment shows that In KaT is approximately independent of exchanger composition. However, the composition dependence of the selectivity coefficient KaT does not give information directly as to whether the exchanger phase is an ideal solid solution. Indeed, KaT will depend on E B as (v + (u - v)EB)U-V, according to Eq. 5.59, if the exchanger phase is ideal (i.e., if K; is composition independent). Moreover, a set of activity coefficients cannot be defined by the equations (5.60)

j

i'

in analogy with Eq. 5.10, because the equivalent fraction concentration scale, may not be used to define rational activity coefficients (any more than the normal concentration scale can be used for activity coefficients in aqueous solutions). Unfortunately, the parameters gA and gB' introduced by Gaines and i Thomas, often have been used in Eq. 5.44 (with KaT replacing Kv), instead of the activity coefficients fA and fB' in order to produce equations analogous with Eqs. 5.46, 5.48, 5.50a, and 5.50b. Numerical values of gA and gB are of ther- , modynamic significance, however, only as they may be related to fA and fa . through the combination of Eqs. 5.10, 5.25a, and 5.60.

5.5. SPECIFIC ADSORPTION

Although the term "adsorption" is used frequently in the literature of exchange reactions, it should be emphasized that a necessary relationship between surface phenomena and the thermodynamic theory of ion exchange does not exist. The theoretical development that commences with either Eq. 5.1 or 5.7 involves the strictly macroscopic concepts of activity and activity coefficient as well as the properties of the chemical potential and in no sense requires a knowledge of the underlying mechanism that governs the exchange reaction. The thermodynamic

theory ofion exchange is a theory ofmixtures that is independent ofany mechanistic consideration.

THE THERMODYNAMIC THEORY OF ION EXCHANGE

151

Consider again Eq. 5.1. This chemical equation is mathematically but a special case of Eq. 3.23a, which describes the replacement of one cation by another in a homogeneous solid mixture. In Section 3.7 a thermodynamic theory of coprecipitation was developed that is exactly analogous with the theory of cation exchange presented in Section 5.4 (see Eqs. 3.33 and 5.50a, Eqs. 3.34 and 5.50b, and Eqs. 3.35 and 5.50c). However, coprecipitation refers to a mixture of two or more elements in a solid phase under conditions wherein adsorption processes are ruled out. For example, coprecipitation of the feldspars, albite (NaAISi 3 0 s) and anorthite (CaAI 2Si 20 s), may occur under hydrothermal conditions to form a plagioclase solid solution. The relevant chemical reaction, with respect to the composition of the bulk plagioclase phase, is

Since (SiOls)) = 1.0 when quartz is present, the equilibrium constant for this reaction is identical with Kex in Eq. 5.24 after the identifications u = 1, v = 2, B = anorthite, and A = albite have been made. Equations 5.46 (with n; = 0) and the remainder of the theory given in Section 5.4 then can be applied to this problem in which surface phenomena play no role. These remarks should make clear that a connection between adsorption processes and ion exchange equilibria cannot be established by means of purely thermodynamic data. Other kinds of experimental information (e.g., the rates of diffusion of the exchangeable ions through the solid phase under consideration, the kinetics of the exchange reaction, and spectroscopic data on the species at the surface of the solid phase) are required before an ion exchange reaction can be classified as a surface phenomenon. For certain exchangeable ions and certain soil exchangers, the degree to which the ions can penetrate into the exchanger is small, the exchange reaction is rapid, and the exchanger maintains its integrity as a solid phase during the exchange reaction. In this case an adsorption mechanism may be operating. If, in an ion exchange experiment, the magnitude of the conditional separation factor, 'D, is observed to be very large, then the ion B is commonly said to be adsorbed "specifically" by the exchanger. "Specific adsorption," which might better be termed highly selective adsorption, actually is supposed to refer to the selectivity, for a given cation or anion, of certain complexing functional groups at the surface of an inorganic or organic solid substance. If these surface functional groups form very stable complexes with a given ion, the functional groups are said to be selective for that ion, and the interaction between the ion and the solid that bears the functional groups is called specific adsorption. Usually specific adsorption phenomena are reflected in a H-curve type of exchange isotherm for the selectively adsorbed ion. This isotherm will show little sensitivity to changes in the ionic strength (or total normality) of the equilibrium aqueous solution contacting the exchanger. Examples of cations that typically are considered to be adsorbed selectively by soil minerals and organic matter are the bivalent transition metal and Group IB and liB metal cations. Examples of

152

THE THERMODYNAMIC THEORY OF ION EXCHANGE

selectively adsorbed anions include the oxyanions, SO~-, PO~-, AsO~-, BOj-, and MoO~-, and their protonated forms. Because specific adsorption phenomena are expected to produce H-curve exchange adsorption isotherms, there has been an unfortunate tendency in the soil chemistry literature to regard H-curve isotherms as definitive evidence for highly selective surface complexation. This point of view reflects a misunderstanding of the relationship between molecular mechanisms and thermodynamic data. The mechanism of surface complexation may be employed to interpret exchange isotherms, but the mechanism itself must be established by separate experiments. The exchange isotherm is a strictly macroscopic concept that is; consistent with several different kinds of exchange mechanism at the molecular, level. For example, consider the secondary precipitation reaction between hydrous ferric oxide and tri-sodium phosphate: Fe(OH)3(s)

+

Na3POiaq)

= FeP04(s) + 3 NaOH(aq)

If the equilibrium state involved a homogeneous mixture of the two iron-containing solid phases, this chemical reaction could be regarded as a special case ) of Eq. 5.7 (with C = OH, Y = Fe 1/ 3' D = P04, W = 3, and x = 1) and could be described with the help of the theory developed in Sections 5.2 and 5.4. In particular, Eq. 5.26 could be applied to calculate the exchange isotherm. This macroscopic description would be no different were there an adsorption mech- " anism operating for the o-phosphate interaction with hydrous ferric oxide instead of the chemical reaction indicated above.

NOTES J. This point was made first by K. L. Babcock, L. E. Davis, and R. Overstreet, Ionic activities in:: ion-exchange systems, Soil Sci. 72:253-260 (1951). The problem of the Reference State for ion'> exchangers is reviewed by L. W. Holm, On the thermodynamics of ion exchange equilibria. I. The i, thermodynamical equilibrium in relation to reference states and components, Arkiv for Kemi 10:151-156 (1956). 2. G. L. Gaines and H. C. Thomas, Adsorption studies on clay minerals. II. A formulation of the' thermodynamics of exchange adsorption. J. Chem. Phys. 21 :714-718 (1953). 3. H. Laudelout, R. van Bladel, G. H. Bolt, and A. L. Page, Thermodynamics of heterovalentr cation exchange reactions in a montmorillonite clay, Trans. Faraday Soc. 64:1477-1488 (1968). d

4. H. E. Jensen and K. L. Babcock, Cation-exchange equilibria on a Yolo loam, Hilgardia 41:475488 (1973).

~

5. For a review of the concept and application of this general kind of selectivity coefficient, see D. i Reichenberg, Ion-exchange selectivity, Ion Exchange and So/vent Extraction 1:227-276 (1966). ~ Selectivity coefficients for the cation exchangers found in soils are discussed by M. M. Reddy, Ion. ; exchange materials in natural water systems, Ion Exchange and So/vent Extraction 7:165-219 'I (1977). 6. The four classes of exchange isotherm are adapted from a scheme for solute adsorption proposed by C. H. Giles, T. H. MacEwan, S. N. Nakhwa, and D. Smith, Studies in adsorption. Part XI. A ' system of classification of solution adsorption isotherms. and its use in diagnosis of adsorption mechanisms lind in measurement of specific surface areas of solids. J. Chl'm. Soc. (London)

THE THERMODYNAMIC THEORY OF ION EXCHANGE

153

1960:3973-3993. See also C. H. Giles, D. Smith, and A. Huitson, A general treatment and classification of the solute adsorption isotherm, J. Colloid Interface Sci. 47:755-765 (1974). 7. R. M. Barrer and J. Klinowski, Ion-exchange selectivity and electrolyte concentration, J. Chern. Soc. Faraday I 70:2080-2091 (1974). 8. H. E. Jensen, Cation adsorption isotherms derived from mass-action theory, Royal Veterinary and Agricultural University Yearbook. Copenhagen, Denmark, pp. 88-103, 1971. 9. See R. M. Barrer and J. Klinowski, Cation exchangers with several groups of sites, J. Chern. Soc. Faraday I 75:247-251 (1979) for a review of Eq. 5.33 and other general relationships. 10. H. E. Jensen, Selectivity coefficients of mixtures of ideal cation-exchangers, Agrochirnica XIX:257-261 (1975). II. A discussion of mixed-exchanger systems that show extrema in the selectivity coefficient KGT in Table 5.1 has been given by R. M. Barrer and J. Klinowski, Ion exchange involving several groups of homogeneous sites, J. Chern. Soc. Faraday I 68:73-87 (1972). 12. G. L. Gaines and H. C. Thomas, Adsorption studies on clay minerals. V. Montmorillonitecesium-strontium at several temperatures. J. Chern. Phys. 23:2322-2326 (1955); H. Laudelout and H. C. Thomas, The effect of water activity on ion-exchange selectivity, J. Phys. Chern. 69:339-341 (1967); G. H. Bolt and C. J. G. Winkelmolen, Calculation of the standard free energy of cation exchange in clay systems, Israel J. Chern. 6:175-187 (1968); H. Laudelout, R. van Bladel, and J. Robeyns, The effect of water activity on ion exchange selectivity in clays, Soil Sci. 111:211-213 (1971); and H. E. Jensen, Potassium-Calcium exchange equilibria on a montmorillonite and a kaolinite clay. I. A test on the Argersinger thermodynamic approach, Agrochirnica 17:181-190 (1973). Equations 5.48 and 5.49 are extended to mixed aqueous-organic solvents by A. M. Elprince and K. L. Babcock, Thermodynamics of ion-exchange equilibria in mixed solvents, J. Phys. Chern. 79:1550-1554 (1975). 13. A. Cremers and H. C. Thomas, The thermodynamics of sodium-cesium exchange on Camp Berteau montmorillonite: An almost ideal case, Israel J. Chern. 6:949-957 (1968). 14. R. G. Gast, Standard free energies of exchange for alkali metal cations on Wyoming bentonite, Soil Sci. Soc. Arner. Proc. 33:37-41 (1969); R. G. Gast, Alkali metal cation exchange on Chambers montmorillonite, Soil Sci. Soc. Arner. Proc. 36: 14-19 (1972); R. J. Lewis and H. C. Thomas, Adsorption studies on clay minerals. VIII. A consistency test of exchange sorption in the systems sodium-cesium-barium montmorillonite, J. Phys. Chern. 67: 1781-1783 (1963); and H. Laudelout, R. van Bladel, M. Gilbert, and A. Cremers, Physical chemistry of cation exchange in clays, Trans. 9th Int. Congress Soil Sci. 1:565-575 (1968).

FOR FURTHER READING K. L. Babcock, Theory of the chemical properties of soil colloidal systems at equilibrium, Hilgardia 34:417-542 (1963). Section IV of this classic review provides a fine introduction to the theory of ion exchange in soils. n J. Greenland and M. H. B. Hayes, The Chemistry ofSoil Constituents, John Wiley, Chichester, U.K., 1978. Every chapter of this excellent treatise may be read with profit to understand more about the structure of the soil exchange complex. J. Grover, Chemical mixing in multicomponent solutions: An introduction to the use of Margules and other thermodynamic excess functions to represent non-ideal behavior, in Thermodynamics in Geology. ed. by D. G. Fraser, pp. 67-97, D. Reidel, Dordrecht, The Netherlands, 1977. This review article provides a complete introduction to the thermodynamic theory underlying the power series expansion of rational activity coefficients, all in Eq. S.I I.

154

THE THERMODYNAMIC THEORY OF ION EXCHANGE I

F. Helffericb, Ion Exchange. McGraw-Hill, New York, 1962. Chapter 5 of this standard monograph deals exhaustively with ion exchange equilibria. M. B. King, Phase Equilibrium in Mixtures. Pergamon Press, Oxford, 1969. Chapter 6 of this comprehensive book provides a thorough discussion of model expressions for the rational activity coefficients of components in solid solutions.

6 THE MOLECULAR THEORY OF CATION EXCHANGE

6.1. FUNDAMENTAL PRNCIPLES The objective of the molecular theory of cation exchange is to derive the thermodynamic properties of exchange reactions from the principles of statistical mechanics. Thus the molecular theory seeks to explain how the underlying atomic configurations and interactions in a soil exchange complex produce the observed values of exchange equilibrium constants and the observed composition dependence of the rational activity coefficientsof exchanger components. Section 1.1 emphasized that thermodynamics itself makes no reference to the atomic structure of matter and, therefore, that it may be applied with confidence to describe the equilibrium states of any soil system, regardless of what is known (or unknown) about the molecular nature of that system. On the other hand, thermodynamic descriptions of soil systems can do no more than prescribe general relationships among intensive and extensive variables of state, since they must introduce quantities such as equilibrium constants and activity coefficients solely as empirical parameters. The molecular behavior in a soil that produces the actual, measured values of these quantities is properly the object of statistical mechanics to describe. This underlying behavior can never be described by thermodynamic reasoning alone. Equilibrium statistical mechanics is a well-developed discipline of physical chemistry whose basic concepts will be summarized in this section in a form that leads into molecular models of cation exchange in a straightforward way. No attempt will be made to discuss the motivation and mathematical development of the principles in great detail. As the name implies, statistical mechanics is the result of combining the methods of the theory of probability with those of either quantum or classical mechanics. The key mathematical concept in statistical mechanics thus is probability, and the fundamental postulates on which statistical mechanics is based prescribe how probability is to be interpreted for physical systems at equilibrium. In the case of a closed system consisting of a single component, the basic postulate of statistical mechanics may be stated as follows. The relative probability that a system containing N identical molecules has the total energy E(V. N) is given by the expression P( E) - cxp[ - H(

v. N)/ k 71 R

(6.1 ) 155

THE MOLECULAR THEORY OF CATION EXCHANGE

156

where k B is the Boltzmann constant absolute temperature.

(kB

= 1.3807 X 10- 23 J K-Jj and T is the

The function P(E) is called a Boltzmann factor. The Boltzmann factor gives the relative likelihood of finding a physical system consisting of N molecules enclosed in a volume Vand in equilibrium with a thermal reservoir at temperature T in a state whose total mechanical energy is E( V, N). P(E) can be converted to an absolute probability through the following process of normalization. If, as is customary, many states are accessible to a physical system, many P(E) exist that may be summed to give the partition Junction, QN( T, V):

QN( T, V)

=

"1:,IP(E 1)

=

"1:,1 exp[ - Et< V, N)/ k o T]

(6.2)

The sum in Eq. 6.2 is over each state of the system characterized by a distinct molecular configuration. The normalized Boltzmann factor, or absolute probability, then is defined by the equation (6.3) Clearly, the sum of the Pabs(E1) equals 1.0. Because the system is macroscopic, Et< V, N) often will have the same numerical value for many different values of I and, in general, will form a set of closely spaced energy values, so that the] P(E1) and Pabs(E 1) form nearly a continuum of probability values. i,I In the case of an open, single-component system, the basic probability pos-, I tulate of statistical mechanics is as follows. The relative probability that a system has the total energy E(V, N) and the\, total number oj identical molecules N is given by the expression " ,j P(E, N) = exp[(N#t - E)/ k o T]

(6.4)

where #t is the chemical potential per molecule in the system.

LL

N=O

jj

exp[(N#t - E,)/ koT] I

00

=

L

QN( T, V) exp(N#t/ k o T)

,i l

00

=

1

"

The function P(E, N) is called a Gibbs factor. The Gibbs factor gives the relative likelihood of finding a physical system with a total mechanical energy, E( V, N), and a total number of particles, N, when it is enclosed in a volume V and is in equilibrium with both a thermal reservoir at temperature T and a matter reservoir at chemical potential #to P(E, N) may be converted to an absolute probability by defining the grand partition Junction: Z( T, V, #t)

l' 1.[

(6.5)

N-O

and writing down the analog of Eq. 6.3: (6.6)

THE MOLECULAR THEORY OF CATION EXCHANGE

157

The generalization of Eqs. 6.1 and 6.4 to systems that contain more than one kind of molecule is straightforward. For example, in a multicomponent system, Eq. 6.4 becomes

P(E, N 1,

••• ,

Nd

=

exp

{[~ N,Jtn -

E(V, N 1,

. · · ,

Nd ]lkBT}

(6.7)

where C is the total number of different kinds of molecule in the system. Once the probability expressions have been established, statistical mechanical equations for the thermodynamic properties of closed and open systems can be derived. In each instance these properties will be expressed as average values according to the prescription (6.8) for closed systems and 00

(A) =

LL

N-O

At< V, N)Pabs(E" N)

(6.9)

/

for open systems, where At< V, N) is some mechanical property of the system in its lth state and (A) is the average value of this property [i.e., the thermodynamic quantity associated with At< V, N)]. Consider, for example, a closed, single-component system. The entropy of this system in its lth state is given by the Boltzmann definition:

(6.10) The thermodynamic entropy of the system is the average value of S(E/) computed according to Eq. 6.8:

(6.11) Similarly, the internal energy of the system is the average value of Et< V, N): (6.12) Since

In Pabs(E/) = -EdkBT - In QN(T, JI) according to Eq. 6.1, it follows that Eq. 6.11 may be expanded in the form:

S = - k B 'L/Pabs(E/)[ - E/I kB T - In QN( T, JI)] = (1 I n'L/E/p/abs + kB In QN( T, JI) = VIT + k B In QN(T, JI)

(6.13a)

where Eq. 6.12 has been employed in the last step. However, the Euler equations in Table 1.4 indicate that

S -

Vir + (PV

-

IJ,N)/r - Vir - Air

(6.l3b)

158

THE MOLECULAR THEORY OF CATION EXCHANGE

for a single-phase, single-component system, where A is the Helmholtz energy. Equations 6.13 thus lead to the conclusion that (6.14) The Helmholtz energy, A( T, V, N), is the natural thermodynamic potential with which to describe a closed system, since the variables of state are T, V, and N (see Section 1.4). This fact is reflected in the simple relationship between A and the partition function QN that appears in Eq. 6.14. If all possible values of the total energy of the system are known, QN can be calculated readily with the help of Eq. 6.2, and A( T, V, N) is determined for any thermodynamic state by Eq. 6.14. The thermodynamic properties of the system that are conjugate to T, V, and N then can be calculated as outlined in Section 1.4. For example, the entropy is given by the equation S = -(iJAjiJnv,N = k B In QN

+

kBT(iJ In QNjiJnv,N

(6.15a)

which, upon explicit calculation of the temperature derivative of In QN, is found to agree precisely with Eqs. 6.13a and 6.11. The pressure on the system is (6.15b) and the chemical potential is Jl = (iJAjiJNh.v = - kBT(iJ

In QNjiJNhv

(6.15c)

It should be remembered that Jl in Eq. 6.15c is on a per molecule basis and, therefore, is related to the thermodynamic chemical potential by the equation Jl(thermodynamics) = HAJl(statistical mechanics)

(6.16a)

on a molar basis and by the equation Jl(thermodynamics) = .NAJl(statistical mechanics)j m

(6.16b)

on a mass basis, where H A is the Avogadro constant (HA = 6.0,22 X 1023 " mol :") and m is the mass per molecule. The use of the same symbol for the j chemical potential in thermodynamics and statistical mechanics is conventional I and does not result in confusion. For an open, single-component system, the Boltzmann definition of the entropy results in the expression co

S = - kB

L L Pabs(E/, N) In Pabs(E/, N) N-O

(6.17)

/

The internal energy is given by the equation 00

U -

L L E~ V, N)Pabl(E/. N) N-O

/

(6.18)

THE MOLECULAR THEORY OF CATION EXCHANGE

159

in complete analogy with Eq. 6.12. The average number of molecules in the system is 00

N

= (N) = L L NPabs(E" N) N=O

(6.19)

/

Now, according to Eqs. 6.4,6.5,6.6,6.17,6.18, and 6.19, 00

S =

r

k« L

N~O

= -NIt/T

L

[(Nit - EMkBT - InZ(T, V, 1t)]Pabs(E/, N) I

+

+

V/T

k« In Z(T, V, It)

(6.20a)

and, according to the first row in Table 1.4,

S = -NIt/T

+

V/T

+

(6.20b)

PV/T

It follows from Eq. 6.20 that PV

=

(6.21)

k 8T In Z( T, V, It)

The product PV is the natural thermodynamic potential for an open system, in that PV is proportional to the logarithm of the grand partition function, Z( T, V, It). Sometimes - Pt/, for an open system, is denoted O( T, V, It), which is called the grand potential. This thermodynamic potential was not considered in Section 1.4, but it can be shown through an application of the methods in that section that dO

=

(6.22)

-SdT - PdV - Ndlt

Therefore: -(ao/aTh.~

=

-[a(-kBT In Z)/aT]v,~ k B In Z + kBT(a In z/aT)v.~ -(ao/avh~ = kBT(a In Z/aV)T.~ -(ao/althv = kBT(a In z/althy

S = P = N =

(6.23a) (6.23b) (6.23c)

where (6.24) lquations 6.23 were derived on the basis of the thermodynamic expression in 1':4· 6.22, but they are verified readily with the help of Eq. 6.4. For example, by "4. 6.23c, N = k Btis In z/althv

k T ",

= ~-.

L

N-II

=

zkBT (az/althv

N

QN( T. V)

k1' cxp (NIt/ k II

R

T)

=

(N)

160

THE MOLECULAR THEORY OF CATION EXCHANGE

in complete agreement with Eqs. 6.4 and 6.19. The same procedure may be used to relate Eqs. 6.17 and 6.23a. The most important equations developed in this section are Eqs. 6.1, 6.2, and 6.14, as well as Eqs. 6.4, 6.5, and 6.24. These expressions are the basis for computing the thermodynamic properties of closed or open systems in terms of the set of values allowed for the total mechanical energy of those systems. The total mechanical energy, of course, depends on the configurations and mutual interactions of the molecules in the system and, in general, is a quantity that is difficult to calculate from first principles. The purpose of molecular theory is to construct simplified models that make the calculation of the total energy a tractable mathematical problem while still exhibiting some degree of realism in their physical content. This approach will be illustrated now for the specific example of cation exchange phenomena.

6.2. EXCHANGE ISOTHERMS

The four types of exchange isotherm discussed in Section 5.2 can be derived for one-to-one cation exchange reactions on the basis of a simple molecular model that is essentially a direct application of Eq. 6.4. In Section 5.2 it was pointed out that the L-curve isotherm represents primarily the effect of affinity and steric factors, whereas the S-curve and H-curve isotherms represent primarily the ' effect of affinity factors alone in the exchange process. These factors appear prominently in the following set of model assumptions pertaining to a cation exchange process. 1. The exchange reaction occurs on identical, discrete sites. 2. The exchange process is the replacement of one cation by another single cation. 3. The exchange sites are covered at all times by the exchangeable cations, with one cation per site. 4. The cations on the sites do not interact with one another. Assumption (1) requires the electrical charge on the exchanger that is balanced by an exchangeable cation to be confined to the vicinity of a definite point in the exchanger structure. This requirement is met by cation exchangers in soils that exhibit either protonated complexing functional groups (e.g., OH and COOH) or localized points of positive charge deficiency created by isomorphous substitution. The discrete nature of exchange sites produced through isomorphous substitution has been demonstrated in studies on the adsorption of organic cations of known charge distribution and molecular structure. Assumption (2) limits the exchange process to homovalent cation exchange; the model is not applicable, for example, to the replacement of Ca 2 t by two Na ". Assumption (3) is fundamentally a requirement of electroneutrality. The system comprising the exchanger and the exchangeable cations must be electrically neutral overall and. therefore. cannot support vacant exchange sites or sites

THE MOLECULAR THEORY OF CATION EXCHANGE

161

with two exchangeable cations. This stipulation does not rule out some degree of separation of the exchangeable cation from a site, since that situation corresponds to an outer-sphere complex between the cation and the exchange site (see Section 3.3). Both inner-sphere and outer-sphere complexes are consistent with assumption (3). However, the protonat ion of a surface hydroxyl group in an exchanger to produce OHi is not consistent with assumption (3) [or with assumption (2), since cation replacement is not involved in the process]. Assumption (4) requires that the coulomb interactions between exchangeable cations be negligible when compared to the interaction of the cations with the exchange sites. If the exchangeable cations form inner-sphere complexes with the sites, assumption (4) should be a very good approximation; if they do not, the neglect of interactions between the cations can be justified only if the exchange sites are widely spaced enough to diminish significantly the strength of the coulomb field orginating at each site. The principal effect of assumption (4) is to make the model independent of the geometry of the exchanger structure. Because the exchangeable cations do not interact with one another, they are not affected by the spatial configuration of the exchange sites. Each site is an isolated unit during the exchange process, regardless of whether the exchangeable cations are arranged along a line of sites, in a surface array, or at points in a three-dimensional framework. Neither the distances separating the exchange sites nor the dimensionality of their arrangement in space will enter as parameters in the modeL Because of assumption (4), the cation exchanger is regarded as an assembly of M independent sites, each of which bears the valence u - and contributes additively to the thermodynamic potential chosen to describe the exchanger. It is convenient, then, to picture a given exchange site as a small, open subsystem of the exchanger. Each site is imagined to be in equilibrium with a thermal reservoir at temperature T and to be in contact with matter reservoirs that maintain the exchangeable cations in aqueous solution at fixed chemical potentials (see the discussion of cation exchange in Section 1.2). The aqueous solution that bathes the exchanger in a typical cation exchange experiment is a good physical example of these matter reservoirs. According to Eq. 6.4, the relative probability that an exchange site contains a cation AU+ in a state of energy E is given by the expression

To allow for the possibility of several states for the cation bound to the site, PA(E, 1) may be summed to give a total relative probability: PA(site) - 'L/PA(E" 1) = 'L,exp[(/lA - E/)/koT] = QA exp(/lA/ k o

n

(6.25a)

The corresponding relative probability for a cation S" '. which competes with AU', is (6.25b)

THE MOLECULAR THEORY OF CATION EXCHANGE

162

By assumption (3), the relative probability that the site is occupied by AU+ or BU+ must be the sum of the relative probabilities in Eq. 6.25, since no cations other than A u+ and r~~u+ are permitted on the site and only one cation is permitted at a time. Therefore

H T,

JLA' JLB)

PA(site) + PB(site) = QA exp(JLA/ k B + QB exp(JLB/ k B

=

n

n

(6.26)

The function HT, JLA' JLB) is the grand partition function for a single-cation exchange site. It could have been obtained directly from summing the probability in Eq. 6.7 (for the case C = 2) as indicated in Eq. 6.5, but under the restriction imposed by assumption (3). The grand partition function for the entire exchanger is a product of H T, JLA> JLB): (6.27) The parameter M replaces the volume, V, which appears in Eq. 6.5, because M is more relevant to exchange reactions and, in fact, is proportional to V, since all of the exchange sites are assumed to be identical. The product form in Eq. 6.27 ensures that the grand potential, D( T, M, JLA' JLB)

= -

k B TIn Z( T, M, JLA' JLB)

(6.28)

will receive the same additive contribution from each exchange site. The exchange isotherm for A u+ can be calculated by applying Eq. 6.23c to Eq.6.27: N A = kBT(a In z/aJLAhM,I'B

= MkBT(a

= MQA exp(JLA/ k Bn/ H T, M, JLA' JLB)

In ~/aJLA)T,M'I'B (6.29)

At equilibrium, (6.30a) and (6.30b) according to Eq. 2.35, where JLD is the Standard State chemical potential of a cation in aqueous solution and a is the activity of the cation in the aqueous solution bathing the exchanger. The combination of Eqs. 6.29 and 6.30 produces the expression (6.31 ) where (6.32) for both exchangeable cations, Fquation (1..11 is the exchange isotherm for the: model developed on the basis of assumptions (I) to (4),

THE MOLECULAR THEORY OF CATION EXCHANGE

163

If aA/a B is assumed equal to [Au+]/[BU+], Eq. 6.31 has the same overall mathematical form as the well-known Langmuir isotherm. Note, however, that a graph of N A versus a A or [A U+] will not be a Langmuir isotherm unless a B is maintained constant. A failure to control a B in an exchange experiment thus could lead to the hasty conclusion that Eq. 6.31 does not apply, if it were assumed erroneously that the coefficient of a A in the Langmuir equation is constant regardless of the variation of aB' On the other hand, a graph of NA/M = E A , the equivalent fraction of AXu(s) in the exchanger, against EA , the equivalent fraction of A u+ in the aqueous solution phase, will have precisely the general Langmuir form. This fact is demonstrated by rewriting Eq. 6.31 in the form

E

A

(KA / KB)EA / EB = 1 + (KA / KB)E A / EB (KA / KB)E A

(6.33)

where the first step comes from replacing aA/a B in Eq. 6.31 with [Au+]/[Bu+] = EA / EB (see Eq. 5.25b for the case u = v) and the second step follows after multiplying the numerator and denominator by EB = 1 - EAo Equation 6.33 shows that a graph of E A versus EA can produce an L-curve exchange isotherm, as illustrated in Fig. 5.2. The affinity parameters, KA and K B, defined in Eq. 6.32 are of the same mathematical form as P A (site) and P B (site) in Eq. 6.25. Each of the affinity parameters, therefore, can be interpreted as a relative probability that an exchangeable cation occupies an exchange site that is in equilibrium with an aqueous solution containing the cation at unit activity. If such an aqueous solution were brought into contact with the exchanger, Kin Eq. 6.32 would give the relative probability that the cation will bind to the exchanger. The ratio K A / KB is a comparative measure of this probability for A u+ and BU+ and, therefore, it is related to the affinity of A u+ for the exchanger as compared to that of BU+. If AU+ has a much greater affinity than BU+, K A » KB and the relationship in Eq. 6.33 will graph as an H-curve exchange isotherm. On the other hand, if KA « K D, a graph of E A versus EA according to Eq. 6.33 will have the character of the S-curve isotherm. If K A = K B, the graph will be the linear C-curve isotherm. Finally, note that neither K A , K B , nor their ratio can be interpreted as exp ( - Eads! k B 1), where E ads is an "energy of adsorption," unless two conditions are fulfilled. If 1J.1 = IJ.g and each cation can be in only one state when on an exchange site:

where (E A - ED) could be termed a "relative energy of adsorption." However, these two conditions arc very unlikely to be satisfied by any soil exchange complex and pair of exchangeable cations. Assumption (I) of the molecular model that led 10 hi. (1.3 I required each of the exchange sites to he identical. This condition may he too restrictive for

164

THE MOLECULAR THEORY OF CATION EXCHANGE

soil exchange complexes because of the reasons given in Section 5.3. Therefore it is of interest to consider the model further in the more general case that the exchange sites can be grouped into classes according to their affinities for the exchangeable cations. Each class of sites is assumed to be described accurately by assumptions (1) to (4) and to produce an exchange isotherm having the Langmuir form, as in Eq. 6.31. The overall exchange isotherm then will be a weighted sum of the isotherms for all of the classes of sites. To facilitate the calculation, let the parameter q be defined by the equation (6.34) Ths parameter can take on any value between - 00 and + 00 as is necessary to produce the value of K A/ KB according to Eq. 6.34. Now let m(q)dq be the number of exchange sites associated with a value of 1n(K A / K B) between q and q + dq, such that

J

m(q)dq = M

00

(6.35)

-00

where M is the total number of exchange sites. Then the exchange isotherm that generalizes Eq. 6.31 for a nonuniform exchanger is

J

N =

00

A

-00

m(q) exp(q)(aA/aB)dq 1 + exp( q)( aA/ aB)

(6.36)

Suppose that m(q) could be represented mathematically by a single Dirac delta- "function"; m(q) = M o(q -

qo)

The Dirac delta-"function," o(x - x o), exhibits a single, extremely sharp peak at x = Xo that leads to the defining property:

J

00

fix)o(x -

xo)dx

=

fixo),

-00

where fix) is any smooth function that remains finite. Therefore Eq. 6.36 would, reduce to I N = A

M exp( qo)( a A/ aB) 1 + exp( qo)( aA/ aB)

(6.37)

This equation is the same as Eq. 6.31, since qo refers to a specific, single value. of 1n(KA / K B ) , according to Eq. 6.34. Thus, if m(q) is a very sharply peaked function at some value of q, all of the exchange sites belong in the class of relative affinities corresponding to that value of q, and the generalized isotherm reduces to Eq. 6.31, as it should. If there are two distinct classes of sites, one corresponding to the value qo and one to q, then m(q) has the mathematical. form: m(q) = M[Poo(q - qo) + (1 ~ Po)o(q - ql)] where Po is the fraction of sites associated with the value qo. In this case Eq. 6.36 would reduce to a sum of two terms, each of which is like that on the right side of Eq. 6.37. The generalization of m( q) to n classes of sites is. evidently.

THE MOLECULAR THEORY OF CATION EXCHANGE

165

n

m(q) = M

L

pl)(q -

qk)

k=O

where the P« are weighting coefficients that are subject to the constraint }:,kPk = 1. The exchange isotherm corresponding to this particular form of m(q) would be a weighted sum of Langmuir isotherms. I These examples indicate the manner by which m(q) determines the overall exchange isotherm. Actually, it is possible to regard Eq. 6.36 as an integral equation whose solution is m(q), given the explicit dependence of N A on (a A / aB ) as measured in a cation exchange experiment. It can be demonstrated that a unique relationship exists between N A (as a function of aA/ aB) and m( q). An expression for m(q) that is of particular interest because of the overall exchange isotherm it produces is _ M m(q) 7r 1

+

a sin(7r13) exp( -(3q) 2a cos(7r{3) exp( -(3q) + a 2 exp( -2{3q)

(6.38)

where a and (3 are related to the value of q at the maximum in m(q):

I

qmax

= ~ In a

m(qmax)

=

M

= 27r tan(7r{3/2)

mmax

(6.39)

A graph of m(q) given in Eq. 6.38 shows that this function is very similar to a gaussian curve centered about the value of qmaX" If the value of (3 remains between -1 and + 1, m(q) obeys Eq. 6.35; otherwise, the integral does not converge. Since mmax > 0, {3 must be positive also, and 0 < (3 < 1. Moreover, lim m(q) = M o(q -

qmax)

(j-l

where qmax is given in Eq. 6.39. The chemical interpretation of m(q), then, is .hat it represents essentially a gaussian distribution of exchange site affinity classes about the affinity class corresponding to qmaX" The sharpness of the maxmum in m(q) and, therefore, the degree of nonuniformity in the exchange sites ire determined by the parameter {3. The nearer {3.is to 1, the more uniform is .he exchanger. The combination of Eqs. 6.36 and 6.38 produces the overall exchange sotherm:

NA =

Ma( aA/ aB){j

(6.40)

---'--'-"---='--

1

+ a( aA/ aB){j

Ihis isotherm reduces to a form of the well-known van Bemmelen-Freundlich .quation when (a A/ aB){j « 1 (i.e., N A « M): NA

;:::::

Ma(aA/aB){j

(N A

«

M)

(6.41)

\n expression of the same general form as Eq, 6.41 often is observed to describe race adsorption data quite well in soil exchangers. This fact may be interpreted x evidence for nonunilormity among soil exchange sites.

166

THE MOLECULAR THEORY OF CATION EXCHANGE

6.3. THE DISCRETE SITE MODEL

In its simplest form, the discrete site model of cation exchange is a direct elaboration of assumptions (1) to (4) in Section 6.2. The objective of the model is to calculate the exchange equilibrium constant and the activity coefficients of the exchanger components in terms of the interactions between exchangeable cations and the exchange sites.' For the case of uni-univalent exchange, the discrete site model can be developed solely from Eq. 6.27 which, in this case, gives the grand partition function for a set of M univalent exchange sites that contain two kinds of univalent exchangeable cation. The parameter M is, then, directly proportional to the cation exchange capacity. Equation 6.29 expresses N A in terms of ILA- The equation (6.42) which follows from applying Eq. 6.23c to Eq. 6.27, does the same for N B• Equations 6.29, 6.30, and 6.42 now can be employed to calculate the Vanselow selectivity coefficient according to Eq. 5.3 (with u = v = 1):

K; __ NBaA __ QB exp(ILU k B 1) _ KB NAa B QA exp(ILUk B 1) KA

(6.43)

where KB and K A are defined in Eq. 6.32. According to the discrete site model, the Vanselow selectivity coefficient is independent of the exchanger composition and is equal to the ratio of relative probabilities, K B / KA- Therefore, according to Eq. 5.50, K; is the same as the exchange equilibrium constant, and the rational activity coefficients ofAX(s) and BX(s) in the exchanger phase both have unit value. If the exchanger is selective for B+, KB > K A ; if it is selective for A+, K A > KB (see the discussion in Section 5.1). That the exchanger phase forms an ideal solid solution is not surprising, given assumptions (l) and (4) in Section 6.2. In fact, this same result would be found even if assumption (4) were generalized somewhat to permit the cations to interact, as long as the energy of interaction were the same between any two cations, regardless of their species.' However, experimental data on uni-univalent exchanges in soils usually are not described well by a K; that is independent of the exchanger composition. This characteristic may reflect either nonuniformity among the exchange sites, as discussed in Section 5.3, or nonuniformity among the interactions between the exchangeable cations (i.e., the energy of interaction between A+ and B+ differs from that between A+ and A +, and between B+ and B+, on the exchange sites). Either kind of nonuniformity would produce a variable Ky. Quite possibly both kinds exist in soils. The discrete site model can be extended readily to uni-bivalent cation

THE MOLECULAR THEORY OF CATION EXCHANGE

167

exchange after dropping assumption (2) and making a slight modification of assumption (4) in Section 6.2. Instead of permitting only one cation per exchange site, each site will be assumed to be univalent and, therefore, each cation of valence + u will be assumed to occupy u sites. Thus a univalent cation will occupy one site and a bivalent cation will occupy two sites. This stipulation has the effect of introducing a "configurational interaction" between two cations of different valence, despite the assumed lack of coulombic interaction between them, indicated in assumption (4) of Section 6.2. If a bivalent cation occupies two neighboring sites, not all of the sites that are neighbors of one of the two occupied sites are available with the same probability to other cations. The double occupation of exchange sites "shuts off" some of the options open to the cations in a uni-univalent exchange to the extent that the exchanger phase will no longer be an ideal solid solution, in spite of assumption (4). Consider now an array of exchange sites in which each site has z nearest neighbors. For example, if the array is two dimensional and the sites are in equally spaced rows, z = 4 (square planar array); if the array comprises sites arranged in equilateral triangles, z = 6 (triangular planar array), as it does also for a three-dimensional array of simple cubes; if the array is a face-centered cubic lattice, z = 12. Whatever the value of z, a consideration of specific examples will demonstrate that, for the exchangeable cation A + occupying a single site, there are z nearest-neighbor sites whereas for a bivalent cation B2+ occupying two sites, there are 2(z - 1) nearest-neighbor sites. For example, if z = 4, A + has four nearest-neighbor sites and B2+ has six. Because of assumption (4) in Section 6.2 and because the exchange is unibivalent, the largest set of univalent exchange sites that needs to be studied in order to work out probabilities of occupation is a set containing two sites. Let PAA(sites) be the probability that a neighboring pair of sites is occupied by two A +. This probability is equal to the probability that any site is occupied by A + times the probability that any neighbor of the first site also is occupied by A +. The probability that both the first site and the second one are occupied by A + equals a product of independent probabilities. Therefore NA

zN X - - - - ' - 'A- - - M zN A + 2(z- l)N B

PAA(sites) = -

(6.44)

where [zN A + 2(z - I)NBl is the total number of nearest-neighbor sites available for occupation, and the second factor on the right side of Eq. 6.44 is the fraction of sites that are nearest neighbors of one occupied by A +. Now let PH(sites) be the probability that a pair of neighboring sites is occupied by B2+. This probability is equal to the probability that any site has B2+ on it times the probability that anyone of its z nearest neighbors has "the rest" of B2+ on it. Therefore I'II(sitcs)

(6.45)

"

,

," THE MOLECULAR THEORY OF CATION EXCHANGE

168

where the factor of 2 occurs because each 82+ has two positive charges with which to cover a univalent site. The factor 1/ z is the chance that the second positive charge on 82+ occupies a nearest neighbor of the site where the first positive charge has been placed. From Eqs. 6.44 and 6.45 it follows that PB PA A

2N B[zNA + 2(z - 1)N Bl = (zN A)2

(6.46)

However, according to the basic postulates of statistical mechanics, as embodied in Eq. 6.25, it also is true that

PB PM

n

QB exp(#LB/ k B [ QA exp(#LA/ k B

-

(6.47) ,

nP

with #LB and #LA given in Eq. 6.30. The combination of Eqs. 6.46, 6.47, 6.30, and 6.32 produces the expression 2N B[zNA + 2(z - 1)N Bl (zN A)2

i

(6.48a)

Equation 6.48a may be employed to yield NBai KB Z2 Ky = - - = -.,---------.,Nia B 2Ki [zNA + 2(z - I)NBl

(6.48b)

for the Vanselow selectivity coefficient pertaining to the exchange reaction in ' Eq. 5.1 (with u = I, v = 2). In this case, K; will vary with the exchanger. composition and, therefore, the rational activity coefficients, fA and fB' cannot have unit value. The rational activity coefficients, fA and fB' and the exchange equilibrium . constant, KeX' can be calculated directly for the model by inserting K; in Eq, 6.48b into Eq. 5.50. The results of this calculation are: 2 In fA

=

2 In.( 2 - 2

In fB = In (2 - E B ) In Kex

=

In

Z 2K B) (

Ki

E) B -

z In

(z -

+ (z

-

(z - E) E) z z

1) In ( z -

B

I

B

(6.49a)

i'

(6.49b)

1) In (z - 1) - z In z

where E B is the equivalent fraction of BX 2(s) in the exchanger phase. The cal- . cuiation is facilitated by writing Eq. 6.48 in the alternate form K

_

z 2 K B (2 -

y - 4 x; (z -

E B)

EB)

(6.50)

The value of K.. and the exact variation with composition of the rational activity' coefficients depend on the magnitude of the parameter z. Often, in applications

THE MOLECULAR THEORY OF CATION EXCHANGE

of Eq. 6.48, the value z to the expressions:

=

169

4 is chosen. In that case, Eqs. 6.49 and 6.50 reduce

_ (2 - E B ) _ 4 (2 - N A ) fA - 8 (4 - E B)2 (3 - N A) 2 I'

= 27 (2 -

E B) = 27 (1 (4 - E B)3 4 (2

JB

Ky

K

ex

=

4KB(2 - E B ) K1(4 - E B )

-

+ +

N B) 2 N B)3

4KB -----=-K1(2 + N B)

27 K B = 16 K1

The model expressions for fA and fB bear some resemblance to those for the rational activity coefficients pertaining to the Gaines-Thomas exchange constant, listed in Table 5.l. The discrete site model can be applied to an arbitrary uni-multivalent cation exchange as described in Eq. 5.1 (with u = I). The generalizations of Eqs. 6.44, 6.45, and 6.47 appropriate to uni-multivalent exchange are:"

A ( N N AN·)V-I

N PA ...A(sites) = M X

A

PB(sites) =

NB

a -

M

+q

(6.51)

B

1

X P

(6.52)

n

QB exp(~B/ k B [QA exp(~A/ k B

nr

(6.53)

where q = v - (2/ z)( v-I), a is the number of permutations of the positions of the sites within the set of v sites occupied by W+ that leave the set unchanged in its geometric arrangement (e.g., a = 2 for a pair of sites and a = 6 for an equilateral triangle of sites), and p is the number of ways BV+ can be placed on a set of nearest-neighbor univalent sites after it has been placed on one site (e.g., p = z for B2+). If W+ is assumed to occupy a linear array of v sites on the exchanger, a = v! (the factorial function) and p = z, regardless of the overall configuration of the sites. In the special case v = 2, it can be verified directly that Eqs. 6.51 to 6.53 reduce to Eqs. 6.44, 6.45, and 6.47, respectively.

6.4. THE DIFFUSE DOUBLE LAYER MODEL

The diffuse double layer model of cation exchange is based on four assumptions that are quite different from those on which the discrete site model was founded. These assumptions may be stated as follows.

170

THE MOLECULAR THEORY OF CATION EXCHANGE

1. The exchange sites form a continuum of negative charge of density (J on a uniform, planar surface that is effectively infinite in extent. 2. The exchangeable cations are dissociated completely from the exchanger and interact through the coulomb force both among themselves and with the surface. The same condition applies to the anions in the system. 3. The mean electric potential at the position of an ion near the exchanger surface is proportional to the average energy required to bring the ion from infinite distance to that position. 4. The water in the aqueous solution phase is a continuum liquid characterized by a uniform dielectric constant. Assumption (1) indicates that an exchanger is to be modeled as a continuous, planar distribution of charge, without consideration of "edge effects" in the electric field, which would be created by this charge distribution if it had finite dimensions. Thus the exchange sites are smeared out uniformly in an infinite plane instead of being discrete entities. Assumption (2) stipulates that complex formation with the exchanger is not permitted and that covalency, or any other cause for a noncoulombic interaction between the exchanger and an exchangeable cation, must be absent. Therefore the valence of an exchangeable cation is the only property that can playa role in the interaction of the cation with the exchanger surface. Assumption (4) models the aqueous solution phase as a continuum into which a swarm of ions is embedded. No direct account of the molecular nature of water is considered, nor is there any reference to a change in the dielectric properties of the water near the exchanger surface or in the vicinity of an ion. Assumption (3) can be interpreted more readily after the basic tenets of the model are developed in terms of fundamental electrostatic theory. Because of assumption (1), the electrostatic properties of the diffuse double layer model need be studied only along a line that extends perpendicularly from the exchanger surface. The behavior in any plane parallel with the surface is the same at all points of the plane because the charge on the exchanger is uniformly distributed. For example, the total charge density on the exchanger surface is related to the volumetric charge density in the aqueous solution phase according to the equation (J

= -

J

d

p(x)dx

(6.54)

o

where p(x) is the net number of coulombs of positive charge per cubic meter between x and x + dx in the vicinity of the exchanger and d is the point along the x-axis (whose origin is at the exchanger surface) where charge balance between the aqueous solution phase and the exchanger is achieved. If there is only one exchanger surface whose charge is to be neutralized. d is the point at infinity. If there arc two exchanger surfaces in a parallel arrangement and lying

THE MOLECULAR THEORY OF CATION EXCHANGE

171

perpendicularly to the x-axis, d is the point midway between the surfaces.' An expression similar to Eq. 6.54 can be written for the density of surface charge neutralized by ions of species i in the aqueous solution phase: a,

= -

J

d

p;(x)dx

(6.55)

o

where p;(x) is the volumetric charge density contributed by ions of species i. Equation 6.55 can be written for the density of surface charge neutralized by ions of any chosen valence. According to the basic principles of statistical mechanics, p(x) may be expressed mathematically by the equation" p(x) = ~jPI(x) = ~jc,ZjF exp[ - W';(x)/ RIl

(6.56)

where c, is the number of moles of ion i per cubic meter in the aqueous solution phase at the point x = d, Z, is the valence of ion i, F is the Faraday constant, and W';(x) is the energy per mole required to bring ion i from the point at infinity to point x, averaged over all configurations of the other ions on solution. The sum in Eq. 6.56 is over all distinct species of ions in the aqueous solution phase. The exponential terms in the sum look very similar to the Boltzmann factor in Eq. 6.1. Actually, each one is proportional to an integral of a Boltzmann factor over the position coordinates of all the ions except ion i; hence W';(x) is an average energy per mole for ion i. According to Assumption (3) of the diffuse double layer model, W';(x) = ZjFif;(x)

(6.57)

where if;(x) is the average electric potential at x. Equation 6.57 would be exact if the effects of non-coulombic interactions (e.g., short-ranged repulsive forces) between the ions could be neglected entirely and if the actual electric potential at a given point did not fluctuate about its average value. Since W';(x) refers to the total energy of interaction, it includes both coulombic and non-coulornbic contributions, whereas if;(x) is only a mean coulombic potential. Moreover, to bring an ion through the aqueous solution phase requires adjustments in the configurations of neighboring ions which, in turn, produce fluctuations in the electric potential, at a given point, about the mean electric potential at that point. These effects are included in W';(x) but not in if;(x). However, the errors of omission involved with Eq. 6.57 turn out to be partially compensating, such that the net error committed in making assumption (3) often may be small.' According to Eqs. 6.54 to 6.57, the computation of (J and a, requires a knowledge of if;(x). This function, in turn, can be found as the solution of the Poisson equation: p(x)

where

to

= IU<5 X 10

I]

C].J

I

m

I

(6.58)

is the permittivity of vacuum and IJ is the

THE MOLECULAR THEORY OF CATION EXCHANGE

172

dielectric constant in the aqueous solution phase. The combination of Eqs. 6.56 to 6.58 produces the differential equation

d 2¥i 1 2 = - - "J:.jcjFZ j exp[ - ZjF¥i(x)/ Rll dx foD

(6.59)

which is called the Poisson-Boltzmann equation. When Eq. 6.59 has been integrated once to obtain d¥i/ dx, the surface charge density (J can be calculated. This relationship can be demonstrated by combining Eq. 6.58 with Eq. 6.54: d Idd2¥i (J = - 0 p(x)dx = foD 0 dx2 dx I

= -foD

Cd¥!) dx

(6.60) x=o

The last step is obtained by imposing the condition that the electric field, - d¥i/ dx, vanish at x = d. Consider now the case of uni-univalent exchange, described by setting u - v = 1 in the reaction in Eq. 5.1. In this case,

p(x) = (cA + cB)F{exp[ - F¥i(x)/ Rll - exp[F¥i(x)/ Rll} = -2F(cA + cB) sinh [F¥i(x)/ Rll

(6.61)

according to Eqs. 6.56 and 6.57, where the fact that CA + CB - Ccl has been used in the first step and the definition 2 sinh y = eY - e ? has been used in the second step. Equation 6.59 then becomes the differential equation: ~¥i

dx

2

2F

= -

foD

(CA

. + CB) smh[F¥i(x)/ Rll

(6.62)

The manipulation of Eq. 6.62 is facilitated by introducing into it the transformations (6.63)

y(x) = F¥i(x)/ RT Then

(6.64) which may be integrated on both sides once to give the result:

d2y dy . dx'

r

=

2y

r d dy dx = ~ . dx' dx

= ,\l(C" -+

en)

I

y)2 d (d . dx .1·

sinh y dy

= ~ (dy)2

= ,\2(C" -+

dx

en) cosh y

-+

K

THE MOLECULAR THEORY OF CATION EXCHANGE

173

or

(~~)2 =

2X 2(CA + cB) cosh Y

+ 2K

(6.65)

where K is a constant of integration. The assumption that dy j dx vanishes at x = d can be used to evaluate the constant Kin Eq. 6.65: (6.66) o = 2X2(CA + cB) cosh Yd + 2K where Yd

== y(d).

Therefore

dy dx

=

{2X2(CA + cB)(cosh Y - cosh Yd)}1/2

(6.67)

and -~2F

= -X- [2(cA + cB)(cosh Yo - cosh Yd)] 1/2

(J

(6.68)

where Yo = y(O). The evaluation of the amount of surface charge density neutralized by an excess of A + and one-half the deficit of Cl- proceeds similarly to the derivation of Eq. 6.68, except that Eq. 6.60 cannot be used. According to Eqs. 6.55 and 6.67, (JA

= -

I

d

o

= 2FcA

{FcA exp[ - Fy;(x)j Rl1 - FC A exp[Fy;(x)j Rl1}dx

I I

d

sinh[Fy;(x)j Rl1dx

o

= 2Fc A

Yd

sinh y

dx

d

Y

Yo

2Fc A [2X2(CA + CB)] 1/2 -

dy

I

Yd

YO

sinh y dy (cosh y - cosh Yd)I/2

-4FcA(cosh Yo - cosh Yd)1/2j[2X 2(CA + CB)] 1/2

(6.69)

where the result d(cosh y) = sinh y dy has been employed to obtain the fourth step. The mole fraction of surface charge neutralized can be calculated by dividing (J in Eq. 6.68 into (J A: (6.70a) The corresponding result for N B is: a -

all

Nil = --

a

=

--=-('"

+

('II

(6.70b)

THE MOLECULAR THEORY OF CATION EXCHANGE

174

If it is assumed that (1) the concentrations CA and CB are equal to the millimolar concentrations of ACI and BCl in the aqueous solution phase, and (2) the activity coefficients of AC1(aq) and BC1(aq) are equal, then the Vanselow selectivity coefficient may be calculated directly with the help of Eqs. 5.3 (with u = v = 1) and 6.70: (6.71) According to the diffuse double layer model, K; is a constant equal to 1.0. Thus the exchanger phase forms an ideal solid solution, and uni-univalent exchange does not exhibit selectivity of one cation over another. The same thing can be demonstrated, in fact, for any homovalent exchange reaction simply by replacing F with ZF in Eqs. 6.61 to 6.69, where Z is the valence of cation A or B. The physical basis of this result comes from assumptions (2) and (3), which permit only coulomb interactions among the exchangeable cations and provide only the valence as a measure of the strength of the ion-exchanger interaction. In the discrete site model, on the other hand, an exchangeable cation is permitted to interact with the exchanger in any fashion possible and, therefore, differences in selectivity among cations of the same valence are consistent with that model. In both the discrete site and the diffuse double layer models, ideal behavior in the exchanger phase is the direct result of assuming uniformity in the interactions among the exchangeable cations. The case of uni-bivalent exchange, as described in Eq. 5.1 with u = 1 and v = 2, can be studied by specializing Eq. 6.59 to the expression d 2y;

-

dx'

- F

= -

~oD

{CA exp[ -

Fy;(x)j Rl1

+ 2c B exp[ -2Fy;(x)j Rl1 -

Cel

exp[Fy;(x)j Rl1}

(6.72a)

It is customary to simplify this equation by replacing the last term on the right side with the two terms CA

exp[Fy;(x)j Rl1

+ 2c B exp[2Fy;(x)j Rl1

The second exponential should be exp[Fy;(x)j Rl1, but it is assumed conventionally that its replacement by exp[2Fy;(x)j Rl1 has a negligible effect on Y;(x). Therefore Eq. 6.72a is written in the form

d~~x

= 2F {cA sinh [Fy;(x)j Rl1 ~oD

+ 2c

B

sinh[2Fy;(x)j Rl1}

(6.72b)

The transformations in Eq. 6.63 bring Eq. 6.72b into the form

~~

=

,\2(£,,,

sinh y

+

2£'n

sinh 2y)

This equation may be integrated on both sides once

10

(6.72c)

produce an expression

THE MOLECULAR THEORY OF CATION EXCHANGE

175

analogous to Eq. 6.67: dy dx = {2A 2[cA(cosh y - cosh Yd) = {2A2[cA(cosh Y -

cosh Yd)

+ cB(cosh 2y + 2cB(cosh2 Y -

cosh 2Yd)]}I/2 cosh" Jd)]}I/2

(6.73)

where the identity 2 cosh? Y - 1 = cosh 2y has been used in the second step. The surface charge density follows from combining Eqs. 6.60 and 6.73: a

-2F

= -A- {2[cA(cosh Yo

+ 2coCcosh 2 Yo

- cosh Yd)

- cosh' Yd)]}1/2

(6.74)

The amount of surface charge neutralized by an excess of A + and one-third of the deficit of CI- is calculated by the same procedure as was carried out to obtain Eq. 6.69: U

J

A = 2Fc A

sinh Y

Yd

J

2FcA

Ud

= V2 A

~x dy y

Yo

Uo

du c- -( u -U 2)] 1/2 ( U 2-- -U) 2-CB-

-[

(CA/2)

+

d

A

(u -

coshy)

d

+ 2c BuO +

Y2c B [CA(U O

-

Ud )

+ 2CB(U~

- U~)]1/2 (6.75)

This complicated expression can be simplified with the help of Eq. 6.74. First, it may be noted that -Au 2F

--=

(6.76a)

which can be substituted into the denominator of the logarithm in Eq. 6.75. Second, Eq. 6.74 can be written as a quadratic algebraic equation in Uo: 2 2

2 CAU O I fA8P U Uo + 2c - 2c B B

+

CAUd

+ 2C BUd2J

= 0

The solution of this equation is -

CA

Uo = 4CB

1

CA

2

I

+ f16 (C) + 2CB B

(

Au 2 2

8P

+

2)

CAUd

+ 2C BUd

J1/2

(6.76b)

(The positive square root is chosen in order that Uo = I when Ud = I and = 0.) The introduction of Eq. 6.76 into Eq. 6.75 produces the expression a/\

=

F('/\

A~

In

ArT ~ I

I P( 1'/\ (1'/\

I 4('1111,,))

I 41'1111,,)1,'

+ I'IIA)a) I I

U

I!)

!

(6.77)

THE MOLECULAR THEORY OF CATION EXCHANGE

176

Finally, the identity - sinh-I y-ln[-y

+ (y2 +

1)1/2]

may be used to rewrite Eq. 6.77 in the form" UA

=

A FC .r

•

Sin

h- I

AVCB

l

(cA

Au0:

+ 4C B U d)F

J

(6.78)

The division of U A in Eq. 6.78 by U yields an expression for the equivalent fraction ofAX(s) in the exchanger phase. This expression, in turn, may be transformed into one for the mole fraction N A with the help of Eq. 5.25a. Then the mole fraction N B can be calculated as the difference, 1 - NA- These results may be employed to calculate K; according to Eq. 5.3 (with u = 1 and v = 2). The final expression is complicated and cannot be evaluated numerically without data on a, U d, CA , CB, and the activity coefficients of AC1(aq) and BCIlaq). A straightforward, alternate procedure for making this computation has been presented.? First, Eq. 6.78 is written in the form

EA where E A

=

=

SAR Sin 'h- I

v73

U

l

SAR

v73

+

U

4Ud

0:

J

(6.79)

uA/u is the equivalent fraction ofAX(s) in the exchanger phase,

is the sodium adsorption ratio,

fJ

= P/~oDRT = 1.084 X 1016 m mol:"

(T = 298.15 K)

and U is expressed in units of moles per square meter. Next, a plot of E A versus SAR according to Eq. 6.79 is made at fixed ionic strength, for chosen values of the parameters a and U d. Generally, this plot will be slightly different for different ionic strengths. By extrapolation, one can obtain the relation between E A and SAR that applies for zero ionic strength. Then the value of N A corresponding to each value of E A can be calculated, as can the value of Ns = 1 - N A • With this information and the fact that SAR is proportional to aA / alf2 in the limit of zero ionic strength, the values of K; can be computed according to Eq. 5.3 (with U = 1, v = 2). Some illustrative results of this method of computation are given in Table 6.1. 9 The values of K; apply to the exchange reaction 2AX(s)

+ BClbq) = BXls) + 2AC1(aq) + (0). The values of chosen are characteristic of the

tr and to U d = 1.0 (d = clay minerals montmorillonite (u = - 1.1 X 10- 6 mol m") and kaolinite (0' = -5 X 10- 6 mol rn"). It is apparent that K; increases with increasing EA , according to the diffuse double layer model. Thus the model predicts nonideal behavior in the exchanger phase for uni-bivalcnt exchanges, Moreover, a change

I

THE MOLECULAR THEORY OF CATION EXCHANGE

177

TABLE 6.1. The Vanselow and Gapon selectivity coefficients for uni-bivalent exchange, calculated according to the diffuse double layer model" (Y

(mol m")

-1.1 X 10- 6

EA

Ky

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.210 0.257 0.318 0.405 0.481 0.581 0.685 0.813 0.926

K,,:

0.459

~G~x

(kJ mol "):

+ 1.908

-5.0 X 10- 6

KG

Ky

KG

0.826 0.826 0.826 0.826 0.800 0.763 0.694 0.599 0.441

4.76 5.88 7.14 9.09 11.11 12.50 16.67 20.00 20.00

4.00 4.00 4.00 4.00 3.85 3.57 3.33 2.86 2.13

10.00 -5.786

in selectivity is predicted with an increase in 0". At the higher value of 0", the exchanger is selective for A +, as indicated by the facts that K; and Kex (computed with the help of Eq. 5.50c) are less than 1.0 and that AG~x > O. At the lower value of 0", the exchanger is selective for B2+, since K; > 1.0 and AG~x < O. The surface charge density at which AG~x = 0 is, in fact, -1.7 X 10- 6 mol m". These predictions usually are not in agreement with experimental data. Values of the empirical exchange constant KG, which was defined in Table 5.1, also are listed in Table 6.1. Although KG does not remain constant as the composition of the exchanger is varied, the diffuse double layer model does predict that KG will undergo no change for E A -< 0.4, regardless of the value of 0". In the particular case of Na-Ca exchange, this result is of great practical significance. since good agricultural soils are maintained with EN. -< 0.20. For these soils, if the diffuse double layer model is applicable, a constant value of KG may be expected to describe the Na-Ca exchanges of interest according to the equation

which follows from combining the expression in Table 5.1 with Eq. 5.25a. However, as can be verified with the data in Table 6.1, the numerical value of Kb is not equal to that of K w nor is there a proportional relationship between the two parameters. The Gapon selectivity coefficient must not be accorded the significance of an equilibrium constant simply because its value remains fixed over a certain range of exchanger compositions.

178

THE MOLECULAR THEORY OF CATION EXCHANGE

6.5. THE SURFACE COMPLEX MODEL

It is accepted generally that both the discrete site model and the diffuse double layer model contain elements of truth, insofar as the molecular mechanism of cation exchange is concerned. The discrete site model has the positive attributes of local exchange sites and allowance for complex formation between an exchangeable cation and the exchanger surface. The local sites are consistent with experiments concerning the nature of exchange sites on clay minerals and soil organic matter. The possibility of complex formation leads to a reasonable explanation of differences in selectivity among cations having the same valence. On the other hand, the discrete site model does not include interactions between cations bound to exchange sites (except for the "configurational interaction" that occurs in heterovalent exchanges), nor does it account for the neutralization of exchanger charge by anion deficits. These characteristics, however, are features of the diffuse double layer model. Through Eqs. 6.54 and 6.56, which include contributions from both cationic and anionic species in the aqueous solution phase, the negative surface charge of the exchanger is required to balance the sum of local cation excesses and anion deficits. The excess cations, moreover, are allowed to interact among themselves to the extent implied in Eq. 6.57 (i.e., through coulomb forces with an averaged interaction). What is missing from the diffuse double layer model is a basis for selectivity other than cationic valence and, of lesser importance, a possibility for short-ranged interactions among neighboring exchangeable cations. Since these two characteristics either are, or could be, natural features of the discrete site model, it is inevitable that a combination of the diffuse double layer and discrete site models should be given consideration. Perhaps the simplest type of surface complex model is the type that pictures two distinct modes of binding between a given species of exchangeable cation and an exchanger surface. One of these modes of binding entails the formation of an inner-sphere complex, while the other involves the development of a diffuse double layer. Thus, in a binary cation exchange reaction, each type of exchangeable cation can be found both closely bound to the exchanger surface and dissociated from the surface as part of a swarm in the aqueous solution phase. This concept forms the basis for the well-known Stern model of the electrical double layer. The elaboration of the Stern model to include a calculation of the Vanselow selectivity coefficient involves the definition of surface complex stability constants for each type of exchangeable cation and the assumption that only the ions in the diffuse double layer can interact mutually. 10 This approach is limited by its need for independent estimates of the surface complex stability constants and by its exclusion of outer-sphere surface complexes from consideration. It is probable that the surface complexes formed by the alkali metal and alkaline earth metal cations will involve solvated species. Recent versions of the surface complex model I J contain methods for esti-

THE MOLECULAR THEORY OF CATION EXCHANGE

179

mating stability constants as curve-fitting parameters and permit the formation of both inner-sphere and outer-sphere complexes. These models have been applied principally to oxide minerals that are assumed to form inner-sphere complexes with protons and outer-sphere complexes with all other cations as well as with anions. Unfortunately, in most cases, the derivation of the model equations has involved an arbitrary division of the electrochemical potential of a cation into "purely chemical" and "purely electrical" parts, as prescribed in Eq. 4.23. This step cannot be justified thermodynamically but must be taken as a definition within the model whose only justification is the success with which the model parameters can be adjusted to fit equations to experimental data. The development of a surface complex model in quantitative terms can be illustrated through a consideration of one of the so-called site-binding models of the electrical double layer." The model calculations will be applied to an exchanger that is selective for protons (e.g., insoluble organic matter, an oxide surface, or the edge surface of kaolinite or montmorillonite), but this application should not be interpreted as a limitation of the scope of surface complex models. The fundamental assumptions in the site-binding model are as follows. 1. The exchanger surface contains functional groups that form innersphere complexes with protons and outer-sphere complexes with both cations and anions. 2. The complexed protons do not interact with other complexed ions at the surface. Complexed cations, other than protons, and complexed anions interact with the electric charges at exchange sites that are either occupied by complexed ions or neutralized by local excesses and deficits of the ions in the aqueous solution phase. 3. The exchange sites that do not participate in surface complexation bear negative charges that, together with the ion swarm in the aqueous solution phase, form a diffuse double layer as described by assumptions (1) to (4) in Section 6.4. Assumptions (1) and (3) can be translated readily into conditions of mass and charge balance on the exchanger. If M is the total number of exchange sites, the mass balance condition can be expressed by the equation M = N(SOHi)

+

+ N(SOH;"

- A -) + N(SOH) N(SO- - M+) + N(SO-)

(6.80)

where SO- refers to a proton-selective functional group on an exchanger surface (i.e., an exchange site), N(SOH) is the number of surface proton complexes of the type SOH, with similar meanings for the other terms on the right side of Eq. 6.80, and SOH;" - A ~, SO- - M+, and SO- refer to outer-sphere anion complexes, outer-sphere metal cation complexes, and uncomplexed functional groups, respectively. Equation 6.HO is limited to complexes involving only simple univalent ions in order to kccp thc mathematics uncomplicated. Outer-sphere

180

THE MOLECULAR THEORY OF CATION EXCHANGE

complexes that contain multivalent ions or hydrolytic species, such as MOH~-m) can be included in the model without difficulty. The condition of charge balance that corresponds to Eq. 6.80 has the form (10

+ (1~ + (1dilf

(6.81)

= 0

where

.N~.A

(10 =

(1~

=

(1dilf

=

[N(SOHi)

+ N(SOHi

- A -) - N(SO- - M+)

- N(SO-)]

(6.82a)

.N~.A

(6.82b)

[N(SO- - M+) - N(SOHi - A -)]

+2[EiJDRT(cH+ cM)(coshYdilf - 1)]'/2

(6.82c)

and .A is the total surface area occupied by the exchange sites. The surface charge density (10 is that created solely by the gain or loss of protons from the exchanger and the surface charge density (1~ is that created solely by the formation of outer-sphere complexes. The sum, (10

+ (1~ = .N~.A

[N(SOHi) - N(SO-)]

represents the surface charge density that must be neutralized by ions in the aqueous solution phase. Thus (1diff is the surface charge density attributed to local excesses or deficits of these ions and is expressed mathematically as an application of Eq. 6.68. It is assumed that the only electrolytes present in the aqueous solution phase are HA(aq) and MA(aq) and that the parameter din Eq. 6.54 is equal to +00. The quantity Ydiff = F1/;(Xd)/ RT, where 1/;(xd) is the mean electric potential just outside the region near the exchanger surface that contains the outer-sphere complexes. This electric potential is related to that at the exchanger surface by the equation (10

(1dill'

=---

(6.83)

The parameters C, and C2 are integral capacitances, respectively, of the region 1 between the exchanger surface and the locus of centers of the outer-sphere complexes and of the region 2 between the locus of centers and the plane through the point x = Xd' Equations 6.80 to 6.83 constitute a set of algebraic equations that can be solved uniquely for the numbers of all of the surface species, given input values of C, and C2, which are treated as empirical curve-fitting parameters, and given a method to relate N(SOHi), N(SO- - M+), and so forth. to concentration data pertaining to the aqueous solution phase. The statistical mechanical aspects of surface complexation can be developed on the basis of assumption (2). To make the calculation as clear as possible. M I will be considered. Thus it is assumed only the complexes SOil and SO

THE MOLECULAR THEORY OF CATION EXCHANGE

181

that experimental conditions have been arranged so that Eqs. 6.80, 6.82a, and 6.82b reduce to the expressions M = N(SOH) ITo

=

-F .NA

+ N(SO-

- M+)

.:Ii N(SO- - M+) =

-lTp

(6.84) (6.85)

Because of Eq. 6.85, there is no contribution from the diffuse double layer to the balance of surface charge (Eq. 6.81). Let z be the average number of exchange sites that are nearest neighbors of a given exchange site. (The modifier "average" is employed because the array of sites need not be regularly spaced). Then (6.86) where N H = N(SOH), N M = N(SO- - M+), and Nap(a, (3 = H, M) is the number of pairs of nearest-neighbor sites in which one site is occupied by species a, the other by species {3. Equations 6.86 are mass-balance equations for the total number of nearest neighbors of a site containing the complex SOH (i.e., zN H) and of a site containing the complex SO- - M+ (i.e., zN M) , respectively. The partition function for the assembly of M sites is given by the equation Q(T, M, N M) = [qM(T)]NM[qH(T)]M-NM

L

g(NM, N MM, M)e-NMM'MM/kBT

NMM

(6.87) where q( T) is the same as QA in Eq. 6.25a (i.e., the partition function for a cation bound to an exchange site and fMM is the energy of interaction between M+ on a site and the positive charge on a nearest-neighbor site that also contains M+ in a complex). The factor g(NM, N MM, M) is the number of distinguishable arrangements of H+ and M+ on the sites that are possible for chosen values of N M, N MM, and M. The sum in Eq. 6.87 is carried out over all values of N MM that are consistent with Eq. 6.86 and with the condition

~

M! g(NM, N MM, M) = NM!(M - N M)!

(6.88)

where M! = M(M - I) ... I is the factorial function. The term on the right side of Eq. 6.88 is the total number of distinguishable arrangements of NM M+ ions and N H = (M - NM ) H+ ions on M sites. Equation 6.87 is the special case of Eq. 6.2 that results for a two-component assembly subject to Eq. 6.84 and assumption (2), provided that the states of H+ or M+ bound in a surface complex can be separated from the states that are produced by different arrangements of the cations on the sites. This separation of the two types of states should be valid as long as the surface complexes SO- M+ are not perturbed seriously by the interactions between M+ and its neighboring exchange sites. Upon substitution of both the second expression in Eq. 6.86 and the definition

THE MOLECULAR THEORY OF CATION EXCHANGE

182

into Eq. 6.87, it becomes

x

q~

-

NM

L flN M, N MM,

M)e NMM'MM/2k BT

(6.89)

NMM

Equation 6.89 presents a formidable computation problem because of the sum over NMM. However, the problem can be made tractable by evaluating Q( T, M, N M ) in the van der Waals limit. This limit is represented by the expression (6.90)

lim

where w is a finite constant. In the van der Waals limit, the number of nearest neighbors of M+ on a site becomes arbitrarily large, while the energy of interaction between M+ and these neighboring sites goes to zero in such a way that the product UMM remains finite. The physical significance of this limiting process is that a cation M+ interacts equally with all of the other sites containing M+ (i.e., they all become nearest neighbors) and the average energy of interaction per cation pair with these sites becomes w. The cation M+ is, then, subjected to a kind of average electric field created collectively by all of the other sites containing M+ cations in surface complexes. This situation is very much like what would be expected in an array of interacting outer-sphere complexes because of the long-ranged character of the coulomb force. If the coulomb interaction dominates the short-ranged interactions between the complexed M+, the van der Waals limit of Eq. 6.89 should be a very good approximation. In this limit, the last factor in Q( T, M, N M ) becomes lim z-oo

L flN M, N MM,

M)e N HM'MM/2k BT

=

e WN M(M-NM)/2MkBT

tMM---O NMM

and Eq. 6.89 reduces to

M' Q( T , M , N M) = NM!(M _. N )! M _

M'• NM!(M - N M)!

(q

e-w/2kB1\NMqM-NMeWNM(M-NM)/2MkBT J H

M

(q M

e-wNM/2MkB1\NMqM-NM J H

(6.91)

All of the thermodynamic properties of the assembly may be calculated using Eq.6.91. The proper thermodynamic potential with which to describe the assembly is the Helmholtz energy, whose total differential is given in Eq. 1.7:

dA = - SdT - ¢dM

+ IJMdN M + IJlldN II

This expression differs from /.:1.1. 1.7 in having (/>dM instead of PdV. where ¢ = PV M and VM is the average volume per exchange site. Because of Fq. 6.X4,

,

THE MOLECULAR THEORY OF CATION EXCHANGE

183

N M and N H cannot both be treated as independent variables and, therefore, the total differential of A is written in the form (6.92) According to Eqs. 6.14 and 6.92: JLM -

=

JLH

(6.93)

-kBT(alnQlaNM)T,M

Equation 6.93 is the only statistical thermodynamic expression required to calculate the exchange equilibrium constant. With the help of this expression and the Stirling approximation, In N! = N(In N - I), which is excellent for any N> 10, Eqs. 6.91 and 6.93 can be combined to yield JLM -

JLH

=

kBT [ In (

q:

q )

+

In

(N~

eWNM/MkBT)

_ N

1

(6.94)

M

According to thermodynamics, JLM -

JLH

=

JL~ -

JL~

+

k B TIn (aMI aH)

and, therefore, Eq. 6.94 may be written in the alternate form Ky

= (M

KM

NMa H

_ N)

M aM

= -K

exp( - wNM/MkBT)

(6.95)

H

where K H and K M are defined in Eq. 6.32 [with Q - q( T)]. These two parameters may be interpreted in probabilistic terms in the same way as were K A and K B in Section 6.2. The rational activity coefficients.ji, and 1M, and the exchange equilibrium constant may be calculated by introducing Eq. 6.95 into Eq. 5.50 (with u = v = 1, A = H+, B = M+, and E B = NM/M): (6.96a) (6.96b) (6.96c) Equations 6.96a and 6.96b are of the same mathematical form as the Margules expansions in Eq. 5.1Ic and 5.IId, with a, = -w12k B T and a, = a, = 0. Since only a, is nonvanishing, the present model is a special case of the regular solution model, which was mentioned in Section 5.1. With the help of Eqs. 6.84 and 6.85, Eq. 6.95 can be written in the logarithmic form

pH - log where

(I

a

-(~

)

+

log(M I)

=

-log

(K

M)

KIIJ

+

W

(InIO)knT

(~(6.97)

THE MOLECULAR THEORY OF CATION EXCHANGE

184

If the left side of Eq. 6.97 is plotted as a function of a, a straight line should result whose slope is w/(ln 10)kBT and whose y-intercept is -log(KM / K H ) . Thus measurements of pH and (M+) in the aqueous solution phase, along with measurements of rIo under the conditions stipulated in Eqs. 6.84 and 6.85, can lead to a determination of the parameters wand K M / K H . Once the parameters in Eq. 6.95 have been measured, that equation can be written in the form

N(SO- - M+) =

N(SOH)(M +) K M (+) - exp(w.A.N ArIo/MFkBT)

H

KH

(6.98)

which then may be substituted into Eqs. 6.80, 6.82a, and 6.82b. The same procedure can be followed for all of the terms on the right side of Eq. 6.80. Then, for the determined values of parameters such as K M / K H and w, the mass and charge balance equations can be solved to obtain N(SOH), 1/;(0), and 1/;(xd ) , subject to Eq. 6.83 and a chosen value for C2•

NOTES I. The statistical mechanical theory of adsorption onto nonuniform surfaces has been discussed in a number of articles. See (in the order listed) G. Halsey and H. S. Taylor, The adsorption of hydrogen on tungsten powders, J. Chern. Phys. 15:624-630 (1947); R. Sips, On the structure of a catalyst surface, J. Chern. Phys. 16:490-495 (1948); T. L. Hill, Statistical mechanics of adsorption. VI. Localized unimolecular adsorption on a heterogeneous surface, J. Chern. Phys. 17:762771 (1949); R. Sips, On the structure of a catalyst surface. II., J. Chern. Phys. 18:1024-1026 (1950); U. Landman and E. W. Montroll, Adsorption on heterogeneous surfaces. I. Evaluation of the energy distribution function via the Wiener and Hopf method, J. Chern. Phys. 64: 1762-1767 (1976); and G. Sposito, Derivation of the Freundlich equation for ion exchange reactions in soils, Soil Sci. Soc. Am. J. 44:652-654 (1980). 2. C. Krishnamoorthy, L. E. Davis, and R. Overstreet, Ionic exchange equations derived from statistical thermodynamics, Science 108:439-440 (1948); C. Krishnamoorthy and R. Overstreet, Theory of ion exchange relationships, Soil Sci. 68:307-315 (1949); C. Krishnamoorthy and R. Overstreet, An experimental evaluation of ion-exchange relationships, Soil Sci. 69:41-53 (1950); L. E. Davis, Ionic exchange and statistical thermodynamics. 1. Equilibria in simple exchange systems, J. Colloid Sci. 5:71-79 (1950); L. E. Davis, Ionic exchange and statistical thermodynamics. II. Equilibria in irregular systems, J. Colloid Sci. 5:107-113 (1950); and L. E. Davis and J. Riblc, Monolayers containing polyvalent ions, J. Colloid Sci. 5:81-83 (1950). 3. See the first paper by L. E. Davis, cited in note 2, for a demonstration of this conclusion. 4. For details of the calculation, see Chapter 10 in E. A. Guggenheim, Mixtures, Oxford University Press, London, 1952. 5. If an exchanger is regarded as a dielectric plate of finite thickness that bears two charged surfaces, and if two of these plates are parallel, the charge on the two facing surfaces need not be neutralized solely by the ion swarm between them. This is true because the ions in the aqueous solution phase near the outer surfaces of the plates create neutralizing electric fields that can also act on the inner surfaces. However, in most cases of interest, d may be set equal to the midway point between the two facing surfaces without serious error. For additional discussion, see E. C. Childs, The space charge in the Gouy laycr between two planc, parallel non-conducting particles, Trans. Faradav SOl', 50: 1.156 1.162 (1954),

THE MOLECULAR THEORY OF CATION EXCHANGE

185

6. L. Onsager, Theories of concentrated electrolytes, Chern. Rev. 13:73-89 (1933) and J. G. Kirkwood, On the theory of strong electrolyte solutions, J. Chern. Phys. 2:767-781 (1934). 7. For a discussion of this point, see Chapter 7 in M. J. Sparnaay, The Electrical Double Layer. Pergamon Press, Oxford, England, 1972. 8. The equation for (J A was originally derived by E. Eriksson, Cation-exchange equilibria on clay minerals, Soil Sci. 74: 103-113 (1952). See also G. H. Bolt, Ion adsorption by clays, Soil Sci. 79:267-276 (1955), and G. H. Bolt and A. L. Page, Ion exchange equations based on double-layer theory, Soil Sci. 99:357-361 (1965). 9. H. E. Jensen, Potassium-calcium exchange on a montmorillonite and a kaolinite clay. II. Application of double-layer theory, Agrochimica XV11:191-201 (1973). 10. A review of this early form of the surface complex model is given in Chapter 9 of S. A. Rice and M. Nagasawa, Polylectrolyte Solutions. Academic Press, New York, 1961. The Rice-Harris model was adapted to soil exchangers in a simplified fashion in W. R. Heald, M. H. Frere, and C. T. deWit, Ion adsorption on charged surfaces, Soil Sci. Soc. Arn. Proc. 28:622-627 (1964). A more complete adaptation was developed by I. Shainberg and W. D. Kemper, Ion exchange equilibria on montmorillonite, Soil Sci. 103:4-9 (1967). See also the discussion of this paper in Soil Sci. 104:444-453 (1967). 11. For comprehensive reviews of some of these models, see R. O. James, P. J. Stiglich, and T. W. Healy, Analysis of models of adsorption of metal ions at oxide/water interfaces, Faraday Disc. Chern. Soc. 59:142-156 (1975); J. W. Bowden, A. M. Posner, and J. P. Quirk, Ionic adsorption on variable charge mineral surfaces. Theoretical-charge development and titration curves, Aust. J. Soil Res. 15: 121-136 (1977); T. W. Healy and L. R. White, Ionizable surface group models of aqueous interfaces, Advan. Colloid Interface Sci. 9:303-345 (1978); and J. Westall and H. Hohl, A comparison of electrostatic models for the oxide/solution interface, Advan. Colloid Interface Sci. 12:265-294 (1980). 12. J. A. Davis, R. O. James, and J. O. Leckie, Surface ionization and complexation at the oxide/ water interface. I. J. Colloid Interface Sci. 63:480-499 (1978), II. J. Colloid Interface Sci. 67:90107 (1978).

FOR FURTHER READING J. O'M. Bockris and A. K. N. Reddy, Modern Electrochemistry. Plenum Press, New York, 1970. An excellent conceptual description of an electrical double layer is given on pages 623-639 of this comprehensive introductory textbook. R. H. Fowler and E. A. Guggenheim, Statistical Thermodynamics, Cambridge University Press, London, 1956. This celebrated treatise presents an exhaustive discussion of the principles of equilibrium statistical mechanics and their applications in physical chemistry. E. A. Guggenheim, Mixtures. Oxford University Press, London, 1952. Chapters I to IV. X, and XI of this book provide a comprehensive discussion of the statistical mechanics of discrete-site models. T. L. Hill, An Introduction to Statistical Thermodynamics. Addison-Wesley, Reading, Mass., 1960. This book should be on the shelf of every soil chemist who is interested seriously in molecular models. Chapters 7 and 14 discuss adsorption phenomena. H. van Olphen, An Introduction to ('lay Colloid Chemistry. John Wiley, New York, 1977. Chapters J, 4, and 7 and Appendix III give a complete survey of the diffuse double layer model as il is applied 10 clay suspensions.

186

THE MOLECULAR THEORY OF CATION EXCHANGE

A. L. Smith, Electrical phenomena associated with the solid-liquid interface, in Dispersion ofPowders in Liquids, ed. by G. D. Parfitt, pp. 86-131, Applied Science Publishers, London, 1973. This excellent survey of the diffuse double layer model covers both theoretical aspects (including site-binding models) and experimental methodologies. G. R. Wiese, R.O. James, D. E. Yates, and T. W. Healy, Electrochemistry of the colloid-water interface, in Electrochemistry, ed. by J. O'M. Bockris, pp. 53-102, Butterworths, London, 1976. This advanced survey presents a detailed review of sitebinding models of the electrical double layer.

~

7 THE THERMODYNAMIC THEORY OF WATER IN SOIL

7.1. VARIABLES OF STATE AND THERMODYNAMIC POTENTIALS

A thermodynamic soil system is a multicomponent, multiphase assembly consisting of inorganic and organic solids, water and dissolved solutes, and constituents of the soil atmosphere, including water vapor. The detailed, macroscopic description of the chemical reactions that may occur in this system has been the principal subject of the previous chapters. The kind of thermodynamic analysis developed in those chapters was germane to a complicated, heterogeneous system whose many chemical components need to be identified and investigated as to their roles in a web of reactions. However, if chemical reactions are not of principal concern and if the chemical composition of each phase in the soil need not be known in detail, the type of thermodynamic analysis required is of a different and much simpler character than what has been considered thus far. In particular, it is this simpler type of analysis that pertains to the study of soil water phenomena as it is undertaken conventionally in soil physics. Ordinarily, both the equilibrium behavior and the transport of water in soil are regarded mathematically as problems in field theory. Thus most of the soil water properties of interest are intensive and are associated with points in space within the volume occupied by a soil sample. These points in the soil are centroids of volume elements that contain many solid grains and fluid-containing interstices, as explained in Section 1.1. If the soil sample is relatively small, its intensive properties may not vary from point to point and, therefore, it is classified as a homogeneous system. This perspective can be taken, however, only if the detailed chemical composition of the solid phases in a soil sample is ignored and if the fluid species in the sample are partitioned only grossly into an aqueous component and a dry air component. The aqueous component then is referred to as soil water and the dry air component is referred to as soil air, with no further consideration being given to the chemical nature of the two. A soil sample, then, is regarded as a thermodynamic system comprising just three components: soil solids, soil water, and soil air. Since these three components arc assumed not to react chemically and not to segregate at any time into macroscopic regions of the soil sample, the soil is a homogeneous mixture (i.c.. a solution). This point or vicw is taken normally in soil physics. IK7

., THE THERMODYNAMIC THEORY OF WATER IN SOIL

188

A soil sample will not be a solution, even under the simplification of its composition just described, if the properties of its components vary spatially on a macroscopic scale. For example, an undisturbed sample of field soil may be a heterogeneous mixture because of spatial variability caused by depositional stratification in its solid component. In this case, a smaller unit than the whole soil sample may still be a solution if the properties of the subunit are sufficiently uniform. This same conclusion applies to the stratification in the soil water and soil air components that may be induced by the gravitational field of the earth. The smaller, homogeneous subunits of the soil sample defined under this condition are called gravitational phases. They will be discussed in Section 7.6. The thermodynamic variables of state for a soil solid-water-air solution are listed in Table 7.1. These variables are just special cases of those discussed in Section 1.3. Following the suggestion in that section, one can define a number of additional intensive variables simply as ratios of the extensive variables in Table 7.1. Among the most important of these are the gravimetric water content: (7.1) the volumetric water content:

(7.2) where o; is the mass density of the water component and Pb is the dry bulk density: Pb

= m.j V

(7.3)

The variable Pb is to be distinguished from the wet bulk density: Pbw = (m,

+

m;

+

(7.4)

ma)/ V

,

TABLE 7.1. Summary of thermodynamic variables for a soil solid-water-air solution (soil-water system) Variable Temperature Entropy Pressure Volume Chemical potential of soil solids Chemical potential of soil water Chemical potential of soil air Mass of soil solids Mass of soil water Mass of soil air

Symbol

Extensive

Intensive

Units

X

J.L.

X

K J K- ' N m- z ml J kg-I

J.Lw

X

J kg"

J.L.

X

J kg:" kg kg kg

T S P V

X X X

m, mw

X X

m.

X

,

THE THERMODYNAMIC THEORY OF WATER IN SOIL

189

Each of these secondary intensive variables is a point function of position in a sample of soil conceptualized as a solid-water-air solution. If anyone of the variables has different values at different points in the soil sample, the soil is not a solution but is, instead, a heterogeneous mixture comprising subunits that themselves are solutions exhibiting uniform values of the variables. The principles underlying the construction of thermodynamic potentials that were discussed in Section 1.4 may be applied without modification to a soil solid-water-air solution. For example, if the independent variables chosen to describe the system are T, P, m.; m.; and m as the natural thermodynamic potential is the Gibbs energy, given in differential form in Eq. 1.9. In the present case:

dG = -SdT

+

VdP

+ u.dm, + J.lwdmw + u.dm,

(7.5)

Equation 7.5 describes thermodynamic processes in a soil contacting thermal and volume reservoirs. None of the three components is permitted to enter or leave the soil. Since soil samples often are permitted to be open to the atmosphere during experiments on soil water behavior, a useful alternative to G is the Groenevelt-Parlange potential, G*( T, P, m; m.; J.l.}. The differential form of this potential is:I

dG* = -SdT

+

VdP

+ J.lsdm s + J.lwdmw

- m.du;

(7.6)

According to the discussion in Section 1.4, G* describes a soil in which T, P, m.; m.; and J.la are the controlled variables. The soil is thus imagined to be in contact with thermal, volume, and air reservoirs that control the values of T, P, and J.la' The air reservoir, of course, may be simply the atmosphere. In practice, there may be some difficulty in preparing a soil system whose independent variables include m; and J.la' since the thermodynamic wall surrounding the system would have to be permeable to air and impermeable to water at the same time. Some means to prevent the evaporation of water from the soil, or any other net transfer of soil water across the wall, that did not inhibit the passage of dry air would be required. This practical difficulty, on the other hand, does not arise in applications of the thermodynamic potential <1>( T, P, m.; /-Lw, J.la), whose differential form is:

The potential may be employed to describe thermodynamic processes in soil systems that contact thermal, volume, water, and air reservoirs. An experimental arrangement of this set of reservoirs might be simply a sample of soil in a dessicator containing a saturated, aqueous salt solution, with the dessicator placed in a constant-temperature bath. The atmosphere in the dessicator is both the air and the water reservoir because it contains water vapor at a fixed relative humidity and, therefore, fixed chemical potential (see the discussion following Eq. 2.34), which has been established by the salt solution. Another possible experimental arrangement would be a soil in contact through a porous plate with a reservoir containing liquid water under a chosen pressure. If the pressure applied to the water in the reservoir were less than thaI of (he atmosphere. the

190

THE THERMODYNAMIC THEORY OF WATER IN SOIL

chemical potential of the water would be less than its value in the Standard State and the soil would remain unsaturated, with its water component at the chemical potential so selected. Regardless of which thermodynamic potential is employed to describe the phenomena of interest in a soil solid-water-air solution, the Gibbs-Duhem relation acts as an additional equation of constraint among the intensive variables T, P, f.L" f.Lw, and f.La. According to Eq. 1.24, this equation of constraint is:

Because a soil solid-water-air solution is a single, three-component phase in which no chemical reactions take place, it has five variable intensive properties. Equation 7.8 makes evident the fact that only four of these properties can be varied independently, in accordance with the Gibbs phase rule.

7.2. THERMODYNAMIC STABILITY CONDITIONS

A necessary and sufficient condition for a soil solid-water-air solution, henceforth to be called a soil-water system, to be in a state of equilibrium is that the internal energy of the system be a minimum for a given maximum value of the total entropy (see Section 2.1). This condition may be expressed mathematically in the two equations:' dU

=

0

(7.9)

Equations 7.9 state that first-order, infinitesimal variations in U(S, V, m; m.; mw ) must vanish and that second-order infinitesimal variations must be positive for an equilibrium state. The vanishing of dU was shown in Section 2.1 to be necessary in order that the temperature, pressure, and chemical potentials in a system be uniform and equal, respectively, to the values imposed by controlling thermal, volume, and matter reservoirs. Thus dU = 0 is the criterion for homogeneity in an equilibrium state. The second condition in Eq. 7.9, on the other hand, requires that the next higher-order change in U never decrease the value of U. This condition insures that a state produced by such a change will never exhibit a lower value of U than does the equilibrium state. Thus ~U > 0 is the criterion for stability in an equilibrium state. The implications of the stability criterion for soil-water systems may be investigated once an expression for ~U is available. In order to simplify the mathematical analysis, the extensive character of U as epitomized in the equation (see Eq. 1.17) (7.10) will be employed. where A > 0 is a scale factor of arbitrary magnitude. The

THE THERMODYNAMIC THEORY OF WATER IN SOIL

191

special case of Eq. 7.10 where A = 1I m, leads to the definition

u(s, v, 8., 8) -

U(Slm" Vim" 1, maim" mwlm,) = Olms)U(S, V, m.; m a, m w)

(7.11 )

where s = Slm" v - Vim" and 8a = mal m; The new independent variables can in turn be manipulated more easily with the help of the definitions (7.12)

In terms of Eqs. 7.11 and 7.12, the stability criterion is:' 1

d'u =

4

"2 L i=l

4

L

uijdx;dx j > 0

(7.13 )

j= I

where (i,j

=

1,2,3,4)

(7.14a)

The inequality in Eq. 7.13 is required as long as the trivial condition dX1 = dX2 = dX3 = dX4 0 does not obtain. The coefficients uij in Eq. 7.13 may be arranged to form a four-by-four square matrix. The leading principal minor determinants of this matrix are:

=

D2

-

I

~~: ~~: I U l2

Un

U 22

U 23

U 32

U 33

U 42

U 43

(7.15a)

(7.15b)

These determinants figure importantly in a theorem of matrix algebra that, when applied to Eq. 7.13, states that d 2u > 0 if and only if D, > 0 (i = 1, 2, 3, 4).4 The thermodynamic meaning of this theorem may be seen as follows. According to the techniques described in Section 1.5 that pertained to the Maxwell relations, Eq. 7.14a can be rewritten in the form (i,j = 1,2,3,4)

(7.14b)

where Pi = aulax; is an intensive variable that is conjugate to the variable Xi according to Eq. 1.3. In particular, PI = aulax! = T

P3 = aulax 3 =

Ila

P2 = aulax 2 = -P P4 = aulax4 = Ilw

(7.16)

Each of these intensive variables represents an equation of state, as explained in Section lA, and each has the differential form (i= 1,2,3,4)

(7.IH)

THE THERMODYNAMIC THEORY OF WATER IN SOIL

192

which involves the coefficients u.; Consider now the special case dP = dP 2 = dP 3 0 in respect to Eq. 7.18. The set of simultaneous linear equations for the dx, then becomes: j

=

u11dx + u21dx, + U31dx1 + u4,dx, + j

u 12dx2 + u22dx 2 + u32dx 2 + U42dx2 +

UJ3 dX3 + U23dx3 + U33dx3 + U43dx3 +

U'4dx4 = U24dx4 = U34dx4 = U44dx4 =

0 0 0 dP 4

The solution of these equations for dX4 may be found with the help of Cramer's Rule:

dX4 =

D D4

- 3 dP4

(7.19a)

Since Eq. 7.19a is the same as the differential expression

dX4 = (ax 4/ ap4) PI.P2. P,dP4

(7.19b)

it follows that (7.20) where the inequality comes from the condition on the D, that guarantees that Eq. 7.13 will hold. Equation 7.20 is one of the four criteria for stability in a soil-I water system. The corresponding result for any x j can be derived by choosing a value for i. then writing out Eq. 7.18 for the case dP, = ... = dPj _ , = 0, dXj + 1 = ... = dX4 = 0, and solving them for dx, with the help of Cramer's Rule. The generalized form of Eq. 7.20 then is found to be (i=I,2,3,4)

(7.21)

Equation 7.21 is a general, thermodynamic criterion for stability in a soil-water system. The case i = 1 in Eq. 7.21 produces the inequality

(as/anv.o•..:» 0

(7.22a)

according to Eqs. 7.12 and 7.16. Because c, = T(as/anv.o•.o

(7.23a)

is the heat capacity at constant volume per unit mass of solid component, the stability criterion in Eq. 7.22a is the same as requiring the heat capacity of the system to be positive. Thus the addition of heat to a stable soil-water system will always increase its temperature. The case i = 2 in Eq. 7.21 produces the condition

-(eJv/eJPh.o.,o> 0

(7.22b)

I

THE THERMODYNAMIC THEORY OF WATER IN SOIL

193

according to Eqs. 7.12 and 7.16. Since KT

=

-~ ( : ; )

Teo

(7.23b)

• a.

is the isothermal compressibility of the system, the stability criterion in Eq. 7.22b is the same as requiring the isothermal compressibility to be positive. Thus the application of pressure to a stable soil-water system will always decrease its volume. The cases i = 3 and i = 4 in Eq. 7.21 produce the stability criteria

(aOa/allah,P.8> 0

(ao/aIlw) T.P.~a >

0

(7.22c) (7.22d)

These inequalities require any increase in the chemical potential of a fluid component of a stable soil-water system to be accompanied by an increase in the mass (per unit mass of solid) of that component. Conversely, an increase in the mass of a fluid component must produce an increase in the chemical potential of the component. 5 Equation 7.22d may be of particular value in the examination of moisture characteristic data (Section 7.5) for self-consistency and in the development of model expressions that relate Ilw to O.

7.3. THE CHEMICAL POTENTIAL OF SOIL WATER

The chemical potential of soil water, Il"n represents one of the equations of state that correspond to the thermodynamic potential chosen to describe a closed soilwater system (see Section 1.4). Therefore the variables on which Ilw depends will themselves be dependent on the choice of thermodynamic potential. If the controlled variables for a soil-water system are T, P, m.; m.; and m., then G in Eq. 7.5 is the appropriate thermodynamic potential and the equation of state for Ilw is (see Eq. 1.1 0): (7.24a) If the controlled variables are T, P, m.; m.; and Ila instead, G* in Eq. 7.6 is the thermodynamic potential and

(7.24b) In each of Eq. 7.24, a different functional dependence of Ilw is specified, but the quantity on the left side is the same, regardless. In the soil physics literature, this quantity, when referenced against the value of Il~, is called the water potential and is given the symbol tfw' Thus tfw = Ilw - Il~. The water potential, according to Table 7.1, is expressed in units of joules per kilogram. Quite often equivalent units are employed in place of the Sf units. For example, the quotient,

THE THERMODYNAMIC THEORY OF WATER IN SOIL

194

Ilw/ M w, where M; is the mole weight of water, is called the "molar water potential" (joules per mole); the product P~llw, where P~ is the mass density of bulk liquid water, is called the "soil water pressure" (J m -3 = N m -2), and the quotient Ilw/ g, where g is the gravitational acceleration, is called the "soil water head" (J sec' kg::' m:' = m). These equivalent units, of course, make no change in the physical interpretation of Ilw, which follows from the discussion in Section 1.3. The chemical potential of soil water is no more a pressure physically after multiplication by the mass density of pure liquid water than it was before. Likewise, Ilw/ g is no more a length physically than is Ilw by itself. According to Eq. 7.24b, the water potential may be written in the differential form: du.;

= (allw/aT)P,m.,mw.#.dT + (allw/aph,m.,mw.#.dP

+ (allw/amS)T.P.mw.#.dm + (allw/amw)T,P,m,,#.dmw + (allw/allah,P.m.,mwdlla s

(7.25)

The coefficients before the differentials on the right side of Eq. 7.25 can be related directly to experimentally determined quantities with the help of the Maxwell relations that are associated with the Groenevelt-Parlange potential. The method of calculation of these relations is the same as that illustrated for the Gibbs potential in Eqs. 1.12 and 1.13. The coefficient of dT in Eq. 7.25 thus can be expressed (7.26a) upon taking note of Eq. 7.6. The partial derivative, Sw, is called the partial specific entropy of soil water. It is equal to the slope of a graph of the total entropy versus the mass of soil water, with the variables T, P, m; and u, held constant. The total entropy may be determined for chosen values of T, P, ni; m.; and III with the help of data on the heat capacity at constant pressure for a soil-water system. Alternatively, Sw may be determined using data on the heat of immersion of soil in water or, of course, through measurements of the temperature dependence of Ilw at constant P, m.; m.; and Ila' The coefficient of dP in Eq. 7.25 is

allw a2 G* a2 G* ap = apam w = amwap

=

av am w

_

= Vw

(7.26b)

The partial derivative Vw is called the partial specific volume of soil water. It is equal to the slope of a graph of the total volume versus the mass of soil water, with the variables T, P, m.; and Il a held constant. If a soil-water system does not shrink or swell as its water content changes, Vw = 0 by definition, For a soil that does undergo volume changes, Vw usually approaches zero as the value of m; goes to zero, When a soil is completely saturated with water, Vw = 1/ Pw'

THE THERMODYNAMIC THEORY OF WATER IN SOIL

195

The third and fourth terms on the right side of Eq. 7.25 are treated most conveniently by combining them into a single term that incorporates Eq. 7.1:

dm ( allw ) a m; T.P.". w

+ (a llw)

llw) dm = (a dO am, T.P.".' ao T.P.".

(7.26c)

Equation 7.26c can be justified as follows. Let Aw(m w, m.) and A,(m w, m.) be the coefficients in the exact differential (T, P, Ila constant)

These coefficients are subject to the condition (T, P, Ila constant)

(7.27)

=

If Eq. 7.26c is valid, the quantity A (allw/aOh,p.". must be a function of 0 only, and this property must follow directly from Eq. 7.27. Now

aA Bm;

dA ao dO Bm;

1 dA m, dO

--=---=---

aA am,

dA ao dO am,

-0 dA m, dO

-=--=--

(7.28)

according to the chain rule for derivatives. The elimination of d A] dO from Eq. 7.28 produces the expression 1 aA ---

(7.29)

e am,

Equation 7.29 is the requirement that must be met if A - A(O). When dm, = 0, Eq. 7.26c can be written in the form

ao = A Bm; w

(7.30a)

A -

and, when dm; = 0, it can be written in the form

ao A-=A am, s

(7.30b)

The partial derivative of Eq. 7.3Da with respect to m, is

a~, (~,) ~, (:~,) ~; :~: =

-

=

and the partial derivative of Eq. 7.30b with respect to m; is

a (-AO) aA 0 A aA, amw ----;;;: = - amw m, - m; = amw Since the right sides of thcsc equations arc equal, by Eq, 7.27, it follows that Eq. 7.29 is true also. Therefore A = A(f1) and Eq. 7.26c is valid.

THE THERMODYNAMIC THEORY OF WATER IN SOIL

196

The coefficient of dv; in Eq. 7.25 may be expressed af.Lw

a2 G*

-

af.La

a2 G*

-

af.Laamw

Bm,

=---

amwaf.La

(7.26d)

Bm;

according to Eq. 7.6. The partial derivative of m, with respect to m; is calculated while T, P, m.; and f.La are held fixed. This partial derivative is closely related to Vw , as can be seen in the following sequence of equations:

v=

Vw + Vs

v = (av

w

+

Va

(av +

a

)

amw T,P.m•./la

w

)

amw T.P.m•.

~.

_~+~(ama) Pw

Pa

(7.31)

Bm; T.P,M',/la

where Vw = mwlPw is the volume of water in the system, Vs is the volume of solids, Va = mal Pa is the volume of air, and Pa is the mass density of air. Equation 7.31 can be rearranged to give the expression ama ) ( -am;

V - -1 )

=p a

T,P,ms,J.La

(7.32)

(-

Pw

w

If the soil air component is assumed to exhibit ideal gas behavior (a good approximation), Eq. 2.33 and the discussion of gases in Section 2.2 lead to the equation

du,

=

RT

M din P, a

=

RT

M P di', a

1

=-

a

Pa

dr,

(7.33)

where M, is the average mole weight of dry air and P, is the air pressure. The combination of Eqs. 7.26d, 7.32, and 7.33 yields the differential term af.Lw) ( 1 -) Pw - Vw dP a ( af.La dii; =

(7.26e)

which may be substituted into Eq. 7.25. This term vanishes for any water-saturated soil and reduces to (1 I Pw) dPa for any unsaturated soil that does not shrink or swell. In general, Vw -< p;;!, so the coefficient of di', in Eq. 7.26e is nonnegative. With the help of Eq. 7.26, Eq. 7.25 now may be written in the form

du; = -Sw d T

+

VwdP

+ (af.LwlaO)T.P.p.dO +

(:w -

Vw)dPa

(7.34)

Equation 7.34 may be employed to describe the effects of changes in temperature, applied pressure, gravimetric water content, and soil air pressure on the water potential. The second term on the right side of Eq. 7.34 is expressed (7.35)

THE THERMODYNAMIC THEORY OF WATER IN SOIL

197

where 1/;p is called the envelope-pressure potential. This component potential represents the effect on the water potential of external pressure applied to a soil. The applied pressure may be created by a load on the soil surface or by the overburden of soil above a point in the soil profile where 1/;p is to be evaluated. When Eq. 7.35 is integrated, 1/;p must be referenced to a zero value at P = 1 atm. It may be noted in passing that d1/;p vanishes identically for a nondeformable, unsaturated soil and that #p = (1/ Pw) dP for any saturated soil. 6 The third term on the right side of Eq. 7.34 usually is expressed (7.36) where 1/;m is called the matric potential. This component potential represents the effect of changes in the water content, an effect usually attributed to the presence of the solid portion of a soil-water system. Equation 7.36 includes the effect of dissolved solid components of the soil-water system on Jl.w' As indicated in Section 7.1, because the soil-water system is a single phase comprising nonreacting components, there can be no separate term in Eq. 7.25 that represents the contribution of dissolved solutes. In order to have such a term, the thermodynamic soil system would have to be treated as a three-phase, heterogeneous system wherein each of the phases (solid, aqueous, and gaseous) is a solution containing chemically differentiated species. Some of these species would be able to undergo dissolution-precipitation reactions. Thermodynamic potentials then could be written for the aqueous solution phase, and equations of state analogous to Eq. 7.24 could be developed for the chemical potential of water in that phase. The functional dependence of Jl.w on the masses or the mole numbers of dissolved solutes could be specified by differential terms, such as those in dm; and dm, in Eq. 7.25. However, the measurement of these terms representing the dissolved solutes would be difficult because of the need to determine actual solute concentrations in the soil solution. The term in dl', in Eq. 7.34 can be written in the form

#0

=

C:w -

Vw)dPa

(7.37)

where 1/;0 is called the pneumatic potential. This component potential represents the effect of changes in the soil air pressure on Jl.w' It must vanish when P, = 1 atm. Moreover, if air pressure is employed to exert a load on a sample of soil and this is the only cause of the load, then P = P, and Eqs. 7.35 and 7.37 can be combined to produce the term (7.38) In this case, the net contribution of 1/;p and 1/;n to Jl.w is of the same form for all soils, regardless of whether they shrink or swell or whether they are saturated or unsaturated with water. Two important general properties of the terms in Eq. 7.34 should be mentioned. First, each of the cocllicicnts on the right side of the equation is a func-

THE THERMODYNAMIC THEORY OF WATER IN SOIL

198

tion of the same independent variables as the ones on which f.lw depends. This characteristic is a purely mathematical aspect of exact differentials. Since f.lw = f.lw (T, P, 0, f.la)' it follows that Sw, Vw, (af.lwjaO) and, of course, (p;l Vw) depend on T. P, 0, and f.la' If Eq. 7.34 is to be integrated in order to describe a thermodynamic process, it is important to remember that each coefficient of a differential term in the equation depends on more than just the single independent variable in the differential. The second important property of the component potentials in Eq. 7.34 relates to their units. Each potential like Y;p and Y;m has the same units as f.lw, (i.e., joules per kilogram). However, as with f.lw, the component potentials can be multiplied by p~ to make their units pascals (or atmospheres) instead, or they can be divided by g to make their units meters. These manipulations have nothing to do with the physical interpretations of the component potentials. For example, P~Y;m is no more a pressure than was Y;m' Equivalent units exist for practical reasons having no relation to the intrinsic physical meaning of a component of the water potential.

7.4. THERMODYNAMIC THEORY OF THE MEASUREMENT OF THE WATER POTENTIAL

The differential expression for f.lw given in Eq. 7.34 may be applied to describe the measurement of the water potential by means of conventional techniques in soil physics. Three of these techniques will be discussed in this section, both as an illustration of one important use of Eq. 7.34 and as an investigation of the thermodynamic processes involved in the measurement of f.lw' From the point of view of thermodynamics, a tensiometer is a system comprising liquid water in contact with a volume reservoir and in , communication with a soil-water system through a membrane that is permeable ' only to fluid phases. When a tensiometer is placed at some point in a soil profile, equilibrium with respect to the transfer of water (and dissolved solutes) is established across the semipermeable membrane. This equilibrium is expressed by the equation The Tensiometer.

f.lw(soil-water system) = f.lw(tensiometer)

(7.39)

according to the principles discussed in Section 2.1. The water potential in the soil will differ from the chemical potential of water in the Standard State if T 298.15 K, P 1 atm,O +00, and P, 1 atm. It will be assumed that T and P, have their Standard State values. The variables P and 0 will differ from their Standard State values because of the pressure applied by the soil material overlying the point in the profile where the tensiometer has been placed (i.e., the overburden pressure) and because of the presence of the soil solid and soil air components. To account for these differences. a thermodynamic process may be envisioned in which the water content is lowered from I IX) to 0, while

*

*

*

*

THE THERMODYNAMIC THEORY OF WATER IN SOIL

199

T, P, and P, retain their Standard State values, followed by an increase in P to the appropriate value of overburden pressure, while 0, T, and P, remain the same. This process would bring /-Lw from the value /-L~ to the desired value, /-L w (soil-water system). The corresponding integration of Eq. 7.34 is

or /-Lw(soil-water system) = /-L~

+ 1/;p(P, 0) + 1/;m(pO, 0)

(7.40)

where pO = 1 atm and Eqs. 7.35 and 7.36 have been employed. Note that 1/;m in Eq. 7.40 is evaluated under zero load pressure. The water potential in the tensiometer may be described by Eq. 7.8 once du; in that expression is set identically equal to zero. It can be assumed also that du, vanishes (no passage of air through the semipermeable membrane) and that T remains fixed at 298.15 K. The chemical potential of water in the tensiometer then can differ from /-L~ because of a decrease in applied pressure prompted by a movement of water from the tensiometer into the soil. The pressure decrease usually is measured by a vacuum gauge or a manometer attached directly to the tensiometer. The corresponding integration of Eq. 7.25 is:

or .

/-Lw( tensiometer)

=

/-L~

+ 01

Pw

(P, -

pO)

(7.41 )

where P, is the pressure applied to the water in the tensiometer, and it has been noted that Vw = 1/ P~ in this case. The value of P~ is assumed to remain unchanged over the range of pressure between P, and The combination of Eqs. 7.39 to 7.41 produces the equation

r:

(7.42) where T wis the reading on the tensiometer gauge and is called the soil water tension. Equation 7.42 shows that a tensiometer measures the sum of the envelope-pressure and matric potential components of the water potential, with these two components expressed in the equivalent units of pascals or atmospheres (i.e., pressure). If the soil-water system under study does not shrink or swell, Vw o and 1/;p( P, 0) will vanish, regardless or the value or the overburden pressure, P. In that case, the tensiometer would measure only the mat ric potential component or P-w' The presence or dissolved solutes in soil water will not affect I':q. 7.42

200

THE THERMODYNAMIC THEORY OF WATER IN SOIL

because these are assumed to pass freely into the tensiometer. At equilibrium, the contribution of dissolved solutes to each side of Eq. 7.39 will be the same. In terms of thermodynamic concepts, a pressure membrane apparatus consists of two matter reservoirs: an air reservoir and a water reservoir. Both reservoirs contact a sample of soil (the air reservoir on top, the water reservoir below) through the appropriate semipermeable membranes and both are maintained at T = 298.15 K. The pressure in the air reservoir is increased from the value PJ to some value Pm' and equilibrium is established between the soil-water system and the water reservoir by virtue of the transfer of water from the soil to the water reservoir. When the transfer of water has ceased, The Pressure Membrane Apparatus.

JJ.w (soil-water system)

= JJ.R =

JJ.~

(7.43)

since the water in the reservoir is in the Standard State. The initial value of JJ.w in the soil-water system differs from JJ.Z because the initial values of P and P, are not equal to Pm and because the initial value of 0 is greater than its final value. Since dP = dl', in this case, Eq. 7.38 may be applied to Eq. 7.34 to yield (at constant 1) (7.44) The integration of Eq. 7.44 proceeds in two steps. First, the air pressure is brought from PJ to Pm while 0 remains fixed, then 0 decreases from OJ to Of while p. = Thus

r:

#w(OjJ

J

#w(O,)

dJJ.w =

J

Of

(aJJ.w/ ao) T.P, dO

0,

= Y;m(P m, Of) = JJ.w(Of)

1

J r;

Pw

P'

+-

- Y;m(Pm, OJ)

+ -1

Pw

d I', (Pm - PJ)

- JJ.w(O;)

or

The combination of Eqs. 7.43 and 7.45 produces the expression t:..Pm = P; - PJ = Pw[Y;m(Pm, OJ) - Y;m(PJ, 0;) - Y;m(Pm, Of)]

(7.46)

where t:..Pm is the reading on the pressure membrane gauge and Y;m(PJ, 0,) = JJ.w( T, PJ, 0;) - JJ.Z. Equation 7.46 shows that a pressure membrane apparatus measures differences in the mat ric potential. If the soil initially is saturated with 0,) - 0; then Eq. 7.46 reduces water, it may be true that Y;m(P m, 0;) = to the equation

«sr.

(7.47)

THE THERMODYNAMIC THEORY OF WATER IN SOIL

201

In this case, the apparatus measures a matric potential directly. According to Eqs. 7.46 and 7.47, the pressure membrane apparatus always measures Y;m at the pressure Pm. If the application of pressure alters Y;m [i.e., if (alJ-w/aO)T.P. depends significantly on Pal, the apparatus can give no information about the p = po state of the soil water. Since Pm » P' in almost all cases, this fact may limit seriously the interpretation of data obtained with the help of Eq. 7.47. A comparison of Eq. 7.42 with Eq. 7.47 indicates that, if a soil water system is deformable, T w '1= /:,.P m- The tensiometer and the pressure membrane apparatus do not measure the same components of IJ-w' except for perfectly rigid soils. The vapor pressure of soil water can be measured precisely with thermocouple devices that have been calibrated against electrolyte solutions of known water activity. Thus a thermocouple reading corresponds to a certain value of IJ-w in a reference electrolyte solution (usually KCl), and this value of IJ-w must also be that of the soil water in the sample. In other words, at equilibrium, Soil Water Vapor Pressure.

IJ-w (soil-water system) = IJ-w (KCI solution)

(7.48)

where the right side of the equation could be read from a thermocouple calibration curve. According to Eq. 2.33, IJ-w(soil-water system)

= IJ-~ +

RT In ~ M; J »

(7.49)

where M; is the mole weight of water and fw is the fugacity of soil water. As pointed out in the discussion in Section 2.3 that followed Eq. 2.34, the fugacity of soil water is equal, at equilibrium, to the fugacity of soil water vapor. Moreover, since soil water vapor is a low-pressure gas, its fugacity is very nearly equal to its partial pressure, p.; Therefore Eq. 7.49 can be reduced to the expression IJ-w(soil-water system) =

IJ-~ +

RT In (Pw/ p~)

Mw

(7.50)

where p~ is the vapor pressure over pure water at T = 298.15 K and P = I atm. If a thermocouple device is calibrated to read Pw/p~ directly, the water potential can be calculated immediately with the help of Eq. 7.50. Thus a measurement of soil water vapor pressure is equivalent to a measurement of all of the components of the water potential. Data obtained from a vapor pressure measurement will not necessarily be equivalent to the data obtained using a tensiometer or a pressure membrane apparatus.

7.5. THE MOISTURE CHARACTERISTIC: HYSTERESIS !\ graph of the matrie potential. Y;II\' against either the gravimetric or the volu-

metric water content is called a

11/0;,I'/U'I'

characteristic. The moisture charac-

THE THERMODYNAMIC THEORY OF WATER IN SOIL

202

teristic of a soil-water system provides information about one of the most important components of the water potential in Eq. 7.34. Since (aJ,Lw/ aO) T.P.P, in that equation is, in principle, a function of T, P, and Po, as well as of 0, it is important that moisture characteristics be determined under isothermal, isobaric conditions. A representative moisture characteristic for a clayey soil is shown in Fig. 7.1. The most remarkable feature of the graph is that 1/;m appears to be a multivalued function of O. The curve labeled "primary drying curve" is the set of (1/;m, 0) pairs observed when the soil was taken from a condition of water saturation at OSO! = 0.35 kg water/kg soil to a condition of dryness at 0 < 0.15 kg water/kg soil. The curve labeled "primary wetting curve" is the set of (1/;m, 0) pairs corresponding to the return from the dry state to that of saturation. The data points that begin at the positions labeled A, B, and C on the primary wetting curve show the (1/;m, 0) sets found when drying of the soil commenced at

FIGURE 7.1. The moisture characteristic for a clay loam soil. Scanning curves leave the primary wetting curve at the points, A, B, and C.

i:-

-1.00

~

Primary drying curve

-0.50 Primary

....

O'------~-----.L..---------'------'----"""":::O

0.15

0.20

0.25

e(kg water/kg soil)

Q30

0.35

THE THERMODYNAMIC THEORY OF WATER IN SOIL

203

those three positions. These sets of data always remain within the region of the 1/;m - (J plane defined by the primary curves and are known as scanning curves. The multivalued character of 1/;m shown in Fig. 7.1 is referred to as hysteresis. The existence of hysteresis in the moisture characteristic is in violation of the requirement, stated in Section 1.1, that the values taken by thermodynamic properties not depend in any way on the previous history of the system under study. If the soil-water system represented in Fig. 7.1 is brought (at fixed T, P, and Pa) from saturation to a water content of 0.30 kg water/kg soil, the value of 1/;m will be about -0.5 m. On the other hand, if the system is brought from a condition of saturation to that of dryness at (J < 0.15 kg water/kg soil and then is brought up to the same final water content of 0.30 kg water/kg soil, the value of 1/;m will be about - 0.15 m. Clearly, the value of the matric potential depends on the previous wetting-and-drying history of the soil-water system. This behavior certainly makes the application of ordinary thermodynamics questionable. In particular, since what are presumably infinitesimal changes in the water content are not reversible changes, the states of the soil-water system represented along the moisture characteristics in Fig. 7.1 cannot be equilibrium states (see Section 1.2), even though they seem to correspond to a continuum of (J values. In mathematical terms, this fact means that du; in Eq. 7.34 and, therefore, d1/;m in Eq. 7.36 cannot be exact differentials. If the right side of Eq. 7.36 is integrated between some (Jj and a water content of interest, (J, the result will depend on whether (a/-Lw/ a(J) T,P,P. is computed entirely along either a wetting curve or a drying curve, or along both a wetting curve and a drying curve, between the two water contents. Therefore the result depends on the path of integration in the space of the variables T, P, PO' and (J, and it follows that d1/;m must be an inexact differential. The difficulties caused by hysteresis do not mean that thermodynamics cannot be applied to study the moisture characteristic. In particular, the thermodynamic potentials may be employed to some extent to describe the curves in Fig. 7.1. For example, even though the states of the soil-water system along a wetting or a drying curve are not equilibrium states, they are still stable states because they are observed always to meet the stability criteria summarized in Eq. 7.22, especially the criterion involving (a/-Lw/a(Jh,P... that appears in Eq. 7.22d. This fact is consistent with the well-known persistence of any state of a soil-water system for long periods of time. Another useful thermodynamic result can be derived by studying the variation of the Groenevelt-Parlange potential along the primary wetting and drying curves. With T, P, and P, fixed, any decrease in G* is equal to the work delivered by the soil-water system in the process under consideration (see Section 1.4). If the variable m; also is constant, then, according to Eq. 7.6, dG* = /-Lwdmw, which is the infinitesimal transfer of chemical energy caused by a change in the mass of water in the system. Consider now the two points in Fig. 7.1 where the primary wetting curve joins the primary drying curve, One of these points is at saturation. (J"'l = (1..15 kg water/kg soil. and the other is at a water content

204

THE THERMODYNAMIC THEORY OF WATER IN SOIL

somewhat less than 0.15 kg waterjkg soil; call it 0min' The chemical energy delivered by the system as it is brought from 0min to O,at along the primary wetting curve is given by the expression (7.51a) where J.Lw(p.w.c.) is the chemical potential of soil water evaluated along the primary wetting curve. The right side of Eq. 7.51a is proportional to the area in Fig. 7.1 that lies between the O-axis and the segment of the primary wetting curve that starts at 0mi. and ends at O,at. The constant of proportionality is equal to minus the weight of soil solid component, m,g. Therefore ~G:

= -

m,g X area below p.w.c.

(7.51b)

The negative sign of ~G: refers to the loss of chemical energy from the soilwater system during the wetting process. On the other hand, the chemical energy delivered to the system as it is brought along the primary drying curve is given by the equation (7.51c) where J.Lw(p.d.c.) is the chemical potential of soil water evaluated along the primary drying curve. The right side of Eq. 7.51c is proportional to the area in Fig. 7.1 that lies between the O-axis and the segment of the primary drying curve that starts at O,at and ends at Omin: ~G~

= m,g X area below p.d.c.

(7.51d)

If the wetting-drying process were reversible, the sum of ~G: and ~G~ would equal zero and there would be no net chemical energy delivered to the system in going from Omin to O,at and back to 0min' However, because of irreversible processes that occur during the wetting-drying cycle, the sum of ~G: and ~G! is a positive quantity given by the expression (7.52) which follows from Eqs. 7.51a and 7.51c. The integral on the right side of Eq. 7.52 is proportional to the area enclosed by the hysteresis loop between 0min and O,ato according to Eqs. 7.51b and 7.5Id. This area multiplied by m,g is equal to the extra chemical energy that must be added to the soil-water system in order to bring it around the hysteresis loop once, starting from 0mi.' Alternatively, the right side of Eq. 7.52 also is equal to the absolute value of the irreversible loss of chemical energy from the system, as heat, when it is brought around the hysteresis loop once, starting from (J.n"

THE THERMODYNAMIC THEORY OF WATER IN SOIL

205

The deficit in chemical energy that is expressed in Eq. 7.52 may be interpreted to mean that a soil-water system does not really undergo infinitesimal changes in state as its water content is varied. If, instead, on some small but still macroscopic scale, the system undergoes natural processes that cannot be made to approach reversible processes in some limiting sense (see Section 1.2), changes in water content will be irreversible processes. In this case, a change in water content that seems to be very small actually will induce, in tiny subunits of the system, finite transitions that will not be detectable except for the hysteretic behavior they produce in the system as a whole. These small subunits of the system are called domains. The domain concept of hysteresis has proven to be capable of modeling the moisture characteristics of soil-water systems. In its most recent versions, the domain theory can be used to calculate scanning curves, given only the primary wetting and drying curves as input information. As with any thermodynamic theory, however, it must be remembered that the domain model is a conceptual structure, not an explanation of the microscopic physical phenomena underlying hysteresis. 7

7.6. SOIL WATER IN A GRAVITATIONAL FIELD

The presence of a uniform gravitational field such as that produced near the surface of the earth will require a modification in the equation for the total differential of the internal energy of a soil-water system. In the absence of gravitational effects,

dU

=

TdS - PdV

+ J.L,dm, + J.Lwdmw + J.Ladma

(7.53)

according to Eq. 1.2 as applied to a soil water system. Because the gravitational potential, ¢ = gz (where z is the vertical coordinate, measured positive outward from the surface of the earth), is much larger than the gravitational potential created by any sample of soil and is much smaller than the energy per unit mass associated with the thermal radiation field in a soil, the modification to be made in Eq. 7.53 is quite simple. The additional energy required to change the mass of a component by dm while that component is influenced by the gravitational potential ¢ is just ¢dm. Therefore Eq. 7.53 can be altered to

dU = TdS - PdV + (J.L, + ¢)dm, + (J.Lw + ¢)dmw + (J.La + ¢)dma

(7.54)

The coefficient of dm; in Eq. 7.54, (7.55) is known as the gravichemical potential of soil water. I n the literature of soil physics, J.LKw, referenced to J.l~, is called the total potential and is denoted 1/;,. Thus 1/;, = J.lKw - J.l~. The gravichcmical potential, unlike the electrochemical potential discussed in Section 4.5, always can be scparntcd into a "purely chcnucal"

206

THE THERMODYNAMIC THEORY OF WATER IN SOIL

part and a "purely gravitational" part because the field described by ¢ is so much stronger than that created by any sample of soil and, therefore, is completely unaffected by the presence of chemical species. Moreover, since ¢ is not large enough to alter the thermal radiation field in a soil, it appears only in the coefficients of dm in Eq. 7.54 and not in the coefficient of dS. 8 The introduction of the gravichemical potential into Eq. 7.54 has two important general consequences. First, since ¢ depends on the vertical coordinate, z, it may be expected that, at equilibrium, some of the intensive properties of a soil-water system also will depend on z. Therefore a soil-water system in a gravitational field will become a heterogeneous system whose properties are nonuniform along the vertical direction. The system as a whole then may be regarded as a sequence of macroscopically thin, horizontal slices called gravitational phases. A gravitational phase will exhibit equilibrium properties that depend on z, which is also the position of the centroid of the phase relative to the surface of the earth. If matter is transferred vertically from one place to another, it undergoes a gravitational phase transition and its properties will change by amounts determined solely in terms of the shift in the value of z. Equations 7.54 and 7.55, then, refer to the internal energy change and to the gravichemical potential within a gravitational phase. The second important consequence of introducing J.lgw is that this quantity now becomes the criterion for equilibrium with respect to the transfer of soil water. This fact can be verified directly by employing Eq. 7.54 instead of Eq. 7.53 in the derivations that led to Eqs. 2.12 and 2.15. Thus, for example, Eq. 2.15 would be modified to the form (7.56) where Aand a denote two different phases in which soil water can exist. The two phases can be solid and liquid, liquid and gas, or two gravitational phases. The analog of Eq. 7.34 for soil water in a gravitational field is

dJ.lgw = - Sw d T

+

+

C:w -

VwdP

+ (aJ.lw/ aO) T.p,p.,zdO

Vw)dPa

+

(7.57)

gdz

Equation 7.57 may be applied either to processes within a gravitational phase or to a gravitational phase transition. Consider now a soil profile in which there are no gradients in the temperature or in the pressure of soil air. Then Eq. 7.57 reduces to the expression (7.58) If there is equilibrium with respect to the transfer of soil water within the profile, dJ.lgw = 0 according to Eq. 7.56. This condition may be applied to Eq. 7.58 to produce an expression for the equilibrium moisture gradient, dO/ dz:

dO - Vw(dP/dz) ~ K dz =--(;j~-:Fiii)/~,~,:,:-

(dJ.l~w

=

0)

(7.59)

THE THERMODYNAMIC THEORY OF WATER IN SOIL

207

The derivative dl'[dz may be evaluated readily if it is assumed that hydrostatic equilibrium exists in the soil profile: dP -dz

(7.60)

where Pbw is the wet bulk density (Eq. 7.4). With the help of Eq. 7.60, Eq. 7.59 becomes dO dz

-=

(VwPbW -

l)g

(aJ-lwlaO) T,P,P"z

(dJ-lgw = 0)

(7.61 )

Equation 7.61 describes the equilibrium moisture profile in a soil. If a soil is rigid, Vw - 0 and Eq. 7.61 reduces to the Buckingham equation:

-g dO - -------'''--dz (aJ-lwl aO) tr.r,»

(7.62)

This equation states that the moisture gradient at equilibrium in a soil is determined solely by the slope of the moisture characteristic. Moreover, dO I dz is predicted always to be negative because of Eq. 7.22d; that is O(z) should decrease with an increase in the vertical coordinate, z. This behavoir is expected in any rigid soil. In general, there are three different possibilities for the sign of dO I dz in Eq. 7.61, depending on the relative magnitude of the product, VwPbw' These possibilities may be listed as follows:" VwPbw> 1 VwPbw = I VwPbw

<

1

de] dz > 0 dOldz = 0 dOldz < 0

hydric profile pycnotatic profile xeric profile

(7.63a) (7.63b) (7.63c)

The hydric profile occurs usually under wet conditions, when both V w and Pbw tend to be large.'? Indeed, if a deformable soil is saturated with water, Vw = P:;! and VwPbw = Pbwl Pw > 1 according to Eq. 7.4. The pycnotatic profile occurs when VwPbw just happens to equal 1.0. In that case, O(z) will not change with depth in the soil profile. The xeric profile is expected under dry conditions, since Vw normally tends to zero as 0 decreases. Thus relatively dry soils should follow the Buckingham equation approximately, regardless of their ability to shrink or swell. In these soils O( z) will increase with depth in the profile. However, when a swelling soil is wet enough (e.g., close to water saturation), O(z) will decrease with depth, according to Eq. 7.63a. It is important to remember that (aJ-lwl aO) T,P,P, z in Eq. 7.61 is the slope of the moisture characteristic evaluated at a point z in the soil profile. The value of P, the overburden pressure, at that point will be different from P' and, therefore, it may be expected that >./;",( P, 0) will be different from >./;",( PO, 0). If the partial derivatives of these two mutric potentials with respect to 0 also arc different, errors will result in an attempt to predict dOld=: using measurements of the moisture characteristic in the absence or an applied load,

208

THE THERMODYNAMIC THEORY OF WATER IN SOIL

NOTES I. G* was introduced by P. H. Groenevelt and J.-Y. Parlange, Thermodynamic stability of swelling soils, Soil Sci. 118:1-5 (1974). 2. The development of Eq. 7.9 is discussed carefully in Chapters II and VI in A. MUnster, Classical Thermodynamics. John Wiley, London, 1970. 3. See, for example, Chapter 8 in H. B. Callen, Thermodynamics. John Wiley, New York, 1960. The method of analysis presented here is suggested in Appendix G of this book. 4. For a proof of this theorem, see page 260 in F. E. Hohn, Elementary Matrix algebra. MacMillan, New York, 1958. For a good discussion of the properties of determinants, see Chapter 10 in H. Margenau and G. M. Murphy, The Mathematics of Physics and Chemistry. D. Van Nostrand, Princeton, N.J., 1956. 5. Equations 7.22 were introduced by P. H. Groenevelt and J.-Y. Parlange in the article cited in note 1. 6. The measurement of if;p is discussed in T. Talsma, Measurement of the overburden component of total potential in swelling field soils, Aust. J. Soil Res. 15:95-102 (1977). 7. For details of the domain model as applied to soil-water systems, see (in the order listed) A. Poulovassilis, Hysteresis of pore water, an application of the concept of independent domains, Soil Sci. 93:405-412 (1962); J. R. Philip, Similarity hypothesis for capillary hysteresis in porous materials, J. Geophys. Res. 69: 1553-1562 (1964); Y. Mualem, Modified approach to capillary hysteresis based on a similarity hypothesis, Water Resour. Res. 9: 1324-1331 (1973); and Y. Mualem, A conceptual model of hysteresis, Water Resour. Res. 10:514-520 (1974). The earlier article by Y. Mualem gives a review of the development of the domain model. 8. The case where

FOR FURTHER READING A. MUnster, Classical Thermodynamics, John Wiley, New York, 1970. Chapters II. VI, and VII give a fine introductory survey of the thermodynamic theory of stability. D. R. Nielsen, R. D. Jackson, J. W. Cary, and D. D. Evans, Soil Water, Soil Science Society of America, Madison, Wis., 1972. This monograph presents a complete survey of the equilibrium and transport properties of water in soil. Theoretical, laboratory, and field investigations are included. S. A. Taylor and G. L. Ashcroft, Physical Edaphology, W. H. Freeman, San Francisco, 1972. Chapters 7 to 10 in this excellent textbook give a thorough introduction to soil-water systems from the perspective of the soil physicist. This book provides good preparation for the present chapter.

SELECTED PHYSICAL CONSTANTS* Avogadro constant Boltzmann constant Faraday constant Molar gas constant Permittivity of vacuum

6.02205 X 1023 mor ' 1.38066 X 10- 23 J K- I 9.64846 X 104 C mor ' 8.3144 J K- I mol" 8.85419 X 10- 12 C 2 J- I m- I

*M. L. McGlashan, M. A. Paul, and D. H. Wiffen, Manual of symbols and terminology for physiochemical quantities and units, Pure and Appl. Chern. 51:1-41 (1979).

)()11

\

INDEX OF PRINCIPAL SYMBOLS Symbol

A A

am aq

.A a a

B BH Bs B2( 1) (:3 (:3 C

C C CEC

C(1)

D D

cD 210

Name Helmholtz energy Debye-H tickel parameter differential coefficient hard Lewis acid chemical symbol of jth species in a reaction oxidized chemical species reduced chemcal species soft Lewis acid total molar ligand concentration thermodynamic activity activity of water in an exchanger phase X-ray amorphous solid phase aqueous solution phase total surface area of an exchanger Freundlich parameter fraction of ionized exchange sites distribution coefficient (free metal cation) distribution coefficient (I: 1 complex) distribution coefficient (2:2 complex) distribution coefficient (free ligand) distribution coefficients for aqueous NO), NO;, and NH: empirical parameter in an expression for 'Y +, 'Y-, or 'Y hard Lewis base soft Lewis base second virial coefficient Freundlich exponent

")..2/2 number of components cation exchange capacity double layer integral capacitance cation exchange capacity difference between Infand IJ./ RT concentration of cation A at x = 00 heat capacity at constant volume separation factor dielectric constant conditional separation factor

Section 1.4

2.6 7.3 3.3 2.1 4.1 4.1

3.3 3.6 2.3

5.4 2.5 1.6

6.5 6.2 6.5 3.4 3.4 3.4 3.4 4.3

2.6 3.3 3.3 2.2

6.2 6.4 1.6 2.4

6.5 5.3 2.2

6.4 7.2 3.7 6.4 5.1

211

INDEX OF PRINCIPAL SYMBOLS Symbol

Narne

Section

c/>

determinant point of charge balance in the diffuse double layer extent of reaction parameter Dirac delta-"function" EMF of a Galvanic cell total mechanical energy equivalent fraction of component A in an exchanger phase equivalent fraction of electrolyte AC1 u (aq) in an aqueous solution equivalent fraction of component A in the ith exchanger phase equivalent fraction of solid solution component B electrode potential Standard electrode potential liquid junction potential electron interaction energy between two cations M permittivity of vacuum Faraday constant number of independently variable intensive properties rational activity coefficient of solution component A rational activity coefficient of exchanger phase component A fugacity of soil water fugacity fugacity of an ideal gas Standard State fugacity thermodynamic potential electrical potential (inner potential)

5.1 7.4 2.2 2.2 2.2 7.1 4.5

c/>

PVM

6.5

gravitational potential Gibbs energy Groenevelt-Parlange potential Gibbs energy of ath phase standard free energy change standard free energy of an exchange reaction standard free energy of formation gas phase combinatorial factor gravitational acceleration Gaines-Thomas parameter for component A in an exchanger phase ratio of mcan ionic activity coctlicicnts

7.6 1.4 7.1 1.5 2.3 5.4 2.4 2.2 6.5 7.5

D, d d~

o(q -

E E Ell

f. f

fd 1"

c/>

G G*

Ga so~G~,

~Gf

g g g gil

I'

qo)

7.2 6.4 2.1 6.2 4.5 6.1 5.1 5.2 5.3 3.7 4.2 4.2 4.5 4.1 6.5 6.4 4.2 1.6 2.6

5.4 5.2

212

INDEX OF PRINCIPAL SYMBOLS Symbol

1',+ 1',-

I'± 1'+ 1'-

H !1H<

HO I

lAP K K

s, KG KGT KR KRK K,

K,., *K,.,

Ky KY i

CKcx -s, cK", ko

kH k'H km X KT

1

In log A A

Name molal activity coefficient Macinnes activity coefficient of cation MacInnes activity coefficient of anion mean ionic activity coefficient single-cation activity coefficient single-anion activity coefficient enthalpy standard enthalpy change standard partial molal enthalpy ionic strength ion activity product thermodynamic equilibrium constant affinity factor equilibrium constant for a dissolution reaction of a solid exchange equilibrium constant exchange equilibrium constant for the ith class of exchanger formation constant Gapon selectivity coefficient Gaines-Thomas selectivity coefficient equilibrium constant for a reduction half-reaction Rothmund-Kornfeld selectivity coefficient complex stability constant solubility product constant solubility product constant for an hydroxide reacting with H 20 Vanselow selectivity coefficient Vanselow selectivity coefficient for the ith class of exchanger conditional exchange equilibrium constant conditional complex stability constant conditional solubility product constant Boltzmann constant Henry's Law constant (mole fraction basis) Henry's Law constant (molal basis) salting coefficient gas phase activity coefficient isothermal compressibility liquid phase natural logarithm common logarithm scale factor double layer parameter

Section 2.6 2.6 2.6 2.6 2.6 2.6 1.4 2.3 2.3 2.6 2.3 2.3 6.1 3.1 2.3 5.3 2.3 5.1 5.1 4.1 5.1 2.3 2.3 3.1 2.7 5.3 2.7 2.7 2.7 6.1 2.2 2.2 2.6 2.6 7.2 1.6 2.2 2.3 1.5 6.4

213

INDEX OF PRINCIPAL SYMBOLS Symbol

AO M M MT MTS

m m mO m± m+ m~

mi;

ma m(q) m, mw /.L /.L /.L /.L0 /.Lgw /.LR /.Lw

N N

N' NA NA NA NaP

v Vj

n P P

PA P(E) P(E,N)

Name "purely chemical part" of ji, in the Standard State molarity total number of exchange sites total molar concentration of a metal total molar concentration of a metal in aqueous solution mass molality Standard State molality mean ionic molality ca tion molality anion molality mass of ith component in the ath phase mass of air in soil affinity class density function mass of solid phases in soil mass of water in soil chemical potential electrochemical potential chemical potential per molecule Standard State chemical potential gravichemical potential chemical potential of a matter reservoir chemical potential of water number of chemical reactions total number of particles Standard State mole fraction mole fraction of component A mole fraction of exchanger phase component A number of adsorbed cations of type A number of nearest-neighbor sites occupied by ions a and 11 Avogadro constant mole number mole number of the ith cherncal species in a reaction moles H 20 per equivalent of CEC sum of stoichiometric coefficients in a binary electrolyte stoichiometric coefficient of jth reacting species grand potential applied pressure number of phases probability that cation A is adsorbed on a site Boltzmann factor Gibbs factor

Section 4.5

2.2 6.2 3.5 3.5 1.3

2.2 2.2 2.6 2.6 2.6 1.4 7.1

6.2 7.1 7.1 1.3 4.5 6.1

2.2 7.6

2.1 1.5 1.6 6.1

2.2 2.2 5.1 6.2 6.5 6.1 1.3

2.4 5.4 2.2 2.1 6.1 1.3 1.6 6.1 6.1 6.1

214

INDEX OF PRINCIPAL SYMBOLS Symbol

Pco, pE

Pk Pi Y;m

Y;" Y;P Y;, Y;w y;(X) Q

Q QA

QN q q R

R P

Pa Pb Pbw Pw

P~ p(X)

S I::.SO

SO

Name intensive variable pressure applied to water in a pressure membrane pressure applied by a volume reservoir pressure on water in a tensiometer vapor pressure negative log molar concentration partial pressure of CO 2 negative log electron activity weighting coefficient of kth class of exchange sites fraction of the CEC carried by the ith class of exchanger matric potential pneumatic potential envelope-pressure potential total potential water potential diffuse double layer electric potential stability parameter (TN/ uv)U-' site partition function partition function affinity class parameter site partition function molar gas constant Pauling radius of ion exchange site symmetry factor mass density of soil air dry bulk density wet bulk density mass density of soil water mass density of bulk water volumetric charge density entropy standard entropy change standard partial molal entropy

Section

7.2 7.4 2.1 7.4 1.2 3.5

3.2 4.2 6.2 5.3

7.3 7.3 7.3 7.6 7.3 6.4 2.4 5.2 6.1 6.1 6.2 6.5

2.2 2.4 6.3

7.3 7.3 7.1

7.3 7.3 6.4 1.3

2.3 2.3

SAR s

entropy of ath phase partial specific entropy of soil water sodium adsorption ratio solid phase

7.3 6.4

s

Sf m, -

7.2

Sa

Sw

X,

exchange site permutations surface charge density surface charge density ncutrnlizcd by i\ I and C I surface charge density of outer-sphere surface complexes

1.5

1.6

6.3 6.4

6.4 6.:'i

INDEX OF PRINCIPAL SYMBOLS

215

Symbol (J diff

(5j

(50

T TN TR Tw

() ()a ()v

V u u uij

u(x) V V ~VJ

po VM

Vw Va V V

W;(x) W

X XM X

x; ~

y y(x) Z Z

ZHC z z

Name surface charge density of diffuse layer standard deviation of J.l0 for the jth chemical species in a reaction surface charge density absolute temperature total normality absolute temperature of thermal reservoir soil water tension gravimetric water content gravimetric air content X3 volumetric water content internal energy valence of exchangeable cation A u+

=

ut»; 2ujax/Jx

a j cosh y(x) volume partial molar volume

standard volume change standard partial molal volume average volume per exchange site partial specific volume of soil water volume of ath phase valence of exchangeable cation 8"+ Vjms=x 1 energy of cation i at point x mean interaction energy anionic part of an exchanger phase electronegativity mole fraction of Mg in a magnesian calcite product of A and m.; grand partition function for an adsorption site molar activity coefficient Ff(x)j RT valence of ion grand partition function hydrozincite solid phase number of nearest neighbors of an exchange site vertical coordinate

Section 6.5 2.5 6.5 1.3 5.2 2.1 7.4 7.1 7.2 7.1 1.4 5.1

7.2 7.2 6.4 1.3 2.6 2.3 2.3 6.5 7.3 1.5 5.1 7.2 6.4 6.5 1.6 2.4 3.7 1.5 6.1 2.6 6.4 2.4 6.1 3.2 6.3 7.6

INDEX OF AUTHORS

Amjad, Z., 101 Ashcroft, G. L., 208

Babcock, K. L., 18, 125, 152, 153 Baes, C. F., 41, 63 Balazs, N. L., 208 Barrer, R. M., 153 Bartlett, R., 125 Basset, R. L., 101 Bates, R. G., 124 van Bladel, R., 153, 154 Bockris, J. O'M., 184 Bohn, H. L., 124 Bolt, G. H., 153, 185 Bowden, J. W., 186 Brauner, P. A., 101 Brown, W. E., 63

Caiserrnan, A., 125 Callen, H. B., 18,208 Caroli, S., 41 Cary, J. W., 208 Childs, E. C., 185 Christ, C. L., 64 Connell, W. E., 125 Cremers, A., 154

Davies, C. W., 64, 124 Davis, J. A., 186 Davis, L. E., 153, 185 Dawson, J. M., 208 Dixon, J. B., 101 Durst, R. A., 125 Dutt, G. R., 100

Elprince, A. M., 153 Eriksson, E., 185 Evans, D. D., 208 Fowler, R. II., IK4 Frere, M. II., I K5 lullcr, W. II., 100

Gaines, G. L., 153 Garrels, R. M., 64 Gast, R. G., 154 Gibbs, J. W., 18 Gilbert, M., 154 Giles, C. H., 153 Greenland, D. J., 152 Groenevelt, P. H., 208 Grover, J., 152 Guggenheim, E. A., 18,64, 124, 184, 185

Halsey, G., 185 Hamer, W. J., 64 Harned, H. S., 64 Harvie, C. E., 64 Hatsopoulos, G. N., 18 Hayes, M. H. 8., 152 Heald, W. R., 185 Healy, T. W., 184, 185, 186 Helfferich, F., 152 Hemingway, B. S., 41, 64 Hendricks, D. M., 100 Herington, E. F. G., 64 Hill, T. L., 184, 185 Hohl, H., 186 Hohn, F. E., 208 Holm, L. W., 153 Huitson, A., 153

Jackson, R. D., 208 James, A. M., 18, 124 James, 8., 125 James, R. 0., 184, 185, 186 Jenne, E. A., 101 Jensen, H. E., 153, 185 Jensen, W. 8., 100 Jorgensen, H., 125 Jurinak, J. J., 100

Keenan, J. H., 18 Kemper, W. D., 185 Kcstin, .I., 19 Khurukn, Y. K., 101 Kin!!, M. II., 152

",

INDEX OF AUTHORS

218 Kittrick, J. A., 63, 64, 100 Klinowski, J., 153 Koutsoukos, P., 101 Krishnamoorthy, C., 185

Lafon, M. G., 100 Landman, Y., 185 Langmuir, D., 101 Laudelout, H., 153, 154 Leckie, J. 0., 186 Leggett, D. J., 100 Lewis, G. N., 19, 63, 65 Lewis, R. J., 154 Lindsay, W. L., 100, 101 Lippmann, F., 64 Loy, T., 125

MacEwan, T. H., 153 Macinnes, D. A., 125 Margenau, H., 208 Marion, G. M., 100 Martell, A. E., 41 Mattigod, S. Y., 64, 100 McBryde, W. A., 64 McCauley, J. W., 100 McDuff, R. E., 101 McGlashan, M. L., 63, 209 Mesmer, R. E., 41, 63 Milazzo, G., 41 Misono, M., 77, 100 Montroll, E. W., 185 Morel, F., 101 Morgan, J. J., 101, 125 Mualem, Y., 208 MUnster, A., 208 Murphy, G. M., 208

Nagasawa, M., 185 Nakhwa, S. N., 153 Nancollas, G. H., 101 Nieboer, E., 64 Nielsen, D. R., 208 Nordstrum, D. K., 100 Nriagu, J. 0., 64

Parlange, J. Y, 208 Parsons, R., 125 Patrick, W. H., 125 Paul, M. A., 209 Pearson, R. G., 100 Philip, J. R., 208 Pitzer, K. S., 64 Plummer, L. N., 100, 101 Ponnamperuma, F. N., 125 Posner, A. M., 186 Poulovassilis, A., 208 Prue, J. E., 65 Pytkowicz, R. M., 65 Quirk, J. P., 186 Rai, D., 100 Randall, M., 19, 63, 65 Reddy, A. K. N., 184 Reddy, M. M., 153 Reichenberg, D., 153 Rible, J., 185 Rice, S. A., 185 Robeyns, J., 153 Robie, R. A., 41, 64 Robinson, R. A., 63, 65 Roy, R., 100 Sadiq, M., 41 Saito, Y, 100 Santillan-Medrano, J., 100 Schwertmann, u, 125 Shainberg, I., 125, 185 Sips, R., 185 Smith, A. L., 184 Smith, D., 153 Smith, R. M., 41 Sparnaay, M. J., 185 Sposito, G., 18,64, 100, 185 Stiglich, P. J., 185 Stokes, R. H., 63, 64, 65 Stumm, W., 101, 125 Suarez, D. L., 100

Ochiai, E., 100 van Olphen, H., 184 Onsager, L., 185 Overbeek,J. Th. G., 125 Overstreet, R., 125, 153, 185 Owen, B. 8., 64

Talsma, T., 208 Tardy, Y, 64 Taylor, H. S., 185 Taylor, R. M., 125 Taylor, S. A., 208 Thomas, H. C, 153, 154 Thorstenson, D. C, 100 Tianco, E. M., 125 Turner, F. T., 125

Page, A. 1.., 15.1, 1!l5 Parkhurst, D, 1.., 101

VCilh,

r.,

1>4

INDEX OF AUTHORS Walton, A. G., 101 Weare, J. H., 64 Weed, S. B., 101 Westall, J., 186 White, L. R., 186 Wiese, G. R., 184 Wiffen, D. H., 209 Wigley, T. M. L., 101

219 Winkelrnolen, C. J. G., 153 Winland, H. D., 100 de Wit, C. T., 185

Yates, D. E., 184 Yoneda, Y., 100

INDEX OF SUBJECTS

activity, 34, 36, 118 electron, 107 ion, 37 notation, 36 relation to equilibrium constant, 34, 69 relation to solubility, 67 activity coefficient, 54, 57, 127 exchanger component, 127, 129, 133, 145, 168, 183 gas, 54 MacInnes, 57 mean ionic, 57 molal, 57,60 molar, 57 rational, 55,97, 127, 129, 145, 168 single-ion, 59 activity ratio diagram, 88, 90, 92, 93 anion exchange, 127 Boltzmann factor, 156 bulk density, 8, 188 wet, 188, 207 cation exchange, 52, 96, 126, 160 chemical potential, 8, 25, 32, 118, 158, 193 relation to solid dissolution, 66 soil water, 193 Standard State, 25, 32, 41 statistical mechanical, 156 chemical reaction, 24 cation exchange, 52, 126 precipitation, 48, 66 weathering, 50 complex, 38, 46, 75, 78, 80, 83, 86 inner-sphere, 76, 79 outer-sphere, 76, 79 relation to solubility, 80, 83, 86 component, 4, 16, 17, 18 coprecipitation phenomena, 94 mixed solids, 95, 96 adsorption, 95 inclusion, 95

Davies equation, 60 diffuse double layer model, 169 uni-bivalent exchange, 174 uni-univalent exchange, 172 discrete site model, 166 uni-bivalent exchange, 167 uni-univalent exchange, 166 distribution coefficient, 81, III double-layer potential, 171, 180 electrochemical potential, 118 electrode phenomena, 119, 121 irreversible, 122 reversible, 119 suspension effect, 124 electrode potential, 110 standard, 110 electron activity, 107 relation to electrode potential, 110 EMF, cell, 120, 121, 124 enthalpy, II entropy, 8, 20, 157 relation to equilibrium, 20 statistical mechanics, 157 equation of state, 10 equilibrium, 6, 20 chemical, 24 exchange, 126 mechanical,21 redox, 102 phase, 22, 70 soil water, 206 thermal, 20 equilibrium constant, 34, 36, 48 conditional, 62, 133 dissolution, 66 exchange, 36, 38, 53, 63, 126, 133 formation, 36, 38, 49 redox, 104 solubility product, 36, 38, 49, 62, 66 stability, 38, 46, 62 variation with pressure, 40 variation with temperature, 40 lulcr equation. 1,1, J 57

22/

222 exchange isotherm, 134, 160, 164 molecular model, 160 nonpreference, 136 nonuniform exchanger, 164 extent of reaction, 24 Freundlich isotherm, 165 fugacity, 26 gas, 26 liquid,27 relation to activity, 34 soil water, 201 solid,27 solute, 28, 29 fundamental relation, 10, 20, 205 Gibbs-Duhem equation, 15,97,146,190 exchanger, 146 Gibbs energy, 12, 189 Gibbs factor, 156 Gibbs phase rule, 16 grand potential, 159 Helmholtz energy, II, 158, 182 Henry's Law, 28, 30 HSAB principle, 75, 78 hysteresis, 203 domains, 205 internal energy, 10, 20 relation to equilibrium, 20 ion activity, 37, 123 controlled by solid, 67 mean, 37 measured by electrode, 123 ion activity product, 27, 66, 69, 80, 85 variation with time, 69 ionic strength, 59 Langmuir isotherm, 163 Lewis acid, 75, 76 softness parameter, 77 Lewis base, 75, 77 LFER,42 complexes, 46 oxides, 42 smectites, 44 MacInnes convention, 58 Margules expansion, 129 mass, 8 matric potential, 197 Maxwell relation, 13 mixed exchanger, 139

INDEX OF SUBJECTS moisture characteristic, 20 I moisture profile, 205, 207 molality, 30 mean ionic, 57, 58 molarity, 30 mole fraction, 28 mole number, 8 nitrate reduction, 111 oxidation number, 102 oxidation-reduction, 102 sequential, 109 solubility, 109, 115 partition function, 156, 181 van der Waals limit, 182 pE, 107 phase, 4, 16, 17, 18,23 equilibria, 23, 27, 34 physical constants, 209 Poisson equation, 171 Poisson-Boltzmann equation, 172 potential, 10, 189 chemical,8 electrochemical, 118 electrode, 110 enthalpy, 11 envelope-pressure, 197 Helmholtz energy, 11 Gibbs energy, 12, 189 grand, 159 gravichemical, 205 Groenevelt-Parlange, 189 matric, 197 pneumatic, 197 total,205 water, 193 predominance diagram, 69, 114 effect of water activity, 73 pE-pH diagram, 114 pressure, 8, 22, 194 pressure membrane, 200 process, thermodynamic, 6 adiabatic, 6 infinitesimal, 6 isobaric, 6 isochoric, 6 isothermal, 6 natural, 6 reversible, 6 property, thermodynamic, 3 extensive, 4 fundamental, 3 intensive, 4 relation to variable of state, 4 psychrometer, 201

223

INDEX OF SUBJECTS Reference State, 53, 62, 127 aqueous solution, 56 Constant Ionic Medium, 56 exchanger component, 127 gas, 54 Infinite Dilution, 56 solid solution, 55 reservoir,S, 7, 20 matter, 22 thermal,20 volume, 21 selectivity coefficient, 63, 127, 132 Gaines-Thomas, 133, ISO Gapon, 132, 177 Vanselow, 63,126,177 separation factor, 96 conditional, 130 sodium adsorption ratio, 176 solubility diagram, 92, 94 solubility equilibria, 66 solution, 4, 28, 95 regular, 130 speciation calculation, 83, III computer, 84, 86 redox, Ill, 113 relation to lAP, 85 specific adsorption, 150 stability, thermodynamic, 190 soil-water system, 192 standard enthalpy change, 40 standard entropy change, 39 standard free energy change, 34, 149, 106 microbial catalysis, 106 relation to equilibrium constant, 36 variation with pressure, 40 variation with temperature, 40

standard free energy of formation, 41 Standard State, 25, 33, 68, 128 gas, 26 exchanger component, 128 liquid,28 solid,28 solute, 28, 30, 32, 61 solid,28 standard volume change, 40 state, thermodynamic, 4, 6 variables, 4, 7, 8, 9, 188 surface complex model, 178 activity coefficients, 183 swelling soil, 194, 207 temperature, 8, 21, 188 tensiometer, 198 thermodynamic soil system, 3, 20, 21 adiabatic,S closed,S heterogeneous, 4 homogeneous, 4 open, 5 total potential, 205 van Laar equation, 130 variables of state, 4, 6, 7, 8,9, 188 volume, 8, 188 wall, thermodynamic, 3, 5, 6, 119 diatherrnal, 5, 20, 21 insulating,S, 20, 21 permeable,S, 22 semipermeable,S, 23, 119 water content, 188 water potential, 193

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