Kinetics of Multistep Reactions (Comprehensive Chemical Kinetics)

Kinetics of Multistep Reactions (Comprehensive Chemical Kinetics) Friedrich G. Helfferich 2nd Ed. 2004 Elsevier ISBN 0444516530 ISSN 00698040

COMPREHENSIVE

CHEMICAL

ADVISORY

Professor

C.H. BAMFORD

Professor

S.W. BENSON

Professor

G. GEE

Professor

G.S. HAMMOND

Professor

K.J. LAIDLER

Professor

SIR HARRY

Professor

S. OKAMURA

Professor

Z.G. SZABO

Professor

0. WICHTERLE

BOARD

MELVILLE

KINETICS

Table of Contents Chapter 1. Concepts, definitions, conventions, and notation Chapter 2. Fundamentals Chapter 3. Experiments and their evaluation

Chapter 4. Tools for reduction of complexity Chapter 5. Elementary combinations of reaction steps Chapter 6. Practical mathematics of multistep reactions Chapter 7. Network elucidation Chapter 8. Homogeneous catalysis Chapter 9. Heterogeneous catalysis Chapter 10. Chain reactions Chapter 11. Polymerization Chapter 12. Mathematical modeling Chapter 13. Unusual thermal and mass-transfer effects Chapter 14. Instability, periodic reactions, and chaos

This book addresses primarily the engineer in industrial process development, the research chemist in academia and industry, and the graduate student intending to become a reaction engineer. In industry, competitive pressures put a premium on scale-up by large factors to cut development time. To be safe, such development should be based on "fundamental" kinetics that reflect the elementary steps of which the reaction consists. The book forges fundamental kinetics into a practical tool by presenting new, effective methods for elucidation of mechanisms and reduction of complexity without unacceptable sacrifice in accuracy: fewer equations (lesser computational load), fewer coefficients (fewer experiment to determine them). For network elucidation, new rules relating network configurations to observable kinetic behaviour allow incorrect networks to be ruled out by whole classes instead of one by one. For modelling, general equations and algorithms are given from which equations for specific networks can be recovered by simple substitutions. The procedures are illustrated with examples of industrial reactions including, among others, paraffin oxidation, ethoxylation, hydroformylation, hydrocyanation, shape-selective catalysis, ethane pyrolysis, styrene polymerization, and ethene oligomerization. Many of the rate equations have not been published before. The expanded edition of the 2001 title, Kinetics of Homogeneous Multistep Reactions includes new chapters on heterogeneous catalysis and periodic and chaotic re-actions; new sections on adsorption, statistical methods, and lumping; and other new detail. * Contains new chapters on heterogeneous catalysis, oscillations and chaos * Includes new sections on statistical methods, lumping adsorption and software and databases * Provides a better understanding of complex reaction mechanisms

Preface Grey is all theory, but green is life's golden tree. Johaiin Wolfgang v. Goethe, Faust For, his reason's razor slant rules: what must not happen, can Y. Christian Morgenstern, Galgenlieder"^

May the reader who studies this book, or goes as far as trying to work with it, keep in mind what these two wise poets had to say: one reminding us that nature weaves an infinitely finer, more intricate, more colorful tapestry than the best of all theories can project, the other whimsically warning against doctrinaire conclusions from what we have come to perceive as right. In no other field of science and engineering are their words more to the point than in reaction kinetics. Even so, I have written a book full of theory of reaction kinetics. I have done so in the firm belief that sound theory can at least serve as a sturdy framework, ready to be fleshed out with all the vagaries we encounter; that it can help us to acquire insight, a "feel" for what is apt to happen and why, a subconscious knowledge and perspective that springs from familiarity. If we throw a stone into a quiet pond, we don't even have to look: In our mind we will see the picture of expanding rings of waves the stone's impact has set in motion. Ideally, the kineticist will have learned to "read" a network and see in his mind how the reaction will evolve, how it will respond to changes in conditions, much like a conductor can read the score of a symphony and in his mind hear the orchestra play it. I believe such subconscious comprehension of complex reaction kinetics is within our grasp, and hope my book will help to bring it closer. The origins of this book date all the way back to the 1960s and 70s, when I worked in, and for a time directed, grass-roots development of large-scale processes in chemical industry. I spent untold hours, days, and weeks struggling to unravel mechanisms, derive rate equations, understand cause and effect, finally telling myself there had to be a better way. Ever since, I have worked on and off trying to find better, shorter, easier ways in practical reaction kinetics, and this book by an old man is the culmination of my efforts. It is the book I dearly wish I had had at hand when still in the front lines of development. Yet, I see it as only a step toward true mastery of its subject, and am hoping others will carry on where I left off. Songs from the gallows, translation by Walter Arndt, Yale University Press, 1993.

Preface This book tries to go beyond collecting and compiling accepted wisdom. In every instance it attempts to evaluate prior art critically, and in large parts it presents new methodology that has yet to withstand the acid test of extensive practical application. All this has required judgment calls. With much own experience and good advice to draw on, I am confident the big picture is correct. If there are errors in detail, I must accept sole responsibility, even where other sources are quoted. To expedite publication and reduce cost, this book has been reproduced photographically from the manuscript. I beg the reader's forbearance if the visual appearance of its pages does not in all places meet exacting standards. I am deeply indebted and grateful for gracious help, expert advice, and invaluable suggestions, first and foremost to Phillip E. Savage (University of Michigan), who has closely and patiently worked with me throughout most of the years this project has taken, and furthermore to Shao-Tan Hsieh (Mitsubishi Chemical), Yng-Long Hwang (Union Carbide), Jia-Ming Chern (Tatung Technical University), Robert L. Albright (Albright Consulting), Joe E. Hightower (Rice University), and many others, too many to name them all. Special thanks are also due to George Selembo for preparation of the illustrations and formulas, to my friends at Elsevier for their unwavering patience, to the editors for their encouragement and constructive criticism, and last but not least to the Pennsylvania State University and its Department of Chemical Engineering for lending their resources. This book has been almost ten years in the making. It will be my last such endeavor. If I ever take up the pen again, it will be to write science fiction, a field for which I now feel well equipped. Friedrich G. Helfferich State College, Pennsylvania September 2000

Preface to Second Edition It has been gratifying to see that large interest in this book has called for a second edition after only three years. The first edition was restricted to homogeneous reactions because, for these, a practical, yet reasonably rigorous treatment not found in standard text could be presented. Readers have commented that this unduly discourages practitioners of heterogeneous catalysis, who would also profit from the book's basic concepts and methods. In response, a brief chapter on heterogeneous catalysis has been inserted. This has been done although the complications are such that a balanced and practical approach must seriously simplify the kinetics of the surface reaction. I hope this chapter will nevertheless be welcomed as a supplement to existing, more detailed texts in showing ways how key results can be obtained more easily and modeling of complex kinetics be simplified. The extension to heterogeneous catalysis has necessitated a short section on adsorption equilibria and rates as well as some adjustments in other places. Another, very brief chapter on instability, oscillations, and chaotic behavior has been added. This topic is absent from most standard texts on reaction engineering and kinetics. Without delving into the intricacies of mathematics, the attempt is made to show the reader how and why these phenomena may arise, so he can avoid them. Inclusion of another chapter on heterogeneous noncatalytic reactions was contemplated, but decided against when a first draft threatened to grow to the size of a book all by itself. Not only is the variety of possible situations staggering, but a satisfactory treatment calls for greater detail on mass transfer effects than this book can accommodate. I hope an expert will take up the challenge of addressing this so far short-changed topic. In the chapter on experiments and their evaluation, the discussion of research reactors has been extended and an overview of statistical methods has been added. Also, the presentation of the principle of microscopic reversibility, the Delplot method, and lumping in mathematical modeling have been expanded. Other changes do not go beyond minor tightening and updating. It is a pleasure to thank Nancy Ohmer for contributing the sketches of Goethe and Morgenstern that grace the Preface in this second edition. Friedrich G. Helfferich State College, Pennsylvania April 2004

Introduction Chemical reaction kinetics differs in character from all other disciplines of engineering. The fundamental equations in those others are laws of nature— Newton's, Darcy's, Coulomb's, the laws of thermodynamics, etc.—and are always the same, though applied in ever different ways and combinations. In contrast, the fundamental laws of reaction kinetics, that is, the rate equations of chemical reactions, differ from case to case depending on the particular combinations of molecular events of which the reactions consist. Moreover, the rate equations of a new reaction are not even known at the outset, and experiments must be designed to establish them. Every new reaction poses its own challenges, opportunities, and pitfalls. Standard recipes are of only limited value, and ingenuity is at a premium. This is a source of unending fascination—and frequent frustration—and sets reaction kinetics apart from the rest of engineering. Reaction kinetics is unique. Reaction kinetics used to be one of the foremost topics of physical chemistry. Then, about halfway through the twentieth century, physical chemists let themselves be lured away to more glamorous pursuits in emerging fields such as nuclear magnetic resonance, neutron activation, electron microscopy, molecular beams, and quantum mechanics. As a result, much of old-fashioned traditional physical chemistry, including reaction kinetics, fell to chemical engineers by default. However, reaction kinetics did not fit the engineering mold. The engineer is trained to think in terms of dimensionless numbers between which theoretical or empirical correlations can be established, a procedure that is inappropriate for reaction kinetics with its ever different rate equations. Many current texts on reaction engineering accord reaction kinetics only a relatively rudimentary treatment. Typically, the most recent, 7th edition of Perry's Chemical Engineers' Handbook devotes only 13 out of its 2646 pages to reaction kinetics [1], and that although no other single facet has as much impact on the conception of a new chemical process and the design and operation of a chemical plant. In its new home of chemical engineering, reaction kinetics has remained a stepchild to this day. The word kinetics stems from the Greek Ktvelv, to move, and reaction kinetics is the science of how fast chemical reactions proceed. Beyond that broad definition, reaction kinetics means different things to different practitioners. Ask a chemical physicist, and he may think of molecular beams, potential-energy profiles along pathways, or ab initio calculations of rates of which he is proud if their results are correct within an order of magnitude. Ask a development chemist, and he might see in his mind tabulations of rates under a variety of conditions, and of

Introduction power-law or polynomial equations that best fit the data. Ask a physical organic chemist, and he is apt to conjure up Woodward-Hoffmann exclusion rules or electrons that pair in different ways. Ask a plant engineer, and he will think of how yield and purity in his reactor's effluent respond to changes in control settings. This book is devoted to still a different facet of kinetics, to what is sometimes called "fundamental kinetics," that is, the study of reactions as composites of elementary molecular steps and the mathematics reflecting them. The pioneer work in fundamental reaction kinetics—^by Bodenstein, Michaelis, Lindemann, Hinshelwood, Rice, Christiansen, and Semenov, to name only the most prominent—was done in the first six decades of the twentieth century. Since then, surprisingly few advances have been made in the state of the art of fundamental kinetics, with notable exceptions mostly in heterogeneous catalysis, polymerization, and on esoteric topics such as periodic and chaotic reactions. Perhaps this can be attributed to our preoccupation with thermodynamics. In any other field of science and engineering, the excitement is in dynamics, and statics is left to the more pedestrian minds. Only in chemical engineering and physical chemistry have we let our technical thinking and education be dominated by thermodynamics, which is not dynamics by any stretch of the imagination and should rightly be called thermostatics. Just because this wonderful and enormously successful tool exists, we have even tried to use it for dynamic phenomena, an application for which it was not designed and is not too well suited. In a way, we are now paying the price for the genius of Gibbs, Clausius, and their peers, who created for us this admirable edifice that has placed dynamics in its shadow for a century. Today, however, we see a resurgence of interest in reaction kinetics. Chemical industry has matured and its competitive pressures keep increasing. A chemical plant must produce to pay for its construction, its operation and maintenance, the raw materials it consumes, the disposal of the by-products and wastes it generates, the development of its process, the attempted developments of maybe half a dozen other processes that came to naught, the salaries of the company's managers and business staff, dividends for the stockholders, and taxes. The plant generates income only while it is in operation. Unlike a car, a plant does not die from old age or corrosion, it is shut down because a better or cheaper process has been invented, the need for the product has disappeared, a raw material has become too expensive, or some other event has made its operation unprofitable. That point in time is quite independent of when the plant was taken on stream. The only way to prolong the plant's productive life is to move its start-up date forward by shortening the time span between conception of the process and start of production. Accordingly, there is a great incentive to cut process development time by replacing traditional scale-up through several intermediate stages—demonstration units and pilot plants of increasing sizes—^by a direct scale-up from the laboratory bench to the eventual, full-sized commercial plant. To be safe, any scale-up by a

Introduction very large factor cannot rely on empiricism, it must be based on mathematics that correctly reflect the individual molecular phenomena, among them the elementary steps of which the chemical reactions consist. This requires a sovereign command of fundamental reaction kinetics. Even routine operation of a plant is safer if the fundamental kinetics of its chemistry is fully understood. To be sure, the fundamental approach to process development cannot obviate demonstration units and pilot plants. They are still needed as final proof of operability and to ascertain long-term effects such as catalyst life or build-up of minor impurities, effects that cannot be measured in short-duration bench-scale experiments. They also serve well for producing representative samples ahead of time for potential future customers. Moreover, they are invaluable for fine-tuning and provide excellent opportunities for corrosion tests and piloting envisaged process control. However, fundamental kinetics can free them of the obligation to scan wide ranges of potential operating conditions for optimization and design. This is not to say that the fundamental approach to reaction kinetics is automatically the best in every situation. At least today, if the scale is small, the process likely to be short-lived, the chemistry complicated, and timing more important than cost, the work to elucidate the mechanisms may not be warranted or entail unacceptable delay. An empirical scale-up then is preferable. In industry, the fundamental approach is at its best and fundamental kinetics in greatest demand if the scale is large and construction of successive plants over years to come is envisaged. This book has been written chiefly with such applications in mind. Almost every chemical reaction of practical interest consists of a network of elementary steps, each with its own contribution to kinetics. Single-step reactions are most often found in textbooks. Until fairly recently, computers were not efficient enough to permit reactor design and optimization to be based on rate equations reflecting individual steps, except in quite simple cases. Today, computation has become so fast and cheap that capacity and execution time are no longer limiting. The problem is not how to program and solve the simultaneous equations for the reaction steps, mass transfer, heat transfer, etc., but to verify the presumed reaction network and obtain numerical values for all its rate coefficients and their activation energies. To make fundamental reaction kinetics a practical tool, it must be streamlined without unacceptable sacrifices in accuracy. Chemical engineers are known for loving to construct complicated theories of simple phenomena. Here, the opposite is needed: a simple theory of a complicated phenomenon. In the words of Ian Stewart: "Science is not about devising hugely complex descriptions of the world. It is about devising descriptions that illuminate the world and make it comprehensible" [2]. Progress is the progression from the primitive to the complicated to the simple, to a clarity that arises from true comprehension of the subject. Nowhere is this more true than in practical reaction kinetics, where much of that last step is still to be taken. May this book help to speed us along.

Introduction All emphasis on simplification notwithstanding, the development of reliable reaction mathematics for design is an exacting job. Because scale-up by large factors may be involved, mechanisms that are merely plausible working hypotheses will not do; the basis must be established beyond reasonable doubt. Incorrect kinetics is worse than none. In process development, kinetics is not a game for amateurs. Even in research, misinterpretation of kinetic observations may result in futile efforts and missed opportunities. The presentation here is geared toward the demand for exactitude. This book is intended as an aide and guide for the hands-on chemist and engineer in development. While stressing accuracy, it is kept as simple as possible. It addresses methodology rather than science, and glosses over many of the finer points of kinetic theory. Its goals are those of reaction kinetics in practice and can be summarized as follows: • establishment of reliable mathematics of reactions by means of short-duration bench-scale experiments, and • construction of simple, but sufficiently accurate mathematical models of reaction kinetics for design, scale-up, optimization, on-line control, and trouble shooting. The book is on kinetics, not reaction engineering: It focuses on reactorindependent behavior, that is, on reaction rates under given momentary and local conditions (concentrations, temperature, pressure). Reactor-dependent, global behavior is included only to the extent necessary for evaluation of kinetic experiments, which, of course, require reactors, and in a few instances in which vagaries of multistep kinetics produce uncommon behavior or impact reactor choice. I have chosen to place the emphasis on homogeneous reactions because these are free of the many complications of heterogeneity, thus allowing actual reaction kinetics to be treated with more rigor and in greater detail. However, the principles, concepts, and methodology are applicable to heterogeneous reactions as well, notably to surface reactions in heterogeneous catalysis. A brief chapter on this very important topic is therefore included, even though the additional complications of heterogeneity are usually so massive that a balanced and manageable treatment must simplify the aspects of the actual surface reaction. On the other hand, the book deals in depth with gas-liquid reactions such as air oxidation, hydrogenation, hydrocyanation, hydroformylation, etc., in which the reaction occurs in the liquid phase although a reactant must be resupplied from the gas phase. The chemist may miss a sharp distinction between catalytic and non-catalytic homogeneous reactions. This is because, often, the mathematics is the same: As long as its concentration remains constant, the catalyst can simply treated as being both a reactant and a product. Important is whether or not the catalyst concentration varies with conversion, as it does in enzyme catalysis. The more complex mathematics for such situations and ways to simplify it are covered in detail.

Introduction The book also includes a chapter on principles of polymerization kinetics, to serve as an introduction and guide that will facilitate the use of special texts. The book addresses mostly the concerns of the industrial chemist and engineer. It does not include an in-depth coverage of very fast reactions of biochemistry or methods for their study. The book is structured to supplement modern texts on kinetics and reaction engineering, not to present an alternative to them. It intentionally concentrates on what is not easily available from other sources. Facets and procedures well covered in standard texts—statistical basis, rates of single-step reactions, experimental reactors, determination of reaction orders, auxiliary experimental techniques (isotopic labeling, spectra, etc.)—are sketched only for ease of reference and to place them in context. Emphasis is on a comprehensive presentation of strategies and streamlined mathematics for network elucidation and modeling suited for industrial practice. While concentrating on methods, the book uses a number of reactions of industrial importance for illustration. However, no comprehensive review of multistep reactions is attempted, simply because there are far too many reactions and mechanisms to present them all. Instead, the book aims at providing the tools with which the practical engineer or chemist can handle his specific reaction-kinetic problems in an efficient manner, and examples of how problems unique to a specific reaction at hand can be overcome. Some examples drawn from my own laboratory and consulting experience have been construed or details have been left out, in order to protect former employers' or clients' proprietary interests. In particular, the omission of information on exact structure and composition of catalysts is intentional. Each chapter concludes with a summary. Before he delves into the main text, the user may want to check it to see whether what he seeks is indeed covered. I expect a rapid evolution of fundamental reaction kinetics in the years to come and a growing awareness of its enormous practical value. I hope and trust that this book will contribute its share. Although the publisher does not agree, I wish it will soon be made obsolete by swift advances in practical kinetics it will help to stimulate.

References 1. 2.

S. M. Walas, Reaction kinetics, Chapter 7 in Perry's chemical engineers' handbook, 7th ed., D. W. Green, and J. O. Maloney, eds., McGraw-Hill, New York, 1997, ISBN 0070498415, pp. 3-15. I. Stewart, Life's other secret: The new mathematics of the living world, Wiley, New York, 1998, ISBN 0471158453, p. 9.

Chapter 1 Concepts, Definitions, Conventions, and Notation For ease of reference this chapter outlines and explains the essential concepts and the formalism of presentation.

1.1.

Classification of reactions

Reactions can be classified as homogeneous or heterogeneous, depending on whether they occur in a single phase or involve two or more of phases. Established custom is to count reactions still as homogeneous if they require transfer of a reactant or reactants from another phase. An examples is liquid-phase hydrogenation with a dissolved catalyst. In contrast, reactions at interfaces such as surfaces of solids are regarded as heterogeneous. The tide of the book refers to multistep reactions, defined as all kinds of reactions that involve more than a single molecular event such as rearrangement or break-up of a molecule or transformation resulting from a collision of molecules. Some standard texts speak instead of complex reactions and multiple reactions, depending on whether or not the mechanism involves trace-level intermediates. The term multistep reactions comprises both these categories. On the other hand, a distinction exists between multistep and simultaneous reactions. The latter are independent reactions that occur side by side in the same reactor. For example, a reaction in which one and the same reactant A can undergo two different reactions leading to different products ^ p

A qualifies as multistep because the same reactant is involved in two different molecular events. In contrast, two reactions A —•

P

occurring side by side in the same reactor are simultaneous (but each may be multistep, namely, if it involves one or more intermediates).

Chapter L Concepts, definitions, conventions, and notation 1.2.

Steps, pathways, networks, and cycles

Almost every chemical reaction in industrial and laboratory practice results not from a single rearrangement or break-up of a molecule or collision of molecules, but from a combination of such molecular events called elementary steps, or steps for short. The steps of a reaction may occur in sequence, reactants reacting to form intermediates which subsequently react to form other intermediates and ultimately a product or products. The sequence of steps then is called a pathway. Almost always, however, one or several of the reactants or intermediates can also undergo alternative reactions that eventually lead to undesired by-products or different, but also desired co-products. The combination of steps then is called a network with branches. Pathways from specific reactants to specific products can be defined within networks. Points at which pathways branch are called nodes, and linear portions between nodes or between a node and an end member are called segments. The network may contain parallel pathways from one node to another or to an end member, involving conversion of the same reactants (or intermediates) to the same products (or other intermediates); such pathways form a loop. In catalysis, the catalyst is first consumed and then reconstituted by a later step. The resulting circular pathway is called a cycle. Such a cycle is not a loop. In principle, every chemical reaction is reversible, and so are all of its steps. This is because the decrease in free energy accompanying a totally irreversible reaction or step would have to be infinite. In practice, however, a reaction or step is said to be irreversible if, at equilibrium, its reactants are almost completely converted to products. It is left to the practitioner to decide on the merits of the case how strict he wants to be in interpreting this "almost." A common way of representing a multistep chemical reaction, used in many texts on physical organic chemistry, is by listing all of its steps in succession, with arrows pointing forward for irreversible steps and double arrows for reversible steps. * For example: A<

•K

K

• P

K

•Q

* The "single line-double arrow" notation, <—>, for reversible steps is employed here with apologies to the organic chemist who likes to see it reserved for resonance structures and prefers "double line-double arrow," ^^P, for reversible reactions. The latter notation, however, causes problems in depiction of reversible catalytic cycles: Since the arrowheads along the inner and outer circles in the diagram of a cycle point in opposite directions, either all reactants or all products would have to be crowded into the interior of the circle, or cross-overs would occur. For a book in which the distinction between reversible and irreversible steps of cycles is essential and resonance is not an issue, the <—^ notation appears more practical.

1.2. Steps, pathways, networks, and cycles

9

However, a mechanism is easier to visualize if a network notation is used, as is done in this book. In that form, the reaction above appears as ^ ^ P A <—• K . . ^ ^ Q A step may have more than one reactant or more than one product. In such cases, a practical shorthand notation is to write the co-reactants above, and the coproducts below the arrow of the step, each with its own arrow connecting it to the pathway. For example, the sequence of steps A

• K

K+ B

• L

L

• P+Q

can be shown in network notation as A

B • K -^^==-> L^=;^ P

(11)

Q The convention of placing co-reactants above and co-products below the line of the main reaction is arbitrary, but has the advantage of providing a distinction between the former and the latter that is apparent at a glance; as will be seen in examples in later chapters, this is convenient when rate equations are to be compiled because the concentrations of co-reactants and co-products enter these in different ways. However, if the network is complex, adherence to this convention may not be practicable. Also, for catalytic cycles a slightly different convention is preferable: The catalyst is shown as a member of the cycle rather than as a co-reactant, and all reactants and products are placed outside the cycle and connected to it by arrows. For example, the cycle of a reaction A + B —• P, catalyzed by cat and with intermediates K and L, might be

10

Chapter 1. Concepts, definitions, conventions, and notation

1.3. Rates The central topic of the book is the rates of chemical reactions or elementary steps. A rate r^ states the number of moles of species i formed (if positive) or consumed (if negative) by a chemical reaction or reactions per unit volume and unit time (unit catalyst weight and unit time in heterogeneous catalysis). It is a process rate as distinct from a rate of change. The distinction is important. A rate of change is an observable phenomenon of nature, e.g., a change of concentration with time, and is the combined result of all contributing process rates.* For an example from everyday life, we might think of "Home on the Range" where breakfast eggs are being soft-boiled. For the rate of change of the water temperature, the contributing process rates are heat transfer from the hot element via the pot to the water and from the water to the eggs, as well as heat loss to the atmosphere and evaporative cooling; once the boiling point is reached, the process rates compensate one another to produce a zero rate of change. Typical rates of change and process rates in chemical engineering include: rates of change variation of concentration with time concentration with distance pressure with time temperature with time

process rates rate of chemical reaction diffusion convective mass transfer heat conduction

An ideally mixed continuous stirred-tank reactor at steady state may serve as an example. The process rate -r^ of consumption of a reactant A is finite, but is compensated by the inequality of the reactant flow rates into and out of the reactor. The result is a zero rate of change of the reactant concentration, d Q /dr, in the reactor and its effluent. Figure 1.1 illustrates the connection between process rates and rates of change. The reaction engineer works in this scheme either from left to right or from right to left. In the elucidation of a reaction mechanism, the rate of change of a concentration of a participant is measured and, with the help of theory or a model, the process rate of formation or consumption by reaction is deduced by accounting for the rates of all other contributing processes. In reactor design, the rate of change of a concentration is calculated as the resultant of the rates of the respective reaction or reactions and all other contributing processes. * The distinction between process rates and rates of change was pioneered by Stuart W. Churchill [1]. Many current textbooks on reaction engineering still use dQ Idt for the process rate of a chemical reaction. This is a left-over from a time when reaction kinetics was a laboratory endeavor practiced by physical chemists with their batch reactors. It is ironic that this habit should have survived the take-over of reaction kinetics by mathematically-minded chemical engineers who mostly deal with flow reactors.

1.4.

Rate equations and activation energies

11

PROCESS RATE RATE of CHANGE

/ THEORY \ -4— — •

or 1 y MODEL J

PROCESS RATE PROCESS RATE

Figure 1.1. Connection between rate of change and process rates (from Helfferich and Savage [2]). 1.4.

Rate equations and activation energies

A rate equation states the rate r, of formation (if positive) or consumption (if negative), in moles per unit volume and unit time, of a participant i in a reaction as a function of the local and momentary concentrations (for gas-phase reaction see farther below). For example, the rate equations of A, B, and P in a single-step reaction A + B —• P are ''A = " ^ A P ^ A ^ B '

^B = ~ ^ A P ^ A ^ B '

^P = ^AP ^A ^B

^

'^

As required by stoichiometry, r^ = —r^ = —r^. According to eqns 1.2, the rate is proportional to the concentrations of the reactants A and B. The proportionality factorfc^p^^ ^ rate equation such as 1.2 is the rate coefficient. In this book, rate coefficients are identified where necessary by double indices; the first member refers to the reactant of the step, the second to the product. For example, the forward and reverse rate coefficients of a step A <—• K are fc^K and /:KA» respectively. Any co-reactants or co-products are ignored in the indices. Thus, the coefficient of the step K-hB —• L in the pathway 1.1 appears simply as k^n^. The dimension of the rate coefficient depends on the reaction order (see Section 1.5) and the choice of concentration units. For example, the coefficient in eqns 1.2 has the dimension [V mol "^ r ~^]. Here, rate equations are given in terms of concentrations Q , in units of moles per unit volume. This is the most convenient choice for liquid-phase reactions. For gas-phase reactions, partial pressures p, suggest themselves as an alternative and can be substituted without loss of generality. However, the dimensions and numerical values of the rate coefficients change accordingly. In this book, rate coefficients are assumed to depend only on temperature, not on concentrations. This is an idealization, as will be discussed in Section 2.2. The temperature dependence of a rate coefficient is characterized by the activation energy E^ as given by the Arrhenius equation [3] k = Atx^i-EJRT)

(1.3)

The activation energy is not necessarily independent of temperature (see Sections 2.3 and 13.1 for details).

12 1.5.

Chapter 1. Concepts, definitions, conventions, and notation Orders, molecularities, and ranks

The order (reaction order, kinetic order) of a reactant or other participant in a reaction is defined as the exponent of the concentration of that species in the rate equation, written as a power law. The overall order is defined as the sum of the exponents of all concentrations in the rate equation. For example, a reaction with power-law rate equation

is second order in A, first order in B, and third order overall. Reaction orders are in general empirical quantities, deduced from observed behavior. Only if a reaction is known to be single-step can they be derived from the stoichiometry. The molecularity of a step indicates how many reactant molecules participate. For example, a step A—• P is unimolecular, steps 2A—• P and A -f- B—• P are bimolecular. Trimolecular steps are rare, and quadrimolecular steps are unheard-of. Molecularities can only be stated if the pathway or network of the respective reaction is known. They refer to individual steps. For a multistep reaction as a whole, no molecularity can be defined. The rank of an intermediate or product of a multistep reaction indicates, on a relative scale, how soon the respective species is formed [4]. Primary intermediates or products arise from the original reactant or reactants either directly or via pathways in which all steps but one are very fast. Secondary intermediates or products are formed from primary ones; tertiary, from secondary ones, etc. Ranks of participants provide qualitative information about reactions whose pathways or networks are still entirely unknown. Ranking requires judgment calls because the choice of the time scale is left to the practitioner (see Section 7.1.2 for details). 1.6.

Conversion, yield, and selectivity

The progress of a chemical reaction can be characterized by several quantities. Conversion indicates consumption of the reactant or reactants. ThQfi'actionalconversion /A of a reactant A is the ratio of the amount (in moles) of A consumed to the amount of A charged. At constant fluid density:

/^ = ^LLS^

= 1 - cjc:

(1.4)

^A

where superscript o refers to the initial charge to a batch reactor or the feed to a flow reactor of any type. If the fluid density varies with conversion, concentrations Cj must be replaced by molar amounts A^j in batch reactors and by molar flows F^ in flow reactors.

1.6. Conversion, yield, and selectivity

13

If several reactants are charged in non-stoichiometric amounts, their fractional conversions differ. As a convention, reference then is to the limiting reagent, that is, the reactant that could be consumed completely (barring thermodynamic limitations). Yield describes product formation. The yield y^ of a product P is defined as the ratio of the number of moles of reactant (or limiting reagent) converted to P to the number of moles of reactant (or limiting reagent) charged. At constant fluid density: (Cp - Cp )mp

.. -.

;;

^^-^^

yv =

c:in^ (stoichiometry is n^A —• /ipP, and the n, are the stoichiometric numbers, that is, the mole numbers in the stoichiometric equation of the reaction). The (cumulative) yield ratio Y^ of two products P and Q is defined as the ratio of the yields of P and Q. At constant fluid density: y

^ "''

y^

^

(Cp- c;)in^

>'Q

'

(CQ

^^^^

- Q)ln^

In both eqns 1.5 and 1.6, the concentrations must be replaced by molar amounts in batch reactors or molar flows in flow reactors if the fluid density varies. The instantaneous yield ratio FpQ of the two products is defined as the ratio of the momentary rates of conversion to P and Q: r^ln^

In eqns 1.6 and 1.7, the stoichiometrics of conversion to P and Q must be written in such a way that AZA has the same value for both. For example, if the parallel reactions are 2A —• P and A —• Q, then AZp = 1 and HQ = 2 (with ^A = 2), or /2p = 1/2 and HQ = 1 (with n^ = 1); in either case, n^ln^ = 1/2. Selectivity refers to different extents of product formation by a^ reaction with side reactions. In this book, the (cumulative, fractional) selectivity Sp of a product P is defined as the ratio of the number of moles of reactant A converted to P to the number of moles of that reactant consumed:

If conversion is complete (/A = 1), the cumulative selectivity and the yield are equal. At less than complete conversion (/A < 1), the yield (based on reactant charged) is lower than the selectivity (based on reactant converted).

14

Chapter 1. Concepts, definitions, conventions, and notation

The instantaneous selectivity Sp is the ratio of the momentary rate of reactant conversion to P to the rate of reactant consumption: 5p -

^ I ^

(1.9)

The distinction between instantaneous and cumulative yield ratios or selectivities becomes immaterial in gradientless reactors (continuous stirred-tank at steady state or differential once-through reactors) or if the instantaneous values do not vary with conversion. A variety of other definitions of selectivity are used in standard texts on kinetics and reaction engineering. The most common among them is the ratio of the numbers of moles of two products formed. However, this definition is less useful than that chosen here if the reaction yields more than two products. Extent of reaction. A quantity not used in this book is the extent of reaction, defined as the number of moles formed or consumed, divided by the respective stoichiometric coefficient, v^: ^ = i;^:

1!L

(1.10)

where i is any reactant or product (note the stoichiometric coefficients of reactants are negative). Whereas the fractional conversions of reactants in the same reaction differ unless stoichiometric amounts are used, the extent of reaction is the same for all reactants and products. This is an advantage in reactor design, but purchased with the inconvenience of having to work with a dimensional quantity whose value depends on the size of the initial charge. It is ironic that, here, the engineer prefers a dimensional quantity while the kineticist finds the dimensionless fractional conversion more to his liking. Note that the extent of reaction is proportional to the fractional conversion of the limiting reactant, so conversion from one to the other is not a problem. 1.7. Phase-equilibrium and transport properties Equilibria. Equilibrium of adsorption on solids is usually represented by an adsorption isotherm, which shows the concentration of the respective species on or in the solid as a function of its concentration in the contacting fluid at constant temperature (see Section 2.6 for details). A convenient quantitative measure of uptake of a gaseous species by a liquid is the distribution coefficient (partition coefficient), defined as the ratio of the concentration of the respective species in the gas to that in the liquid. If the concentration of the species in the liquid remains low, the distribution coefficient often remains independent of concentration. This is known as Henry's law, and the constant distribution coefficient is called the Henry coefficient, H{.

Summary

H,

^

CJp,

= const.

15

(1.11)

Rates. In this book, the rate of diffusion of a reactant or product within a solid catalyst is characterized by an effective diffusion coefficient, D,, defined in accordance with Pick's first law: J,

=

-D,{dCJdr\

(1.12)

(/j = flux of i in moles per unit cross-sectional area and unit time, r = radial space coordinate in solid particle). The typically porous solid is treated as a quasihomogeneous phase, and the retardation of diffusion by the solid matrix is accounted for by a correspondingly lower value of that coefficient. Transfer to or from the surface of a solid is expressed in terms of a masstransfer coefficient, /:mt, the flux being the product of that coefficient and the concentration difference across the respective layer as the "driving force:" J,

= -KAC.

(1-13)

[The relationship to the diffusion coefficient is k, = DJAz, where Az is the thickness of the layer.]

Summary Homogeneous reactions occur within one phase, here taken to be fluid. Included are reactions in which a reactant is supplied from another phase by mass transfer. Heterogeneous reactions involve two or more phases, as in catalysis on solids. Multistep reactions consist of a combination of elementary steps. No distinction is made between complex reactions (with trace-level intermediates) and multiple reactions (with intermediates at higher than trace concentrations). An individual molecular event such as a collision, break-up, or rearrangement is called a step. A sequence of steps is a pathway. If reactants or intermediates can react in different ways, the combination of steps is called a network, with nodes at which pathways branch. Linear portions between nodes, or between nodes and end members, are called segments. A network with parallel pathways is said to contain a loop. Catalytic reactions contain closed sequences of steps in which a catalyst reappears after having been consumed. Such sequences are called cycles. A network notation is used in which the sequence of steps is shown directly, and co-reactants and co-products are connected to the main sequence by their own arrows. A distinction is made between rates of change and process rates. The former are observable, measurable phenomena of nature; the latter, abstractions expressing the contributions of physical processes to such rates of change. For example, variations of concentrations with time or distance are rates of change, whereas rates of chemical reactions or mass transfer are process rates.

16

Chapter 1. Concepts, definitions, conventions, and notation

Rate equations state rates of formation (if positive) or consumption (if negative) of species in terms of moles per unit volume (or unit catalyst weight) and unit time as functions of the local and momentary concentrations of the participants. For gas-phase reactions, partial pressures may be substituted for molar concentrations. Where necessary, rate coefficients are identified by double indices, the first member for the reactant, the second for the product (co-reactants and co-products are disregarded). The temperature dependence of rate coefficients is characterized by Arrhenius activation energies. Molecularities, defined only for single elementary steps, state the number of reactant molecules involved. The reaction order with respect to a participant is the exponent of the concentration of that species in the (possibly empirical) power-law rate equation. The overall reaction order is the sum of all such exponents. The rank of a product is an empirical quantity derived from observed rate behavior and indicating whether that species is formed directly from reactants or indirectly from intermediates. Fractional conversion of a reactant is defined as the ratio of the amount consumed to that charged. In this book, the following definitions of yield, yield ratio, and selectivity are used: The yield of a product is the ratio of the amount of reactant (or reactants) converted to the product to the total amount of reactant (or reactants) charged. The cumulative yield ratio of two products is the ratios of their yields. The instantaneous yield ratio is the ratio of the momentary rates of conversion to these products. The cumulative selectivity to a product is the ratio of the amount of reactant (or reactants) converted to that product to the amount consumed. The instantaneous selectivity is the ratio of the momentary rate of reactant conversion to the product to that of reactant consumption. Equilibrium of adsorption on a solid is characterized by an adsorption isotherm, which shows the concentration on the solid as a function of the concentration in the contacting fluid. A quantitative measure of uptake of a gaseous species by a liquid is the distribution coefficient, defined as the ratio of the concentration on the solid to that in the' contacting fluid. If concentration-independent, the coefficient is also called Henry coefficient. Diffusion of a species in a porous solid is expressed in terms of an effective diffusion coefficient, whose value accounts for the retardation by the solid matrix. Mass transfer to or from a solid is expressed in terms of a mass-transfer coefficient, the flux being the product of that coefficient and a concentration difference as "driving force."

References 1. 2. 3. 4.

S. W. Churchill, The interpretation and use of rate data: The rate concept, Scripta, Washington, DC, ISBN 0070108455, 1974, p. 8. F. G. Helfferich and P. E. Savage, Reaction kinetics for the practical engineer. Course #195, AIChE Educational Services, New York, 7th ed., 1999, Section 2.1. S. Arrhenius, Z. Physik. Chem., 4 (1889) 226; see also any text on physical chemistry or reaction engineering. N. A. Bhore, M. T. Klein, and K. B. Bischoff, Ind. Eng. Chem. Res., 29 (1990) 313.

Chapter 2 Fundamentals The picture we have formed of our world is based on reproducible observation. This is how we have discovered and established our laws of nature. In fact, it has been said that human intelligence could not have developed in a world without such observable, reproducible regularities. In this respect, reaction kinetics poses a problem: It has no universal laws. Rather, reactions differ widely in their kinetics and so resist the attempt to formulate equations that are generally valid. Indeed, even one and the same reaction may obey different rate equations in different media. At the root of this difficulty are differences in mechanism, that is, in the combination and interplay of reaction steps. For an orderly approach to kinetics we are therefore best served by one of the engineer's favorite methods of problem solving: If a problem is difficult, to forget it and, instead, solve a simpler one of the same kind and then try to generalize. In the case of reaction kinetics, a simpler problem is that of elementary reaction steps, that is, conversion as a result of a single rearrangement or break-up of a molecule or of a single collision of two or more molecules. The kinetics of single elementary reaction steps nicely follow the laws of statistics, provided a sufficiently large population is observed. In this chapter, the kinetics of single steps, extensively covered in standard texts on reaction kinetics and reaction engineering [Gl-Gll], is briefly reviewed and then used to formulate in general terms the mathematics of multistep reactions. Also discussed are consistency criteria as well as adsorption equilibria and rates as needed in heterogeneous catalysis.

2.1.

Statistical basis: molecularities and reaction orders

The relationships between stoichiometry, molecularity, and reaction orders of elementary steps arise from statistics and are illustrated in Table 2.1. In a unimolecular elementary step, a reactant molecule undergoes rearrangement or break-up. The reaction is a matter of probability. At a given temperature, each reactant molecule has the same probability to react (however, see a qualifying comment at the end of this section). Accordingly, the reaction rate—defined as the number of molecules converted per unit volume and unit time—is proportional to the number of reactant molecules per unit volume at the respective time, that is, to the local and momentary reactant concentration. The proportionality factor reflecting the probability of the event is the rate coefficient, denoted k.

18

Chapter 2. Fundamentals

Table 2.1.

Molecularities, rate equations, and reaction orders of elementary steps.

molecularity

step

rate equation*

unimolecular

A —• product(s)

-/-A = kC^

bimolecular

A + B —• product(s)

~^A ~ f^Cf^C-Q

2A —• product(s)

-/•A = 2A:Q2

A + B + C —• product(s)

-r^ =

KCP^C^CQ

V in A V in B P' in C 3"^ overall

2A + B —> product(s)

~^A ~ 2A:CA C B

2"" in A V in B y^ overall

3A —• product(s)

-r, = 3kC,'

trimolecular

reaction order

V'

1

V' in A V in B 2"^ overall 2nd

ord

* Numerical factor (stoichiometric number) indicates relative amount of A consumed; see Section 2.4.

In a bimolecular step, two molecules of the same or different reactants collide and form one or several products. For any one reactant molecule, the probability of collision with another is proportional to how many of the other are around, that is, to the concentration of the other reactant. The overall probability of the event occurring, summed over all molecules of the first reactant, is, in addition, proportional to the concentration of that reactant. If the step involves two molecules of the same reactant, the overall probability is proportional to the square of the reactant concentration. Not every collision of molecules capable of reacting is successful. Both the number of collisions and the probability of their success are reflected in the rate coefficient. A trimolecular step requires three molecules of the same or different reactants to collide simultaneously to form one or several products. If molecules were ideal billiard balls, the time of contact of two colliding partners would be infinitesimal, and so would be the probability of a third partner making contact at exactly the same time. However, molecules are somewhat soft and deformable. Moreover, bonds and bond angles are distorted in the force field of a near-by other molecule. In a collision, this happens before actual contact is made and lasts for a time after contact has been broken. Two colliding molecules thus are in states of contact and distortion for a finite time, during which a third may join in. However,

2,1. Statistical basis

19

the probability is low. This makes trimolecular steps rare because, more often than not, faster uni- or bimolecular steps consume the reactant molecules before three of them collide simultaneously. As in bimolecular steps and for the same reasons, the rate is proportional to the respective reactant concentrations or their powers. Again, not every qualifying ternary collision is successful, and the rate coefficient reflects both the number of such collisions and the probability of their success. A quadrimolecular step would require four molecules to collide with one another. Such an event is even much more improbable than a collision of three. Indeed, no quadrimolecular elementary step has ever been identified. Provided the sample is large enough for statistics to be reliable, the rate equations in Table 2.1 are valid under all circumstances, regardless of the presence or absence of other molecules not involved in the respective step, and regardless of whether other reactions occur simultaneously in the same volume element. However, these "ideal" rate equations in terms of concentrations and with concentrationindependent coefficients are only approximations. Deviations must be expected, but are relatively minor in most cases of practical interest and will be disregarded in this book, except for a brief discussion of nonideality in the next section. As seen in Table 2.1, the overall order of an elementary step and the order or orders with respect to its reactant or reactants are given by the molecularity and stoichiometry and are always integers and constant. For a multistep reaction, in contrast, the reaction order as the exponent of a concentration, or the sum of the exponents of all concentrations, in an empirical power-law rate equation may well be fractional and vary with composition. Such apparent reaction orders are useful for characterization of reactions and as a first step in the search for a mechanism (see Chapter 7). However, no mechanism produces as its rate equation a power law with fractional exponents (except orders of one half or integer multiples of one half in some specific instances, see Sections 5.6, 9.2, 10.3, 11.3, and 11.4). Within a limited range of conditions in which it was fitted to available experimental results, an empirical rate equation with fractional exponents may provide a good approximation to actual kinetics, but it cannot be relied upon for any extrapolation or in scale-up. In essence, fractional reaction orders are an admission of ignorance. Except for reactions known to be single-step, molecularities or mechanisms cannot be deduced from observed reaction orders, nor can orders be predicted from stoichiometrics. The statement that each reactant molecule in a unimolecular reaction has the same probability to react is an acceptable and convenient simplification. In reality, molecules of the same species can exist at different energy levels, and only those "activated" to occupy a high energy level are capable of reacting. Energy is exchanged between molecules when they collide. If collisions are frequent, as in dense fluids, the energy distribution is practically in equilibrium. The simplified statement then applies to the average molecule, that is, it includes the probability of the molecule being activated. In contrast, in gases at very low pressures, collisions

20

Chapter 2.

Fundamentals

may be so rare that activated molecules react before having had a chance to lose their high energies in subsequent collisions. If so, activation by collision (a bimolecular event) becomes rate-controlling [1-3]. The simplified statement then applies only to the activated molecules, and the reaction should be viewed as a two-step event.

2.2.

Nonideality

Rate equations in terms of concentrations and with presumably concentrationindependent rate coefficients, as used in this book, are idealizations. In the real world, matters are more complex. For example, a change in polarity of the medium with progressing conversion may cause a variation of rate coefficients. Such effects are hard to predict and, as a rule, not overly serious. For practical purposes they can often be disregarded. Where this is not so, an experimental determination of the composition dependence of the coefficients is usually the best way to proceed. At first glance one might think that deviations from ideality could be accounted for by substitution of thermodynamic activities for concentrations in the rate equations. However, this is not so. In fact, it can even make matters worse, as can be shown with the theory of absolute reaction rates [4-9]. According to this theory, the rate is the product of a universal frequency factor and the concentration of the activated complex or transition state, M* (the system in the transient state of highest potential energy), crossing the energy barrier in the direction toward the products. The activated complex, in turn, is postulated to be in equilibrium with the reactants. Say, for a single-step reaction A + B —• P: = K^

where AT* is the thermodynamic equilibrium constant of the process A + B <—^ M*, and the 7; are the activity coefficients. With the rate proportional to C^t, it follows that TATB ^AP

ex. "A^B

(2.1)

where the coefficient /r^p is the product of the frequency factor and the equilibrium constant K^. A comparison of eqn 2.1 with the ideal rate equation r^ = k^pCj^C^ shows the deviation from ideality to appear as the factor 7^73 /7M+- A replacement of the reactant concentrations Q and Q by thermodynamic activities 7 A Q and y^C^ thus does not necessarily constitute an improvement. Rather, if the activity coefficient ratio 7A7B/7M* is close to unity but the coefficients themselves are not, the ideal rate

2.3. Temperature dependence

21

law is more accurate than that with thermodynamic activities. Only if the activity coefficient of the activated complex, 7^*, is closer to unity than are the activity coefficients of the reactants does the substitution of reactant activities for concentrations lead to a better approximation. This may or may not be the case. Nonidealities are important for the study of kinetics as a science, However, activity coefficients of activated complexes cannot be measured directly and are difficult to predict with any degree of certainty. Therefore, in practice, rate equations are almost always stated in terms of concentrations, as is done in this book. This may be frowned upon by thermodynamic purists, but we could counter that, here, the pot calls the kettle black, that their science deals with equilibrium, a state that in our world is only approached, nowhere and never exactly attained. 2.3.

Temperature dependence

Reactions differ from human beings. We humans tend to slow down when it gets hot, reactions speed up. At least this is true in general for elementary steps. With increase in temperature, bond vibrations in molecules become stronger and collisions occur more often and with greater vigor, increasing the probability of molecules to react. As a rule, the rates of single-step reactions therefore increase with increasing temperature. In the great majority of cases this is also true for rates of multistep reactions, but not without exceptions: The overall rate may decrease with increasing temperature if the rates of reverse steps increase more sharply with temperature than those of forward steps. Such anomalies will be discussed in detail in a later chapter (see Section 13.1). An approximate formula for the temperature dependence of the rate coefficient, k, is given by the Arrhenius equation slope -E^/R In^ k = A txp(-EJRT) (1.3) or, in differential form dink d(l/7)

-El R

(2.2) VT

These equations apply regardless of the reaction order. If the activation energy E^ is taken to be constant, integration of eqn 2.2 yields k(T) = k(r)

exp

X"

Figure 2.1. Arrhenius plot.

R

J_ 'T'O

T

(2.3)

Chapter 2. Fundamentals

22

where r° is a reference temperature. If the activation energy is constant, a plot of ink versus \IT (-E.U / gives a straight line with slope —E^/R, as shown in Figure 2.1. >. The activation energy can be (£ viewed as an energy barrier which 2 A.H the reaction must overcome. This is illustrated in Figure 2.2, in O ^' • which the potential energy of the reaction system is plotted versus a "reaction coordinate," an imaginary exothermic — • <— — endothermic quantity characteristic of how far the reaction has progressed. If the reaction is exothermic, the system reactio n coordinate moves from a higher initial to a lower final potential-energy level Figure 2.2, Potential-energy profile of a (left to right in Figure 2.2); if the single-step reaction. reaction is endothermic, the system moves in the opposite direction. As the illustration shows, the activation energy of an endothermic reaction must be higher than the standard-enthalpy change A//° of the reaction; in a reversible reaction, the activation energy must be higher in the endothermic than in the exothermic direction; and the A//° value must equal the difference between the activation energies of the forward and reverse reactions. Figure 2.2 is for a single elementary step; the plot for a multistep reaction would show a local potential-energy maximum for each step. In contrast to the formally analogous van't Hoff equation [10] for the temperature dependence of equilibrium constants, the Arrhenius equation 1.3 is empirical and not exact: The pre-exponential factor A is not entirely independent of temperature. Slight deviations from straight-line behavior must therefore be expected. In terms of collision theory, the exponential factor stems from Boltzmann's law and reflects the fact that a collision will only be successful if the energy of the molecules exceeds a critical value. In addition, however, the frequency of collisions, reflected by the pre-exponential factor A, increases in proportion to the square root of temperature, at least in gases. This relatively small contribution to the temperature dependence is not correctly accounted for in eqns 2.2 and 2.3. [For more detail, see general references at end of chapter.] For an elementary step, the straight-line Arrhenius plot with negative slope is a reasonably good approximation, corresponding to a positive and approximately constant apparent activation energy (E^ in eqn 1.3) and an increase in rate with increase in temperature. A multistep reaction, however, may show a different behavior. Its Arrhenius plot may have a positive slope, corresponding to a negative A

f

1J

/

I

k

agendo

D

1

2.4. Compilation of rate equations ofmultistep reactions

23

activation energy and a decrease of the reaction rate with increasing temperature. In other cases, the activation energy may "jump" within a narrow temperature span from one value to another, giving rise to an Arrhenius plot with two connected, almost straight-line portions of different slopes. In quite exceptional instances, the Arrhenius equation may fail entirely. All these deviations from normal behavior are rare. Their nature and mechanistic causes will be discussed in Section 13.1. Because deviations can occur, extrapolations with the Arrhenius equation or plot to temperature ranges not covered by experiments may not be reliable. 2.4.

Compilation of rate equations of multistep reactions

In unabridged form, the kinetics of a multistep reaction is described by a set of simultaneous rate equations r„ one for each participant (reactant, intermediate, product, catalyst, silent partner). Mathematics is simple, but cumbersome because of its bulk if the reaction has more than a few steps. Much of the chapters to come deals with how to reduce it to a more manageable size without unacceptable loss of accuracy. The full set will hardly ever be used, but a working knowledge of the ground rules by which the equations are compiled will be helpful. The steps of a reaction are statistically independent, coupled only through their mutual dependence on the concentrations of the participants they have in common. Accordingly: •

The algebraic form of the rate equation of a step and the value of its rate coefficient are not affected by any other steps; they are the same as they would be if the step were the only reaction occurring.

Each rate equation may consist of contributions from several steps: •

The rate equation r, of a participant i is obtained by summation over the contributions from all steps involving that participant.

Here, the forward and reverse directions of a reversible step are counted as two steps. Thus, each arrowhead in the network, except if pointing at a co-product or co-reactant, corresponds to one step. Steps in which a species i does not participate as reactant or product do not contribute to the respective r,. For example, in the network

'!s^

-^' 2Q

the rate equation r^ for the intermediate K is obtained as the sum of the contributions from the steps 2A—> K, K—• 2A, K + A—#• L, and L—• K -h A; the steps L—• P and L—• 2Q do not contribute.

24

Chapter 2. Fundamentals

The contributions of the respective steps to the rate equations r, for participants are compiled according to the following rule: The contribution of a step to a rate equation r, is the product of three factors: • the stoichiometric coefficient v-, of species i for that step, • the rate coefficient of that step, and • the concentration (or concentrations) of the reactant (or reactants) of that step raised to the power that corresponds to the respective molecularity (or molecularities).

For example, for a step K —• L, the stoichiometric coefficients are ^K = ~1 ^^^ Vi^ = + 1 , the rate coefficient is /C^L , and the step is first order in its reactant K; Thus, according to the rule above, the contribution of the step to r^ is -/:KLQ» ^^^ that to TL is +/:KLQIf a species appears in more than one location in the network, the contributions from all steps involving it are additive. Special care must be taken with participants of which more than one molecule is consumed or formed in a step (i.e., with \v^\ > 1). To avoid inconsistencies: •

All rate coefficients must be in terms of rate equations of the form {llv-^ r^ = ..., even if the reported data are not.

For example, according to the rule above, the contribution to r^ of a step 2A —• K is -2kj^j^C^. The numerical value of the coefficient must correspond to the rate equation r^ = —2k^^C^. This value is only half of that of the coefficient in the alternative rate equation r^ = -kj^^C^, commonly used for a single-step, secondorder reaction. The need for a convention as used here becomes immediately apparent when the stoichiometry of the step is examined. The procedure above gives r^ = -Ik^YiC^ and r^ = kf^^C^. As the stoichiometry requires, - r ^ is twice as large as r^ (two moles of A consumed per mole of K formed). If, instead, r^ were taken as —k^^^C^, this requirement would be violated (unless r^ were now taken to be Vik^^^^C^). Example 2.1. Compilation of rate equations for a hypothetical network. For a given network 2A <

• K ^t^^-> L ^..^^^^ 2Q

the rate equations, compiled according to the rules stated above, are

2.4. Compilation of rate equations ofmultistep reactions

ite jualtion

step 2A-~>K

Step K-^2A

step K+A—i-L

Step L-^K+A

^A

=

""^^AK^A

H-2/:KAQ:

— ^KL^A^K

+ ^LK^L

^K

=

•^^AK^A

— kyj^Cy^

~%LQ\^K

"'"^LK^L

^L

=

^"%L^A^K

~^LKQ

^p

=

'-Q

=

25

Step L-^P

Step L--^2Q

•~^LP^L

~^LQQ

' ^LP^L "^2KLQCL

It is wise to check such a compilation for consistency. All of the following conditions must be met: There must be as many columns as there are arrowheads in the network (not counting arrowheads pointing to co-products or co-reactants). Each column must have positive and negative entries: positive ones for products, negatives ones for reactants. For each step there must be one entry for each participating species. Each entry must have a numerical factor corresponding to the stoichiometric coefficient of the species considered (index of r). In each entry the indices of the rate coefficient must correspond to the reactant and product of the step. Each entry must have the concentrations of all reactants of the respective step as factors. Each concentration must have an exponent corresponding to the stoichiometric number of that reactant in that step. Matrix notation. For convenience in computer programming, the set of rate equations can be written in matrix form. For example, the matrix equation for the network in the example above is:

{r}

^^AK^A

"*"^^KA"~^KL^A

+^LK

0

0

^^AKQ

~^KA"~^KL^A

^^KL

0

0

0

"^^KL^A

_^ ^h —h ^Ui '^LP '^LQ

0

0

0

0

^Kv

0

0

0

0

+2t^

0

0

{C}

26

Chapter 2. Fundamentals where {r} and {C} are vectors with the r, and Q, respectively, as elements (i = A, K, L, P, Q). While useful for computer programming, the matrix notation does not contribute to better understanding.

2.5. Consistency criteria Rate equations and their coefficients in networks are not entirely independent. They are subject to two constraints: those of thermodynamic consistency and so-called microscopic reversibility. For reversible reactions, the algebraic form of the rate equation of the forward reaction imposes a constraint on that of the rate equation of the reverse reaction. In addition, the requirements of thermodynamic consistency and microscopic reversibility can be used to verify that the postulated values of the coefficients constitute a self-consistent set, or to obtain a still missing coefficient value from those of the others. 2.5.1. Thermodynamic consistency At equilibrium, there is no net formation or consumption of reactants and products, that is, the forward and reverse reaction rates must be equal. This is true no matter how many steps the reaction involves. Therefore: Equating forward and reverse rates must lead to an expression that is compatible with the mass-action law of equilibrium. This fact can be used as a self-consistency check of postulated equations for the forward and reverse rates and their coefficients; or as a help in deriving the reverse rate equation from the forward one; or to calculate the reverse rate coefficient from the forward one and the equilibrium constant, or the forward rate coefficient from the reverse one and the equilibrium constant [11,12]. Example 2.2. Thermodynamic consistency of an association reaction. To see how the thermodynamic consistency criterion can help in the search for a reverse rate equation compatible with a given empirical forward rate equation, consider the following reaction: stoichiometry: 2A M-> P (2.4) equilibrium requirement: Cp/C^ = const. = ^AP (2.5) empirical forward rate: r^ = k^ C^^^ (2.6) A likely reverse rate equation is reverse rate: -rp = LCp/CP-^^

2.5. Consistency criteria

27

It is compatible because equating the forward and reverse rates gives kjk^

= const. = CP/CA''CA''

= Cp/C^ = /^AP

meeting the equilibrium requirement 2.5. However, this reverse rate equation is not unique. Rather, any equation of the form reverse rate: -% = ky,C? IC^""-^-^^ with constant n chosen at will meets the equilibrium requirement 2.5 because it gives kjk^ = const. = CpVC"^''*^'' = Cp"/Q'" = (Cy/ClY = Kl^

(2.7)

(if ^AP is constant, so is A'^p). If the forward rate equation contains an expression with additive terms, the reverse rate equation must contain that same expression. For example, if the forward rate of the reaction 2.4 were forward rate

r^ =

only a reverse rate equation of the form 1r

reverse rate

/^n^2-2n

-rp =

(same denominator with additive terms) could meet the equilibrium requirement. If the equilibrium constant and the algebraic forms of both the forward and reverse rate equations are known, the reverse rate coefficient k^ can be calculated from the forward coefficient k^ or vice versa: K = K/K:,

(2.8)

with n in accordance with eqn 2.7. Equation 2.8 can also be applied to forward and reverse rate equations with denominators containing additive terms; this is so because the denominators cancel when the ratio is formed. Moreover, of course, eqn 2.8 is equally valid for singlestep reversible reactions. 2.5.2. Microscopic reversibility "Microscopic reversibility" as used in chemical kinetics is a classic misnomer. The name stems from a complicated derivation based on Onsager's axiom of reversibility at the molecular level [13-17]: At that level there is no preferential direction of time and, therefore, all events are in principle reversible. What is called microscopic reversibility in chemical kinetics is the statement that there can be no net circular reaction in a loop at equilibrium. For example, at equilibrium there can be no net circulation A —• B —• C —• D —• A in the loop 2.9 (next page):

Chapter 2.

28

Fundamentals

( V,

D,

3

(2.9)

Without invoking reasoning at the molecular level, that fact can easily be proved with the following argument. For a net forward reaction to occur (i.e., for the forward rate to exceed the reverse rate), the Gibbs free energy of the product or products must be less than that of the reactant or reactants. This makes a circular reaction in a loop impossible because the free energy would have to decrease from step to step as on a spiral staircase, yet reach its starting level again with the step that Figure 2.3. Waterfall by M. C. Escher, ® 2004 closes the loop, a feat that can be accomplished only in the world The M. C. Escher Company, Baarn, Holland. of M. C. Escher (see Figure All rights reserved. 2.3). The principle of microscopic reversibility can be used to check a set of postulated rate coefficients for self-consistency or to calculate the still unknown value of one rate coefficient from those of all others. To this end, most texts on kinetics prescribe a procedure called detailed balancing. However, a much simpler rule will do:

The product of the "clockwise" rate coefficients in a closed loop must equal the product of the "counter-clockwise" rate coefficients.

Derivation. The equilibrium constants of the reversible steps of the loop are related to the standard Gibbs free energies AG° of formation of the members. For the fourmembered loop 2.9:

2.5. Consistency criteria - RT In K^

= iAG;\

-(AGf°),

- RT InK,^ = {AG,% -RTlnK^

29

(AG;\

= (AGf% - {AG°)^

-RT\nK^, = (AG;)^ - ( A G , % Adding these equations to one another one obtains - RT (InK^^ + InATgc + InT^^D + I^^DA) = ^ so that ^AB^BC^CD'^DA

~

^

Each equilibrium constant equals the ratio of the forward to the reverse rate coefficient of the respective reaction step: K, = k,,/k,

(2.10)

Accordingly: ^AB^BC^CD^DA

^

^BA^CB^DC^AD

(2.11)

as stated in the rule above. The argument leading to the conclusion that there can be no net circular reaction in a closed loop is based on the free energies of the members. As thermodynamic quantities, these are independent of v^hether or not the species involved also undergo other reactions. Accordingly, the rule is valid also for loops that are parts of larger networks, say, for ABC in

-• D -I—•A

(2.12)

Likewise, the rule that the product of the clockwise coefficients must equal that of the counter-clockwise ones, based on the standard free energies, also holds even in such cases. The only restriction is that a full-circle reaction, if it were to occur, may not entail any net conversion of co-reactants to co-products. In contrast to the traditional derivation based on microscopic reversibility, that given here does not invoke equilibrium. It thus shows that no net circular reaction is possible even under non-equilibrium conditions. However, there is one very important qualification: A net circular reaction does occur if it entails the conversion of co-reactants to co-products of lower Gibbs free energy. For example, the cycle 2.13 (next page) converts reactants A and B to product P while undergoing a net circular reaction K—•L—>-M—>N—•K. This is a typical catalytic cycle. Such a cycle is not 2i loop, a term to be reserved for circular pathways in which any

30

Chapter 2. Fundamentals

A , ; ^

K P^r N

L \ "l/^B

(2.13)

M

co-reactants are restored (a loop of two parallel pathways converting same reactants to same products meets this condition). Net circulation in the catalytic cycle occurs only if it entails a drop in free energy, that is, if the reactant or reactants (A and B in the cycle 2.13) are not in equilibrium with the product or products (P in 2.13). If so, the cycle is also not at equilibrium. At equilibrium, there is no net conversion of reactants to products to cause circulation. The principle of no net circular reaction at equilibrium thus applies to catalytic cycles as well as loops. However, the rule describing the constraint on the rate coefficients must be modified: The product of the forward rate coefficients in a catalytic cycle must equal the product of the reverse rate coefficients multiplied with the equilibrium constant of the catalyzed reaction.

For example, for the catalytic cycle 2.13:

where ^AP is the equilibrium constant of the reaction A + B <—• P. Derivation. At equilibrium of the four reversible steps in the cycle 2.13:

so that Replacing Cp/QCe by the equilibrium constant K^^ of the reaction A + B —• P and the other equilibrium constants by the respective ratios of the forward and reverse rate coefficients (eqn 2.10) and then multiplying both sides by the product of the reverse rate coefficients one obtains eqn 2.14. It is often loosely said that a catalyst "drives" a reaction. As the consideration here demonstrates, a picture more true to nature shows the reaction driving

2.5. Consistency criteria

31

the catalyst. With its free-energy decrease, the reaction puts the catalyst system through its paces, much like water drives a water wheel. Rather than "driving" the reaction, the catalyst makes it possible by providing a pathway. The principle of no net circular reaction in a loop, even under non-equilibrium conditions, has an important corollary:

Forward and reverse reaction must occur along the same pathway.

The reverse reaction is not allowed to take a different path, even if only partially, because that would create a loop with net circular reaction. The rule also applies to catalytic reactions, preventing the reverse reaction from taking a different pathway with no or a different catalytic cycle: The step sequence must be the same as for the forward reaction, but in reverse order. No matter that under non-equilibrium conditions there is net circulation in any catalytic cycle along a pathway, there can be none in a loop of two parallel pathways. Any catalytic cycles in the pathways of a loop are just "wheels within wheels." A network may have more than two parallel pathways. In such cases, several loops can be constructed from pairs of pathways, and the rules for selfconsistency of the rate coefficient values are more convenient when reformulated:

The ratio of the products of forward and reverse rate coefficients must be the same for all pathways.

This is true even if some or all pathways are catalytic. Derivation. In any loop of two parallel pathways, clockwise is forward along one pathway and reverse along the other. Thus, the clockwise-to-counter clockwise product ratio of the rate coefficients is a forward-to-reverse product ratio in one pathway and a reverse-to-forward one in the other. The equality of the total clockwise and counter-clockwise products of the loop therefore requires the forwardto-reverse product ratios to be the same along the two pathways. The rule can also be obtained as a consequence of the fact that the thermodynamic equilibrium constant is pathway-independent and equals the ratio of the products of the forward and reverse rate coefficients or a power thereof (see Section 6.3). However, for multistep reactions that equality would first have to be established. Conversely, the proof shown above could be used to derive that equality.

32

Chapter 2, Fundamentals

2.6. Adsorption equilibria and rates Adsorption isotherms. Adsorption is an important facet of kinetics of heterogeneous catalysis. Adsorption isotherms, showing the equilibrium concentration of a species on a solid as a function of the concentration in the contacting fluid at constant temperature, may have different shapes, usually classified as Types I through V [18] (see Figure 2.4). Type III

Figure 2,4. Types of adsorption isotherms (according to Brunauer etal. [18]). Isotherms of Type I arise from competition of molecules for sites on the solid: The higher the coverage, the smaller is the chance for a molecule still to find an empty site. In contrast. Type III reflects synergism: Molecules already adsorbed attract others to the surface. Type II, more common than III, shows competitive behavior at low concentrations and synergism at high concentrations. It is found, for example, in adsorption of water vapor on many drying agents. As the relative humidity is increased, pore walls become covered with water films, and capillary condensation begins to become effective and produces synergism. Like Type III, Types IV and V are relatively rare. Type I isotherms can be approximated by simple equations. The two most common ones are Freundlich's [19] and Langmuir's [20,21] (see also any textbook on physical chemistry): Freundlich Langmuir

q. = cpt

{a < 1)

(2.14) (2.15)

^J

1 +E(^./^.)

2.6. Adsorption equilibria and rates

33

where q, is the concentration of species i on the adsorbent (per unit weight of adsorbent); p^ is its partial pressure in the fluid phase; ^^ is the total concentration of adsorption sites; c, a, and the a, and b, are temperature-dependent coefficients, and n is the number of adsorbates (the a, and b, are not independent, and Langmuir's assumptions and a strict application of thermodynamic consistency requires a, = b, [22,23]). The Langmuir isotherm is shown here in its form for multicomponent adsorption [21]. Freundlich's isotherm is empirical, but can be rationalized with the assumption of a logarithmic distribution of adsorption strengths of sites [24]. Langmuir derived his with the postulate that each molecule occupies one site, with idealized assumptions about adsorption and desorption rates (see farther below), and with the argument that these rates are equal at equilibrium [20]. Derivations from thermodynamics [25,26] and statistics [27] have also been given. In the multicomponent Langmuir isotherm 2.15, each denominator term b,p, is proportional to the number of sites occupied by the respective species i, scaled so that the leading "1" is proportional to the number of unoccupied sites. Also, if a species dissociates upon adsorption, as does H2 on many catalyst surfaces, its terms in the Langmuir equation appears as square roots. For example, for adsorption of (dissociating) hydrogen in the presence of another adsorbable species A q^ =

'—^.

1 . (b^p^r111 - b,p^

(2-16)

Multilayer adsorption and other complications are accounted for in the more elaborate Brunauer-Emmett-Teller isotherm (BET isotherm) [28], which involves the heats of adsorption of the first and subsequent layers and with which not only Type I behavior can be approximated. In its simplest form, for an unlimited number of adsorption layers and partial pressures well below saturation, the BET equation can be written q. =

'P^'i{Po -P)[^ + (^ - ^)PJP.]

where c is a function of the heats of adsorption and liquefaction, and P^ is the saturation pressure. A detailed derivation and discussion can be found in the book by Holland and Anthony [29], who also give the more complicated equations for limited numbers of layers and any partial pressures. Still another isotherm has been derived by Temkin [30] with the assumption of a decrease of adsorption strength with increasing surface coverage: q, = ^ln[aexp(e„//?7)pj

34

Chapter 2.

Fundamentals

where Qo is the heat of adsorption at zero coverage, and a. and a are constants. Isotherms other than Langmuir's are not readily generalized for multicomponent adsorption and are therefore rarely used in practice in kinetics of heterogeneous systems. An exception is the use of the Temkin isotherm in ammonia synthesis [31]. The isotherm equations shown here are for adsorption from a gas phase. They can also be used for adsorption from a liquid or for liquid-liquid distribution (with concentrations replacing/?i or q, andpj). Distribution coefficients. At any given conditions, adsorption or distribution equilibrium can be characterized by a distribution coefficient, given by the ratio of the concentrations in the two phases (see Section 1.7). Adsorption rates. The most common assumption for adsorption and desorption rates is that used by Langmuir in the derivation of his isotherm equation [20]. These are proportionality of the adsorption rate to the partial pressure or concentration of the species and the number of still unoccupied adsorption sites; and of the desorption rate to the number of sites occupied by the respective species. This is as for a reversible reaction of the species with an unoccupied site: adsorption

(r^)^,^ = k,p.q^

(2.17)

desorption

(r)^^^ = k_.q.

(2.18)

where k, and k^, are the respective rate coefficients. For single-component adsorption, their ratio equals the adsorption equilibrium constant.

Summary Although reactions differ widely in their kinetics in a manner that is unpredictable unless their mechanisms are known, rate equations of elementary steps obey simple laws of statistics: They are power laws with integer reaction orders that can be directly deduced from molecularities. Rate equations are conventionally written in terms of concentrations. This is an idealization, but substitution of thermodynamic activities for concentrations is not a proper way to account for nonidealities and may even make matters worse. The temperature dependence of rate coefficients of elementary steps generally follows the Arrhenius equation in good approximation. That of apparent rate coefficients of empirical rate equations of multistep reactions, however, may deviate in several ways: The activation energy may be negative (slower rate at higher temperature) or have different values in different temperature regions. The kinetics of a multistep reaction is described by a set of simultaneous rate equations, one for each participant. The equations are independent of one another in their

References

35

algebraic forms and values of their coefficients. Each equation is the summation of the contributions from all steps in which the respective species participates. Each such contribution is the product of the stoichiometric coefficient of the species, the rate coefficient of the step, and the concentration (or concentrations) of the reactant (or reactants) of the step, raised to the power corresponding to the molecularity. Self-consistency of postulated forward and reverse rate equations and their coefficients can be tested with the principles of thermodynamic consistency and so-called microscopic reversibility. The former invokes the fact that forward and reverse rates must be equal at equilibrium. The latter is for loops of parallel pathways and for catalytic cycles. Thermodynamic consistency allows the reverse rate equation to be constructed from the forward one if at least one of its reaction orders is known, and requires the ratio of the products of the forward and reverse rate coefficients to be equal to the thermodynamic equilibrium constant. Microscopic reversibility leads to several useful conclusions: The products of the clockwise and counter-clockwise rate coefficients of a loop must be equal; the product of the forward rate coefficients of a catalytic cycle must be equal that of the reverse rate coefficients multiplied with the equilibrium constant of the catalyzed reaction; forward and reverse reaction must occur along the same pathway; and the ratio of the products of forward and reverse rate coefficients must be the same along all parallel pathways from same reactants to same products. The latter two rules apply regardless of whether or not any of the reactions are catalytic. Adsorption on solids is characterized by adsorption isotherms that show the concentration of the respective species on the adsorbent as a function of its concentration in the contacting fluid phase. Isotherms may have different shapes, depending on whether adsorption is competitive (adsorbed molecules discourage additional adsorption) or synergistic (they encourage it). The most commonly used isotherm is Langmuir's, applicable to multicomponent adsorption, but restricted to competitive behavior. Adsorption and desorption rates are usually taken as postulated by Langmuir, that is, as for a reversible reaction of the molecule in the fluid phase with an unoccupied adsorption site.

References General references Gl. G2.

G3. G4.

M. Boudart, Kinetics of chemical processes, Prentice-Hall, Englewood Cliffs, 1968. K. G. Denbigh, The principles of chemical equilibrium: with applications in chemistry and chemical engineering, Cambridge University Press, 4th ed., 1981, ISBN 0521236827. H. S. Fogler, Elements of chemical reaction engineering, Prentice-Hall, Englewood Cliffs, 3nd ed., 1999, ISBN 0135317088. G. F. Froment and K. B. Bischoff, Chemical reactor analysis and design, Wiley, New York, 2nd ed., 1990, ISBN 0471510440.

36

Chapter 2.

Fundamentals

G5.

C. G. Hill, h.. An introduction to chemical engineering kinetics & reactor design, Wiley, New York, 1977, ISBN 0471396095. 06. C D . Holland and R. G. Anthony, Fundamentals of chemical reaction engineering, Prentice-Hall, 2nd ed., 1989, ISBN 0133356396. G7. K. J. Laidler, Chemical kinetics, McGraw-Hill, New York, 3rd ed., 1987, ISBN 0060438622. G8. O. Levenspiel, Chemical reaction engineering, Wiley, New York, 3nd ed., 1999, ISBN 047125424X. G9. J. W. Moore and R. G. Pearson, Kinetics and mechanism. A study of homogeneous chemical reactions, Wiley, New York, 3rd ed., 1981, ISBN 0471035580. GIO. C. N. Satterfield, Heterogeneous catalysis in practice, McGraw-Hill, New York, 1980, ISBN 0070548757. Specific references 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

J. A. Christiansen, Reaktionskineticske studier, thesis, University of Copenhagen, 1921, p. 58. F. A. Lindemann, Trans. Faraday Soc, 17 (1922) 598. C. N. Hinshelwood, Proc. Roy. Soc, A 113 (1927) 230. H. Eyring, J. Chem. Phys., 3 (1935) 107. W. F. K Wynne-Jones and H. Eyring, J. Chem. Phys., 3 (1935) 492. S. Glasstone, K. J. Laidler, and H. Eyring, The theory of rate processes, McGrawHill, New York, 1941, Chapter VIII. H. Eyring, S. H. Lin, and S. M. Lin, Basic chemical kinetics, Wiley, New York, 1980, ISBN 0471054968, Chapter 4. Moore and Pearson (ref. G9), Chapter 5. Froment and Bischoff (ref. G4), Section 1.6. J. H. van't Hoff, Etudes de dynamique chimique, Muller, Amsterdam, 1884; see also any text on physical chemistry or chemical engineering. Denbigh (ref. G2), Section 15.5. Hill (ref. G5), Section 5.1.3. R. C. Tolman, Phys. Rev., 23 (1924) 693. R. C. Tolman, The principles of statistical mechanics. Clarendon Press, Oxford, 1938 (reprint Dover, New York, 1979, ISBN 0486638960), p. 163. K. G. Denbigh, The thermodynamics of the steady state, Methuen, London, 1951, p. 31. Denbigh (ref. G2), p. 448. Hill (ref. 05), Section 4.1.5.4. S. Brunauer, L. S. Deming, W. E. Deming, and E. Teller, /. Am. Chem. Soc, 62 (1940) 1723. H. M. F. Freundlich, Kapillarchemie, Leipzig, 1909; Colloid and capillary chemistry, Dutton, New York, 1926.

References 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31.

37

I. Langmuir, J. Am. Chem. Soc, 38 (1916) 2221. I. Langmuir, /. Am, Chem. Soc, 40 (1918) 1361. C. Kemball, E. K. Rideal, and E. A. Guggenheim, Trans. Faraday Soc, 44 (1948) 948. E. J. Franses, F. A. Siddiqui, D. J. Ahn, C.-H. Chang, and N.-H. L. Wang, Langmuir, (1995) 3177. G. D. Halsey and H. S. Taylor, J. Chem. Phys., 15 (1947) 624. M. Volmer, Ann. Phys. Chem., 115 (1925) 253. F. G. Helfferich, Chem. Eng. Educ, 26 (1992) 23 and 51. R. H. Fowler, Proc. Cambridge Phil. Soc, 31 (1935) 260. S. Brunauer, P. H. Emmett, and E. Teller, J. Am. Chem. Soc, 60 (1938) 309. Holland and Anthony (ref. G6), Section 8.LB.3. M. L Temkin, Zh. Fiz. Kim., 15 (1941) 296. M. L Temkin and V. Pyzhov, Zh. Fiz. Khim., 13 (1939) 857.

Chapter 3

Experiments and Their Evaluation

The first order of business in the study of a new reaction in the context of process research and development is to measure reaction rates, establish approximate reaction orders for empirical power-law rate equations, and obtain values of their apparent rate coefficients. This chapter presents a brief and critical overview of laboratory equipment, design of kinetic experiments, and evaluation of their results, including an overview of statistical methods. It is intended as a tour guide for the practical chemist or engineer. More complete and detailed descriptions can be found in standard texts on reaction engineering and kinetics [G1-G8]. 3.1. Research reactors The four principal types of reactors used for bench-scale kinetic studies are batch, continuous stirred-tank (CSTR), tubular, and differential reactors. Which of these to choose is essentially a matter of the reaction conditions, available equipment, and the chemist's or engineer's predilections. The discussion here will focus on facets that pertain specifically to quantitative kinetic studies. A table summarizing the advantages and disadvantages of the different reactor types can be found on page 47. It is a good idea to leave reactor choice to the person to perform the experiments: He'll do best with what he is familiar with. Whatever reactor is chosen, the reaction should be conducted at constant temperature, if at all possible. A variation of temperature in the course of the reaction makes the evaluation much more difficult and less reliable. 3.1.1.

Batch reactors

Typical batch reactors for kinetic studies of liquid-phase reactions at ambient and elevated pressures are shown in Figures 3.1 and 3.2, respectively. They are equipped with stirrer, heating or cooling arrangement, temperature sensor, reflux condenser, and sampling ports. Batch kinetic results are easiest to evaluate if the reactor is operated at constant volume. A variation of reaction volume with conversion contributes to the variation of concentrations with time (even the concentrations of inerts will vary!).

40

Chapter 3. Experiments and their evaluation

Oxygen ft Volitilei Outlet (to condenser)

Coolant Mechtnical Agitator Inlet and Outlet

Oxygen Inlet

Figure 3.1. Batch reactor for kinetic studies of oxidation at ambient pressure (from Blaine and Savage [1]). This not only complicates the evaluation, but also introduces a source of error (see also Section 3.3.4). Volume variations with conversion are large for constant-pressure gas-phase reactions with change in mole number. Here, as a rule, operation at constant volume poses no difficulties. Liquid-phase reactions may also entail volume contraction or expansion. However, these are not related to changes in mole number and can be predicted only if information on partial molar volumes is at hand. Because liquids are essentially incompressible, even at elevated temperature, it is unsafe to conduct liquid-phase reactions without a gas cap in a rigid, closed reactor. Some variation of liquid-phase volume with conversion therefore is apt to occur. Fortunately, the variation at constant temperature is usually so small that it can be neglected in the evaluation or accounted for by a minor correction.

3.1. Research reactors

41

An advantage of batch over continuous stirred-tank and tubular reactors is that a single experiment with samples taken at frequent intervals in effect scans the entire conversion range. A disadvantage is that samples must be taken from the reactor itself. For elucidation of mechanisms, rate data at very low conversions—that is, initial rates—may be highly desirable. They can be obtained more easily from a batch reactor than from a continuous stirred-tank or plug-flow tubular reactor. The reason is that for both of the latter, very high flow rates would be required, or the tubular reactor would have to be equipped with a sampling port near its inlet, a mechanical complication apt to perturb the flow pattern. On the other hand, if the reaction can be confined to a very small flow reactor, a differential reactor operated once-through may be an even better choice for obtaining initial rates (see also Sections 3.1.3 and 3.1.4 farther below). The two main problems in the design of batch reactors for quantitative kinetic studies are establishment of a sharp zero time and effective removal or supply of heat to keep the temperature constant.

Figure 3.2. Commercial batch reactor for kinetic studies at elevated pressures (courtesy Autoclave Engineers, Div. of Snap-tite, Inc., Erie, Pennsylvania, reproduced with permission).

42

Chapter 3. Experiments and their evaluation

A sharp zero time calls for fast attainment of reaction conditions. No significant conversion can be tolerated before the reactants are completely mixed and the mixture is at reaction temperature and pressure. If reaction conditions are reached too slowly, rate coefficients can no longer be calculated accurately. Moreover, while the reactant ratios, temperature, and pressure are not yet on target, reactions different from those under the aimed-for reaction conditions may occur at least to some extent and falsify the results. Especially for reactions at elevated temperatures and pressures and for studies using unstable intermediates as starting materials, fast attainment of reaction conditions usually requires special set-ups. For example, the starting materials may be divided into two unreactive portions, which are pressured and heated separately in different vessels and then mixed as quickly as possible to initiate the reaction. An intermediate that is unstable under reaction conditions may be held in a cooled and pressured vessel, to be injected quickly into a much larger amount of a medium already at reaction temperature and pressure or at a temperature such that mixing of the cold injected fluid with the hot reactor contents produces the desired reaction temperature (see Figure 3.3). Heat transfer is more of a problem in batch reactors than in other types of equipment because of their small surface-to-volume ratio and because rates are high initially. Effective stirring is essential. If the reaction is highly exothermic or endothermic, cooling or heating coils are usually needed. Batch reactors can also be used for studies of gas-liquid reactions. A common procedure, sometimes called "semibatch," is to conduct the reaction as batch with respect to the liquid and bubble the injection gas through at constant composition and vessel pressure. Effective gas-liquid contacting is essential in order to avoid mass-transfer limitation with respect to the reactant supplied by the gas phase. Good ways of introducing the gas are through a fmereactor pore sieve plate or through the hollow shaft and arms of a stirrer. In a constant-volume batch reaction, the concentration variations with time are caused exclusively by chemical reaction, Figure 3.3. Arrangement for inso that jection at zero time (schematic). r = dC/dr (process rate equals rate of change, see Section 1.3). However, the rate cannot be measured directly. What can be measured are concentrations at different times, and a finite-difference approximation is needed to obtain the rate:

3.1, Research reactors r, =

AC /Ar

43

(3.1)

This has important implications for the evaluation of results (see Section 3.3). If a gas-phase reaction with change in mole number is conducted at constant pressure rather than constant volume, the governing equation is (l/y)cW./dr or, in finite-difference form: r. ^

(l/V)AN./At

(3.2)

where V is the (varying) reaction volume. 3.1.2,

Continuous stirred-tank reactors

Continuous stirred-tank reactors (CSTRs, see Figure 3.4) have the advantage over batch reactors that they are easier to keep at constant temperature. This is because they do not see the high initial reaction rate of batch reactors, and because continuous flow into and out of the reactor helps to consume or supply heat. For example, if the reaction is highly exothermic, the entering fluid may be introduced at a temperature below that of the reactor, so that it consumes heat as it is heated up. Another advantage is that samples can be taken from the effluent rather than the reactor itself. A disadvantage is that any single experiment in a CSTR at steady state provides information only at one conversion level, whereas a batch experiment in effect scans an entire conversion range. Reactors like those shown in Figures 3.1 and 3.2 can be used as CSTRs for kinetic studies if equipped with entry and exit ports. The main problems in the design of CSTRs for quantitative kinetic studies are to provide effective mixing and excellent control of the flow rate. As a rule, the evaluation presumes that the entering fluid is instantaneously mixed with the reactor contents, so that the latter is uniform. Incomplete mixing Figure 3.4. Continuous stirred-tank can falsify results. Corrections are reactor (schematic). complex and require detailed knowledge

44

Chapter 3. Experiments and their evaluation

of fluid dynamics in the reactor at hand, information that is hard to come by. Effective mixing of gases is especially difficult. CSTRs are therefore not well suited for gas-phase reactions. However, they are excellent for gas-liquid reactions because the effective mixing also provides good gas-liquid mass transfer. The flow rate enters into the evaluation of the results and should be accurately known and constant. This calls for an excellent flow meter (mass flow meter for gases) and pump (for liquids). The transient behavior of CSTRs is mathematically complex. As a rule, the evaluation is therefore based on operation at steady state. In that state, the difference between inflow and outflow of a reactant (or product) equals the amount consumed (or formed) by reaction. For a reaction without change in fluid density (no volume expansion or contraction), a material balance for species i gives r, = iV/V){Q - O

= (q - O/r

(3.3)

where Vis the reactor volume. Vis the volumetric flow rate, r = V/V'is the reactor space time, Q is the concentration in the reactor and its effluent, and C° is that in the entering fluid. If the fluid density changes significantly, eqn 3.3 must be replaced by r, = (l/V)(F; - F,^)

(3-4)

where F^ = QV is the molar flow rate of species i. As eqns 3.3 and 3.4 show, the rate is obtained directly from the effluent concentration at given reactor volume, feed concentration, and flow rate, without a finite-difference approximation as for a batch reactor in eqn 3.1 or 3.2. This has important implications for the choice of the evaluation method (see Section 3.3). 3,1.3.

Tubular reactors

The principal use of tubular reactors for kinetic studies is as catalytic fixed-bed reactors in heterogeneous catalysis. They are rarely used for quantitative studies of homogeneous reactions because these are difficult to confine sharply to reactors of this type. Compared with batch reactors, tubular reactors have the advantage of easier heat removal or supply: Heat release or consumption at the entry of a tube is as great as in a batch reactor at start, but the surface-to-volume ratio is more favorable, and the entering fluid can help to cool or heat. A disadvantage compared with a batch reactor is that a tube at steady state, like a CSTR, gives information only on the conversion achieved at the conditions of the respective experiment, whereas one single batch experiments with samples taken at frequent intervals scans the entire conversion range.

3.1. Research reactors

45

Unless the reaction involves a solid catalyst, the main problem in the design of tubular reactors for quantitative studies is to confine the reaction sharply to the reactor itself. This requires rapid mixing of the reactants at the entry, and equally excellent quenching of the exiting fluid. Both are easier to achieve for liquid-phase than for gas-phase reactions. Tubular reactors are not suited for gas-liquid reactions because gas sparging would disrupt the flow pattern. Confinement to the reactor is automatic in heterogeneous catalysis, and easily achieved for radiation-induced reactions, e.g., gas-phase chlorination of alkanes initiated by ultraviolet light. The reactor, usually a differential one, then is a cell in which the fluid passes through a beam of light. For quantitative studies, a tubular reactor, like a CSTR, must be operated at a constant and accurately known flow rate. Also, as a rule, the evaluation presumes negligible pressure drop and ideal plug flow. The first of these poses a problem only in gas-phase reactions at very low pressures. The second is an idealization and calls for a reasonably large reactor diameter and, in heterogeneous catalysis, for uniform packing and a diameter of at least twelve catalyst particle diamaters. At constant pressure and granted ideal plug flow, the behavior of a tubular reactor at steady state is mathematically analogous to that of a batch reactor: A volume element of the reaction mixture has no means of knowing whether it is suspended tea bag-style in a batch reactor or rides elevator-style through a tubular reactor; being exposed to the same conditions it behaves in the same way in both cases. As in a batch reactor, what is measured directly are concentrations—here in the effluent—, and a finite-difference approximation is needed to obtain the rate from experiments with different reactor space times and otherwise identical conditions. For a reaction without fluid-density variation: r, = dC./dr

=

AC./Ar

(3.5)

For reactions with fluid-density variations, especially gas-phase reactions with change in mole number, the basic material balance r, = dF./dV with r = V7y° (y"" = volumetric flow rate at inlet) gives, in finite-difference form r, = 3.1.4.

(l/y°)AF./Ar

(3-6)

Differential reactors

In essence, a differential reactor is a tubular reactor operated so that the difference in composition between the entering and exiting fluids is minimal (very small reactor size or very high flow rate). Since the reactor is, in effect, gradientless, its behavior equally resembles that of a continuous stirred-tank reactor at minimal conversion: no difference between tube and CSTR at infinitesimal conversion!

46

Chapter 3. Experiments and their evaluation

Differential reactors are primarily used for studies of heterogeneous catalysis, Homogeneous reactions are very difficult to confine as sharply as necessary to a very small flow reactor. An exception are radiation-induced reactions (see comment in preceding section on tubular reactors). A differential reactor operated with "once-through" flow has the advantage of functioning, like a CSTR, at one well-defined conversion level. Moreover, other than in a CSTR, this level is directly set by the chosen composition of the entering fluid. The principal problem is how to measure the minimal composition difference between the entering and exiting fluids with the required accuracy. Differential reactors with once-through flow thus are especially suited for determination of initial reaction rates, with an entering fluid as yet free of products, but much less so for measurements at moderate to high conversions. For evaluation of results from a once-through differential reactor, the rate is calculated from the difference AQ between the entering and exiting concentrations: r, = (V7V)AC

(3.7)

Since conversion across the reactor is minimal, no correction for possible fluiddensity variation with conversion need be applied. Because of the difficulty of measuring minimal conversions accurately, differential reactors are often operated with recycle instead of once through. A large, constant amount of fluid is rapidly circulated through the reactor /^ ^—^^ and its composition is monitored continuously or at frequent time intervals (see Figure 3.5). Although conversion with each pass through the reactor is reactor Q minimal, it is cumulative and so progresses steadily with time. Thus, like a batch reactor, a differential recycle reactor operated over a prolonged time span scans a wide conversion range. In performance, a differential sample recycle reactor resembles a batch report actor in that conversion progresses with time. The difference is that in a batch reactor the reaction occurs in all of the fluid, whereas in a differential recycle pump reactor it does so only in a small fraction because at any time most of the Figure 3.5. Differential recycle reactor fluid is in the recycle loop rather than (schematic). the reactor itself. The rate is given by

J

Al

3.1. Research reactors Table 3.1. Advantages and disadvantages of types of research reactors. Advantages

Disadvantages

Batch Reactors simple, inexpensive, easy to operate can scan wide conversion ranges in single experiment can be operated at constant fluid density results easy to evaluate reactors for elevated temperature and pressure commercially available

sharp zero time calls for fast attainment of reaction conditions (mixing, heating, pressuring, etc.) samples must be taken from reactor demands for heat transfer very with conversion

Continuous Stirred-Tank Reactors easy temperature control sampling from reactor effluent efficient gas-liquid contacting possible

excellent mixing required exact, constant flow rate required new steady state must be attained for each measurement at different conversion level not well suited for gas-phase reactions

Tubular Flow Reactors excellent for heterogeneous catalysis minimum attrition of solid catalyst sampling from reactor effluent temperamre control easier than in batch

exact, constant flow rate required new steady state must be attained for each measurement at different conversion level correction for any deviations from plug flow complicates evaluation not well suited for homogeneous reactions

Differential Reactors Once-Through Operation excellent for measurement of initial rates sampling from reactor effluent easy temperature control minimum attrition of solid catalyst

accurate measurement of minimal product concentrations required exact, constant flow rate required not well suited for homogeneous reactions uniform flow pattern difficult to achieve

Recycle Operation sampling from reactor effluent easy temperature control minimum attrition of solid catalyst can scan wide conversion ranges in single experiment

exact, constant flow rate required not well suited for homogeneous reactions uniform flow pattern difficult to achieve long running times required

48

Chapter 3. Experiments and their evaluation r, = (V'.o.a/^.acJdq/df

=

(V,,,/V,,,„„)Aq/Ar

(3.8)

for batch and tubular reactors, a finite-difference approximation is required for calculation of rates from concentrations observed in the recycle loop. 3.1.5,

Techniques for fast reactions

The great majority of reactions of practical interest have half times of the order of minutes or hours. Occasionally, however, faster reactions must be studied. Special reactor designs then are called for to ensure conditions that permit an accurate quantitative evaluation. Reactions with half times Solutions from syringes of the order of fractions of a second can still be handled with fairly simple laboratory equipment. A convenient setup for liquid-phase reactions is shown in Figure 3.6. The Outlet key component is a small mixing chamber into which the reactants enter at high velocity. The mixture then passes into a capillary. Its composition is monitored (e.g., by its spectrum) at a series of distances from the Cathode-ray tube mixing chamber. A calibraFigure 3.6. Flow reactor for study of fast reactions tion correlates distance along the capillary to time elapsed (schematic, from Caldin and Trowse [2]). from entry into the mixing chamber. Each distance thus corresponds to a reaction time as in a batch reactor, and a single experiment with scans at different distances along the capillary can be used to cover a wide conversion range. Since the technique's introduction by Hartridge and Roughton in 1923 [3], miniaturization of equipment and sensors has pushed its limits down into the millisecond range. More popular for reactions in the millisecond range is the "stopped-flow" technique [4], which consumes less fluid and for which conmiercial equipment is available (e.g., see Figure 3.7). Over an extremely short time span, liquids are injected into and mixed in a small reaction chamber, and the composition of the mixture is then monitored continuously or analyzed after short, preset reaction times. In contrast to the Hartridge-Roughton reactor, a stopped-flow reactor functions essentially as a micro-batch reactor.

3.2. Analytical support

49

IGNITER MONOCHROMATOR

ARC LAMP

jf

5Q

ARC LAMP POWER SUPPLY

Motor Drive

Syringe Chamber

COMPUTER

KinTek STOPPED-FLOW CONTROL UNIT

Valve

PMT

n:

KEYBOARD

OBSERVATION CELL

M-'-''^^imm%: Figure 3.7. Commercial stopped-flow system for fast liquid-phase reactions (courtesy KinTek Corporation, reproduced with permission).

50

Chapter 3. Experiments and their evaluation

Reactions with very short half times, of the order of or less than microseconds, call for special techniques. Most of these are based on relaxation, pioneered by Eigen [5]: A reaction mixture at equilibrium is exposed to a practically instantaneous pulse or step change in conditions, and the response is monitored by recording an instantaneously measurable quantity such as electric conductivity or a spectrum. For example, the response to a sudden pressure pulse can be observed in a shock tube [6,7], the temperature of a liquid can be raised stepwise by application of a very strong electric field [5,8] or a laser pulse [9], a dissociation might be triggered by flash photolysis [10], and nuclear magnetic resonance allows relaxation in response to a magnetic field to be measured [11,12]. Today, laser pulse and photon-echo techniques allow events on time scales as short as femtoseconds to be observed [13-15]. A comprehensive account of older work has been given by Bernasconi [16]. Shorter surveys can be found in Laidler's and Connors' books [G6,17]. 3.2.

Analytical support

Most industrial laboratories have excellent analytical groups. The bottleneck in a kinetic study program thus is almost always in the kinetic experiments themselves, not in the analytical work-up. Analysis usually is routine and conducted by technicians without higher academic training, the kinetic experiments are not. In the design of an experimental program, the experienced development chemist or engineer is stingy with experiments, but lavish with their analytical evaluation. Thorough analytical support can help to get the job done with fewer kinetic experiments and thus in a shorter time. It is not unusual for a single batch experiment to involve sampling at a dozen different conversion levels and analysis of each sample for ten to fifteen components. What particular analytical method is best suited depends on the chemistry at hand, and only very few general suggestions can be offered. The most broadly applicable technique is gas chromatography [18-21]. Gas chromatography also has the advantage that it can usually be set up side-by-side with the kinetic equipment, requires only very small samples, can provide results within minutes or less, and can be handled by a laboratory technician. It shares one disadvantage with most other techniques: Sampling, whether automatic or manual, removes material from the reaction environment and exposes it to usually quite different conditions. Effective quenching and quick processing is essential but, even so, the composition of the sample might change before its analysis is complete. Spectrophotometric methods [22-26], where applicable, have the advantage of allowing a sample to be studied under reaction conditions. Ultraviolet, visible, or infrared spectra can be taken at elevated temperatures and pressures [27]. If a reaction is so fast that sampling is not practical, a pump-around loop with spectrophotometric cell can be hooked to the reactor or, even better, a probe can

3.3. Reaction orders and apparent rate coefficients

51

be inserted into the reactor [28]. Spectra taken under reaction conditions have proved invaluable in the elucidation of catalyst chemistry of homogeneous reactions catalyzed by complexes of transition-metal ions (see also Section 7.5). As a quantitative tool, however, spectrophotometry is not on a par with chromatography. Another excellent method of quantitative analysis makes use of tagging with radioactive or stable isotopes [29,30]. Detection is by radiation, mass spectroscopy [31], or nuclear magnetic resonance [32]. Unfortunately, experiments involving radioactivity require elaborate precautions and special laboratory rooms to avoid contamination and meet legal requirements, and mass-spectroscopic and NMR equipment is expensive and may not be available. It is always desirable to have an immediate and convenient indication pressure reducing how a reaction is progressing. In this valve way, unproductive experiments can be aborted before much time is lost, and equipment failures etc. become inmiediately apparent. In a reaction that convessel sumes a gas, this can be done by monitoring the gas consumption. In a gas reaction in a constant-volume batch reactor reactor, the pressure drop is a direct measure of the reaction rate and can be followed at a glance on the pressure gauge or the strip chart of a pressure Figure 3.8. Set-up for monitoring gas recorder. For a reaction at constant consumption in a reaction at constant pressure, a set-up as shown in Figure pressure (schematic). 3.8 can be used: A surge vessel at higher pressure is connected to the reactor via a pressure-reducing valve that admits gas on demand to the reactor to keep its pressure constant. Gas consumption is observed as pressure drop in the surge vessel. The latter is repressured from time to time as needed to keep its pressure above that set at the reducing valve. Other useful techniques include mass spectroscopy [31], spin-state labeling [33], nuclear magnetic resonance [32,34,35], X-ray fluorescence [36], neutron activation [37], and electron spin resonance [38,39], among others. The interested reader is referred to an excellent review by Tolman and Faller [40].

3.3.

Reaction orders and apparent rate coefficients

Most reactions of practical interest do not obey power-law rate equations, and so have no exact and constant reaction orders. Nevertheless, their observed rates can be fitted more or less well to empirical power laws as a first step toward the

52

Chapter 3. Experiments and their evaluation

elucidation of their mechanisms and establishment of more accurate rate equations. On the other hand, since the results of such fitting are not expected to be more than approximations for use as temporary expedients, any attempt at high accuracy would be wasted (see also Section 7.1.1). Recommended plots for determination of reaction orders and apparent rate coefficients are summarized in Table 3.2 (page 54). Instead of the plots, corresponding numerical methods can be used, as will be examined in more detail with examples in the next sections. A guideline for choosing a suitable method is to avoid approximations as much as possible. Thus, plots of concentrations, or functions of concentrations, versus time or reactor space time are preferred for evaluation of experiments with batch, tubular, and differential recycle reactors, in which concentrations are directly measured and rates can only be obtained by a fmite-difference approximation (see eqns 3.1, 3.2, 3.5, 3.6, and 3.8). On the other hand, plots of rates, or functions of rates, versus concentrations or functions of concentrations serve equally well for evaluation of results from CSTRs or differential reactors without recycle (gradientless reactors), where concentrations and rates are related to one another by algebraic equations that involve no approximations (see eqns 3.3, 3.4, or 3.7). It has become customary to classify evaluation methods as "differential" or "integral." These terms stem from a time when practically all experiments were conducted in batch reactors, so that rates had to be found by differentiation of concentration-versus-time data, and the calculation of concentrations from postulated rate equations required integration. The terms do not fit the work-up of data from gradientless reactors such as CSTRs, in which rates and concentrations are related to one another by algebraic equations requiring no calculus, and are therefore avoided here. 3.3.1.

Irreversible reactions with single reactant reaction: A —• product(s) (3.9) power-law rate equation: -r^ = Kp^C^ (3.10) Rate methods. The simplest and most common method suited for evaluation of data from gradientless reactors such as CSTRs is based on trial and error. The rate -r^ is calculated with eqn 3.3 or 3.4, and results from different runs are plotted versus CA with some likely value of the order, n. If the resulting plot is "concave up," the guess of n was too low; if it is "convex up," the guess was too high (for a guess of « = 1, see Figure 3.9). The value of n is adjusted accordingly until a reasonably straight line with zero intercept is obtained. The apparent rate coefficient, k^^^, is found as the slope of that line. Since accurate values of n are not sought, very few tries will suffice. A second rate method found in most texts is based on the rate equation 3.10 in its logarithmic form:

3.3. Reaction orders and apparent rate coefficients In(-r^)

= ln/:^pp + n\nC^

53

(3.11)

so that a plot of ln(—r^) versus In Q gives a straight line with slope n and intercept In /Capp. Because of the logarithmic scales, this method is much less reliable. Note that there also is a concentration method for evaluation of CSTR results (see farther below). Rate methods are much less suited for evaluation of results from batch, tubular, and differential recycle reactors because, for these, the rate must be obtained by a finite-difference approximation (see eqns 3.1, 3.5, and 3.8). In particular, the method based on eqn 3.11 should not be used for such a purpose because of its high sensitivity Figure 3.9. Straight-line plot for firstorder reaction in CSTR, also showing to even minor experimental errors. curves given by reactions of higher and For ease of reference, plots lower orders. are summarized in Table 3.2. Concentration methods. For results from batch, tubular, or differential recycle reactors at constant fluid density, the most common procedures are based on integrated forms of the rate equation. For a first-order, constant-volume batch reaction A —• P:

= -K^^t

(3.12)

CA = CAexp(-A:,ppO

(3.13)

\n{CJCl) or so that

(3.14)

-r. and for the product: ~K,,t

(3.15)

Cp = C ( l - exp(-^,ppO)

(3.16)

ln(l-Cp/CA°)

=

For constant-volume batch reactions with orders other than first, one has (l/Q)"-' - ( 1 / C ) " - '

= {n-\)K^t

(3.17)

(not applicable to first order, error-sensitive for orders close to first). For tubular reactors and reactions with no fluid-density variation, the reactor space time, r = V/F, takes the place of the actual time, t.

Chapter 3, Experiments and their evaluation

54

Table, 3.2. Common straight-line plots for determination of reaction orders and apparent rate coefficients. RATE PLOTS (applicable to all reactor types) react ion rate

A—• products ^A

^

^

^ ^

-''A

reaction rate

%pp^A

Slope

w^A+«gB 4-... —• produc

^ ^

/c^pp

intercept 0

^y^

-^A

Slope k^^ intercept 0

0

0 ()

0

c:

^A ^B

CONCENTRATION PLOTS constant-volume batch reac tion rate

reaction rate

A —• products "''A ~

continuous stirred tank

^app^A

A—• products "''A ~

^app^A

slope -k

reaction

A ~ ^ products

intercept 1

reaction rate

A—• products

slope /r^pp intercept 0

slope (/i-l)/:3pp

intercept l/CQ)"'

reaction rate

A + B + ... —> products '''A

%PP^A^B

slope k^ic; - c;) intercept InCQ/Q)

plug-flow tube If fluid density does not vary with conversion, use same plots as for constant-volume batch, with reactor space time r substimted for time t.

3.3. Reaction orders and apparent rate coefficients

55

Equations 3.15 and 3.16 presume that no product is present initially; otherwise, Cp must be replaced by Cp - Cp. Also, for stoichiometrics AZ^A —• WpP, multiply CA in eqns 3.15 and 3.16 with In-

A2p/«J.

Straight-line plots are summarized in Table 3.2. Reactions of higher or lower order than assumed give concave-up or convex-up curves, respectively, as shown for a first-order plot in Figure 3.10. In eqn 3.12 and Figure Figure 3.10. First-order concentration 3.10, \n{C^IC^) can be replaced plot for constant-volume batch reactor, by In CA- If so, the intercept is at also showing curvatures given by reIn CA , but the slope of the curve actions of higher and lower orders. remains the same. A more general form of eqn 3.12 for first-order reactions in batch or differential recycle reactors is ln(l - A ) =

(3.18)

^app^

where/A = 1 - Nj^/N^ is the fractional conversion (see definition 1.4). In this form the equation is valid regardless of any fluid-density variations. For a continuous stirred-tank reactor at steady state and a reaction with no fluid-density variation, the evaluation can be based on concentrations instead of rates. Replacement of -r^ by KppC^ in eqn 3.3 for A yields (C; - C,)/C^

= k^^^T

(3.19)

For a first-order reaction A —• P, eqn 3.19 reduces to C/CA

=

(3.20)

1 + ^appT

The respective straight-line plots are included in Table 3.1. From eqn 3.20 and Cp = CA - CA one can obtain

c^ A so that

^A

^

1 -^ ^.nn-^ ' app

-r' A

^ A ^app ^

(3.21)

1 + K.J app

= r' P = 1 ^appCA + ^.nn^ app

(3.22)

56

Chapter 3. Experiments and their evaluation

Incorrect guesses of the reaction order lead to curved plots. As in Figures 3.9 and 3.10, orders higher or lower than assumed give concave-up or convex-up curves, respectively. Fractional-life methods. If a reaction is known to be first order and at constant fluid density, its apparent rate coefficient can be found very quickly. For batch and differential recycle reactors, the relationship between the rate coefficient and the time ty required for all but a fraction y of the reactant to be consumed is ^app =

-dny)/^,

(3-23)

Specifically, for the half-time t^i2 and the time t^,^ (e = 2.71828): ^app = (ln2)/r.,,

and

k^^ = \lt,,^

(3.24)

For tubular reactors and reactions with no fluid-density change, the same equations apply with reactor space time substituted for real time. For a CSTR at steady state: ^app = V r

""'^

^=PP = ^''^"

^^-^^^

yj

as follows from the material balance -^A = ^appQ = ( Q - C^)!? with Q = yC^. Some texts describe fractional-life methods for reactions other than first order and with more than one reactant. However, the effort their application requires is not in proportion to the limited objectives of the evaluation in the present context. 3.3.2.

Irreversible reactions with two or more reactants

reaction:

AZ^Q + n^C^ + ... —• product(s)

power-law rate equation:

— r^ M^ = /Capp Q'" Q" • • •

(3.26) (3.27)

Rate methods. As for reactions with a single reactant, the most primitive and convenient rate method is to guess and then adjust the reaction orders until an approximately straight-line plot of -r^ versus the rate versus C^ Q ... is obtained (see Table 3.2). For example, if a reaction is expected to be second order in A and first order in B, the rate would be plotted versus C^C^. For first guesses of reaction orders, the available data can be subdivided so that the concentration of all but one participant are the same within each group. The dependence of the rate on the concentration of that participant then is an indication of the respective order. Often, the evaluation need not be carried beyond this stage. If the development chemist or engineer has the opportunity to design the kinetic experiments, the evaluation can often be simplified by establishment of pseudo-orders, as shown farther below.

3.3. Reaction orders and apparent rate coefficients

57

As for reactions with a single reactant, rate methods are not well suited for evaluation of results from batch, tubular, and differential recycle reactors. For complex mechanisms, however, no convenient concentration methods might exist, so that a rate method could be the lesser of two evils. Concentration methods. Results from constant volume-batch or differential recycle reactors for a reaction n^A + n^B —• product(s) give a linear plot of ln(CB / Q ) versus t with slope k^p^in^^^C^ - n^Cj^) and intercept ln(C^/C^) if the reaction is first order in both A and B [41]. However, this procedure is inapplicable to experiments with stoichiometric amounts of A and B (i.e., n^C^ = n^^C^), and is highly error-sensitive for amounts that are close to or extremely different from stoichiometric. For tubular reactors, the same procedure can be used with reactor space time r substituted for time t. Pseudo- and overall reaction orders. Kinetic textbooks describe other, more complicated methods applicable to other forms of proposed rate equations, mostly for evaluation of results from batch reactors. However, if the development chemist or engineer can commission experiments—as opposed to having to evaluate existing data—he can often save himself much effort by determination of pseudo- and overall reaction orders. For example, for a reaction A + B —• product(s) with power-law rate equation —r^ = KppC/^C^, three series of experiments suggest themselves: (1) with a very large excess of A over B, (2) with a very large excess of B over A, and (3) with stoichiometric amounts of A and B. In Series (1), the concentration of A decreases only insignificantly from its starting value while B becomes depleted, so that the rate equation 3.27 can be reformulated with a new, approximately constant coefficient k^: if C3 « C^,

- r , = k,C^

(k, ^ k^^^ {ClT = const.)

Under these condition, the kinetic behavior is analogous to that of «'th order with single reactant B, and n can be determined as for such a reaction. The reaction is said to be pseudo-n'th order in B. Similarly, under the conditions of Series (2), if C^ « C„

-r^ ^ KC^

(k^ ^ k^^^ (C^r = const.)

The reaction is pseudo-m'th order in A, and m can be determined as for a reaction with A as the sole reactant. Lastly, in Series (3), the concentrations of A and B remain equal because both reactants were charged in equal amounts and are consumed at the same rate. With Cj^ = C^, the rate equation 3.27 becomes

58

Chapter 3. Experiments and their evaluation

n+m

1

^ n-^m

The reaction behavior now is analogous to that of a reaction of order m-\-n with A (or B) as the sole reactant, and the overall order m+n can be determined as for such a reaction. Thus, two of the three series of experiments suffice to determine all reaction orders and the apparent rate coefficient. Pseudo-orders can also be established by semi-batch experiments in which one or several reactants are replenished to keep their concentration or concentrations constant. In this way, the reaction order or orders with respect to the other reactant or reactants can be established. The order with respect to a replenished reactant can then be found by comparison of the rates obtained with different concentrations of that reactant. This method is particularly convenient for gas-liquid reactions such as homogeneous oxidation, halogenation, hydrogenation, hydroformylation, and hydrocyanation. Here, the gaseous reactant or reactants can be admitted to the reactor on demand as they are consumed, or by bubbling an excess of the gas at constant pressure and composition through the reactor. Some later examples of network elucidation are of cases in which this method was used. 3.3.3.

Reversible reactions

reaction:

^t^Q 4- n^C^ + ... <—• /?pCp + nQC^ -f- ... (3.28)

power-law rate equation:

-r^ /«A = K Q C^--• - K C/ CQ^..

(3.29)

Reversible reactions are best considered as consisting of forward and reverse steps. Such step combinations will be examined in Section 5.1. Simple plots suited for evaluation exist only for first order-first order reactions A <—• P (see Figure 5.2). In more complex situations, the best approach often is to isolate the forward from the reverse reaction. This can be done to some extent with experiments that measure initial rates. Initial rates. As long as conversion has still remained insignificant, the reverse reaction contributes little to the net rate: For a short initial time span the reaction behaves as though it were irreversible. This allows the previously described methods for irreversible reactions to be used to establish the reactions orders and rate coefficient of the forward reaction. Often, initial reverse rates can be measured in the same fashion by using the reaction products as starting materials. Otherwise, a likely reverse rate equation and its coefficient can be established with the principle of thermodynamic consistency (see Section 2.5.1). The procedure based on initial rates is not entirely reliable. The dependence on reactant concentrations is not necessarily the same for the initial rate as for the rate in the course of the reaction. This is because products building up during the reaction may slow the rate, but initial rates are measured in their absence.

3.4. Numerical work-up, error recognition, and reliability

59

3.3.4. Gas-phase reactions at constant pressure The chemist or engineer designing his experiments to establish quantitative kinetics of gas-phase reactions will do his best to look for constant-volume equipment. However, occasionally he may have to work with data obtained at constant pressure. The complication here is that a change in mole number affects the reaction volume and, thereby, the concentrations of the participants, distorting their histories from which reaction orders and rate coefficients are deduced: Fluid-density variation disguises kinetics and must be corrected for. Reaction time and rate in a batch reactor are related by [42,43] t

=

""1( - 0^(v/v^)

(3.30)

where f^ is the fractional conversion of the limiting reactant, V is the (variable) reaction volume, and y° is that volume at start. A strictly analogous equation with reactor space time r instead of reaction time t applies to an ideal plug-flow reactor. Standard practice is to postulate a linear variation of the volume with conversion; this is correct if the ideal gas laws are obeyed, any intermediates remain at very low concentrations, and the reaction does not switch pathways. If so: V/V° = 1 + 5^/^

(3.31)

where 5^ = (N^^__^ - A^°)W°

(3.32)

is the relative variation in mole number or volume resulting from complete conversion of the limiting reactant A [44,45]. For example, if the mole number N increases from 2 to 3, causing a 50% increase in volume, d^ is +1/2; or if the mole number decreases from 3 to 2, 5^ is - 1 / 3 . The partial pressures of the participants, corrected for volume variation, are 0 PA

-

1 - / A ^—7^ » 1 ^ ^A/A

PA -J

Pi

_ -

P-^ ~ (^i/^A)PA/A :; ^—7 1 + ^A/A

Equation 3.30 may have to be integrated numerically, but that is no problem at a time when even pocket calculators can integrate. Since reaction orders hardly ever need to be determined exactly, the following general rule may come in handy: Volume expansion, if not corrected for, lets the reaction order appear farther from first than it is; volume contraction has the opposite effect.

60

Chapter 3. Experiments and their evaluation

For example, if the mole number doubles, the concentration histories up to about 50% conversion look much the same for a reaction of order 1.5 at constant pressure as for one of order 1.8 at constant volume; or for order 0.5 at constant pressure as for order 0.2 at constant volume. If the reaction is first order, a volume variation does not distort the concentration history. Unfortunately, the conditions that guarantee the volume to vary linearly with conversion are often not met in reactions with complex mechanisms. This makes the evaluation harder and less reliable and provides an even stronger incentive to conduct the work at constant volume. 3.4.

Numerical work-up, error recognition, and reliability

Today, easy-to-use computer software for evaluation of experimental kinetic data to obtain reaction orders and apparent rate coefficients by regression is commercially available and will be discussed in the next section. Short of resorting to computerized evaluation, the chemist or engineer may want to stay with more pedestrian long-hand numerical methods that are perfectly adequate for the immediate purpose of obtaining approximate orders and coefficients in power-law equations that serve only as temporary expedients. In essence, these methods are the numerical equivalents of the plots that were shown earlier in Table 3.2. Two detailed examples will illustrate typical work-up, including error recognition and relative reliability of experimental data. Example 3.1. Decomposition of hexaphenylethane in constant-volume batch. The decomposition of hexaphenylethane into two triphenyimethyl radicals (QH5)3C-C(QH5)3

•

2(QH5)3C-

in liquid chloroform at 0° C has been studied in a batch reactor [46]. The volume variation is negligible. The results are shown in the first two columns of Table 3.2 (next page). A quick first inspection of the experimental data suggests that the times required to reduce the fractional reactant concentration by one half from 1.0 to 0.5, from 0.5 to 0.25, and from 0.25 to 0.125 are about the same: close to 190 seconds. This indicates that, at least in good approximation, the reaction is first order, as its chemistry makes one expect. Application of eqn 3.23 to a data point in the midconversion range, say, at 209 seconds, gives ^' app = -(ln0.471)/209 = 3.6*10-3s-^ Since the reaction may indeed occur in a single step and so be accurately first order, a more thorough evaluation with eqn 3.12 may be desired. This is done in columns 3 and 4 of Table 3.3. In each row, the logarithm of the fractional concen-

3.4.

Numerical work-up, error recognition, and reliability

61

Table 3.3. Decomposition of hexaphenylethane in constant-volume batch reactor; global evaluation. time

concentration

k

-IiKC^/C/)

=

'^app

-\n(CJC^yt

s

1 ^^ '^ •' 35.4 54.0 174 209 313 367 434 584 759

0.941 .883 .824 .530 .471 .324 .265 .206 .118 .059

3.49*10-2 1

0.0608 .1244 .1936 .6349 .7529 1.1270 .3280 .5799 2.1371 .8302

3.51 3.59 3.65 3.60 3.60 3.62 3.64 3.66 3.73

Table 3.4. Decomposition of hexaphenylethane in constant-volume batch reactor; point-by-point evaluation.

time

In ( C , / C / )

Aln(Q/Q°)

s 0 17.4 35.4 54.0 174 209 313 367 434 584 759

Ar s

1.000 0.941 .883 .824 .530 .471 .324 .265 .206 .118 .059

=

k

concentration

1

0 - 0.0608 .1244 .1936 .6349 .7529 - 1.1270 .3280 .5799 - 2.1371 .8302

- 0.0608 .0636 .0692 .4413 .1180 .3741 .2010 .2519 .5572 .6931

17.4 18.0 18.6 120 35 104 54 67 150 175

app

-Aln(Q/C/) Ar

3.49*10-^ 3.53 3.72 3.68 3.37 3.60 3.72 3.76 3.71 3.96

tration is calculated and divided by the time to give the apparent first-order rate coefficient. The coefficient is seen to be constant within reason, confirming that the reaction is indeed first order. This "global" evaluation procedure is the numerical equivalent of plotting \n{C^IC^) versus t and obtaining a value of ^^pp for each data point as the negative slope of the straight line from the origin to that point. The values

62

Chapter 3. Experiments and their evaluation so found are characteristic of the time intervals from start of the experiment to the moments the respective samples were taken. Confirming the quick first estimate, the average value is 3.61*10"^ s"^ Alternatively, the original data can be evaluated point by point by calculating the differences between successive values of InCQ IC^) and dividing them by the differences between the respective times, as is done in Table 3.4. This procedure is the numerical equivalent of connecting successive points in the plot of InCQ /C/) versus t by straight lines and obtaining k^^^ values for the respective time intervals from the slopes of the connecting lines. The average value found by this procedure is a little higher, 3.65*10"^ s"^ For comparison, the interested reader might want to evaluate the experimental data with the method based on eqn 3.11. He would find a reaction order that varies between 0.24 and 1.24 in point-by-point evaluation, and between 0.52 and 1.06 in global evaluation. This demonstrates the high sensitivity of this method to even minor experimental errors.

It is instructive to compare the two evaluation methods used in Tables 3.3 and 3.4. The global method gives a more nearly constant value of the rate coefficient because each value is, in effect, averaged over the entire time span from start to sampling. In contrast, the point-by-point method is particularly suited for spotting suspect data points. For instance, the coefficient value for the time interval from 174 to 209 seconds is out of line, making one suspect an error in one of the two times or concentrations used for calculating it. Apart from that value, the greatest deviations from the average are at the lowest and highest conversions. In general, these results are the least reliable and might be given less weight. Batch data at very short reaction times suffer most from a lack of ideal sharpness of the zero time. In fact, if all listed sampling times were decreased by half a second, the coefficient values from the global method would vary less and be more in line with those from the point-by-point method (in the latter, only the first value would change, to 3.60*10"^ s~^). A half-second delay might well have resulted from the finite time needed to bring the reactant from liquid-nitrogen temperature to 0° C. At the other end of the scale, results at very high conversion suffer most from analytical inaccuracies. This is because the relative error of analytical methods is largest at low concentrations. If the product concentration had been determined, the situation would be no better because the evaluation at high conversion then would depend on small differences between large concentration values. Even if the time scale were adjusted by half a second and suspect values omitted, a very slight tendency of the rate coefficient to increase with conversion remains. A reaction order slightly less than one would give a better fit. A deviation in the opposite direction could be explained with the assumption that the reaction is not entirely irreversible (see also Section 5.1.1), but the trend observed

3.4. Numerical work-up, error recognition, and reliability

63

here has no plausible mechanistic explanation. If real at all, which is doubtful, it might reflect a slight variation of the rate coefficient with composition of the medium. It might seem as though averaging the coefficient values obtained with the point-by-point method, without the first and last, would give the best final value. This is not true. In fact, with evenly spaced data points, the result would be the same as that of global evaluation of the entire respective time interval [47]. This is because such an evaluation in effect averages the data of the time span automatically. A simple way to get a good value is to apply the point-by-point method to two data points that are not suspect and are far apart, at moderately low and moderately high conversions; in our example, say, to those for 54 and 584 seconds. The value so obtained is 3.67*10"^ s~^ Example 3.2. Decomposition of a herbicide in a CSTR."^ Accelerated life tests of an experimental herbicide in oleic solution in a 500-ml CSTR at 125° C give the results shown in the first three columns of Table 3.5. The analytical method is said to have a possible error of ±0.005 M; however, the feed concentration C^ is not affected because a stock solution of exactly known composition was used. The flow rate is accurate to ±0.02 mL s~^ Fluid-density variation is negligible. Table 3.5. V mLs"' 1.0 3.0 6.0 12.0

Decomposition of herbicide in 500-ml CSTR at 125° C. -r, M

M

0.951

0.660 .828 .884 .912

= (V/V)AC, Ms-' 5.82*10-' 7.38 8.04 9.36

J. (1st) '^app

_

h (2nd) '^app

_ -

-rACd' M-'s-' 8.82*10-'* 8.91 9.10 10.26

1.34*10-3 1.08 1.03 1.13

The rates calculated with eqn 3.3 are listed in the fourth column, and rate coefficients for first- and second-order reaction in the fifth and sixth columns, respectively. It may seem difficult to distinguish between first and second order on the basis of the coefficients alone. This is because all data points except the first are in a narrow range of low conversion. Distinction between orders hinges on linearities of plots, and, over narrow ranges, even curved plots are not far from linear. However, the results for 12 mL s"^ flow rate are the least reliable because the coefficients are calculated from a small difference between large concentration values.

To emphasize error recognition, the conditions and data of this example have been altered.

64

Chapter 3. Experiments and their evaluation

Taking the analytical error into account, the first-order coefficient at that flow rate could be as small as 8.90*10"'^ s~^ (and as large as 11.64*10"'^ s"^), bringing it more in line with the others. On the other hand, even maximum analytical and flow rate errors cannot account for a second-order coefficient of less than 1.27*10"^ M~^ s"^ at the lowest flow rate. The reaction thus appears to be first rather than second order, as is more plausible anyway, with a rate coefficient of about 9*10"^^ s"^ Alternatively, the rate coefficients in Table 3.5 could be calculated from eqns 3.20 and 3.19, with identical results. This would be the equivalent of graphical evaluation with a concentration rather than rate method. The interested reader might also want to use the method based on eqn 3.11 for comparison. With point-by-point evaluation he would find reactions orders ranging from 0.55 to 1.41, pointing toward first order but demonstrating a sensitivity to experimental errors even in evaluation of results from gradientless reactors. Results at longer reactor space times (higher conversions) would have been desirable. If limitations of the pump did not allow flow rates of less than 1.0 mL s~^ a larger reactor should have been used. General conclusions from the two examples can be summarized as follows: •

In batch reactions, results at very low and very high conversions are the least reliable. For very low conversion, the zero-time error is large. For very high conversion, evaluation involves either very low concentrations, for which the relative analytical error is largest, or very small differences between large concentration values. If at all possible, experiments should therefore be designed to include good coverage of the mid-conversion range.

•

Clear distinction between reaction orders requires results from a broad conversion range.

•

Point-by-point evaluation is best for spotting suspect data because it does not average over broad conversion ranges.

•

If quick inspection shows a reaction at constant volume to be first order, an approximate value of the rate coefficient can be found immediately by application of the fractional-life equation 3.24 or 3.25 to a data point in the mid-conversion range.

•

Evaluation of a pair of points at moderately low and moderately high conversions gives a good average value of the rate coefficient.

•

In flow reactors, most pumps operate with largest relative error at low flow rates.

•

Confidence limits should be considered in the work-up.

3.5. Overview of statistical methods

65

3.5. Overview of statistical methods * Statistics is an invaluable tool, though one that has no intelligence of its own. It is fast and does to perfection and almost any desired accuracy what it is designed for and instructed to do, but it does not think. Apply it with caution. Useful sources are the books by Box et al. [48], Draper and Smith [49], Cook and Weisberg [50], Chatterjee et al. [51], and DeCoursey [52] and the programs SOLVER [53] and SYSTAT [54].

Statistics can effectively be used to provide a best estimate of the value of a repeatedly measured variable, establish the reliability of such an estimate (confidence interval), estimate parameter values of a model from experimental data, help to discriminate between rival models on the basis of goodness of fit, and guard against acceptance of a model whose superior fit may well be due to chance. It can also help to design experimental data gathering to be most efficient [48]. On the other hand, statistics alone cannot be relied upon to identify or verify reaction pathways or mechanisms. Mean Value, Standard Deviation, Variance, and Confidence Intervals. Every experiment is subject to uncertainty. Statistical methods are the best tool for determining these uncertainties, which, in turn, can be used for parameter estimation. Consider a variable that is accessible to experimental measurement. It could be the mass of a catalyst or reactant, a flow rate, an electronic signal from an analytical instrument, etc. Measurements, subject to error, cannot deliver the true value of the variable. All they can do is to provide an estimate. If the same variable y is measured n times, the best estimate is the arithmetic mean, y, of the measured values

= l.j^y.

(3.33)

(This assumes all measured values are equally reliable. Otherwise, the values can be assigned different weights, Wj.) How close is this best estimate expected to be to the true value? The statistic most commonly used to assess the variability of the measurements is the standard deviation, s, calculated as (3.34)

_n-lti The square of the standard deviation, s^, is termed the variance. This section is largely based on material provided by P. E. Savage, 2002.

66

Chapter 3. Experiments and their evaluation

It is often desirable to state the variability in the data in terms of a quantitative confidence interval, C.I., calculated as C.I. =

(3.35)

±t^^sn-

where ^st is Student's t, which can be read from a table for a specified level of confidence and number of degrees of freedom [55]. For the purpose at hand, the number of degrees of freedom, is n - 1, where n is the number of experimental measurements. The level of confidence can be chosen at will, but 95% is customary. This level of confidence implies that there is a 95% chance that the true value lies within the indicated range. Table 3.6. Student's t distribution for 95% confidence interval. degrees of freedom

2

4.303 Student's t J

3

4

5

6

8

10

12

3.182

2.776

2.571

2.447

2.306

2.228

2.179

Example 3.3. The rate coefficients k in the adjacent table were calculated by Savage and Smith [56] from six independent measurements of the oxidation of acetic acid in supercritical water at 440 °C in a batch microreactor. The mean value of k and its 95% confidence interval are obtained as follows. The mean value of k is

15 2.131

yt(mol/Lys-^ 7.99*10-' 8.26 8.26 6.75 7.50 9.17

lA , -Y^k.^ - 7.99 + 8.26+8.26 +_6.75+7.50+9.17 *,^.4 1 0 ' 7.99 * 10-' Accordingly, the best estimate of the true value of k is 7.99*10"' (mol/L)"^^s-^ To determine the 95% confidence interval, the standard deviation must first be calculated: ^{k.,-kY s =

i:(^j-7.99* 10-')2 i=l

n- 1

= 0.814*10-'

The confidence interval can now be determined from Student's table. For five degrees of freedom (six measurements) and 95% confidence, the entry is Tst = 2.571. Therefore the 95 % confidence interval is

3.5.

(2.571)(0.814 * IQ--^)

C.I..

=

67

Overview of statistical methods

^st^«"

= 0.85 * 10-^

The rate coefficient is then reported as k = (7.99±0.85)*10-^ (mol/L)-^^s-^ With the widespread availability of hand-held calculators, computer spreadsheets, and statistical software packages for PCs, there is no need to do the calculations outlined above by hand. Propagation of Errors. A quantity of interest may be a function of several variables subject to measurement. The quantity could be, say, the reaction rate in a CSTR, calculated from measurements of a concentration and a flow rate. Uncertainties in both measurements affect the uncertainty of the calculated rate. For a quantity 3^ that is a function of m variables x^, X2, ..., x^, whose errors are uncorrelated, independent, and known, the variance can be calculated from the variances of the variables x'

<^ = E

dX:

ol

dlny dlnx

or

(3.36)

(a is used instead of s because this is a "true" variance, calculated from known experimental uncertainties). The variance is the square of the absolute uncertainty. Ay or AJCJ, in the value of the variable. The relative uncertainties Axjx^ and Ay/y thus are a^j /x-, and Oy ly. The propagation of errors in x-, can be expressed as 2

Ay

dlny

-E

dlnXj

^1 . ^' J

(3.37)

Example 3.4. The relative uncertainty of the rate -r^, calculated from the catalyst mass W, feed flow rate F^, and fractional conversion/^ in a CSTR, is to be established. The relative uncertainties in W, F^, and/A are ±0.5%, ±5%, and ±2%, respectively. The rate is calculated from the design equation

ln(-rj

or Thus

ain(-o ainF;

= 1,

ain(-rj d\nf.

1,

InF^ + In/,

InW

ain(-rj _ dlnW

and the relative uncertainty in the rate is found to be A(-0 -r.

((-0.005)2+ (0.05)2+(0.02)2)

^ (0.00293)^

0.05

= 5

68

Chapter 3. Experiments and their evaluation

Parameter estimation, objective functions, and maximum likelihood. A common problem is to find the set of values of the parameters (e.g., rate coefficients, equilibrium constants, etc.) that best fit the observed values of a measurable variable (e.g., a rate). Parameter estimation entails the calculation of the most likely value from a set of experimental values which are subject to random error. This is achieved by minimizing the value of an objective function that expresses the discrepancy between observed values (yXhs of the variable and the values (yi)caic calculated with sets of values of the model's parameters. The best set is that which gives the minimum value of the objective function. The most widely used objective function involves the sum of squares of differences between observed and calculated parameter values i=l

where Wj is a statistical weight that can be assigned to incorporate the relative magnitude of the experimental uncertainty of the respective measurement. Leastsquares fitting employs this objective function and estimates the most likely parameter values, provided the measurement errors are independent and normally distributed with constant standard deviation. A least-squares fit is a maximum likelihood estimation. Given the experimental data, the likelihood of the set of parameters values is linked to the probability of having obtained the data with that set. That probability can be calculated, and maximum likelihood estimation varies the parameter values to determine the most likely ones by maximizing this probability. The probability of having obtained the entire set of m data points data with any given set of parameters is given by the product of the probabilities of having obtained each individual data point. If the error is random and normally distributed, the individual probability that the chosen set of parameters gives the respective data point within an error Ay is \2

Pj =

((yi)obs -

exp

(yi)a

2af

Ay

The product, P, of these individual probabilities is maximized when the sum of their negative logarithms is minimized. This sum is

_ i ^ p ^ j . [(yXbs - (yi)caic]' ^ ^^^^ Ay i=i

2 a?

i=i

o.yjlif

The second sum contains only quantities not affected by the variation of the model parameters, so that - I n P is minimized by minimizing the objective function

3.5. Overview of statistical methods

[v^i/obs

p '- E

vi/calcj

69

/'2 2Q)

o'

This is the familiar least-squares form, with the statistical weight of each point being the reciprocal of its variance. The value of F is a measure of the goodness of fit. [For constant o^, F is the chi-square statistic]. If the measurements have constant absolute error, say, ± 0.1 °C at temperatures of 20 as well as 1000°C, the standard deviation and the variance are the same for each, regardless of the magnitude of the measured value. It is often more realistic to assume a constant relative error, especially for chemical analyses. In this case, the standard deviations are proportional to the measured values. The statistical weight of any point—that is, the reciprocal of the variance—then is the reciprocal of the square of the measured value. Thus, the objective functions for constant and relative error are: m

for constant absolute error

^ = Y^ [O'i)obs ~ 0'i)caic]^

for constant relative error

^ "" J^

O'i)obs -

(yi) ca

(3.40)

(3.41)

(yi)oi

The behavior of the error may not be known. Instead of postulating constant absolute or relative error, one can then determine the variances experimentally at some extra effort. This requires each experiment to be replicated n times under the same conditions, so that a mean value of y and variance s^ can be calculated for those conditions with eqns 3.33 and 3.34. The objective function then is

i=l

j=l

S^

Deviations from normal distribution are common and require different objective functions. Often, however, the shape of the error distribution is unknown. This is why the least-squares (chi-square) estimation of parameter sets is so frequently used. Example 3.5. Rate data for a liquid-phase reaction A + B—• P have been obtained in a continuous stirred-tank reactor with various concentrations of A and B. The rate is found to increase with increasing concentrations of A and B, but less than proportionately (see "observed" columns in Tables 3.7 and 3.8, next page),

70

Chapter 3. Experiments and their evaluation

Table 3.7. Nonlinear least-squares fit of coefficients in rate eqn 3.41, assuming constant absolute error (objective function [ratCobs ~ ratCcaJ^). parameter estimates k^ = 0.00986, itb = 1.12242, it, = 1.97536

1 ^'

CB

rate* 10^ [n[lolL-'s-^]

[mol L-']

[mol L']

observed

calculated

0.1 0.1 0.1 0.3 0.5 0.1 0.7 1.0 0.1 0.1 2.0 10

0.1 0.3 0.5 0.1 0.1 1.0 0.1 0.1 2.0 10 0.1 0.1

7.5 18 23 20 29 31 35 42 40 47 56 80

7.528 17.356 23.477 19.280 28.031 31.934 34.802 42.501 38.950 47.254 57.286 79.377

square error *10^^ 0.0008 0.4149 0.2276 0.5189 0.9383 0.8727 0.0393 0.2508 1.1035 0.0647 1.6538 0.3881 E = 5.3499

Table 3.8. Nonlinear least-squares fit of coefficients in rate eqn 3.41, assuming constant relative error (objective function [(rate^bs - ratecaic)/rateobs]^)parameter estimates k^ =0.010079, k^ = 1.157419, yt, = 2.037194

1 ^"

CB

rate* 10^ [nlolL'^s"^]

[mol L-']

[mol L-^]

observed

0.1 0.1 0.1 0.3 0.5 0.1 0.7 1.0 0.1 0.1 2.0 10

0.1 0.3 0.5 0.1 0.1 1.0 0.1 0.1 2.0 10 0.1 0.1

7.5 18 23 20 29 31 35 42 40 47 56 80

calculated 7.639 17.509 23.611 19.496 28.273 31.967 35.042 42.687 38.839 46.906 57.291 78.878

square error *10' 0.3421 0.7428 0.7067 0.6354 0.6281 0.9731 0.0014 0.2676 0.8423 0.0040 0.5311 0.1966 E = 5.8712

3,5.

71

Overview of statistical methods

that is, the apparent reaction orders are between zero and plus one. No reasonable mechanism produces a rate equation with fractional exponents. Preferably, rate data should be fitted with equations having integer exponents and a denominator with additive terms and a leading " 1 . " Such rate behavior is common for multistep reactions (see one-plus rate equations, Section 7.2). This suggests a trial rate equation of the form ^a ^ A

-r.

(3.42)

^B

1 ^ k,C^ ^ k^C^

The results of nonlinear regression to fit its three coefficients k^, k^, and k^ with commercial software are shown in Tables 3.7 and 3.8, respectively (opposite page). A comparison of observed and calculated rates shows reasonable agreement. The best-fit coefficient values differ slightly, depending on whether constant absolute or constant relative error is assumed. This is because high values of the variable carry more weight for constant absolute error, and low values do so for constant relative error. Its much smaller sum-of-squares error does not indicate that constant absolute error is the better assumption. Such a comparison is not possible because it hinges on the accuracy of the rate equation and because the numerical values of the two errors sums have different dimensions. For example, if the rates were given per minute rather than second, the numbers in the square-error column would be 3600 times as large for absolute error, which has the dimension of a rate squared, while remaining the same for relative error, which is dimensionless. The coefficients of the rate equation can also be fitted by multiple linear regression. For this purpose, eqn 3.42 is brought into its linear form: ^A^B

1

(3.43)

^c. ^ ^Q

Regression versus Q and Cg gives \lk^ as intercept and k^lk^ and k^lk^ as coefficients. A comparison of the best-fit coefficient values obtained with the three different methods is shown below. Table 3.9. Best-fit values of coefficients in eqn 3.42, obtained with nonlinear and linear regression. nonlinear regression parameters

\

K K

constant absolute 1 error 0.00986 1.12242 1.97536

constant relative error

linear regression

0.010079 1.157419 2.037194

0.00994 1.124

2.002

1

72

Chapter 3. Experiments and their evaluation

Error correlation and co-variance. Parameters may affect one another, as do the pre-exponential factor A and the activation energy E^ in the Arrhenius equation. If so, the errors are correlated, and a co-variance must be included in the errorpropagation formula. For a quantity y that is a function of two parameters x^ and ^2, the CO-variance is denoted o^^^^, and the variance in y is 2

^

0^ '^1

dy dx^

r

2

+ a" ^2

dy dx^

+ laXX

dy dx^

•\

dy dx^

(3.44)

Caveats. In complicated situation, statistics may zero in on a false optimum, that is, a local rather than absolute minimum of the objective function. Moreover, statistics by itself delivers uncritically the set of variables that gives the best numerical fit to the not entirely accurate input data, and it may happen that the set of correct coefficient values is different and is disregarded because its fit is not quite as good. Therefore, it is good practice to run regression programs with any constraints on the variables that the physics of the system may suggest. Lastly, the quality of a statistical correlation alone cannot be taken as an indication of correctness of the assumptions. For example, a model with slightly larger, but random error is more likely correct than its rival with smaller, but systematic error, and primitive statistics programs do not take this into account. As Connors puts it, "the human eye, in combination with chemical knowledge, is a more subtle qualitative judge of data than is regression analysis" [57]. Sumimary Bench-scale kinetic experiments can be conducted in batch, continuous stirred-tank (CSTR), tubular plug-flow, or differential reactors. The last of these can be operated with once-through flow or recycle. Advantages and disadvantages of the various types are discussed. In particular: Batch reactors are inexpensive, but require attention to rapid attainment of reaction conditions at start; CSTRs are excellent for gas-liquid, but less so for gas-phase reactions; tubular reactors are especially suited for reactions of heterogeneous catalysis; and differential reactors operated "once through" are best for measurement of initial rates. Fast reactions (in the millisecond to second range) require special reactors with efficient mixing chambers. Faster reactions (down to the microsecond range and below) call for special techniques; most of these are based on relaxation after an equilibrium state has been disturbed by an instantaneous pulse or step variation of conditions. With laser and photon-echo techniques the range has been extended down to femtoseconds. Analytical support is essential. A wise program shifts its work load as much as possible from kinetic experiments, which required highly skilled personnel, to analytical routine work that can be performed by technicians. Gas chromatography and spectra are the most useful tools.

References

73

Reactions orders and rate coefficients can be established with methods that use either rate or concentration data. Batch, tubular plug-flow, and differential recycle reactors yield concentrations as directly measured quantities, and calculation of rates requires finite-difference approximations. To avoid these, concentration methods should be used. In contrast, continuous stirred-tank reactors allow rates to be calculated from material balances without approximation. Here, evaluation based on rates is equally suited. In reactions with two or more reactants, reaction orders are best established by experiments with stoichiometric initial concentrations (to give the overall reaction order) and with all but one of the reactants in large excess (to give the order with respect to that minority reactant). For reversible reactions, measurement of the initial rate allows the forward reaction to be studied largely unencumbered by the reverse one. Rate equations of multistep reactions often are not power laws. Reaction orders therefore may vary with concentrations, and attempts at accurate determination would be futile. Unless reaction orders are integers, or integer multiples of one half in special cases, only their ranges (such as between zero and plus one) are of interest. Long-hand numerical work-up, error recognition, and reliability is illustrated with two examples: thermal decomposition of hexaphenylethane and an accelerated decomposition test of an herbicide. An overview of statistical methods covers mean values, standard deviation, variance, confidence intervals, Student's / distribution, error propagation, parameter estimation, objective functions, and maximum likelihood. References General references Gl. G2. G3. G4. G5. G6. G7. G8.

S. W. Churchill, The interpretation and use of rate data: the rate concept, revised printing, Hemisphere, New York, 1979, ISBN 0891162348. K. A. Connors, Chemical kinetics: the study of reaction rates in solution, VCH Publishers, New York, 1990, ISBN 3527218223, Chapters 2 and 3. J. H. Espenson, Chemical kinetics and reaction mechanisms, McGraw-Hill, New York, 2nd ed., 1995, ISBN 0070202605, Section 2-8. H. S. Fogler, Elements of chemical reaction engineering, Prentice-Hall, Englewood Cliffs, 3nd ed., 1999, ISBN 0135317088, Chapter 5. C. G. Hill, Jr., An introduction to chemical engineering kinetics and reactor design, Wiley, New York, 1977, ISBN 0471396095, Chapter 3. K. J. Laidler, Chemical kinetics. Harper & Row, New York, 3rd ed., 1987, ISBN 0060438622, Chapter 2. O. Levenspiel, Chemical reaction engineering, Wiley, New York, 3nd ed., 1999, ISBN 047125424X, Chapters 2 and 3. M. J. Pilling and P. W. Seakins, Reaction kinetics, Oxford University Press, Oxford, 1995, ISBN 0198555288, Chapter 2.

74

Chapter 3. Experiments and their evaluation

Specific references 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

13. 14.

15. 16. 17. 18. 19. 20.

21. 22.

23.

S. Blaine and P. E. Savage, Ind. Eng. Chem. Res., 30 (1991) 2185. E. F. Caldin and F. W. Trowse, Disc. Faraday Soc, 17 (1954) 133. H. Hartridge and F. J. W. Roughton, Proc. Roy. Soc, A 104 (1923) 376. B. Chance, J. Franklin Inst., 229 (1940) 455, 613, and 737. M. Eigen, Disc. Faraday Soc, 17 (1954) 194; 24 (1957) 25. J. N. Bradley, Shock waves in chemistry and physics, Methuen, London, 1962. A. G. Gay don and I. R. Hurle, The shock tube in high-temperature chemical physics, Chapman & Hall, London, 1963. M. Eigen and L. de Mayer, Z. Elektrochem., 59 (1955) 986. B. Marcandalli, G. Stange, and J. F. Holzworth, /. Chem. Soc, Faraday Trans. I, 84 (1988) 2807. R. G. W. Norrish, Proc. Chem. Soc, 1958, 247. E. D. Becker, High resolution NMR: theory and chemical applications. Academic Press, San Diego, 3rd ed., 2000, ISBN 0120846624. T. J. Swift, Investigation of rates and mechanisms of reactions. Vol. 6 of Techniques in chemistry, G. G. Hammes, ed., Wiley, New York, 3rd ed., 1974, ISBN 0471830968, Part II, Chapter XII. G. R. Fleming, Chemical applications of ultrafast spectroscopy, Oxford University Press, 1986, ISBN 0195036441. V. Sundstrom, ed., Femtochemistry andfemtobiology: ultrafast reaction dynamics at atomic-scale resolution, Nobel Symposium 101, Imperial College Press, London, 1997, ISBN 1860940390. C. Rulliere, ed.. Femtosecond laser pulses: principles and experiments, Springer, Berlin, 1998, ISBN 3540636633. C. F. Bemasconi, Relaxation kinetics. Academic Press, New York, 1976, ISBN 0120929503. Connors (ref. G2), Section 4.2. W. Jennings, E. Mittlefehldt, and P. P. Stremple, Analytical gas chromatography. Academic Press, San Diego, 2nd ed., 1997, ISBN 012384357X. H. M. McNair and J. M. Miller, Basic gas chromatography, Wiley, New York, 1998, ISBN 0471172618. C. F. Poole and S. K. Poole, Gas chromatography, in Chromatography, E. Heftmann, ed., Elsevier, Amsterdam, 5th ed., 1992, Vol. A, ISBN 0444882367, Chapter 9. R. P. W. Scott, Introduction to analytical gas chromatography, Dekker, New York, 2nd ed., 1998, ISBN 0824700163. D. A. Harris and C. L. Bashford, eds.. Spectrophotometry and spectrofluorimetry, in The practical approach series, Oxford University Press, 1987, ISBN 0947946691. K. Feinstein, Guide to spectroscopic identification of organic compounds, CRC Press, Boca Raton, 1995, ISBN 0849394481.

References 24. 25. 26. 27. 28. 29.

30. 31. 32. 33. 34. 35. 36. 37.

38.

39. 40.

41. 42. 43. 44. 45. 46. 47.

75

W. O. George and P. S. Mclntyre, Infrared spectroscopy, Wiley, New York, 1987, ISBN 0471913839. B. C. Smith, Infrared spectral interpretation: a systematic approach, CRC Press, Boca Raton, 1999, ISBN 0849324637. D. H. Williams and I. Fleming, spectroscopic methods in organic chemistry, McGraw-Hill, New York, 5th ed., 1995, ISBN 0077091477. R. V. Kastrup, J. S. Merola, and A. A. Oswald, Adv. Chem. Ser, 196 (1981) 43. G. Dotzlaw and M. D. Weiss, Chem. Eng. Progr., 89(9) (1993) 42. D. Gibbons and D. A. Lambie, Radiochemical methods in analysis, in Comprehensive analytical chemistry, C. L. Wilson and D. W. Wilson, eds., Elsevier, Amsterdam, ISBN 0444417354, Vol. IIC, 1971, Chapter II. M. F. L'Annunziata, Radionuclide tracers: their detection and measurement, Academic Press, London, 1987, ISBN 0124362524. R. M. Smith, Understanding mass spectra: a basic approach, Wiley, New York, 1999, ISBN 0471297046. R. K. Harris and B. E. Mann, eds., NMR and the periodic table. Academic Press, London, 1978, ISBN 0123276500. J. W. Faller, Adv. Organomet. Chem., 16 (1977) 211. P. S. Pregosin, ed., Transition metal nuclear magnetic resonance, Elsevier, Amsterdam, 1991, ISBN 044488176X. U. Weber and H. Thiele, NMR spectroscopy: modem spectral analysis, WileyVCH, New York, 1998, ISBN 3527288287. R. Jenkins, X-ray fluorescence spectrometry, Wiley, New York, 2nd ed. 1999, ISBN 0471299421. D. de Soete, R. Gijbels, and J. Hoste, Neutron activation analysis, Wiley, New York, 1972, ISBN 0471203904 (out of print, reprint Books on Demand announced, ISBN 0608176125). C. P. Poole, Jr., Electron spin resonance: a comprehensive treatise on experimental techniques, Wiley, New York, 1967, ISBN 047069386X; 2nd ed. reprint Dover, New York, 1997, ISBN 0486694445. J. E. Wertz and J. R. Bolton, Electron spin resonance: elementary theory and practical applications, McGraw-Hill, New York, 1972, ISBN 0070694540. C. A. Tolman and J. W. Faller, in Homogeneous catalysis with metal phosphine complexes, L. M. Pignolet, ed., Perseus, Cambridge, New York, 1983, ISBN 03064121IX, Chapter 2. Hill(ref. G5), p. 29. Levenspiel (ref. G7), Chapter 5. Hill (ref. G5), Section 8.1. Levenspiel (ref. G7), pp. 71-72. Hill (ref. G5), Section 3.1.2. Hill (ref. G5), p. 66. W. E. Roseveare, /. Am. Chem. Soc, 53 (1931) 1651.

76 48.

49. 50. 51. 52. 53. 54. 55.

56. 57.

Chapter 3. Experiments and their evaluation G. E. P. Box, W. G. Hunter, and J. S. Hunter, Statistics for experimenters: an introduction to design, data analysis, and model building, Wiley, 1978, ISBN 0471093157. N. R. Draper and H. Smith, Applied regression analysis, Wiley, New York, 3rd ed., 1998, ISBN 0471170828. R. D. Cook and S. Weisberg, Applied regression including computing and graphics, Wiley, New York, 1999, ISBN 04713171IX. S. Chatterjee, A. S. Hadi, and B. Price, Regression analysis by example, Wiley, New York, 3rd ed., 2000, ISBN 0471319465. W. J. DeCoursey, Statistics and probability for engineering applications with Microsocft Excel, Elsevier, Burlington, 2003, ISBN 0750676183. B. S. Gottfried, Spreadsheet tools for engineers: EXCEL 2000 Version, McGrawHill, New York, 2000, ISBN 0072321660, 5o/ver program. SYSTAT (from Systat, Inc., Evanston, IL, USA). R. A. Fisher and F. Yates, Statistical Tables, Oliver & Boyd, Edinburgh, 1938; 3bndgtdi2ib\esmHandbookof Chemistry and Physics, D. R. Lide, editor-in-chief, CRC Press, Boca Raton, 80* ed., 1999-2000, ISBN 0849304806, p. A-105, and any handbooks of mathematical table or statistics. P. E. Savage and M. A. Smith, Environ. Sci. TechnoL, 29 (1995) 216. Connors (ref. G2), p. 52.

Chapter 4 Tools for Reduction of Complexity Chemical reactions may involve large numbers of steps and participants and thus many simultaneous rate equations, all with their temperature-dependent coefficients. The full set of rate equations is easily compiled as shown in Section 2.4, and to obtain solutions by numerical computation poses no serious problems. With a large number of equations, however, it may become too much of a task to verify the proposed network and obtain values for all its coefficients. Therefore, every available tool must be brought to bear to reduce the bulk of mathematics, and that without unacceptable sacrifice in accuracy. The present chapter critically reviews the principal tools for such a purpose: stoichiometric constraints and the concepts of a rate-controlling step, quasi-equilibrium steps, and quasi-stationary states. Other tools useful in catalysis, chain reactions, and polymerization will be discussed in the context of those reactions (see Sections 8.5.1, 9.3, 10.3, and 11.4.1). In essence, the use of these tools allows rate equations to be eliminated from the set. In the best of all worlds, mathematics can be reduced to a single rate equation and simple algebraic relationships between the concentrations of the participants. More often, several simultaneous rate equations remain, but the reduced set is nevertheless much more convenient for network elucidation and modeling. For ease of reference, the present chapter explains the nature of the tools and the errors introduced by their use. Applications are a major topic in later chapters. 4.1.

Stoichiometric constraints

The use of stoichiometric constraints is so elementary that standard texts don't even mention it. Yet, the application to large networks with many intermediates warrants a brief review. To start with an almost trivial example, let us look at a single-step reaction A + B
78

Chapter 4, Tools for reduction of complexity

Each of these two equalities allows one rate equation to be expressed in terms of another, so that any one of the three equations—that for A, B, or P—suffices to describe the reaction. In reactions with intermediates, the constraints are usually best formulated in terms of conservation of atoms or groups. Hydroformylation, examined in examples in several later chapters, can illustrate this. The reaction is olefin

+ H2, CO

• alcohol

+ aldehyde, IHj -H.O

+ H2

paraffin

+ H2

• aldehyde

heavy ends

(arrows are multistep pathways with trace-level intermediates not shown here). Four constraints suggest themselves. Three are conservation requirements: skeletal carbon

-r^,^ = r^^^ + r^,, + r^,^ + 2r,^

hydrogen

-r^^ = V ^ ^aid + 2r^,^ + 4r,^ - r^^o

oxygen

-r^o

=

^ald + ^alc ^ ^ e ^

^

and the fourth expresses that heavy ends and water are formed in equimolar amounts:

The four constraints can be used to eliminate four rate equations, say, those of olefin, H2, CO, and water, leaving only those of paraffin, aldehyde, alcohol, and heavy ends. If all reactants, intermediates, and products are accounted for, the stoichiometric constraints are exact. They become approximations only if any participants are disregarded. This is done routinely with trace-level species, where it entails no significant loss of accuracy. 4.2.

Rate-controlling steps

A concept traditionally held in highest esteem, especially by chemists, is that of a rate-controlling step. The idea is that the overall rate is determined by the slowest step in the mechanism, the "bottleneck." For a linear pathway in which one step is much slower than all others, this may allow the set of simultaneous rate equations for all participants to be reduced to one single rate equation of formation of the product or products.

4.2. Rate-controlling steps

79

Let us be specific about what is meant when steps are loosely called "slow" or "fast." These terms refer not to rates (number of molecules reacting per unit time and unit volume), but to how soon a given reactant molecule can be expected to react {reaction probability). "Hard" and "easy" would be better terms, but the slow-fast notation has become too ingrained to be changed. As an example, consider a pathway A —^ K —• P

(4.1)

in which the second step has a much larger rate coefficient than does the first. Just about every molecule of K will then decay to P almost as soon as it is formed from A. The rates of the steps A —• K and K —• P then are practically equal. Yet, the second step is said to be much "faster" than the first because the molecules of K have a much higher probability to react than do those of A. Highway traffic provides a good analogy. A bottleneck caused, say, by road work, is "slow," the unobstructed road is "fast"—this although only the cars that have passed the bottleneck are actually traveling on the road beyond, so that traffic flow in cars per unit time is no heavier anyplace there than at the bottleneck. 4.2.1.

Pathways of irreversible steps

The simplest example of a chemical reaction with rate-controlling step is the pathway 4.1 with very different rate coefficients of the two steps. Solving the rate equations for A, K, and P in a constant-volume batch reaction with only A present at start and without assumptions about the rate coefficients, one obtains for the rate of product formation: ^AK^KP ^ A

.(exp(-/:AKO - exp(-/:KpO)

(4-2)

and for a continuous stirred-tank reactor at steady state with only A in the feed: ^AK^KP^A''" (1

+ ^AKT)(1 +

(A

Vi

^KPT)

(see Section 5.4 for more detail). In the limiting cases of extremely different rate coefficients and at not too short times or reactor space times, this reduces to: Case I (k^ « k^p)

Case II (k^^ « k^^)

batch:

r^ = /^AK^A^^PC-^AKO

batch:

r^ = kj^pC^Qxpi-k^^t)

CSTR:

rp = k^^C^/d ^ k^^r)

CSTR:

r^ = ^KPCAV(1 + /TKPT)

Chapter 4, Tools for reduction of complexity

80

Case I first step slow

Case II second step slow (/CKP

K

« k^nj

K

0 0

Liquid from container A dribbles right through K into P. Rate of flow into P is controlled by rate of flow out of A (slow step).

Container A empties quickly into K, from where liquid dribbles slowly into P. Rate of flow into P is controlled by rate of flow out of K (slow step).

Figure 4.1. Hydrodynamic analogue of batch reaction A - Kfirst or second step.

P with slow

A comparison of the rate equations in the two limiting cases with eqns 3.14 and 3.22 shows them to be the same as for a single step A —• P with rate coefficient /r^K ii^ Case I and k^^ in Case II, that is, with the rate coefficient of the slow step in both cases. A hydrodynamic analogue for the batch reaction is shown in Figure 4.1. The error incurred with the approximation of a rate-controlling step can be estimated by comparing the approximate rates above with the exact ones given by eqns 4.2 and 4.3. For the two-step reaction 4.1, the relative error—difference between approximate rate and exact rate, divided by the latter—turns out to be for batch reactor for CSTR at steady state

(4.4)

4,2. Rate-controlling steps

81

The approximation for the batch reactor at low conversion involves an additional error caused by the omission of the exponential involving the coefficient of the fast step. Provided the rate coefficients differ by at least an order of magnitude, this error is below 1% for reaction times longer than 5lkf^^{. A„, < 0.01

if

t > 5lk,^^,

(4.5)

The approximation overestimates the rate if based on true rate coefficients (but not, of course, if empirical coefficients in the approximate rate equation are fitted to experimental results). The argument can be extended to pathways with any number of irreversible steps. In general terms: If one step in a sequence of irreversible steps is much slower than all others, its rate coefficient alone controls the rate of product formation.

However, this rule is valid only if the steps are sequential and irreversible. Moreover: • If the steps have widely different activation energies, rate control may shift fi'om one step to another with change in temperature (see Section 13.1.2). •

If the steps are of different reaction orders, rate control may shift from a high-order step at low conversion to a low-order step at high conversion. This is because higher-order steps slow down more strongly with decreasing reactant concentrations.

In a multistep irreversible reaction, each disregarded fast step introduces an error as discussed above for the two-step pathway: The errors are cumulative. 4.2.2.

Pathways with reversible steps

The concept of rate-controlling steps can be formulated more generally for pathways that include reversible steps. The idea here is that the slow reaction of the ratecontrolling step gives all others time to approach equilibrium very closely: If the forward and reverse rate coefficients of a step in the pathway of a reversible reaction are much smaller than all others, the other steps are practically in equilibrium.

82

Chapter 4. Tools for reduction of complexity

In loose parlance, steps that are much faster than others are ^ often said to be in equilibrium. ^ ^~ However, as long as the overall g^gp 3 .^ reaction proceeds, net conversion in the forward direction must step 4 =±: occur through each of its steps: In each, the forward reaction must Figure 4.2. Forward and reverse rates of outweigh the reverse reaction by steps in a four-step reaction with rate control what amounts to the net conby second step (schematic). Forward direcversion of the overall reaction, tion left to right; lengths of arrows indicate ^^^^ precludes ideal equilibrium, magnitudes of rates. . f. ,_ ^ j j ^ m which forward and reverse rates are exactly equal. However, if a step is very fast in both directions, its forward and reverse rates differ very little on a relative basis, and so can be equated in good approximation. This situation is illustrated in Figure 4.2 for a four-step reaction with rate control by a slow, second step. For precision's sake, the fast steps are said to be in quasi-equilibrium (see also next section). Contrary to what is true for pathways with no reversible steps, fast reversible steps preceding the rate-controlling step do affect the rate of product formation. The rate depends on the equilibrium constants of such steps and thus on the ratios of their forward and reverse rate coefficients. Specifically, equilibria favoring the reverse reaction reduce the rate. As for pathways of irreversible steps, the more general rule allowing for reversibility remains restricted to sequential steps, and rate control may shift to a different step with temperature or concentration. Each quasi-equilibrium step introduces an error into the approximation as will be discussed in the next section. In its more general form, the concept of the rate-controlling step is often used in catalysis (see Sections 8.5.2 and 9.2). ^^^P ^

-

Example 4.1. Nitration of aromatics. The Gillespie-Ingold mechanism [1-3] of nitration of aromatic compounds according to ArH + HNO3 — • ArNOj + H2O

is HB

ArH

HNO, ^3 ^ ^ v^» ^ H2NO3+ < B-

^

NO2+ -^^H,0 ^2^

B• ArN02H+ — ^ - v^^ ArNO, ^.x.v.2

(4.6)

HB

where ArH is the aromatic, and HB and B" are an added strong acid and its anion. (HB may be nitric acid itself, but usually sulfuric acid is added; the step in which ArNOsH"^ is formed may actually involve radicals as intermediates [4].)

4.2. Rate-controlling steps

83

For aromatics of low reactivity, the reaction of the nitronium ion, NOj^, with the aromatic is the slow, rate-controlling step. The rate then is

where /:23 is the rate coefficient of the slow, third step. (Note that the subsequent steps are irreversible, so that their quasi-equilibrium is shifted entirely to the end product, ArNOj, as though the slow step were producing it directly.) The nitronium ion is formed in the first two steps with combined stoichiometry HNO3 + HB < • NOj" + B- + H2O The quasi-equilibrium condition (mass-action law) for the reaction with this stoichiometry is

c

c c .

— = Al- = const.

C

C

°^

Solved for the nitronium ion concentration:

r.

(4.8)

= KJ^!^

NO2

02 ^

/^

^Hp^B-

The dissociation equilibrium of the acid HB with dissociation constant ^HB is: CH-CB-/C„B

=

^HB

(4.9)

and can be used to express the ratio Q B /Cg- in terms of the hydrogen ion concentration. Combining eqns 4.7 to 4.9 and taking the concentration of water (the solvent) as constant, one finds: Accordingly, the reaction is first order with respect to each the aromatic, undissociated nitric acid, and hydrogen ion. On the other hand, aromatics of high reactivity react with the nitronium ion much more rapidly than the latter is formed. Here, the formation of that ion becomes the slow, rate-controlling step: ^ArNO, =

^n^n^no;

(4.11)

where ki2 is the rate coefficient of the slow, second step. The protonated nitric acid is formed in the first, fast step, with stoichiometry HNO3 + HB <

• H2N03^ -f B-

and quasi-equilibrium condition C C —L-J C C ^HNO,

^HB

=

Kf,, = const. ^^

(4.12)

84

Chapter 4. Tools for reduction of complexity Solved for the concentration of the protonated nitric acid: ^H,No;

=

^OI^HNO^^HB'Q-

(4-13)

The resulting rate equation in this case is, with eqn 4.9, '-ArNo, = KCnmC^-

(K - KiKoJK^y^)

(4-14)

This rate equation differs from eqn 4.10 in that the reaction is now of zero order with respect to the aromatic. The other orders are the same as before. For aromatics of intermediate reactivity the rates of the second and third steps may be comparable, so that no single step is rate-controlling. A better tool then is needed to obtain a closed-form rate equation. This case will be examined in Example 4.4 in Section 4.4 and Example 6.1 in Section 6.3. The postulate of quasi-equilibrium of all steps except a single one that controls the rate is very powerful. It reduces the mathematical complexity of kinetics even of large networks to quite simple rate equations and has become a favorite tool, employed today in a great majority of publications on kinetics of multistep reactions, sometimes uncritically. In many cases, a sharp distinction between fast and slow steps cannot be justified. A more general approach that avoids the postulate of a single rate-controlling step and contains the results obtained with it as special cases will be described in Sections 4.4 and 6.3 and widely used in later chapters. 4.3.

Quasi-equilibrium steps

The concept of the rate-controlling step singles out one step as much slower than all others. The concept of a quasi-equilibrium step does the opposite: It singles out one or several steps as much faster than all others. If this is so, the slowness of the other steps gives the fast steps time enough to come essentially to equilibrium: If the forward and reverse rate coefficients of one or more steps are much larger than all others, reactants and products of the fast steps are practically in equilibrium.

The use of the equilibrium condition for a fast step can greatly simplify mathematics, as will be seen in various examples in later chapters. Prominent among the fast steps to which the approximation can be applied are dissociation reactions in the gas phase and ionic reactions such as electrolytic dissociation, neutralization, and complex formation, as well as loss, addition, and exchange of

4.3, Quasi-equilibrium steps

85

ligands in the liquid phase. With few exceptions, such steps are fast compared with other typical chemical transformations. As discussed in the preceding section and illustrated in Figure 4.2, even a very fast step cannot attain ideal equilibrium as long as the overall reaction proceeds. The term quasi-equilibrium indicates that the forward and reverse rates of the step differ very little on a relative basis, so that the use of the equilibrium condition equating these rates is justified as an approximation. The quasi-equilibrium approximation can be applied to more than one step. As the preceding section has shown, the postulate of a rate-controlling step implies that all others are in quasi-equilibrium. The error incurred with the approximation of quasi-equilibrium of a single step is readily estimated. For the first step in a pathway A <,_• K —• P

(4.15)

with or without co-reactants or co-products (not shown), the approximation amounts to postulating that the step K—• P has only a negligible effect on the concentration of K, that is, ^K-P << ''K-* A • The relative error is ^e,

^

'-K^p/'-K^A

(4.16)

[stated in terms of rates rather than rate coefficients because the latter do not reflect the effects of possible co-reactants; alternatively, Ard = XKP /XKA, with X coefficients as defined in Section 6.2]. In more complex situations, only the largest of such errors need be considered. If the approximation is applied to several steps, the errors are cumulative. As in the case of a rate-controlling step, the approximation overestimates the rate if based on the true rate coefficients of the steps. Example 4.2, The hydrogen-iodide reaction. The reversible formation of hydrogen iodide from hydrogen and iodine in the gas phase, with stoichiometry Hi + I2 ^^—• 2HI

nicely follows an empirical rate equation ^Hi = KPYi,P\, - Kpii

('^•^'^)

as one would expect for a single-step, reversible reaction that is bimolecular in both directions. Indeed, this mechanism had become the classical textbook example of kinetics, even featured in an animated educational movie produced by a Nobel laureate for the American Chemical Society. However, the true mechanism is [5,6] H, \

"

^ 2HI *• 21 ^ >»^--^

(4.18)

The dissociation of I2 in the first step is very fast, the trimolecular second step is slow. The quasi-equilibrium condition for the first step is

86

Chapter 4. Tools for reduction of complexity

where ^oi is the dissociation constant of I2. This expression can be used to eliminate Pi from the rate equation of product formation in the second step: ''HI

=

2^I2AVH,

-

2^2IPHI

= IknKo.p^p,^-

Ik^xPm

^^'^^"^

where k^i and ^21 ^^e the forward and reverse rate coefficients, respectively, of the second step. The rate equation 4.19 is seen to be of the same algebraic form as eqn 4.17 for the single-step mechanism, with 2^12^01 corresponding to k^, and 2k2i to k^. Bodenstein, the first to study the reaction, had pointed this out as early as 1898 and had suggested both mechanisms as possibilities [7], but this was long ignored. More than half a century later, the single-step mechanism was finally questioned because the analogous HCl and HBr reactions proceed via quasi-equilibrium dissociation of CI2 or Br2 as the first step. For the HI reaction, such a step had to be even more favored because the I2 molecule is more strongly dissociated. However, on the basis of the observed concentration dependence of the rate alone, no discrimination between the single-step and two-step mechanisms was possible. Eventually, compelling evidence for the two-step mechanism was provided by Sullivan [5] with experiments in which iodine atoms were produced in the reactor by flash photolysis at temperatures at which thermal dissociation is negligible; he showed that Arrhenius extrapolation of the rate coefficient of the photolytically induced reaction accounted fully for the thermal rates at higher temperatures. Even so, Sullivan's conclusion was questioned on the grounds that the two-step mechanism requires too high an activation energy and therefore should not contribute [8]. However, an examination in terms of molecular orbital theory shows that the single-step mechanism can be ruled out because it violates the Woodward-Hoffmann exclusion rules [9] (see Example 7.8 in Section 7.4). Once again, nature has served notice that to draw conclusions about kinetics from thermodynamic arguments is to skate on thin ice! Beyond illustrating the application of the quasi-equilibrium approximation, this example also strikingly demonstrates that a mechanism or pathway cannot be deduced with certainty from the rate equation. An empirical rate equation is little more than a symptom. It can disprove a mechanism as incompatible with its algebraic form, but it does not prove a compatible mechanism to be correct. There will always be a number of other potential mechanisms that give the same rate equation in approximations so good that a distinction through experiments that measure the concentration dependence of the rate in a conventional reactor is impossible. If there is any doubt, the experienced kineticist seeks conclusive discrimination by other means (as did Sullivan, see above). Failing this, he applies Occam's razor, settling for the simplest of such rival mechanisms. However, as the example of the hydrogen-iodide reaction has shown, this might be the wrong choice. Unlike HI, its sisters HCl and HBr are formed by chain reactions (see Chapter 10). The reason for this difference in mechanisms is that the trimolecular step 2X + H2 — • 2HX (X = halogen) is too slow for CI and Br because of the

4.4. Quasi-stationary states

87

weak dissociation of chlorine and bromine. That step, being second order in halogen atoms, needs a sizeable partial pressure of these for the two-step mechanism 4.18 to outrun the competing chain reaction, and only iodine dissociates strongly enough to provide it. Even so, at very high temperatures a chain mechanism takes over [10]. No doubt both the chain and two-step mechanisms play a role in all three hydrogen-halide reactions, the difference lies in which of them dominates. This demonstrates another facet of kinetics: Very often, a multistep reaction proceeds via several pathways in parallel but, fortunately, most of the time one of them is so much faster than the others that it completely dominates the kinetic behavior. Where there is an eight-lane freeway, parallel dirt tracks do not contribute much to traffic flow! 4.4.

Quasi-stationary states: the Bodenstein approximation*

The concept of quasi-stationary states is an extremely powerful and widely applicable tool for reduction of mathematical complexity in kinetics. It will be used extensively in the chapters to follow. If a reaction intermediate X is so unstable that it decomposes practically as soon as it is formed, its concentration necessarily remains quite small. The same must be true for its net rate of formation r^ : If that rate were large and positive, the concentration would rise to large values, which it is known not to do; if that rate were large and negative, the concentration would have to become negative, which it cannot. Accordingly, provided the intermediate is and remains at trace level, its net formation rate r^ is small compared separately with its rates of formation and decay: The net rate of formation of an intermediate that is and remains at trace level is negligible compared with its contributing formation and decay rates.

In terms of mathematics, for a pathway A ^1—• X ^e->

the rates are net rate:

^x

~

P

^AX^A + ATp^Cp

formation rate

(4.20) + -fexp) decay rate

^x(^XA

* Suggested independently by Chapman [11] and Bodenstein [12] in 1913, implemented and developed by the latter the same year. Thorough discussions have been presented by Bowen et al. [13], Aris [14], and Turanyi et al. [15], the last of these with copious references to earlier work.

88

Chapter 4. Tools for reduction of complexity

If the intermediate is and remains at trace level: r XI '' ^Ax^A "^ ^px^p' ^x(^xA "^ ^xp) net formation decay or ^x = ^Ax^A - ^PxCp - C^(K, - ^xp) =

0

(4.21)

as though the net rate rx of the intermediate were zero. This is called the Bodenstein approximation. In reality, of course, rx usually is not zero, merely negligibly small compared with the formation and decay rates. Equation 4.21 can be solved for the concentration of the intermediate: Cx

^

^AX^A

^

^PX^P

^XA "^ ^XP

The approximation of negligible net formation rate of a trace intermediate can be used to replace a rate equation by an algebraic equation, with which the concentration of the intermediate can be eliminated.

The condition expressed by the Bodenstein approximation rx = 0 is often misleadingly called a steady state. It is not. It is not a time-independent state, only a state in which a specific chemical process rate is small compared with the others. In fact, some textbooks apply what they call the steady-state approximation to batch reactions in order to derive the time dependence of the concentrations, unwittingly leading the incorrect presumption of a steady state ad absurdum. And a continuous stirred-tank or tubular reactor may, and usually does, come to a true steady state, even if the Bodenstein approximation is and remains inapplicable. [The approximation compares process rates r{, it is irrelevant for its validity whether or not the reactor comes to a steady state, that is, whether the rates of change, dQ/dr, become zero.] The Bodenstein approximation can be applied repeatedly to different tracelevel intermediates in succession. Each application removes one rate equation and the concentration of one trace-level intermediate. This makes the Bodenstein approximation especially useful because trace-level intermediates are difficult to detect and their concentrations can rarely be measured accurately. Example 4.3. Radioactive decay. A long-lived radioisotope A decays; the n intermediates X^, Xj, ..., X^ are very short-lived; the final product P is stable: /\

•

yvj

—•

2\.2 — ^

...

^

A-jj

•

r

Setting the r-^ of the intermediates approximately equal to zero:

4.4. Quasi-stationary states

'•i = K^f,-^ - KMA = 0

89

(i = l,...k)

(X in indices suppressed here and below) one finds

C^:C,:q:...:q

=

(llkj:{llk,^):(\lk^,):...:i\lkj

that is, the well-known rule that the concentrations of the unstable intermediates are inversely proportional to their decay constants, and so proportional to their half-lives. An important corollary of quasi-stationary behavior is The net rate through each of the steps of a pathv^ay at quasi-stationary state is practically the same.

Any inequality would lead to concentration increases or decreases of intermediates, which stationarity does not allow. Accuracy. The Bodenstein approximation is accurate within reason, provided the intermediate is and remains at trace level, and with the exception of a very short initial time period in which the quasi-stationary state is established [13-15]. It is left to the practitioner to decree how low a concentration must be to qualify as "trace." The more generous he is, the less accurate will be his results. For the pathway 4.20 with one trace-level intermediate, the error introduced can be estimated in the same way as for rate control by a slow step (see Section 4.2.1): A., ^

C,IC,

(4.22)

(concentration ratio at quasi-stationary state). In repeated applications of the approximation to successive intermediates, the errors are cumulative. In addition, in a batch reaction, time is needed at start to "fill the pipeline" in the approach to quasi-stationary conditions. In a pathway with k trace-level intermediates, this requires a time of about t qss

= HCJ^-r^)

(4.23)

where the Q are the concentrations of the intermediates under quasi-stationary conditions. The rate of product formation is overestimated if the Bodenstein approximation is based on the true rate coefficients of the steps. The error estimates do not apply to chain carriers in chain reactions since these are regenerated by the reaction (see Chapter 10).

90

Chapter 4. Tools for reduction of complexity

The following points should be observed when the Bodenstein approximation is applied: •

The approximation r-^ = 0 should always be applied to eliminate the concentration Q of the intermediate Xj, not that of any other intermediate Xi_i or Xi^.1 . [If this rule is not observed, two small quantities may inadvertently be equated to one another.]

•

If the approximation is applied several times in succession^ the easiest way is to begin at the end of the pathway and work one's way toward its start one step at a time. The approximation is not applicable to short initial time spans and may give inaccurate results in large networks in which quasi-stationary states may be attained at different times in different portions.

•

To illustrate the first two points above, the following example shows how the Bodenstein approximation can be applied repeatedly to a reaction that includes reversible steps. Note, however, that the same result can be obtained much more easily with a general formula for "simple" pathways, to be developed in Section 6.3. Example 4.4. Nitration of aromatics of intermediate reactivity. In Example 4.1 the concept of a rate-controlling step was used to obtain simple rate equations for nitration of aromatics of either low or high reactivity. For aromatics of intermediate reactivity, no single step is rate-controlling. However, if the concentrations of H2N03'*", N02^, and ArN02"*" in the pathway 4.6 remain at trace level—this is a judgment call—the Bodenstein approximation can be applied repeatedly to obtain an explicit, closed-form rate equation. The Gillespie-Ingold pathway 4.6 can be written HB

ArH

HNO3 N ^ v^ X, ^

^

B-

B-

X2 -^^^^^—• X3 ^ v »

ArN02

Hp

^^'^^)

HB

(Xi = H2NO3", X2 = N02^, X3 = ArN02H^, and indices 0 and 4 will refer to HNO3 and ArNOs, respectively). The rate of product formation is '-ArNO,

=

^34^3^8-

(4-25)

(X suppressed in subscripts). The Bodenstein approximation for X3 is r^

-

k^^C^C^^^

^34^3^6-

Solved for C C3

Substitution in eqn 4.25 gives

—

^23^^ArH'^^34^B-

—

^

4.4.

Quasi-stationary states

'-ArNo. =

91

K^C^C,^^

(4.26)

The Bodenstein approximation for Xj gives ^2

"=

^12^1 " ^ 2 ( ^ 2 3 ^ A r H ' ^ ^ 2 l Q , o )

^

^

Solved for C2: ^1

=

^12^l''(^23^ArH "^ ^ 2 1 ^ H p )

(4.27)

Substitution in eqn 4.26 gives ^ArNO,

=

^12^23^1^ArH/(^23^ArH "^ ^21^H3o)

(4.28)

The Bodenstein approximation for Xi gives ^\

^

^Ol^HNOj^HB "^ ^ 2 1 ^ ^ H , 0 ~ S (^12 "^ ^ l O ^ B " )

=

^

Solved for Q : Cj

=

^Ki^wmpm "^ ^2i^2^Hp)/(^i2 "^ ^lo^B-)

(4.29)

This equation re-introduces C^, which must be eliminated with eqn 4.27. Solving again for Q after that substitution one finds after considerable rearrangement r

=

^0l^HNO,^HB(^23^ArH "*" ^21 ^ H p )

"1

^12^23^ArH "^ ^10^23^ArH^B" "^ ^ 1 0 ^ 2 1 ^ - ^ H , O

Using this to replace Q in eqn 4.28 one obtains k k k C

C

'^Ol '^12'^23^HNO,

C HB^ArH

^12^23^ArH "^ ^10^23^B-^ArH

(4.30)

"^ ^10^21^B-^H,0

With the equilibrium condition 4.9 for dissociation of the acid HB, the assumption that the concentration of water can be taken as constant, and identification of the participants, eqn 4.30 in a form with fewest empirical coefficients becomes _

r

^ a ^ArH

ArNOj

c

c

(4.31)

1 + /c,C,,„(l + ^,/Ce-)

where k

'^a

=

A- A^ J^ ^10^21'^HB^H2O

K = ^b

^23

k C

'

-^01 ^12^^23 ^ H B ^HjO

)t = ^^^ '^

k

92

Chapter 4. Tools for reduction of complexity (Ratios of forward and reverse rate coefficients of a step can be replaced by the respective equilibrium constants.) According to eqn 4.31, the rate is first order each in undissociated nitric acid and hydrogen ion and of an order between zero and one in the aromatic. Moreover, at given pH the rate is of an order between zero and one in the concentration of the anion B" of the added strong acid.

The rate equations 4.10 and 4.14, derived previously with the assumption of a single rate-controlling step, turn out to be special cases of eqn 4.31, obtained with the Bodenstein approximation: If k23 is very small (third step slow), eqn 4.31 reduces to eqn 4.10; if ki2 and /:2i are very small (second step slow), it reduces to eqn 4.14. The Bodenstein approximation leads to results of greater generality! The derivation of eqn 4.31 shows that the application of the approximation to an intermediate consumed by a reversible step—the second step in the example above—re-introduces the previously eliminated concentration of another intermediate (C2 in eqn 4.29), so that an implicit equation results. With each repetition of such an application, the amount of algebra required increases steeply and soon gets out of hand. No doubt this is why it has become established practice in kinetics to postulate a single rate-controlling step instead of using the Bodenstein approximation. However, a general formula for "simple" pathways, to be given in Section 6.3, allows the rate equation for quasi-stationary conditions to be written down immediately, without need for a lengthy derivation. This will be shown in Example 6.1 (Section 6.3) by application to the nitration pathway 4.24. Extensive examples of applications of the Bodenstein approximation, with or without the general formula for simple pathways, will be found in later chapters.

Summary The principal tools for reduction of mathematical bulk by elimination of rate equations are the use of stoichiometric constraints and the concepts of a rate-controlling step, quasiequilibrium steps, and quasi-stationary states of trace-level intermediates. Each stoichiometric constraint, such as the conservation of carbon or oxygen atoms, allows one rate equation to be expressed in terms of others and so be eliminated. If a step in a pathway is much slower than all others, it constitutes a bottleneck that controls the overall rate. Fast, reversible steps in the pathway have enough time to come to quasi-equilibrium. Such equilibria also affect the overall rate. In either case, all steps must be sequential for the rules to apply. If one or several steps in a pathway or network are much faster than all others, they attain quasi-equilibrium, and their (algebraic) equilibrium conditions can be used to replace rate equations. Any highly reactive intermediate that is and remains at trace level attains a quasistationary state in which its net chemical rate is negligible compared separately with its formation and decay rates. This is the basis of the Bodenstein approximation, which

References

93

allows the rate equation of the intermediate to be replaced by an algebraic equation for the concentration of the intermediate, an equation which can be used in turn to eliminate that concentration from the set of rate equations. The approximation can be applied in succession for each trace-level intermediate. It is the most powerful tool for reduction of complexity. It is the basis of general formulas to be introduced in Chapters 6 and 8 and widely used in others. Examples include hydroformylation of olefins, nitration of aromatics, the hydrogeniodide reaction, and radioactive decay. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

R. J. Gillespie, E. D. Hughes, C. K. Ingold, D. J. Millen, and R. I. Reed, Nature, 163 (1949) 599. C. K. Ingold, Structure and mechanism in organic chemistry, Cornell University Press, Ithaca, 2nd ed., 1969, ISBN 0801404991, Chapter VI. A. A. Frost and R. G. Pearson, Kinetics and mechanism: a study of homogeneous chemical reactions, Wiley, New York, 2nd. ed., 1961, Section 12.E. C. L. Perrin, J. Am, Chem. Soc, 99 (1977) 5516. J. H. Sullivan, /. Chem. Phys., 46 (1967) 73. J. W. Moore and R. G. Pearson, Kinetics and mechanism, a study of homogeneous chemical reactions, Wiley, New York, 3rd ed., 1981, ISBN 0471035580, p. 210. M. Bodenstein, Z. Physik. Chem., 13 (1894) 56; 22 (1897) 1; 29 (1898) 295. K. J. Laidler, Chemical kinetics, McGraw-Hill, New York, 2nd ed., 1965, p. 121. R. Hoffmann, J. Chem. Phys., 49 (1968) 3739. J. H. Sullivan, J. Chem. Phys., 30 (1959) 1292; 36 (1962) 1925; 39 (1963) 3001. D. L. Chapman and L. K. Underbill, /. Chem. Soc, 103 (1913) 496. M. Bodenstein, Z. Physik. Chem., 85 (1913) 329. J. R. Bowen, A. Acrivos, and A. K. Oppenheim, Chem. Eng. Sci., 18 (1963) 177. R. Aris, Am. Scientist, 58 (1970) 419. T. Turanyi, A. S. Tomlin, and M. J. Pilling, J. Phys. Chem., 97 (1993) 163.

Chapter 5

Elementary Combinations of Reaction Steps

In his search for the pathway or network of a reaction, the chemist or engineer finds himself in the same position as a physician who must diagnose a patient's illness: He must evaluate symptoms. To do so competently, he must learn to recognize the patterns of kinetic behavior produced by the most common combinations of reaction steps, to be reviewed in the present chapter. Any one of these combinations might conceivably constitute a complete network, although many such networks would be no more than improbable and uninteresting interconversions of isomers. However, even if embedded in larger networks, the simple combinations to be examined here are apt to produce their characteristic kinetic effects—or can be made to do so by appropriate choice of experimental conditions. In fact, most of the examples to be used for illustration will be of this type, that is, their real networks are combinations of multistep pathways rather than of single steps. This chapter is intended primarily to provide information useful for network elucidation rather than modeling or design. For this reason, emphasis is on behavior in constant-volume batch reactors. Although these are the least common type of reactors in industrial plants, they are advantageous for the purpose at hand. In particular, they make it relatively easy to obtain results at low conversions that may be crucial for a distinction between rival mechanisms (see Section 3.1.1; note that the mathematics of plug-flow tubular and differential recycle reactors is analogous). Also, the focus is on simplicity, on symptoms that will stand out clearly when experimental results are interpreted. Complex or lengthy mathematics has its place in modeling and design, but is of little help in network elucidation. Most of the step combinations examined in this chapter are covered in advanced texts on kinetics and reaction engineering [Gl-GlO], with the exception of coupled parallel steps and reactions with fast pre-dissociation (Sections 5.3 and 5.6, respectively) and the equations for continuous stirred-tank reactors. The material is reviewed here for the user's convenience and for ease of reference. 5.1, Reversible reactions The simplest example of a step combination is a reversible reaction A <—• P composed of a forward step A —• P and a reverse step P —• A. Although such reactions are commonly referred to as single-step reversible, it is more expedient for reaction mathematics to treat them as composed of two separate steps.

Chapter 5. Elementary combinations of reaction steps

96 5.1.1.

First order-first order reactions

The most common first order-first order reversible reactions are isomerizations, although many of them are catalytic and involve intermediates. Often they are parts of larger networks. Many other reversible reactions are also first order-first order. stoichiometry:

A <-

rate equation:

-r.

(5.1)

-• P

rp =

k^pC^

^PA^p

(5.2)

It proves convenient to define new quantities k

C = C^ + Cp

— k^p + ATp^,

In terms of these, the rate equation becomes

Figure 5.1. Concentrations vs time for first order-first order reaction A <—• P in batch.

at time t:

r^ - kC^ kpj^C

at equilibrium:

0 = kC^ - k^^C

(superscript oo indicates equilibrium). Subtraction of these equations from one another yields r, =

k(C^-C:)

or, in words: The rate is proportional to the distance from equilibrium. The proportionality factor is the sum of the forward and reverse rate coefficients.

The decrease of distance from equilibrium thus is a first-order process. To obtain a more general formula, define: X

=

Q

(i = A,P)

(5.3)

c - c: This is thQ fractional distance from equilibrium, varying fromx = l at start tox=0 at equilibrium and having the same value for A and P. In terms of x, the rate is •kx

(5.4)

where the process rate r^ is the contribution of the reaction to the rate of change of X.

97

5.1. Reversible reactions

Specifically for constant-volume batch reactions, in which the reaction rate is the only process rate contributing to the rate of change: dx/dt

kx

A plot of Inx versus t yields a straight line with slope —k (see Figure 5.2). It makes no difference whether x is calculated with i = A or i = P, the lines obtained are identical. Moreover, the method remains applicable even if some product P was already present at zero time. The individual rate coefficients k.p and ^PA can be calculated from k and the equilibrium concentrations C^ and Cp* with

and

Inx = -kt

\nx

(5.5)

slope -k

Figure 5.2. First-order plot for reversible reaction in batch reactor.

kcric

/C^P = K U p / C/,

(5.6)

The rate equation 5.5 and the first-order plot remain valid in terms of any physical variable f that is a linear function of the concentrations. This is usually true for properties such as refractive index, electric conductivity, specific gravity, and rotation of the plane of polarized light. In all such cases, x is defined as before by eqn 5.3, but with f instead of C, . Equation 5.5 remains valid even if the variable changes its sign in the course of the reaction, as it might with rotation of the plane of polarized light. Equation 5.5 and the first-order plot also remain valid regardless of volume variations if the fractional distance from equilibrium is defined in terms of amounts A^i instead of concentrations Q : X

=

N, - K N- - K

(i = A, P)

(5.7)

The relationship between the characteristic rate coefficient k and the time t^ required to reduce the distance from equilibrium to the fraction x is analogous to that for irreversible first-order reactions (see eqn 3.23): {\nx)lt^

(5.8)

In a continuous stirrer-tank reactor at steady state, a first order-first order reversible reaction with no fluid-density variation describes a straight line with slope k and intercept 1 in a plot of \lx versus r (see Figure 5.3). Again, it makes no difference whether x is calculated from the concentration of A or P or any physical

98

Chapter 5. Elementary combinations of reaction steps property that varies linearly with concentrations. If the reaction involves a fluid-density variation, the same straight-line plot is obtained if X is calculated with eqn 5.7 in terms of amounts rather than concentrations. Moreover, as in a batch reactor, the relationship between the characteristic rate coefficient k and the reactor space time r^ required to reduced the distance from equilibrium to a fraction x is analogous to that for an irreversible first-order reaction (see eqn 3.25):

slope k

Hx A, P

0

Figure 5.3. First-order plot for reversible reaction in continuous stirred-tank reactor. k =

1 -X

(5.9)

TX

Example 5.1. Reversible formation of a lactone. Results obtained for the conversion of 7-hydroxybutyric acid into its lactone

0=o

.COOH

Table 5.1. Conversion of 7-hydroxybutyric acid into its lactone in aqueous solution in batch reactor at 25° C [1]. t

min 0 21 36 50 65 80 100 120 160 220 47 h 62 h

Q M

x ~

1 1 - Cp/Cp" 1

0 0.0241 .0373 .0499 .0610 .0708 .0811 .0900 .1035 .1155 .1328 .1326

LOOO 0.819 .720 .624 .541 .467 .389

1

.322 .221 .130

0 0

H,0

in aqueous solution at 25° C in a batch reactor and with an initial acid concentration of 0.1823 M are listed in the first two columns of Table 5.1. The product concentration remained constant within experimental error from 47 to 62 hours. The value at 47 hours, 0.1328 M, can therefore be taken as the equilibrium concentration of lactone, C^. The corresponding values of x (fractional distance from equilibrium) are listed in the third column. A plot of \nx versus t is shown in Figure 5.4 (left diagram) and is seen to be nicely linear, indicating that the reaction is first order-first order (the reverse step actually is pseudo-first order as it involves H2O in large excess as a

5.1. Reversible reactions

99

Injc

120 180 t [min]

120 180 t [min]

240

Figure 5.4. Conversion of y-hydroxybutyric acid into its lactone, plotted as reversible first order-first order reaction (left) and as irreversible reaction of order 1.5 (right). co-reactant). With k = 9.42*10"^ min"^ from the slope of the plot in Fig. 5.4, left diagram, the rate coefficients are: k^^ = kC^'ICt

= 6.84* 10-3 min'^

/CpA

— K ~ K AP

2.58 * 10-3 min-

The example of this reaction demonstrates another important facet of kinetics. Figure 5.4 shoves side by side the experimental data plotted as a first order-first order reversible reaction and as an irreversible reaction of order 1.5. Over a limited conversion range (here about tv^o thirds of the way to equilibrium) the second plot is linear within the scatter of the data points. Although evaluation of the full conversion range leaves no doubt that the reaction is indeed reversible and first order-first order, its rate up to a rather high conversion is approximated surprisingly well by the equation for an irreversible reaction of higher order, in this instance of order 1.5: rp = 0.016 CA' Mmin"' (5.10) Unless results at conversions close to equilibrium are available, a reversible reaction can be mistaken for an irreversible reaction of higher order. This is a striking example of the dangers of working by recipe with data that do not fully cover the entire conversion range, and of the importance of considering what is chemically reasonable. If the experiment had been terminated at or before 2 hours (about 50% conversion of the acid), eqn 5.10 might have been accepted as adequate, but would have given wrong predictions upon extrapolation to higher conversions. However, a competent chemist would have rejected a reaction order

100

Chapter 5. Elementary combinations of reaction steps

of 1.5 as requiring a kind of mechanism that is highly improbable for a simple ring closure with water loss under mild conditions, while he would gladly have accepted the proposition that the reaction might be first order and reversible. 5.1.2.

First order-second order reactions

Reactions in which a molecule dissociates into two different or equal fragments are very conamon, although many are so fast that they are practically at equilibrium (e.g., see dissociation of I2 in the hydrogen-iodide reaction in Section 4.3). stoichiometry:

A ^e-> P + Q

rate equation:

""^A

= ^p =

(5.11) =

^Q

^APQ

-

^PAQCQ

(5.12)

Integration eqn 5.12 for constant-volume batch and Cp° = CQ° = 0 gives [G8] I

/A

2 -

In

+

/A

JK

/A

~

/A(1

~

/A

)

.

= k^^t

JK

where/A = 1 — Q IC^ is the fractional conversion of A. Since this equation is in terms of reactant A, it also holds for reactions of the type A <—• 2P. For a CSTR, the relationship between concentrations and reactor space time is too unwieldy to be of practical use. 5.7.5.

Second order-second order and higher-order reactions

Single-step reactions of this type, with different or identical reactants, are found mostly in textbooks. The mathematics of the HI reaction fits this formalism although the reaction actually proceeds by a different, two-step mechanism (see Section 4.3). stoichiometry:

A+B<

rate equation:

-''A

=

• P+ Q

""^B

= ''p =

(5.13)

^Q

=

^AP^AQ

-

^PAQCQ

(5.14)

A solution simple enough to be at least of some use in practice exists only for constant-volume batch with stoichiometric amounts of A and B and no P and Q present initially [G7,G8]: , m

/A

1 ~

/A

/A

"

(2/A /A

-

1)/A

^r^oi ^ = ZL^K^^t

"/A

The mathematics of reversible reactions of higher order than second are cumbersome. Rather than struggling with them, the chemist or engineer interested in their kinetics will design his experiments to circumvent this obstacle by determining pseudo-orders or evaluating initial rates (see Section 3.3.2 and 3.3.3).

5.2. Parallel steps 5.2.

101

Parallel steps

Steps or pathways are csMed parallel (or concurrent) if they originate from the same reactant, but lead to different products. Practically every reaction yields byproducts to some extent, and most of these arise from parallel steps at some points in the network. A good understanding of the kinetics of parallel steps is therefore highly important. 5,2.1.

Parallel first-order steps

In its literal form, this reaction is only of academic interest because a molecule is unlikely to break up or isomerize irreversibly in two or more different ways. However, situations frequently encountered in practice are those of multistep parallel first-order decomposition reactions and of parallel reactions that involve coreactants but are pseudo-first order in the reactant A. An example of the first kind is dehydrogenation of paraffins, examples of the second kind include hydration, hydrochlorination, hydroformylation, and hydrocyanation of olefins and some hydrocarbon oxidation reactions. All these reactions are multistep, but the great majority are first order in the respective hydrocarbon, and pseudo-first order if any co-reactant concentration is kept constant. ^^^ A

network:

P •

Q

(5.15)

rate equations: (5.16) Figure 5.5. Concentrations vs time for parallelfirst-ordersteps A <—• P and A —• Q in batch.

where

k ^ k.p +"'AQ k,^ + ... ''AP

(5.17)

There may be any number of parallel steps. According to eqns 5.16, all rates are proportional to the concentration of A and thus, since A reacts to completion, to the distance from the state at complete conversion. Moreover, this being so, the decay or formation rate of each participant is proportional to tht fractional distance, x, from the state at complete conversion: r^ = - kx

102

Chapter 5. Elementary combinations of reaction steps

where x is given by eqn 5.3, and k by eqn 5.17. In a network of irreversible parallel first-order steps, the fractional distance from complete conversion at any conversion level is the same for all participants. The process rate —r^ is proportional to the fractional distance jc. The characteristic rate coefficient is the sum of the rate coefficients of all steps.

Since the ratios of the rates remain constant regardless of the conversion level, the concentrations at complete conversion are independent of the reactor type, and so are the distances from that state at any given conversion. Specifically in constant-volume batch reactors, in which the process rate r^ equals the rate of change dx/dt: dx/dt

= - kx

Inx

- kt

or, after integration. (5.18)

where x can be expressed in terms of any participant:

X = cjci

= 1 - Cp/Cp" = 1 - Co/c; Q/^Q

Accordingly, a plot of Inx versus time gives a straight line with slope —k, as in Figure 5.6. It makes no difference whether x is calculated with i = A, P, Q, or any other product, the straight lines obtained are identical. If such behavior is found experimentally, it provides a very strong indication for a network of general type of 5.15 with all reactions first order in A. The coefficients k^^ , /T^Q , etc., can be calculated from

A, P, Q

Inx

slope -k

Figure 5.6. First-order plot for parallel steps in batch reactor. ^AP

"

(^p

/^A)^»

(5.19)

"AQ

(C^/C^)k,

etc.

(5.20)

with k obtained from the slope in Figure 5.6 or a corresponding numerical or statistical method. Alternatively, the individual coefficients can be calculated from data on incremental conversions: (5.21) fcACp/(-AC), = kACJ(-ACJ, etc. "AQ

5.2, Parallel steps

103

Equations 5.19 assume that no products are present initially. Otherwise one has X = (Cp"-Cp)/(Cp"-Cp) = ... ^AP = k{C;~C;)ICl,

etc.

It may appear strange that all products give first-order plots with the same slope although their individual formation rate coefficients may differ widely. It is true that the formation rates of products with larger rate coefficients are higher at any time. However, their concentrations at complete conversion are also higher by the same factors. The result is that, in a given time span, the concentration of each product increases by the same fraction of the final concentration, regardless of the disparity of the rate coefficients. In a continuous stirred-tank reactor, eqn 3.20 and the corresponding straightline plot in Table 3.1 remain valid, and eqn 3.21 for the products becomes _ CA/:APT

_ Clk^^T

\ + kT

^

(5 22)

\ + kr

Example 5.2. Hydroformylation of propene [2]. Hydroformylation converts an olefin to an aldehyde of next higher carbon number by addition of carbon monoxide and hydrogen. The reaction is catalyzed by dissolved hydrocarbonyl complexes of transition-metal ions such as cobalt, rhodium, or rhenium. The carbon atom of the carbon monoxide can attach itself to the carbon atom on either side of the olefinic double bond, so that two aldehyde isomers are formed. If the catalyst also has hydrogenation activity, the aldehydes are converted to alcohols and paraffin is formed as by-product. For propene and such a catalyst the (simplified) network is:

propene x ' ^

V

CO, 2H^

^ /-v^OH

CO 2H • ^

•

^

^OH I

«-butanol isobutanol propane

Results obtained in a constant-pressure semi-batch reactor in which synthesis gas was supplied on demand are listed in Table 5.2. The analytical accuracy is said to be within ± 0.002 M. [Before alcohol analysis the samples were hydrogenated to convert any residual aldehydes to the corresponding alcohols.] At the resulting constant partial pressures of CO and Hj in the reactor and with very good gas-liquid mass transfer, the CO and Hj concentrations in the liquid remain constant and the rate depends only on the propene concentration (see pseudo-orders. Section 3.3.2).

104

Chapter 5. Elementary combinations of reaction steps Table 5.2. Hydroformylation of propene in 2-ethylhexanol with phosphine-substituted cobalt hydrocarbonyl catalyst, HCo(CO)3Ph (Ph = organic phosphine) and synthesis gas of 2:1 Hj-to-CO ratio at 50 atm and 130° C in semi-batch reactor. time

concentration, M

min

propene

«-butanol

isobutanol

propane

0 10 20 60 120 20 h

.807 .679 .571 .285 .101 < .002

0 .107 .196 .437 .592 .677

0 .009 .017 .038 .051 .058

0 .010 .019 .042 .055 .063

Most homogeneous hydrogenation, hydrohalogenation, halogenation, hydroformylation, and hydrocyanation reactions are first order in the olefinic reactant. A test whether this is the case here suggests itself. A numerical work-up is shown in Table 5.3. The fractional distances from complete conversion are calculated with eqns 5.19 for all participants at all times, and values of the characteristic coefficient k are then obtained for each from eqn 5.18 (global method), with the concentrations at 20 hours taken as C^ Table 5.3. Data work-up for hydroformylation of propene. (A = propene, P = w-butanol, Q = isobutanol, R = propane.) t

X

=

min

CA/C/

10 20 60 120

.8414 .7076 .3532 .1252

X

=

= 1 - Q/Cr

k == -(lnx)/r*10^ min'^

P

Q

R

A

P

Q

R

.8419 .7105 .3545 .1256

.8448 .7069 .3448 .1207

.8413 .6984 .3333 .1270

1.73 1.73 1.73 1.73

1.72 1.71 1.73 1.73

1.69 1.73 1.77 1.76

1.73 1.79 1.83 1.72

The consistency of the values for k shows that plots of Inx versus t will be linear and have the same slope for each participant, and indicates strongly that all three reactions are indeed first order in olefin. A network of the general type of 5.15 can therefore be accepted. An average value for k is 1.73*10"^ min"^ Not unexpectedly, the scatter is largest for the lowest concentrations, where the relative analytical error is greatest. The individual rate coefficients kj^p, k^q, and k^^ can be calculated with eqns 5.20. The results are:

5.2. Parallel steps

105

1.73 * 0.677 0.807

^AP

. l-^^*0-05S ^AQ

; ^AR = /:^_

^ ^Q.2 ^ Q^2 * 10-2 j^i^-i^

0.807 1.73 * 0.063

0.807

.^_2

n 1/f

in-2

• -

* 10 "^ = 0.14 * 10 ^ mm

The astute reader will notice that the material balance fails to close within the analytical error: The sum of the fmal product concentrations leaves more than 1.2% of the initial propene unaccounted for. Factors contributing to this discrepancy are a slight increase in the volume of the liquid phase as C4 alcohols are formed from C3 olefm, and a small amount of by-products of higher molecular weights that have escaped the gas-chromatographic analysis. For use in modeling, the rate coefficients would also have to be corrected for gas-liquid distribution at least of propene and propane. In particular, some propene initially in the gas phase enters the liquid and is converted (making the materialbalance discrepancy worse). The analyses of the liquid do not reflect this, and the calculated value of k therefore is slightly too low. Another consequence of the presence of volatiles in the gas phase is that the sum of the partial pressures of CO and H2 was a little below the 50 atm total pressure, only approaching that level more closely as propene was converted to less volatile products. Such nuances notwithstanding, the symptom of a network with parallel firstorder reactions to all three products emerges quite clearly from this single experiment. Even so, however, the question remains open where exactly the reaction path branches and how the rate depends on the partial pressures of CO and H2. We shall return to this interesting type of reaction in later examples (Examples 5.3, 6.2, 6.5, 7.5, 12.1, 12.2, 13.1, and 13.2).

5.2.2.

Parallel second-order steps

The simplest case of parallel second-order steps is that of formation of two different dimers of a reactant A, corresponding to the network 5.23 and rate equations 5.24 (see next page). At all times, both products are formed in the same ratio r^'.rQ = ^AP-^AQ» so that the decay of A is an ordinary second-order reaction with rate coefficient k = k^^ + kp^Q. Likewise, the product formations are ordinary secondorder reactions. (One could think of the initial amount of A as divided into two portions in the ratio A:AP'^AQ ^hat react independently of one another, one to P and the other to Q.) All equations and plots for irreversible second-order reactions thus are valid (see Section 3.3.1).

106

Chapter 5. Elementary combinations of reaction steps

network

^^ ^ ^

P

(5.23)

^^^ Q rate equations:

r^ -

-2kC^

rp = k^,C^

where

(5.24)

A: = /c^p + ^Q

(5.25)

Mathematics becomes more complicated if co-reactants are involved [GIO], as for instance in networks such as B ^ ^

^ ^

P

A+B

^—^

P

A

V ^ ^ ^

^ «

Q

However, if the co-reactant is the same in both reactions, as in the network above left, experiments for network identification can be conducted with stoichiometric amounts of the reactants, so that Q = C^ at all times. The mathematics then reduces to the same simple form as for the network 5.23. With different coreactants as in the network above right, that technique can also be employed, but the Cg-to-Q ratio at which equal fractions of these two reactants are consumed would first have to be determined. 5.2.3.

Parallel steps of different orders

Combinations of parallel steps of different orders are common in practice. The most frequently encountered situation is the formation of a heavy by-product by higher-order dimerization or oligomerization parallel to a first-order main reaction. In other common cases, the side reaction is a first-order decomposition of a reactant parallel to a main reaction of higher order. network A

(5.26)

5.2. Parallel steps rate equations:

= ~~ ^/CAPQ^ ~ ^AQQ rp = k^,C^

107

^A

(5.27)

Unfortunately, most combinations of steps of different orders lead to rather unwieldy mathematics as far as reactor performance is concerned [G1,G10]. However, one important general facet of such networks warrants special attention and is easily demonstrated with the simplest case, the network 5.26. Of interest are the yield ratio of the two products and the selectivity of conversion to one of them. For the network 5.26, the instantaneous yield ratio is seen to be FpQ ^

2r,/r^

= ^(kJKc)C^

(5.28)

(see definition 1.7 in Section 1.6). The yield ratio (second-order to first-order product) is proportional to the reactant concentration. A consequence is: Higher reactant concentration favors the parallel step of higher reaction order.

This qualitative rule is not restricted to the simple network 5.26 and has important consequences for reactor selection in plant design. In a batch or plug-flow tubular reactor, the reactant concentration is at its highest initially—in the batch at start, in the tube at the inlet—and then decreases as conversion progresses to its final level. In contrast, in a well-mixed continuous stirred-tank reactor, the reactant concentration everywhere and at all times is at that final, lowest level because the effluent has the same composition as the reactor contents. Accordingly, if the composition of the initial charge or feed stream and the conversion to be attained are given and if P, formed by the higher-order step, is the desired product, a tubular or batch reactor gives the better selectivity; and if Q is desired, a continuous stirred tank reactor does. [In the first case, if for some overriding reason a stirred tank is the only acceptable reactor configuration, a cascade of stirred tanks in series should be considered because the reactant concentration will be higher at least in its earlier stages.] Also, whatever the reactor type, formation of the product of the higherorder step is more favored at lower final conversion, so there may be a trade-off between selectivity and size of a recycle stream. For the simple network 5.26 and a reaction with no fluid-density variation, the magnitude of the effect is easily calculated: The cumulative selectivity of conversion to P (moles of A converted to P per mole of A consumed, see definition 1.8) in batch and continuous stirred-tank reactors as a function of fractional conversion, /A, is

108

Chapter 5. Elementary combinations of reaction steps

5p

batch:

= 1 - -^r-^ In

/,$

1+(1-A)$

(5.29)

(5.30)

CSTR:

i-(i-A)^ where #

=

(5.31)

2fcApC/^AQ

Results for # = 2 are listed in Table 5.4. fractional conversion /A =

selectivity to P

S, =2Cp/(C- CJ

1

^A

1-

—

1

batch

CSTR

0.642 0.612 0.574 0.524 0.491 0.472

0.615 0.545 0.444 0.286 0.167 0.091

Table 5.4, Cumulative selectivity of conversion to higher-order product in first- and second-order parallel steps with no fluiddensity variation in batch and CSTR, (calculated for

1

^ 0.20 0.40 0.60 0.80 0.90 0.95

Derivation. The instantaneous selectivity to P (moles of A forming P per mole of A reacting) at given conversion level /^ is 5p =

2r,/(-0

(see definition 1.9). With eqns 5.27 and 5.31 this becomes

(CJCl)^ 2^AP^A

"'' ^AQ

1+

iCJCl)^

(1 - A)$

(5.32)

1+(1-/A)*

In a CSTR at steady state, the compositions of reactor content and effluent are the same, so that the cumulative selectivity 5p (moles of A converted to P per mole of A reacted) equals the instantaneous selectivity Sp. In a batch reactor, eqn 5.32 describes the incremental fractional conversion of A to P at the respective conversion level: 2dCp

^dc:

i + d-A)*

With d Q = CA°d(l - / A ) , integration over 1 - / ^ from start to final conversion level /A yields

5.3. Coupled parallel steps

109

/A=0

As defined by eqn 1.8, the cumulative selectivity for the case at hand is 5p = 2(Cp-Cp°)/AC° With the integral above this gives eqn 5.29. This derivation demonstrates that yield ratios and selectivities are more easily obtained by integration over fractional conversion instead of time or reactor space time. Equations for selectivities in other situations with parallel steps of different orders can be derived in the same way, but are more complex. 5.3.

Coupled parallel steps

Many reactants in organic chemistry can exist in the form of two or more isomers that give rise to different products. The most typical examples are reactions in which an olefin adds water, halogen, or any compound other than hydrogen under conditions that also promote migration of the double bond. Many reactions of industrial importance belong to this class. The simplest network of this type is Ai

•

P

(5.33) A2

•

Q

where Ai and A2 are the reactant isomers forming products P and Q, respectively. The rate equations are ^21S

- c^iK^.k,,),

''1

-

f-l

= k^^ C^-

(5.34)

C^ik^^ + k^f^,

("A" is suppressed in the indices to avoid clutter). The behavior depends strongly on the relative magnitudes of the rate coefficients and the initial isomer ratio. Since all steps are first order, an analytical solution can be given with matrix algebra, but is not easy to interpret. An examination of special cases provides more insight:

110

Chapter 5. Elementary combinations of reaction steps

Case I:

^12 ' Kl

'^ K? ' ^2Q

Case IL

^12 ' ^21

''

Case III:

'^12

isomerization very fast compared with conversion isomerization very slow compared with conversion

^IP ' ^2Q

'^21

'^IP

all rates comparable

'^2Q

Case I. Here, isomerization quasi-equilibrium is established quickly, before any significant conversion to products has taken place. From this short initial time span on the isomers remain in quasi-equilibrium, that is, the fractions of total A present as Ai and A2 remain constant: ^1

_

^1

1

1-^.2

CA

^n

^

(5.35)

l.K,,

C,

where CA = C, + Cj is the concentration of total A, and K12 is the equilibrium constant of isomerization Aj •*—*• kj. As a result, all rates can be expressed in terms of the concentration of total A: k

•-

-

k

"• . C

K,,'^'

r

=

"^

K

_22—-C

TVY;,^'

(5.36)

This set is of the same algebraic form as eqns 5.16 for conversion of a single reactant to two products in parallel steps. Because quasi-equilibrium of isomerization is maintained, the reactant acts as a "homogeneous source" despite being split into two isomers, and the mathematics of product formation is the same as for the simpler reaction with network 5.15. Results from constant-volume batch plotted as Inx versus r for P, Q, and total A give identical straight lines with slope —k, as in Figure 5.6, where k

^

(k,,^k,^K,,)/(l^

K,,)

(537)

(see Figure 5.7, top left). If no analytical method to distinguish between the isomers and no information on the very short initial time span before attainment of isomerization quasi-equilibrium were available, the existence of the reactant in the two isomeric forms might go unnoticed. A good way to determine the rate coefficients ^ip and kjQ from experimental results is first to find k from the slope of a first-order plot for the respective reactant or an equivalent numerical or statistical method, then to calculate k^^ and k2Q from it, the isomerization equilibrium constant K12, and the product concentrations at complete conversion:

5.3. Coupled parallel steps

111

Case I A, P, Q

\nx

\nx

Figure 5.7. First-order constantvolume batch plots for two coupled parallel first-order steps. Top left: very fast isomerization; top right: very slow isomerization; bottom left: comparable rates with k^^ ^ ^21 ^ ^ip ^ ^2Q ^^^ initial charge of A J (schematic).

Injc

k,, =

k(UK,,)

CI

•^20

= k

^

(5.38)

c:

(granted no products are present at start; otherwise replace the final product concentrations by Cp~ - Cp° and CQ°° - CQ°). Note that information on the individual isomer concentrations is not needed. The isomerization rate coefficients k^ and k2i cannot be calculated unless accurate data on the short transient before attainment of isomerization quasi-equilibrium are at hand. Case II. Here, conversion to products nears completion before any significant isomerization has occurred. The steps A^ —• P and A2 —• Q are uncoupled, proceeding side by side without noticeably affecting one another. First-order plots of Inx versus / for P and A^ give identical straight lines with slope —k^p , those for Q and A2 give identical straight lines with different slope -^20 » and that for total A is curved, with a slope that flattens with increasing time and approaches that of the curve for A2 and Q (see Figure 5.7, top right). The shape of the plot for total A depends not only on the relative reactivities of the isomers, but also on their relative amounts in the initial charge. If the reactivities differ substantially and the amounts charged are comparable, the early portion of the curve is strongly affected by the fast decay of the more reactive isomer (assumed to be Aj in Figure 5.7). The late portion, stemming from a time span in which that isomer has all but disappeared, then reflects the slow decay of the less reactive isomer in the absence of the other. Because of the decreasing slope

112

Chapter 5. Elementary combinations of reaction steps

of its first-order plot, the decay of total A resembles a reaction of higher order although the decay rates of the individual isomers are first order. An example from the early days of kinetic reaction engineering is Mobil Oil's modeling of catalytic cracking in the 1950s: Cracking of typical gas oil and naphtha feedstocks is second order although that of their individual components is first order (with allowance for catalyst deactivation) [3,4] (see also Section 12.4). Optimization with the model showed that the conventional cracking reactors were larger than needed, and a lot of money was saved by making new ones smaller from then on, a remarkable success that helped to put fundamental reaction-kinetic modeling in process design on the map. Case III. Here, the rates are comparable. This is the most complex, most interesting, and in practice most important situation. The behavior depends strongly on the relative reactivities and the initial isomer ratio. Let us say, isomerization is somewhat faster than conversion, and the initial charge is the more reactive isomer Aj (this is assumed in Figure 5.7, bottom left). Isomerization then produces a steady-state isomer ratio, but not until significant conversion to products has occurred. Loosely said, the system swings from a transient behavior dictated by the initial charge to one resembling that of Case L At start, the formation rate of P is very high because the parent of P, isomer Aj, is present in high proportion. In contrast, the formation rate of Q is zero initially because the parent of Q, isomer A2, is absent and must first be formed from Aj by isomerization. Accordingly, in the first-order plot the curve for P starts with a steep slope that subsequently flattens, and that for Q starts with zero slope and becomes steeper. Because initially all A is present in the form of the more reactive isomer, the curve for total A also starts with a steep slope and then flattens. As time progresses, the isomers asymptotically approach a steady-state distribution that remains constant from then on: The reactant becomes a "homogeneous source." As a result, the curves for P, Q, and total A become parallel straight lines. An experimentally observed behavior of this kind is very strong evidence for coupled, irreversible, parallel steps that are first order in the reactant isomers. If the initial charge consists of A2, the less reactive isomer, the opposite behavior results: The curve for Q starts with steeper slope; that for P, with zero slope; and that for total A, with gentler slope. If the initial charge consists of an equilibrium isomer mixture, little shift in isomer distribution occurs during the reaction, and the curves for P, Q, and total A almost coincide, as they do in Case I. The asymptotically approached constant isomer distribution is not necessarily close to equilibrium. If the conversion rate coefficients differ greatly and the equilibrium fraction of the more reactive isomer is small, the latter becomes depleted because resupply by isomerization cannot quite keep pace with the high consumption rate, and the fraction of A present as the more reactive isomer is significantly smaller than at isomerization equilibrium. Theoretically, the Ai-to-A2

5.3.

Coupled parallel steps

113

ratio under reaction conditions can be calculated from the steady-state condition -rJCy = -r2/C2 or, if the "drain effect" is large, be estimated with the Bodenstein approximation of quasi-stationary behavior of Aj. In practice, neither method can be recommended. Both require iteration because the values of the coefficients in the equations depend in turn on the magnitude of the drain effect, and both are highly sensitive to the value of the isomerization rate coefficients, usually known only in rough approximation. A better way is to use the coefficient values obtained assuming isomerization equilibrium as initial guess for regression or, if possible, to determine the isomer ratio under reaction conditions directly by analysis. An observation from everyday life may help to visualize the drain effect. A common type of office hot-water dispenser for instant coffee or tea consists of a container with show glass and stopcock at the A, bottom of the show glass (see Figure 5=8). It can serve as a hydrodynamic model of our reaction if : we drill a small hole into the bottom of the container (better, just imagine what would happen if such a hole were there). The water in the container is Aj, that in the show glass is Ai , the water dribbling out of the small hole goes to H ^ Q Q, that draining from the show glass goes to P. If the stopcock Figure 5.8. Hydrodynamic model for and hole are closed, the water level reaction with coupled parallel first-order in the show glass is the same as in steps, widely different conversion rate the container; that corresponds to coefficients, and small equilibrium fraction isomerization equilibrium. How- of more reactive isomer. ever, when the stopcock is opened, the level in the show glass falls far below that in the container because resupply from the latter is not fast enough to make up for the quick drain. That the hole in the container allows a small amount of water to dribble out makes little difference. [If we close the stopcock, the level in the show glass overshoots equilibrium and oscillates until settling. This is caused by the inertia of flow through the connection between container and show glass and has no equivalent in reaction-kinetic systems: The hydrodynamic model is not perfect in every detail.]

Ln

Just like the approach to isomerization equilibrium in a system without conversion to products, the approach to the steady-state isomer distribution is a firstorder process whose characteristic rate coefficient is the sum of the forward and reverse coefficients (see Section 5.1):

114

Chapter 5, Elementary combinations of reaction steps Ko -

K2-k,,

(5.39)

As the steady state is approached, the curves in the first-order plots become parallel straight lines. Accordingly, the fractional-life equation 5.8 can be used to obtain an estimate of k^^^ from the time required to approach this condition closely. The individual coefficients k^2 ^nd k2i can then be obtained from their sum k^.^ and the equilibrium constant K12 — ^12 ^^21 • ^la = k,^KJ(l.K,,),

k,, = kJil.K,,)

(5.40)

This is a typical example showing how the kinetic effect of a simple step combination—a first order-first order reversible reaction in the case at hand—can be clearly recognized and evaluated even if the steps are embedded in a larger network. The coefficient k characteristic for the rate of conversion to products can be determined graphically from the slope of the straight-line portions of the curves in first-order batch plots of In x versus t, or numerically from k = - (Alnx)/At

(5.41)

with differences taken from within the range in which the curves are straight and parallel. The conversion rate coefficients k^p and k2Q can then be calculated from k and incremental conversions within that range: ^ip —

CJC,

^^^

k

-

q/c.

-AC,

(5.42)

Here, the isomer fractions Q / Q and C2/CA are those at steady-state distribution, given by eqns 5.35 unless the drain effect is appreciable and calls for a correction. Example 5.3. Hydroformylation of l-pentene with phosphine-substituted cobalt hydrocarbonyl catalyst [2]. Results of hydroformylation of l-pentene in a semi-batch reactor are shown in Table 5.5. Since the catalyst and the reaction conditions are similar, one might expect pentene to behave much like propene (see Example 5.2). However, a work-up of the results as for propene in Table 5.3 does not give consistent values of the characteristic coefficient k. Plots of \nx versus t for pentene, «-hexanol, 2-methyl pentanol, 2-ethyl butanol, and pentane are shown in Figure 5.9. The pattern resembles that of Case III in Figure 5.7 in that the curves start out with different slopes but later become parallel straight lines. This behavior is typical for coupled parallel first-order reactions with comparable rates of isomerization and conversion. Indeed, such a mechanism is eminently plausible because l-pentene, in contrast to propene, can undergo migration of the double bond from the terminal to an internal position. Attachment of CO to the carbon atom on either side of the double bond would lead to production of nhexanol and 2-methyl pentanol from l-pentene (double bond in terminal position), and 2-methyl pentanol and 2-ethyl butanol from 2-pentene (double bond between second and third carbon atom). This suggests the network 5.43, shown on page 116.

5.3. Coupled parallel steps

115

Table 5.5. Hydroformylation of 1-pentene in /i-dodecanol with phosphinesubstituted cobalt hydrocarbonyl catalyst and synthesis gas of 2:1 C0:H2 ratio at 50 atm and 150° C in semi-batch reactor.

concentration, M

time min

pentene

nhexanol

2-methyl pentanol

2-ethyl butanol

0 10 20 60 120 180 22 h

.659 .233 .210 .168 .127 .098 < .002

0 .350 .368 .393 .417 .436 .499

0 .030 .032 .039 .046 .051 .068

0 .0008 .002 .006 .011 .015 .027

D A V • o

pentane

0 1 .039 .041 .044 .046 .048 .053

w-hexanol 2-methyl pentanol 2-ethyl butanol pentane pentene

Injc

Figure 5.9. First-order batch plots of total olefin and products in hydroformylation of 1-pentene (date from Table 5.5)

116

Chapter 5. Elementary combinations of reaction steps

(A,) 1-pentene

^ ^

_ /-./-^.^oH

"-^^''^'"'^ (^)

OH 2-methyl pentanol (Q) (5.43)

«-pentane (S) 2-pentene /AS ^^^

^^

/^vv^^\ T ^OH

2-ethyl butanol (R)

[Actually, 2-pentene exists in cis- and rra/25'-configurations, but this complication is ignored here as relatively unimportant.] Although more complex than the prototype 5.33, this network includes the essential features of coupled parallel first-order reactions and can account for the curves in Figure 5.9. To estimate the isomerization rate coefficients, eqn 5.8 is applied to the time required for close approach to the straight-line behavior of the first-order curves. Judging this time to be about 25 minutes for a 90% approach to the steady-state isomer distribution, eqn 5.8 yields k-i^^ « 0.1 min"^ With this value and an isomerization equilibrium constant K12 = 20 at 150°C calculated from thermochemical data [5,6] (with 2-cis and 2-trans pentene lumped into a single pseudo-component), eqns 5.40 give as rough estimates k^^ ^ 0.1 min-^

k^^ ^ 0.005 min'^

The calculation of the conversion rate coefficients k^^, k^Q, /CIR, k^s, ^2Q> ^2R ^nd kzs (P = A2-hexanol, Q = 2-methyl pentanol, R = 2-ethyl butanol, S = pentane) from the limited data available requires some judgment calls. First, the steep initial slope of the plot for pentane indicates that this product is formed practically exclusively from 1-pentene, so that ^2s = 0 and the entire pentane yield is attributable to k^^Second, for lack of better information, one may tentatively assume that the ratio k^Q :/:iP of branched- to straight-chain alcohol formation from 1-olefin is the same as for propene in Example 5,2, namely 0.058:0.677 = 0.086, and to attribute 2-methyl pentanol produced in excess of this to /CJQ- Experiments with other initial isomer distributions and sampling at short reaction times would be needed to confirm these two guesses. If they are accepted, a value of A: = 0.0045 min'^ from the straight-line slopes of the first-order plots and the data on incremental conversions in the range beyond 60 minutes give the following results with eqns 5.42:

5.3. Coupled parallel steps

k^p = 0.059 min-\

k^^ = 0.005 min-\

it^Q = 0.0007 min-\

111

k^^ = 0.006 min-\

^^R = 0-0006 min-^

The values of ^^p, k^Q, and k^^ for conversion of the highly reactive isomer Ai still await regression to account for a possible drain effect, which in this case turns out to be rather severe: The corrected coefficients are almost three times as large as the estimates above. Otherwise the fit is fairly reasonable, with the following exception. The only unsatisfactory feature of the fit is that the first-order plot of the observed results for pentane dips farther down than that for n-hexanol, the product whose initial rate should be boosted most strongly by the initial abundance of 1pentene, its only parent (see Fig. 5.9). This behavior is at odds with any fit based on the network 5.43. A better fit would be obtained with a physically impossible negative value of kjs or inclusion of a mechanistically inconceivable direct pathway from 2-pentene to n-hexanol. The most likely explanation of the effect is that mass transfer from the gas phase could not quite keep pace with the extremely high initial rate of consumption of CO and H2 in the liquid, so that the concentrations of these reactants in the liquid decreased temporarily. The effect would be stronger for CO as the larger molecule, and thereby cause a temporary increase in the H2-to-CO ratio in the liquid. A higher ratio favors paraffin production at the expense of the alcohols (see Example 7.5 in Section 7.3.2) and so can explain the observed behavior. (For more detail on mass-transfer effects in this reaction, see Section 13.3.2). This has been a typical example of how coupled parallel steps or reactions can be recognized in a practical situation, how isomerization embedded in a larger network manifests itself, and how the results of a single batch experiment can yield a wealth of kinetic information and lead to a workable model. However, a good fit to the results of one experiment does not prove the network to be correct, even if backed up by a very plausible mechanism. Confirmation from experiments starting with other isomer compositions is needed, quite apart from the fact that each of the arrows in a network such as 5.43 represent multistep pathways rather than single steps and that the branches may be at nodes along these pathways (for more detail, see Section 7.3.2). In hydroformylation, convincing evidence that the basic structure of the network 5.43 is correct has come from experiments with long-chain olefin isomers [7]: If the starting material is a 1-olefin, the first-order plots for the alcohols as in Figure 5.9 are such that that for the straight-chain product dips deepest, that for the alcohol with the -CH2OH group at the second carbon atom dips second deepest, etc., and that for the alcohol with the group attached to the middle of the chain dips least; and if the starting material is the isomer with double bond in the center of the molecule (say, the 6-position in dodecene), the sequence is the exact opposite. This confirms the essential features of the network 5.43, namely, that conversion occurs in concert with double-bond migration along the carbon chain and that the alcohols arise from attachment of CO and H2 to a carbon atom on either side of the double bond.

118 5.4.

Chapter 5. Elementary combinations of reaction steps Sequential steps

In almost every chemical reaction, the product or products may react further or decompose. Optimization with respect to yield and product purity then relies on evaluation of sequential steps (also called consecutive steps or steps in series). Other situations involving sequential steps are those in which a substitution or transformation such as chlorination, oxidation, dehydrogenation, etc., occurs successively at different carbon atoms of a reactant molecule, or those involving repeated addition of a reagent to the original reactant, as in ethoxylations of ethanol and other low alcohols to glycol ethers [8,9] (e.g., Union Carbide's Cellosolve and Carbitol family of solvents) and of long-chain primary alcohols in manufacture of detergents [10]. Lastly, polymerization is a sequential reaction/?(2r excellence and so important that it has been assigned its own chapter (see Chapter 11). 5,4.1.

Sequentialfirst-ordersteps

pathway:

A —• K —> P

(5.44)

rate equations: ^A ~

'•K

~ ^AK ^A '

= ^AKCA-^KPQ.

(5.45)

This is the simplest case of sequential steps. Integration for constant-volume batch Figure 5.10. Concentrations vs ^j^,^ ^^j ^ ^^^^^^ jj^i^i^n j ^ ^ , time for first-order sequential steps A — • K — • P i n batch. c^ ^ C^exp(-k^t) (5.46) ^AK

k

c. = c:

- k

1 -

(exp(-/:^j^O - exp(-/:j.pO)

(5.47)

- ^AK^XP(-V) - k

(5.48)

K?^^V{-KJ)

k

The concentration of the intermediate K reaches its maximum at the time ^™> "

and the height of this maximum is

"it

nt

(5.49)

5.4. Sequential steps

119

(Q).ax = Clik^lkJ'-'^'--'^''

(5.50)

The latter two equations are useful for determination of the rate coefficients from experimental results. Equations 5.47 to 5.50 are not applicable if /:KP = ^AK- In that case they must be replaced by Q

= Clkttx^{-kt)

(5.51)

Cp = C;(l - (I + kt)Qxp(-kt)) t^ = Ilk (QU

= C;/e

(5.52) (5.53) (5.54)

where k = ^^K = ^KP» ^^d ^ = 2.71828 is the basis of natural logarithms. If K is also present initially, eqn 5.47 contains an additional term CKexp(-/:KpO on the right-hand side; and eqn 5.48, a term CK(1 - exp(-/:KpO). Also, eqns 5.49 and 5.50 no longer apply. The product P is inert; accordingly, any P present initially has no effect on A and K and only adds to the amount of P formed in the reaction as given by eqn 5.48 and 5.52. For a continuous stirred-tank reactor, the concentrations as a function of the reactor space time r are: CA = C;/(1 + ^AKT) -'K

(5-55)

(1 + k^yj){\ + fcj^pT)

(1 + k^r){\

+ fc^pT)

The maximum concentration of the intermediate K is attained at the reactor space time

and the height of the maximum is =

(^AK/fcKp)CA° [(1 + (k^/k^p)

.555. ]

Often, the pathway consists of more than two steps: A —• K, —• Kj —• ... —V Ki —.• ... —• ...

(5.60)

120

Chapter 5. Elementary combinations of reaction steps

The equations stated above for A and the first intermediate (now K^ instead of K) then apply unchanged, and those for P give the sum of the concentrations of all participants beyond the first intermediate. This is because once a molecule has reacted to form K2, it never reverts to Kj or A. Mathematics for products beyond Ki has been developed [8,11-13], but is complex except if all rate coefficients are equal. If so, the concentration of the i'th intermediate is given by [14,15]: (5.61)

c:^txp(-kt)

Example 5.4. Ethoxylation of alcohols [10]. In the presence of base as catalyst and at moderate temperature, liquid alcohols react with gaseous ethene oxide, which is inserted as -OC2H4~ between the bulk of the molecule and the - O H group. The resulting alcohol successively inserts further — OC2H4— blocks to form higher adducts:

V R-OH

V RO'^'^^-v^OH

V

V

RO-'^^^Cr^v^OH ^^==^^ RO'''"^^0'^^0'\/OH

This reaction is important for the manufacture of household detergents, many of which are sulfated ethoxy adducts of straight-chain Cjo to C16 primary alcohols averaging two to six ethoxy blocks per molecule.

t [min] Figure 5.11. Concentration histories of alcohol and first four successive ethoxy adducts in batch ethoxylation at constant partial pressure of ethene oxide, calculated fox k^ = 0.012 min"^ for reaction of alcohol, k2 = 0.015 min~^ for reaction of adducts (ale = alcohol; 1-EO, 2-EO, 3-EO, 4-EO = first, second, third, and fourth adduct, respectively).

5.4. Sequential steps

121

All steps from the second on amount to insertion of an ethoxy block between a previously inserted block and the -OH group, and so have very similar rate coefficients. Usually, the original alcohol reacts at a slightly lower rate. If the reaction is carried out at constant partial pressure of ethene oxide, each insertion including the first is pseudo-first order in the alcohol or ethoxy alcohol reactant. With increasing reaction time in batch, successive adducts reach maximum concentrations and then decay to form higher adducts, as shown for a calculated case in Figure 5.11. The variation in yield structure with reactor space time in a continuous stirred-tank reactor is similar, but with less pronounced concentration maxima. Often, the first product in a pathway of sequential steps is the desired one. In such cases, the yield ratio of the first product (K) to the subsequent ones (lumped into P) is of special interest. The instantaneous yield ratio is

At very low conversion, when the concentration of A is still high and little K has as yet been formed, the yield ratio is favorable; with progressing conversion it declines and at some point becomes negative as the decay rate of K starts to outrun the formation rate. Batch and plug-flow tubular reactors give better yields at same conversion than does a continuous stirred-tank reactor. This is because in a batch or tubular reactor the yield ratio is favorable at least initially—in the batch early on, in the tube near the inlet—and deteriorates only as conversion progresses, whereas in the stirred tank it is at the worst, final-conversion level all the time and in all of the reactor because the composition in the latter equals that of the effluent. The situation is much the same if an early intermediate is the desired product.

If the desired product is the first or an early intermediate in a pathway of sequential steps, batch or plug-flow tubular reactors provide better selectivity than do continuous stirred-tank reactors.

The (cumulative) selectivity for K (fraction of reacted A that is converted to K, see definition 1.8) in a batch reactor is \

-

^AK(exp(-^.KO - exp(-/:KpO) (^KP - W ( l - exp(-/:AKO) \

= ^

/^^^P(-^;>^ 1 - exp(-fe)

jf ,^^ ^ ,^^

iik,^^k^,{=k) ""^ ^^

(5.63)

(5.64)

122

Chapter 5. Elementary combinations of reaction steps

and in a continuous stirred-tank reactor: (5.65)

1 + k^^r

Equations 5.63 to 5.65 give the selectivity as a function of time or reactor space time. More helpful in practice, however, is the dependence on fractional conversion, /A. For a batch reactor: 1

^AK ''KP

(1 - f \^^'^^^

-/A

(5.66) ^ f '^AK ' ^ ^ K P

A

^AK

^ -/A

In

if k AK

1

/A

"•KP

(5.67)

-/A

and for a continuous stirred-tank reactor: 1 ^K

-/A

(5.68)

=

1 -/A(^KP/^AK-

1)

This reduces to 5K = 1 - / A if the two coefficients are equal. While the selectivity to K decreases monotonically with conversion, the concentration of K is at its maximum in a batch reactor at I

_

^AK AK

it - k

if kj^p 9^

(^A), A^K^

k^

(5.69)

[ KP

(f^)^^^

= 1 - 1/e

= 0.6321

if kj^p - k^^

(5.70)

and in a continuous stirred-tank reactor at 1 (/A)K.

1 +

1/2

(5.71)

(k^^/k^)

or (/A)Kmax ~ 1/2 if thc rate coefficients are equal. Example 5.5. Oxidation ofparaffins to secondary alcohols (see also Section 10.6.2). Alcohols can be produced by oxidation of paraffins with air or oxygen at moderate temperatures (typically 120 to 180° C) in the presence of borate esters or boroxines [16-19]. These intercept the alkyl hydroperoxide, the first oxidation product, preventing it from generating radicals that would cause further degradation including scission of carbon-carbon bonds and produce aldehydes, ketones, and acids. The peroxy borates so formed then are hydrolyzed to yield the alcohol. The carbon atoms at the chain ends are largely immune to oxidation, so the product consists predominantly of isomeric secondary alcohols.

5.4. Sequential steps

123

The reaction does not stop at the mono-alcohol. Rather, subsequent oxidation at other carbon atoms introduces additional ~0H groups: 1/20,

paraffin

1/2O2

mono-alcohol

V2O,

di-alcohol

V2O2

tri-alcohol

Since each step consists of the same chemical event, the oxidation of a secondary carbon atom, the rate coefficients of all steps are almost the same. [With each attack, the number of still unoxidized secondary carbon atoms in the j ^ ^ / ^ 5 5 Cumulative selectivity to first molecule decreases, and so intermediate in pathway of first-order steps does the statistical probability A-~^ K - > ... with equal rate coefficients, of further oxidation; however, for the first few products of a selectivity toK fractional long-chain paraffin, this effect conversion remains minor.] At constant cj(c:- -CA) partial pressure of oxygen, the batch CSTR A steps are pseudo-first order in the respective organic reactant. 0.892 0.20 0.800 The dependence of the selec0.50 j 0.693 0.500 tivity to mono-alcohol, usually 0.462 0.75 0.250 the desired product, on reactor 0.256 0.90 0.100 type and conversion level is dramatic, as the data in Table 5.6 demonstrate. 5.4.2.

Sequential steps of other orders

As long as the steps of the sequence are irreversible, the original reactant behaves as in single-step decay, regardless of the reaction orders. Concentration histories of later members in sequences that include steps of orders other than first have been derived for only a few simple cases and are unwieldy even for these [20,21]. Here, numerical solution on a computer is preferable. However, two general rules can be stated: •

•

If the desired product is the first or an early intermediate, batch and plugflow tubular reactors provide better selectivity than does a continuous stirredtank reactor. Higher concentration of the original reactant favors the intermediates whose formation rates are of higher reaction orders than are their decay rates.

The first of these rules had already been stated for pathways of first-order steps, the second is a consequence of the stronger concentration dependence of reactions of higher order (see also Section 5.2.3).

124

Chapter 5. Elementary combinations of reaction steps

The rules have implications for reactor choice and operating conditions in situations in which a desired product undergoes subsequent decay. If the desired reaction is of higher overall order than the decay, selectivity is higher in batch or plug flow reactors than in CSTRs, and is higher at higher charge or feed concentrations in reactors of any type. If the decay is of higher order, the opposite is true. 5.5.

Competing steps

The term competing steps (or series-parallel steps) is used if one and the same component participates as reactant in more than one step of the pathway or network. [The idea is that such steps "compete" with one another for the reactant they have in common.] The simplest such case is [22] A P^^^^yA . K^ ^ P More interesting is the slightly more complex pathway with reversible first step: pathway:

A ^^ V ^ p

^ ^

rate equations:

(5.72)

r^ = -k^^C^ + k^^C^ - k^.C^C^ ^X

^

^AX^A ~ ^XA^X ~ %P^A^X

Tp

=

^^XP^A^X

(5.73)

Many condensation reactions are of this general type, though the actual mechanism may be more complex (see also Section 8.2.1). If the intermediate X in the pathway 5.72 remains at trace level, the Bodenstein approximation can be used (see Section 4.4) and gives ^x

=

^AX^A'(^XA ^ ^XP^A)

so that ^AX%P^A

^^

""

k

/g

J^\

+ k C

'^XA ^ '^XP'^A

Although all exponents are integers, the rate equation is in general not a power law, so the reaction order may vary with conversion. Two special cases are noteworthy: Case I (/:xA « ^XP^A) rp

=

k^x^A

reaction is first order in A

Case II (k^A » ^xpQ) ^P

=

(^AX%P

'kxAJ^A

reaction is second order in A

5.6. Reactions with fast pre-dissociation

125

A reaction 2A —• P with reversible step A -4—• X preceding conversion A + X —• P may be first order or second order or have any order between these two, depending on the relative magnitudes of the coefficients.

In Case I, the reverse step X —• A is negligible because it is outrun by the much faster second step A + X —• P. Here, the slow first forward step controls the rate. In Case II, the step A + X —• P is so slow that quasi-equilibrium is established in the first step. Here, the reaction rate is proportional to the rate coefficient of the slow, second step, but is also affected by the equilibrium in the first (equilibrium constant ^^x = ^AX^^XA)As this examination shows, the order of a reaction with pathway 5.72 depends on the relative magnitudes of/TXA andfcxp^A.Cases I and II, with first- and second-order behavior, respectively, are the extremes. In between, with the two terms in the denominator of eqn 5.74 of comparable magnitude, the order is between first and second and increases with conversion. It is even possible for the same reactant to have different reaction orders under different conditions, for instance, in different solvents. The pathway 5.72 is also the simplest that can produce other anomalies. If the equilibrium constant ^^x of the first step decreases with increasing temperature, and does so more strongly than the rate coefficient k^? increases, the rate of the overall reaction may decrease with increasing temperature (negative activation energy, see Section 13.1.1.). Moreover, If the intermediate builds up to substantial concentrations, the reaction may be self-accelerating (see Section 8.9.1). The pathway 5.72 is the simplest in which a reversible step precedes a competing irreversible one. Such behavior is quite common, especially in catalysis. A reaction order that varies between one and two with respect to the reactant for which the steps compete is a typical symptom, even though the actual pathway usually is more complex. Two examples—aldol condensation and ethyne dimerization—will be examined later (see Examples 6.4 in Section 6.4.3 and 8.2 in Section 8.2.1). 5.6.

Reactions w^ith fast pre-dissociation

A number of reactions involve dissociation of a reactant as the first step. In many such cases, this step is reversible and fast enough compared with subsequent conversion of the dissociation products to be at quasi-equilibrium. Typical examples are gas-phase reactions of halogens or hydrogen halides. The simplest network of this type is

126

Chapter 5, Elementary

combinations

of reaction

steps

(5.75)

With the quasi-equilibrium condition 1/2

2

= ^d SO that p^ = (K.p^) (Ki = dissociation constant of Aj) the rate equation becomes /'A/PA,

If conversion of A is reversible, a term for the reverse step appears: r, ^

K,p^{K,p^r

- k,,p,

{5.11)

Conversion of the dissociation product may occur via a more complex pathway rather than a single step, or may involve other co-reactants or co-products. However, as long as that conversion is first order in A, the square-root factor {KdPk2f^ in the rate equation remains unchanged. The reaction may start with dissociation of a reactant into unequal fragments. The rate equation then becomes complex unless dissociation equilibrium is highly unfavorable so that the dissociation products remain at trace level: B X—^^=^=^

•P

C

(5.78)

If so, the formation rates of P and Q are necessarily equal by virtue of the overall stoichiometry, A —• P + Q (amounts present as intermediates remain negligible). This leads to a simple rate equation: rp = r^ = iK.k^k^^p.p^pf'

(5.79)

If conversion of the dissociation products is reversible, an explicit rate equation can still be obtained, but involves the root of a quadratic equation. Derivation of eqn 5.79. From the equality of the formation rates of P and Q ^p

one finds

= ^X?PXPB

^ ^Q "

^YQPYPC

5.7.

General solution for first-order networks

PY

=

127

PX^XPPB^^YQPC

Using this relationship to replace py in the quasi-equilibrium condition PXPY^PA

=

^d

and solving for p^ one obtains Px =

(K,P^k^,Pc^k^pPsf'

Substitution of this expression for Px in the rate equation rp = k^^PxPh gi^^s eqn 5.79. Characteristic of the rate equations 5.76 and 5.79 is their one-half order with respect to the dissociating reactant, in the case of eqn 5.79 with respect to the coreactants B and C as well. This is an exception to the rule that a reasonably simple mechanism does not give a rate equation with fractional exponents. Conversely, an observed, conversion-independent order of one half is an indication that the reaction might involve fast pre-dissociation. On the other hand, a power-law rate equation with integer reaction orders cannot be taken as evidence against fast pre-dissociation. An example is the hydrogen-iodide reaction, which involves fast pre-dissociation although the reaction orders are integers (see Example 4.2 in Section 4.3). Here, conversion 21 + H2 —• 2HI of the dissociation products is second order in these, so that the square-root factor is squared, making the reaction first order in I2. Positive or negative fractional exponents of one half or integer multiples of one half are also common in rate equations of chain reactions, where, however, they are caused by bimolecular termination steps rather than fast pre-dissociation (see Chapter 10). Apart from chain reactions, the most common occurrence of fractional exponents of one half or integer multiples of one half is in heterogeneous catalysis. Such a case, hydrogenation of propanal over nickel, will be discussed in detail in Example 9.4 in Section 9.3.1. 5.7.

General solution for first-order networks

An elegant, general solution for first-order networks has been provided in a classic publication by Wei and Prater [23].* In essence, the mathematics are developed for a reaction system with any number of participants that are all connected with one another by direct first-order pathways. For example, in a system with five participants, each of these can undergo four reactions, for a total of twenty first-order steps. Matrix methods are used to obtain concentration histories in constant-volume batch reactions, and a procedure is described for determination of all rate coefficients from such batch * An abstract mathematical solution without the concept of reaction paths and no guidance for determination of coefficients from experiments had been given earlier by Matsen and Franklin [24].

128

Chapter 5. Elementary combinations of reaction steps

experiments. The method is clearly presented in the original publication [23] and in several advanced texts on kinetics [25,26], to which the reader wishing to apply it is referred for details. Only a brief outline is given here. In principle, all the reactions in a completely interconnected network are coupled with one another. The key to obtaining explicit solutions is to uncouple them. Wei and Prater achieved this by defining and identifying "pseudo-components" which are combinations of real components in ratios such that they do not interconvert into one another. If the amounts of the real components are divided up among the pseudo-components in this way, the latter react independently of one another, that is, their reactions are mathematically uncoupled. This allows the rate coefficients to be determined from experiments and provides for a better understanding of reaction behavior. The determination of the rate coefficients from experimental results ^ is best described with an example. The simplest case is that of interconversion of three isomers. Figure 5.12 shows histories of several batch experiments, plotted in a composition diagram (triangular diagram with the isomer mole fractions as coordinates). The composition variation with time appears as a curved path. The path of each experiment ends up at the equilibrium point, e. A first experiment starts with pure isomer A^ (point 1) and traces a curved path, and so do experiments starting with mixtures of Ai and A3 corresponding to points 2, 3, and 4. Determination of the rate coefficients calls for identification of the point from which a straight-line Figure 5.12. Reaction paths in a three- path originates. By trial and error, component, first-order reaction system (from point 5 is found to meet that requireFroment and Bischoff [26], schematic). ment. For confirmation, an experiment is started with an A1-A2 mixture corresponding to point 6, on the linear extension of the straight-line path from 5 to e, and its history is found to follow that extension as required. In a like manner, the end points g and h of a second straight-line path across the diagram are identified. Matrix manipulations can then be used to calculate the values of the six rate coefficients from the compositions 5, 6, g, and h and the rates at which the composition points move from these points toward the equilibrium point. [The points of origin of the straight-line paths are the pseudo-components. ] The method is noteworthy for its generality and mathematical elegance and for the insight into reaction behavior it has contributed. However, it is more often quoted than applied in practice. The reason is not mathematical complexity; today, the required

5.7.

General solution for first-order networks

129

matrix operations are easily performed on a computer. Rather, as the authors themselves pointed out, a relatively large number of experiments must be performed and evaluated. Also, if the number of participants is large, the very generality of the method becomes a disadvantage because many straightline paths in a multidimensional space must be determined, and the possibility of converging on a false solution becomes real. Lastly, the method, K although general in all other respects, has one limitation in that it implies a Figure 5.13. Reaction paths in a threefully reversible system with unique component, first-order reaction with pathequilibrium composition. This may be way A—•K—•? (schematic). illustrated by two examples of simple cases in which the method fails to give complete results. Figure 5.13 shows typical reaction paths of a three-component system with sequential, irreversible steps A—^K—•? (see also Section 5.4). There is only one straight-line reaction path, from K to P. Any initial mixture containing A produces a curved path. Here, the method yields the coefficient /TKP (from the rate of advance along the straight-line path A—•K) and the information that the coefficients /:pK, ^PA, ^KA> ^^^ kfi^p must be zero or close to zero, but no value of ^AKThe second example is that of a network 5.33 with coupled parallel steps (see Section 5.3):

(5.33)

Here, too, only one straight-line reaction path can be found, namely, that for initial mixtures containing the isomers A^ and A2 in their steady-state ratio [i.e., as homogeneous source as defined in Section 5.3). This mixture is a pseudo-component. Unless isomerization is very fast compared with conversion, there is no unique equilibrium point because the relative amounts of P and Q at complete conversion depend on the initial isomer distribution. The single straight-line path and the rate of progress along that path are insufficient for determination of the four non-zero rate coefficients of the network. The method is at its best when applied to interconversion of not too large a number of isomers such as the butenes. For networks involving combinations of multistep pathways, techniques to be described in the next chapters are usually easier to handle.

130

Chapter 5. Elementary combinations of reaction steps

Summary This chapter surveys the kinetic behavior of the following step combinations: reversible steps sequential steps parallel steps competing steps coupled parallel steps reactions with fast pre-dissociation. The various combinations may show their characteristic symptoms, or be made to do so by choice of experimental conditions, even if they are composed of multistep pathways rather than single steps or are embedded in larger networks. In first order-first order reversible reactions, the rate of approach to equilibrium is proportional to the fractional distance from equilibrium, measured in terms of concentrations or any other quantity that is a linear function of the concentrations. Unless carried to high conversion, the rate of a reversible reaction may be indistinguishable from that of an irreversible reaction of higher order. The rates of product formation in parallel first-order steps are proportional to the fractional distance from equilibrium, which is the same for all participants. In reactions with parallel steps of different reaction orders, the selectivity to the product formed by the parallel step of higher order is higher in batch or plug-flow than in continuous stirred-tank reactors, and decreases with progressing conversion in any type of reactor. Typical examples of coupled parallel steps are isomerizations in concert with conversion of the isomers to different products. If isomerization is fast compared with conversion, the isomers are at quasi-equilibrium, producing a kinetic behavior like that of a single reactant. If isomerization is slow compared with conversion, the reactions of the isomers are essentially uncoupled; the apparent reaction order of decay of total reactant (sum of the isomers) is higher than that of the individual isomers. If the rates of isomerization and conversion are comparable, the isomer distribution approaches a steady state that is not necessarily close to isomerization equilibrium, and from then on the rates become proportional to the fractional distance from equilibrium. In reactions with sequential steps, the selectivity to the first intermediate is higher in batch and plug-flow than in continuous stirred-tank reactors, and decreases with progressing conversion in any type of reactor. Reactions in which a step competes for the same reactant with an earlier reversible step are the simplest that may show features not found in single-step reactions: The rate equation may not be a power law, the apparent reaction order may be fractional and vary with conversion, the rate may be lower at higher temperature, and the reaction may be self-accelerating. Reactions with fast pre-dissociation may, but need not, lead to fractional reaction orders of one half or integer multiples of one half or non-power law rate equations involving such exponents. A brief overview of the Wei-Prater general mathematical solution for arbitrary networks consisting entirely of reversible first-order steps is provided. Examples include reversible lactone formation, hydroformylation of propene and 1-pentene, ethoxylation of aliphatic alcohols, and oxidation of paraffins to secondary alcohols.

References

131

References General references Gl.

S. W. Benson, The foundations of chemical kinetics, McGraw-Hill, New York, 1960, updated and corrected reprint, Krieger, Melbourne, 1982, ISBN 0898741947, Chapters III and V. G2. J. J. Carberry, Chemical and catalytic reaction engineering, McGraw-Hill, New York, 1976, ISBN 0070097909, Section 2-9. G3. N. H. Chen, Process reactor design, Allyn & Bacon, Boston, 1983, ISBN 0205079032, Sections 3.12 to 3.15. G4. K. A. Connors, Chemical kinetics: the study of reaction rates in solution, VCH Publishers, New York, 1990, ISBN 3527218223, Section 3.1. G5. J. H. Espenson, Chemical kinetics and reaction mechanisms, McGraw-Hill, New York, 2nd ed., 1995, ISBN 0070202605, Chapters 3, 4, and 5. G6. C. G. Hill, Jr., An introduction to chemical engineering kinetics and reactor design, Wiley, New York, 1977, ISBN 0471396095, Chapter 5. G7. O. Levenspiel, The chemical reactor omnibook, OSU Book Stores, Corvallis, 1989, ISBN 0882461648, Chapters 2 and 8. G8. J. W. Moore and R. G. Pearson, Kinetics and mechanism: a study of homogeneous chemical reactions, Wiley, New York, 3rd. ed., 1981, ISBN 0471035580, Chapter 8. G9. J. I. Steinfeld, J.S. Francisco, and W. L. Hase, Chemical kinetics and dynamics, Prentice-Hall, Englewood Cliffs, 2nd ed., 1999, ISBN 0137371233, Chapter 2. GIO. Z. G. Szabo, in The theory of kinetics, Vol. 2 of Comprehensive chemical kinetics, C. H. Bamford and C. F. H. Tipper, eds., Elsevier, Amsterdam, 1969, ISBN 0444406743, Chapter 1. Specific References 1.

2. 3. 4. 5. 6. 7.

N. M. Emanuel' and D. G. Knorre, Kurs khimicheskoi kinetiki (gomogennye reaktsii), Izdatel'stvo Vysshaya Shkola, Moscow, 2nd ed., 1969; English translation Chemical kinetics—homogeneous reactions, Wiley, New York, 1973, ISBN 0706513185, pp. 167-169. F. G. Helfferich, unpublished, 1965. F. H. Blanding, Ind. Eng. Chem., 45 (1953) 1186. V. W. Weekman, Jr., Ind. Eng. Chem., Proc. Des. Dev., 16 (1968) 90. Hill (ref. G6), Appendix A. Handbook of chemistry and physics, 80th ed., D. R. Lide, ed. CRC Press, Boca Raton, 1999, ISBN 0849304806, Section 5, pp. 40 and 47. F. G. Helfferich, Technical Report 203-65, Shell Development Company, 1965.

132 8. 9.

10.

11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.

Chapter 5. Elementary combinations of reaction steps B. Weibull and B. Nycander, Acta Chem. Scand., 8 (1954) 847. J. E. Logsdon, Ethanol, in Kirk-Othmer, Encyclopedia of chemical technology, 4th ed., J. I. Kroschwitz and M. Howe-Grant, eds., Wiley, New York, Vol. 9, 1994, ISBN 0471526770, p. 846. J. L. Lynn, Jr., and B. H. Bory, Surfactants, in Kirk-Othmer, Encyclopedia of chemical technology, 4th ed., J. I. Kroschwitz and M, Howe-Grant, eds., Wiley, New York, Vol. 23, 1997, ISBN 0471526924, p. 501-502. E. Abel, Z. Physik. Chem., 56 (1906) 558. Szabo (ref. GIO), Section 1.IV.3. A. E. R. Westman and D. B. DeLury, Can. J. Chem., 34 (1956) 1134. P. J. Flory, J. Am. Chem, Soc, 62 (1940) 1561. Benson (ref. Gl), Section III.6. A. N. Bashkirov, Dokl Akad. Nauk SSSR, 118 (1958) 149. F. Broich and H. Grasemann, Erdol Kohle Erdgas, Petrochem., 18 (1965) 360. L. C. Fetterly, C. W. Conklin, K. F. Koetitz, F. G. Helfferich, P. W. Gilderson, and S. F. Newman (to Shell Oil Co.), USP 3,375,265 (1968). N. Kurata and K. Koshida, Hydrocarbon Process., 57(1) (1978) 145. Benson (ref. Gl), Section III.7. Szabo (ref. GIO), Section 2.IV. M. Frenklach and D. Clary, Ind. Eng. Chem., Fundam., 22 (1983) 433. J. Wei and C. D. Prater, Adv. CataL, 13 (1962) 203; AIChE J., 9 (1963) 77. F. A. Matsen and J. L. Franklin, J. Am. Chem. Soc, 72 (1950) 3337. M. Boudart, Kinetics of chemical processes, Prentice-Hall, Englewood Cliffs, 1968, Section 10.2. G. F. Froment and K. B. Bischoff, Chemical reactor analysis and design, Wiley, New Work, 2nd. ed., 1990, ISBN 0471510440, Sections 1.4 and 1.7.

Chapter 6 Practical Mathematics of Multistep Reactions This chapter addresses the estabUshment of practical mathematics of kinetics for given pathways and networks. The complementary problem of establishing pathways or networks from observed kinetic behavior will be taken up in the next chapter. The discussion of special aspects of catalysis, chain reactions, and polymerization is deferred to Chapters 8 to 11. In principle, a multistep reaction with any network is fully described by the complete set of rate equations of all participants, compiled as shown in Section 2.4. However, if the reaction is complex, the experimental work required to verify the assumed mechanism and to determine all its coefficients and their activation energies can get out of hand. A reduction of complexity then is imperative, and is also desirable for both a better understanding of reaction behavior and more efficient numerical modeling. The present chapter describes ways of achieving this. The key elements of this approach are • the use of pseudo-first order rate coefficients, • a reduction of the number of simultaneous rate equations by use of the Bodenstein approximation, and • the establishment of general equations and algorithms for rates and yield ratios applicable to pathways, network segments, and networks regardless of their number of steps, locations of nodes, and points of co-reactant entries and co-product exits. 6.1.

Simple and non-simple pathways and networks

For convenience, a distinction is made between "simple" and "non-simple" pathways or networks. A pathway, network, or any portion of one of these is called simple if it meets both of the following criteria: • all intermediates are and remain at trace level, and • no step (forward or reverse) involves two or more molecules of intermediates as reactants."^ * Networks which meet this second condition have been called "linear" [1,2], This term is avoided here because of the seeming self-contradiction if a three-dimensional structure with branches is declared linear.

134

Chapter 6. Practical mathematics ofmultistep reactions

The first of these conditions ensures that the Bodenstein approximation of quasistationary behavior (see Section 4.4) can be used for all intermediates, the second guarantees that the algebra is linear. If both conditions are met, explicit equations or algorithms for rates and yield ratios of all reactants and products can be given, regardless of the actual complexity of the pathway or network. To be sure, a simple pathway or network, or any portion of these, may be of any size and topology and contain any number of steps with higher molecularities as long as the co-reactants or co-products are not themselves intermediates: allowed steps

disallowed steps

Xj + A —• Xk, X. — X, + Q, Xj + A -f B —• Xk

2X3 —• Xk, Xj + X, — X,, 2Xj + A —• Xk

where A, B, and Q are bulk reactants and products, and the Xj are intermediates. A non-simple pathway or network can be broken at the offending point or points into piecewise simple portions (see Section 6.5). The notation Xj will remain reserved for trace-level intermediates, with sequential indices. In indices of concentrations, rates, rate coefficients, etc., X is suppressed for simplicity. For instance, the concentration of Xj is given as Cj; the formation rate of Xj, as r,; and the rate coefficient of a step Xj—• X^, as k^^. For consistency in indices of sums and products, rate coefficients of steps A —• X^ are given as k^^, those of steps X^.i —• P as k^.^^ (^ "= ^^^ product). Intermediates that are not, or do not remain, at trace level are designated K, L, etc.

6.2.

Pseudo-first order rate coefficients

To use linear algebra as much as possible, pseudo-first order rate coefficients are introduced for steps with co-reactants. Each such coefficient is the product of the actual, higher-order rate coefficient and the concentration of the respective coreactant (or concentrations of the co-reactants, if there are several) raised to the power that corresponds to the respective molecularity. For example, for a step Xj + B — •

Xk

the rate contribution (

^j ) j - k

~

(^k)j-k

~

^jk^j ^ B

is written instead (~^j)j-k

=

(^k)j-k

=

^jkCj

6.3. General formula for simple pathways

135

where V

-

k^,C^

(for Xj + B — X J

(6.1)

This convention is used for all forward and reverse steps with more than one reactant (provided no more than one molecule of intermediates functions as reactant). For steps without co-reactant, e.g., Xj—• X^ or Xj—• X^ + Q, the X coefficient is identical with the actual, first-order k coefficient: XJ, -

k^,

(forXj-> ...)

(6.2)

With this convention, and provided no step involves two or more molecules of intermediates as reactants, any reaction with or without co-reactants and coproducts and with actual rate coefficients k^^ becomes mathematically formally equivalent to one composed exclusively of pseudo-first order steps with pseudo-first order rate coefficients Xj,.* As an example, a pathway B A A < ^ — > X, ^ - ^

X2 ^ ^

» X3 <-

Q is equivalent to the pseudo-first order pathway A

<

•

Xi

<

•

X2 <

•

X3

with coefficients Xoi = ^01 ^B »

X12 = ki2 ,

X23 = k23C^ ,

X34 =

\o

X21 = k2iCQ ,

X32 = ^32 ,

X43 = k42 .

= ^10 »

^34 ,

6.3. General formula for simple pathways The keystone in the reduction of mathematical complexity is a general rate equation for simple (linear) pathways with arbitrary number of steps and with or without coreactants and co-products [6-9]. For the arbitrary simple pathway A <-^

Xi <^—• X2 <—•

... <—> X^.i ^i—• P

(6.3)

with or without co-reactants and co-products (not shown) and, by definition, with all intermediates Xj at trace level, the full set of rate equations is * The pseudo-first order rate coefficients Xj^ used here were first introduced by Christiansen [3] as "reaction probabilities" Wj. Equivalent quantities are also standard in generalized treatments of chain reactions [4,5].

Chapter 6. Practical mathematics ofmultistep reactions

136

^A

=

-\l^A

+ ^10<^1

'"i = ^^i-u^M - (\i-, ^P

~

- KJc.,

+ \^,,q.,

(i = l,...,k-l)

^k-l,k^k-l ~ ^k,k-l^P

(Indices 0 and k in X coefficients are used for A and P, respectively, to avoid complications in later formulas with sums and products.) If the end members A and P additionally act as co-reactants and co-products, the respective rates r^ or r^ must be replaced by il/nj^)r^ or (l/AZp)rp to account for the stoichiometry. After elimination of the concentrations of all the intermediates by repeated application of the Bodenstein approximation, the set of rate equations can be reduced to the single rate equation -r.

=

r„ =

A.,C. - A . C .

where

(6.4) k-l

A„^

=

n K. Ko =

D, j-1

D.Ok

i=0

D.Ok

k-l

n K. n K i=i

(6.5)

(6.6)

i=j

(products n to be taken as unity if lower limit exceeds upper). As eqn 6.4 shows: A multistep simple pathway with or without co-reactants or co-products can be reduced to a pseudo-single, pseudo-first order step

The forward and reverse segment coefficients AQ^ and Aj^o of that pseudo-single step, however, are not in general concentration-independent. Given by eqns 6.5 and 6.6, they are functions of the true rate coefficients k^^ of all steps and of the concentrations of any co-reactants and co-products, but are independent of the concentrations of the intermediates. The problem of accounting for the effects of co-reactants and co-products has not been solved, but has been deferred to a time when it is more easily taken care of.

6.3. General formula for simple pathways

137

Equations 6.5 and 6.6 for the segment coefficients look rather formidable, but are in fact quite simple and very easy to remember and apply. The numerator of Aok, the forward coefficient, is the product of all forward X coefficients; similarly, the numerator of A^o, the reverse coefficient, is the product of all reverse X coefficients. The denominator Dok, common to both segment coefficients, is easily obtained with the following recipe: •

Construct a square matrix of order k, with elements 1 along the diagonal, with forward X coefficients of the m'th step in the m'th column in all rows above the diagonal, and with reverse X coefficients of the m'th step in the m'th column in all rows below the diagonal; then obtain DQ^ as the sum of the products of the elements in each row. forward X coefficients

reverse X coefficients In detail, the matrix is: 1

\ 3

K

1

\-2,k-l

\-lX

(6.7)

\-l,k

K

32

\-l,k-2

1

For example, for a pathway with three intermediates (k=4) and with DQ^ compiled as described, the procedure yields

138

Chapter 6. Practical mathematics ofmultistep

K

^12^23^34 ^ ^ 1 0 ^ 3 ^ 4

"^ ^ l o \ l ^ 3 4

reactions

+ ^10^1^32

^10^1^32^3

\o ^12^3^34

+ ^10^3^34

^^ ^ 1 0 ^ 1 ^ 4

+

^lo\l^3:

If any step in the sequence is irreversible, the equations apply with the respective reverse X coefficient equated to zero. Automatically, this also makes the reverse segment coefficient zero since the latter contains the zero X coefficient as a factor: If one step is irreversible, the entire pathway is irreversible! To obtain the rate equation for the pathway in terms of true rate coefficients, X coefficients in eqns 6.5 and 6.6 are replaced by the corresponding true coefficients /Cjk, multiplied by any co-reactant or co-product concentrations, in accordance with eqns 6.1 and 6.2. In this fashion, the set of rate equations of any simple pathway (unless it is part of a network) can be reduced to a single rate equation and the algebraic equations expressing the stoichiometry. To illustrate how much work can be saved in this way, let us return to the Gillespie-Ingold mechanism of nitration of aromatics, for which a repeated application of the Bodenstein approximation provided a rate equation in Example 4.4 in Section 4.4 Example 6.1. Nitration of aromatics of arbitrary reactivity. The pathway of the reaction is HB ArH BHNO3 ^

v» Xj < B-

^

X2 -^=^=^—• X3 ^

^

Hp

ArN02

(4.24)

HB

(see Example 4.4). Application of eqns 6.4 to 6.6 yields ^

'^01^12^23734^HNO3 ~ ^^12^23^34 "^ K^KlflA

J^^^Si^^'^I^rNOj

"^ ^10^21^34 "^ i^t^^'ST^

Here, terms that drop out because the reverse coefficients X32 and X43 are zero are shown as struck out, and so is the coefficient X34 that then cancels. Using eqns 6.1 and 6.2 to replace the X coefficients by the true rate coefficients and co-reactant or co-product concentrations one obtains k k k C _

C C

'^0l'^12'^23'^HNO,^HB^ArH

,A ^ Q N

^12^23 ^ArH ^ ^10^23 ^ ' ^ A r H "^ ^10^21 ^ B " ^ H p

that is, the same result as was derived rather more laboriously in Example 4.4 with repeated application of the Bodenstein approximation to the three intermediates.

139

6.3. General formula for simple pathways

An application to a considerably more complex reaction is shown in the next example, that of hydroformylation of olefins with a cobalt hydrocarbonyl catalyst. Example 6.2. Hydroformylation with "oxo" catalyst. Hydroformylation of olefins olefin + H2 + CO —• aldehyde

(6.8)

is catalyzed in solution under synthesis-gas pressure by a dissolved cobalthydrocarbonyl complex, HCo(CO)4, known as the "0x0" catalyst in distinction from the phosphine-substituted catalyst in Examples 5.2 and 5.3. The most likely mechanism, shown here for 2-d5-butene, is OC CO HCo OC CO

quasi-equilibrium

(cat)

2-methyl butanal

cobalt acyl dihydride

(6.9)

OC CO (X2)

H3C0.

OC

V-

trihydride

OC CO Co OC OC CO Co OC CO

cobalt alkyl

cobalt tetracarbonyl alkyl The catalyst (cat) loses one CO ligand to form Co(CO)3 (cat') and then binds the olefin (ole) as a 7r-complex (Xj) at the vacated coordinative site. Addition of H2 produces a trihydride (X2), which loses H2 to form a cobalt alkyl with metal-carbon a-bond (X3). Next, a new CO ligand adds to the newly vacated coordinative site (X4).

140

Chapter 6. Practical mathematics ofmultistep reactions

One of the CO ligands is then inserted between metal and alkyl carbon to form a cobalt acyl (X5), which is hydrogenated to a dihydride (X^). Finally, the latter splits into aldehyde (2-methyl butanal) and the CO-deficient hydrocarbonyl (cat') to complete the cycle. At very high CO pressures, the cobalt acyl reversibly adds another CO ligand to form a catalytically inactive tetracarbonyl-acyl species (Y). All members of the cycle are at trace level; the tetracarbonyl catalyst is not, nor is the tetracarbonyl acyl at very high CO pressures. Ligand dissociation cat <—^ cat' + CO and olefin addition cat' + ole <—^ Xi, involving only coordinative bonds, can safely be taken as fast, so that cat, cat', olefin, and Xi are at quasi-equilibrium (shown in a box in the network 6.9). Under reaction conditions, formation of the carbon-carbon bond (X4—• X5) is irreversible, but all preceding steps are reversible. With only a slight modification to be discussed farther below, this is the mechanism originally postulated by Heck and Breslow [10,11] and still found in current texts such as the most recent edition of Kirk-Othmer's encyclopedia [12]. If for the time being the side reaction to the tetracarbonyl acyl is disregarded, the pathway from 7r-complex to aldehyde in the network 6.9 can be written H2

CO

H2

Xj X2 ^ ^ - v ^ X3 <^=^-> X4

• X5 < ^ - >

X^ —=;:;>

aid

cat'

H2

Application of eqns 6.4 to 6.6 to this pathway gives (6.10) I

K3^34K5h6hl

~^ ^21^34^45^56^67 "^ ^21^32^45^56^67

l^+ A.2i?t32>^43^56f67 + A 2 t ^ ^ 2 ^ ^ 4 ^

]

"^ -^^^'32^^^43^^^54^ j

As in the previous example, terms containing zero-valued reverse coefficients (here X54 and X76) are struck out, and so are the forward coefficients that then cancel. The condition of quasi-equilibrium of cat, cat', olefin, and 7r-complex (X^) ^'^^°

C C, cat

=

K, = const.

ole

(validity of Henry's law is assumed) gives C.

=

1

cat

ole

Pco With this substitution, replacement of the X by /: coefficients multiplied with coreactant or co-product concentrations, and collection of terms, eqn 6.10 becomes _

^ 1 ^12^23^34^45 ^ole^cat/^H,

(^23^34^45 "^ ^21^34^45)^^00 "^ (^21^32^45 "^

/ ^ ^^x

KlKl^A^^Pw,

6.3. General formula for simple pathways

141

After division of numerator and denominator by the second term of the latter, a very simple rate equation with only two phenomenological coefficients is obtained: k C, C

(6.12)

1 + KPCO^PH^

where ^

^1^12^23^34^45 ^21^32(^45 "^ ^43)

^

^

(^23 "^ ^2l)^34^4: ^21^32(^45 ^ ^43)

Equation 6.12 is in very good agreement with the observed dependence on total pressure and H2-to-CO ratio. In fact, an empirical rate equation of this form was established by Martin [13] well before the mechanism had been elucidated. The effect of a "dead-end" side reaction such as that to the tetracarbonyl acyl will be examined in detail in Section 8.7. In a nutshell, the rate is reduced, but the form of the rate equation 6.12 remains unchanged because both dead ends, to HCo(CO)4 and the tetracarbonyl acyl, involve addition of CO to a cycle member. The key intermediates in the mechanism, the cobalt alky Is and acyls, are firmly established by independent experiments and have been synthesized [11]; the trihydride has not. The original Heck-Breslow mechanism [10,11] lacked that intermediate, and so do the networks found in current texts. Heck and Breslow postulated rate control by hydrogenation of cobalt acyl. Various minor modifications, most of them concerning rate control, were suggested by others [14-18]. However, neither Heck and Breslow's original mechanism nor any of these modifications can account for the observed rate behavior according to the Martin equation 6.12* or the fact that double-bond migration in the olefin, presumably occurring via the cobalt alkyl (X3) but hardly the acyl dihydride (X6), is accelerated by an increase in H2 pressure as much as is conversion to aldehydes. Moreover, all must make the unlikely assumption that, under reaction conditions, CO insertion with carbon-carbon bond formation is reversible as otherwise the rate would be independent of hydrogen pressure. Barring new evidence to the contrary, the network 6.9 that includes the elusive and still controversial trihydride [19] appears to be the most likely candidate. The example was chosen to illustrate pathway reduction and the resulting simplification of mathematics. So as not to distract from that message, complications encountered in practice with this particular reaction were disregarded. These include isomerization of 2-cis- to 2-trans- and 1-butene, conversion of 1-butene to 2-methyl butanal and Az-pentanal, and aldol condensation of the latter (see also Example 12.1 in Section 12.2). * This is true even if it is admitted that intermediates may contain significant fractions of total cobalt. Christiansen mathematics (see Section 8.4) shows that all then appearing additional denominator terms of the rate equation contain as factors a zero-value reverse coefficient of an irreversible step or the olefin concentration and must therefore be negligible (since the reaction is first order in olefin, the denominator may not include additive terms containing its concentration as a factor).

142

Chapter 6. Practical mathematics ofmultistep reactions In this example, the unreduced network (without the side reaction to the tetracarbonyl-acyl) has twelve rate equations, one for each participant, and fourteen coefficients. Four rate equations can be replaced by stoichiometric constraints, namely, the carbon skeleton, cobalt, hydrogen, and CO balances. This still leaves eight rate equations with ten coefficients. Pathway reduction has reduced this to a single rate equation, eqn 6.12, with two coefficients. The rate equation 6.12 was derived without use of any short-cuts. With rules to be shown in Section 7.3.1, algebra could have been reduced. Specifically, the steps following the irreversible formation of the cobalt acyl (X5) could have been omitted (Rule 7.12) and the pairs of steps from Xi to X3 and from X3 to X5 been consolidated each into a single step (Rule 7.24). This would have reduced the mathematics of conversion of the 7r-complex to the aldehyde to that of a pathway with only two instead of six steps.

The general formula 6.4 to 6.6 for simple pathways is so important that a closer examination of its properties is warranted. First, we can see that it meets the requirement of thermodynamic consistency (Section 2.5.1): Setting rp = 0 for equilibrium one finds i=0

i=0

The forward X coefficients contain the concentrations of all co-reactants as factors; the reverse X coefficients, those of all co-products (the concentrations include those of A and P if these species additionally appear as co-reactants or co-products). Accordingly, after replacement of the X coefficients by means of eqns 6.1 and 6.2: k-l

Ir

TT-^

^"P^"Q

= const.

=

" "^ -

(6.13)

where A, B, ... are the reactants and P, Q, ... are the products. This is in accordance with the mass-action law, so that the requirement of thermodynamic consistency is met. Incidentally, eqn 6.13 also shows that, for multistep pathways, the thermodynamic equilibrium constant equals the ratio of the product of the forward rate coefficients to that of the reverse rate coefficients, or a power thereof. Moreover, eqn 6.13 can be applied to a reversible catalytic reaction in which the catalyst acts as a co-reactant in an early step and is restored as a co-product in a later one. Its concentration, Qat, then appears in both the numerator and denominator on the right-hand side of eqn 6.13, and so cancels, in accordance with the requirement that the presence of a catalyst does not affect equilibrium. In the rate equation developed from eqns 6.4 to 6.6, Q^^ appears as co-factor in both terms of the numerator. This makes the rate first-order in the catalyst, provided Qat does not also appear in some terms of denominator, as it will unless the catalyst reacts in the first step and is reformed in the last, as is usually true.

143

6.3. General formula for simple pathways

Lastly, the rate equation of a reversible reaction A <—• ... <—• P with linear simple pathway must have a symmetry property: In principle, one could declare P instead of A as the reactant, and A instead of P as the product. The rate equation then obtained must be the "mirror image" of the original one. Specifically, eqns 6.4 to 6.6 must be invariant to the exchange of all indices i for k - i (with 0 < i < k), including A for P. Equation 6.4 is seen to have this symmetry property when the A coefficients are expressed in terms of the X coefficients. Derivation of equation 6.4. The general formula 6.4 to 6.6 for simple pathways can be derived in several different ways. First, the equation is a special case of a much more complex formula for catalytic reactions, derived by Christiansen [3] and later independendy by King and Altman [20] (see Section 8.4). A proof along these lines, but specifically for the simple pathway 6.3, is as follows [9]. A necessary corollary of the Bodenstein approximation in a pathway is that the net rates of conversion are the same for all steps (an intermediate with higher formation than decay rate would not remain at trace level). In matrix form this condition is: 1

r

M

\lr

1

CJr

1

1 ^i'^ '

1 1

[C,Jr

1 J

Here, r is the net forward rate of each step: r

=

(i = 0,. ..,k-l)

K,A - KuA.

a square matrix of order k given by \l^Q

M =

~\o

0

0

0

0

-K

0

0

0

-X32

0

0

0

\ 2

0

0

\ i

0

0

0

0

\-2,k-l

[-\,k-l<^k

0

0

0

0

-\-, \-,

(Note indices 0 and k indicate A and P, respectively.) With Cramer's rule one finds: 1/r = detM°/detM

(6.14)

144

Chapter 6. Practical mathematics of multistep reactions

where

1

-^10

0

0

0

1

Xj2

-X.

0

0

1

0

0

X,•k-2,k-l

1

0

0

0

M° =

The two determinants in eqn 6.14 can be evaluated by expansion into their minors along their first columns: det M = n

K.x^o - n K^f,^

det M° =

E

j=l

j=0

Replacement of the determinants in eqn 6.14 by these expressions gives eqns 6.4 to 6.6. Alternatively, eqns 6.4 to 6.6 can be derived without matrix algebra by conclusion from n to nH-1. This involves proof that the formula is correct for n-f 1 steps, provided it is correct for n steps. Correctness for two steps then ensures correctness for any number of steps. For two steps

the Bodenstein approximation for Xi

is solved for Q to give

Substitution of this expression into the rate equation for formation of P by the second step ^12^1

^^21^?

yields ^

^01^12^A

^10^21^P

In

X,2 +

\Q

in accordance with eqns 6.4 to 6.6. For the pathway

6.4. Simple networks and granted correctness of eqns 6.4 to 6.6 for the portion A

145 X-

After replacement of the A coefficients by means of eqns 6.5 and 6.6, followed by collection of terms and cancellations, one finds

i=0 'P

—

n+1

E fiv.riN,. i= l

i=j

This is equivalent to the general formula for the pathway with n+1 steps. Proved to be correct for two steps, and for n+1 steps if correct for n, the formula is correct for any number of steps.

6.4. Simple networks Networks differ from pathways in that they contain branches or loops. This usually makes it impossible to reduce mathematics to a single rate equation. However, a significant reduction in complexity can be achieved with the formulas and procedures introduced in the previous section, as will now be shown. 6.4.1. Network reduction The general formula for simple pathways, eqns 6.4 to 6.6, is also applicable to any (linear) simple segment between two nodes in a network or between a node and a network end member [8]. In this way:

Application of the general formula for simple pathways allows any simple network to be reduced to one with only pseudo-single, pseudo-first order steps between adjacent nodes or end members and adjacent nodes.

146

Chapter 6. Practical mathematics ofmultistep reactions For example, the network

I

I

Q

R

with arbitrary co-reactants and co-products (not shown) and arbitrary numbers of steps within the individual segments can be reduced to A

< ! — • Xj

<-^

Xk

I

I

Q

R

^—•

P

with pseudo-first order segment coefficients A^j, Aj^, AJQ, A^R, and A^p for its pseudo-single steps. A network may also contain two or more parallel pathways between the same two nodes. The simplest example of such a loop is path 1

" ^ ...

. — -

X.

X, ^

path 2

- — •

...

^

The contribution of the loop to the rates -r^ and r^, is the sum of the contribution of the two pathways:

Accordingly, loop coefficients i£j|< and ^^j can be defined ^jk

-

(Ajk)pach,

- (Ajk)pa*2.

^k,

-

(Akj)pach,

+ (\Ppa„2

^^.IS)

SO that

C-kVk = ^>.C. - X,f,

(6.16)

In effect, like a linear pathway, the loop is reduced to a pseudo-single step Xj

^ - ^

X,

6.4. Simple networks

147

with pseudo-first order loop coefficients iSjj, and iS^j that are the sums of the respective segment coefficients of the contributing pathways. Usually, loops occur in combination with linear pathways. A further reduction may then be possible. For example, the rate Xj <—• X^ through a network portion

X. ^

... ^

X,

X, - -

...

- -

X,

is found with eqns 6.4 for Xj •*—• X^ and X, •*—• X^, and eqn 6.16 for X^ -*—• X,, followed by application of eqns 6.4 to 6.6 to the resulting reduced pathway:

With collective coefficients T^^ and T^j defined by

the contribution is (r)

= T C - r C

^ k''j**m

jm j

mj m

In this way, the entire network portion between Xj and X^, has been collapsed into a pseudo-single step Xj

<

•

Xjn

with pseudo-first order collective coefficients Fj^, and F^j. 6.4.2. Rate equations Rate equations r^ for the end members of simple networks can be given either in explicit form or as an algorithm. Such equations are useful for mathematical modeling in that they are much simpler to handle than the sets of simultaneous rate equations for all participants in a reaction. This is because the concentrations of all intermediates have been eliminated and the number of simultaneous equations is much smaller. However, even for networks of only moderate complexity, the concentration dependence of the rates is rather involved. The principal application

148

Chapter 6. Practical mathematics of multistep reactions

of rate equations therefore is in modeling rather than network elucidation. For elucidation, yield-ratio equations to be discussed later in this section are more convenient. Single-node simple networks. Rate equations for the end members of single-node networks can be given in explicit form. The reduced single-node network is

^

" ~ " ^^ ^

(6.18) Q

(co-reactants and co-products not shown). The rate equation for product P is ^

Kv(K^C^

+

AQ.CQ)

- Ap,(A,^ +

A,Q)CP

(6.19)

^ K? ^ KQ

\A

Because of complete symmetry, the equations for Q and A are analogous, with obvious interchange of indices. Derivation of equation 6.19 [8]. The Bodenstein approximation for the node intermediate Xk is 'k = A.,C. •^Ak'^A + Ap,Cp ''^'Pk'^P + A^.C^ ''^Qk'^Q - (A,. ^''*^kA + A.p ''*kP + A,„)C, ''^kQ^'^k

=

0

Solved for Q: ^

^

^Ak^A + \k<^P + V ^ Q

'

KK

•" K

-^ ^kQ

Substitution of this expression into the rate equation for P Tp

-

A^pC^

Ap^Cp

yields eqn 6.19. If both segments X^ <—^ P and X^ <—• Q contain at least one irreversible step, as is often the case in practice, the terms involving Cp and CQ do not contribute and eqn 6.19 reduces to r

=

^AkKpC^

^^20)

As a rule, this simpler equation is also a good approximation for initial rates in networks of reversible reactions.

6.4. Simple networks

149

For single-node networks with additional branches, eqn 6.19 is readily extended by addition of respective terms. For example, for the reduced network P

I X, ^ ^

Q

I R the numerator receives the additional terms A^PAJ^CR and -ApkA^RCp ; the denominator, the term A^R. However, such situations are rare. Multinode simple networks. An algorithm derived by Chern [9] for the compilation of the rate equations of the end members of multinode simple networks can be given as follows: •

The rate equation r^. for an end member Pj is composed of additive positive contributions from all pathways Pj <—• Pj (i?^j), each expressing the rates at which Pj is formed from the respective Pj, and negative contributions expressing the rates at which Pj reverts to the respective Pj.

•

Each contribution Pj —• Pj consists of a factor ^^C^^, a factor det Mj j , and a factor 1/det M. Here, '^jj is the product of all A coefficients (if or Y coefficients in the case of loops) on the path from Pj to Pj. M, characteristic of the entire network, is a square matrix of order n, where n is the number of nodes, with elements S,i\ = l,...,n) along its diagonal, with elements A,,^ in row k and column ( for all X^ and X^ directly connected with one another, and elements zero in all other positions. Mjj = Mjj, characteristic of the pathway Pj <—• Pj, is a square matrix similarly compiled, but with the m'th rows and columns omitted, where the m are the indices of all node intermediates on the path from Pj to Pj. 5j is the sum of the A coefficients of all segments originating from the node intermediate Xj and leading to the directly adjacent node intermediates or end members.

The following example will illustrate the application of the Chern algorithm.

150

Chapter 6. Practical mathematics of multistep reactions Example 6,3: Rate equations for hypothetical "snowflake" network. In the eightnode, ten-ended, fully reversible, reduced "snowflake" network p,

P:

/

\ X1

I

X2

\

/

\

/ '\

X3

I

\ X

n

^

(6.21)

.

\

/

^ ,

•

p.

P7

the rate rp. is composed of nine contributions of forward minus reverse reactions between P: and the other nine end members: Z-r ^'^P,-*?,

(6.22)

^?-*v}

= 1

For example, the contribution to r^^ from the pathway Pi (rp)p^p^ =

P5 has the form

("^is^P, - '^5i^P,) detMi5/detM

(6.23)

where ^ = A A A A A

^

\,5^53^32^21^1.P

(6.24)

(products of segment coefficients along respective pathways);

0

M =

0

0

0

0

0

0

0

^26

0

0

A34

A,,

0

0

0

0

0

^43

^4

0

0

0

0

0

^53

0

0

0

0

0

A• 6 2

0

0

0

0

0

0

0

0

0

0

0

0

0

A,.

(6.25)

6.4.

Simple networks

151

(elements S^ along the diagonal, A coefficients for all existing direct connections (1**2, 2**3,3*>4, 3**5, 2**6, 6**7, and 6**8) as elements in the respective rows and columns, and all other elements zero);

M,3

=

s.

0

0

0

0

Ki

0

s. K

s.

0

0

A„.

0

s.

\

(6.26)

(rows and columns m = 1, 2, 3, and 5 omitted from M because the respective X„ are on the path connecting Pi and P5); and S^ = A^, =

Ajj

+ A 3 , + A35

•^5

=

^5,P,

S,

= A,p^ + A, p^ + A„

+ ^5.P. +

^23

+

^26

A4,p_ + A , p^ + A , 3

53

^62 + \ l

^53

Kf,

(6.27)

+ A^8

+ ^8,P,. + ^ 8 6

(Si = sum of A coefficients of segments originating from node XJ. Derivation of Chem algorithm [9]. The simplest way to derive the Chern algorithm is by stepwise evolution from simpler networks, outlined as follows: The rate equation for the single-node, reduced network j+i

P

•*—*•

(6.28)

X„

obtained by extension of eqn 6.19, can be written in terms of ^ij and S,: \

= E

(^ijCp, - ^ , C p ) / s „

i=j+i

or, instead. j^2

(6.29) with "P coefficients T".

ij

m'

'P.. =

ji

m

(6.30)

corresponding to a still further reduced network j+i

(6.31) J+2

152

Chapter 6, Practical mathematics ofmultistep

reactions

For the rate equations of the end members, Pj+i and Pj+2, it makes no difference whether or not the node species, Pj , can undergo additional reactions. Equation 6.29 thus also holds if the network portion 6.28, reduced to 6.31, is itself only part of a larger network, with a node intermediate Xj replacing Pj. This allows a network consisting of two single-node portions linked by a common intermediate P..1

P..3

X

^ ^ X. <-> X^

(6-32)

P

P

^j+2

^j + 4

to be further reduced to >

X.

^

N

>J + 2

(6.33)

P ^j + 4

with rates \.

=

E

(4i.kCp, - 3 j . , , C p J / ^

(k = l,...,4)

(6.34)

where 4 i . h . = ^MJ^j^K'

(i,k = l,...,4;

i^k)

(6.35)

and 4

g'. = Y'P

(6.36)

i=l

In general, the 3 coefficients are defined as the products of the !P coefficients along the respective paths from one end member to another; and the ^ , as the sums of the T coefficients of all segments connecting the respective intermediate with its immediate neighbors in the further reduced network. In this fashion, network portions can be linked to one another and rate equations in terms of A coefficients be obtained by appropriate substitutions as in eqns 6.34 to 6.36 and 6.30. For example, in the snowflake network 6.21, two pairs of single-node portions 6.28 are linked at their respective common intermediate X^ and the two resulting larger portions are linked to a fifth single-node portion. This procedure eliminates all node intermediates and leads to the described Chern algorithm.

6.4. Simple networks

153

6.4.3. Yield-ratio equations [8] For elucidation of networks, yield-ratio equations are much more convenient than rate equations. The (instantaneous) yield ratio YpQ is defined as the ratio of the molar conversion rates of a reactant to products P and Q: y,

-

^

<••')

(see Section 1.6). For a single-node reduced network (i.e., network 6.18) the yield ratio, found by dividing the formation rates 6.19 of P and Q by one another, is ''

"

Ko(K.C,

- Ap,)Cp -

AQ,(A,,

. A,p)C,

(6.37)

If the reaction to P and Q is irreversible, a much simpler expression is obtained by use of eqn 6.20 instead of 6.19: Y^

= AJ\Q

(6-38)

Usually, this simple equation is also a good approximation for the initial yield ratio of a reversible reaction because the terms containing the product concentrations as factors remain negligible until conversion has become significant. In multinode networks, eqns 6.37 and 6.38 remain valid for any two products originating directly from the same node intermediate, that is, by pathways that contain no further nodes, as in a reduced network with a portion

\

If the reaction is irreversible, relatively simple expression are obtained also for products that originate from different node intermediates, as long as these are not too far apart. For example, in a reduced network with a portion ... <—> Xk ^—• X, — • P

I

i

R

Q

the yield ratio of products P and R (two nodes apart) is Y'PR

=

^"^'^ \R(AI^

+ A,P + A PQ-'

(6.39)

154

Chapter 6. Practical mathematics ofmultistep

reactions

Similarly, in a reduced network with portion

t I

I

Q

R

the yield ratio of P and S (three nodes apart) is Y

^

'PS

\f^fp(^mk

^ ^mR ^ ^ m s )

^km^ms(^flc + \ v

(6.40)

^ ^ Q )

Yield ratios for other situations can be derived from the Chern algorithm for rates in multinode simple networks, given previously in Section 6.4.2. Equations 6.39 and 6.40 usually are also good approximations for the respective initial yield ratios in reversible reactions. Note, however, that the concentration dependence of the initial rate may not coincide with the rate equation (see Section 3.3.3). Also, if the nodes are far apart, quasi-stationary behavior may not yet have been attained in one portion when the reverse rate has already become significant in another. Example 6.4. By-product formation in ethyne dimerization. Dimerization of ethyne (acetylene) to vinyl ethyne (vinyl acetylene) 2HC = CH

•

H2C = CH-C = CH

(6.41)

is catalyzed by Cu"^ ions in dilute aqueous hydrochloric acid [21-23]. The principal by-product is divinyl ethyne, formed by subsequent addition of another ethyne molecule to vinyl ethyne. However, if some Cu^^ is produced from Cu"*^ by oxidation, it induces a side reaction producing chlorovinyl ethyne [22,23]. Vinyl ethyne (VE) and chlorovinyl ethyne (CVE) arise from the same node intermediate, the former by a reaction first order in H"^, the latter by one first order in Cu^"^:

„

^ ^

^

>H,C = C - C = CH

2HC = CH < N — ^ ^ ^ III- C u - C ^ C H Q

V 2Cu2+,Cl-

H

"^ ^

(arrows represent multistep pathways)

3Cu'

- ^ ^ H X = C - C = CH

6.5. Non-simple pathways and networks

155

Both reactions are irreversible, so that the VE-to-CVE yield ratio obeys eqn 6.38 and is proportional to the H"^-to-Cu^^ ratio: y,,,,,, = const. *(C,./Q,.) (6.42) in agreement with experimental observation [23]. (Strictly speaking, since vinyl ethyne also forms divinyl ethyne as another by-product, the yield ratio in eqn 6.42 is that of vinyl plus divinyl ethyne to chlorovinyl ethyne.) The principal application of yield ratio equations is in network elucidation, to be discussed in Section 7.3.2. An additional example will be given in that context. Further examples for establishment of yield ratio equations and their application in network elucidation can be found in O. N. Temkin's book on reaction networks [24], although not under that name. Also, in mathematical modeling, simple algebraic yield ratio equations can sometimes be substituted for rate equations, which may be differential (see Section 12.2). 6.5.

Non-simple pathways and networks

For reactions with non-simple pathways or networks, the formulas and procedures described so far are not valid. Any step involving two or more molecules of intermediates as reactants destroys the linearity of mathematics, and any intermediate that builds up to higher than trace concentrations makes the Bodenstein approximation inapplicable. Such non-simple reactions are quite common and will be examined later in this chapter. The discussion of reactions with abnormal nonsimple behavior—detonations and periodic or chaotic reactions—will be sketched in Chapter 14. However, a discussion of more than only their most elementary facets is beyond the scope of this book. The reader interested in more than this is referred to the literature cited in that chapter. If a majority of the steps of a reaction are non-simple, there is at this time no substitute to traditional "brute force" modeling of the rate equations of all participants except those that can be replaced by stoichiometric constraints. This is so, for example, in hydrocarbon pyrolysis and combustion, where, fortunately, an extensive data base on rate coefficients and activation energies has been assembled [25-29]. However, in a large number of non-simple reactions of practical interest, only one or a few steps out of many are non-simple. In such cases, the complexity of mathematics can be significantly reduced. In a few other instances with only one or two offending steps, additional approximations may make it possible to arrive at explicit rate equations. If only a small minority of reaction steps are non-simple, much benefit can be had by breaking the pathway or network down into "piecewise simple" portions and then applying the methods described in the preceding sections to these [8]. To this end the pathway or network is cut at the offending intermediates or steps, as will now be shown.

156

Chapter 6. Practical mathematics ofmuhistep

reactions

A network may be non-simple because an intermediate builds up to higher than trace concentration, or because an intermediate (at trace or higher concentration) reacts with itself. In both these cases, the network is cut into portions of formation and decay of that intermediate (Cases I and II, respectively, in Table 6.1). If an intermediate (at trace or higher concentration) reacts with another intermediate, the network is cut into portions of formation and decay of these intermediates and, if both are on the same pathway within the network, the portion of formation of one from the other (Cases III and IV in Table 6.1). Table 6.1. Examples of break-up of non-simple reduced networks into piecewise simple portions (co-reactants and co-products of other steps not shown, and other parts of network assumed to be simple).

case

non-simple reduced network K

simple portions AT^

K

^Si.

K

X.

II

X. ^

2X.,

P

X;

Ill

IV

V

A

; > p

X

\ ^

X,

B p

X. X

K+X, X;

X.

x,+x,

p

A

x,+x,

Q

p

X,

Loss of simplicity may or may not result from a step in which an intermediate dissociates into two others. If the step is reversible, its reverse and thus the network are non-simple. The latter then is broken into portions of formation and decay of the dissociation products (Case V in Table 6.1). On the other hand, if the step is irreversible, it does not offend simplicity. Mathematics then is the same as that of a fictitious (and stoichiometrically inconsistent) network in which two parallel steps with identical rate equations replace the single dissociation step:

6.5. Non-simple pathways and networks

157

X, X. —
is replaced by X, ^

... '^-*' X.

...

^--> X, ^

...

If the networks in Table 6.1 contain other non-simple features, the affected portion would have to be subdivided accordingly. As a rule, a reduction to a single, explicit rate equation (plus algebraic equations for stoichiometric constraints and yield ratios) is not achieved. Rather, the equations for the end members of the piecewise simple network portions must be solved simultaneously. Nevertheless, The concentrations of all trace-level intermediates that do not react with one another have been eliminated by this procedure and, in many cases of practical interest, the reduction in the number of simultaneous rate equations and their coefficients is substantial. The reduction in mathematical complexity of non-simple networks is mainly of value in modeling. For network elucidation, a better approach usually is to study as many network portions as possible separately, say, by experiments that use synthesized intermediates as starting materials or by appropriate lumping, as the following example will illustrate (see also Section 7.3.3). Example 6.5. Olefin hydroformylation with phosphine-substituted cobalt hydrocarbonyl catalyst [30]. The pathway 6.9 of olefin hydroformylation with the "oxo" catalyst, HCo(CO)4, has been shown in Example 6.2 in Section 6.3. For phosphinesubstituted catalysts, HCo(CO)3Ph (Ph = organic phosphine), the pathway olefin—• aldehyde is essentially the same. However, these catalysts also promote hydrogenation of aldehyde to alcohol (Examples 7.3 and 7.4) and of olefin to paraffin (Example 7.5). Moreover, straight-chain primary aldehydes under the conditions of the reaction undergo to some extent condensation to aldol, which is subsequently dehydrated and hydrogenated to yield an alcohol of twice the carbon number (e.g., 2-ethyl hexanol from Az-butanal; see Section 12.2). The entire reaction system is olefin

+ H2, CO

• aldehyde

+ H2

• alcohol (6.43)

+ H2

+ aldehyde

+ 2H2

paraffin

aldol

• heavy alcohol — H2O

All arrows represent multistep, irreversible pathways, and aldehyde and possibly aldol build up to higher than trace concentrations. The network is non-simple for that reason and also because the aldehyde, an intermediate, acts in addition as a co-reactant in the pathway to the aldol.

158

Chapter 6. Practical mathematics ofmultistep

reactions

The network can be cut into four piecewise simple portions: olefin —• aldehyde, I • paraffin

aldehyde —• alcohol, (6.44) 2 aldehyde —• aldol,

aldol —• heavy alcohol (H2 and CO co-reactants and H2O co-product not shown). Since aldol dehydration and hydrogenation is fast, the portion from aldehyde to heavy alcohol can be treated without much detriment as a single, simple portion, at least if heavy-alcohol formation is a minor side reaction. The two pathways from olefin to aldehyde and paraffin cannot be separated because they must be expected to go through a common intermediate. Since the portions are irreversible, none of them feeds back into a preceding one. Accordingly, the portions are not coupled except through the concentrations of their common members (aldehyde and aldol). The network 6.43 still fails to include olefin isomerization, the distinction between isomeric products, and the effects of catalyst equilibria. The last of these will be examined in Section 8.2.2. A more complete network for the total reaction with a typical industrial catalyst and including isomerization will be discussed in Section 12.2.2. In a number of reactions of practical interest, the offending non-simple step is a reversible dissociation of a reactant or intermediate, as in Case V in Table 6.1. Often, such a step is fast compared with the others and thus is at quasi-equilibrium. If so, the quasi-equilibrium approximation (see Section 4.3) can greatly simplify mathematics, in some instances even lead to an explicit rate equation. This has been discussed in Section 5.6. Another case in point is chain reactions (see Chapter 10). As a rule, these are non-simple because their termination step involves two intermediates as reactants. Nevertheless, explicit rate equations can often be derived with the Bodenstein and long-chain approximations. The classical example is that of the hydrogen-bromide reaction (see Section 10.3). An extensive, though abstract treatment of networks based on graph theory and including non-simple configurations has been provided in two recent books by Russian authors [1,2].

Summary This chapter describes practical, simplified mathematics for ordinary multistep reactions, excluding for the time being catalysis, chain reactions, and polymerization. The concept of "simplicity" of a pathway or network is introduced. For a pathway to be "simple," all its intermediates must be and remain at trace level, and no step may involve two or more molecules of intermediates as reactants. The first condition ensures

References

159

applicability of the Bodenstein approximation to all intermediates, the second guarantees mathematics to be linear. In addition, pseudo-first order rate coefficients Xj,, are introduced, defined as the products of the real rate coefficients k^^ and the concentrations of any co-reactants. The principal tool for simplifying mathematics is network reduction: Any simple pathway or simple network segment between nodes or between a node and an end member can be reduced to a pseudo-single, reversible, pseudo-first order step with first-order "segment coefficients" that are functions of all the real rate coefficients and the concentrations of any co-reactants and co-products, but are independent of the concentrations of the intermediates. An easily remembered general formula for these coefficients allows the rate equation for any simple segment to be written down immediately, regardless of the number of steps and participation of co-reactants and co-products. The procedure does not eliminate the complications of co-reactant and co-product involvement, but postpones it from the hard task of establishing a rate equation to simple substitutions in that equation. For simple pathways, a single, explicit rate equation in terms of the concentrations of the bulk participants is immediately obtained. For simple networks of any degree of complexity, rate behavior is given by simultaneous rate equations for the network end members. An algorithm for compilation of these rate equations is provided. For elucidation of all but quite small networks, rate equations are too unwieldy even if the network is simple. Here, the much more compact yield ratio equations are a better tool. These can also profitably be used to replace rate equations in modeling. Many reactions of practical interest have non-simple pathways or networks, i.e., the concentration of an intermediate rises above trace level or a (forward or reverse) step involves two or more molecules of intermediates as reactants. If the majority of the steps meet the simplicity conditions, a significant reduction in mathematical complexity can still be achieved by cutting the network into piece wise simple portions at the offending steps. In some other instances, the quasi-equilibrium or long-chain approximations can be invoked in order to obtain explicit rate equations although the network is non-simple. On the other hand, if a majority of steps in a network are non-simple, the tools described here are of little use. Practical examples include nitration of aromatics, olefin hydroformylation with cobalt hydrocarbonyls and phosphine-substituted hydrocarbonyls as catalysts, and ethyne dimerization.

References

G. S. Yablonskii, V. I. Bykov, A. N. Gorban', and V. I. Elokhin, Kinetic models of catalytic reactions. Vol. 32 of Comprehensive chemical kinetics, R. G. Compton, ed., Elsevier, Amsterdam, 1991, ISBN 0444888020. O. N. Temkin, A. V. Zeigarnik, and D. Bonchev, Chemical reaction networks: a graph-theoretical approach, CRC Press, Boca Raton, 1996, ISBN 0849328675. J. A. Christiansen, Adv. CataL, 5 (1953) 311.

160 4. 5. 6. 7. 8. 9. 10. 11. 12.

13. 14. 15. 16. 17.

18.

19. 20. 21. 22. 23. 24. 25.

26.

Chapter 6. Practical mathematics ofmultistep

reactions

A. A. Frost and R. G. Pearson, Kinetics and mechanism, a study of homogeneous chemical reaction, Wiley, New York, 2nd. ed., 1961, Chapter 10. J. J. Carberry, Chemical and catalytic reaction engineering, McGraw-Hill, New York, 1976, ISBN 0070097909, Section 2-12. F. G. Helfferich, Detergent-range alcohols via hydroformylation; V. Kinetics, Technical Report 203-65, Shell Development Co., 1965. F. Wilkinson, Chemical kinetics and reaction mechanisms. Van Nostrand-Reinhold, New York, 1979, ISBN 0442302487, Chapter 3. F. G. Helfferich, J. Phys. Chem., 93 (1989) 6676. J.-M. Chem and F. G. Helfferich, AIChE J., 36 (1990) 1200. D. S. Breslow and R. F. Heck, Chem. Ind., 1960, 467. R. F. Heck and D. S. Breslow, J. Am. Chem. Soc, 83 (1961) 4023. E. Billig and D. R. Bryant, Oxo reaction, in Kirk-Othmer, Encyclopedia of chemical technology, 4th ed., J. I. Kroschwitz and M. Howe-Grant, eds., Wiley, New York, Vol. 17, ISBN 047152686X, 1996, p. 902. A. R. Martin, Chem. Ind., 1954, 1536. J. Falbe, Carbon monoxide in organic synthesis (translated from German), Springer, Berlin, 1970, Section 1.2. M. van Boven, N. H. Alemdaroglu, and J. M. L. Penniger, Ind. Eng. Chem., Prod. Res. Dev., 14 (1975) 259. N. H. Alemdaroglu, J. M. L. Penniger, and E. Oltay, Monatsh. Chem., 107 (1976) 1153. J. P. CoUman, L. S. Hegedus, J. R. Norton, and R. G. Finke, Principles and applications of organotransition metal chemistry. University Science Books, Mill Valley, 2nd ed., 1987, ISBN 0935702512, p. 622. G. W. Parshall and S. D. Ittel, Homogeneous catalysis: the application and chemistry of catalysis by soluble transition metal complexes, Wiley, New York, 1992, ISBN 0471538299, Section 10.3. J. E. Mahler, unpublished, 1963. E. L. King and C. Altman, J. Phys. Chem., 60 (1956) 1375. Parshall and Ittel (ref. 19), p. 197. O. N. Temkin, G. K. Shestakov, and Yu. A. Treger, Acetylene chemistry, reaction mechanisms, and technology, Khimiya, Moscow, 1991 (in Russian). Temkin et al. (ref. 2), p. 267. Temkin et al. (ref. 2), Chapter 5. Landolt-Bomstein, New Series, Radical reaction rates in liquids, H. Fischer, ed., Springer, Berlin, Part II Vol. 13 (5 subvolumes), 1983-1985, ISBN 0387126074, 0387132414, 0387117253, 0387121978, 0387136762; Vol. 18 (5 subvolumes), 1994-1997, 3540560548, 3540560556, 3540560564, 3540603573, 3540560572. D. L.Baulch, C. J. Cobos, R. A. Cox, C. Esser, P. Frank, T. Just, J. A. Kerr, M. J. Pilling, J. Troe, R. W. Walker, and J. Warnatz, J. Phys. Chem. Ref Data, 21 (1992)411.

References 27.

28.

29. 30.

161

D. L. Baulch, C. J. Cobos, R. A. Cox, P. Frank, G. Hayman, T. Just, J. A. Kerr, T. Murrells, M. J. Pilling, J. Troe, R. W. Walker, and J. Warnatz, /. Phys. Chem. Ref. Data, 23 (1994) 847. W. G. Mallard, F. Westley, J. T. Herron, R. F. Hampson, and D. H. Frizzell, NIST chemical kinetics data base: Windows Version 2Q98, National Institute of Standards and Technology, Gaithersburg, MD, 1998. Handbook of chemistry and physics, D. R. Lide, ed.-in-chief, CRC Press, Boca Raton, 80th ed., 1999-2000, ISBN 0849304806, Section 5, pp. 505 to 519. F. G. Helfferich and P. E. Savage, Reaction kinetics for the practical engineer, Course #195, AIChE Educational Services, New York, 7th ed., 1999, Section 8.2.

Chapter 7 Network Elucidation Network elucidation starts with the identification of the participants in the reaction: reactants, products, known intermediates, and possibly catalysts and any other species that affect the rate. From there on approaches differ. Access to powerful computers has made it possible in principle to begin with a general network in which each participant in the reaction is connected to all others by reversible steps that conform with stoichiometry {maximum model [1]). Values of the rate coefficients of this network are then fitted to experimental results. Steps with very small coefficients are dropped, and the values of the remaining coefficients are refined. The entire procedure is easy to learn and can be automated. However, in a reaction of any degree of complexity, the number of possible combinations of reaction steps is staggering, and this "blind" approach not only requires a lot of calculation, but also is in danger of converging on a false optimum. Moreover, there may be unknown intermediates. On the other hand, if the reaction is relatively simple, theoretical chemistry, experience, and common sense almost always allow a few plausible mechanisms to be singled out from improbable or sterically or energetically impossible ones, making the comprehensive screening of all a waste of time. For these reasons, maximum models are rarely advantageous, and no further space will be given to them here. In most cases, a more cost-effective and reliable approach is to undertake a preliminary sorting by reaction orders and, if called for, product ranks and then proceed in either of two ways: •

compile empirical rate equations from extensive experimental results over a wide range of conditions; then find a plausible mechanism and network that can produce such rate equations, or

•

compile all networks that make sense from the point of view of theoretical chemistry and experience (stereochemistry, thermodynamics^ molecular orbital theory, selection and exclusion rules, analogies with other reactions of the same type or with same catalyst, etc.); then devise bench-scale experiments to discriminate most effectively between these rival networks.

Usually, the first approach is preferable if a large amount of reliable quantitative kinetic data is already available; the second, if the kinetic behavior of the reaction is still largely unknown and the reaction engineer has a say in the design of the

164

Chapter 7. Network elucidation

kinetic experiments yet to be carried out. Several examples of compilation of rate equations for postulated networks have already been given in Chapters 4, 5, and 6. Examples of deduction of networks from empirical rate equations will be provided in the present chapter. Regardless of what approach is taken, the ultimately postulated network and its mathematics should be validated by verification of their predictions, preferably counterintuitive ones, under conditions not previously studied (see also Chapter 12). Large networks are common in industrial practice. Of course, the larger the network, the harder is its elucidation. The best approach to large networks is to break them down as much as possible into portions that can be studied separately and independently. Non-trace intermediates can usually be synthesized and used as reactants in kinetic studies. Also, sometimes, side reactions or product decomposition can be suppressed in laboratory studies by additives or by structural modifications of the reactants in a way that will not affect the pathway of the reaction of principal interest. For this reason, emphasis in this chapter is on the analysis of relatively small and primitive networks. 7.1.

Order and rank

Once the identities of the participants in a reaction have been established, a logical next step toward proposing and testing networks is the determination of reaction orders and, in the case of reactions with many products, of product ranks. The present section examines these preliminaries. 7.1.1.

Reaction orders

Reaction orders and experimental techniques to establish them are discussed in great detail in texts on kinetics and reaction engineering (see general references in Chapter 3). A brief survey concentrating on practical aspects of equipment and data evaluation has been given in Chapter 3. In network elucidation, the determination of reaction orders is a preliminary step whose results are intended mainly for orientation. Constant, integer reaction orders may not exist in the case at hand. A fractional and possibly varying reaction order, while reflecting ignorance of the mechanism and true rate equation, is none the less a key symptom that can prove important in the subsequent work to establish the latter. On the other hand, just because orders are apt to be variable, any effort to determine their precise values is a waste of time: All the information sought at this stage is a range—say, an order between plus one and plus two—and, if the order is variable, in which direction it changes with conversion. Often, plots for guessed reaction orders and conclusions from any curvatures that become apparent are all that is needed. Examples in the present chapter will illustrate such methods.

7.1. Order and rank

165

Most reactions of practical interest have orders with respect to several participants. The methods to determine individual orders in such cases are therefore especially important. In multistep reactions in principle, the rate equation even of only the forward reaction may involve the concentrations of any participants, not only those of the reactants. This includes catalysts, products, and "silent partners" whose presence affects the rate although they are not catalysts nor are formed or consumed by the net reaction and so do not appear in the stoichiometric equation. The determination of reaction orders can therefore not remain restricted to reactants, even if the reaction is irreversible. Unusual reaction orders are found in product-promoted or reactant-inhibited ("autocatalytic") reactions, the former with positive apparent order with respect to a product, the latter with negative apparent order with respect to a reactant (see Section 8.9). An example of a product-promoted reaction is acid-catalyzed ester hydrolysis. An example of a reactant-inhibited reaction has already been encountered, namely, olefin hydroformylation, whose order with respect to CO is negative (see eqn 6.12 in Section 6.3). Such behavior is also not uncommon in heterogeneous catalysis (see Section 9.3.2) and enzyme catalysis ("substrate-inhibited" reactions in biochemistry lingo. Section 8.3). A reaction having an order with respect to a silent partner—CO in a homogeneous hydrogenation—will be examined in some detail later in this chapter (see Examples 7.3 and 7.4). The self-accelerating nature of a reaction can have profound consequences for reaction engineering and can be design-limiting (see Section 13.3 and Chapter 14). 7.1.2. Ranks and Delplot * The rank (primary, secondary, etc.) of an intermediate or product of a multistep reaction reflects the provenance of the species. A primary species is formed directly from the original reactant or reactants; a secondary species, from a primary one; etc. The formation from a species of next lower rank may involve more than one step, but only if all but one of them are very fast. , ^, . reactant(s)

•

primary . : products

•

secondary , , products

•

tertiary . \ products

•

...

(For practical reasons, "products" is defined here as including intermediates.) Rank-ordering of products on the basis of experimental results is one of the tools for elucidation of networks of reactions about which very little is known as * The rank is a relatively recent concept, developed and formalized at the University of Delaware [2,3], based in part on earlier work by Myers and Watson [4]. The name "Delplot" for the plots to determine it alludes to this origin. The concept and procedure have not yet found their way into standard texts on reaction engineering.

Chapter 7. Network elucidation

166

yet. The method is most useful for degradation reactions of large molecules, reactions that typically yield a host of different products, but rarely include steps in which molecules of the original reactant or their fragments react with one another. Ranking often involves judgment calls, especially if the rate coefficients differ greatly or if the network is large and includes higher-order steps. Ranking is largely a qualitative tool and, in practice, is rarely carried beyond a distinction between primary and higher-rank participants. Ranking requires detailed information on behavior at very low conversions, best obtained with batch reactors. Identification ofprimary products. Primary products arise directly and exclusively from the original reactants, initially present at finite concentrations. In contrast, products of higher ranks arise from parents whose initial concentrations are zero. Thus, the initial formation rates of primary products are finite, those of products of higher ranks are zero. This allows the two types of products to be distinguished by their concentration histories. This principle is formalized and sharpened in the Delplot method. In a firstrank Delplot [3], selectivities 5p = y^ //A are plotted versus/A , where y^ is the yield of product P and/A is the fractional conversion of the original (limiting) reactant A. primary product

higher-rank product

(Cp >^P

/A

Figure 7.1. Concentration histories and first-rank Delplots of primary and higher-rank products.

^

C,yn,

(1.5)

c:in. -

1 -

CJCl

(1.4)

(see Section 1.6). The plots are extrapolated to zero conversion. Primary products, having finite initial formation rates, give plots with finite intercepts; products of higher ranks, having zero initial formation rates, give plots with zero intercepts. This is true regardless of reactor type and reaction orders of the steps, and even if the steps are reversible.

Primary products give first-rank Delplots with finite intercepts. Products of higher ranks give first-rank Delplots with zero intercepts.

7.1. Order and rank

167

Example 7.1. High-temperature hydrolysis of butyronitrile [5]. Nitriles in toxic aqueous wastes, e.g., from manufacture of explosives, can be hydrolyzed at high temperature and pressure. Amide, acid, and ammonia are formed, and nitrile conversion can be carried to extinction. For butyronitrile as a model compound: CN

+

CN

+ 2H2O

.0 'NH2

H2O

At high conversion, yields of butyric acid and butanamide are about 0.8 and 0.2, respectively (see Figure 7.2). First- and second-rank Delplots are shown in Figure 7.3. In the firstrank plots the extrapolated intercept for butyric acid appears to be close to zero, and that for butanamide close to one (the selectivities must sum up to 1.0). This indicates that butanamide is a primary product and butyric acid is not. The first step must be irreversible because butyronitrile reacts to extinction. The second step must be reversible because the concentration of butanamide, the intermediate, remains finite as conversion becomes complete.

OH 1.0 r

butyronitrile

0.8

butyric acid

0.4 butanamide

-^X.

1

0^

60

120

180

t [min]

Figure 7.2. Concentration histories in batch hydrolysis of butyronitrile in water at 330°C and 472 bar (Adapted from Iyer and Klein [5]).

Figure 7.3. First- and second-rank Delplots of butanamide and butyric acid in hydrolysis of butyronitrile at 330°C (from Iyer and Klein [5]).

168

Chapter 7. Network elucidation

Nitrile hydrolysis is known to be catalyzed by acids and bases. None were added in the study whose Delplots are shown here. However, butyric acid is formed and must be assumed to accelerate the reaction autocatalytically. This suggest a network with a catalytic reaction parallel to the first thermal step:

(7.1)

Although the catalytic reaction nitrile —• amide almost certainly proceeds in more than one step (which would be trimolecular), concentration histories in good agreement with observation were obtained with this trial network [5]. Distinction between higher ranks. A distinction between higher ranks of products, difficult to make by mere inspection of concentration histories, is possible, but may remain ambiguous if the network contains steps of higher molecularities, as must be so if reactants combine to form larger product molecules. However, if the carbon skeleton of the original reactant remains intact or is fractured, the network may well consist exclusively of first-order or pseudo-first order steps. To distinguish between higher ranks, the quantity yj(f/d^ is plotted for each product i versus/A successively with/? = 1,2, etc. {R-rank Delplots) and extrapolated to zero conversion. Delplots with /? = 2 are called second-rank; those with R — 3, third-ranked; etc. Provided all steps are first or pseudo-first order, ranks of products can be identified as follows:

/?-rank Delplots give finite intercepts for products of rank /?, zero intercepts for products of ranks higher than R, and diverge for products of ranks lower than R.

For example, in butyronitrile hydrolysis, the second-rank Delplots for butanamide diverges, and that for butyric acid has a finite intercept (see Figure 7.3). This confirms butanamide as the primary product and butyric acid as the secondary one. For reactions with steps of higher molecularities, the same plots are used. However, a distinction between the Delplot rank and network rank of a product must now be made. The Delplot rank R is determined by the plots as for networks of firstorder steps only. The network rank A^ indicates the order of appearance: A^ = 1 for primary products, A^ = 2 for secondary products, etc. The Delplot rank of species L of a step n^K —• L (n'th order in K) is related to the Delplot rank of its parent K by

7.1. Order and rank

R,

=

169

(7.2)

1 + «K^K

More generally, for a step ny_^K^ + n^^Kj + ... +

«K„K„

—•

«LL

+

n

(7.3)

For example, if butanamide in network 7.1 would dimerize instead of forming butyric acid, theDelplot rank of the dimer would be 1 + 2*1 = 3, although the network rank as that of a secondary product is 2. Limitations and opportunities. The Delplot method reflects the mathematics of the network, not its structure. If two different networks produce the same approximate rate behavior, the Delplot sees no difference. If the rate behavior of a multistep pathway can be approximated by that of a single step, the Delplot sees it as a single step. The words "step" and "directly" must not be taken literally: A "step" may be a sequence of steps of which all but one are fast. The Delplot user sets a time scale by his sampling for analysis, and steps fast on that scale fall by the wayside. This is so, for example, if the concentration of a first intermediate has already approached a plateau or gone through a maximum when the first sample is taken. Example 7.2. Paraffin autoxidation. The primary, but unstable products in autoxidation of paraffins are hydroperoxides, which quickly decay to ketones and more slowly to alcohols and acids. Typical selectivities to hydroperoxides and ketones as functions of fractional conversion are shown in Figure 7.4 (first-rank Delplots). The selectivity to hydroperoxides as sole and unstable primary products is 1.0 at zero conversion and decays steeply. The selectivity to ketones as quickly formed secondary products is zero at zero conversion and goes through an early maximum of about 0.35, other products also being formed. However, if no samples were taken during the first 0.5% conversion, the selectivity to ketones would be judged to extrapolate to about 0.4, as though ketones were primary products: The Delplot may miss a fast first step and declare products arising from the first intermediate as primary.

hydroperoxides

0.01

0.02

0.03

0.04

/A

0.4 •

• •/

ketones

0.2 •1

0

0.01

0.02

1

1

0.03

0.04

/A

Figure 7.4. Typical selectivities to hydroperoxides and ketones in autoxidation of «-octane at 145° C (from data by Garcia-Ochoa et at. [6]).

170

Chapter 7. Network elucidation This example can also illustrate how such an erroneous conclusion from Delplots can be avoided even if data at very low conversion do not allow unambiguous extrapolations. The selectivities to all products must sum up to 1.0, so if the bestlooking extrapolations of the plots for hydroperoxides and ketones sum to a larger value, one of them must be incorrect and, instead, that plot must dip down as zero conversion is approached. Its relatively gentle slope at the lowest observed conversions makes it likely that this is the plot for ketones, so that these should be secondary rather than primary products. Furthermore, direct formation of ketones would require the very tight oxygen-oxygen bond in O2 to be broken (498 kJ mol"^), making such a reaction step energetically much less likely than the pathway through a hydroperoxide.

Reactions with other than unimolecular steps pose still another difficulty. Formula 7.3 yields Delplot ranks if the network is known, but the user wants the reverse, to deduce networks from Delplot ranks. Ambiguities as to provenance may arise. For example, in both the two networks below, the Delplot ranks are 1 for P and Q, so that no distinction is possible without additional information.

A

• P

A

B -^=^ Q

B

• P

In so simple a case, stoichiometry can settle the issue: If P is not an isomer of A, the network on the left is incorrect, and if it is not an isomer of B, the network on the right is incorrect. However, stoichiometry cannot decide between the following two networks, both with Delplot ranks 1 for P and R, and 2 for Q: A

• P —p^ Q B

• R

A B

• P

• R -^^^^-> Q

The question here is whether P or R is a final product. Results at high conversion can show this, but only if the final steps are irreversible. That, however, cannot be taken for granted. Another indication of provenance can sometimes be gained from a comparison of Delplot slopes. Regardless of rank, a steep downward slope of a plot indicates that the respective product decays rapidly. Such decay must be matched by rapid formation of one or more other products. The latters' Delplots of same rank must show comparable steep upward slopes in the same conversion range. Matching steep upward to steep downward slopes can thus help to assign provenance. As an example, see the slopes of the plots for hydroperoxides and ketones in Figure 7.4.

7.2. "One-plus" rate equations

111

These examples show that an evaluation of extrapolated Delplot intercepts alone may be inconclusive or even lead to misinterpretations. Slopes as well as intercepts should be considered. Moreover, Delplots should not be used by themselves. Their evaluation should be combined with whatever can be gleaned from stoichiometry, chemical energetics, stereochemistry, analogies with similar reactions, and any other information that can brought to bear. 7.2.

"One-plus" rate equations

For mathematical convenience and economy of effort, rate equations in network elucidation and modeling are best written in terms of the minimum necessary number of constant "phenomenological" coefficients, which may be composites of rate coefficients of elementary steps. This not only simplifies algebra and increases clarity, but also lightens the experimental burden: With fewer coefficients, fewer experiments are required to determine their numerical values and their temperature dependences, without which a model is worthless for process development and design. Rate equations of multistep reactions with reverse steps contain additive terms in the denominator, but not the numerator. Examples from previous chapters include nitration of aromatics and olefin hydroformylation (see Examples 4.4 in Section 4.4 and 6.2 in Section 6.3, respectively). In all such cases, the number of phenomenological coefficients can be reduced by one if numerator and denominator are divided by one of the terms of the latter. The result is a "one-plus" rate equation, with a "1" as the leading term in the denominator. (This procedure is superfluous if all terms in the denominator consist only of coefficients, or of coefficients multiplied with the same concentration or concentrations, so that they can be combined to produce a true power-law rate equation.) One-plus equations are common in many fields of science and technology, the most notable being the Langmuir adsorption isotherm (see Section 2.6). One-plus rate equations play a key role in network elucidation. Typically, the most difficult step in that endeavor is the translation of a mathematical description of experimental results into a correct network of elementary reaction steps. The observed behavior can usually be fitted quite well by a traditional power law with empirical, fractional exponents, at least within a limited range of conditions. This has indeed been standard procedure in times past, and to some extent is still so today. However, such equations cannot be expected to result from combinations of elementary steps. Their acceptance may be expedient, but as far as network elucidation is concerned they are a dead end. In contrast, one-plus rate equations can result from step combinations, and their establishment therefore is an important stepping stone in the course of network elucidation, as the present chapter will demonstrate.

Chapter 7. Network elucidation

172

If fitting a power law requires fractional exponents, a one-plus rate equation with integer exponents should be tried instead.

Moreover, being more likely to reflect the true mechanism, the one-plus rate equation is also more likely to remain valid upon extrapolation to still unexplored ranges of conditions. There are important exceptions: Most chain reactions and some reactions of heterogeneous catalysis or polymerization or involving pre-dissociation produce exponents of one half or integer multiples of one half in power-law or one-plus rate equations (see Sections 5.6, 9.2, 10.3, and 11.3.1). Such exponents should be accepted if found not to vary with conversion and if there is good reason to believe that a mechanism of this kind may be operative. 7.2.7.

Types of one-plus rate equations

For an example of the simplest type of one-plus rate equation, let us return to the reaction A (5.72) A <—• X ^=^—> P with rate equation A-p

=

^AX % P ^ A

(5.74)

'^XA ^ '^XP^A

Dividing numerator and denominator by the first term of the latter one obtains

h = 1

kc:

(7.4)

"-b^A

with the two phenomenological coefficients

K

-

k Ik

The number of coefficients, three in eqn 5.74, has been reduced to two. Another, already encountered example of a one-plus rate equation is that of olefin hydroformylation (see Example 6.2 in Section 6.3). Here, the rate equation after cancellations but before reduction was

7,2. ''One-plus" rate equations

Ylli

K^ ^12 ^23 ^34 ^45 ^ole ^cat/^H, (^23^34^45 ^ ^21^34^45)/^CO "^ (^21^32^45 "^

(6.11) KlKl'^^^^Pu,

with seven rate coefficients of steps and one equilibrium constant. Collection of terms and division of numerator and denominator by the second term of the latter gave % ^ole ^cat

(6.12)

1 . KP^JP^^ with only two phenomenological coefficients =

^1^12^23^34^45 ^21^32V%5 "^ ^43)

j^

^

(^23 "^ ^2l)^34^45 ^21^32(^45 "^ ^43)

This example also demonstrates another point. The denominator of a rate equation obtained from the general formula (eqns 6.4 to 6.6) may contain several terms that involve the same combination of concentrations of co-reactants. Such terms can, of course, be lumped. Thus, in the original rate equation 6.10, the first two denominator terms both contained the co-factor j[?co' the third and fourth both contained p^^. In eqn 6.11, obtained by lumping the terms of these pairs, the number of denominator terms had been reduced from four to two. The one-plus form 6.12 was then obtained by division by the second of these two, actually the sum of the third and fourth terms in the original equation. [Rule 24 in Section 7.3.1 allows such lumping even before the rate equation is derived.] Even after such lumping, the denominator may be left with more than two terms. The one-plus equation will then have more than two phenomenological coefficients. 7.2.2. Establishment of one-plus rate equations from experimental data The principle of establishing a one-plus rate equation and the values of its phenomenological coefficients is very simple. If the reaction is irreversible and found to be of an order between zero and one with respect to a participant i, the simplest one-plus equation contains the respective concentration Q (or/?;) as a factor in the numerator and in some but not all terms of the denominator. More generally, if the order is between n (positive integer) and n + 1, the simplest equation contains the factor C^^^ in the numerator and Q in some but not all terms of the denominator. Many other combinations are possible, but less likely. For instance, an order between zero and plus one might also result from a numerator with factor C^ and a denominator with Q in some terms and Q^ in the others. Occam's razor suggests the best policy: to try the simplest option first.

174

Chapter 7. Network elucidation

The approach for negative reaction orders is similar. For example, the simplest equation giving an order between zero and minus one contains the respective concentration in one or more terms of the denominator, but not in the numerator. To summarize the forms of some of the simplest one-plus equations:

between between between between

factor in some but not all denominator terms

factor in numerator

order 0 and +1: 0 and - 1 : +n and +/24-1: 0 and —n:

Q -

c,

cr'

Q

Q

—

C"

As an example, the four simplest one-plus rate equations for an irreversible reaction of first order in A and orders between zero and plus one in B and C are: "•aQ^sQ

^a^A^B^C ^P

=

r?

1 + fc^Cg + k^C^ '

=

1 . k,C^C^ (7.5)

^a^A^sQ

r

'V

1

r

^B^A^BQ

-

1

•^ ^b^B '^ ^c^B^C

+ % ^ B ^ C "^ ^c^C

A convenient method for testing a tentative one-plus equation and obtaining values of its coefficients is nonlinear regression (see Section 3.5). Alternatively, the equation can be brought into linear form and tested by linear regression or plotting. The rate is a nonlinear function of the concentrations because of the additive terms in the denominator, but the concentration dependence becomes linear when the reciprocal is taken. For example, if the rate is TT.

'p

KC,

=

1 - KC^

its reciprocal is 1 a

A

a

so that a plot of 1/rp versus l/C^ gives a straight line with slope l/k^ and intercept k^ Ik^ if the coefficients are constant. If the plot of the experimental results does not give a straight line, a different one-plus form may have to be tried.

7.2.

"One-plus" rate equations

175

The following example illustrates the procedure step by step as applied to a moderately complex reaction. Example 7.3. One-plus rate equation for hydrocarbonyl-catalyzed hydrogenation of aldehyde [7]. Homogeneous liquid-phase hydrogenation of aldehydes to alcohols H. aldehyde J:-^

(7.6)

alcohol

is catalyzed by dissolved phosphine-substituted cobalt hydrocarbonyl, HCo(CO)3Ph, where Ph is a tertiary organic phosphine. The reaction requires the presence of CO as well as H2 in order to keep the catalyst stable. Table 7.1 lists experimental results of hydrogenation of 2-ethylhexanal at different aldehyde concentrations and partial pressures of H2 and CO. Table 7.1. Rates of hydrogenation of 2-ethylhexanal in w-dodecanol at 165°C, measured in a continuous stirred-tank reactor [8] (concentration of catalyst, HCo(CO)3Ph, same in all runs).

run

1 2 3 4 5 6 7 8 9 10 11 12 13 14

aldehyde concentration

partial pressure atm

M

of H2

of CO

0.100 .109 .105 .101 .100 .111 .129 .136 .050 .051 .061 .020 .024 .010

80 80 40 20 80 40 20 10 80 40 20 40 20 40

80 40 40 40 20 20 20 20 40 20 20 20 20 20

rate M min'^ 1.83*10-3 3.90 3.33 2.42 7.20 6.87 6.18 4.44 1.81 3.17 2.90 1.20 1.16 0.62

Most homogeneous hydrogenation reactions are first order in the organic reactant. A cursory inspection of the results suggest that this may be the case here, too. To test this hypothesis, a tentative rate equation is written: ale

-

^app(/^H,'/^Co)^ald

176

Chapter 7. Network

elucidation

whose apparent first-order rate coefficient, k^^^, still is a function of the partial pressures of H2 and CO. For each run, ^^pp = r^ic /Qi^ is calculated. Comparison of the values (fifth column in Table 7.2) shows them to be the same within experimental error for runs with different aldehyde concentrations but same partial pressures of H2 and CO (runs 2 and 9 at 80 atm H2 and 40 atm CO; runs 6, 10, 12, and 14 at 40 and 20 atm; runs 7, 11, and 13 at 20 and 20 atm). Thus, k^^^ is independent of the aldehyde concentration, i.e., the reaction is first order in aldehyde as expected. Table 7.2. Work-up of data on aldehyde hydrogenation in Table 7.1 [7].

1 2 3 4 5 6 7 8 9 10 11 12 13 14

U

=

Qld

PH2

Pco

M

atm

atm

0.100 .109 .105 .101 .100 .111 .129 .136 .050 .051 .061 .020 .024 .010

80 80 40 20 80 40 20 10 80 40 20 40 20 40

1.83*10-2 3.58 40 1 3.17 2.40 40 7.20 20 6.19 20 4.79 20 3.26 20 3.62 40 6.22 20 4.75 20 6.00 20 4.83 20 6.20 20

run

80 40

rale M min '

|

spp ''ale ' Q l d

min"'

observed

calculated

1.83*10-3 3.90 3.33 2.42 7.20 6.87 6.18 4.44 1.81 3.17 2.90 1.20 1.16 0.62

1.81*10-3 3.95 3.24 2.41 7.24 6.86 6.16 4.46 1.81 3.15 2.91 1.24 1.15 0.62

To establish the order with respect to CO, values of k^^^ at same partial pressure of H2 but different partial pressures of CO are compared. Within each of these sets (runs 1, 2, 5, and 9 at 80 atm H2; runs 3, 6, 10, 12, and 14 at 40 atm; runs 4, 7, 11, and 13 at 20 atm) the coefficient is seen to vary in good approximation in inverse proportion to the CO pressure. The order with respect to CO thus is minus one within experimental error. In a like manner, to establish the order with respect to H2, values of ^^pp at same partial pressure of CO but different partial pressures of H2 are compared. At both 40 atm CO (runs 2, 3, 4, and 9) and 20 atm CO (runs 5 to 8 and 10 to 14) the coefficient is found to increase with increasing pressure of H2, but clearly less than in proportion to that pressure: The order with respect to H2 thus is between zero and plus one. The simplest one-plus equation with the established apparent reaction orders and first order in catalyst is

7,2,

177

''One-plus'' rate equations

(7.7) To test it, eqn 7.7 is rearranged into a form with reciprocal rate on the lefthand side and simplest possible additive terms on the right: C C

1

(7.8)

KPH,

^alcPcO

If eqn 7.7 reflects the data correctly, a plot of the left-hand side of eqn 7.8 versus the reciprocal partial pressure of H2 will give a straight line. Figure 7.5 shows that to be true. 1.6h

1.2

''ale Pco

0.8

[min atm0.4

0.02

0.04

0.06

0.08

0.10

Figure 7.5. Plot to test rate equation 7.7 (C^^^, same in all runs, not included in calculation) [7]. According to eqn 7.8, the slope of the straight line in Figure 7.5 is l/k^ and the intercept is k^lk^. Evaluation by linear regression yields k

= (0.105 mm-')IC

k^ = 0.060 atm-^

The last two columns of Table 7.2 show a comparison of observed rates with those calculated with eqn 7.7 and these coefficients. The agreement is excellent. Even so, eqn 7.7 should not be viewed as established beyond doubt until a plausible mechanism leading to it has been found and predictions made with it have proved correct. The search for a mechanism will be described in Example 7.4 in the next section.

178

Chapter 7. Network elucidation

A fit to the experimental results may require a one-plus rate equation with three or more terms in the denominator (e.g., see eqns 7.5). If so, the coefficients can be determined by nonlinear regression, multiple linear regression, or, longhand, by cross-plotting. Say, the one-plus equation is 1 + k^C^ + k^C^ One of several possible graphical procedures then is to group the rate data into different sets, each with different Q but same (constant) C^. The rate equation is rearranged to _ ^P

= —i + — ^A ^a

where

k^ =

i—i = const, at const. Cg K

and k^ and kjk^ are obtained from slope and intercept of a plot of C^/r^ versus 1/Q. The k^ values from the different sets are then plotted versus Cg to obtain k^ Ik^ and \lk^ from slope and intercept. The only still unknown coefficient, k^, can now be calculated from the previously determined ratio k^lk^.

7.3.

Relationships between network properties and kinetic behavior

At the very basis of chemical kinetics as we know it is the knowledge of how the mechanics of a molecular event is reflected in observable kinetic behavior: the knowledge that the spontaneous decay or rearrangement of a molecule occurs at a rate that is proportional to the concentration of the species, that the formation of a product by collision of two molecules does so at a rate proportional to the concentrations of both reactants, etc. To every chemist and chemical engineer this has been self-evident for as long as he or she can remember. Unfortunately, this core of elementary knowledge tells us only what happens in single-step reactions and proves woefully inadequate when we face the complications of real-life chemistry. Efficient handling of kinetic problems in practice, and especially of network elucidation, calls for a broadening of that basis to include multistep reactions. The present section addresses this problem with the deduction of additional rules [7,9]. While covering additional ground, the set of rules in the present section still leaves important areas of kinetics untouched. Four of these—trace-level catalysis, heterogeneous catalysis, chain reactions, and polymerization—will be addressed in the next four chapters. A fifth, reactions with periodic or chaotic behavior, will be avoided by the practical engineer if at all possible, and only a brief outline of when and how they may arise will be given in Chapter 14.

7.3. Network properties and kinetic behavior 7.3.1.

179

Simple pathways [7,9]

The rules deduced in this section are exclusively for simple pathways. A pathway or network is "simple" if all its intermediates are and remain at trace level and if no step involves two or more molecules of intermediates as reactants (see definition in Section 6.1). In its most general form a simple pathway is X,

X,.

(6.3)

with arbitrary number of reversible steps and any number of co-reactants and coproducts (not shown) participating in any or all of the steps. If a step is irreversible, its reverse coefficient is set to zero. The effects of co-reactants and coproducts are accounted for in the pseudo-first order rate coefficients Xy of the steps, defined in Section 6.2. Suppose the pathway has six steps (the generalization to any number of steps will be obvious). The rate of product formation then is \y\2\^\^\^\(,C^

(7.9)

D^

where DQ^, is the sum of the products of the elements of the rows of the matrix

K

1 1

Ms ^34

(7.10)

M5

K ^32

''43

^56

^32

\3

1

(see eqns 6.4 to 6.6 and matrix 6.7). The i'th column contains coefficients of the i'th step, the forward coefficient above the diagonal, the reverse coefficient below the diagonal. Equation 7.9 with matrix 7.10 (or 6.7) permit the following rules to be deduced. Irreversible steps A pathway is irreversible if one or more of its steps are irreversible.

(7.11)

180

Chapter 7. Network elucidation

Even if only one of the reverse X coefficients in eqn 7.9 is zero, the second term in the numerator and thus the rate of the reverse reaction are zero. Also: Steps following an irreversible step have no effect on the rate equation.

(7.12)

If, say, the third step is irreversible, its reverse coefficient X32 is zero. This coefficient appears in the second, negative term of the numerator of the rate equation, making it zero. It also appears in all matrix rows from the fourth on down, so that the products of the elements of these rows are zero. The denominator of the rate equation thus contains only three terms: the products of the elements of the first three rows. These terms contain the forward coefficients of the steps from the fourth onward as common factors, and so does the numerator; accordingly, these coefficients cancel and the rate equation is reduced to

^12^23

•" ^ 1 0 ^ 2 3

•"

^10^1

as though the pathway ended at the irreversible third step. Reaction order plus one (forward reaction) A reaction is first order with respect to any reactant that participates (with one molecule) in only the first step.

(7.13)

The concentration of the original reactant A is a factor in the numerator of the rate equation. It does not appear as a co-factor in any of the X coefficients. This makes the rate proportional to that concentration, so that the reaction is first order in A. The first reaction step may involve a co-reactant, B. If so, the concentration of B is a co-factor in the forward coefficient XQJ of the first step. This coefficient is a factor in the numerator, but does not appear anywhere in the matrix and so is absent from the denominator. Accordingly, the rate is proportional to the concentration of B, so that the reaction is first order in B. In the case of reversible overall reactions, the rule above and the following ones for reaction orders are for the forward reaction only. They do not apply to orders obtained by fitting a forward power law to a limited conversion range of a reversible reaction, as in Figure 5.4, right diagram, in Section 5.1.1.

7.3. Network properties and kinetic behavior

181

Reaction order plus two (forward reaction) A reaction is second order with respect to any reactant that participates with two molecules in only the first step.

(7.14)

The same argument as for first-order behavior applies, except that the reactant concentration now appears as factor Cl in the numerator or, if the participant is a co-reactant B, as co-factor C^ in XQI in the numerator. Reaction order between zero and plus one (forward reaction) A reaction is of order between zero and plus one with respect to any reactant that participates (with one molecule) in a step other than the first.

(7.15)

The concentration of a co-reactant involved in a step other than the first is a cofactor in the X coefficient of that step. This coefficient appears as a factor in the numerator and as an element in some but not all matrix rows. For example, X12 is present in only the first row, X23 in only the first two rows, etc. Accordingly, the co-reactant concentration is a factor in some but not all denominator terms. As a result, the order with respect to the co-reactant is between zero and plus one. Both limits of that range—reaction orders zero and plus one—may be realized. The order is plus one if all denominator terms containing the X coefficient with the co-reactant concentration as co-factor are negligible. It is zero if all other denominator terms are negligible. Thus, an observed positive order less than one is proof that the respective reactant participates in a step other than the first, but an observed order of plus one does not prove its participation in the first step. Other positive reaction orders (forward reaction), The line of argument is easily extended. For example: A reaction is of order between one and two with respect to any reactant that participates (with one molecule each) in the first and a later step.

(7.16)

If the reactant is the original reactant A, its concentration appears as factor in the numerator and as co-factor in a forward X coefficient in both the numerator and at

182

Chapter 7. Network elucidation

least one denominator term, but not in all of these. If the reactant is a co-reactant B, its concentration appears as co-factor in XQI in the numerator and in another forward X coefficient in both the numerator and at least one denominator term, but not all of these. As before, both limits of the range may be realized. Similarly, the order is between n and « + 1 if the respective reactant participates in the first and n subsequent steps with one molecule in each. Negative reaction orders (forward reaction) For a reaction order to be negative, the respective participant (reactant, product, or silent partner) must be a product in a reversible step that is neither the last nor preceded by an irreversible one.

(7.17)

For the order of the forward reaction to be negative with respect to a participant, a X coefficient in which the concentration of the latter is a co-factor must appear only in the denominator. Since the numerator contains all forward X coefficients, this can only be a reverse coefficient. Accordingly, the participant must be a coreactant in a reverse step (or, in other words, a co-product in a reversible forward step). That step may not be the last because the denominator does not contain the last reverse X coefficient, and may not be preceded by an irreversible one because in that case it would have no effect on the rate (see Rule 7.12). The condition stated is necessary, but not sufficient: The denominator terms containing the X coefficient of the reverse step in question could be negligible even if the step is not preceded by an irreversible one, and the order would then be zero. For a reaction order to be negative with respect to a reactant or silent partner, the step in which that participant is a product must precede the step or steps in which it is a reactant.

(7.18)

This condition is in addition to the earlier one and can be deduced as follows. If the participant is a silent partner, it must function as a reactant in one step and as a product in another. The forward X coefficient of the step in which it is a reactant appears in both the numerator and denominator. The reverse X coefficient of the step in which it is a product appears only in the denominator. Both coefficients contain the concentration of the participant as co-factor. A negative order with respect to a participant requires the concentration of the latter to appear raised to higher power in the denominator than in the numerator. Therefore, at least one denominator term must contain both coefficients containing the

7.3, Network properties and kinetic behavior

183

concentration of the participant. As evident from the matrix 7.10, the only such combinations are those in which the forward coefficient is that of a later step. If the participant in question is a reactant, two of its molecules must re-enter after one is split off. A negative reaction order then is possible if one or both reentry steps (but not the split-off step) are preceded by an irreversible step and so have no effect on the rate equation. Negative orders with respect to products are fairly common. A negative order with respect to a reactant (reactant-inhibited reaction) is unusual. An example is olefin hydroformylation with a reaction order between zero and minus one in CO, a reactant (see eqn 6.12). This behavior results from a first step in which one CO ligand is displaced from the catalyst, and the fact that one of the two later steps in which CO re-enters the pathway occurs after an irreversible step. An example of a negative reaction order with respect to a silent partner is the hydrocarbonyl-catalyzed hydrogenation of aldehydes, with a rate of order minus one in CO (see eqn 7.7) and a pathway still to be explored later in this section (see Example 7.4). Reaction order zero (forward reaction) A reaction is of order zero in any participant involved exclusively in steps preceded by an irreversible step.

(7.19)

This is a corollary of the rule that steps preceded by an irreversible one have no effect on the rate (see Rule 7.12). It is a sufficient condition, but not a necessary one: Even if the respective species participates in a step not preceded by an irreversible one, its order may be zero. This is because a zero order is possible as a special case of an order between zero and plus one, or between zero and minus one, as discussed in the context of those orders. Positive order with respect to a product (product-promoted reaction) A positive reaction order with respect to a product requires a step in which the product acts as reactant.

(7.20)

Self-acceleration of a reaction as conversion progresses (often called autocatalysis) usually stems from promotion by a product or early major intermediate. A classical example is acid-catalyzed ester hydrolysis, where the acid formed adds to the amount of catalyst initially present. Rule 7.20 above has long been recognized as self-evident, and applies even if the reaction occurs in a single step. However, its converse is not necessarily true: A step in which a product of the overall reaction

184

Chapter 7. Network elucidation

functions as a reactant fails to produce self-acceleration if it is preceded by an irreversible step and thus does not affect the rate (see Rule 7.12). Reaction orders and sequence of co-reactant entries. If several co-reactants enter the pathway at different steps (with one molecule each), the sequence of entries is that of decreasing reaction orders: The later a co-reactant enters a pathway, the lower is its reaction order.

(7.21)

This can be gleaned from the matrix 7.10 or 6.7: The co-reactant concentration appears as co-factor in the forward X coefficient of the step of entry. This coefficient is a factor in the numerator of the rate equation and in some denominator terms (except if the step is the first). As the matrix shows, the later the step, the greater is the number of rows containing that coefficient and, therefore, the greater is the number of denominator terms containing the coefficient and thus the coreactant concentration. With more denominator terms containing the concentration, the reaction order is lower. Say, co-reactant B enters in the second step, and co-reactant C in the fourth, which is irreversible. The rate equation then is k k k k C C C ^12^23^34 ^ B ^ C "^ ^10^23^34 ^ C ^ ^10^21 ^34 ^ C "^ ^10^21^32

Its numerator is proportional to the concentrations of both B and C while its denominator, containing the concentration of C in its first three terms and that of B in only the first, increases more strongly with the former concentration than with the latter. This makes the order closer to first for B than it does for C. While a lower reaction order indicates later entry of the respective coreactant, equal reaction orders do not allow a distinction with respect to sequence of entry to be made because the denominator terms containing only the concentration of the later entrant could be negligible. Pathways with rate-controlling step If the forward and reverse coefficients of a step are much smaller than all others, all other steps are at quasi-equilibrium if reversible, and at complete conversion if irreversible.

(7.22)

This rule, too, is of long standing. The formalism introduced here permits a very simple, yet rigorous proof to be given, as shown below for a specific case. The generalization will be obvious.

7.5.

Network properties and kinetic behavior

185

Proof. Suppose the slow step is the third in a six-step pathway, with coefficients X23 and X32. All matrix rows except the third contain either of these two coefficients, so that the products of their elements become negligible. The denominator thus consists of only one term, with the coefficients in the third matrix row as factors. Accordingly, the rate equation becomes ^

^01 ^12^23r34f 45^56 ^10^21^34^5^56

r 10^21^32^43^54^65 ^

fl0f21^34'^45'^56

In the first term, the last three forward coefficients appear in both the numerator and denominator and cancel, and \)\\2I\Q^I\ = (^01^12/^oi^i2)%2 = f^oi^oi^ where ^02 is the equilibrium constant of A -h ... <—> X2 + ... and 5Ro2 is the ratio of the product of the concentrations of any co-reactants to that of any co-products in that reaction. In the second term, the first two reverse coefficients cancel, and X43A54A55/A34X45A56 =

(^43^54^65

lk,,k,,k,,)/%, = {K,,%,r\ with K,, and %e as the equilibrium constant and co-reactant to co-product concentration ratio, respectively, of X3 -f- ... <—• P -f- ... With these substitutions: Tp = ^^KQ^'^Q^C^ - X32(/(:36 9?36) Cp According to the mass-action law, ^02%2Q is the concentration of X2 in equilibrium with A and the co-reactants and co-products of the first two steps, and (K^^^di^^y^Cp is that of X3 in equilibrium with P and the co-reactants and co-products of the last three steps. This completes the proof. One-plus rate equations A one-plus rate equation requires a pathway with at least one reverse step.

'

(7.23)

For the general rate equation 7.9 to reduce to a one-plus form rather than a power law, the denominator must have at least two non-zero terms. However, only the first row of the matrix contains no reverse coefficients, so the first term alone would survive if all reverse coefficients were zero. Condition 7.23 is necessary, but not sufficient. All significant denominator terms may have the same concentration or concentrations as co-factors, so that they can be lumped into a single term. Moreover, despite reverse steps in the pathway, a disparity of other coefficients can make all denominator terms but one negligible. In either case, the rate equation is a power law rather than a one-plus equation. In the great majority of cases, an observed fractional reaction order expresses a rate that in reality obeys a one-plus equation. The exception are orders of one half or integer multiples of one half produced by some types of reactions with nonsimple pathways (see Sections 5.6, 9.3, 10.3, 11.3.1, and 11.4.1).

186

Chapter 7. Network elucidation

Insensitivity of rate equations to step consolidations The algebraic form of the rate equation is not changed if any of the following step sequences is consolidated into a single step: • a step with co-reactant entry, followed by a step with coproduct exit or a rearrangement step, and • a rearrangement step, followed by a step with co-product exit or another rearrangement step. At most the last step of such a sequence may be irreversible.

(7.24)

Specifically, the permissible consolidations are: B

Xj^^x,^^-^x,

to

X. V^ v^ X

X, 5^=-> X, <—^ X,

X, ^

Xj^^-^x^^^^x,

x.^—^x,

Q

• X,

Q

X.<->X,<—X,

X^^^

•X,

Note that a step with co-product exit may not be consolidated with any subsequent step, nor may a step with co-reactant entry be consolidated with any preceding step. The procedure may be repeated to consolidate more than two successive steps. Consolidation saves work in establishing a rate equation and in modeling. On the downside, the concentration dependence of the rate provides no clue as to whether the actual mechanism involves a step sequence or a single step into which such a sequence could be consolidated. Proof. Shown below are the matrix portions relevant for a step sequence (left) and its consolidated step (right). original sequence column rowj-1

j-1 ...

1

j

consolidated k

>^^t hi

£

-'

h4'i

^

hi

KM

-

rowk

..,

Xj^.j

\^

1

\^^j

,..

row£

... X..,,

1

...

Xj^ X^

j-1

jk

£

1

Xjg

h,e+i

Xjj.,

1

h^i

... x.^.

hi

1

KMI •••

^<>WJ

step

...

7.5.

Network properties and kinetic behavior

187

The two matrices contain the same elements except in rows and columns j and k (shaded). In columns j and k the sequence matrix has elements Xj^ and X,,^ in rows 1 to j — 1, and elements X^j and X^^ ^^ rows k+1 to last; the corresponding step matrix has instead a single column jk with elements Xj^ in rows 1 to j — 1 and X^j in rows £ to last. The concentration co-factor is the same in Xj^ as in Xj^X^^, and in X^j as in XkjX^k' so from each of these rows the sequence matrix and the step matrix produce denominator terms of the same algebraic forms. Not yet accounted for are the sequence-matrix rows j and k, containing elements 1 and X^^ in row j , and X^j and 1 in row k. Consolidation replaces them by a single row with element 1 in the consolidated column jk. However, \f\i and X^j have no concentration co-factors, the denominator terms from the two rows j and k can be lumped into one, which is of the same algebraic form as that from the consolidated row jk. This condition is met unless an entry step is consolidated with a preceding step, or an exit step with a subsequent step. Steps following an irreversible one have no effect on the rate. Consolidation with subsequent ones would thus be pointless and could introduce spurious co-factors. Note of caution: As stated at the outset, all rules in this section are for simple pathways and do not necessarily apply to other types of networks. Application. The set of rules in this section is an invaluable tool in pathway elucidation. It identifies observable features of kinetic behavior as consequences of pathway configurations. This makes screening of rival pathways more effective. No pathway that includes a configuration producing a behavior contrary to observation can be correct. Thus, incorrect pathways can be rejected as whole groups instead of one by one. A relatively simple example will illustrate how the rules can be applied. Example 7.4. Pathway elucidation of hydrocarbonyl-catalyzed aldehyde hydrogenation [7,9]. In Example 7.3 in the previous section, a one-plus rate equation for hydrocarbonyl-catalyzed aldehyde hydrogenation was established: ^ a ^ald/^H^ ^cat

/-j

j\

'"' ^ /'cod - KP^) A plausible mechanism that would produce this kinetic behavior is to be found. The reaction is first order in aldehyde and catalyst, of order between zero and plus one in H2, and of order minus one in CO, a silent partner. Rule 7.13 for first order suggests a first step in which aldehyde reacts with catalyst. Rule 7.15 for orders between zero and plus one shows that H2 cannot be a reactant in the first step. Rule 7.18 for negative order with respect to a silent partner shows that CO must be split off in a reversible step before being re-incorporated. Thus, at first glance it seems the simplest pathway may consist of a reversible first step in which aldehyde displaces one CO ligand from the catalyst to form an adduct that is subsequently hydrogenated in an irreversible step, yielding alcohol and a CO-deficient catalyst cat' that later recovers its missing CO:

188

Chapter 7. Network

elucidation

Trial pathway I cat

H2

aid N - ^

X ^ v ^ ale CO

cat'

(step cat' + CO —• cat not shown). The analogy with the Heck-Breslow mechanism of hydroformylation (network 6.9), in which a CO ligand is lost from such a catalyst and its place is taken by the reactant, lends credence to such a pathway. For pathway I above, the general rate equation reduces to ^Ol^n^ald^H^^cat

(125)

The reaction can be of order minus one in CO as required, but only if the first denominator term is negligible compared with the second. The reaction can also be of positive order less than one in H2 as required, but only if the first denominator term is not negligible compared with the second. These are mutually exclusive demands. Accordingly, this pathway is incorrect. Moreover, as the matrix 7.10 or 6.7 shows, the first denominator term of the general rate equation contains only forward X coefficients, and must therefore be negligible to produce the order minus one in CO. Without its first term, the denominator contains neither Xoi nor X12 (first matrix row now negligible). But to produce the observed order less than one in H2, the denominator must contain the H2 pressure as co-factor in at least one additive term. Accordingly, no pathway in which H2 reacts in either the first or second step can be correct. To get around this problem, we might try to insert a reversible rearrangement step into the pathway so as to make hydrogenation the third step. Such a step could be the rearrangement of a 7r-complex of the keto or enol form of the aldehyde with the catalyst to a a-bonded species more susceptible to hydrogenation [10]: Trial pathway II cat

H2

aid ^ ^ ^

X, <

• X2 ^ v ^ ale

CO

eat'

(step cat' + CO —• cat not shown). For this new pathway, the general rate equation reduces to ^01 ^12^23 ^ald/^Hj^cat

^12^23/^H, "^ ^XQKZPW^PCO

/y 26^

"^ ^lo^iiP CO

If the first denominator term is negligible, as it will be if the first step is at quasiequilibrium, the denominator still contains /?H2 i^ ^^^ ^^ the two remaining terms and Pco in both, and so produces a rate equation of the algebraic form of eqn 7.7 as required.

7.3.

Network properties and kinetic behavior

189

Pathway II is not the only one to produce the observed kinetic behavior. Although the reaction is first order in aldehyde and catalyst, either or both of these might react in a step later than the first. An equally simple pathway is: Trial pathway III aid

cat <—^

H2

cat' -4-=^=*—• X2 ^^>» ale CO

cat'

with rate equation ^01 ^12^23 ^aldPn^^cat ^12^23 ^ald/^H, "*" ^ 1 0 ^ 2 3 P H / C O

^

/y 2 7 ) ^\oK\PcO

(index 1 stands for cat'). Like eqn 7.26, this equation reduces as required to the form of eqn 7.7 if the first denominator term is negligible. Equations 7.26 and 7.27 differ only in the first denominator term, which must be negligible to fit the observed reaction order with respect to CO. A discrimination between the pathways II and III on the basis of the evidence at hand is therefore not possible. In fact, the most probable pathway is slightly more complex than either: Pathway IV aid

cat <—^

H

cat' "^^^^^^—• X^ <

• X3 " ^ ^

CO

ale

(7.28)

cat'

with rate equation ^01 ^12^23^34 ^aldPn, ^a ^12^23^34 ^Id/^H, "^ ^10^34(^23'^^2l)/^H3/^CO "^

(7.29) ^W^ll^SlPcO

Like eqns 7.26 and 7.27, this equation reduces to the required form if the first denominator term is negligible. Like pathway II, pathway IV incorporates the mechanistically probable rearrangement from a 7r-complex to a a-bonded species, but it avoids the need to assume quasi-equilibrium between aldehyde and catalyst. (Note that by Rule 7.24, pathways III and IV have the same rate behavior.) In any event, no fewer than three pathways giving the observed rate behavior have been found. This places the rate equation 7.7 on firmer ground, even though the question which pathway is correct has not been settled. In the context of process development and design, differences between mechanisms matter little as long as the mathematics of the reaction are the same. In the case at hand, the design engineer would only have to make sure that a range of conditions will never be entered in which the first denominator term would become significant.

190

Chapter 7. Network elucidation

7.3.2. Simple networks The procedure of arriving at a probable mechanism via an empirical rate equation, as described in the previous section, is mainly useful for elucidation of (linear) pathways. If the reaction has a branched network of any degree of complexity, it becomes difficult or impossible to attribute observed reaction orders unambiguously to their real causes. While the rate equations of a postulated network must eventually be checked against experimental observations, a handier tool in the early stages of network elucidation are the yield-ratio equations (see Section 6.4.3). This approach relies on the fact that the rules for simple pathways also hold for simple linear segments between network nodes and end products. The (instantaneous) yield ratio Y^Q of two products P and Q is defined as the ratio of the conversion rates of a reactant to these products: r^ln^

^^

TIT

Yield ratio equations assume simple forms if product formation is irreversible. If it is not, the equations for irreversible formation are reasonable approximations at very low conversions. Equations for products arising from the same or different network nodes were given in Section 6.4.3. The procedure of application will be illustrated here. The yield ratio of two products arising irreversibly (or at very low conversion) from the same node intermediate X^ is ^PQ = K'K^

(6-38)

Here, A^p and A^Q are the segment coefficients of the pathways from the node to the two products, given by eqns 6.5. Rival networks differing in the way co-reactants participate in steps along the two segments give rise to A coefficients and yield ratios with different dependences on co-reactant concentrations. Table 7.3 (next page) lists such dependences for a typical irreversible segment. Usually, quite a number of combinations of configurations of the two segments give results compatible with all experimental observations. The method thus chiefly serves to rule out possibilities. Example 7.5. Olefin hydroformylation with paraffin by-product formation [7,9]. Hydroformylation of olefins to aldehydes, catalyzed by a phosphine-substituted cobalt hydrocarbonyl, HCo(CO)3Ph (Ph = tertiary organic phosphine), has been used for illustration in examples 5.2 and 5.3 in Sections 5.2 and 5.3. The catalyst also promotes hydrogenation, so aldehyde produced from olefin is converted to alcohol, and paraffin is formed from olefin as by-product:

7.3.

Network properties and kinetic behavior

191

Table 7.3. Algebraic forms of segment coefficient A^p of segment with stoichiometry X^ + B —• P and different configurations (from Helfferich and Savage [7]). configuration of segment

algebraic form of A^

entry of B preceded by irreversible step ' : ^

—•

p

A,p = const.

V

X.

[^/(CB)]

entry of B at first irreversible step X . ^ ^

AkP = KC^ ^ X

^ ^ ^

K, = KCJi^+KC^)

entry of B within reversible portion preceding irreversible step X, ^ ^ - . . . - * p B^

A.P = KC^

\p = + H „ CO

KCjii+k,c^)

+H2

• aldehyde olefin

• alcohol

(7.30)

paraffin The catalyst differs from the "0x0" catalyst, HCo(CO)4, only in that one CO ligand is replaced by a tertiary organic phosphine, and rates obey the Martin equation 6.12 (see Example 6.2 in Section 6.3). Therefore, it seems a safe bet that the olefinto-aldehyde pathway is the same as with the 0x0 catalyst (network 6.9). Also, almost certainly, paraffin arises from an intermediate along that pathway rather than from a different olefin-catalyst adduct. Logical candidates for this node intermediate are the TT-complex (Xi in network 6.9), the trihydride (X2), and the cobalt phosphino-di- and tricarbonyl alkyls (X3 and X4, respectively) as paraffin can no longer be expected to form once irreversible carbonylation to cobalt acyl (X5) has occurred. Granted these premises and the additional simplest assumption that paraffin forms from the respective node intermediate in either a single step or in steps that can be consolidated with Rule 7.24 into a single step, four networks are in contention. They are shown on next page with their yield ratio equations. To avoid clutter, only the pathway portions from 7r-complex to cobalt acyl and paraffin are shown (since carbonylation to X4 —• X5 is irreversible, steps after X5 have no effect).

192

Chapter 7. Network elucidation

Table 7.4. Trial networks for hydroformylation with paraffin by-product formation. H,

cat

\

(I)

^

par

CO X, ^^^==^—• X , <

^>

X , ^^^^^

•

X,

•

(7.31) H, H ^k X, ^ ^ — • Xj ^

(11)

cat

^

^

•par /CO X3 ••^:^^—- X,

•

X5

H, _ j V _ par/aid

Pn,

^3par(^4S +^43)

A

(7.32)

^34^45

cat ^

(HI) X,

^1^^:^-—•

X2 < — ^

^2par _ A

par/aid

par X3

^ ^

K^zv^^iJ^AsPCO

•

X,

•

"^ (^32^43 '^^^l^^^^Pu

X5

\

^23^34^45/^CO

(7.33) -

^^P^^

^2par^32(^45 ''"^43) ^23^34^3

H2

cat'

>>—^^-> par /H2 Xi ^ ^ > X2 <—^

(IV)

y

_ ^""^

^Ipar _

CO X3 ^ ^

X,

•

X3

^lpar(^23^34^45/^CQ "^^21 ^34^45^^00 "^ ^21^32^45 Pw ^ ^21^32^43 Pw \

^Idld

^lpar(^23 + ^ 2 l ) ^12^23

•

^12^23^34^45/^CO

,

^lpar^21^32(^45 "^^43) ^12^23^34^43

(7.34) Pco

7.3.

Network properties and kinetic behavior

193

The A coefficients of the segments leading to paraffin and aldehyde are obtained from the general formula 6.4 to 6.6 in terms of X coefficients, which then are replaced by the true rate coefficients multiplied by co-reactant concentrations where called for. In network I, paraffin is formed by hydrogenation of the tricarbonyl alkyl, X4. Here, the segment coefficients are ^ 4 par ~ \ p a r ~ ^4par/^H,'

^ 4 aid ^ ^ 5 ^ ^45

In network II, paraffin is formed by hydrogenation of the dicarbonyl alkyl, X3, and the segment coefficients are A \ ^ n ^^3 par - ^3 par ^^3 par/^H,'

A ^ h aid

T

^34^5 — y

_

^SAKSPCO h ^ k

In network III, paraffin is split off from the trihydride, X2, and the segment coefficients are ^2par ^ '^2par ~ ^2 par

X23 X34 X45

A2 aid

k^^ ^34 ^45/?co

^34^5 + ^32^5 + ^32^3

KK5PCO

+ ^32(^45 + ^ 4 3 ) P H ,

Lastly, in network IV, paraffin is formed by hydrogenation of the 7r-complex, Xi, and the segment coefficients are ^Ipar ~ '^Ipar ~ ^Ipar/^H^

^12^23^34^5

^lald

^ 3 ^ 4 ^ 5 + \ \ ^34 ^ 5 + \ \ \ l \ 5 - ^ \ \ ^32 ^ 3 ^12^23^34^45/^H,/^CO (^23 "^ ^2l)^34%5/^CO ^ ^21^32^^45 "^ '^A^^Pw^

The yield ratio equations, obtained as ratios of the segment coefficients according to eqn 6.38, are shown with the respective networks in Table 7.4 (facing page). For network I, the yield ratio is proportional to the partial pressure of H2. In the other three cases, the yield ratio is seen to depend only on the H2-to-CO ratio, not on total pressure at same H2-to-CO ratio. However, the dependence on that ratio differs. For network II it is of the form V/a,d

= KiPnJPco)

(7.35)

= K - KiPn'Pco)

(7.36)

whereas for networks III and IV: V/a.d

194

Chapter 7. Network elucidation This suggests experiments in 0.12 which the yield ratio is measured at different total pressures and H2-to-CO ratios. Results of such experiments 0.08 are shown in Figure 7.6, in which the yield ratio is par/aid plotted versus the H2-to-CO ratio. The yield ratio is seen 0.04 o 25-30 atm to be independent of total D 50-60 atm pressure at same H2-to-CO A 75 atm ratio, to vary linearly with that ratio, and to remain finite when extrapolated to a zero 2.0 3.0 1.0 value of that ratio; that is, PH;'PCO eqn 7.36 is obeyed but eqns 7.31 and 7.35 are not. The Figure 7.6. Molar paraffin-to-aldehyde yield agreement with eqn 7.36 is ratios in hydroformylation of /i-dodecene catagood confirmation of the lyzed by HCo(CO)3Ph at 185° C as function of initial assumptions, and rules H2-to-CO ratio at different total pressures [11]. out networks I and II. A discrimination between networks III and IV is not possible on the basis of the information at hand. Although the exact location of the node in the network has remained ambiguous, eqn 7.36 appears reliable as it represents all experimental results and can be explained with an eminently plausible mechanism.

7.3.3. Non-simple pathways and networks Many reactions in industrial practice have non-simple networks. Their variety is so great that standard recipes for elucidation cannot be stated: What works in one case will not in another. Only a few strategies that might be useful can be suggested. Some of the more common ones will be shown in this section and be illustrated with examples. Non-simplicity is caused by intermediates whose concentrations rise above trace level, or by steps in which two or more molecules of intermediates function as reactants. Non-simplicity caused by the first of these possibilities usually becomes apparent immediately, when the known participants in a reaction are sorted into reactants, products, intermediates, and possibly catalysts and silent partners. Where this is not so, say, because the number of participants is very large—not uncommon in hydrocarbon processing and combustion—, Delplot rank ordering can help to distinguish intermediates from end products (see Section 7.1.2). Nonsimplicity caused by reactions of trace intermediates with one another may not be apparent at the outset, only to turn up as the mechanism becomes clearer. If so, the kineticist will have to cross that bridge when he comes to it.

7.3. NeWork properties and kinetic behavior

195

In practice, many reaction systems involve non-trace intermediates, but no obvious non-simple reactions of intermediates. A good strategy in such situations is to cut the overall reaction network into portions at the non-trace intermediate or intermediates (see Section 6.5), then reduce the portions as described for simple networks in Section 6.4.1. Network reduction makes it unnecessary to keep track of trace intermediates (except those reacting in a non-simple manner) and so obviates much of the hard work: Trace intermediates are the more troublesome ones in network elucidation because they are difficult or impossible to detect, identify, analyze for, or synthesize, tasks that usually do not pose problems with intermediates that rise above trace level. Often, the network portions will turn out to be "piecewise simple" (see Section 6.5). If not, further cutting at additional nonsimple steps is called for when these become apparent. If the pathway or segment of a portion producing a non-trace intermediate is irreversible, no subsequent portion of the overall network feeds back into it. As a rule, this allows the subsequent portion or portions to be studied independently by using the separately synthesized non-trace intermediate as starting material. It also allows the portion yielding the non-trace intermediate to be studied independently: For this purpose, all subsequent intermediates and products are lumped with the intermediate produced by the portion (i.e., the concentrations are summed up) to obtain the total production of the portion. Alternatively, before analysis, all intermediates are converted to end products, and only these then need to be analyzed for and lumped. Piecewise simple portions feeding into others may, of course, be reversible. This complicates network elucidation significantly. Often, however, such backreactions can be blocked by an additive, the omission of a catalyst or co-catalyst, or some other experimental stratagem. Alternatively, the intermediate produced by the portion can be trapped in some fashion; this allows at least the forward reaction through the portion to be studied without interference. The following example illustrates the application of such stratagems with a relatively complex network. Example 7.6. Olefin hydroformylation with phosphine-substituted cobalt hydrocarbonyl catalyst [7]. The overall reaction system of olefin hydroformylation with a phosphine-substituted cobalt hydrocarbonyl catalyst to produce alcohol, paraffin, and a heavy alcohol has been shown in Example 6.5 (Section 6.5): -f H2, CO

olefin + H,

+ H2

• aldehyde

• alcohol

+ aldehyde

(6.43)

+ 2H2

paraffin

aldol

• heavy alcohol — HiO

196

Chapter 7. Network elucidation Aldehyde and aldol are the only non-trace intermediates. If the network is cut at these, the portions are olefin —• aldehyde, I • ^

aldehyde —• alcohol, (6.44) 2 aldehyde —• aldol, aldol —• heavy alcohol

and can be investigated separately as follows. The study of the first portion, olefin to aldehyde and paraffin, can be conducted under normal reaction conditions. For the mathematical evaluation, aldehyde and all products arising from it—alcohol, aldol, and heavy alcohol—are lumped into one single pseudo-species (counting aldol and heavy alcohol each as two aldehyde molecules). This is possible because, in the network, the aldehyde marks a point of no return: Once a molecule has been converted to aldehyde, it can only remain aldehyde or react on to alcohol, aldol, or heavy alcohol, but not revert back to olefin or paraffin. Aldol condensation is catalyzed by base, but not by the cobalt catalyst. This allows the pathway from aldehyde to alcohol to be studied separately with aldehyde as starting material and under conditions that preclude aldol condensation, in the case at hand in the absence of a base or with an aldehyde whose carbon skeleton has a branch adjacent to the aldehyde group. The pathway from aldehyde to aldol can be studied in the presence of a base and absence of the cobalt catalyst. If need be, the last pathway, from aldol to heavy alcohol, can be studied separately under normal reaction conditions with synthesized aldol as starting material. Actually, the network 6.43 also contains a step in which two molecules of intermediates react with one another, namely, the step of entry of the second aldehyde molecules along the pathway from aldehyde to aldol. However, the cutting of the network at the aldehyde has removed that non-simplicity, too. If a pathway or network should turn out to contain a step in which two or more molecules of the same or different intermediates react with one another, it can be cut at the offending species into piecewise simple portions as discussed in Section 6.5. However, it will rarely be possible to study these portions independently because, more often than not, synthesis of the respective trace intermediates for use as starting materials will be too difficult. If many or even a majority of the steps are non-simple, the network reduction methods described here are of little use in network elucidation. This is typically the case in hydrocarbon pyrolysis and combustion, where reactions of radicals with one another are common. Fortunately, an extensive data base of rate coefficients and activation energies of reaction steps of species in this field of chemistry has been compiled over the years and can be of help in network elucidation [12-16] (see also Section 12.4).

7,4. Other criteria and guidelines

197

7.4. Other criteria and guidelines The rules and regularities developed in the preceding section and their adaptations to catalysis in Section 8.6 provide a good deal of information about what features a pathway or network may have, and definitely cannot have. As is generally true in kinetics, they allow possibilities to be ruled out as incompatible with observed behavior, but cannot serve to prove a compatible mechanism to be correct. Other considerations, well covered in standard texts, are called for to narrow the field. A brief survey is given here. Stereochemistry. On the most primitive level, mere steric considerations can often suggest plausible features of a network. Common sense tells us that conversion must take place at some reactive group or configuration of the reactant molecule, most likely with no or only minimal changes elsewhere. Homogeneous catalytic hydrocyanation of mono-olefinic compounds may serve as a simple example. It stands to reason that HCN will add to the olefinic double bond, H going to one of the latter's carbon atoms, and CN to the other: CN HCN

(7.37)

That this is indeed so has been confirmed by experiments with pentenenitrile isomers and deuterium-labeled hydrogen cyanide, DCN [17]: DCN

+

(7.38)

D

CN

NC

D

(Arrows represent multistep pathways; for details of these, see Example 8.7 in Section 8.5.5)

198

Chapter 7. Network elucidation

Under typical hydrocyanation conditions, double-bond migration along the carbon chain of the olefinic reactant is relatively slow, so that reactant isomerization remains unimportant. This makes the example above almost trivial. A similar but more complicated situation is encountered in hydroformylation of olefins (see Example 5.3 in Section 5.3). Here, the CO carbon atom can be expected to attach itself to either of the two double-bonded carbon atoms:

HCO

(7.39)

HCO

Depending on what catalyst is used, double-bond migration may be so fast that the product isomer distribution depends only little on what olefin isomer or isomer mixture was used as starting material. This, of course, makes it harder to identify the olefin isomer parents of any given product isomer, so much so that at one time a single, common intermediate was postulated from which all product isomers were said to arise [18]. In contrast, if CO attachment to either double-bonded carbon atom is accepted, the (simplified) network to be expected for a straight-chain monoolefin is

CHO

HCO

HCO

HCO

(7.40)

7.4. Other criteria and guidelines

199

(H2 and CO reactants not shown; arrows represent multistep pathways; for mechanistic details, see Example 6.2 in Section 6.3). This basic network structure of coupled parallel steps has been verified as described in Section 5.3, in refutation of the earlier postulate of a common intermediate. Molecularities. An equally elementary criterion is the fact that a great majority of reaction steps are uni- or bimolecular; trimolecular steps are rare and slow, and steps of still higher molecularities are unheard of (see Section 2.1). A trimolecular forward or reverse step in a postulated mechanism calls for an explanation why its reactants are not consumed by bimolecular steps before they have a chance to undergo the trimolecular one (e.g., see the Example 7.8 farther below). No mechanism involving a forward or reverse step of even higher molecularity should ever be considered. Thermodynamics. Despite its name, thermodynamics is a science of equilibrium, not of dynamics. It compares energies of states of matter, and such a procedure by itself does not allow rates and mechanisms to be predicted. Nevertheless, thermodynamics can often help to decide which of various conceivable mechanisms are the more probable ones. The activation energy of an endothermic step is necessarily at least Table 7.5. Approximate average bond as high as the standard molar reaction energies of importance in organic enthalpy, A//° (see Figure 2.2 in Secchemistry. tion 2.3), and a step with very high activation energy is apt to be quite slow. Accordingly, a pathway with energy energy one or more highly endothermic steps bond bond kJ mol-^ kJmol-' is suspect if there are alternatives without these. 260-300 C-C C-N 285-430 Thermochemical data for possible reaction intermediates may not H=H C=C 436 320-740 be available. However, approximate 0 =0 210-280 C-0 498 average bond energies [19-21] can serve as a rough guide (see Table 7.5; C-H 360-500 0-H 360-450 the actual energies depend on the substituents). The skilled kineticist will first look for pathways in which no high-energy bond is broken or, if none such is possible, for those in which a new high-energy bond is formed as the other is broken. Unfortunately, this procedure is not conclusive because an apparently plausible mechanism that avoids highly endothermic steps may well be blocked for other reasons. The hydrogen-iodide reaction

200

Chapter 7. Network elucidation 2 HI

provides a telling example (see Section 4.3). The single-step, bimolecular reaction mechanism is thermodynamically plausible and compatible with observed rate behavior, and the true mechanism H, I2 <

•

(4.18)

21 ^ ^

» 2HI

involves the fairly highly endothermic dissociation of I2 as well as a trimolecular forward step. This is because the single-step mechanism violates the WoodwardHoffmann exclusion rules (see also Example 7.8 farther below). Tolman's 16- or IS-electron rule [22,23]. Loosely related to thermodynamics is the 16- or 18-electron rule, suggested by Tolman. This rule is a very helpful guideline for postulating intermediates in reactions involving transition-metal complexes, particularly in homogeneous catalysis. The rule states that a great majority of such reactions proceed through intermediates with 16 or 18 valence electrons. This is because such species are energetically favored over those with 14, 20 or odd-numbered electrons. Cobalt hydrocarbonyl-catalyzed olefin hydroformylation with network 6.9 may serve as an example. Cobalt, with atomic number 27, contributes nine valence electrons to its complexes (the other eighteen occupy the inner \-s, 2-5", 2-p, 3-5, and 3-p orbitals); H, the alkyl group, and the acyl group contribute one each, CO contributes two (of its fourteen electrons, four are shared by C and O in the double bond, an additional four each complete the inner octets of C and O), and an olefin ligand contributes the two 7r-electrons of its double bond. The contributions and totals for some key participants are: valence-electron contributions Co

H

HCo(CO),

9

1

8

18

HCo(CO)3

9

1

6

16

•C?(CO)3

9

1

6

V-Co(CO)3

9

6

V-C-Co(CO)3

9

6

>

CO

-C=C-

-alkyl

-acyl

2

total

18 1

16 1

16

7.4. Other criteria and guidelines

201

There are exceptions to Tolman's rule, however [24,25]. For example, if the ligands are very bulky, the 16-electron complex may be sterically hindered, making a 14-electron species the more stable one. The complex Pd[P(r^rr-Bu)3]2 is a case in point [26]. Also, a solvent such as benzene can act as electron donor and thereby stabilize a nominally 14-electron complex as a 16-electron solvate [27]. A few reactions appear to proceed through paramagnetic, 17- or 19-electron complexes as intermediates [28,29]. 20-electron species are believed to be formed as intermediates in some associative ligand substitution reactions [30,31]. All such species are relatively unstable. Woodward-Hoffinann exclusion rules. The Woodward-Hoffmann exclusion rules are based on the principle of conservation of molecular-orbital symmetry upon reactions, put forth by Robert B. Woodward and Roald Hoffmann in their landmark book on this topic [32,33]. Symmetry of the wave function is conserved while the valence electrons involved in the breaking and forming of bonds rearrange themselves from their orbitals in the reactants to those in the product or products. If this symmetry constraint permits formation of the product or products in the ground state from the reactants in the ground state, the reaction is "allowed;" if it requires either to be in an excited state, the reaction is "forbidden." In keeping with this picture, the exclusions are not iron-clad: Like traffic laws, they may be broken, but not too often and only for a good reason. An energy input high enough to overcome the energy barrier of a "forbidden" reaction can make it go. Specifically, a photochemical reaction in which a reactant is elevated to an excited state is "allowed" if the same thermal reaction is "forbidden," and is "forbidden" if the thermal reaction is "allowed." Moreover, the rules apply to "concerted" reactions, that is, to single reaction steps, and do not preclude the possibility that a multistep alternative pathway might exist. Though only the highest occupied ground-state orbitals need be considered, a proper application of the rules requires familiarity with molecular-orbital theory (e.g., see [34]), a topic well beyond the scope of this book. However, two simple examples may serve to show how the Woodward-Hoffmann rules, apart from providing a better understanding of molecular mechanisms, can rule out seemingly plausible pathways or at least show them to be highly improbable, and lead to general predictions. Example 7.7. Cyclo-addition reactions and the 4n + 2 rule. The predictive power of the Woodward-Hoffmann principle becomes apparent, for example, with the application to cyclo-addition reactions [32,35,36]. Consider first the dimerization of ethene to cyclobutane:

n

202

Chapter 7. Network elucidation The four 7r-electrons of the two ethene double bonds must redistribute themselves to the (T-orbitals of the newly formed cyclobutane single bonds. As it turns out in this case, the requirement of symmetry conservation does not allow two ground-state ethene molecules to combine in a concerted reaction to form a ground-state cyclobutane molecule: The reaction is "forbidden." The only allowed interactions are destabilizing. In contrast, for the addition of ethene to butadiene to form cyclohexene

o the result is the opposite. Symmetry conservation allows the stabilizing interactions, but not the destabilizing ones: The reaction is "allowed," with only a minor energy barrier to be overcome. A more general rule emerges from these considerations: Concerted cycloaddition reactions are "forbidden" if involving 4n 7r-electrons, as in ethene dimerization (n is any integer), and are "allowed" if involving 4n + 2, as in cyclohexene formation [32,35,36]. Example 7.8. Hydrogen-halide reactions. The hydrogen-iodide reaction is known to proceed through the fairly highly endothermic dissociation of Ij and a subsequent trimolecular step (see above and Section 4.3) rather than in a single, bimolecular step that avoids these normally unfavorable conditions. However, the single-step mechanisms violates the orbital symmetry requirement [37]. The situation regarding the molecular orbitals of the valence electrons is similar to that in ethene dimerization (see Example 7.7 above): Symmetry conservation does not allow ground-state products to be formed from ground-state reactants. The reaction would have to proceed through an exited state, and the estimated height of the energy barrier is greater than for iodine dissociation. An analogous argument applies to the hydrogen-bromide and hydrogen-chloride reactions and explains why these also proceed through dissociation of the halogen molecule (here followed by a chain reaction) rather than in a seemingly energetically favored single step.

7.5. Auxiliary techniques In addition to the simplification of mathematics and the criteria and guidelines described so far, many other techniques can be brought to bear in network elucidation. Excellent literature on these is available, and only a brief overview will be given here. For quick orientation the reader is referred to a comprehensive review with copious references [38].

7.5, Auxiliary techniques

203

Determination of isomer distribution. A must in the elucidation of complex networks is to keep track of the isomer distribution of reactants and products wherever isomers are involved. Often, a reactant can add in different ways to another, giving rise to different product isomers. Also, a reactant might isomerize before reacting, with the same result. Today's analytical techniques, foremost among them gas chromatography, are sufficiently advanced to distinguish between isomers. In many instances, the results provide excellent evidence for or against postulated pathways. A case in point is hydroformylation of mono-olefins, where the carbon atom of CO adds to either of the two double-bonded carbon atoms (see network 7.39 and Example 5.3 in Section 5.3). The discussion in Section 5.3 has shown how the basic structure of the network can be inferred from the dependence of the product isomer distribution on which reactant isomer is originally charged. Isotope techniques. The labeling ("tagging") of reactants with distinguishable radioactive or stable isotopes is a powerful tool for the elucidation of mechanisms. Isotopes often used are deuterium, carbon-13, carbon-14, oxygen-17, and oxygen18. In essence, selected atoms in a reactant are replaced by their isotopes, and the product or products are analyzed for the positions which these isotopes have taken up. Suitable methods of analysis for positions of stable isotopes are, among others, nuclear magnetic resonance and gas chromatography with mass-spectroscopic detection. The classical example of such a study is that of ester hydrolysis by Polanyi and Szabo [39]. The authors used ^^0H~ for the reaction and found that the majority of the heavy oxygen turned up in the acid anion rather than the alcohol: OR' RC O

,« + ^^OH"

•

^'ORC -h R'OH O

This allowed them to conclude that the bond broken is that between oxygen and carbonyl carbon rather than that between oxygen and alkyl carbon. An example already encountered in this section is the use of DCN in hydrocyanation of olefmic compounds, confirming expectations as to the addition of D and CN to the double-bond carbon atoms. The result of isotopic tagging can be obscured by fast self-exchange of the tag and the regular isotope between positions on the molecule or between different molecules, say, between reactant and solvent. Such "scrambling," if it occurs, makes the technique useless. Another use of isotopically labeled reactants is for study of kinetic isotope ejfects [40,41]. The difference in zero-point energies between isotopes results in a difference in bond energies and thus in a difference in activation energies and reaction rates. The largest difference is that between hydrogen and deuterium. The effect can be of help especially in the identification of a rate-controlling step.

204

Chapter 7. Network elucidation

However, the interpretation of results is not straightforward as zero-point energies in the activated complex also play a role. Synthesis of intermediates. An excellent technique for confirming or refuting a postulated pathway is to synthesize intermediates and use them as starting materials. Often, a key intermediate that is reactive enough to remain at trace level under reaction conditions is stable at very low temperatures (e.g., that of liquid nitrogen) and can be synthesized. If the reaction starting with the postulated intermediate yields the same products in the same ratios, this can be taken as evidence in favor of the presumed pathway. For example, the essential features of the Heck-Breslow mechanism of hydroformylation (see Example 6.2 in Section 6.3) with cobalt hydrocarbonyl catalysts have been verified in this way by synthesis and use of the alkyland acyl-cobalt species [42]. The main problem with this approach is that the intermediate must be brought to reaction conditions almost instantaneously. A transient of any length of time, say, to reach the elevated temperature and pressure of the reaction under study, is apt to falsify the results seriously. Equipment of the type described in Figure 3.3 in Section 3.1.1 or some other injection mechanism is needed. Spectrophotometry. The theory of spectra is far advanced. In many cases, compounds can be unambiguously identified by their ultraviolet, visible, or infrared spectra (e.g., see Smith's book [43]). As an example, the double bond of a CO ligand in a complex has a strong characteristic infrared vibration frequency whose exact value depends on the electronic properties of the coordinating metal; these, in turn, are affected by the other substituents. In homogeneous catalysis by transition-metal complexes in particular, foremost among them hydrogenation, hydroformylation, and hydrocyanation, spectra have contributed much to the identification of reaction intermediates and thus of pathways. It is essential that the spectra be taken under reaction conditions, usually involving elevated temperatures and pressures. Especially in homogeneous catalysis by metal complexes, ligand exchange and oxidation-reduction reactions are usually so fast that complex rearrangements keeps pace with equilibrium shifts as a sample is depressured and cooled. Until high pressure-high temperature cells were developed, information from spectra often was entirely misleading. An amusing incident highlights this important point. In the 1960s, Shell Oil's patent position on phosphine-substituted cobalt hydrocarbonyls as hydroformylation catalysts was practically air-tight. A competitor, however, obtained a patent on a catalyst of this type, differing only by replacement of a CO ligand by a crotyl group, on the strength of the evidence that, at ambient conditions, its IR spectrum differed from those of the Shell-patented catalysts. Shell chemists thereupon took IR spectra of the

Summary

205

competitor's catalyst under reaction conditions and found it to convert within seconds to what is present in the systems patented by Shell [44]. A costly lawsuit was avoided by the simple expedient of sending a preprint of the respective journal publication to the competitor to put him on notice that his patent would not stand up in court if he ever dared to use it. Nuclear magnetic resonance. Spin-state and saturation labeling can be used to investigate relatively fast reactions [38,45]. In essence, the results show whether a reaction is fast or slow relative to the characteristic NMR frequency. For example, spin-state labeling can detect whether or not a fast exchange of ligands such as organic phosphines with ^^P nuclei occurs between different coordinative sites or between the complex and the solution [46]. Spin saturation transfer experiments have been used to clarify e.g. the mechanism of exchange between a methyl group and a methylene hydride: HOs3(CO)ioCH3 <

• H20s3(CO)ioCH2

Saturation of the methyl group in the reactant leads to a decrease in the intensity of only one of the two hydride signals of the product, showing that the mechanism is hydride abstraction from methyl and that only one of the hydride sites is involved to a significant degree [47]. Electron spin resonance. Because of its extremely high sensitivity, electron spin resonance has been used in mechanistic studies of reactions of radicals [48,49]. Direct observation of radicals is possible. Often, however, radical traps such as nitroso compounds are used instead to catch the radical in form of a stable species [50]. Care must be taken to exclude other possible sources of radicals [49]. Also, it may not be possible to analyze a sample under reaction conditions. This brief survey does not include the many strictly analytical methods that can be used to for quantitative determination of concentrations of participants.

Summary Networks can be effectively elucidated in either of two ways: (1)

establishment of empirical rate equations that correctly reflect all available rate data, followed by deduction of a plausible network that can produce such rate equations; or

(2)

compilation of all plausible networks, followed by experiments to eliminate those that prove incompatible with observation.

206

Chapter 7. Network elucidation

Whether the first or second approach is better suited depends largely on how much quantitative kinetic data are already at hand. In either case, before the postulated network is accepted, it should have established a good track record of correctly predicting behavior under conditions not previously studied. Correct counterintuitive predictions are the most convincing. The two most general features of a reaction are the apparent kinetic orders with respect to the participants (reactants, products, intermediates, catalysts, and silent partners) and the ranks of the intermediates and products. Reaction orders may vary with conversion, so accurate values are not sought. Ranks, established by Delplots, provide an indication of the sequence in which the respective species are formed, and are useful primarily in the study of reactions with many participants and about whose networks little is known to start with. The conventional procedure of fitting a rate equation to experimental data is to use a power law reflecting the observed reaction orders. However, while fractional reaction orders may provide an acceptable fit, they cannot be produced by reasonable mechanisms. A better way is to fit the data to "one-plus" rate equations, that is, equations containing concentrations with integer exponents only, but with denominators composed of two or more additive terms of which the first is a "one." Such equations behave much like power laws with fractional exponents but, in contrast to these, can arise from reasonable mechanisms and therefore are more likely to hold over wide ranges of conditions. As an exception, rate equations with constant exponents of one half or integer (positive or negative) multiples of one half can result from chain reactions and reactions initiated by dissociation, and are acceptable if such a mechanism is probable or conceivable. The general formula for the rate in simple pathways, derived in Section 6.3, can be used for deducing a large number of rules that relate observable kinetic behavior, such as reaction orders, to properties the network may have or definitely cannot have. (Catalytic reactions require qualifications; see Section 8.6.) These rules greatly facilitate network elucidation: Pathways or networks that include a feature producing behavior contrary to observation can be ruled out as whole groups rather than one at a time. If a pathway or network turns out to be non-simple, a good strategy is to try to break it up into piecewise simple portions that can be studied independently. Whether and how this can be done depends on the reaction at hand. The job is easiest if the portions are irreversible, so that none of them feeds back into a preceding one, and if the non-trace intermediates can be synthesized. Criteria and guidelines useful in network elucidation and supplementing the rules derived in this chapter include considerations of steric effects, molecularities of postulated reaction steps, and thermodynamic constraints, as well as Tolman's 16- or 18-electron rule for reactions involving transition-metal complexes, and the Woodward-Hoffmann exclusion rules based on the principle of conservation of molecular orbital symmetry. Auxiliary techniques that can be brought to bear include, among others, determinations of isomer distribution, isotope techniques, and spectrophotometry. Examples include high-pressure hydrolysis of a nitrile; paraffin autoxidation; homogeneous aldehyde hydrogenation; olefin hydroformylation to alcohol with paraffin by-product formation, aldehyde condensation to heavy ends, and olefin isomerization; cyclo-addition reactions; and hydrogen-halide reactions.

References

207

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

13.

14.

15.

16.

17. 18. 19. 20. 21. 22. 23.

V. I. Dimitrov, Kinet. Katal. Lett,, 7 (1977) 81. M. T. Klein and P. S. Virk, Ind. Eng. Chem., Fundam., 22 (1983) 35. N. A. Bhore, M. T. Klein, and K. B. Bischoff, Ind. Eng. Chem. Res., 29 (1990) 313. P. S. Myers and K. M. Watson, Nat. Petr. News Tech. Soc, 38 (1946) 388. S. D. Iyer and M. T. Klein, J. Supercrit. Fluids, 10 (1997) 191. F. Garcia-Ochoa, A. Romero, and J. Querol, Ind. Eng. Chem. Res., 28 (1989) 43. F. G. Helfferich and P. E. Savage, Reaction kinetics for the practical engineer. Course #195, AIChE Educational Services, New York, 7th ed. 1999, Section 7.3. J. L. Van Winkle, unpublished, 1977. F. G. Helfferich, J.Phys. Chem., 93 (1989) 6676. L. Marko, Proc. Chem. Soc, 1962, 67. J. L. Van Winkle, personal communication, 1974. Landolt-Bomstein, New Series, Radical reaction rates in liquids, H. Fischer, ed., Springer, Berlin, Part II Vol. 13 (5 subvolumes), 1983-1985, ISBN 0387126074, 0387132414, 0387117253, 0387121978, 0387136762; Vol. 18 (5 subvolumes), 1994-1997, 3540560548, 3540560556, 3540560564, 3540603573, 3540560572. D. L. Baulch, C. J. Cobos, R. A. Cox, C. Esser, P. Frank, T. Just, J. A. Kerr, M. J. Pilling, J. Troe, R. W. Walker, and J. Warnatz, /. Phys. Chem. Ref Data, 21 (1992)411. D. L. Baulch, C. J. Cobos, R. A. Cox, P. Frank, G. Hayman, T. Just, J. A. Kerr, T. Murrells, M. J. Pilling, J. Troe, R. W. Walker, and J. Warnatz, J. Phys. Chem. Ref Data, 23 (1994) 847. W. G. Mallard, F. Westley, J. T. Herron, R. F. Hampson, and D. H. Frizzell, NIST chemical kinetics data base: Windows Version 2Q98, National Institute of Standards and Technology, Gaithersburg, MD, 1998. E. T. Denisov and T. Denisova, Handbook of antioxidants: bond dissociation energies, rate constants, activation energies, and enthalpies of reactions, CRC Press, Boca Raton, 2nd ed., 2000, ISBN 0849390044, Chapter 2. J. D. Druliner, Organometallics, 3 (1984) 205. I. J. Goldfarb and M. Orchin, Adv. Catal., 9 (1957) 609. T. L. Cottrell, The strengths of chemical bonds, Butterworths, London, 2nd ed., 1958. N. N. Semenov, Some problems in chemical kinetics and reactivity (English translation), Vol. I, Princeton University Press, 2nd ed., 1958. Handbook of chemistry and physics, D. R. Lide, ed.-in-chief, CRC Press, Boca Raton, 80th ed., 1999-2000, ISBN 0849304806, Chapter 9, pp. 51-74. C. A. Tolman, Chem. Soc. Rev., 1 (1972) 337. J. P. CoUman, L. S. Hegedus, J. R. Norton, and R. G. Finke, Principles and applications of organotransition metal chemistry. University Science Books, Mill Valley, CA, 2nd ed., 1987, ISBN 0935702512, p. 33.

208 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50.

Chapter 7. Network elucidation S.-Y. Chu and R. Hoffmann, J. Phys. Chem., 86 (1982) 1289. Collman et.al. (ref. 21), Section 2.3. S. Otsuka, T. Yoshida, M. Matsumoto, and K. Nakatsu, J. Am. Chem. Soc, 98 (1976) 5850. M. H. J. M. de Croon, P. F. M. T. van Nisselrooij, H. J. A. M. Kuipers, and J. W. E. Coenen, /. Mol. CataL, 4 (1978) 325. J. Halpern, in Fundamental research in homogeneous catalysis, M. Tsutsui, ed., Plenum, New York, Vol. 3, 1978, ISBN 0306401991, p. 25. A. E. Stiegman and D. R. Tyler, Comments Inorg. Chem., 5 (1986) 215. G. Yagupsky, C. K. Brown, and G. Wilkinson, J. Chem. Soc, A 1970, 1392. Collman et al. (ref. 22), Section 4.5.C. R. B. Woodward and R. Hoffmann, The conservation of orbital symmetry, VCH Publishers, Weinheim, 1970, 5th printing 1989, ISBN 0895731096. R. B. Woodward and R. Hoffmann, Angew. Chem. Int. Ed., 8 (1969). 781. R. McWeeny, Coulson's Valence, Oxford University Press, 1979 (3rd ed. of C. A. Coulson, Valence), ISBN 0198551452. R. Hoffmann and R. W. Woodward, J. Am. Chem. Soc, 87 (1965) 2046. A. Pross, Theoretical and physical principles of organic reactivity, Wiley, New York, 1995, ISBN 0471555991, Section 10.4.1. R. Hoffmann, 7. Chem. Phys., 49 (1968) 3739. C. A. Tolman and J. W. Faller, in Homogeneous catalysis with metal phosphine complexes, L. H. Pignolet, ed. Plenum, New York, 1983, ISBN 03064121IX, Chapter 2. M. Polanyi and A. L. Szab6, Trans. Farad. Soc, 30 (1934) 508. L. Melander and W. H. Saunders, Jr., Reaction rates ofisotopic molecules, Wiley, New York, 1980, ISBN 0471043966. K. B. Wiberg, Physical organic chemistry, Wiley, New York, 1964, pp. 351-363. R. F. Heck and D. S. Breslow, /. Am. Chem. Soc, 83 (1961) 4023. B. C. Smith, Infrared spectral interpretation: a systematic approach, CRC Press, Boca Raton, 1998, ISBN 0849324637. W. W. Spooncer, A. C. Jones, and L. H. Slaugh, J. Organomet. Chen., 18 (1969) 327. L. M. Jackman and F. A. Cotton, eds., Dynamic nuclear magnetic resonance spectroscopy, Academic Press, New York, 1975, ISBN 0123788501. J. W. Faller, Adv. Organomet. Chem., 16 (1977) 211. R. B. Calvert and J. R. Shapley, J. Am. Chem. Soc, 100 (1978) 7726. J. K. Kochi, Ace Chem. Res., 7 (1974) 351. T. L. Hall, M. F. Lappert, and W. P. Lednor, /. Chem. Soc, Dalton Trans., 1980, 1448. S. Terabe and R. Konaka, /. Chem. Soc, Perkin Trans. 2, 1972, 2136.

Chapter 8 Homogeneous Catalysis Homogeneous catalysis is one of the most interesting fields of chemistry, especially for its mechanisms and kinetics. Unencumbered by the complications of nonuniform and easily contaminated surfaces and of adsorption equilibria and rates that make heterogeneous catalysis more difficult to interpret, homogeneous catalysis provides excellent opportunities for the study of the molecular causes of reactivity, of what makes reactions go. Homogeneous catalysis, by enzymes, can produce both very high rates and very high selectivities. Life itself could not have evolved without it. Moreover, homogeneous catalysis can be translated into heterogeneous catalysis by anchoring the catalytic species onto a solid polymeric support or, if it is ionic, having it held by electrostatic forces in a gel with fixed groups of opposite charge sign (see Section 9.6). Lastly, a thorough understanding of the molecular effects in homogeneous catalysis can serve as a guide in the development of new or more efficient metal and metal oxide catalysts and their supports. From the point of view of kinetics and mechanisms, homogeneous catalysis can be classified into single-species and complex catalysis. In the former, a single molecule or ion combines with a reactant molecule to initiate the reaction and eventually reappears in its original form. An example is catalysis by hydrogen ions. In complex catalysis, the catalyst exists in several different forms, thus constituting a chemical system of its own that interacts with the reactants. Acid-base catalysis and catalysis by metal-organic complexes are the most prominent members of this category. A further complication arises if a significant fraction of the total catalyst material may be present in the form of reaction intermediates rather than as the free catalyst. If the catalyst is highly active, a minute amount suffices to produce a high reaction rate, and even a trace-level intermediate may then contain a large fraction or possibly most of the catalyst material. Such behavior is typical for enzyme catalysis, but not confined to it. In such cases, the concentration of free catalyst may vary with conversion, may not be known, and may be very difficult to measure. Rather, what is known is the total amount of catalyst material added, and rate equations in terms of the latter are therefore needed. Such systems will be discussed in the later sections of this chapter. Regardless of these complications, the Bodenstein approximation of quasistationary states of trace-level intermediates remains the principal tool for reduction

210

Chapter 8. Homogeneous catalysis

of mathematical complexity. In fact, it is more often applied to catalytic than to thermal reactions. This is because, usually, the amount of catalyst needed to produce the desired reaction rate is small, and the catalyst-containing intermediates thus are apt to remain at very low concentrations, as the validity of the approximation requires. In trace-level catalysis, of course, this requirement is automatically met. It will be taken for granted in this chapter. Moreover, this chapter remains restricted to "simple" reactions (no steps with two or more molecules of intermediates as reactants). Non-simple reactions are relatively rare in homogeneous catalysis. They can be handled as outlined in Section 6.5. A more detailed discussion is beyond the scope of this book. 8.1. Single-species catalysis Many homogeneous reactions are catalyzed by a single chemical species, usually by solvated protons or metal ions. A typical example is the hydrolysis of sucrose in aqueous solution: sucrose -h H2O + H"^

•

glucose + fructose -h H^

alleged to be the first reaction whose kinetics was studied [1], and with its simple and easily observed rate behavior one of the staples of the undergraduate chemical laboratory. Many other hydrolysis, hydration, dehydration, and esterification reactions are of this type. Certain metal ions such as copper and zinc in aqueous solution can function in a similar fashion. Like the proton, they act as Lewis acids, initiating the reaction by inducing a positive charge on the reactant [2,3]. A typical example is decarboxylation of organic acids. To arrive at rate equations of catalysis by a single species that is present almost exclusively as free catalyst, the formalism developed for noncatalytic reactions can be used with the catalyst appearing as both a reactant and a product. In fact, many of the examples used in earlier chapters to illustrate the deduction and application of rate equations were from catalytic reactions of this type. Thus, all the rules derived in Chapter 7 for network elucidation remain valid in such cases. Typically, this kind of catalysis involves a single free catalyst (cat) which converts a reactant A to a product P in a single cycle with arbitrary number k of members, and possibly with co-reactants and co-products (not shown): X,

<

^

...

<

•

X,.,

<

(8.1) cat

8.1. Single-species catalysis

211

The rate equation, obtained with eqns 6.4 to 6.6, is k-l

k-1

i=0

i=0

n \ u. - n Ku

(8.2)

D.00 where both indices 0 and k refer to the free catalyst, and D^ is given by eqn 6.6. The concentrations of A and P do not appear explicitly, but are co-factors in the coefficients Xoi and X^^k.i, respectively. In general, as the appearance of Qat as a factor in the numerator of eqn 8.2 shows: Forward and reverse rates in single-species bulk catalysis are first order in free catalyst.

This is in keeping with the rule that a reaction is first order in a reactant that participates with one molecule in only the first step of a simple pathway (see Rule 7.13 in Section 7.3.1). However, even in single-species, single-cycle, bulk catalysis there are exceptions to this rule, as will be shown later in this section. Example 8.1. Acetal hydrolysis [4,5]. The catalytic cycle of acetal hydrolysis in aqueous solution, catalyzed by dilute acid R' OR C R" OR

+ H.O + H^

R' C = 0 + 2 R 0 H + H^ R"

(8.3)

is believed to be R' OR C ^ R" OR

R' C= 0 ROH

ROH

(8.4) R' C^ (X,) R" OR

HP

212

Chapter 8. Homogeneous catalysis The second step is commonly assumed to be irreversible and rate-controlling, so that the first is in quasi-equilibrium: ~ ^acetal

^1

=

^

^12 ^l

-^01 ^acetal ^ H *

(indices 1, 2, ... refer to intermediates Xi, Xj, ..., and 0 to cat; KQ^ is the equilibrium constant of the first step). Accordingly, the rate in terms of bulk participants is -'•aceu,

^

^C^ceulQ-

(k

^ K,,k,,)

(8.5)

and is first order in acetal and hydrogen ion, in agreement with experimental observation. The assumption that the second step is rate-controlling follows tradition, but is needlessly restrictive. Granted that the intermediates remain at trace level, as they almost certainly do, the observed kinetic behavior results even if the first step is not close to equilibrium. For the cycle 8.4 with irreversible second step as assumed, the general formula 6.4 to 6.6 gives ^

^

^01^12

^^cetalQ^10

(8.6)

•'" ^ 1 2

This rate equation is of the same algebraic form as eqn 8.5, but with different physical significance of the coefficient k. This is another example of the fact that an observed rate behavior can usually be explained in several different ways. A distinction between the two rival explanations may be possible. An Arrhenius plot of IniKQik^:^ should give a reasonably straight line, whereas a plot of In [kQiki2 l{kiQ + /:i2)] may be curved. This is because the first plot is for the logarithm of z.product of two constants giving straight-line logarithmic plots, whereas the second is for a logarithm of an expression involving a sum of such quantities (see Section 13.1.4). Accordingly, a distinctly curved Arrhenius plot would be evidence against quasi-equilibrium in the first step. However, straight-line behavior would still allow both interpretations because the plot of In [^1^12/(^10 + ^12)] would be straight if the activation energies of k^Q and k^ were equal. In acid-catalyzed reactions, the distinction between single-species and complex catalysis is not always clear-cut. The actual catalyst is the solvated proton, HgO^ in aqueous solution, and H2O (or a molecule of the nonaqueous solvent) may thus appear as a co-product in the first step and as a co-reactant in the step reconstituting the original solvated proton, possibly also in other additional steps, e.g., if the overall reaction is hydrolysis or hydration. Moreover, the acid added as catalyst may not be completely dissociated, and its dissociation equilibrium then affects the concentration of the solvated proton. At high concentrations this is true even for fairly strong acids such as sulfuric, particularly in solvents less polar than water. Such cases are better described as acid-base catalysis (see Section 8.2.1).

8.1. Single-species catalysis

213

At high acid concentrations, hydrolysis or hydration reactions may no longer be first order in hydrogen ion as in eqns 8.5 and 8.6. Consider a hypothetical hydration reaction with catalytic cycle

(8.7)

with protonated intermediates Xj"^ and Xj^ at trace level and irreversible third step. The general formula 6.4 to 6.6 (with index 0 for HgO"^) gives for this case ^01 ^12 ^20 (^np

'•P

=

* ^A

CHJO

.2 ( ^ 1 2 + ^ 1 0 ) ^ 2 0 ^ 1 1 ^ 0 + ^10^21 ^H^O

or, in one-plus form after cancellations, k C r, where

k ^ Mi!^ ^

k k /VJQA,2I

C C

(8.8)

1 - KCup and

fc

= ^

^Ki-Ko)Ko k k '^10'^21

The rate is first order in A and YL^O^ and of order between zero and one in H2O (becoming first order in H2O if the first two steps are at quasi-equilibrium). At very high acid concentration, more water is bound in the form of hydration shells of the ions if more acid is added. This reduces the concentration oi free water (CH20 in eqn 8.8). Such a reduction of availability of water as reactant has a retarding effect and may even overcompensate the increase in catalytic hydronium ion concentration. If it does, the rate as a function of acid concentration goes through a maximum [6]. The effect just described produces an apparent reaction order in HgO"^ that is less than one and may even become negative at very high acid concentrations. The opposite behavior, an apparent order higher than one, is also found in a large

Chapter 8. Homogeneous catalysis

214

12

F~ h-

H^soy

10

[-

HCIO,

^

8 h

HF^^

/

\6h -

HNO^

/Mcy''^ A-^HJO,

H^

-f 0 HP

number of instances. At very high concentrations of a strong acid, species other than HjO"^ may function as additional proton donors. A semiempirical procedure to account for this effect is to correlate the rate to the Hammett acidity function H^ instead of the /?H of the solution [7-10]. In brief review and without discussion of the theoretical basis, H^ is given by

\y\

1

1 L

1

0.2

1

1

\

0.4 0.6 0.8 HA mole fraction HA Figure 8.1. Hammett acidity functions for various strong acids (from Olah [8]).

pK^ - \og{C^^JC^)

and characterizes the tendency of the solution to protonate a neutral basic indicator B. Hammett acidity functions as a function of acid concentration are shown in Figure 8.1 for various strong acids (see also a detailed treatment by Connors [11]).

8.2. Complex catalysis The term complex catalysis indicates that the catalyst exists in several different forms that interact with one another and play different roles. Usually, interconversion of the catalyst species is by dissociation-association or ligand exchange, reactions that are orders of magnitude faster than the catalyzed formation or cleavage of covalent bonds within molecules. If so, the assumption that the catalyst species remain very close to equilibrium is justified. 8.2.1. Acid-base catalysis The most common and most thoroughly studied type of homogeneous catalysis is acid-base catalysis. It includes hydrolysis, alcoholysis, esterification, and condensation reactions among many others. It is characterized by the fact that the equilibrium between base and conjugate acid, or between acid and conjugate base, is coupled with the actual catalytic cycle. Several examples in previous sections fall under the heading of acid-base catalysis. Nitration of aromatics with catalyst acid HB and its conjugate base B~ is one of these (see reaction 4.6 in Section 4.2). Also, acid-catalyzed hydrolysis and hydration reactions such as 8.3 and 8.7 can be viewed as belonging to this category because the original catalyst actually is U^O^ rather than H"^ and is reconstituted in a step that involves its conjugate base, H2O, as co-reactant.

8.2. Complex catalysis

215

A classical example of acid-base catalysis is base-catalyzed aldol condensation of aldehydes [12-14]. It may serve here to illustrate typical facets of reactions of this kind. The catalytic cycle is

(aid)

.-C

(x,-)

-c-c O H

(aldol)

H -C-C H (aid)

(8.9)

H ..-C-C;

..-C-C H •C-C H

H I -C-C H

(x^-)

where aid is aldehyde, B~ and HB are the catalyst base and its conjugate acid, respectively, and the intermediates are at trace level. The second step, Xf -f aid —• X2", is commonly assumed to be irreversible. In such a pathway, aldehyde is a reactant in both the first and second steps. Accordingly, this is a typical case of competing steps, discussed in Section 5.5. With Xf at trace level, the general formula for simple pathways (eqns 6.4 to 6.6.) gives KQI ki2 Caid Cg ' aldol

^12 Q d

(8.10)

k C

with two possible limiting cases (see Section 5.5): Case I first step rate-controlling

Case II second step rate-controlling ^10 ^HB * ^12^ald

aldol

—

'^OlWld^B-

rate first order in aldehyde and first order in B~ general base catalysis

^aldol

^

(^01 ^12/^10) Qld(CB-/C^HB)

rate second order in aldehyde and first order in OH" specific base catalysis

(Note that CB-/CHB is proportional to CQH- because of the dissociation equilibrium B -h H9O with /ir„R = CYCCU^O ICUXKCC^U- = const, and = HB + OHB"^H20 ' '^HB^OH" -'H2O Cu.o — const.)

216

Chapter 8. Homogeneous catalysis

If the first step is rate-controlling (Case I), the rate is first order in the respective base, B", not in hydroxide ion. If several bases are present, their contributions to the rate are additive. This is called general base catalysis. In contrast, if the second step is rate-controlling (Case II), the rate is first order in hydroxide ion, regardless of what base or bases are used as catalyst. This is called specific base catalysis. Intriguingly, mathematics produces first order in 0H~ even though that ion is not a reactant in any of the kinetic steps. While the two limiting cases of general and specific base catalysis are encountered reasonably often, other types of behavior are possible. Obviously, the two terms in the denominator of the rate equation 8.10 may be comparable. The reaction then is of varying order between one and two in aldehyde and between zero and one in 0H~ [13]. A more interesting deviation from textbook behavior is found in some organic solvents. The rate may be strictly second order in aldehyde, but not first order in either OH" or B". The catalytic pathway 8.9 cannot be questioned even in such cases. Rather, the results can be quantitatively explained if the second step, the addition of aldehyde to the carbanion Xf, is made reversible. Instead of eqn 8.10, the rate equation then is ^

K\^n^20^i\d^B-^HB .-^t2fz(r%id^nHB

(8.11)

"*" ^10^20 ^HB "^ ^10^21 ^HB

If the rate is second order in aldehyde, the first term of the denominator must be negligible (this implies quasi-equilibrium in the first step). After cancellations and rearrangement to one-plus form, eqn 8.11 then becomes ^a^ald^B-

(g ]^2)

^aldol

1 + ^b^HB

where K -

'^~Y^

and

k, =

kjk,,

Example 8.2. Aldol condensation of n-butanal in alcohol solvent [15]. Results obtained for aldol condensation of n-butanal in n-octanol are shown in Table 8.1. The reaction is second order in aldehyde. Its apparent second-order rate coefficient, k^^^ = Taidoi /Qjd, is tabulated as a function of buffer composition. Although the reaction is second order in aldehyde, its rate is not proportional to the B"-to-HB ratio and so is not first order in OH". Equation 8.10 thus cannot represent the results. Equation 8.12, more general for reactions second order in aldehyde, can be rearranged: Qid Q-

1 K

K f^ K

8.2. Complex catalysis

in

Accordingly, if eqn 8.12 is applicable, a plot of Qi^CB-Zr^idoi versus Table 8.1. Observed second-order rate coefficients for aldol condensation of nCHB should give a straight line with butanal (ll%wt in n-octanol, potassiumslope k^j/k^ and intercept l/k^. Such octoate buffer, 171°C) [16]. a plot, shown in Figure 8.2, is found to be linear with only some buffer composition [M] random scatter. The values of the «-app coefficients calculated from slope BM~^min"^ HB and intercept or by linear regression are 0.0237 0.011 0.099 0.099 .0173 0.033 k^ = 0.257 M-^min-^ A:, = 10.9 M-^ Equation 8.12 with these values of the coefficients provides a reasonably good fit to the observed rates.

0.099 0.099 0.030 0.059 0.171

0.068 0.090 0.030 0.059 0.171

.0139 .0119 .0065 .0093 .0157

8.2.2. Catalysis by metal complexes Many highly interesting and important reactions of organic chemistry are catalyzed by complexes of transition-metal ions in solution. These include hydrogenation, hydrocyanation, hydroformylation, and some oxidation and polymerization reactions, many of them practiced in industry on a large scale [Gl ,G3,G5,G10]. The most useful metal ions for this purpose are those of Group VIII, in particular cobalt, nickel, palladium, platinum, and rhodium. The most common ligands are carbon monoxide, organic phosphorous compounds, chloride, and the nitroso and cyano groups. A common feature of catalysts of this type is that they exist as different complexes in or close to equilibrium with one another, some of which are able to accept reactants such as olefins, aldehydes, etc., as ligands. A knowledge of the catalyst equilibria is essential for an understanding of catalyst reactivity, selectivity, and kinetics. Moreover, a proper model of the reaction must Figure 8.2. Straight-line plot of data from include the effect of catalyst equiTable 8.1 with eqn 8.12 [15]. libria.

218

Chapter 8. Homogeneous catalysis Example 8.3. Phosphine-substituted cobalt hydwcarbonyls as hydroformylation catalysts. Extensively studied catalyst systems with complex equilibria include phosphine-substituted hydrocarbonyls of cobalt, HCo(CO)3Ph, where Ph stands for a tertiary organic phosphine. They are modifications of the original oxo catalyst, HCo(CO)4. Like the latter, they catalyze the oxo or hydroformylation reaction of olefins to aldehydes one carbon number higher: olefin + H2 + CO

•

aldehyde

(see Example 6.2 in Section 6.3). Unlike the 0x0 catalyst, they also have hydrogenation activity, so that the aldehyde produced is immediately converted to alcohol. This is an advantage because, usually, alcohol rather than aldehyde is the desired product. The phosphine stabilizes the hydrocarbonyl. This reduces the reaction rate under comparable conditions, but makes it possible to carry out the reaction at much lower pressure and to separate the products from the catalyst by short residence-time vacuum distillation. Moreover, the phosphine-substituted hydrocarbonyls give a much higher selectivity to the usually desired straight-chain over branched-chain aldehydes. However, their hydrogenation activity also results in formation of paraffin as byproduct [17-19]. The catalyst forms spontaneously from a cobalt salt or dicobalt octacarbonyl under reaction conditions (synthesis gas pressure and elevated temperature) in the presence of the phosphine and exists in a ligand-exchange equilibrium with the hydrotetracarbonyl, HCo(CO)4 [20,21]. Even only a minute amount of the much more reactive HCo(CO)4 contributes significandy to the rate and adversely affects selectivity, apart from being the main cause of catalyst loss through cobalt deposition. Both hydrocarbonyls are acids, HCo(CO)4 a much stronger one than HCo(CO)3Ph [22]. As the stronger acid, the undesired HCo(CO)4 can be preferentially neutralized by addition of the right amount of base. Its anion, Co(CO)4~, is stable under hydroformylation conditions, but is catalytically inactive, and so is Co(CO)3Ph". With base added to maintain selectivity and catalyst stability, the key cobalt species under reaction conditions thus are the two hydrocarbonyls and their anions, linked by their ligand-exchange and neutralization equilibria as shown in the scheme 8.13 on the facing page. In this catalyst system, addition of phosphine or decrease in partial pressure of CO (or of total pressure at constant synthesis-gas ratio) shifts the phosphine-carbon monoxide ligand-exchange equilibrium toward the phosphine-substituted species (right to left in scheme 8.13). A decrease in phosphine concentration or an increase in CO pressure shifts it in the opposite direction (left to right). Addition of base shifts equilibrium downward, from the acids to the anions. In the presence of phosphine in an at least moderate excess, thermodynamics strongly favors HCo((I!0)3Ph over HCo(CO)4, but Co(CO)4" remains very strongly favored over Co(CO)3Ph" under all conditions of practical interest. The equilibria 8.13 of the cobalt-containing species explain a puzzling observation: The addition of base turns the effects of CO pressure (or total pressure at constant synthesis-gas ratio) and phosphine concentration on rate into their opposites! In the absence of base, an increase in CO pressure or decrease in phosphine

8.2. Complex catalysis CO

HCo(CO),Ph ^ ^

219

Ph

•^>

HCo(CO),

•

B-

B-

HB

co(co)3Ph-

CO

."^

Ph

y

,

k

(8.13) HB

Co(CO),

(predominant cobalt-containing species in larger font) concentration shifts cobalt from HCo(CO)3Ph to HCo(CO)4, to a more active catalyst, and so increases the rate. In contrast, in the presence of base, the shift is primarily from HCo(CO)3Ph to Co(CO)4", from a catalytically active to an inactive species, and so decreases the rate. The decrease in rate with increase in pressure (in the presence of base) in a reaction consuming gas is counterintuitive all by itself.* Lastly, in the absence of base, the shift toward HCo(CO)4 caused by an increase in pressure or a decrease in phosphine concentration results in poorer selectivity and higher catalyst losses. If the amounts of the catalyst ingredients (cobalt, phosphine, and base) are increased by the same factor without change in pressure and H2-to-CO ratio, the increase in phosphine concentration uncompensated by an increase in CO pressure shifts the ligand exchange equilibrium toward HCo(CO)3Ph. Without base, this shift is at the expense of the more active HCo(CO)4, producing an apparent reaction order lower than one in "catalyst." In the presence of base, the shift is mostly at the expense of the inactive Co(CO)4~, resulting in an apparent order higher than one. Granted quasi-equilibrium of ligand exchanges and neutralizations and the fact that Co(CO)3Ph" is catalytically inactive and accounts for only an insignificant fraction of total cobalt, the system 8.13 can be modeled with two of the three equilibrium constants linking the concentrations of HCo(CO)3Ph, HCo(CO)4, and Co(CO)4~. This will be shown in more detail in Example 8.11 in Section 8.8.2. The species shown in the system 8.13 are those present under reaction conditions. At lower temperatures and pressures, a wealth of different complexes are present, including dicobalt octacarbonyl and cationic complexes of cobalt that differ depending on what cobalt compound was initially charged [17,23-25]. Equilibria between different complexes are the rule in catalyst systems of this kind. However, the co-existence of two or more different complexes that catalyze the same reaction, but have different activities and selectivities, as do HCo(CO)3Ph * There is no violation of the Le Chatelier principle, which applies to equilibria, but not to rates. Equilibrium conversion is indeed higher at higher pressure, but is close to 100% anyway even at lower pressure.

220

Chapter 8. Homogeneous catalysis

and HCo(CO)4 in the example above, is less common (another example of such behavior are phosphine-substituted rhodium hydrocarbonyl catalysts for hydroformylation [26]). In such cases, the rate is given by additive rate equations, one for each active species (see eqn 8.96 in Example 8.11). In some important metal-organic catalyst systems, a significant fraction of the catalyst metal may be bound in the form of reaction intermediates rather than as free catalyst. Reaction mathematics under such conditions will be discussed in the next two sections of this chapter. Apart from these two complications, the formalism developed in Chapter 6 for noncatalytic reactions remains applicable as far as rate equations are concerned, but must be combined with modeling of the catalyst equilibria: Where rate equations of transition-metal catalysis appeared in examples in earlier chapters, the quantity Qa^ is the concentration of the free catalyst and may have to be supplied by a catalyst equilibrium model. 8.3. Classical models of enzyme kinetics As has already been pointed out, any rate equation containing the concentration of the free catalyst is of little practical use if that concentration is unknown, is difficult or impossible to measure, and may vary with conversion, as is the case if a significant fraction of the total catalyst material is present in the form of intermediates of the reaction. This is often true in catalysis by enzymes or other tracelevel catalysts. To be sure, the equations in terms of free-catalyst concentration remain correct. However, unless practically all the catalyst material is present as free catalyst, they no longer reflect the actual reaction orders. This is because the concentrations of the participants affect the rate not only directly as expressed explicitly in those equations, but also indirectly and implicitly through their effect on the free-catalyst concentration: As the reactant concentration decreases, so do those of the intermediates; in turn, this produces an increase in the free-catalyst concentration that boosts the rate and, thereby, decreases the apparent reaction order. To reflect this facet correctly, what is needed are rate equations in terms of the total amount of catalyst material, a quantity that is constant and known. The Bodenstein approximation is not invalidated by this complication: As long as the total amount of catalyst is extremely small compared with those of the reactants, as is largely true in enzyme kinetics, the catalytic cycle attains quasistationary conditions. (Exceptions are fast biological reactions of substances themselves at very low concentrations, a topic beyond the scope of this book.) The earliest quantitative theory of enzyme kinetics dates back to 1913, when Michaelis and Menten [27] succeeded in explaining a key feature of enzyme reactions with a very simple model. As an introduction and to establish the relationship between trace-level and bulk-species catalysis, this classical work and its subsequent refinements will now be reviewed.

8.3. Classical models of enzyme kinetics

221

8.3.1. Michaelis-Menten kinetics The catalytic cycle considered by Michaelis and Menten [27,28], the simplest to exhibit the most characteristic feature of enzyme catalysis, is

(8.14)

with quasi-equilibrium in the step A + cat <—• X: Cx'C.C^.

=

^Ax = const.

(8.15)

The catalyst is distributed over free catalyst (cat) and the intermediate (X), so that Cca. - C ,

= C,„,

(8.16)

where Qcat is the total amount of catalyst per unit volume. The initial rate of product formation is that of the second step: r^ = kxpC^

(8.17)

(reverse reaction still insignificant). Expressing Cx in eqn 8.17 in terms of Qcat by means of eqns 8.15 and 8.16 one obtains: 0

^^

•*»• A Y ^ Y D ^

A ^ ^ AX'^XP'^A'^Lcat

(g ][g\

1 - ^AXC,

An empirical equation of this form had been suggested as early as 1903 by Henri [29]. Ten years later, Michaelis and Menten provided its theoretical basis [27]. According to eqn 8.18, the initial rate is first order in total catalyst material and of order between zero and one in the reactant, A. Note that Rule 7.13 for simple thermal pathways, which states that a reaction is first order in a reactant that participates in the first step, is not obeyed (it will be modified in Section 8.6). Limiting cases of eqn 8.18 are ^P — ^AX^XP^A^cat

if

^AX^^A « 1

(first order in A)

Tp =

if

^AX^A * ^

(zero order in A)

/^xpQcat

222

Chapter 8. Homogeneous catalysis

The first extreme, at low reactant concentration, corresponds to a first-step equilibrium very strongly in favor of the free catalyst, so that Qcat = QatJ the second, at high reactant concentration, to an equilibrium very strongly in favor of the intermediate, X, so that Qcat = Q- Between the two extremes, the rate varies nonlinearly with the reactant concentration, approaching a limit at very high concentrations (see Figure 8.3). In a Michaelis-Menten cycle, the initial rate is first order in the reactant at low reactant concentration and approaches a limiting value when that concentration is increased. Such behavior is generally referred to as Michaelis-Menten kinetics or saturation kinetics. As will be seen, it is a rather common feature of trace-level catalysis, not restricted to cycles as simple as 8.14. How saturation kinetics comes about is easy to see. If the reactant concentration is very low, most of the catalyst material is present as the free catalyst, and the behavior is the same as in bulk-species catalysis. saturation limit At very high reactant (rX concentrations, however, most of the catalyst material gets converted to the intermediate X. Regardless of how much farther the reactant concentration •a is raised, the rate, given by eqn 8.17, can at most approach the limit (rp)^^ = ^xpQcat because Cx reactant concentration, C^ (catalyst bound as interFigure 8.3. Rate as function of reactant mediate) cannot possibly concentration in reaction with Michaelisexceed Qcat (total catalyst Menten saturation kinetics (schematic). material). Equation 8.18 for Michaelis-Menten kinetics has been shown here in a form consistent with the formalism used elsewhere in this book. Usually, however, the equation is written in a different, but equivalent form [27,28]: V p)max ^ A ^VA

+

C.

(8.19)

8.3, Classical models of enzyme kinetics

223

where (rp)max = ^xpQcat is the maximum possible rate. The so-called Michaelis constant ^XA is the dissociation constant of X <—• A + cat and is the reciprocal of ^Ax ill cqn 8.18. 8.3,2. Briggs-Haldane kinetics As first pointed out by Briggs and Haldane [30], the assumption of quasiequilibrium in the first step is inconveniently restrictive. They relaxed that postulate by replacing the quasi-equilibrium condition 8.15 with the Bodenstein approximation for the trace intermediate X: '•x = ^AxC.,C, - C^(k^ . fc,p) ^

0

(8.20)

so that ^

^

^AX ^ a t

^A

-'X

'^XA

"^

^XP

With the catalyst balance 8.16 and rate equation 8.17 this gives o

_

^'^AX'^XP^A^Ecat AX^XP^A^Ecat ^XA +

^XP "*" ^AX

(8 2 1 )

c.

or, in one-plus form, rp

k

where

k„ =

k

=

K ^A ^-Lcat

1 .

k

^ ^+ ^'k

'^XA

(8.22)

k,C,

^

K -

and

^AX

k '^XA

'^XP

+ k '^XP

The one-plus rate equation 8.22 is of the same algebraic form as the Michaelis-Menten equation 8.18, only the physical significance of the coefficients is different [instead of the constant JST^X* the expression k^x /(^XA + ^xp) ^^^ appears]. Accordingly, the behavior is the same as for Michaelis-Menten kinetics, and that name is often used for Briggs-Haldane kinetics as well. 8.3.3. Reversible cycles As a rule, enzyme reactions are reversible. To account for the reverse reaction at significant conversion and so remove the restriction to initial rates, the reverse step P + cat—• X in the cycle 8.14 must be included. Both the rate equation 8.17 and the Bodenstein approximation 8.20 then contain an additional term to account for this reverse step. The resulting rate equation is (^AX ^XP ^ A

^XA ^PX ^ p ) Q cat

^XA "^ ^XP "^ ^ A X ^ A

"^

Sx^P

(8.23)

224

Chapter 8. Homogeneous catalysis

or, in one-plus form ^

^

v'^a'^A

'^c'^P^^Eca/

(8.24)

1 + k,C^ + k,C^

where _,

'^XA'^PX

^XA

7.

'^XP

_^

^AX

'^XA

'^XP

'^XA

'^XP

Equation 8.23 is the most general rate equation for a trace-level catalyst cycle A <—• P with one intermediate. It reduces to the Briggs-Haldane equation 8.21 if A:px -^ 0 or Cp = 0, that is, if the second step is irreversible or only the initial rate is considered. It reduces further to the Michaelis-Menten equation 8.18 if, in addition, kxp « /^XA* that is, if the first step is in quasi-equilibrium. A comparison of eqn 8.23 for arbitrary distribution of catalyst material with the formula 8.2 developed for bulk-species catalysts is instructive. Application of the latter to the cycle 8.14 yields (^AX ^ X P ^ A '^XA

^XA ^PX ^ p )

^ci

(8.25)

'^XP

Equation 8.23 derived in the present section differs by having the total instead of the free catalyst concentration in the numerator, as well as two additional terms in the denominator. The equivalence of the two equations can be shown as follows: Replacement of Cc^^ in eqn 8.25 with use of eqn 8.16 and subsequent replacement of Cx with the Bodenstein approximation (eqn 8.20 with additional term kpxC^^iC^) yields eqn 8.23. Equation 8.25 is simpler and therefore preferable if one can be sure that all but an insignificant fraction of catalyst is in free form. Equation 8.23 is not subject to this restriction, an advantage one must pay for with the inconvenience of having to handle two more denominator terms. 8.3.4. Common features and plots A common feature of all single-cycle kinetics discussed so far is a one-plus rate behavior with reaction order between zero and one with respect to the reactant, A (and for a possible reverse rate, with respect to the product, P). The MichaelisMenten and Briggs-Haldane rate equations 8.18 and 8.22 have the same algebraic form, and so has the initial rate in the reversible cycle, that is, eqn 8.24 with terms involving Cp still being insignificant. This common one-plus form can be rearranged: Qca,

K .

1

KC,

(8.26)

8.3,

225

Classical models of enzyme kinetics

Accordingly, a plot of C^^^Jr^ versus 1/Q gives a straight line if kinetics is of this type. If so, the coefficients can be calculated from the intercept k^ Ik^ and slope llk^. The often used Lineweaver-Burk plot of l/r^ versus l / Q [31] achieves the same goal, but gives different straight lines for different catalyst concentrations Qcaf Other straight-line plots are those of CA /r^ versus CA (Hanes plot [32]) and r^ versus r^/Cj^ (Eadie-Hofstee plot [33,34]) (see also Segel's book [28]). Reactions orders between zero and one in accordance with one-plus rate equations are very conmion in enzyme catalysis, even if the cycle is more complex and involves additional reactants or products. The plots just described thus are more broadly applicable. On the other hand, straight lines in such plots are only evidence of saturation kinetics, not an indication that the catalyst cycle has only one intermediate. Example 8.4. Phosphate transfer catalyzed by hexokinase. The enzyme hexokinase catalyzes transfer of phosphate to Cg sugars. Table 8.2 shows results reported for the reaction Mg-ATP + glucose < • Mg-ADP + glucose-P (8.27) where Mg-ATP is magnesium adenosine-5'-triphosphate, Mg-ADP is magnesium adenosine-5' -diphosphate, and glucose-P is glucose 6phosphate. The table lists values of C^cat/rp obtained under various conditions (rp = initial rate). A cursory inspection of the results reveals that the reaction orders with respect to both Mg-ATP and glucose are positive, but less than one. For the initial rate this suggests an equation of the form rp

' p

rp°

Qvlg-ATP

^a ^ A ^ B

b

0.473

0.10 .20 .50 1.00 0.10 .20 .50 1.00 0.10 .20 .50 1.00 0.10 .20 .50 1.00

7.48*10-3 4.54 2.90 2.55 6.10 3.90 2.51 2.21 4.87 3.17 2.17 1.83 4.44 2.93 2.03

0.920

K

^a ^ B

_^

c ^a ^ A

C j ^ a t ' ^P

S

+ k^C^ ^h

^

^glucose

mM

1.94 1

c

mM

^a^A^B^cat

1 yk^^

so that:

c

Table 8.2. Initial rates of phosphate transfer reaction 8.27 catalyzed by hexokinase at 25° C [35].

4.00

(A and B are Mg-ATP and glucose). Plots of Qcat/^p° versus l/CMg.AT? and 1.72 1 l/Cgiucose are shown in Figure 8.4 (next page). Both plots give reasonably good straight lines for groups of data for which the concentration of the other reactant is the same, as the hypothetical rate equation demands. The equation thus appears to be acceptable.

226

Chapter 8. Homogeneous catalysis

8.0 _ "

6.0

[s]

o 0.473 mM Mg-ATP

n 0.920 A V

jy^

1.94 4.00

/ ^ >^

r^

-

4.0 -

2.0

1

2.0

1

4.0

1/Cglucose

1

1

1

6.0

8.0

10.0

[mM-i;

"Mg-ATP

Figure 8.4. Plots of C^^Jr^° versus reciprocal reactant concentrations in phosphate transfer reaction (data from Table 8.2). Left: plots versus l/Cgj^^^^g at different C^^_right plots versus I/CjMg-ATP at different Cg,^^„3,. •"Mg-ATP A type of mechanism compatible with the form of the hypothetical rate equation is the cycle

(8.28) glucose-P Mg-ADP with very fast decomposition of Xj into products and catalyst, probably in more than one step. The equation for the initial rate in a cycle of this type is the Briggs-Haldane equation 8.21 with appropriately changed indices and replacement of A:xp (i.e., k^^^) by k^iC^ as the second denominator term: ^0

_

^01 ^12 Q c a t ^ A ^ B ^10 "^ ^12 ^ B

(8.29)

"^ ^01 ^ A

(index 0 = catalyst). Whether glucose or Mg-ATP is first to enter the catalyst cycle (i.e., is reactant A) cannot be decided without additional information. The actual system is more complex than shown here and involves dissociation of the Mg-ATP and Mg-ADP complexes and inhibition by free ATP. However, the cycle 8.28 (with glucose entering first) appears to be essentially correct [35].

8.4. General formula for single catalytic cycles

8.4.

221

General formula for single catalytic cycles: Christiansen mathematics

For catalytic cycles with more than three or four members, the long-hand derivation of rate equations gets out of hand. However, a general formula comparable to that given in Section 6.3 for noncatalytic simple pathways was established as early as 1931 by Christiansen [36-38].* As at the start of this chapter, we consider the single catalytic cycle X,

x„ (8.1) cat

with any number k of members and possibly with co-reactants and co-products (not shown), but this time seeking a rate equation in terms of total catalyst material and admitting that the latter may be distributed in any arbitrary way over free catalyst and intermediates. The rate equation based on Christiansen's mathematics is k-l

n N u, - n Ku i=0

(8.30)

i=0

S

where indices 0 and k refer to the free catalyst, and where the denominator S is the sum of all elements in the "Christiansen matrix," shown here for a four-membered cycle (k = 4) ^^12^3^30

^10^3'^30

^10^1^30

^10^1^32

X23X30X0J

\^\Q\^

^1^32^1

^21^2^3

^3o\l^l2

^32^1^12

^32^3^12

^32^3^10

\l^l2^3

^3^12^3

^3^10^23

^3^10^1

(8.31)

The numerator of the Christiansen rate equation 8.30 is the same as that of eqn 8.2 used earlier for cycles with bulk catalyst, except that the total catalyst concentration, C^cat^ takes the place of the concentration of the free catalyst, C^af However, the

* Christiansen provided all necessary mathematics but omitted to write an explicit equation for the reaction rate. Perhaps for this reason—also no doubt because much of his work was published in Danish or German—he has not received as much credit as he deserved. More often cited later authors, most prominent among them King and Altman [39], apparently unaware of his work, largely reinvented his approach and elaborated upon it.

228

Chapter 8. Homogeneous catalysis

denominator contains additional terms. The terms of the first row of the matrix 8.31 are seen to be those of the denominator DQQ of the equation for bulk-catalyst cycles. The second row is generated from the first by increase of all index numbers by 1 and replacement of any resulting k by 0. Each successive row is obtained from the preceding one by this same recipe, and so is the first row from the last. A rigorous derivation of Christiansen's rate equation is laborious. However, it is easy to see how this formula comes about, as the following argument will show. If all but an insignificant fraction of the catalyst material were present as free catalyst, Q^at would practically equal Qat, so that the Christiansen numerator would equal that in eqn 8.2 for bulk catalyst cycles. The Christiansen denominator must then also equal the denominator in eqn 8.2. It does so if all terms except those of the first row are insignificant, i.e., if ® = DQQ. To generalize from here: Nothing distinguishes the free catalyst mathematically from the other cycle members, and the place where index numbering along the cycle "starts" and "ends" can be chosen arbitrarily. Thus, if practically all catalyst material were present as, say, the first intermediate, we could start numbering at the latter. Since the predominant cycle member then is that with index 0, the rate equation must be the same as it was with the free catalyst as the predominant member and labeled 0. But to restore the original indexing, which we intend to keep, we have to increase all index numbers by 1. The denominator now matches the second row of the Christiansen matrix 8.31, i.e., S = D^ (see definition 6.6, with numbering clockwise around the cycle to and past zero). By the same token, each clockwise shift of the predominant cycle member by one position increases all index numbers by 1 and makes the next matrix row farther down the only one whose terms can be significant. In other words, if one cycle member contains practically all catalyst material, only one matrix row is significant. In this light, the complete Christiansen equation 8.30 can be recognized as the general formula that reduces as required to each of the special cases with practically all catalyst contained in one member of the cycle. This argument also leads directly to one of Christiansen's key conclusions:

In the Christiansen matrix 8.31, the sum of the elements of each row is proportional to the concentration of one of the members of the catalyst cycle: Sum of row 1 proportional to row 2 row 3

free catalyst intermediate X^ intermediate X2

row j

intermediate Xj_i

last row

last intermediate

8,5. Reduction of complexity

229

More specifically:

-'Ecat

sum of row j+1 sum of all elements

(j > 0)

(8.32)

(j = 0 refers to free catalyst). The general catalytic cycle 8.1 in terms of X coefficients allows for any number of reactants and products and any nature of the intermediates Xj, subject only to the condition that no step involves more than one molecule of intermediates as reactant. It thus is applicable to a great number of different types of systems. Only one example will be shown here. Example 8.5. Ping-pong transfer reactions. Some enzymatic transfer reactions proceed by so-caWtd ping-pong mechanisms [40,41]. In these, the conversion of a reactant to a product leaves the enzyme in a different form. The modified enzyme then converts a second reactant to another product while itself being restored to its original form. Enzymatic transaminase reactions interconverting amino and keto acids provide a typical example [40]:

'

H

0=C-C0OH

(amino acids)

(keto acids)

R' H,N-C-COOH ' H

R' 0=C-COOH

(8.33)

Equation 8.30 applies with k = 4, index 2 referring to cat' (the amino derivative of the enzyme), and with the concentrations of the amino and keto acids appearing as cofactors in the respective X coefficients. 8.5. Reduction of complexity Rigorous rate equations for multistep catalytic reactions in terms of total amount of catalyst material are enormously cumbersome. Just the reduction to the level complexity of the Christiansen formula calls for the Bodenstein approximation of quasi-stationary behavior of the intermediates, requiring these to remain at trace concentrations, and that formula still entails a lot more algebra than does the general rate equation for noncatalytic simple pathways: For a reaction with three intermediates, the Christiansen denominator contains sixteen terms instead of four; for a reaction with six intermediates, forty-nine instead of seven! Although the mathematics is simple and easy to program for modeling purposes and, usually, some

230

Chapter 8. Homogeneous catalysis

terms can be consolidated, the equations are too unwieldy for network elucidation if the cycle has more than three or four members and several co-reactants and coproducts. Here, even more than in the case of noncatalytic pathways, further reduction of complexity is imperative, so much so that the practitioner will apply less stringent criteria as to closeness of his approximations. This is true for what is allowed to pass as "trace level" for the Bodenstein approximation as much as for the demands of applicability of additional simplifications. The three principal tools of reduction of complexity, discussed in Chapter 4, are the approximations of a rate-controlling step, of quasi-equilibrium steps, and of quasi-stationary behavior of intermediates. The Christiansen formula has already invoked the last of these three. The other two can be used for additional simplification. A further, new and very powerful tool is the concept of relative abundance of catalyst-containing species. Moreover, much can sometimes be gained if one or several steps can be taken as irreversible. To summarize: Tools for Reduction of Complexity of Christiansen Formula: relative abundance of catalyst-containing species {macs and lacs) rate-controlling step quasi-equilibrium steps irreversible steps

8.5.1. Relative abundance of catalyst-containing species * If the catalyst is present almost completely in the form of one member of the cycle, be it as the free catalyst or an intermediate Xj, that species is called the "most abundant catalyst-containing species," or macs for short. If a macs exists, one row of the matrix 8.31 dominates all others: the first row if the macs is the free catalyst, the (j + l)'th row if the macs is Xj. The Christiansen rate equation 8.30 is thereby reduced to the lower degree of complexity of those of bulk-catalytic and noncatalytic simple pathways, with only k instead of k^ terms in the denominator (k = number of cycle members; see eqns 8.2 and 6.4 to 6.6). A comparison of eqn 8.30 and matrix 8.31 with eqns 6.4 to 6.6 leads to a very important rule. If, say, Xj is the macs, then eqn 8.30 reduces to a form whose denominator consists only of the elements of row j-h 1 of the matrix 8.31; more* The concept of a predominant cycle member was first introduced in heterogeneous catalysis by Boudart [42], who coined the term "most abundant surface intermediate," abbreviated mast (see Section 9.3). See, however, footnote on p. 280 for a difference in definitions.

8.5. Reduction of complexity

231

over, Qcat approximately equals Cy Comparison with the general rate equation for noncatalytic reactions, eqns 6.4 to 6.6, then shows the reduced form of eqn 8.30 to be the same as for a noncatalytic pathway Xj <—> Xj+i <—> ... <—^ Xf. • The rate equation of a catalytic cycle with a macs is the same as that of an imaginary simple pathway of same steps that "starts" and "ends" with the macs. In other words, to obtain the rate equation for a catalytic cycle with a macs, the cycle can be "snipped" at the latter to give a linear pathway. The rate equation of that imaginary pathway approximates that of the cycle (granted the validity of the assumption that practically all of the total catalyst material is present as the macs). This simple rule allows the rate equation of any catalytic reaction with a macs to be written down as quickly and easily as those for linear simple pathways. It will be used extensively in the next section (see also Figure 8.5 in that section). As an example, consider a four-member cycle with a co-reactant B in the second step: X,

X.

cat

(8.34)

X3

If X2 is the macs, the Christiansen matrix retains only its third row: 0

0

0

0

0

0

0

0

\o\l^l2

^32^01^12

^2^3^12

^2^3^10

0

0

0

0

(8.35)

The denominator of the rate equation—the sum of the four elements of the third row—is seen to be the same as for an imaginary simple pathway with same steps as in the cycle 8.34 and starting and ending with X2, the macs. After replacement of the X coefficients and collection of terms, the rate equation is (^01^12^23^30 ^P

^A^B

^10^21 ^32^03 ^ p ) ^Ecat

=

(8.36)

( ^ 3 0 •*" ^ 3 2 ) ^ 0 1 ^ 1 2 ^ A ^ B "•" ^ 3 2 ^ 0 3 ^ 1 2 ^ B ^ P "^ ^ 3 2 ^ 0 3 ^ 1 0 ^ P

(CA is co-factor in XQI, Cg in X12, and Cp in X03). Unless the macs is the free catalyst, the denominator of the rate equation differs from that in eqn 8.2. This affects the reaction orders, as will be discussed in the next section.

232

Chapter 8. Homogeneous catalysis

If one intermediate contains no more than an insignificant fraction of the total catalyst material, it will be called the "least abundant catalyst-containing species," or lacs for short. The matrix row corresponding to that intermediate is negligible. If, say, Xi in the cycle 8.34 is a lacs, the matrix 8.31 reduces to ^12^3^30

^10^23^0

^lO'^l^SO

^10^21^32

0

0

0

0

^3o\l^l2

^32^1 ^^12 ^^32^3^12

^32^3^10

\l^l2^3

^3^12^3

^3^10^21

^3^10^23

(8.37)

and the denominator of the rate equation loses four of its original sixteen terms. More than one intermediate may qualify as a lacs (linguistic purity would demand the abbreviation then to be read as "low-abundance catalyst-containing species"). If so, more than one matrix row becomes negligible. Of course, if there is a macs, all other members of the catalytic cycle are automatically in the lacs category. 8.5,2. Rate-controlling steps Whenever the forward and reverse X coefficients of a step are much smaller than all others, the denominator terms involving those coefficients become negligible (see Section 4.2). Say, the step Xj <—> Xj+i (co-reactants or -products not shown) is rate-controlling. The terms involving Xjj+i and Xj+ij will be then small compared with those which, instead, involve Xjj_i and Xj+ij^.2, respectively. This makes all terms but one disappear in each row of the Christiansen matrix. In the general case of a k-membered cycle, this reduces the number of terms from k^ to k. For example, assume the step X^ + B -4—• X2 in the cycle 8.34 to be ratecontrolling. Inspection of the matrix 8.31 shows that it will be reduced to 0

^10^3^0

0

0

^23^30^1

0

0

0

0

0

0

^ 2 ^ 3 ^10

0

0

^3^10^3

0

(8.38)

After replacement of the X coefficients and collection of terms, the resulting rate equation is _

(^01 ^12^23^30 ^ A ^ B ~ ^10^21^32% ^ p ) Q c a t ^23^30(^10"^ ^01 ^ A ) "*" S 3 ^10 ^ P (^23 ••" ^32)

(8.39)

8.5. Reduction of complexity

233

This equation contains the rate coefficients of all steps including those at quasi-equilibrium. Instead, the rate can be expressed in terms of the more easily accessible equilibrium constants of those steps:

with the respective substitutions the rate equation becomes _^

(^""01"'12 ^ 0 1 ^12 ^ A ^ B '^01 ^ A

^21 ^ 3 2 ^ 0 3 ""P^'^Ecat ^p) Q '^21 *"32^"03 + (1

+

(8 40)

^32)^03^?

8.5.3. Quasi-equilibrium steps If the forward and reverse X coefficients of a step are much larger than all others, that step is at quasi-equilibrium (see Section 4.3). The participants in that step then are present at all times in concentrations related to one another by the thermodynamic equilibrium condition, and so can be lumped into one pseudo-component. Since the number of denominator terms in the rate equation equals the square of the number of the cycle members, this reduces the amount of algebra considerably. With quasi-equilibrium in the step Xj <—• Xj+i and its participants lumped into XL, the general cycle is

cat r

(lumped species shown in box and designated XL). The numerical values of the rate coefficients of formation of the lumped species from Xj_i and Xj+2 are not affected by the lumping, that is, they are the same as those of formation of Xj from Xj_i and of Xj+i from Xj+2» respectively, in the actual cycle: N-UL = ^-l,i

and

X^,,,, = X.,,,..,

(8.41)

This is because, for both the actual and the reduced cycle, the respective rate equations have the concentrations of Xj.^ and Xj+2, respectively, as factors. However, in the rate equations for decay of the lumped species to Xj_i and Xj+2» the concentration of the lumped species appears instead of that of Xj or Xj+i. Therefore, to obtain the coefficients of the reduced cycle, those of the actual cycle must be multiplied by the fractions Cj /CL and Cj+i / Q , respectively, where Q = Cj + Cj+i is the concentration of the lumped species:

234

Chapter 8. Homogeneous catalysis

\ . j - . =(q/c,)x...,,

(C,JC0Ku>.2

LJ+2

(8-42)

The concentration ratios in these equations can be replaced with use of the equilibrium condition for the fast step:

where JRjj+i is the product of the concentrations of any co-reactants of the step divided by that of the concentrations of any co-products. With this condition, the fractions in eqn 8.42 become

C

1

^jj-i^jj^i

C^

1 ^ ^J.i3^.j.i

^

^^

(8.43)

^,.5»jj.

Several steps in a cycle may be fast enough for quasi-equilibrium. If two or more such steps are successive, all the cycle members involved can be lumped into a single pseudo-component. The derivation of the rate equation in final form is best illustrated with a specific example: Example 8.6. Rate for a hypothetical cycle with quasi-equilibrium steps. Assume that the second and third steps in the four-membered cycle 8.34 are in quasi-equilibrium and the fourth step is irreversible (/:o3 = 0):

With lumping of Xi, Xj, and X3 into a single pseudo-species XL (shown in box), the reduced cycle now has only two members: cat and XL- The Briggs-Haldane equation 8.21 applies, without restriction to initial rate and with index X replaced by L: (8.44) ^LA

"^

^LP

"^

^AL^A

(indices A and P instead of 0 are retained for distinction between the two steps of the reduced cycle). With the equilibrium condition Q : C2: C3 = 1 : K^2^^ • ^ B Q ^ the concentration fractions corresponding to eqns 8.43 are

235

8.5. Reduction of complexity

1

^13 ^B

so that

k K C '^30^13^8

^LP

(see eqns 8.42). With these substitutions, eqn 8.44 becomes ^01 ^ 1 3 ^30 ^ A ^ B Q c a t ^10 + ^ 1 3 ^ 3 0 C B ^ ^ 0 I ^ A [ 1 ^ (^12 + ^ 1 3 ) C B ]

Accordingly, the reaction is of order between zero and one in A and B and first order in total catalyst material, showing typical saturation kinetics. It may surprise that the rate decreases with increasing K^2^ a "forward" equilibrium constant. The reason is that an increase in K^2 (^t constant A'13) increases the concentration of X2 at the expense of Xj and X3, making the lumped species more sluggish to react. 8.5.4. Irreversible steps Whenever a step in the catalytic cycle is practically irreversible, the respective reverse rate coefficient Xj+ij and all terms involving it become negligible. This includes the negative term in the numerator. Assume the second step in the four-membered cycle 8.34 is irreversible, so that X21 = 0. The Christiansen matrix then reduces to

\,\,\,

0

0

0

\Q\\\2

W \ \ I

^2^3^12

^2\)3^10

^01^12^3 ^03^12^3 \)3^10^3

(8.45)

^

The number of denominator terms has decreased from sixteen to ten. In general, for a k-membered cycle the reduction is from k^ to k(k4-l)/2. If the irreversible step is Xj —• Xj+i, the (j4-2)th row remains intact, and each successive row loses one more element. More than one step may be irreversible. The approximation can then be applied to each of them, with further simplification of the rate equation. As the reduced matrix 8.45 shows, steps following the irreversible second one in the cycle do affect the rate. This is another example for the fact that the rules for noncatalytic simple pathways do not apply unless the free catalyst is the macs. A more general set of rules will be given in the next section.

236

Chapter 8. Homogeneous catalysis

8.5.5.

Combinations of approximations

The approximations in this section can be combined in many different ways. Significant further simplification may result. For example, for a cycle 8.34 with Xi as the macs and irreversible step Xi + B —• X2, the only remaining row of the Christiansen matrix is the second, and that row has but one single element, X23X30X01 (see reduced matrix 8.45). In this case, the rate equation reduces to r^

_

=

\ l ^12 ^23 \o C^j^cat

~

^12 C^^^^

-

k C C

(8.46)

^^3^30^1

A rather simple rate equation is also obtained from a combination of the approximations of a macs and a rate-controlling step. The existence of a macs leaves only one matrix row, and that of a rate-controlling step ensures that the row has only one contributing term. In many such cases, the rate is controlled by the step consuming the macs. This is because slow consumption of a cycle member favors its attainment of a relatively high concentration. Again returning to the example of the cycle 8.34 with X^ as the macs, if the step X^ + B <—• X2 is ratecontrolling, the only surviving element of the matrix is X23X30X01 (see reduced matrix 8.38), and the rate equation becomes Xj2 A23 A3Q AQJ

\ l ^23 ^ 0 ^21 ^ P

/Cn C-D

^01 ^23 ^30 ^A

c.

(8.47)

Comparison with eqn 8.46 shows that the forward rate is the same as in the previous example, in which the step consuming the macs was irreversible. Many other combinations of the approximations can be used if warranted. Example 8.7. Adiponitrile synthesis via hydrocyanation."^ Hydrocyanation of 4-pentenenitrile (4-PN) to adiponitrile (ADN) in the presence of an arylborane modifier: 'CN

HCN

(8.48)

is an important step in du Font's current Nylon-6 process. The network 8.49 was established by Tolman and co-workers [43-45]. The pathway of the preceding hydrocyanation of butadiene to 3-pentenenitrile is similar. By-product formation, not included in the network 8.49, will be examined later (see Example 8.12 in Section 8.8.3).

* The kinetics analysis in this example is based in part on unpublished work by A. A. Gupte (1992).

237

8.5. Reduction of complexity (cat)

Ph Ph Ni Ph Ph (ADN)

(X,)

(X)

HCN (XJ

Ph Ph Ni Ph

Ph CN HNi Ph Ph

(8.49)

NiPh4 is a complex of zero-valent nickel with tertiary organic phosphine or phosphite ligands, Ph. Under reaction conditions, the ligand-deficient species NiPh3 (Xo), HNiPh2CN (X2), and 4PN-NiPh2 (X5) and apparently also 4PN-NiPh3 (X^) contain only insignificant fractions of total nickel, as evident from IR, UV, and NMR spectra [44,46]. The reaction is irreversible, in all likelihood owing to irreversible carbon-carbon bond formation in the step forming X5. Ligand loss NiPh4 —• NiPhg -h Ph is at quasi-equilibrium, so that these two nickel complexes can be lumped into a pseudo-single species, X (shown in box). In the network 8.49, the principal nickelcontaining species are shown in larger, bold font, and the notation Xj is retained for all intermediates although this requires a more liberal interpretation of "trace level." The reaction is not inhibited by ADN, so that Xx6, which contains the ADN concentration as co-factor, may not appear in the denominator of the rate equation. Therefore, either Xx6 is zero or, more likely, Xg is a lacs so that the entire last Christiansen matrix row, the only one containing Xx6, becomes negligible. Application of the Christiansen formula to this cycle gives the rate equation =

^Xl ^12 ^ 3 ^ 4 Ks ^56 \ x Q c a t ^ ^

(8.50)

(® = sum of all elements of Christiansen matrix). With irreversible carbon-carbon coupling (X54 = 0) and X2, X5, and X^ as lacs, the Christiansen matrix reduces to

238

Chapter 8. Homogeneous catalysis

^12^3^34^45^56^}

0

0

\3X34X45X5gXgxXx1

^

0

0

0

0

0

0

0

0 0

\2i\4\5\6\x\l

0

\l\2^45^56\x^Xl

0

WlKj^ 56\\\\

0

0

^5^56^X^X1 ^12^ 23 \ 3 \ 6 \ x \ l ^ l 2 \ ! 3

0

0

KKxKKKsK

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

With eqns 8.42 and 8.43 for coefficients involving lumped reactants and with Q « Qat and the quasi-equilibrium condition CoCph/Qat = K^^i, one finds Xxi to be

while Xix = \o and X^x = Xgo- With these replacements, cancellation of X56 and Xgo, and the substitutions \ l

~ ^Ol^HCN'

^ 2 ~ ^12'

\ o ~ ^10'

^^23 ~ ^23^4PN '

\ l ~ ^21 ^ P h '

^ 4 ~ ^34'

^32 " ^ 3 2 '

^ 5 ~ ^45'

^ 3 "^ ^43

the rate equation becomes ^ADN

=

^cat^01^12^23^34^45^HCNQpNQcat/®*

^* ^

where 6* = ©Cp^ is given by ©

= (^12^23^34^45 + ^10^23^34^5 ) Q p N ^ P h "•" (^10^21^34^45 ••" ^10^21^32^45 ••" ^10^21^32^43 ) ^Ph + /^23^34^45^01-^cat^4PN ^HCN "•" (^21^34^45^01 ^ ^21^32^45^01 "•" ^21^32^43^01/-^cat^HCN^Ph + (^45^01^12^23 ••" ^45^01^12^23/^cat^HCNQpN

.r. ^^x

+ /Coi^cat^l2^23^34^HCNQpN

The rate equation 8.51, written with phenomenological coefficients, then becomes: kr ^

C

C

'^a'^HCN^4PN^Ecat

/o^ Q \

^bWPN^Ph "•• ^c^Ph + ^ C I ^ H C N Q P N ^ ^e^HCN^Ph

[If Xg is not a lacs, but Xo6 is zero, the last matrix row contributes one single term ^oi^i2^23^34^45^catQcNQpN wlth the samc concentration co-factors as that in the fifth row, so that eqn 8.53 remains valid with only a different significance oik^.\ The first two steps of the cycle are likely to be at quasi-equilibrium. If so, the first denominator term in eqn 8.53 is negligible. The rate equation in one-plus form then has only three phenomenological coefficients. In any event, the reaction orders are: plus one for nickel, between zero and plus one for HCN and 4-pentenenitrile, and between zero and minus two for the organic phosphine.

8.6. Network properties and kinetic behavior

239

Note that the ratio of the first two to all four denominator terms in eqn 8.53 equals the fraction of nickel present as NiPh4 (plus a negligible amount as NiPh3). A knowledge of that fraction—e.g., from spectra—can facilitate the calculation of coefficient values from rate data. The traditional way of handling hydrocyanation networks of this kind has been to postulate rate control by carbon-carbon bonding (X4 —• X5) and quasi-equilibrium in all other steps. On this basis a simpler rate equation has been proposed [44]:

However, this equation is in terms of the concentration of NiPh4 rather than total nickel, and so does not reflect the actual reaction orders. For orders as eqn 8.54 suggests, the denominator in eqn 8.51 would have to consist exclusively of terms with only Cph as co-factor. As eqn 8.52 shows, only three terms meet this condition, and all of them stem from the first matrix row. Accordingly, NiPh4 would have to be the macs, contrary to spectroscopic and NMR evidence. Moreover, the other two terms from the first matrix row would also have to be negligible, requiring the first and second step of the cycle to be at quasi-equilibrium. With cat taken to be total nickel, eqn 8.54 can therefore be only a rough approximation, except for catalyst systems in which NiPh4 is indeed the macs and the two first steps are at quasi-equilibrium. 8.6. Relationships between network properties and kinetic behavior As has already become apparent, some of the rules governing the relationships between network properties and kinetic behavior, derived in Section 7.3.1 for noncatalytic simple pathways, can no longer be relied upon in catalysis, except if the free catalyst is the macs. A re-examination therefore is called for. Rules of general validity. The following rules apply without qualifications: • A pathway is irreversible if one or more of its steps are irreversible (Rule 7.11). This follows from a simple free-energy argument: To be practically irreversible, a step must involve an extremely large free-energy loss. Unless that loss is offset by a comparable free-energy gain (necessitating a practically irreversible reverse step and so being academic), the overall reaction must involve such a loss and be irreversible. This argument is independent of any catalyst effects. • A positive reaction order with respect to a product requires a step in which the product acts as reactant (Rule 7.20, referring to forward reaction). Whether or not the reaction is catalytic, the concentration of the respective product must appear as a co-factor in the numerator of the equation for the forward rate. This happens only if that participant acts as reactant in a forward step.

240

•

Chapter 8. Homogeneous catalysis If the forward and reverse rate coefficients of a step are much smaller than all others, the other steps are in quasi-equilibrium (Rule 7.21).

The general principle that a step much slower than all others gives the latter enough time to attain and maintain quasi-equilibrium applies to catalytic reactions as well. Lastly, Rule 7.24 for step consolidation must be qualified: A step with co-reactant entry can be consolidated with a subsequent step with co-product exit or a rearrangement step, and a rearrangement step can be consolidated with a subsequent step of co-product exit or another rearrangement step, but only if the intermediate is a lacs.

(8.55)

Reaction orders in cycles with macs [47]. The rules for reaction orders in simple pathways do not apply to catalytic cycles with arbitrary distribution of catalyst material over the cycle members. In the general case, any cycle with more than two or three members gives rise to a Christiansen matrix with a profusion of terms, many of which are apt to involve different combinations of reactant and product concentrations as co-factors. This makes it impossible to formulate general rules for such cycles. All that can be said is that a distribution of catalyst material over several species makes for fractional and varying reaction orders as concentrations tend to appear in the numerator and some but not all terms of the denominator. None the less, a set of rules can be given for catalyst systems with a macs. As shown in the previous section, the rate equation for a catalytic reaction with a macs is the same as that for an imaginary (linear) simple pathway that "starts" and "ends" with the macs. Think of the catalytic cycle as being "cut" at the macs to give a linear pathway with the macs at both end (see Figure 8.5). That imaginary equivalent pathway has the same rate equation as the actual catalytic cycle. With this principle, the rules for reaction orders deduced in Section 7.3.1 can be reformulated for catalytic cycles with a macs (as in the earlier section, the rules are for the forward rate if the reaction is reversible). The first such rule is: Steps following an irreversible step, up to and including the step forming the macs, have no effect on the rate. The reaction orders are zero with respect to any species that participate in only those steps.

(8.56)

8.6. Network properties and kinetic behavior (see Rules 7.12 and 7.19). In the equivalent imaginary linear pathway, those and only those steps are preceded by the irreversible one. For example, if X2 is the macs and the step X2 —• X3 is irreversible, all other steps of the pathway are preceded by it (see Figure 8.5). Only that step then affects the forward rate. Among the steps that do not are cat + A <—• Xj and X^ <—• X2, and the orders with respect to any species that participate in them (and not in the irreversible one) are zero. The forward step consuming the macs corresponds to the first step in the equivalent imaginary linear pathway. This correspondence immediately yields the following rules regarding reaction orders from the rules 7.13 to 7.16 in Section 7.3.1:

®-

241

A

"^

cat-^ X

Figure 8.5. Catalytic cycle with intermediate X2 as macs (shown encircled), cut to give equivalent linear pathway starting and ending with macs X^.

A reaction is first order with respect to any reactant that participates (with one molecule) in only the forward step conconsuming the macs.

(8.57)

A reaction is of order between zero and plus one with respect to any reactant that participates (with one molecule) in a forward step other than that consuming the macs. A reaction is second order with respect to any reactant that participates with two molecules in only the forward step consuming the macs. etc.

Rules 7.17 and 7.18 regarding necessary conditions for negative reaction orders must also be reformulated. They are best expressed with reference to the equivalent linear pathway:

242

Chapter 8. Homogeneous catalysis

For a reaction order to be negative, the respective participant (reactant, product, or silent partner) must be a product in a reversible step. In the equivalent linear pathway, this step may not be the last nor be preceded by an irreversible one.

(8.58)

For a reactant or silent partner, the step sequence of the equivalent linear pathway must be such that the step in which the species in question is a product must precede that or those in which it is a reactant. Similarly, Rule 7.21 for the sequence of co-reactant entries becomes The later a reactant enters the equivalent linear pathway, the lower is its reaction order.

(8.59)

Example 8.8. Reaction orders for a hypothetical four-membered catalytic cycle with different most abundant catalyst-containing species. Consider the catalytic cycle

X,

cat

(8.60)

X.

with irreversible step X2 —• X3. The forward rate is Tp

-

(8.61)

^01^12^23^30 ^ A ^ B Q c a / ^

where © is the sum of the elements of the matrix k k k C '^12'^23'^30^B ^23^30^01 ^ A ^30^01^12^A ^ B

^10^23^30

k k k C

^10^21^30

0

0

0

0

0

'^21'^30'^oi^A

0

k k k C r

k k k C C

'^0l'^12'^23^A^B

'^03'^12'^23^B^P

^03^10^23 ^ P

^03^10^

8.7.

243

Cycles with external reactions

(Elements involving the zero reverse coefficient k22 are zero.) The concentrations of the participants other than the catalyst—reactants A and B and product P—appear in some, but not all matrix elements. Thus, in the general case of arbitrary distribution of the catalyst material over the members of the cycle, the forward rate is of order between zero and plus one in A and B, and between zero and minus one in P. However, if a macs exist, only one matrix row contributes, and some or all orders may be at one of their limits. Table 8.3 summarizes the algebraic forms of the rate equations and the reaction orders for the different possible macs. Table 8.3. Algebraic forms of rate equations and reactions orders for catalytic cycle 8.60 with different macs (collective coefficients k^ etc. are defined differently in the different equations). macs

K^k

cat

'

X,

'"p

^

_

'-B Qcal

K^ KC,

^P

X,

^'

reaction orders

rate

^

^a^B^cat

~

^a^Lcat

^aQ^Qcat ^b^A^B •*" ^c^B^P "•" %^P

A: + 1 B: between 0 and + 1 P: 0 A: 0 B: + 1 P: 0 A: 0 B: 0 P: 0 A: between 0 and + 1 B: between 0 and + 1 P: between 0 and - 1

As a comparison shows, the reaction orders derived above with the Christiansen matrix could equally well have been predicted with the rules set forth earlier in this section. The example confirms the usefulness of the rules. It also demonstrates how profoundly kinetics can be affected by the distribution of catalyst material.

8.7. Cycles with external reactions In many reactions of homogeneous catalysis, one or several linear pathways are connected to the actual catalytic cycle. The most common example is catalysis by a ligand-deficient complex, initiated by reversible ligand loss. Also in this category are certain types of inhibition, activation, and poisoning.

244

Chapter 8. Homogeneous catalysis In general terms, such networks are of the type

(8.62)

(no details of the catalytic cycle and external patiiway shown). The cycle member Xj to which the external pathway is connected may be the actual catalyst or an intermediate, and the external pathway may consist of one or several steps. Also, several external pathways may be connected to the cycle at the same or different intermediates [48]; the extensions to such cases will be obvious. For the cycle itself, the general Christiansen formula 8.30 can be used:

ni=0 K. - n \.i i i=0 ^P

=

(8.63)

e

where E° is the total amount of catalyst material in the cycle, and ® is the sum of all elements of the Christiansen matrix for the cycle. However, that equation is in terms of only the catalyst material in the cycle, to the exclusion of that in the external pathway and its end member, S, and must be expanded as will be shown now with practical examples of increasing complexity. 8.7.1. Ligand-deficient catalysts. Catalysis by metal-organic complexes often involves as a first step the loss of a ligand from the catalyst, freeing a coordinative site that can bind a reactant:

(8.64)

The external pathway cat <—^ XQ 4- L involves no species that are formed or consumed by the overall reaction. As only a reversible ligand loss it is also apt to be at least as fast as the cycle, if not very much faster. If so, it will be at quasiequilibrium if XQ is quasi-stationary, and the rate becomes

8.7. Cycles with external reactions

k-l

k-1

i»0

i=0

^="'

245

(8.65)

where D^ is the sum of the first-row elements of the Christiansen matrix for the cycle, and K^^^ = Q Q /Qat is the equilibrium constant of ligand loss. The external reaction is seen to add a term D^Ci^ IK^^^ to the denominator. This new term is proportional to the amount of catalyst metal present as the coordinatively saturated complex, cat, and reflects the fact that the external reaction retards the rate by syphoning metal out of the producing cycle into that inactive form. The ligand acts as a "silent partner" and inhibitor with a reaction order between zero and minus one. Equation 8.65 is an alternative to lumping of the saturated and liganddeficient species, as was done in the hydrocyanation Example 8.7 in the preceding section, and gives identical results with slightly less algebraic handling. Derivation ofeqn 8.65. At quasi-equilibrium of the catalyst: so that C.. = The balance of total catalyst metal is

iCJKJC,

Qca. = Q"-C,a. = C,..(CJKJC, (8.66) Of the metal in the cycle, D°, the fraction present as XQ is given by the ratio of the sum of the first-row elements to the sum of all elements of the Christiansen matrix of the cycle (see eqn 8.32), so that

c„ = (D^/s)q, Substitution into eqn 8.66 gives

Using this to replace C^o in eqn 8.63 one obtains eqn 8.65. The external pathway may consist of more than one step, possibly involving additional co-reactants or co-products. If so, each species Sj along the pathway adds a denominator term that is proportional to its concentration. The new terms are of the form DQQ/^IQKIQ, where K\Q is the equilibrium constant of the partial reaction Si -I- ... <—> XQ + ..., and 5)1 Jo is the ratio of the product of the co-reactant concentrations to that of the co-product concentrations of this reaction (primes indicate that the quantities refer to the external pathway). With m species along the external pathway:

246

Chapter 8. Homogeneous catalysis

k-l

k-1

i=0

i=0

(8.67)

I i=l

(Note that at least one $R|o will contain Q as a factor). external pathway M

For example, with an

(8.68)

K ^—^ ••• <—^ s, ^ ^ s, ^ ^ ^ X.

the additional denominator term proportional to the concentration of S2 would be ^OO(C'L/C'MA'2O).

Example 8.9. Olefin hydrogenation with Wilkinson's catalyst. Wilkinson's catalyst is a dihydrido-chloro-triphosphino complex of rhodium, H2RhClPh3, where Ph is an organic phosphine such as triphenyl phosphine [49-53]. The dominant mechanism of olefin hydrogenation with this catalyst, established chiefly by Halpem [54-56] in detailed studies that included measurements of equilibria in the absence of reactants and of reaction rates of isolated participants, backed by independent NMR studies [57] and ab initio molecular orbital calculations [58], is shown as 8.69 on the facing page (without minor parallel pathways and side reactions). The overall reaction is irreversible and its rate is independent of the concentration of the hydrogenated product. Accordingly, the reverse step Xo + paraffm —• X3 must be kinetically insignificant, so that all terms with X03 can be disregarded. Furthermore, no double-bond migration in the olefin occurs, nor does exchange of H for D on the olefm in experiments with D2 [59], This indicates that metal-carbon bond formation (X2 —• X3) is also irreversible, i.e., terms with X32 are insignificant. Without further assumptions other than quasi-stationary behavior and validity of Henry's law, eqn 8.67 applied to the network 8.69 gives: ^01 ^12^23 ^30 ^ole

Puf'Zc

(8.70)

par

where ® is the sum of all elements of the matrix \ k k k C

'^loriy^io

k k k

0

Kv^vfoxPv^

0

0

^SO^Ol^n/'Hj^ole

0

0

0

|^0l"^12"-23/'H,^ole

0

0

0

'^23'^3
8.7.

247

Cycles with external reactions

(^> E^?5 H, (cat)

pjjRhp^

(8.69)

Ph CU^Ph

pijKh

(X„)

(X,) HRh Ph

^Ph-,

(ole)

(X,)

Also, OQO is the sum of the elements of the first matrix row, and K^^ and K^ are the equilibrium constants of cat -«—• XQ + Ph and M -«—»• XQ + Hj + Ph, respectively. Written in terms of phenomenological coefficients, the rate equation is K^ohPu,^!:

(8.71)

1 + ^b<^ole+ KPn,* kfo^^Pn,-^ ^e<^Ph+ ^f<^o,e<^Ph+ MH,<^Ph+ KCo'J>^C^^

The reaction orders are: plus one in rhodium, between zero and plus one in olefin and hydrogen, and between zero and minus one in phosphine.

248

Chapter 8. Homogeneous catalysis Equation 8.70 is the most general rate equation for the network 8.69. Simplifications may be justified. At high hydrogen pressure, rhodium may exist practically exclusively in the form of its hydrides; if so, the first matrix row (for ClRhPh2) and the term DooCph /K^^t (for ClRhPhs) become insignificant, and the rate equation 8.70 or 8.71 loses all denominator terms not involving Hj and thus becomes zero order in hydrogen. Moreover, if there is reason to assume that the 7r-allyl complex (X2) and the rhodium alkyl species (X3) also contain only an insignificant fraction of the rhodium, the last two matrix rows can be omitted, and eqn 8.71 loses its k^ term as well. Originally, on the basis of batch studies without hydrogen atmosphere, sole rate control by metal-carbon bond formation (X2 —• X3) and rhodium distribution over H2RhClPh3 (M) and H2RhClPh20le (X^) was assumed. This gives a simpler rate equation of the form [55]: r

KC...Q ole-Ecat

(§72)

However, the usually observed dependence on hydrogen pressure remains unaccounted for. In more recent work on hydrogenation of butadiene polymers and copolymers, the attempt was made to explain the dependence on hydrogen pressure with sole rate control by olefin addition to H2RhClPh2 (Xj) and quasi-equilibrium rhodium distribution over the complexes with and without hydrogen [60] instead of kinetic significance of the step XQ + H2 —> Xj. This gives a rate equation for double-bond disappearance of the form

Olefin addition is normally fast, but could conceivably be slow if the olefin is a polymer. However, a fast metal-carbon bond formation is an unlikely assumption. The observed, unusual, close-to-first order dependence on olefin (in this case, degree of unsaturation of the polymer) is accounted for by eqn 8.73, but can be explained in a less contrived fashion with the more general equation 8.70: If X2 and X3 are lacs and the first step of the cycle is at quasi-equilibrium (^I2CH2 « k^^, all denominator terms containing the olefin concentration become negligible and the reaction orders with respect to H2 and phosphine remain unchanged, even with XQ being a lacs in accordance with spectrophotometric evidence. More importantly, a fit of the constants in eqn 8.73 to the observed rates with the assumption of sole rate control by olefin addition would require an equilibrium rhodium distribution favoring the unsaturated and notoriously labile complex ClRhPh2 even at fairly high hydrogen pressures and phosphine concentrations. This is at odds with spectrophotometric evidence, molecular-orbital calculations, and the observation that hydrogen uptake by ClRhPh3 is fast and stoichiometric.

249

S. 7. Cycles with external reactions

It has been said that all members of the catalytic cycle remain at concentrations too low to be detected [61]. However, if all were lacs, ® in the denominator of eqn 9.70 would have to be negligible compared with the other two terms, resulting in a rate that is of order minus one in phosphine. That is contrary to most observations. 8,7.2,

Inhibition, activation, decoy, and poisoning

Inhibition. As seen in the previous subsection, the presence of excess ligand reduces the reaction rate produced by a ligand-deficient catalyst because it syphons catalyst material from the active cycle into the inactive external pathway. This can be viewed as a special case of inhibition: Although a necessary ingredient of the catalyst system, the ligand depresses the rate.* More generally, any substance that reduces the rate through removal of catalyst material from the cycle by reaction with one of the cycle members is called an inhibitor. The inhibitor may react with the true catalyst XQ, i.e., the cycle member that initiates the reaction by binding the reactant A. This is called competitive inhibition because inhibitor and reactant compete directly for a binding site on the catalyst. Alternatively, the inhibitor may react with another cycle member Xj. This is called noncompetitive inhibition. A

X,

inh X.

-> S

noncompetitive inhibition

competitive inhibition

For competitive inhibition with single-step external pathway, eqn 8.65 is directly applicable, except that the inhibitor concentration replaces that of the free ligand. For noncompetitive inhibition by reaction with Xj, DQQ must be replaced by Djj, the sum of the elements of row j + 1 of the Christiansen matrix. Both cases are covered by:

r^

=

k-l

k-l

nx.

i=0

i=0 ^

jj

CEcat (j may be zero) inh

(8.74)

inh

* The conclusion "lowest ligand concentration, highest rate" would be unwarranted, however. At too low a ligand concentration, the catalyst loses more than one of its ligands and either becomes inactive (if another molecule can occupy the coordinative site) or decomposes.

250

Chapter 8. Homogeneous catalysis

Here, K^^ is the equilibrium constant of the inhibition reaction inh + Xj—• S, and index j refers to the cycle member with which the inhibitor reacts (j = 0 for competitive inhibition, and j p^ 0 for noncompetitive inhibition). The extension to external pathways consisting of more than one step is as in eqn 8.67. In both competitive and noncompetitive inhibition, the reaction is of order between zero and minus one with respect to the inhibitor. However, there is a kinetic difference between competitive and noncompetitive inhibition. In the former, the action of the inhibitor can be effectively countered by an increase in reactant concentration; direct competition by the reactant for a catalyst site can "crowd out" the inhibitor. In noncompetitive inhibition, this is not the case; even a large excess of reactant does not impair the inhibitor's access to the cycle member Xj. [Mathematically, in competitive inhibition the new and retarding denominator terms have DQQ as factor, the sum of the first matrix row and only row that lacks the coefficient XQ^, the only coefficient with CA as co-factor. In contrast, in noncompetitive inhibition the terms have Djj as factor and contain XQI and thus C^ as co-factor; the result is that an increase in CA, apart from a direct beneficial effect on the rate, also strengthens the adverse effect of the noncompetitive inhibitor.] If the reaction involves two reactants that enter the cycle at different places, inhibition may be competitive with respect to one and noncompetitive with respect to the other. Also, there may be two or more inhibitors reacting with the same or different cycle members. The formulas given here are easily extended to such cases. Various other forms of inhibition are possible. The simplest of these is reactant inhibition, in which an inhibitor competes with the catalyst for the reactant, reducing the latter's likelihood to enter the catalytic cycle. As an example, consider reactant inhibition in a three-membered cycle whose last step is irreversible: S < -^ » A <(8.75)

cat p

<—

^X,

The quasi-equilibrium condition for the inhibition reaction is K,^ = Cg /C^^Q^ and the material balance for A is Q A = CA + Cg. Combination of these conditions shows that the fraction of total A that is still free to react with the catalyst is

(EA = total A including portion bound in S). In effect, the reaction rate is retarded by a factor 1/(1 + ^inhCinh) ^^d thus is of order between zero and minus one with

8.7. Cycles with external reactions

251

respect to the inhibitor. For example, for a reaction with network 8.75 (threemembered cycle, last step irreversible), the rate is given by k k k C

C

(8.76)

where S is the sum of the elements of the Christiansen matrix ^12^20 ^20^01 ^ A

KKC,

^10^20

^10^21

^21^01 ^ A

^

0

0

A much more comprehensive and in-depth discussion of different types of inhibition can be found in Segel's book Enzyme Catalysis [62]. Activation. Activation is the opposite of inhibition: The reaction rate is increased rather than decreased by a silent partner. The catalyst may require activation to function at all, as in a network

(8.77)

in which the true catalyst is XQ and is produced by reaction of the catalyst species charged, cat, with the activator, act. This is called essential activation. Application of the general equation 8.65 to the network 8.77 gives k-l i=0

k-l

nx,

•'Scat

(8.78)

i=0

S - DJK^C^ The rate is of order between zero and plus one with respect to the activator. The additional denominator term reflects the fact that some catalyst material is present as the not yet activated species, cat. This retarding term is infinitely large if no activator is present, and decreases with increasing activator concentration. A much more detailed discussion of activation mechanisms is given in Segel's book [63].

252

Chapter 8. Homogeneous catalysis

Decay and poisoning. In all situations studied to this point, the external pathway was diat of a reversible reaction fast enough to become quasi-stationary. Of course, the reaction may also be irreversible. If so, catalyst material that has left the cycle does not return and is permanently lost, and the reaction eventually comes to a stand-still unless the catalyst is replenished. If the loss of catalyst material is fast, the system obviously is of little or no practical interest. In contrast, slow irreversible loss is a problem the practical engineer often has to contend with. Any member of the catalytic cycle, or more than one of the members, may be subject to the loss reaction. That reaction may be a rearrangement into an inert form, a breaking apart into inert fragments, or a reaction with another substance to yield an inert species. The first two cases are called decay; the last one, poisoning. Inasmuch as catalyst loss in any practical situation is very slow compared with the catalytic reaction, the quasi-stationary behavior of the latter is not significantly impaired. Accordingly, it makes no difference in kind which of the cycle members is (or are) decaying or being poisoned: The principal result in any event is a slow drain of catalyst material from the producing cycle. A practical approach to quantitative kinetics of catalyst systems subject to decay or poisoning is to model the catalytic reaction as though no loss occurred but, unless the loss is compensated by make-up, to make the total amount of catalyst material time-dependent in accordance with the loss reaction. Often, little is known about the exact cause and mechanism of loss, and an empirical loss rate is the best one can come up with. In the rare instances in which the exact loss reaction has been identified, a more detailed modeling is possible, as a specific example will illustrate. Let us say, the loss reaction is a single step Xj + poi —• S, where poi is the poisoning substance. The loss rate then is and equals -dC^^^Jdt if there is no make-up. The loss rate involves the concentration of the affected cycle member, Xj. According to eqn 8.32 this concentration is C = D..q./S (8.79) J

jj

^

where D^^ is the sum of the elements of row j +1 of the Christiansen matrix for the cycle, S is the sum of all elements of that matrix, and E° is the total catalyst material in the cycle. Proper modeling requires knowledge of the affected cycle member and such an approach. This is because the distribution of total catalyst material over the cycle members varies with conversion. For instance, with saturation kinetics, most of the catalyst material may be in the form of a reaction intermediate as long as the reactant concentration is still high, but will then shift to the free catalyst as the reactant is used up.

8.8. Multiple cycles

253

8,8. Multiple cycles* Catalyst systems with more than one cycle are not unusual. There is a wealth of possible configurations, and only some that are relatively simple and often encountered can be discussed here. The examples are selected for their suitability to illustrate the application of the tools developed in this chapter. With a good understanding of these tools, the reader should have no problems in deriving the mathematics for situations of his that are not covered here. 8.8.1. Competing reactions (cycles with common member) In practice, one and the same catalyst often catalyzes more than one reaction of the same or different reactants. An example is hydrogenation of a mixture of unsaturated compounds. Such networks are of the general type

(8.80)

here shown for just two reactions of different reactants A and B to products P and Q, respectively (details of cycles omitted). If the free catalyst, cat, is the macs, the reactions proceed independently side by side. However, if significant amounts of catalyst material are present in the form of other cycle members, catalyst material in one cycle is not contributing to the reaction in the other. In effect, the two reactants then inhibit one another. The general rate equation for the cycle converting A to P (with m members) can be written m-l

m-l •i,i+l i=0 ^P

^Ecat i=0

=

(8.81)

(^,.iDjD;,)ZD; Here, primes refer to the B —• Q cycle (with n members), S? is the sum of all elements of the Christiansen matrix for the A —• P cycle, and the DQQ are the sums

* The mathematical derivations in this section are based largely on work by Chen and Chern [64].

254

Chapter 8. Homogeneous catalysis

of the first-row elements of the respective matrices. Note that the summation over the DQO /Dj'i does not include D^ (sum of first-row elements of matrix of B —• Q cycle). The second denominator term reflects the inhibition by the other reactant; it becomes negligible if no member of the B —• Q cycle except XQ contains a significant amount of catalyst material. In a network such as 8.80, conversion of one reactant is inhibited by the other if, and only if, at least one intermediate in the cycle involving the former contains a significant amounts of catalyst material. Experimentally observed inhibition thus constitutes evidence that this is indeed the case. Derivation ofeqn 8.81. In an ordinary, single, k-membered cycle, eqn 8.32 can be used to replace Q^at in the general rate equation 8.30 to give: r,

=

EC./D..

(j=0,...,k)

(8.82)

where

s - nx,,^, - u\^,,

(8.83)

After rearrangement, eqn 8.82 gives C. = r^D..IE

(j=0,...,k)

(8.84)

The overall balance of catalyst material is m

n

i=0

i=l

(note the second summation does not include Q ) . With eqns 8.84 and noting that the summations over the Dy^ and DjJ give S and ©', respectively, one obtains Qca, =

'•pS/S + r ^ S ' / S '

(8.86)

Dividing eqn 8.82 for the A —• P cycle by that for the B —• Q cycle and solving for TQ, recognizing that Q (concentration of cat) is the same in both cycles, one finds TQ = rpDooSVDJoS

(8.87)

This allows TQ in eqn 8.86 to be replaced. Solving the resulting equation for r^ yields eqn 8.81. The member which two cycles have in common need not be the free catalyst. Equation 8.81 then applies with the appropriate changes in indices. The example to follow illustrates such a case.

255

8.8. Multiple cycles

Example 8.10. Osmium tetroxide-catalyzed asymmetric dihydroxylation of olefins. Substituted olefins such as styrene can be dihydroxylated to give vicinal diols: ...-CH = CH2 + RO + H2O -CHOH-CH.OH + R The reaction, usually carried out in aqueous acetone and with an amine oxide as the oxidant RO, is catalyzed by osmium tetroxide, activated by an alkaloid ligand, L. To explain the higher enantio-selectivity achieved at lower olefin concentrations, a network with two cycles having a common member has been proposed [65,66]:

(8.88)

-o (X3)

RO The upper cycle is assumed to produce high enantio-selectivity while the lower one does not. The reaction is irreversible because of irreversible diol release. The catalyst, Xi, and X3 have been identified and synthesized, X2 and X4 have not. The step X4 + H2O 4- L —• X2 + diol may actually be two steps in rapid succession. For the network taken at face value, eqn 8.81 with appropriately changed indices, irreversible diol release steps, and X2 and X4 as lacs (so that S = D^ 4- D^ and S' = 1)33) gives

(8.89) k k k

D^^D,, + {D;JD,, )D,,

^34^42^23 ^ L

D,,^

(D,,/D\,HD^^D,,)

c c c c

256

Chapter 8. Homogeneous catalysis

with D

- k k C C

^11

"^

+ k k C

+k k C

^20^01 ^H,0 ^ole "^ ^21 ^01 ^ R ^ole

n

- k k r r

^11

~

'^0l'^12^ole^RO

^22

=

^ 3 4 ^ 4 2 ^ R o Q i p ^ L + ^32 ^42 ^ L Ql^O "^ ^32^43 ^ t O i

^33

"^

^42 ^23 ^HjO Q ^ole "^ ^43 ^23 Q

^ole

The reaction orders are: plus one for osmium; between zero and plus one for olefin, water, and oxidant; between plus one and minus one for the alkaloid ligand; and between zero and minus one for the spent oxidant (zero if oxidation is irreversible). Because only the upper cycle gives high enantio-selectivity, the yield ratio of the two cycles is of interest. From eqn 8.87 with appropriate change in indices and after cancellations: 2^22

_

S*D

~

^20(^34^42^,0^0

"^ ^32^42^L^H,0 "*" ^ 3 2 ^ 4 3 ^ R )

k k k C T

'^ ^ 2 2

(8.90)

'^34N2'^23^oIe'^RO

The yield ratio is inversely proportional to the olefin concentration and increases less than proportionately with increasing concentrations of water, ligand, and spent oxidant and decreasing concentration of oxidant. It has been said that the reaction is first order in olefin. Equation 8.89 shows that, if the network 8.88 is correct, this can be true only if X^ and X3 are also lacs (to make D^ and D33 negligible) and Td-oi « ^^40,. That may well be the case if the olefm concentration is kept very low to achieve high enantio-selectivity.

8.8,2.

Dual- and multiple-form catalysts (connected cycles)

A catalyst may exist in two or more forms with different catalytic activities. In the simplest systems of this type, two such forms interconvert in a quasi-equilibrium step. The conversion may or may not involve other species. It may, for example, be a ligand exchange. The two catalyst species may catalyze the same reaction or different ones. A network of this type, with ligand exchange and different reactions A —> P and B —> Q, is

Li

L2

(8.91)

257

8.8. Multiple cycles The rate equation for the m-membered A • as eqn 8.81, is

P cycle, derived along the same lines

m-l

m-1

i=0

i=0

(8.92)

e ^ K^.(CJQ,)(DJD'

)©'

where ® and S ' are the sums of the elements of the Christiansen matrices of the two cycles, and K^^^ is the equilibrium constant of the ligand exchange reaction cat + Li <—^ cat' + L2. The equation for the other cycle is, of course, analogous. The following example describes a slightly more complicated catalyst system of this type. Example 8.11. Hydroformylation with phosphine-substituted cobalt hydrocarbonyl catalyst. The phosphine-substituted cobalt hydrocarbonyl catalyst used for hydroformylation of olefins has been described in Section 8.2 (see network 8.13). The principal reaction olefin + H2 + CO

aldehyde

follows the Heck-Breslow mechanism 6.9 shown in Section 6.3 (see also Section 7.3.2) and is catalyzed by both HCo(CO)3Ph and HCo(CO)4, where Ph is a tertiary organic phosphine ligand. Stripped of complications—olefin isomerization, formation of isomeric products and paraffin, subsequent aldehyde hydrogenation, and ligand exchange of cycle members—but with the catalyst equilibria, the network is:

(main cobalt-containing species shown in larger font; B~ and HB are added base and its conjugate acid, respectively; both cycles produce both aldehyde isomers, but only the predominant product is shown; for details of the catalytic cycles, see network 6.9).

258

Chapter 8. Homogeneous catalysis According to spectrophotometric evidence, all cobalt-containing species except HCo(CO)3Ph and Co(CO)4" are lacs under typical reaction conditions. Although a lacs, HCo(CO)4 nevertheless contributes to olefin conversion because its catalytic activity is several orders of magnitude higher than that of HCo(CO)3Ph. Within its own cycle, HCo(CO)4 is the macs. The rate is the sum of the rates produced by the two cycles. For each cycle, the rate is given by a one-plus equation of the form k C, C

(6.12)

' oM

1 -^

KPCO'P^.

(Martin equation, see Example 6.2 in Section 6.3). The total catalyst balance, if including only the species containing significant fractions of cobalt, is ~

^HCo(CO)3Ph

(8.94)

••" ^Co(CO);

With this and the equilibrium conditions for HCo(CO)3Ph -h CO <-^ Co(CO)4- + Ph and HCo(CO)3Ph-hCO + B" ^i~^ HCo(CO)4 + Ph +HB C

C

C

= K.

^HCo(CO), ^ P h "HCoCCOjPh

and

C C

^Co(CO),-^Ph^HB

Pco

= K.

(8.95)

•'HCoCCOjPh PCO^B

respectively, one obtains C ^HCo(CO)3Ph

^HCo(co),

c c

= C

^ E C o ^HB^Ph

^ECo ~p—"p

^

^iPcO^h

7~jFT»—ir~

^HB*^Ph ^

^iPcO^B

With these substitutions for HCo(CO)3Ph and HCo(CO)4 and eqn 8.94, the overall rate as the sum of the rates of the two cycles is, after cancellations: ^o ^P1

1 + KPCO^PH,

K^iPc 1 ^

KPCO'PH,

c c c ^HB^Ph ^

(8.96)

^iPcO^B

(primes refer to the catalyst cycle with HCo(CO)4). According to eqn 8.96, the reaction orders are: plus one for total cobalt; between zero and plus one for H2 and HE; between zero and minus one for CO and B~; and between minus one and plus one for phosphine. Although the reaction is first order in cobalt, it may be of higher or lower order than first in "catalyst" if the concentration ratios of the catalyst ingredients cobalt, phosphine, and base are held constant. This has been discussed in Example 8.3 in Section 8.2.

8.8. Multiple cycles

259

8.8.3. Reactions with multiple products (cycles with common pathway segments) Many catalytic reactions form a range of products rather than only a single one. In most such cases, the pathway to a co-product branches off from that to the main product after the first or a few early steps. The network then consists of cycles that have a step or pathway segment in common. Typical examples are the formation of isomeric products in paraffin oxidation and olefin hydration, hydrohalogenation, hydroformylation, and hydrocyanation, as well as paraffin by-product formation in hydroformylation. The simplest networks of this kind consist of two cycles with an initial, common pathway of k steps:

(8.97)

(details of pathway and cycles not shown). In general form, the rate equations are k-l

(\o^K )n\w u - (K+K )n^-i+i'i i=0

i=0

(8.98)

E^n-E (Ako-Du - Aok^oo)Q k

m-l

EA.-E E I i=0

i=k+l

^i^-^kk-^^Ji-^oo ^ik+^iO

(8.99)

^ik-^^io

and an analogous equation for TQ.* Here, the X coefficients are those of the common pathway cat + A <—> ... ^^—^ X^; the A and A' are the segment coefficients * It is not obvious that these equations meet the stoichiometry requirement - ^A = ''p + ^Q- That they do can be verified by replacement of the A and A' coefficients and comparison of the equations for - r^ and rp + KQ after resulting cancellations in the latter.

260

Chapter 8. Homogeneous catalysis

of the pathways X^ <—> ... <—> cat + P (with m members) and X^ <—• ... <—^ cat + Q (with n members), respectively (see eqns 6.5 for segment coefficients); and the Dii are the sums of the elements of the rows of the Christiansen matrix of the "collapsed" network 8.104 (single cycle with loop coefficients for the parallel pathways, see below). For example, for a two-step common pathway (k=2): DQQ

-

Aj2°^20 ^ ^10*^20 "^ ^ 1 0 ^ 2 1

^11

=

' ^ 2 o \ l •*• ^ 2 1 ^ 1 +

^22

~

\ l ^12 "^ "^02 ^12 "^ -^02 ^10

(8.100)

^1^02

with loop coefficients ^20

=

A20

+

A2Q

,

OLQ2

=

AQ2 +

AQ2

(8.101)

In eqns 8.98 and 8.99, the denominator terms are proportional to the concentrations of the catalyst-containing participants: the D^i of the first sum to the members of the common pathway, the terms of the second sum to the intermediates along Xk <—• ... <—^ cat + P, and those of the third sum to the intermediates along Xk <—• ... <—> cat + Q. If any of these participants are lacs, their terms can be dropped. Often, one product (say, P) is desired and the other is a by-product that must be removed and disposed of. If so, the P-to-Q yield ratio, YpQ, and the selectivity to P, 5p, are of interest (see Section 1.6 for yield, yield ratio, and selectivity). From eqns 8.98, 8.99, and the analogous equation for TQ: Y

^ko^kk - ^ok^oo

=

^ko ^kk - K

(8.102)

^00

\o^kk - ^ok^oo

s

»jp .

A '

k-1

k-1 \ TT\

/A

i=0

(8.103)

.^ A ' i=0

Derivation of eqns 8.98 and 8.99. The network 8.97 can be "collapsed" in a single cycle with the k-hl members of the common pathway and a pseudo-single step converting Xj, to cat and P or Q:

P or Q ,^ ^

^ ^

^

^ ^ ^

(8.104)

8.8. Multiple cycles

261

(see Section 6.4.1 for reduction of networks with loops, and eqn 6.15 for loop coefficients). The concentrations of the network members are obtained as follows: Application of eqn 8.84 to the members of the common pathway gives C

=

^L^

'

k-l

k-1

i=0

i=0

(0
(8.105)

^

V

/

/

The concentrations of the members of the cycles can be found with a relationship expressing the concentration of an intermediate in a pathway as a function of those of the end members. Use of the Bodenstein approximation for any intermediate Xj in a pathway Xj <—^... M-^ X^ (reduced to two pseudo-single steps) yields C

=

^y^^-i^ ^^^^^ A , . A,,

(j
(8.106)

(k
(8.107)

Applied to the pathway X^ •*—>•... •*—*• cat + P: ^

^

KjC^^

KA.

With eqns 8.105 to replace Q and Qat this gives C. =

-'-AJD^K, ^

(k
'POQAQJ)

(8.108)

k-l

^.oii ^.i.. - ^o.n K i=0

(A„ - A,„)

The equation for the intermediates in the other pathway, X^ ^<—•... <—• cat + Q, is analogous. The rate —r^ can now be obtained from the catalyst balance and the concentrations. The catalyst balance is k

m-1

n-1

Q.. = Ec.^Ec, ^Ec,' i=0

i=k+l

(8-109)

Using eqns 8.105 to replace the concentrations under the first sum, eqns 8.108 and their equivalents for the other cycle to replace those of the second and third sums, respectively, then replacing the loop coefficients by means of eqns 8.101, and finally solving for —r^ one obtains eqn 8.98. The rate r^ is that of the pathway X,, <--^... <—• cat 4- P: '-P ^

A,q-A„,C^,

With eqns 8.105 for Q and Qat and eqn 8.98 for - r ^ this gives eqn 8.99.

(8.110)

262

Chapter 8. Homogeneous catalysis

This derivation illustrates how, in general, rate equations for more complex networks with common pathway segments can be compiled. The essential steps are: (1) reduction of the cycles other than common pathways to pseudo-single steps with loop coefficients, to obtain a single-cycle collapsed network; (2) use of eqns 8.105 or their equivalents to relate the concentrations of the members of common segments, including the node members, to the overall rate through the collapsed network; (3) use of eqns 8.108 or their equivalents to relate the concentrations of the intermediates in loops to those of the node members and thereby to the overall rate; (4) use of the catalyst balance to obtain the overall rate from the so obtained concentrations of all catalyst-containing participants; (5) establishment of formation rates of individual products from equations equivalent to 8.108 for the respective segments, the concentrations of the segment end members, and the overall rate; (6) replacement of the loop, segment, and X coefficients by the respective true rate coefficients and reactant and product concentrations. Note that it may be possible to consolidate steps within segments in accordance with Rule 8.55. This can reduce the algebra considerably. Example 8.12. By-product formation in hydrocyanation of 4-pentenenitrile [44]. Hydrocyanation of 4-pentenenitrile (4-PN) to adiponitrile (ADN) was examined in some detail Example 8.7. The du Pont process maximizes the yield of adiponitrile, the desired product, by addition of a Lewis acid such as zinc chloride or trialkylborane, but nevertheless some 2-methyl-glutaronitrile (2-MGN) is formed as byproduct:

HCN +

^

(ADN)

(8.111)

To account for the by-product formation, the network 8.49 must be expanded. The branch is at the rearrangement X2 <—> X3 of a 7r-complex to a species with metalcarbon a-bond:

8.8.

Multiple

cycles

263

Ph CN Ph CN HNi Ph \ ^

(Ph = organic phosphine). product formation is:

(8.112) XN

Ph CN Ni^^^CN Ph

Accordingly, the expanded network that includes by-

(cat)

Ph Ph Ni Ph Ph

(8.113)

(X)

Ph (ADN)

O^e)

Ph -> Ni ^ (Y) Ph Ph

Ph Ph Ni ^^NC.

k"

^CN

Ph

(X,) Ph Ni Ph NC.

^CN (X^)

Ph CN Ni ^ ^^,, Ph -^^CN

Ph Ph ,y,. Ni ^^^ ^ Ph ' • NC^-^CN

Y-->Ph Ph ^ Ni Ph •• NC^/^^CN

Ph

Ph CN HNi Ph I

-> (X,)

HCN

(2-MGN)

Ph CN HNi Ph Ph

K

(XJ

NC^^^^-v^CN ( \ -^

HNi Ph (X,)

(x;)

(4-PN)

XN

Ph CN

(x;)

264

Chapter 8. Homogeneous catalysis Equations 8.98, 8.99, 8.102, and 8.103 are directly applicable to the rates of 4-pentenenitrile consumption, adiponitrile formation, yield ratio, and selectivity to adiponitrile, with k = 3 and lumping of cat and XQ into X as in Example 8.7 (with index 0 in the equations replaced by X). The reaction is irreversible, so the negative terms in eqns 8.98 and 8.99 drop out. Species X2, X5, X5, Xg, and Xg are lacs and, therefore, D22 is zero (as are 1)55, D55, D^g, and Z)^, not needed in eqns 8.102 and 8.103). With X54 and X54= 0, the loop and segment coefficients become:

X3X

^4^5

=

^3X

££

^3X + ^3X '

=

A'3X

X45 + X,3

= 0

=

Tp ^'''^''' ^ ^ y A 45 + A43

and the Du are =

^XX

( ^ 1 2 ^ 3 -^ ^1X^2:J + ^ i x \ i ) ^ 3 x -^

^ I 1

=

^Xl ( ^ 2 3 - \ l ) ^ 3 X " ^

D2 2 = 0 '

\ x \ \ \ :

\ l ^ l \ 2 '

^ 3 3 = ^X1^12^32

where ^2>x is given above and the X coefficients are the same as in Example 8.7, with the X' analogous to the respective X (see eqns 8.100 and 8.101). With all these substitutions, explicit equations for 4-pentenenitrile consumption and adiponitrile formation are obtained from eqns 8.98 and 8.99. The equations are more lengthy, but of same algebraic form as eqn 8.53 for the network 8.49 without by-product formation. The adiponitrile-to-2-methyl glutaronitrile yield ratio and the selectivity to adiponitrile are found to be y

_ ADN/MGN

^3x -J-TA 3j^

^3X^33

(A3X+ A'3x)XxiX^2X23

_

^34^45(^5 ^ ^43 ) W h' (k + k ) /C34 AC^g \^K^^ + /C43;

1 +

^34

45 ^ 4 5 "^ ^ 4 3 /

^34^45(^45

••" % 3 )

(see eqns 8.102 and 8.103) and are seen to be concentration-independent. Not surprisingly, the best leverage is an increase of the k23:k23 ratio for the steps at the branch, but increases in ^34:^34 and ^32:^32 also help. In this example, the rates in the networks with and without by-product formation were found to be of same algebraic form and the yield ratio and selectivity to be concentration-independent. This is due to the concentration independence of the segment coefficients of the two parallel pathways, A2X and A2X and is not generally true even if the different pathways consist of strictly analogous steps.

8.9. Self-accelerating reactions (autocatalysis)

265

The situation analyzed here in detail is common in metal-complex catalysis of additions to olefmic double bonds. As a rule, the olefin adds to the metal at a coordinative site, forming a 7r-complex. This is followed by a rearrangement with formation of a a-bond between the metal and either of the two double-bond carbon atoms (see reaction 8.112). In most cases, the rearrangement yields two isomeric species, depending on which of the two carbon atom forms the a-bond. Insertion of another ligand (CN in hydrocyanation, CO in hydroformylation, etc.) between metal and carbon and release from the catalyst then yields two isomeric final products. Moreover, if both 7r-complex formation and a-bonding are reversible, the catalyst also promotes double-bond migration within the olefin (not accounted for in the network 8.113). The new olefin isomers may then give rise to further isomeric final products. A typical example of this is hydroformylation of straightchain olefins (see Example 5.3 in Section 5.3 and network 7.40 in Section 7.4).

8.9. Self-accelerating reactions (autocatalysis) In most reactions, the rate decreases monotonically as conversion progresses. This is not universally true, however. In exceptional cases, the rate increases with conversion, though usually only within a certain range. Of greatest concern in practice is the possibility that such self-acceleration may lead to run-away or an explosion. In large-scale equipment, the most common cause of self-acceleration is temperature increase owing to the inability of heat transfer to keep up with heat generation by the reaction (thermal explosion). This is a problem of reactor design, is well covered in reaction engineering texts, and will be alluded to in Section 14.2. Here, we shall confine ourselves to reactions whose rate increases with progressing conversion even at constant temperature. Self-acceleration of this kind is commonly called autocatalysis. Although genuine catalysis might not be involved, it seems appropriate to discuss such behavior in the context of catalytic reactions. 8.9.1. Product-promoted reactions The typical textbook example of autocatalysis is that of a hypothetical single-step reaction with stoichiometry A <—• P, but mechanism A -h P <-—• 2P

(8.114)

and rate r

- k C C - k C^

(8.115)

[67-72]. The rate increases with progressing conversion up to a maximum and then decays toward zero as equilibrium is approached.

266

Chapter 8, Homogeneous catalysis

Reaction 8.114 is genuine autocatalysis because the product P acts as catalyst. It is a strange reaction, though, because the product could never come into being if it were not present at start in at least a trace quantity (or could arise in some other fashion). The simple, single-step mechanism does not correspond to any known reaction. Its mathematical behavior reflects the most typical symptoms of autocatalysis, but is of little quantitative value. The most common example of genuine autocatalysis in real chemistry is acidcatalyzed ester hydrolysis OT?'

RC^

OH

+ H2O

•

RCQ

+ R'OH

This is a two- or three-step catalytic reaction whose kinetics depends on the type of ester [73-75]. The acid produced and hydrogen ion from its dissociations add to the amount of catalyst initially present, and there may or may not be a slow thermal reaction in parallel [68]. A different and more common type of self-acceleration results from promotion by an intermediate. Although habitually called autocatalysis, this is not genuine catalysis because the intermediate is consumed, whereas a catalyst is defined as a substance that accelerates the reaction, but re-emerges in its original state. The simplest example of this kind is the hypothetical reaction [76]

A

A • K -^^=-> P

in which the intermediate, K, promotes further conversion of the reactant, A. (This can also be classified as competing steps, see Section 5.5.) A real-life reaction involving promotion by an intermediate is oxidation of hydrocarbons. Here, the principal reaction starts with RH + O2

OH • RQ

• products

Formation of the hydroperoxide is a chain reaction requiring radicals as chain carriers. These are formed by an initiation mechanism, but additional ones arise from decomposition of the hydroperoxide and may accelerate conversion. Hydrocarbon oxidation will be examined in more detail in the chapter on chain reactions (see Section 10.6.2). A situation of this general kind—a complex mechanism involving transient species that accelerate the main reaction—often is behind what goes as autocatalytic behavior. "Autocatalysis" as commonly understood, namely, reactions whose rates increase with progressing conversion at least within a certain range, is a broader term than its literal meaning implies. However, like many others in our technical vocabulary, this usage has become too firmly ingrained for change.

Summary

267

The variety of reactions in which a product or major intermediate boosts the rate, possibly while itself decomposing, is great, and each of them has its own mathematics and quirks. It seems impossible to develop universal equations that cover such behavior in a quantitative fashion, and no attempt to do so is made here. Self-acceleration by a product or intermediate is a key factor in periodic or chaotic reactions, a topic to be visited in Chapter 14. 8.9.2.

Reactant-inhibited

reactions

Rather than because of promotion by a product or intermediate, the rate may accelerate because a reactant that acts as inhibitor is consumed. For example, a small amount of inhibitor present initially may depress the rate of a chain reaction until used up (see Section 10.8). More interesting are reactions inhibited by one of the principal reactants (called substrate-inhibited in biochemistry parlance). An example is hydroformylation, in which CO is a reactant with negative reaction order (see Example 6.2 in Section 6.3). There is a subtle but important difference between product-promoted and reactant-inhibited reactions: The rate of a productpromoted reaction builds up to a maximum and then declines as reactant depletion overpowers product promotion. In contrast, the rate of a reactant-inhibited reaction keeps escalating, possibly catastrophically, until the respective reactant is almost completely exhausted. Typically, some other mechanism then takes over. [The negative apparent reaction order of the respective reactant arises from an additive denominator term in a one-plus rate equation, but the other terms may be small or insignificant by comparison.] Possible mass-transfer implications of such behavior will be examined in Section 13.3. Summary Homogeneous catalysis can be classified into single-species and complex catalysis, although the distinction is not always clear-cut. In the former, a single molecule or ion acts as the catalyst; in the latter, the catalyst is a system of several species that interconvert into one another and differ in their catalytic properties. A further complication arises if significant fractions of the total catalyst material may be present in the form of reaction intermediates rather than free catalyst. If so, the concentration of the free catalyst is not known and may vary with conversion. Rate equations that, instead, contain the known, total amount of catalyst material are then needed. In single-species catalysis, the rate laws for noncatalytic reactions apply, the only difference being that the catalyst appears as both a reactant and a product. In catalysis by highly concentrated acids, anomalies may appear: Protonated species other than HjO"^ (or protonated solvent in non-aqueous media) may arise and act as additional catalysts; this can be accounted for with the Hammett acidity function. Also, the rates of reactions such as hydration and hydrolysis may decrease with further increase in acid concentration because of reduced availability of free water as reactant.

268

Chapter 8. Homogeneous catalysis

In acid-base catalysis, both an acid (or base) and its conjugate base (or acid) take part in different reaction steps and are eventually restored. Such reactions are first order in acid (or base) if the link-up with that species controls the rate, or first order in H^ (or OH") if a subsequent step involving the conjugate base (or acid) does so. Traditionally, the first alternative is called "general" acid or base catalysis; the second, "specific" acid or base catalysis. However, this distinction is not always applicable as there may be no clear-cut rate-controlling step, and reversibility of later steps may produce a more complex behavior. Many reactions of organic chemistry are catalyzed by complexes of transition-metal ions, most notably those of Group VIII. Here, the catalyst is a system of different complexes that are linked by ligand-exchange and ligand-dissociation equilibria and differ in their catalytic properties, occasionally with rather counterintuitive results such as a decrease in rate with increase in pressure in a reaction in which gas is consumed. For historical interest and to illustrate a general facet of systems with arbitrary distribution of catalyst material over free catalyst and reaction intermediates, the classical models of enzyme catalysis are briefly reviewed. They show "saturation kinetics:" An increase in reactant concentration causes a shift of catalyst material from free catalyst to an intermediate, so that the rate has an asymptotic limit that can at most be approached even at the highest reactant concentrations. A general formula for single catalytic cycles with arbitrary number of members and arbitrary distribution of catalyst material has been derived by Christiansen. Unfortunately, the denominator of his rate equation for a cycle with k members contains y? additive terms. Such a profusion makes it imperative to reduce complexity. If warranted, this can be done with the concept of relative abundance of catalyst-containing species or the approximations of a rate-controlling step, quasi-equilibrium steps, or irreversible steps, or combinations of these (the Bodenstein approximation of quasi-stationary states is already implicit in Christiansen's mathematics). In some fortunate instances, the rate equation reduces to a simple power law. If significant fractions of the catalyst material may be bound in the form of reaction intermediates, the rules for reaction orders in noncatalytic simple pathways no longer apply. However, if one of the cycle members—^the free catalyst or an intermediate—is a macs (most abundant catalyst-containing species, containing practically all of the catalyst material), the rules for noncatalytic pathways can be adapted: The rate equation and reaction orders for the cycle are the same as for an imaginary equivalent linear pathway that starts and ends with macs. A cycle member that contains only an insignificant fraction of catalyst material is a lacs (low-abundance catalyst containing species), and the denominator terms it contributes can be dropped. In practice, catalytic networks often involve more than a single cycle. In a very common type of network, a linear pathway is attached to the cycle. This is so, for instance, in reactions with ligand-deficient catalysts. Here, the coordinatively saturated "catalyst" has no activity but, rather, must first lose one of its ligands to provide access for a reactant. Although its presence is necessary, the ligand then acts like an inhibitor. Other examples of cycles with attached pathways include systems with inhibition, activation, decay, and poisoning. Also, the network may consist of two or more cycles

References

269

with a common member or pathway. This situation is typical for reactions yielding different isomeric products. The Christiansen formula is extended to cover such cases. In some reactions, the rate increases rather than decreases as conversion progresses. This is loosely called autocatalysis, although no genuine catalysis may be involved. The acceleration may stem from promotion by a product or major early intermediate, or from consumption of a reactant that functions as inhibitor. In product-promoted reactions, the reaction order with respect to a product (or early intermediate) is positive. This causes the rate to increase to a maximum and then to decline as the effect of consumption of the reactant or reactants begins to overcompensate that of promotion by the product. In reactant-inhibited reactions, the order with respect to a reactant is negative. The rate may increase until the respective reactant is almost entirely used up and, in some cases, rise theoretically beyond what is physically possible. Another mechanism or event then necessarily takes over. Examples include acetal hydrolysis, base-catalyzed aldol condensation, olefin hydroformylation catalyzed by phosphine-substituted cobalt hydrocarbonyls, phosphate transfer in biological systems, enzymatic transamination, adiponitrile synthesis via hydrocyanation, olefin hydrogenation with Wilkinson's catalyst, and osmium tetroxidecatalyzed asymmetric dihydroxylation of olefins.

References General references Gl.

G2. G3. G4. G5. G6. G7.

G8. G9.

J. P. Collman, L. S. Hegedus, J. R. Norton, and R. G. Finke, Principles and applications of organotransition metal chemistry, University Science Books, Mill Valley, 2nd ed., 1987, ISBN 0935702512. K. A. Connors, Chemical kinetics; the study of reaction rates in solution, VCH Publishers, New York, 1990, ISBN 3527218223. J. Falbe, Carbon monoxide in organic synthesis. Springer, Berlin, 1970. A. A. Frost and R. G. Pearson, Kinetics and mechanism: a study of homogeneous chemical reactions, Wiley, New York, 2nd. ed., 1961. B. C. Gates, Catalytic chemistry, Wiley, New York, 1992, ISBN 0471517615. L. P. Hammett, Physical organic chemistry: reaction rates, equilibria, and mechanisms, McGraw-Hill, New York, 2nd ed., 1970. C. K. Ingold, Structure and mechanism in organic chemistry: reaction rates, equilibria, and mechanisms, Cornell University Press, Ithaca, 2nd ed., 1969, ISBN 0801404991. O. Levenspiel, The chemical reactor omnibook, O.S.U. Book Stores, Corvallis, 1989, ISBN 0882461648. J. W. Moore and R. G. Pearson, Kinetics and mechanism: a study of homogeneous chemical reactions, Wiley, New York, 3rd ed., 1981, ISBN 0471035580.

270

Chapter 8. Homogeneous catalysis

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20. 21. 22. 23. 24.

L. Wilhelmy, Ann. Phys. Chem. (Poggendorf), 81 (1850) 413 and 499. Moore and Pearson (ref. G9), pp.338-340. Gates (ref. G5), Section 2.2.7. F. A. Long and M. A. Paul, Chem. Rev., 57 (1957) 935. A. M. Wenthe and E. H. Cordes, J, Am. Chem. Soc, 87 (1965) 3173. J. T. Edward and S. C. R. Meacock, /. Chem. Soc, 1957, 2000. L. P. Hammett and A. J. Deyrup, J. Am. Chem. Soc, 54 (1932) 2721. G. A. Olah, Friedel-Crafts chemistry, Wiley, New York, 1973, ISBN 0471653152, p. 368. K. B. Wiberg, Physical organic chemistry, Wiley, New York, 1964, Section 3-8. Hammett (ref. G6), p. 267. Connors (ref. G2), Chapter 7. R. P. Bell, /. Chem. Soc, 1937, 1637. Ingold (ref. G7), pp. 999-1002. Frost and Pearson (ref. G4), Section 12.D. F. G. Helfferich and P. E. Savage, Reaction kinetics for the practical engineer. Course #195, AIChE Educational Services, New York, 7th ed., 1999, Section 7.4. J. L. Van Winkle, personal conmiunication, 1967. L. H. Slaugh and R. D. Mullineaux, /. Organomet. Chem., 13 (1968) 469. Falbe (ref. G3), Section 1.2. E. Billig and D. R. Bryant, Oxo reaction, in Kirk-Othmer, Encyclopedia of chemical technology, 4th ed., J. I. Kroschwitz and M. Howe-Grant, eds., Wiley, New York, Vol. 17, ISBN 047152686X, 1996, p. 902. H. J. Baumgartner, Technical Report 9-67, Shell Development Co., 1967. W. Rupilius, J. J. McCoy, and M. Orchin, Ind. Eng. Chem., Prod. Res. Dev., 10 (1971) 142. Collman et al. (ref. Gl), p. 91. P. Szabo, L. Fekete, G. Bor, Z. Nagy-Magos, and L. Marko, J. Organomet. Chem., 12 (1968)245. W. W. Spooncer, A. C. Jones, and L. H. Slaugh, /. Organomet. Chen., 18 (1969) 327.

References 25.

26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57.

271

C. A. McAuliffe and W. Levason, phosphine, arsine, and stibine complexes of transition metals, Vol. 1 of Studies in inorganic chemistry, Elsevier, Amsterdam, 1979, ISBN 0444417494, Sections III.8.1 and V.6.1. R. V. Kastrup, J. S. Merola, and A. A. Oswald, Adv. Chem. Ser., 196 (1981) 43. L. Michaelis and M. L. Menten, Biochem. Z , 49 (1913) 333. Segel (ref. Gil), Chapter 2. V. Henri, Lois generales de Vaction des diastases, Hermann, Paris, 1903. G. E. Briggs and J. B. S. Haldane, Biochem. J., 19 (1925) 338. H. Lineweaver and D. Burk, /. Am. Chem. Soc, 56 (1934) 658. C. S. Hanes, Biochem. /., 26 (1932) 1406. G. S. Eadie, J. Biol. Chem., 146 (1942) 85. B. H. J. Hofstee, Nature, 184 (1959) 1296. G. G. Hammes and D. Kochavi, /. Am. Chem. Soc, 84 (1962) 2064, 2073, and 2076. J. A. Christiansen, Z. Physik. Chem., Bodenstein-Festband (1931) 69. J. A. Christiansen, Z. Physik. Chem., B 28 (1935) 303. J. A. Christiansen, Adv. Catal., 5 (1953) 311. E. L.. King and C. Altman, J. Phys. Chem., 60 (1956) 1375. Segel (ref. Gil), Section 9.1. P. L. Brezonik, Chemical kinetics and process dynamics in aquatic systems, CRC Press, Boca Raton, 1994, ISBN 0873714318, Section 6.2.2. M. Boudart, AIChE J., 18 (1972) 465. J. D. Druliner, Organometallics, 3 (1984) 205. C. A.. Tolman, R. J. McKinney, W. C. Seidel, J. L. Druliner, and W. R.. Stevens, Adv. Catal., 33 (1985) 1. Parshall and Ittel (ref. GIO), Section 3.6. C. A. Tolman and J. P. Jesson, Science, 181 (1973) 501. Helfferich and Savage (ref. 15), Section 6.8. J.-M. Chem, Ind. Eng. Chem. Res, 39 (2000) 4100. F. H. Jardine, J. A. Osborn, G. Wilkinson, and J. F. Young, Chem. Ind., 1965, 560. J. A. Osborn, F. H. Jardine, J. F. Young, and G. Wilkinson, J. Chem.. Soc, A 1966, 1711. C. O'Connor and G. Wilkinson, Tetrahedron Lett., 1969(18), 1375. Collman et al. (ref. Gl), pp. 530-535. Gates (ref. G5), Section 2.4.2. J. Halpern and C. S. Wong, J. Chem. Soc, Chem. Commun., 1973, 629. J. Halpern, T. Okamoto, and A. Zakhariev, J. Mol. Catal., 2 (1976) 65. J. Halpern, Trans. Am. Cristallogr. Assoc, 14 (1978) 59. C. A. Tolman, P. Z. Meakin, D. L. Lindner, and J. P. Jesson, J. Am. Chem. Soc, 96 {191 A) 2162.

272 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72.

73. 74. 75. 76.

Chapter 8, Homogeneous catalysis C. Daniel, N. Koga, J. Han, X. Y. Fu, and K. Morukuma, 7. Am, Chem. Sac, 110 (1988) 3773. Collman et al. (ref. Gl), p. 534. N. A. Muhammadi and G. L. Rempel, Macromolecules, 20 (1987) 2362; /. Mol CataL, 50(1989)259. e.g., see Gates (ref. G5), p. 82. Segel (ref. Gil), Chapters 3 and 4. Segel (ref. Gil), Chapter 5. T.-S. Chen and J.-M. Chem, Chem, Eng, ScL, 57 (2002) 457. J. S. M. Wai, I. Marko, J. S. Svendsen, M. G. Finn, E. N. Jacobsen, and K. B. Sharpless, /. Am, Chem. Sac, 111 (1989) 1123. T. Gobel and K. B. Sharpless, Angew Chem, Int. Ed,, 32 (1993) 1329. C, G. Hill, h.,An introduction to chemical engineering kinetics & reactor design, Wiley, New York, 1977, ISBN 0471396095, Problem 8.26. Moore and Pearson (ref. G9), p. 26. Steinfeld et al. (ref. G12), Section 5.3. G. G. Froment and K. B. Bischoff, Chemical reactor analysis and design, Wiley, New York, 2nd. ed., 1990, ISBN 0471510440, Example 1.3.1. L. D. Schmidt, The engineering of chemical reactions, Oxford University Press, 1998, ISBN 0195105885, pp. 112-114. R. W. Missen, C. A. Mims, and B. A. Saville, Introduction to chemical reaction engineering and kinetics, Wiley, New York, 1999, ISBN 0471163392, pp. 189191. S. C. Datta, J. N. E. Day, and C. K. Ingold, /. Chem, Soc, 1939, 838. J. N. E. Day and C. K. Ingold, Trans, Faraday Soc, 37 (1941) 686. Frost and Pearson (ref. G4), pp. 316-320. M. Frenklach and D. Clary, Ind. Eng. Chem, Fundam,, 22 (1983) 433.

Chapter 9 Heterogeneous Catalysis The preceding chapter has included many reactions that could be called heterogeneous because one or more gaseous reactants have to enter a liquid phase containing the dissolved catalyst. In all these cases, however, the reaction itself occurs in a single phase, subject only to supply of one or more reactants from another phase by mass transfer. Reactions of this type essentially fit the conceptual and mathematical description of homogeneous catalysis and so are commonly and conveniently classified as such. In contrast, the term heterogeneous catalysis is reserved for reactions occurring on a solid catalyst and involving adsorption of reactants and desorption of products as distinct steps. For example, hydrogenation is "homogeneous" if catalyzed by a transition-metal complex dissolved in a liquid, and "heterogeneous" if catalyzed by a solid metal. Heterogeneous catalysis has to deal not only with the catalyzed reaction itself but, in addition, with the complexities of surface properties (different crystal surfaces, different catalytic sites), possible segregation of adsorbates (so-called island formation), contamination or deterioration of catalytic sites, and adsorption and desorption equilibria and rates. Moreover, mass transfer to and from the reaction site is a factor more often than in homogeneous catalysis.* In practice, these complications may affect behavior more profoundly than does the kinetics of the surface reaction itself. A practical and balanced kinetic treatment therefore uses simplifications and approximations much more generously than was done in the preceding chapters. Excellent textbooks on the subject are available [G1-G7], so coverage here can remain restricted to a critical overview and indications showing when and how concepts and methods developed in the earlier chapters can be useful. Involving adsorption and desorption as distinct steps, heterogeneous catalysis is a multistep phenomenon j[7(3r excellence. It can be approached in either of two ways: In the Langmuir-Hinshelwood-Hougen-Watson approach [1-3], adsorption is regarded as a reversible, single-step "reaction" of the respective species with an unoccupied catalytic site. This allows the same formalism as for homogeneous catalysis to be used. However, any reaction of adsorbed species with one another then appears as a step with two intermediates as reactants that makes the network non-simple and most of the convenient general formulas developed in Chapters 6 * In heterogeneous catalysis, the terms adsorption and desorption refer strictly to attachment to, and detachment from, the solid surface, to the exclusion of mass transfer to and from the surface.

274

Chapter 9. Heterogeneous

Catalysis

and 8 inapplicable. Simplifications are needed to obtain manageable rate equations. The alternative is to handle the surface reaction as though it were homogeneous, and combine it with adsorption equilibria and kinetics. Now, reactions of adsorbed reactants with one another leave the system simple, but reactions of adsorbed intermediates with one another still cause non-simplicity. Moreover, so-called EleyRideal or rake mechanisms, in which adsorbed species react directly with species from the contacting fluid, are excluded. Tackling the real difficulty has only been deferred: Even if the surface reaction is simple, complexity arises when the concentrations of the adsorbed species are expressed in terms of those in the contacting fluid. The need for simplification is as great here as in the more commonly practiced Langmuir-Hinshelwood-Hougen-Watson approach, which will be taken in this chapter. For practical engineering purposes, quasi-stationary behavior can almost always be assumed and will be taken for granted. Also, throughout this chapter, concentrations should be substituted for partial pressures if the contacting fluid is a liquid.

9.1.

Adsorption and reaction: Langmuir-Hinshelwood kinetics

The simplest conceivable example shall illustrate how Langmuir-Hinshelwood kinetics combines adsorption and desorption with a surface reaction. Example 9,1. Langmuir-Hinshelwood kinetics of single-step reversible isomerization. The reaction is A <

• P

(9.1)

Granted Langmuir's assumption of adsorption-desorption as a reversible single-step "reaction," the cycle can be written

(9.2)

( O = catalytic site). Christiansen mathematics can then be applied to the cycle: 0^o\\i\o

"

\o\\\'2)^i:

(^01^12^20^A

k k

+ XojXj2 + Xo2^12 ^

^2^10

'^lO'^ll^OlPp^^L

k k

^10^21

•^ KQKIPK ^ KiKiPk ^ KiKiP? + k^^k^^p^ + Ki^nP? "^ Ki^ioP?

Section 9.2. Rate-controlling steps: the Hougen-Watson formula

275

(see Section 8.4). Here, ih^p, are the fluid-phase concentrations, ^^ is the total of all occupied and unoccupied catalytic sites and, in the indices of the k-,-^, the surface species are numbered clockwise starting with 0 as the unoccupied site. Collecting denominator terms with common concentration co-factors and converting to one-plus form one obtains .

^

(MA -

KPY)^Z

(9.3)

1 + M A + KPV

where k k k

K ^ __-J2L2iZL__ ^12

•^ ^ 1 0 ^ 2 0 "^ ^IC1^21

k k k

k, ^

'^ 10'^21'^02 ^12l'^20 •'' ^ 1 0 ^ 2 0 "*" ^IC1^21

Ki

"^ ^ 0 2 ^ 1 2 "^ ^02

k

^201^01 • ^ ^ 2 1 ^ 0 1 "^ ^01^ 2

^21

^ 1 2 ^ 2 0 "^ ^ 1 0 ^ 2 0 "^ ^10^21

^ 1 2 ^ 2 0 "*• ^ 1 0 ^ 2 0 "^ ^ 1 0 ^ 2 1

Standard texts present lengthier derivations and arrive at more complicated, but equivalent results. We may ask why Christiansen's so convenient formula is hardly ever used in heterogeneous catalysis. Apart from most engineers not being familiar with it, there are two reasons: The formula is for a single catalytic cycle and therefore not applicable to reactions in which adsorbed species react with one another, and the coefficients in the resulting rate equations are composites of rate coefficients of adsorption, desorption, and reaction, making it hard to separate the effects of these phenomena. Nevertheless, for reactions involving only a single catalytic site, Christiansen's formula leads more quickly and simply to a rate equation. 9.2. Rate-controlling steps: the Hougen-Watson formula General solutions for reactions involving several steps or more than one catalytic site are cumbersome, if they can be obtained at all in explicit form. The most common procedure is, instead, to identify or appoint one of the events as ratecontrolling. This may be adsorption of a reactant or desorption of a product, but more often than not it is the surface reaction, taken to be single-step. Example 9.2. Rate control by a single-step, bimolecular surface reaction. Consider a hypothetical reaction A+ B < • P+ Q (9.4) Christiansen's formula cannot be used because two adsorbed species react with one another. If the rate is controlled by the surface reaction, reactants and products are at adsorption quasi-equilibrium, that is, for each the adsorption and desorption rates are practically equal. With Langmuir's equations for these rates: ^iPi^O~^-i^i = ^

^° ^^^^

^i = ^\P\^o

(^ " A,B,P, Q)

276

Chapter 9. Heterogeneous Catalysis

Here, k, and k_, are the rate coefficients of adsorption and desorption of i, respectively; and ATj = kjk_; is its adsorption equilibrium constant. The net rate then is

=

^J^A^BPA/^B-(^P^Q/^>P/^Q]^0

where k^ and k_^ are the rate coefficients of the forward and reverse surface reaction, respectively, and ^^x = Kxl^-rx is its equilibrium constant. This equation still contains the unknown concentration of vacant sites, QQ. TO eliminate it, the site balance

is solved for qQ-. 1 + K^p^ + K^p^ + K^p^ + K^p^ With this equation the rate becomes 2

Kx^^A^hiPAPB - PVPQIK)QZ

(9 5)

(1 + K^p^ + K^p^ + ATpPp + K^p^Y where ^

=

^A^B^rx/^P^Q

is the equilibrium constant of the overall reaction including adsorption and desorption. A general formalism for single-step surface reactions of heterogeneous catalysis has been developed by Hougen and Watson [3]. The rate may be controlled by the surface reaction, adsorption of a reactant, or desorption of a product. Explicitly covered in tabulations are reactions with the following stoichiometrics: A <

• P,

A + B <

• P,

A <

• P + Q,

A + B <

• P + Q

One reactant may dissociate upon adsorption, as does H2 on most catalysts, and one reactant may remain unadsorbed. Hougen and Watson's general formula can be written j.^^g

^

(kinetic group)(driving-force group) adsorption group

^9 g^

The kinetic group contains the relevant rate coefficients, the driving-force group is related to the distance from equilibrium, and the adsorption group reflects the surface coverage by the respective species and has the general form

Section 9.2. Rate-controlling steps: the Hougen-Watson formula

general adsorption group:

(1 + K^p^ + K^p^ + ...)"

277

(9.7)

Each term is proportional to the surface coverage by the respective species, scaled so that the leading " 1" is proportional to the vacant surface, and n is the number of sites involved in the molecular reaction. Some reactions require substitutions in eqn 9.7. A tabulation of groups and substitutions is shown on the next two pages. The Hougen-Watson formula is derived in terms of fractional surface coverage, and so does not explicitly contain the total of catalytic sites, qj^. Otherwise it gives the same results as the long-hand derivation shown previously. Example 9.3. Hougen-Watson rate equation for rate control by a reversible bimolecular surface reaction. In the previous example, a rate equation was derived long-hand for rate control by a surface reaction A+ B <

• P+ Q

(9.4)

The Hougen-Watson formula allows this to be done more quickly. For the stoichiometry 9.4, no dissociation of A, and rate control by the surface reaction, the tabulation on the next two pages lists the various groups as kinetic group

^TX^A^B

driving-force group

p^p^ -

p^p^/K

adsorption group

(1 + K^p^ + K^p^ + K^p^ +

K^PQ)^

With these groups the overall rate is ^

^

K.KAKB(PAPB

(1 4- K^p^ + K^^

- P^PQ^K)

^^ g^

+ ATpPp + K^p^y

and is the same as eqn 9.5, except it lacks the factor q^ and so cannot account for the effect of the concentration of catalytic sites. As already pointed out, this is true for Hougen-Watson rate equations in general. Limitations. The Hougen-Watson formula is widely used, but has its limitations. Specifically, it assumes: all catalytic sites are alike, each adsorbed molecule occupies one site, the surface reaction occurs in, or can be reduced to, a single step, adsorption kinetics and equilibria obey Langmuir equations, a single adsorption, desorption, or reaction step controls the rate, and tabulations cover only reactions with some simple stoichiometrics.

278

Chapter 9. Heterogeneous Catalysis

Table 9.1. Groups in Hougen-Watson rate equations (compiled from a table by Yang and Hougen [4]). stoichi(Dmetry rate control or condition

A ^•-^ P

A .«-• P + Q

dri ving-force group adsorption of A

/7A - PpfK

PA - P?PQ IK

adsorption of B

0

0

A + B <•-• P + Q A + B ^i-^ P

PA ~ PPPQ IKp^

PA - P?IKpf, P? If^Ph PB

PB - PPPQ IKp^

desoq)tion of P

PA -

P?IK

PA/PQ - P?IK

PAPB - PplK

PAPBIPQ - PplK

surface reaction

PA -

P?IK

PA^PQ - P?IK

PAPB - PP IK

PAPB " PPPQ IK

0

PAPB - PplK

PAPB " PPPQ IK

PA - PPPQ If^

PAPB - PplK

PAPB - PPPQ IK

impact of A controls (A not adsorbed) homogeneous reaction

0

PA -

P?IK

repla cements in general adsorption group (eqn 9.7) adsorption of A: replace K^j^ by

K,p,/K

K^P^PQ/K

K^PplKp^

KAPPPQIKP^

adsorption of B: replace K^Ps by

0

0

K^p^lKpj,

KBPPPQIKPA

desorption of P: replace KpPp by

KKpp^

KKpp^ IPQ

KK^PAPB

KKPPJ,PQ/PQ

(K,p,/K)'^

{K,p,p^lK)"'

adsorption of A with dissociation: 1 replace K^j?^ by any rate control and equilibrium adsorption of i with dissociation replace Kp^ by

{K^./Kp^y"

iK,p,p^/Kp^y''

iK,pr

{K,p.r

{Kp.y^

iKpy"

0

0

0

0

1 (i = A,B,P,orQ) any rate control and no adsorption of i replace Kp, by

1 (i = A,B,P,orQ)

Section 9.2. Rate-controlling steps: the Hougen-Watson formula

279

stoichiometry rate control or condition

A ^•-»- P

A^t-^P + Q

A + B -«-• P -f- Q

A + B^«->P

kinetic group adsorption of A

^A

1

adsorption of B

kn

1

desorption of P

k?IK

1

k.

1

adsorption of A with dissociation impact of A (A not adsorbed)

^A^B

homogeneous reaction

"^hom

II

surface reac tion controls rate no dissociation

^rx^A

dissociation of A

^rx^A ^rx-^A

no adsorption of B

kr.K^K^

^x^A^B

Kx^A

^rx^A^B

Kx^\K^

k^K,

krxK,

kfx^A

no adsorption of B dissociation of A

exponen ts of adsoq)tion group adsorption of A, no dissoc iation

n

= 1

1

adsorption of A, with dissociation of A

M=2

1

desorption of P

n =I

1

reaction A + B -*-> P, impact of A controls, no dissociation

n = \

reaction A + B '^—• P + Q, impact of A controls, 1 dissociation of A

n = 2

A2 = 0

homogeneous reaction

1

surface reac tion controls rate no dissociation

n = 1

« = 2

« = 2

1 dissociation of A

rt = 2

n =2

« = 3

« =2 « =3

no dissociation of A no adsorption of B

n = 1

n =2

n = 1

n = 2

dissociation of A no adsorption of B

n = 2

n = 2

n = 2

n = 2

1 1

280

Chapter 9. Heterogeneous Catalysis

9.3. Relative abundance of catalyst-surface species The Hougen-Watson formula presupposes the surface reaction to occur in a single step. This is rarely the case, and may often not even be an acceptable approximation. To reduce the mathematical complexity introduced by a multistep surface reaction, the practical engineer will then look for other simplifications that do not entail undue loss of accuracy. Here, the concept of relative abundance of adsorbed species on the catalyst surface is a very useful tool. Almost the entire surface may be covered by a single species, the macs (most abundant catalyst-surface species). If adsorption is negligible, the free surface is the macs. One or more species may cover only a negligible fraction of the surface. They are termed lacs (low-abundance catalyst-surface species). [See Section 8.5.1 for macs and lacs in homogeneous catalysis.]* 9.3.1. Simplification of rate equations As already mentioned, the denominator of Langmuir-Hinshelwood-Hougen-Watson rate equations is composed of additive terms, each being proportional to one state of coverage: the leading " 1" to the number of vacant sites, and each other term K^ Pi to the number of sites occupied species i. Thus, if a macs exists, all terms but one are negligible; and if any lacs exist, their terms are negligible. Section 8.5 has shown how the existence of a macs or one or several lacs can be used to reduce complexity. Analogous rules can be applied in heterogeneous catalysis. Particularly helpful rules are

All steps following an irreversible step are kinetically insignificant if all their reactants are lacs. If the formation of the macs controls the rate, all preceding steps can be combined into a single quasiequilibrium step. If the consumption of the macs controls the rate, all subsequent steps can be so combined.

(see also Boudart's theorems [5]).

* Note that the free surface may be the macs, and that almost the entire surface is occupied if an adsorbed species is the macs. In this respect the macs differs from Boudart's masi [5], which is the most abundant adsorbed species, regardless of surface coverage.

9.3,

Example 9A. propanal

Relative abundance of catalyst-surface species

Nickel-catalyzed hydrogenation of propanal.

O

2

281

Hydrogenation of

^

is catalyzed by metallic nickel. The rate is believed to be controlled by the surface reaction and is practically irreversible at high pressure and low temperature. Hydrogen is known to dissociate upon adsorption on nickel, and at low temperature the catalyst surface is covered almost completely by atomic hydrogen: Atomic hydrogen is the macs, and the hydrogen term completely dominates the denominator of the rate equation. For the stoichiometry A + B <^—• P, dissociation of hydrogen, and rate control by the irreversible surface reaction, the Hougen-Watson groups are kinetic group

^rx^H^^aid

driving-force group

Pu^Pan

adsorption group

^ +

(ATH^PHJ'"+/^fi^d+i^c)

(insignificant terms are struck out) and give the overall rate equation r,,

= kp-^^PM

{k^kJ^^^K^)

(9.9)

in reasonable agreement with observation [6]. At lower pressures and higher temperatures, almost complete coverage of the catalyst surface cannot be expected. A macs then no longer exists, but aldehyde and alcohol, which are much less strongly adsorbed than hydrogen, still are lacs. Accordingly, the leading " 1 " in the denominator of the rate equation becomes significant, but the aldehyde and alcohol terms remain absent. Widi this change in the adsorption group the overall rate is found to be ^P„,/'a,d ait

y

{k - k^^K^K^,,)

(9.10)

» •?

(l . ( V H . ) " ^ Agreement of the rate equation with observation, however, cannot be taken as evidence supporting the assumed single-step mechanism of the surface reaction. It is highly improbable that two adsorbed hydrogen atoms react simultaneously with adsorbed aldehyde in a single, termolecular step. Much more likely is a two-step mechanism

(aid) + (1?)

• (Hald) + ( ^

(Hdd) + (^

• Ulc) +

( ^

282

Chapter 9. Heterogeneous Catalysis followed by rapid desorption of alcohol. If the first step is rate-controlling, its rate is also that of the overall reaction: '•a. = ^r„9a,d9„

(9.11)

The stoichiometry of the combined preceding adsorption steps is aid + y2H2 + 2 ( ^

<

•

(M)

+ (j?)

with quasi-equilibrium condition K

=

^ ! ! i ^

(9.12)

/^H,Pald^O

where K^ is the equilibrium constant of the combined adsorption steps. The Langmuir isotherm for adsorption equilibrium of dissociating H2, aldehyde, and alcohol gives for the unoccupied sites ^O

(9.13)

=

^+{^H,PH,)

+ jS^Td^d +

^^^c

(see Section 2.6), where the struck-out terms do not contribute if H is the macs. Using eqns 9.12 and 9.13 to eliminate q^ and q^^^ in eqn 9.11 one obtains '•aic = kp„l"Pa>,

(k = K^KqllKn)

(9.14)

The reaction orders are the same as in eqn 9.9, obtained from the Hougen-Watson formula, only the physical interpretation of the coefficient k is different. Agreement with experimental observation, however, does not validate rate control by the surface reaction. For example, rate control by adsorption of aldehyde leads to a rate equation of the same algebraic form [7]. This example brings out a number of facets of heterogeneous catalysis: The existence of a macs, which suppresses all but one of the Langmuir denominator terms, is the simplest possible explanation for an observed rate that fits a power law. The macs can be the unoccupied surface. However, unless the surface reaction is single-step, a mucs does not guarantee that the rate obeys a power law, nor is such rate behavior possible only if there is a macs (see also Section 8.5). Although the Hougen-Watson formula presumes the surface reaction to occur in a single step, it may provide a good approximation in more complex cases. This helps to explain its popularity and success rate.

9.3. Relative abundance of catalyst-surface species •

•

•

283

Dissociation of a reactant produces a reaction order, or exponent in a oneplus rate equation, of one half or a multiple of one half. A rate equation of this type is an indication of reactant dissociation (see also Section 5.6). A reaction order may be negative if the respective reactant covers much of the catalyst surface. As its surface coverage increases, the surface still accessible to other reactants decreases, and does so more strongly on a relative scale. In the our example, if the hydrogen coverage increases, say, by 1% from 98 to 99%, the area accessible to aldehyde (and alcohol) decreases by one half, from 2 to 1 %. A negative reaction order is an indication that the respective reactant covers much of the catalyst surface. Different mechanisms or rate-controlling steps may produce the same rate behavior in good approximation. Additional information is required to distinguish conclusively between rival mechanisms (see Section 9.4).

9.3.2. Self-acceleration A reaction whose rate increases as conversion progresses is said to be self-accelerating. Thermal self-acceleration for lack of sufficient heat transfer is not unusual, but is a matter of reactor design and will only be touched upon briefly in Section 14.2. At constant temperature, the rate of a "normal" reaction decreases with progressing conversion. Such behavior is often taken for granted. However, selfacceleration can occur even at constant temperature (see Section 8.9). In heterogeneous catalysis, it is usually caused by a reactant being the macs. In heterogeneous catalysis, all adsorbed species can be said to act as inhibitors by occupying parts of the catalyst surface, making it inaccessible to other molecules (synergistic adsorption is the exception). Thus, a reactant not only promotes conversion in the normal way, but also inhibits it. Both these opposing effects decrease with conversion, but their balance shifts. Self-acceleration requires the denominator of the rate equation to decrease more steeply with decreasing reactant concentration than does the numerator. In the general Hougen-Watson formula (kinetic group)(driving-force group) (eqn 9.6 with general adsorption group 9.7), the numerator for the forward reaction is proportional to the reactant concentrations while the denominator involves each of these in only one of several terms. For the dependence on the concentration of a reactant to be stronger in the denominator than in the numerator, the exponent n must therefore be greater than 1, implying that the reaction must involve more than one catalytic site. A look at the Yang-Watson tabulation in Table 9.1 shows when this can occur within Langmuir-Hinshelwood-Hougen-Watson kinetics:

284

Chapter 9. Heterogeneous Catalysis

Self-acceleration is possible if the surface reaction is rate-controlling and involves more than one reactant or product, or if a reactant dissociates upon adsorption. This is a necessary condition, not a sufficient one. If its term involving the reactant concentration is small compared with the others, the denominator varies less than proportionately with that concentration even if its exponent is larger than 1: Self-acceleration also requires the reactant to be predominantly adsorbed. Self-acceleration can occur over only a limited conversion range. As the responsible reactant becomes depleted, the numerator of the rate equation approaches zero while the denominator, with its leading "1," always remains finite. Accordingly, the rate decreases again over the last increments of conversion. Propanal hydrogenation over nickel (Example 9.4 with rate equations 9.9 and 9.10) is a typical example of such behavior. 9.4. Model discrimination Different mechanisms and rate-controlling steps may produce rate equations of the same algebraic form, making it impossible to identify mechanism and rate control conclusively on the basis of an empirical rate equation alone. This happens more often in heterogeneous catalysis than in homogeneous reactions. A clearer indication may be gained from studies of the temperature dependence of the coefficients, concentration dependence of initial rates, and tests of model predictions. 9.4.1. Criteria for coefficients The rate equation derived from a trial mechanism contains one or more coefficients that are rate coefficients of steps or composites of these. A study of the temperature dependence of these coefficients can provide valuable clues. It cannot validate a proposed model, but can show it to be inadequate. The coefficients must meet several criteria: • All coefficients must be positive. • Any single-step rate coefficient or product of single-step rate coefficients must increase with increasing temperature. • The temperature dependence of a single-step rate coefficient or of a product, ratio, or ratio of products of single-step rate coefficients must have a reasonably constant activation energy (linear variation of In/: or IniST with \IT according to the Arrhenius or van't Hoff equation). • An adsorption equilibrium constant (ratio of adsorption and desorption rate coefficients) must decrease with increasing temperature, provided adsorption is exothermic, as is almost always the case.

9.4. Model discrimination

285

The Arrhenius behavior of single-step rate coefficients has been discussed in Section 2.3. The Arrhenius behavior of products and ratios arises from the linear dependence of \nk on I IT. Logarithms of products or ratios are the sums or differences, respectively, of the logarithms of their members. Since the logarithms of the individual coefficients are linear functions of reciprocal temperature, so are those of their products and ratios (see also Section 13.1.1). In addition to showing an approximately linear variation of I n ^ with \IT (van't Hoff equation), an adsorption equilibrium constant should meet Boudart's four semi-empirical entropy criteria [8]: AS° < 0,

|AS°| > 10,

|A5°|

|A5°| < 12.2 - 0.0014 Ai/°

< Sg33,

^^^5^

where A5° can be calculated from the temperature dependence of the observed equilibrium constant: , j^

A//° RT

AS° R

Example 9.5. Identification of rate control. A hypothetical rearrangement reaction A <—^ P with favorable equilibrium is found to obey a one-plus rate equation

.

^

KPK

- KP?

1 + ^CPA

in good approximation. For the three possible rate-controlling steps, the rate equations compiled from Table 9.1 are: ~ Pp/^ 1!:— 1 1 -. {KJK + K^)p^ ^A(PA

/n i/:\ (^lo)

adsorption controls rate:

r^ = '

surface reaction controls rate:

r^ = _!L_^!—t 1— 1 + K^p^ + i^ppp

(9.17)

k^K(p^ - p^lK)

^^ ^g^

desorption controls rate:

Rate control by adsorption of A (eqn 9.16) can be ruled out because the denominator does not contain a term involving /7^. Rate control by the surface reaction (eqn 9.17) is possible, provided K^ is small. Rate control by desorption of P (eqn 9.18) is possible without reservation. The temperature dependence of the coefficient k^ in eqn 9.15 might decide the issue: If k^ increases with increasing

286

Chapter 9. Heterogeneous Catalysis temperature, eqn 9.17 can be ruled out because Kp^, an adsorption equilibrium constant, must decrease with increasing temperature. In contrast, in eqn 9.18 an increase in K can outweigh the decrease in ATp and Kj^, leaving rate control by desorption still in contention.

9.4.2. Concentration dependence of initial rates Hougen-Watson rate equations reduce to simpler and more typical forms for initial rates, that is, if the terms involving product concentrations are omitted. This may allow the rate-controlling step to be identified or at least an incorrect one to be ruled out. Table 9.2 shows the equations for the initial rates and their dependences on initial pressure for the four Yang-Hougen stoichiometries and the three possible rate-controlling steps. Three of the plots show the rate to have a maximum at an intermediate fluidphase concentration. This does not necessarily reflect self-acceleration. The plots are for initial rates, and the rate of a progressing reaction may well decrease monotonically with increasing conversion as the effect of product build-up makes itself felt. Also, the maximum of the initial rate may be at a pressure too high to be attained experimentally, so that the concentration depend-ence may appear to be monotonic. Example 9.6. Dehydrogenation ofEthanol [9]. Ethanol can be dehydrogenated over copper-cobalt on asbestos: CH3CH20H

CH3CH0 + H2

r=285''C

The initial rates at different temperatures are found to have maxima at intermediate initial pressures (see Figure 9.1). As Table 9.2 shows, a reaction with stoichiometry A—• P -fQ can only behave in this fashion if the surface reaction controls the rate. Thus, such rate control is conclusively established. [Hj dissociates upon adsorption, but this is irrelevant here because terms Figure 9.1. Initial rate of ethanol dehydroge- involving hydrogen do not nation over Cu/Co catalyst as function of initial appear in the equation for the pressure (from Franckaerts and Froment [9]). initial rate.]

287

9.4, Model discrimination

Table 9.2. Dependence of initial rates on initial total pressure P° (at given PA'-PB ratio where two reactants are involved; compiled from information by Yang and Hougen [4]). rate control stoichiometry

surface reaction .o_

A *-> P

adsorption of A

KJ^AP:

•A

~'*A

"-Ay A

|

i^Kpl

0

_r° - t r.**

l*Kj>°

desorption of P

\^{K^^KK^)pt

-r:

p° O

A -*-• P + Q

-r

A + B •-• P

0 ''A ~

A + B ^•-^ P + Q

-r:-

*

<

KX^A^BPAPB

o 'A

(l^K^^^K^^

KL^A^^PAPB

= k^:

"'A ° « T / ^ ?

^^PA -

Q

r°

^^PAPB

'^A

1 + K^p^

KPIPB

~^A -

^I^Y

* The maximum in the plot for rate control by the surface reaction may occur at a pressure higher than can be experimentally attained.

288

Chapter 9. Heterogeneous Catalysis

9.4.3.

Testing of predictions

Even if a mechanism proves to meet all conditions described so far, its predictions should be tested (see also Section 12.4). The most common technique is to project how additives with known adsorption behavior should affect the reaction. An often cited example from the infancy of catalytic reaction engineering is presented below. Example 9.7. Dehydrogenation of methylcydohexane [10]. Methylcyclohexane (MCH) can be dehydrogenated to toluene over alumina-supported platinum: OCH3

(0>CH3

+ 3H,

To avoid coking of the catalyst, the reaction is conducted with hydrogen in the feed. Initial rates, obtained from fractional conversions across a differential reactor at different temperatures, partial pressures of MCH and hydrogen, and flow rates, were found to be essentially independent of H2 pressure and weakly dependent on MCH pressure (apparent order about 0.15 to 0.2). This suggest a one-plus trial rate equation ^a/^MCH

(9.19)

1 + k^. which was found to be satisfactory. Equation 9.19 is typical for Michaelis-Menten kinetics (see Section 8.3, cycle 8.14), which might arise here from quasi-equilibrium adsorption of MCH followed by a rate-controlling, irreversible surface reaction (possibly in more than one step): MCH "^

"^

(9.20)

^MCH^rr/^MCH^E ~j ~ jp ~

(9.21)

products with rate equation tol

(see eqn 8.18; the authors used a lengthier derivation). This matches eqn 9.19 as desired, with k^Ky^cwQz = K and J^MCH = ^b- The coefficients and reaction entropy met all criteria. However, the mechanism was discarded because its prediction that strongly adsorbed benzene or xylene should significantly depress the rate by effectively competing with MCH for catalytic sites proved incorrect.

9.4. Model discrimination

289

As an alternative, the authors proposed slow desorption of toluene after an irreversible surface reaction (possibly in more than one step): MCH ^ (9.22)

products Instead of the authors' long-hand derivation, Christiansen's formula can be used and gives inmiediately ^01^12^20

r,..

-

^01 ^ 1 2 ^ 2 0 ^ M C H ^ E ^ 1 2 ^ 2 0 "*' ^20^0l/^MCH

^01 ^ 1 2 / ^ MCH

^Ol/^MCH^£ 1

+

(9.23)

(^20^^0l)/^MCH

(second denominator row negligible because adsorbed MCH is a lacs\ terms with reverse coefficients negligible because all steps are irreversible; k^^^ = adsorption rate coefficient of MCH, ^20 = desorption rate coefficient of toluene). The weakness of retardation by added aromatics now becomes understandable: Added benzene or xylene does reduce the surface area accessible to MCH. However, this also reduces the rate of toluene production and, thereby, the toluene coverage, so that the denial of surface to MCH is partially offset. The authors may well be correct when concluding that the surprising weakness of inhibition by aromatics results from slow desorption of a product. However, their model and rate equation appear questionable. Like aromatics, hydrogen is also strongly adsorbed, and so is as likely a candidate as toluene for accumulation on the surface. Also, a single-site mechanism is quite improbable with two strongly adsorbed products. Moreover, the "initial" rates were measured at conversions that entailed a decrease of up to 13 % in fluid density, an effect not corrected for. Lastly, the trial equation was derived only from rates at low conversion and cannot be relied upon to reflect the behavior of the reaction as it progresses. This work was exploratory research, not process development, and predated the Hougen-Watson formula. Despite its shortcomings, it serves admirably to demonstrate that a model meeting all other criteria may nevertheless have to be rejected if its predictions fail. However, its flaws also show that older work, even if reproduced in respected texts, cannot be accepted uncritically.

290

Chapter 9. Heterogeneous Catalysis

9.5. Mass and heat transfer Detailed treatments of mass and heat transfer effects in heterogeneous catalysis can be found in standard texts of reaction engineering and catalysis [11-15]. Here, a brief overview and analysis must suffice. In practice, a solid catalyst is most conveniently modeled as a quasi-homogeneous phase. Even if the catalyst particle is porous, visualize it as a homogeneous, but permeable solid. Mass transfer in its interior is retarded by two effects: obstruction of part of the cross-sectional area by the solid material, and diffusion paths that are longer because molecules have to wind their way around the obstructions (tortuosity effect). In the quasi-homogeneous model, the retardation is accounted for by the use of appropriately smaller "effective" mass-transfer or diffusion coefficients. If this simplified model is accepted, two basic situations must be distinguished: (1)

mass transfer to and from the catalyst particle, with reaction on the particle surface, and mass transfer and reaction in the interior of the catalyst particle.

(2)

Of course, both situations may pertain simultaneously, but for clarity's sake they will be examined separately as extreme cases. In the first case, mass transfer and reaction occur at different locations and are necessarily sequential—what reacts at the surface must first have got there by mass transfer. Mathematically, the equations for mass transfer are the same whether a reaction occurs or not. The reaction merely determines the boundary condition at the catalyst surface. In contrast, in the second case, mass-transfer and reaction occur side by side and simultaneously in the same volume elements. Here, mass-transfer enters as a source-or-sink term in the basic material-balance equation. bulk fluid / /

q"

!

boundary / layer ^

^^^^ ^^ -^

c/

bulk fluid /

catalyst

c,^ ic,)^ reaction

Figure 9.2. Possible mass-transfer situations in heterogeneous catalysis. Left: mass transfer to and from catalyst particle; right: mass transfer and reaction within catalyst particle.

9.5. Mass and heat transfer

291

9.5.7. Mass transfer to and from catalyst particle For simplicity, assume that an irreversible reaction of order n occurs on the surface of a catalyst particle. Granted these conditions, mass transfer across an adherent boundary layer may affect the reaction rate. In general at steady state: -dNJdt

J,S = k^MC^ - C^) =

kS(Cir A;

(9.24)

reaction

mass transfer

(N^ = amount of A, /^ = A^x of A toward catalyst particle, S = catalyst surface area, k^^^ = mass-transfer coefficient, ^^x = rate coefficient of surface reaction; superscripts b and s refer to bulk fluid and catalyst surface, respectively). Case I: slow mass transfer

Case II: slow surface reaction

A arriving at surface reacts almost immediately, so that

A builds up at surface to almost same concentration as in bulk fluid:

CI « C^ -iNJdt

= k^,SC'^

-dNJdt

rate is controlled by mass transfer.

=

K,S(C^r

rate is controlled by reaction.

Moreover, the apparent reaction order of A is one (that of mass transfer) in Case I, and n (that of the reaction) in Case II. Lastly, the apparent activation energy, calculated from the temperature dependence of the observed overall rate -dN^^/dt, is that of mass transfer (very low) in Case I, and that of the reaction in Case II. In either case, the slower of the two sequential processes controls the rate. 9.5.2. Mass transfer and reaction within catalyst particle (Thiele-Damkohler theory [16,17]) For simplicity, assume fluid-phase mass transfer to be fast enough to maintain the bulk-fluid concentration of A up to the catalyst surface, the particle to be spherical, the reaction to be irreversible and first order, and mass transfer in the particle to obey Fick's law of diffusion. With the reaction as source-or-sink term, the differential material balance for A (change of content of a volume element = what enters minus what exits minus what reacts) is

dt

2

D,

+

_

r t

mass transfer

Or

KC, t

reaction

(9.25)

292

Chapter 9. Heterogeneous Catalysis

(r = radial distance from particle center, D^ = effective diffusivity of A). Upon integration over r. -dA^,/dr =

(9.26)

rxrikV(C,y cat ^ A^r

where '

•

*

1 tanh (j>

effectiveness factor

1 <j>

«/> -

r^iKJD.y"

Thiele modulus

Here, (^J^^ is the concentration of A in the catalyst at the surface, and Feat ^nd r^ are the volume and radius of the catalyst particle, respectively. Other reaction orders or one-plus rate equations, other particle shapes, and reversible reactions give more complex equations [18-25], but the behavior is qualitatively the same. The effectiveness factor describes the ratio of the actual rate to the rate that would be observed if the entire particle were in adsorption equilibrium with the contacting fluid (no mass-transfer hindrance). At small Thiele moduli (small particle size or slow reaction) the effectiveness factor is unity (no mass-transfer effect). At large Thiele moduli the effectiveness factor decreases, eventually becoming inversely proportional to the modulus (see Figure 9.3). This correspond to a mass-transfer limitation so severe that only a thin outer layer of the particle is penetrated by the reactant. If there is no masstransfer hindrance at all, rate control is exclusively by the reaction, and the apparent order and activation energy are those of the reaction. If there is severe mass-transfer limitation, the apparent order is (AZ4-1)/2 (n = true order of reaction), and the apparent activation energy is {E,^,^ + E,^J/2 = ViE^,^ (the activation energy of In 7]^ - In 7/2 mass transfer, E^„,i, is typically very small compared with that of reaction, E^,J. In other in In words, both the order and the activation energy are the Figure 9.3. Logarithmic dependence of arithmetic means of those of effectiveness factor on Thiele modulus and reaction and mass transfer construction to determine locations on curve. ("disguised kinetics").

9.5. Mass and heat transfer

293

Granted the premises of the classic Thiele-Damkohler theory, the dependence of the overall rate on the Thiele modulus and thus on particle size can be predicted from experiments at two different particle sizes. In a plot of Inr/ versus \n(j>, a ratio of particle sizes corresponds to a horizontal distance; a ratio of rates, to a vertical distance. A rectangular triangle formed by these two distances can be fitted to the theoretical curve (see Figure 9.3). This allows the Thiele moduli at both particle sizes to be established and, with these, the effectiveness at other particle sizes to be predicted. The method fails if the two rates are equal, or if the ratios of the rates and reciprocal particle sizes are equal (straight-line portions of the graph). 9.5.3. Nonisothermal catalyst particles [26,27] The ideal Thiele-Damkohler theory assumes isothermal behavior of the catalyst particle. However, if the particle is large and the reaction is highly exothermic, heat transfer may not be fast enough to remove the heat of reaction from the interior. The particle center then heats up, causing the reaction rate to be higher than under isothermal conditions. The effectiveness factor can be larger than unity! This may occur if the structural material of the particle and the imbibed fluid are poor heat conductors, as might be the case, for example, in gas reactions in silicaceous particles. The relevant dimensionless parameters are a modified Thiele modulus, the normalized adiabatic temperature rise (jS), and the Arrhenius number -E^ IRT. Plots for a first-order reaction in a spherical particle are shown in Figure 9.4 (next page). For highly exothermic reactions and large values of jS the rate can be multivalued at modified Thiele moduli around 0.5, with two stable and one unstable steady states. At which state the particle performs depends on its prior history. This anomaly requires numerical modeling. It is more often cited than encountered in practice. In real systems the interdependence of heat and mass transfer makes a large effect unlikely. 9.5.4. Forced convection within catalyst particle The ideal Thiele-Damkohler theory also assumes that mass transfer in the particle occurs exclusively by diffusion. In a gas reaction, however, the volume of the reacting mixture expands if the mole number increases, and contracts if the mole number decreases. If it expands, forced convection out of the particle counteracts reactant diffusion into it and thereby slows the reaction down. If the volume contracts, forced convection sucks reactant into the particle and speeds the reaction up [16,28]. Even for gas reactions without change in mole number, forced convection caused by thermal expansion or contraction under nonisothermal conditions can significantly retard or accelerate the rate [29]. Moreover, it can possibly lead to

294

Chapter 9. Heterogeneous Catalysis

nmi I 0.1

I

I I I I ml 0.5

1.0

i

i i I i ml 5.0

= /?.

|

10.0

j I I I Mil 50

100

L 500

1000

Ks,P, 2).

Figure 9.4, Effectiveness factor as function of modified Thiele modulus under nonisotheraial conditions (from Weisz and Hicks [27]).

9.6. "Heterogenized" catalysis

295

oscillations of temperature and rate: Heat generation by fast conversion causes the temperature at the particle center to rise, producing volume expansion and outward convection that counteracts reactant entry. As a result, the rate decreases, allowing the particle center to cool. Cooling produces volume contraction and inward convection that boosts reactant entry and so speeds up the reaction again, letting the cycle start anew. This effect, too, can only be modeled numerically. 9.6. "Heterogenized" catalysis In industrial practice, heterogeneous catalysis offers unique advantages: The use of a fixed-bed flow reactor not only makes the separation of product from catalyst automatic, avoiding an extra processing step that may require undesirable conditions, it also makes it easy to adjust the time of contact with the catalyst to its optimum. This provides an incentive to translate homogeneous into heterogeneous catalysis if possible. Such translations have been achieved in two different fields: in catalysis by ions and by metal complexes. In both cases, the reaction is chemically the same as in homogeneous solution, the heterogeneity being essentially a matter of mechanics rather than chemistry. Catalysis by ion exchangers [30-33]. Many important reactions are catalyzed by ions such as H^, 0H~, Ag"^, or CN~. Among these are hydration and dehydration, esterification and ester solvolysis, etherification, and many condensation reactions. Instead of conducting such a reaction in homogeneous solution with dissolved acid, base, or salt as catalyst, an ion exchanger can be used. Typical ion exchangers are gels in the form of small beads that consist of a polymer skeleton onto which ionic groups are grafted. The beads are either uniform and highly swollen (gel-type resins) or contain relatively large pores for better reactant access (macroporous resins), and can be viewed as polymeric acids, bases, or salts. Electrolytic dissociation within the beads is essentially complete. The "counterions"—e.g., H"^ or Ag"^ in a cation-exchange resin with fixed acid groups—move freely within the gel, and only the constraint of electroneutrality keeps them inside. Most industrial processes of this kind use strong-acid ion exchangers for reactions catalyzed by hydrogen ions. A large-scale example is the synthesis of methyl r^rr.-butyl ether (MTBE) from methanol and isobutene as anti-knock gasoline additive [34,35]. Catalysis by fixed metal complexes [36,37]. Catalytically active metal complexes can be anchored on permeable polymeric supports. This has become a very active field regularly covered at the international conferences on Polymer-Supported Reactions in Organic Chemistry. A single example may suffice here: hydroformylation to produce alcohols from olefins, CO, and H2, catalyzed by phosphinesubstituted metal hydrocarbonyls and often invoked in previous chapters. The metal

296

Chapter 9. Heterogeneous Catalysis

^^ P ^ { CO CO \ .:.

atom of the catalyst (shown here as cobalt) carries as one of its ligands an organic phosphine, one of whose groups, typically an alkyl, can be anchored on the skeleton of a swellable or highly porous polymer [38]. The reaction occurs within the polymer beads in the same way as in homogeneous solution. [However, see Section 13.3 for unusual mass-transfer effects.]

Kinetics. If the kinetics and mechanism of the reaction in homogeneous solution are known, they can be translated at least approximately into the heterogenized system [30,37]. The principal difference between the homogeneous and heterogenized reactions is that the catalyst is distributed uniformly over the entire system in the former, but is concentrated in a part of it in the latter, a distinction of physics rather than chemistry. Provided the reaction is first order in the catalyst, as is usually true, the same amount of catalytically active species—say, hydrogen ions or complexed cobalt metal atoms—could be expected to produce approximately the same rate in both systems. This is a very crude, but useful hypothesis to start with. Complicating facets of the heterogenized reactions are: •

•

In order to reach a catalyst ion or complex, the reactant must diffuse into the resin bead. The environment within the bead may encourage or discourage entry of the reactant. The greater proximity of catalyst species in the bead may affect the yield structure.

Diffusion limitation in general results in a rate that, at the same amount of catalyst species, is lower than in homogeneous solution (for an exception, see Section 13.3). It can be minimized by using beads that swell more strongly (lower degree of crosslinking) or are more porous, but only at the expense of lowering the amount of catalyst species per unit volume of bead. There is an optimum at which diffusion limitation is not serious and catalyst concentration is still reasonably high. To achieve a higher rate and greater selectivity, the polymer can be equipped with additional groups that attract the desired reactant and keep it in place near a catalytically active group [39]. This is how enzymes achieve high rates and selectivities in biological systems, and great progress has been made in imitating them (enzyme-mimicking polymers [40]). A primitive example is the enhancement of H"^-catalyzed olefin hydration by replacement of some of the H"^ ions in a strongacid ion exchanger by Ag^ ions that form a weak complex with the olefin [41]. Unlike biological systems, chemical manufacturing can use solvents other than water. The interior of an ion exchanger is a highly polar medium that strongly prefers the more polar solvent strongly from a mixture [42,43]. Containing a more polar solvent mixture, the bead then attracts polar reactants in preference to less polar ones, and so boosts their rate [44,45].

9.7. Shape selectivity

297

Usually, higher product purity can be achieved by heterogenized than by homogeneous catalysis. This is because the extra separation step and excess contact time with the catalyst are avoided. However, there are exceptions in which the greater proximity of catalyst species in the heterogeneous system promotes side reactions of reactive intermediates with one another. An example is hydration of glycidaldehyde to glycerol, where the formation of traces of C^ byproducts from trace-level reactive C3 intermediates is greater in the heterogenized system [46]. 9.7. Shape selectivity [47-50] Heterogeneous catalysis offers an opportunity to achieve selectivity based on molecular size and shape. Excellently suited for this purpose are zeolites with their crystal structure of regularly repeating unit cells with voids large enough to accommodate reactants. For catalysis, natural zeolites have been large replaced by synthetic products. Table 9.3 lists structural information on some of the most common natural and synthetic materials. Linde A is a typical representative of small-pore zeolites with windows just large enough to let straight hydrocarbon chains pass through. The windows connect larger "supercages." The most popular catalyst is Exxon-Mobil's ZMS-5 with channels wide enough for aromatics. Zeolites are acidic catalysts. Their acidity and pore Table 9.3. Pore structure of zeolites (data from size can be altered to some exChen et al. [47]). tent by replacement of Al or Si atoms by other tri- or quadrizeolite configuration pore size valent metals, respectively, or A by exchanging their cations. 8-membered rings For example, Ca-exchanged Linde A has slightly larger chabazite windows and 3.8 supercages windows. erionite 3.6x5.1 Linde A 4.1 Even for small crystals, the ratio of surface to interior 10-membered rings catalytic sites is so small that one-dimensional 3.5x6.9 partheite surface activity remains channels negligible in most situations. intersecting 5.3x5.6 ZMS-5 channels

Reactant shape selectivity, The most obvious shape selectivity, and the first to be exploited in practice, stems from the fact that the molecules of some reactants can enter the catalyst

12-membered rings faujasite Linde L

7.4 7.4x6.5 7.1

intersecting channels one-dimensional channels

298

Chapter 9. Heterogeneous Catalysis

while others are too large or bulky to do so. The classical example is that Caexchanged Linde A catalyzes dehydration of /i-butanol, but not of iso-hut2ino\ [51]. The windows of this zeolite are large enough for the —OH group, but not for a branched hydrocarbon chain. An early example of industrial importance is preferential cracking of nparaffins by Ca-exchanged Linde A [52]. ^-Paraffins have low octane numbers, and their selective conversion to easily removed lighter Az-paraffins and -olefins can be used to boost octane numbers in gasoline production [53]. Exxon-Mobil's MForming Process goes one step farther in also reacting the light /7-olefins from cracking with benzene and toluene to high octane-number alkyl benzenes [54]. Product shape selectivity. Shape selectivity can also stem from differences in the size or shape of product. An interesting example of industrial importance is the preferential formation of p-xylene, the most desirable xylene isomer, in alkylation of toluene by methanol on Mg- or P-ZMS-5 [55].

. 4
+ MeOH

•

\P}~

-^oy-

np + ^2^ Hfi

3% ^^° 90%

The channels of the modified ZMS-5 are large enough for the para-, but not for the ortho- and m^ra-isomers. SpatiO'Selectivity. A particularly intriguing manifestation of steric hindrance in zeolite catalysts is what is called spatio-selectivity (or transition-state selectivity). The cages or channels may be large enough for the reactants and products, but not for the activated complexes (transition-state) via which some of the reactions must occur. An example of practical importance is the ability of H-ZMS-5 to isomerize xylenes while blocking the undesired disproportionation to toluene and trimethyl benzenes that proceeds through an activated complex with two benzene rings and accompanies isomerization on non-shape selective acidic catalysts [56].

O)-

<

•

\

o)— + --(o)—

<^0

catalyzed

/

•

(2/

"^ ( 2 ) "

blocked

9.8. Catalyst deactivation

299

Shape selectivity and catalyst deactivation. A serious problem in catalytic cracking and other refinery operations is catalyst deactivation by coking. Coke forms on the catalyst from bulky molecules such as polyalkyl benzenes and polycyclic aromatics that are slow or unable to escape from the catalyst [57]. These molecules, in turn are formed mainly from cracked olefins. Coking is severe in zeolites with windowand-supercage structure (chabazite, erionite, Linde A). Zeolites like ZMS-5, with straight channels and no supercages, are much less affected because the formation of bulky coke precursors is sterically inhibited [58]. 9.8. Catalyst deactivation [59-64] Catalyst activity typically declines with time. Decay may be a matter of seconds or years. It is fast in hydrocarbon cracking, and slow primarily in reactions of organic chemistry under relatively mild conditions. Catalyst deactivation is an important factor in process economics and reactor design. Design may have to allow for the fact that the reactor attains a steady or quasi-stationary state only if decay is slow compared with the catalyzed reaction. This section addresses possible pathways and mathematical modeling of catalyst deactivation. Definitions. Traditionally, a normalized catalyst activity, a, or "activity" for short, is defined as the ratio of die reaction rate at the respective time to that when the catalyst is fresh: a{t) ^ j:^te(0_ (927) rate(r=0) The deactivation rate, r^, can then be taken as the decline of that activity with time, usually a function of the activity itself, the concentrations in the fluid phase, and temperature: '•d -

^

= kJi^^P.T)

(9.28)

Causes. The most common causes of deactivation and remedies for them are: Sintering:

Fouling:

At elevated temperature, catalyst crystallites can migrate on the support and aggregate, reducing the catalyst surface area and activity. Inert barriers ("promoters") may obstruct migration Example: K2O on Fe/Al203 catalysts for ammonia synthesis. Deposits may form in catalyst particles and constrict or clog pores or coat catalytic sites. Chemical treatment may achieve clean-up. Example: clean-up of silica-fouled anion exchange resin catalysts with warm NaOH/NaCl solution.

300

Chapter 9. Heterogeneous Catalysis

Coking:

Coking is a special form of fouling, by carbonaceous deposits in hydrocarbon processing. Coke formation can be slowed by addition of hydrogen to the feed. Coke can be removed by high-temperature "burn-off." A poison may deactivate catalyst sites by irreversible adsorption or chemical reaction. A guard bed preceding the reactor may remove or destroy the poison, or a less sensitive catalyst may be substituted. Example: replacement of Pt, susceptible to sulfur poisoning, by V2O5 in SO2 oxidation in manufacture of sulfuric acid.

Poisoning'.

Volatilization:

Catalyst metal atoms may react with gas-phase species to form volatile compounds such as carbonyls, that are then lost by sublimation. Operation at lower temperature can reduce losses, or a less sensitive catalyst may be found. In principle, deactivation can be modeled in either of two ways: mechanistically, based on known chemistry; or empirically with equations that best fit experimental observations. Mechanistic modeling. Mechanistic modeling requires first of all a knowledge of the network of the desired reaction. Beyond that, information is needed about how the deactivation occurs. The four principle pathways of deactivation can be represented schematically: parallel deactivation

A -

o-

-> P

consecutive deactivation simultaneous deactivation independent deactivation where Apoison.

deactivation by a reactant

•©

o^© -> p poi

deactivation by poison in feed

+ 0—© -> P

O,®,® — ©

P is the desired reaction, (j\

deactivation by a product

deactivation with time

is a deactivated catalytic site, and poi is a

Example 9.8. Parallel and sequential deactivation in a hypothetical reaction. The principle of mechanistic modeling can be illustrated by the oversimplified example of a single-step isomerization reaction A <—^ P with Langmuir-Hinshelwood kinetics, rate control by the surface reaction, and slow second-order deactivation.

9.8. Catalyst deactivation

301

For parallel deactivation, the network can be written

S-© (9.29)

(The double-line arrow indicates that catalyst decay is second order). The LangmuirHinshelwood rate equation, for control by the surface reaction and with q^{t) instead of q^ to account for progressing deactivation, is ^ '

^ k^K^ip^ - p,IK)q^a{t) 1 + K^p^ + .STpPp

(9.30)

The concentration of adsorbed A at any time is (9.31) 1 + K^p^ + K^p^ With this and/7p(/) = p^— pS), the deactivation rate is found to be da dt

K,p,{t)a{t) KQI:

(9.32)

\^K,pl^{K^-K,)p^{t)

If second-order deactivation were sequential rather than parallel, its rate would be that of the step ( ? ) = • ( J ) , and the deactivation rate equation would contain K^p^ (t) = Kp \p^ — /7A(0] instead of K/^ p^ (t) in the numerator. Parallel deactivation is relatively fast at low conversion, sequential deactivation is so at high conversion. Whether deactivation is parallel or sequential, its rate equation must be integrated simultaneously with that for the desired reaction. For deactivation, the integration is over time, for the desired reaction in a flow reactor it is over reactor space time. Only in a batch reactor is the independent variable the same. Even so, numerical computation is usually necessary. This example of a very simple situation shows that mechanistic modeling calls for detailed knowledge of pathways and leads to complications even if approximations are used generously. In practice, a lack of required information allows mechanistic modeling only in exceptional cases.

302

Chapter 9. Heterogeneous Catalysis

Time-on-stream modeling. An empirical approach that is simpler and requires less information is more common. It presumes that the activity depends only on temperature and time, not on the composition of the contacting fluid. The simplest models of this kind postulate a power law for the decay raters = k{T)[a{t)r (9.33) This comes closest to reality for aging mechanisms, whose rates are indeed largely independent of the composition of the contacting fluid. For example, in sintering [65,66], the crystallites migrate regardless of any adsorbates, and decay can be represented reasonably well as a reaction of second order in total catalytic sites. Provided deactivation is slow compared with the desired reaction, such models allow the activity to be calculated separately as a function of time, and then used in the rate equation for the desired reaction. For example, for second-order sintering at constant temperature: r^ =

-daldt

= k^[a{t)f

This gives a{t) = 1/(1+it/)

(9.34)

Accordingly, sintering is accounted for if q^ in the rate equation for the main reaction is replaced by q^/{l + kj). Even here, numerical integration is usually required because the integrand contains both t and/?A(0In other time-on-stream models, mosdy used for coking, the decay rate is assumed to be independent of the current activity. An old but still widely used empirical equation by Voorhies [67] relates the coke content, ^^oke. of a catalyst in hydrocarbon processing to temperature and time: ^coke = k{T^^"

(^ - 0.5)

(9.35)

where k and n are empirical coefficients that depend on the nature of the feed stream. The readily measurable coke content can in turn be related to the activity loss by other empirical equations. For dehydrogenation of 1-butene to butadiene over a chromia-alumina catalyst, Dumez and Froment [68] observed a decrease of the coking rate with coke content according to ^coke = ^ln(l + aCeO

(9.36)

where r^oke is the coking rate of a fresh catalyst, and a is an empirical constant. Catalyst deactivation usually parallels coke formation. A detailed discussion of coking can be found in the book by Froment and Bischoff [69].

Summary

303

Nonuniform deactivation. Many catalytically active metals or groups are also good adsorbents for poisons. A typical example is poisoning of Group VIII transition metals by sulfur compounds. Poison entering a fresh catalyst particle may be adsorbed before it can penetrate farther into the interior. After some exposure, the particle then consists of a poisoned shell and a still intact core. Deactivation progresses as more poison diffuses through the shell and becomes adsorbed on still active sites of the core (progressive-shell or shrinking-core poisoning). Coking in catalytic cracking of heavier hydrocarbons tends to occur in a similar fashion: Coke deposits form in the outer layers of the catalyst particles before they do in the interior. In the simplest and quite common situation, poison molecules are adsorbed as soon as they have traversed the shell, and poison concentration in the fluid is so low that deactivation remains slow compared with the catalyzed reaction. If so, the boundary between shell and core remains sharp, and diffusion of poison across the shell is quasi-stationary. The remaining activity as a function of time then is [70] 0 Vi

a(t) = I + sin I s i n - ^ l - 4 T ) a{t) = 0 T s

(9.37)

3DpoiCpoi/ro^^j;

(Dpoi = effective diffusion coefficient of poison in shell, q^^i = poison concentration at particle surface; most sources on shrinking-core mechanisms, including Crank's original derivation for diffusion with rapid immobilization [71], give an equivalent solution for time as a function of conversion rather than vice versa.). The same mathematics apply for coke burn-off in catalyst particles [72]. More detail on nonuniform deactivation is given in Hill's book [73]. Another form of nonuniform deactivation can occur if catalyst sites differ in their activity and their adsorption affinity or chemical sensitivity. Usually, the most active sites are also the most susceptible to deactivation. If so, the activity of a fresh catalyst declines steeply at first and then at a steady, slower pace. This is believed to happen in some petroleum refining operations. Another example is loss of sulfonic acid groups in synthesis of methyl tert.-hntyl ether catalyzed by ion exchangers. In this application, the acid shed by the catalyst causes corrosion problems, and catalysts are therefore "aged" under reaction conditions to eliminate the most labile acid groups before being charged to the reactors. Summary Heterogeneous catalysis is most commonly modeled as a surface reaction with adsorption and desorption as additional steps. Following Langmuir, adsorption is taken to be a reversible "reaction" of a molecule in the contacting fluid with a vacant catalytic site

304

Chapter 9. Heterogeneous Catalysis

(Langmuir-Hinshelwood kinetics). Christiansen's formula is applicable if the reaction involves only a single site. This excludes reactions of adsorbed species with one another. Kinetics of reactions on more than one site are more complex. A general formalism for single-step surface reactions and rate control by the reaction, adsorption of a reactant, or desorption of a product has been developed by Hougen and Watson: ^^tg ^

(kinetic group)(driving-force group) adsorption group

^9^^

The groups have been tabulated for the three possible rate-controlling steps and four simple stoichiometrics. Included are possible dissociation of a reactant upon adsorption and reaction of adsorbed species directly with molecules in the contacting fluid. As in homogeneous catalysis, the existence of a most abundant catalyst-surface species (macs) or one or several low-abundance catalyst-surface species (lacs) can greatly simplify the mathematics of multistep surface reactions. In some cases, mathematics reduces to that of a single reaction step, so that the Hougen-Watson formula becomes applicable. Reaction orders of reactants may be negative, possibly causing self-acceleration within some conversion range. Such behavior requires involvement of at least two catalytic sites and tends to arise if the respective reactants is the macs or at least is very strongly adsorbed. More often than in homogeneous reactions, different mechanisms or rate-controlling steps may give approximate rate equations of the same algebraic form. An empirical rate equation alone thus cannot provide conclusive evidence in support of a mechanism. Additional information that can be used for model discrimination includes the nature and temperature dependence of the coefficients and the concentration dependence of initial rates. In particular, all coefficients must be positive; single-step coefficients and their products, ratios, and ratios of products must obey the Arrhenius or van't Hoff equation; single-step coefficients and their products must increase with increasing temperature; and adsorption equilibrium constants should decrease with increasing temperature. Before a model can be relied upon, it should prove its predictive power. In industrial practice, the most convenient way of accounting for mass-transfer effects is to view the penetrable catalyst particle as a pseudo-homogeneous phase. Obstruction of mass transfer by the solid material in the particle then is reflected by an "effective" intraparticle mass-transfer or diffusion coefficient that is appropriately lower than in the contacting fluid. If this approach is taken, two fundamentally different masstransfer situations appear: Mass transfer to and from the particle across an adherent boundary layer is affected by the reaction only in that the latter sets the boundary condition at the particle. Here, mass transfer and reaction are sequential and occur in different parts of the system, and the slower of the two is the bottleneck and dictates the overall rate and its temperature dependence. Within the particle, however, mass transfer and reaction occur simultaneously and in the same volume element. Here, the reaction introduces a source-or-sink term into the basic differential material balance. If the reaction is slow, it alone controls the overall rate and its temperature dependence. If mass transfer is slow, both reaction and mass transfer affect the rate, and the apparent reaction order and activation energy are the arithmetic means of those of reaction and mass transfer.

References

305

The classic Thiele-Damkohler theory accounts for these effects, but is restricted to isothermal behavior and intraparticle mass transfer only by diffusion. If the reaction is highly exothermic and the particle is a poor heat conductor, the temperature in the particle center may rise above that in the contacting fluid and cause the overall rate to be higher than in the absence of heat- and mass-transfer limitations. Moreover, gas-phase reactions with change in mole number cause forced inward or outward convection that assists or counteracts reactant penetration into the particle and so enhances or depresses the rate. Heterogeneous catalysis offers advantages in industrial processes in that separation of products from catalyst requires no extra step, and time of contact with the catalyst is easily adjusted to its optimum. Reactions of homogeneous catalysis by ions or metal complexes can be "heterogenized" by use of solid ion exchangers or by grafting a ligand of the complex onto a solid polymer. The chemistry then is the same as in homogeneous solution, but the reaction requires mass transfer into and out of the catalyst beads. Moreover, the environment within the beads may affect the rate and yield structure. Heterogeneous catalysts deteriorate in use. The most common causes are sintering (aggregation of catalyst crystallites on the support), fouling (formation of obstructing deposits within the particles), coking (formation of carbonaceous deposits), poisoning (deactivation of catalytic sites by chemical reaction or irreversible adsorption), and volatilization (loss of catalyst material because of reactions forming volatile species). Mechanistic modeling of deactivation is possible if the mechanisms of the desired reaction and of deactivation are known in detail. However, such information is almost never available. More common is empirical time-on-stream modeling which postulates that the decline of catalyst activity depends only on temperature and time, not on the composition of the contacting fluid. Deactivation may be nonuniform if the most active sites are the first to be deactivated, or if poisoning or coating of sites progresses from the particle surface toward the center. In the latter case, a sharp boundary between a deactivated shell and a still fully active core may exist. In both cases, deactivation tends to be relatively fast at first and then settle down to a steady, slower pace. Examples include hydrogenation of propanal over nickel, dehydrogenation of ethanol over copper-cobalt, dehydrogenation of methylcyclohexane to toluene over platinum, hydroformylation of olefins catalyzed by cobalt hydrocarbonyls on solid polymers, hydrogen-ion catalyzed hydration of olefins on ion exchangers, dehydrogenation of 1-butene to butadiene over chromia-alumina, and various hypothetical reactions. References General references Gl. G2. G3.

J. J. Carberry, Chemical and catalytic reaction engineering, McGraw-Hill, New York, 1976, ISBN 0070097909, Chapters 8 and 9. H. S. Fogler, Elements of chemical reaction engineering, Prentice-Hall, Upper Saddle River, 3rd ed., 1999, ISBN 0135317088, Chapters 10-12.G2. G. F. Froment and K. B. Bischoff, Chemical reactor analysis and design, Wiley, 2nd ed., 1990, ISBN 0471510440, Chapters 2, 3, and 5.

306 G4. 05. 06. 07.

Chapter 9. Heterogeneous Catalysis C. G. Hill, h.,An introduction to chemical engineering kinetics & reactor design, Wiley, New York, 1977, ISBN 0471396095, Chapters 6 and 12. O. A. Hougen and K. M. Watson, Chemical process principles, Part HI: Kinetics and catalysis, Wiley. New York, 1947. O. Levenspiel, Chemical reaction engineering, Wiley, New York, 2nd ed., 1972, ISBN 0471530166, Chapter 11. C. N. Satterfield, Heterogeneous catalysis in practice, McOraw-Hill, New York, 1980, ISBN 0070548757.

Specific references 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

I. Langmuir, /. Am. Chem. 5oc., 40 (1918) 1361. C. N. Hinshelwood, The kinetics of chemical change. Clarendon, Oxford, 1940. O. A, Hougen and K. M. Watson, Ind, Eng. Chem., 35 (1943) 529; see also ref. 05, pp.906-907. K. H. Yang and O. A. Hougen, Chem. Eng. Prog., 46 (1950) 146. M. Boudart, AIChE J., 18 (1972) 465. C. C. Oldenburg and H. F. Rase, AIChE J., 3 (1957) 462. C. O. Hill, Jr. (ref. 04), Section 6.3.1.2. M. Boudart, Adsorption and chemisorption, in Surface chemistry of metals and semiconductors (symposium), H. C. Oatos, ed., Wiley, New York, 1960, p. 409. J. Franckaerts and O. F. Froment, Chem. Eng. Sci., 19 (1964) 807; see also Froment and Bischoff (ref. 02), Section 2.4.2. J. H. Sinfelt, H. Hurwitz, and R. A. Shulman, J. Phys. Chem., 64 (1960) 1559; see also Carberry (ref. 01), pp. 414-420. Carberry (ref. 0 1 , Chapter 9). Fogler (ref. 02), Chapters 11 and 12. Froment and Bischoff (ref. 03), Chapter 3. Hill (ref. 04), Chapter 12. Levenspiel (ref. Q6), Chapter 14. O. Damkohler, Chemie-Ingenieur, 3 (1937) 430. E. W. Thiele, Ind. Eng. Chem., 31 (1939) 916. Froment and Bischoff (ref. 03), Section 3.6.1. Hill (ref. 04), Section 12.3.1.4. Levenspiel (ref. 06), p. 475. C. N. Satterfield, Mass transfer in heterogeneous catalysis, MIT Press, Cambridge, 1970, ISBN 0262190621. R. Aris, The mathematical theory of diffusion and reaction in permeable catalysts. Clarendon Press, Oxford, 1975, 2 vols., ISBN 0198519311 and 1198519427. R. Aris, Ind. Eng. Chem. Fundam., 4 (1965), 227, 487. O. W. Roberts and C. N. Satterfield, Ind. Eng. Chem. Fundam., 4 (1965) 286; 5 (1966) 317.

References 25. 26. 27. 28. 29. 30. 31. 32. 33.

34. 35. 36. 37.

38. 39. 40. 41. 42. 43. 44. 45.

46. 47.

48. 49. 50.

51.

307

D. Luss and N. R. Amundson, AIChE /., 13 (1967) 759. G. Damkohler, Z. Phys. Chem., A193 (1943) 16. P. B. Weisz and J. S. Hicks, Chem. Eng. Sci., 17 (1961) 265. V. W. Weekman, Jr., and R. L. Goring, 7. CataL, 4 (1965) 260. V. W. Weekman, Jr., /. CataL, 5 (1966) 44. F. Helfferich, Ion Exchange, McGraw-Hill, New York, 1962 (reprint Dover, Mineola, 1995, ISBN 0486687848), Chapter 11. R. L. Albright, React. Polym., 4 (1986) 155. M. M. Sharma, React. Polym., 26 (1995) 3. F. G. Helfferich and Y.-L. Hwang, Ion exchangers as catalysts, in Ion exchange for industry, M. Streat, ed., Ellis Horwood, Chichester, 1988, ISBN 0745803539, p. 585. H. L. Hoffman, Hydrocarbon Process., 59, No. 2 (1980) 57. Jarvelin, H., Etherification, in Modem Petroleum Technology, 6th ed.. Vol. 2, A. G. Lucas, ed., Wiley, Chichester, 2000, ISBN 0471984116. F. R. Hartley, Supported metal complexes, in Catalysis by metal complexes, R. Ugo and B. R. James, eds., Reidel, Dordrecht, 1985, ISBN 9027718555. P. E. Barroe and B. C. Gates, in Syntheses and separations using functional polymers, D. C. Sherrington and P. Hodge, eds., Wiley, New York, 1988, ISBN 0471918482, p. 123. C. U. Pittman, Jr., and G. O. Evans, Chem. Tech., 3 (1973) 560. K. Jefabek, Z. Prokop, and A. Revillon, React. Polym., 33 (1997) 103. A. Leonhardt and K. Mosbach, React. Polym., 6 (1987) 285. S. Affrosman and J. P. Murray, J. Chem. Soc, B 1966, 1015. C. W. Davies and B. D. R. Owen, J. Chem. Soc, 1956, 1676. F. Helfferich, Ion Echange, McGraw-Hill, New York, 1962, Section 10.2.b; (reprint Dover, Mineola, 1995, ISBN 0486687848). G. G. Thomas and C. W. Davies, Nature, 159 (1947) 372. F. Helfferich, J. Am. Chem. Soc, 76 1954) 5567; Ion Exchange, McGraw-Hill, New York, 1962 (reprint Dover, Mineola, 1995, ISBN 0486687848), Section 11.3.d. F. G. Helfferich, unpublished, 1963. N. Y. Chen, T. F. Degnan, Jr., and C. M. Smith, Molecular transport and reaction in zeolites: design and applications of shape selective catalysts, VCH, 1994, ISBN 0895737655. N. Y. Chen, W. E. Garwood, and F. G. Dwyer, Shape selective catalysis in industrial applications, Dekker, New York, 2nd ed., 1996, ISBN 082479737X. P. B. Weisz, ACS Symp. Ser., 738 (1999) 18. S. M. Csicsery and I. Kiricsi, Shape-selective catalysis—heterogeneous, in Encyclopedia of catalysis, I. T. Horvath ed.-in-chief, Wiley, 2002, ISBN 0471241830, Vol. 6, p. 307. P. B. Weisz, W. R. Maatman, and E. B. Mower, /. Catal, 1 (1962) 307.

308 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73.

Chapter 9. Heterogeneous Catalysis P.B. Weisz and V. J. Frilette, J. Phys. Chem., 64 (1960) 382. C. J. Plank, E. R. Rosinski, and E. N. Givens, U.S.Patents 4,141,859 (1979) and 4,276,151 (1981). H. Heinemann, Catal. Rev.-Sci. Eng,, 15 (1977) 53. W. W. Kaeding, G. C. Barlie, and M. M. Wu, Catal Rev.-Sci, Eng., 26 (1984) 597. D. H. Olson and W. O. Haag, ACS Symp. Ser., 248 (1984) 275. D. E. Walsh and L. D. Rollman, /. Catal., 56 (1979) 195. N. Y. Chen and W. E. Garwood, J. Catal., 652 (1978) 453. Carberry (ref. Gl), Section 8.14. Fogler (ref. G2), Section 10.7. Froment and Bischoff (ref. G3), Chapter 5. Hill (ref. G4), Section 6.6. Levenspiel (ref. G6), Chapter 15. Satterfield (ref. G7), Section 1.3.10. G. C. Kuczynski, ed.. Sintering and catalysis (Materials science research, vol. 10), Plenum, New York, 1975, ISBN 0306385104. S. A. Stevenson, ed., Metal-support interactions in catalysis, sintering, and redisposition. Van Nostrand-Reinhold, New York, 1987, ISBN 0442211600. A. Voorhies, Ind. Eng. Chem., 37 (1945) 318. F. J. Dumez and G. F. Froment, Ind. Eng. Chem., Proc. Des. Dev., 15 (1976) 291; see also Froment and Bischoff (ref. G2), pp. 247-250. Froment and Bischoff (ref. G.2), Section 5.3. F. G. Helfferich, J. Phys. Chem., 69 (1965) 1178, eqn 23. J. Crank, Trans. Faraday Soc, 53 (1957) 1063. P. B. Weisz and R. D. Goodwin, J. Catal, 2 (1963) 397. Hill (ref. G4), Section 12.3.3.2.

Chapter 10 Chain Reactions Chain reactions rely on the existence of highly reactive intermediates, so-called chain carriers, that take part in a repeating sequence of steps in which they are consumed and produced anew. A typical sequence of this kind, with two chain carriers X and Y converting reactants A and B to products P and Q, is X+ A Y -h B

• P +Y • Q+X

(10.1)

Chain carrier X, consumed in the first step, re-appears as a product in the second. Likewise, Y, consumed in the second step, re-appears as a product in the first. If you will (and belong to those who still learned FORTRAN in school), a chain reaction is a "reaction with a DO loop." Like a DO loop in FORTRAN, the chain must be started and eventually be terminated. It is started by an additional reaction that serves as source of chain carriers. This is called initiation. It is terminated by a reaction that, once in a while, consumes chain carriers without generating others. This is called termination. The steps of the DO loop are called propagation. Typically, chain carriers are atoms or radicals with unpaired electrons. They resemble catalysts in that they are reformed after having been consumed, but differ in that they have extremely short life spans and do not survive the overall reaction. Moreover, the initiation and termination steps and their kinetic implications set chain reactions apart from catalysis as well as from any other kinds of chemical reactions. 10.1. General Properties Once a chain has been started, the propagation cycle typically runs through many repetitions before being terminated. This makes chain reactions very sensitive: A minute amount of an initiator that generates chain carriers can set off a massive and fast reaction. Chain reactions are also quite sensitive to any substances that act as radical traps. The presence of a very small amount of such a substance can effectively cause a chain reaction to "flame out."

310

Chapter 10. Chain reactions

Another typical feature occasionally encountered in chain reactions is a time delay upon initiation: When the initiator is added to the reactants, the reaction may start only after an induction period rather than immediately. This happens, for example, if a trace contaminant acting as inhibitor is present: The contaminant catches and consumes the chain carriers produced by the initiator until it has been destroyed in the process. Perhaps the most striking feature of chain reactions is that some of them can result in detonation. If the propagation mechanism includes a step or steps that produce more chain carriers than they consume, the reaction is self-accelerating. This is called chain branching. The result may be an event much more violent than a thermal explosion, in which self-acceleration stems from temperature increase owing to the inability of heat transfer to keep up with the heat production of the reaction. The detonation of a nuclear bomb can be viewed as a chain reaction with neutrons as chain carriers and with chain branching. As highly reactive species with very short life spans, the chain carriers remain at trace level (except in a detonation). The Bodenstein approximation of quasi-stationary behavior thus is applicable to them. In fact, it was first introduced by Bodenstein and his school in their study and mathematical analysis of chain reactions [1]. Its validity for chain carriers will be taken for granted throughout this chapter. 10.2. Initiation Some chain reactions start spontaneously, without prodding by an added initiator because a reactant does that job by itself. The hydrogen-bromide reaction, in which thermal dissociation of Br2 initiates the reaction at sufficiently high temperatures, is an example (see Example 10.1 in next section). In other chain reactions, an initiator need not be added because it forms spontaneously from some intermediate or product. Such a reaction may be quite unintended. For example, upon prolonged standing in contact with air, hydrocarbons accumulate hydroperoxides. The decomposition of the latter into radicals can be set off by trace contaminants (e.g., transition-metal ions, see below) or even the friction of a ground-glass stopper being turned as a flask is opened or closed. This can trigger a self-accelerating reaction that leads to an explosion. Most chain reactions have to be jump-started by an intentionally added initiator. For liquid-phase reactions, the most commonly used initiators are benzoyl peroxide and 2,2'-azo-to-isobutyronitrile (AIBN), unstable compounds that readily decompose into radicals:
^ 2 <0>-C-0.

10.3. Reactions with two chain carriers: the Yi^x reaction H3C ^l^s H3C-C-N=N~C-CH3 NC CN

H-jC 2 H3C-CNC

•

+

311

N2

(dots indicate unpaired electrons). The initiator molecule produces two radicals, R •, that can act as chain carriers (and possibly an additional inert species such as N2):

in

• 2R- + ...

(10.2)

(in = initiator). The rate of chain-carrier production accordingly is r.

= 2k. .C

init

init

(10.3) in

^

At low temperature, decomposition of an initiator may require promotion. Transition-metal ions such as Fe^"^, Co^"^, Ag"*^, etc., that can easily switch valence states can serve this purpose. A likely mechanism of promotion of peroxide initiation by an ion Me^"^ is the Haber-Weiss redox cycle [2]: ROOH

(10.4)

ROO- <

-:^

^

ROOH

Some chain reactions can be initiated photochemically [3]. In fact, most of the early work on kinetics of chain reactions was done with photochemical initiation. For example, in the hydrogen-bromide reaction (see next section), initiation Br2 —• 2 Br can be achieved with ultraviolet light. Such initiation allows the reaction to be conducted at a lower temperature at which thermal initiation is ineffective. This may be an advantage in an industrial process, and also offers opportunities for elucidation of reaction mechanisms. Lastly, some gas-phase chain reactions, among them the hydrogen-oxygen reaction, can be triggered by an electric discharge.

10.3. Reactions with two chain carriers: the hydrogen-bromide reaction The simplest chain reactions are those with two chain carriers as in the sequence 10.1, one or both generated by initiation, and with termination by binary coupling (also called recombination) of chain carriers. The propagation steps may or may not be reversible. In general terms the network can be written:

Chapter 10, Chain reactions

312

propagation initiation

termination X + X Y + ¥•

- • X, Y

"^ -(10.5)

X + Y-

(reactants and products other than chain carrier not shown, and to be accounted for in the respective X coefficients; see Section 6.2). In such a network, the rates of the two propagation steps are 'pi

(10.6)

'p2

(indices pi and p2 are used to avoid ambiguity; minus signs in indices indicate reverse steps). The Bodenstein approximation for X gives '•x = ('•x)i„i, - C,(X.p. + \,) - C , ( \ , . X.p,) . (r,)„^ = 0

(10.7)

Here, (A'x)init and {r^)^,^ are the rates of production and elimination of X by initiation and termination, respectively (the termination rate is negative). Long-chain approximation. Characteristic of most chain reactions is that the propagation cycle is repeated many times between initiation and termination. If so, the initiation and termination terms (first and last in eqn 10.7) are small compared with the propagation terms. The so-called long-chain approximation [4] disregards them. Equation 10.7 can then be used to express the concentration of one chain carrier as a function of that of the other:

c, ^

(10.8)

Cv

With this substitution, eqns 10.6 yield

•pi

'p2

=

Cv

W2 ^-Pl ^

^-plVp2

(10.9)

\ 2

Termination. Equation 10.9 involves the still unknown chain-carrier concentration Cx- This concentration, and with it the final form of the rate equation, depends on the type of termination. In principle, there are three potential mechanisms of termination by coupling: X -I- X —• ..., Y + Y —• ..., and X 4- Y —• .... In reactions

10.3,

Reactions

with two chain carriers:

the HBr

reaction

313

of two radicals with one another to yield an inert product, no bonds need to be broken as the odd electrons link u p , and the great majority of collisions are successful (possible exceptions will be discussed later). The activation energies then are almost nil and the rate coefficients are very similar. The relative rates of the three possible terminations, proportional to Cx^ Cy^, and CxCy, thus depend very strongly on the relative concentrations of the chain carriers: If X outnumbers Y by, say, a factor 100, then X + X termination is roughly times 100 as fast as X + Y , and (100)^ = 10,000 times as fast as Y 4- Y. Termination by coupling tends to be dominated by the most abundant chain carrier.

If the concentrations of X and Y are comparable, all three mechanisms contribute. In general, the overall termination rate r^,^ = (rx)trm + (^Y)trm (consumption of all chain carriers) thus is seen to be: termination

termination rate

if C^ » Cy

X +X

-'^Kx^x

if C^ « Cy

Y +Y

~

if Q « Cy

X + X, X+Y, Y+Y

(10.10)

cY Y

- 2 {k^y^ Cy + A:^y Cy + K^y^ Cy Cy)

where k^y^, k^^, and k^y^^ are the respective rate coefficients (c stands for coupling; the rates are negative because they refer to consumption of chain carriers, and the factor 2 reflects the fact that two carriers are consumed.) The listings 10.10 presume that the reactions of chain carriers with one another proceed unhindered to stable products rather than being blocked or producing substances that in short order decompose again into chain carriers. This is not always the case. Larger radicals may react more sluggishly. Others, among them benzyl, are stabilized to some extent by resonance. Also, for example, two oxy radicals, RO-, react with one another to form a moderately labile peroxide, ROOR, that soon reverts to the two radicals. In cases like the latter:

If the most abundant radical cannot terminate the chain by reaction with itself, it will do so by reaction with whichever radical is the next most abundant.

Chapter 10. Chain reactions

314

In reactions with only two radicals, X and Y: termination:

X +Y

rate:

••^^cXY^X^Y

(10.11)

This is the only situation in which "mixed" termination—that is, coupling of two different chain carriers—can dominate termination. Chain-carrier concentrations. Equation 10.9, derived with the long-chain approximation, shows the rates of the two propagation steps under quasi-stationary conditions to be equal: Each of the two steps consumes as many chain carriers as the other produces. Since the total number of chain carriers is not altered by the propagation steps, the overall rates of chain carrier production and consumption (of X plus Y) must be equal in magnitude: r. + r init

= 0

(10.12)

trm

^

^

The equality applies to the total chain-carrier population, X + Y, but by no means necessarily to each chain carrier separately. For instance, initiation could produce X while termination eliminates Y (granted long chains, the initiation and termination terms in eqn 10.7 are both small compared with the two propagation terms, but are not necessarily equal to one another). Equation 10.12 allows Cx to be calculated from the initiation rate given by eqn 10.3, the appropriate termination rate given by eqn 10.10 or 10.11, and elimination of Cy with eqn 10.8 if necessary: concentration of chain carrier X

termination X + X

^X ^

Y + Y

C, ^

(^init'^cx)

Qn

C\ l

X + Y

C, s

all three

Cx

(10.13)

^ ^-P2

k:.:

c k. . C imt

=

-p2 ^cX

+

^cY ^-Pl

^

in

+ ^.

^ 2

Propagation rate. The rates r^ = r^^ = r^2 of the propagation steps can now be calculated from eqn 10.9 with the appropriate substitution for C^ from eqns 10.13:

10.3. Reactions with two chain carriers: the HBr reaction f

termination X + X

1 1/2 r

KiK2 Ki ^ \2

(10.14)

\l\2 =

'•p

r

^ 1/2 r \ l \ 2

'-p

^-pl>^-p2

=

Pl r

r^ =

^

315

1/2

(10.15)

c

-P2

^

pl p2

^init

[ ^cXY J

-pl -p2

(10.16)

c

[(^-P

all three ^init ( ^ 1 ^ ) 2 '"p

=

^ - p l ' ^ - p 2)^111

/

\1

(^cx(Vpl - \2y - Kyihi - K2y - ^CXY(>^-P1 ^ \ 2 ) ( \ l -

(10.17)

KJ

Equation 10.17, accounting for all three termination mechanisms, is the most general rate equation and contains the other three as special, more common cases. If both propagation steps are irreversible, eqns 10.14 to 10.17 reduce to: termmation X +X

^p

^

v^Jnit / ^ c X )

\

Y +Y

''p

=

(^init/^cY/

\ 2 Q n

X + Y

=

all three

^.

=

{Kinh'kcXY)

(10.18)

1 ^in

(\l\2)

^init \ l \ 2 ^ i

(10.19) (10.20)

Qn 1/2 U/2

(10.21)

(KXK^ + KYKI "- *c: Note that X is the chain carrier that acts as reactant in the first propagation step. For reactions with irreversible propagation steps: • the rate is controlled by the propagation step that consumes the terminating chain carrier (first propagation step if termination is X 4- X; second, if it is Y +Y). The rates of product formation (and reactant consumption) are seen to be of order one half in the initiator; or, if the reaction is initiated by a reactant converted in the propagation cycle, the rate equation involves exponents of one half or integer multiples of one half. For an example, see the hydrogen-bromide reaction below. This is one of the exceptions to the rule that reasonably simple mechanisms do not yield rate equations with fractional exponents. [The other exceptions are reactions with fast pre-dissociation, heterogeneous catalysis with a reactant that dissociates upon adsorption, and some ionic polymerizations (see Sections 5.6. 9.3, and 11.4).]

316

Chapter 10. Chain reactions Example 10,1, The hydrogen-bromide reaction. The elucidation of the mechanism of the gas-phase reaction H2 + Br2

• 2 HBr

(10.22)

by Bodenstein [3,5], Christiansen [6], Herzfeld [7], and Polanyi [8] in 1907-1920 has been one of the most remarkable achievements of classical chemical kinetics. It led to the discovery of chain mechanisms and the introduction of the approximation of quasi-stationary states of trace intermediates by Bodenstein [1]. In the formalism used here, the reaction is H

initiation

j r HBr

2 '^•*^^,^—..^--^

termination

(10.23)

pi

Br2 —• Br- -I- Br-

Br-

HBr'

H-

Br- + Br- —• Br^

^^Br^

Note that the step Br • -f H2 —> HBr -f H • is reversible, but the step H • + Br2 —• HBr 4- Br- is not (see next section). Since termination consumes the chain carrier that acts as reactant in the first propagation step, eqn 10.14 applies. The rates of initiation and HBr production are ^ini. = 2/:.„,/73^,

r^B, = r^, + r^^ = 2r^

(because this is a gas-phase reaction, partial pressures are used in lieu of concentrations). With these rates and the substitutions X = Br- ,

Y = H-,

Xp, = /:p,/?H2,

V

= ^p2PBr2>

^ - p l = ^-plPHBr,

X-p2 = 0

equation 10.14 gives ^((^JKB^PBryKlKlPn^PBr,

^

k-plPmr + KlPBr,

KPH^PB^,

(10.24)

1 + ^HBr/^Br,

where a

d p i '

b

-pi p2

Here, K^ = k^^^^ Ik^^, is the dissociation constant of Br2. A minor complication in the initiation and termination steps will be dealt with in the next section. 10.4.

Identification of relevant steps

Chain carriers, be they atoms or radicals, react with almost any molecules they encounter. As a result, their presence can give rise to a bewildering multitude of reaction steps. Usually, most or all of them occur to some extent, yet the overall behavior of the reaction is dominated by only a few. A central problem in the

10,4. Identification of relevant steps

317

analysis and interpretation of chain reactions, elucidation of their mechanisms, and establishment of their proper rate equations therefore is to identify which of all the potential steps are relevant. Of course, this problem also exists to some degree in other reactions, but it is aggravated by the high reactivity of the chain carriers. Fortunately, at least as long as the reacting species are small, that same high reactivity also opens the door to an approach not generally applicable to other reactions. The key is the estimate of rate-coefficient ratios and relative rates, as will be discussed in this section and illustrated with an application to the hydrogenbromide reaction. Ratios of rate coefficients. Unlike most other kinetic quantities, rate-coefficient ratios of steps involving radicals as reactants can be estimated on the basis of thermochemical data. The estimates are crude, no better than giving approximate orders of magnitude, but this often suffices. However, the estimates are reliable only if the molecular species involved are indeed small. The basis of such estimates is as follows: Bimolecular collisions of chain carriers (atoms or radicals) with others or with small molecules have a very high probability of resulting in reaction, provided the collision partners carry energy at least equaling the activation energy. This is to say that the pre-expoo nential factor A in the Arrhenius ex equation 2.3 for the rate coefficient is of the same or of a similar order exothermic endothermic of magnitude for all of them. As a result, the ratio of two rate coefficients k^ and ki with activation reaction coordinate energies E^^ and E^ is mainly given Figure 10.1, Activation energies and by the ratio of the exponential enthalpy of reactions of radicals. factors: tx^{-EJRT) expi-EJRT)

exp

(10.25) RT

Also, the energy "hump" that such a reaction must overcome is only slightly higher than the energy level on its higher side (see Figure 10.1). As a result: • The activation energies (?/exothermic bimolecular reactions of radicals are low (usually only a few kJmol"^); experience shows them to be lowest for the most strongly exothermic reactions.

318 •

Chapter 10. Chain reactions The activation energies d?/endothermic bimolecular reactions of radicals are only marginally higher than the respective reaction enthalpies, A//°; the difference is found smallest for the most strongly endothermic reactions.

These regularities combined with the approximation 10.25 allow the following order-of-magnitude estimates of ratios k^ Ik2 of rate coefficients of two bimolecular reaction steps 1 and 2 with enthalpies A//i° and A//2° to be stated: if A^i° < A//2° < 0 (both steps exothermic)

kjkj about 1 to 10 (more highly exothermic step faster)

if A / / l ° < 0 < AH2°

, ,,

.^^ 26)

/AL7 0/D7^

(step 1 exothermic, f^ \ ^ ^^P m^^IRT) ^ 27) \ o A .X. '\ (exothermic step faster) step 2 endothermic) ^ ^ ^ if 0 < A//i° < A//2° ^i/^2 « exp[(A//2°-A//i°)//?7)] .^^ 28) (both steps endothermic) (less highly endothermic step faster) ^ ' ^ In practice, the second and third estimates tend to be on the high side. Also, all of them become unreliable if the steps involve large species with many internal degrees of freedom, over which energy can be distributed. The procedure underlying the approximations 10.25 to 10.28 is closely related to, but not identical with, those of Evans and Polanyi [9] and Semenov [10] for reactions of homologous reactants [11,12]. The empirical Polanyi equation for the activation energy is E.^ «

aAE + C

where a (with Q < a < 1) and C are constants, and AE is the difference in bond energies of the bonds formed and broken. AE is approximately equal to the standard reaction enthalpy, — A//°, and for homologous reactants the values of a and C can be taken as very similar. With these assumptions:

Semenov [10] has suggested a « 0.25 for exothermic reactions and a. ~ 0.75 for endothermic ones. Since the pre-exponential factors should be very similar for homologous reactants, the ratio of the rate coefficients should be kjk^

^

exp(a(A//2° -

^)IRT)

differing somewhat from the approximations 10.26 to 10.28. However, the procedure was not specifically designed for reactions of radicals and is not directly applicable to comparisons of exothermic with endothermic steps as needed here. More elaborate and more reliable procedures that can be used for estimates of rate coefficients of reactions of radicals are the bond energy-bond order method (BEBO) of Johnston and Parr [13] and the curve-crossing approach of Pross [14].

10,4, Identification of relevant steps

319

Because of its relative simplicity, the hydrogen-bromide reaction (Example 10.1 in the preceding section) serves well to illustrate the application of the estimates to the identification of relevant steps. The rate equation 10.24, originally established empirically, was attributed to the mechanism 10.23. An obvious first question is, why is the reaction initiated by dissociation of Br2 and not of H2 or, once product has been formed, of HBr? Indeed, thermal dissociation of H2 and HBr also contributes chain carriers, but not to an extent large enough to affect the reaction. At atmospheric pressure and, say, 500 K as a typical reaction temperature, the degrees of dissociation of gaseous Br2, HBr, and H2 are of the orders of 10"^, 10"^^, and 10"^^, respectively. The first two of these values show that thermal dissociation of HBr as a source of bromine atoms is negligible compared that of Br2 under any conditions of interest. This leaves the possibility that hydrogen atoms from thermal dissociation of H2 or HBr, although in a minuscule minority, could dominate the behavior thanks to their much higher reactivity. As will be seen, however, they are vastly outnumbered by hydrogen atoms generated by the propagation cycle and so play only an insignificant role. Application of eqn 10.8 to the (quasi-stationary and long-chain) cycle yields ^-pl^HBr^

(irreversibility of the second step cannot yet be taken for granted). The two coefficients in the numerator are for endothermic steps; those in the denominator, for exothermic ones (see Table 10.1). Dividing numerator and denominator by either of the latter two coefficients and using the approximation 10.27 with A^° values from Table 10.1 one finds

^IPBV,

Table 10.1. Approximate standard enthalpies of steps in hydrogen-bromide reaction. steps

reaction enthalpy

H2 - > H • + H •

-h436kJmor^

Br2 —> Br • -1- Br •

-h 190

HBr-> H- + Br-

+ 360

Br- + H2—• HBr + H-

+

67

H • + Br2 —• HBr + Br •

-

170

io->H + IO-^VH /^Br

Although seen to be outnumbered more than a million-fold by bromine atoms at any compositions of interest, hydrogen atoms from propagation are still more than three orders of magnitude more plentiful than their brothers from dissociation of H2 and HBr. Accordingly, the contribution of the latter to initiation is insignificant.

320

Chapter 10. Chain reactions

This examination also illustrates another facet of chain reactions: Granted quasi-stationary conditions and long chains, the concentrations of the chain carriers are coupled, in reactions like 10.5 through the requirement that the rates of the two propagation steps must be equal. Therefore, only one of the two chain carriers can be in dissociation equilibrium with its source, the other gets boosted by the propagation cycle to a higher than thermal concentration. The answer to a second question, why the step Br- + H2—• HBr -f H- is reversible while H- + Br2—• HBr + Br- is not, is now easy to give. Both reverse steps compete for HBr. The first of them is exothermic ( - 67 kJ mol"^) and so occurs at almost every collision; the second is strongly endothermic ( + 170 kJ mol"^) and therefore at a great disadvantage. Estimated with the approximation 10.27, the ratio of the two rate coefficients is of the order of 10^^ at 500 K. Thus, even though Br - outnumbers H • more than a million-fold, the rate of H • + HBr —• H2 + Br - is still more than ten orders of magnitude higher than that of Br • + HBr —• Br2 + H •, whose contribution to HBr consumption accordingly remains insignificant. The fact that, under typical reaction conditions, Br- outnumbers H- by six or more orders of magnitude also explains why coupling of two Br • alone controls termination, despite the fact that much more energy is gained by coupling of two H • or of H - and Br •. All these couplings being highly exothermic, their enthalpies are not a relevant factor, and the much greater abundance of Br- alone decides the issue (see preceding section). A high bond energy is of no great help in coupling of small radicals. For lack of other effective internal degrees of freedom, much of the released energy must be stored as bond-stretching vibration, making the molecule apt to break apart again in short order. Indeed, even Br- recombination occurs primarily in ternary collisions Br- + Br- -h M—• Br2 -h M, where the collision partner M is some other molecule (or the wall of the reaction vessel) that serves to absorb a substantial portion of the released energy [15]. The algebraic form of the rate equation is not affected because thermodynamic consistency requires M to participate in dissociation if it does so in recombination, and the effects cancel since the two rate coefficients appear only as the ratio /Cjnit/^cBr-* Evidence for the ternary mechanism has been provided by experiments with initiation by ultraviolet light at temperatures low enough for thermal dissociation to be negligible in comparison [3,17,18].

* It has been said that only termination, but not dissociation, involves a collision partner M and that the ratio k,^^ Ik^^, in the rate equation does not equal the dissociation equilibrium constant because the two coefficients are "not linked by detailed balancing" [16]. However, this argument is without merit. In the absence of H2 (or any other species with which Br- can react), thermodynamic consistency and microscopic reversibility clearly require M to participate in dissociation if it does so in recombination. The addition of any species such as H2 that takes no part in the dissociation step may cause the system to deviate from thermodynamic dissociation equilibrium, but can obviously not alter the mechanism of dissociation.

10.4. Identification of relevant steps

321

Activation energies and consistency checks. The discussion so far has shown how the approximations 10.25 to 10.28 can be used to identify which of all possible steps are relevant and which are not. In addition, the approximations provide a means of checking the rate coefficients for consistency. Taking once more the hydrogen-bromide reaction as an example: The oneplus form of the rate equation 10.24 contains two phenomenological coefficients, k^ and k^. The first is given by and thus should have the activation energy (^a)a

=

^((^a)i„U- ( ^ a U ) + (£a)p.

(^0.30)

With (£'a)init aiid (£3)?! ^^ch a little more than the respective Ai/° values of + 190 and + 67 kJmol"^ (see Table 10.1) and (£'a)cBr (highly exothermic) no more than a few kJ mol"\ the activation energy of k^ should be about 165 to 170 kJ mol'^ which is in quite satisfactory agreement with the observed value of 175 kJmor^ The other coefficient is given by

Both /:_pi and k^2 ^^^ coefficients of exothermic steps, the second (p2) being more exothermic than the first. According to the approximation 10.26, the coefficient ratio should thus be roughly in the range between 0.1 and 1 and should depend little on temperature. The observed value of k^ is 0.1 and is essentially temperatureindependent, again in very satisfactory agreement with expectation. Interestingly, because of the factor one half in the first term of eqn 10.30, the estimated overall activation energy of the reaction (ca. 170 kJ mol"^) turns out to be lower than the activation energy of initiation (190 kJ mol"^). Since the factor one half appears whenever termination is second order in chain carriers, as is true with very few exceptions, and since the activation energies of the propagation steps often are relatively low, such behavior is quite common [19]. Some rules of thumb. A few regularities that have their roots in thermodynamics are worth mentioning. They can serve as rough guidelines. Exception must be expected, especially if the propagation steps have different molecularities and if the reaction involves a large entropy change or several reactants at very different concentrations. (This excludes, for example, thermal cracking of hydrocarbons.) The propagation cycle can be viewed as driven by the decrease in free energy that accompanies conversion of reactants to products. Barring overriding entropy effects, two conclusions can be drawn immediately: • At least one of the two propagation steps must be exothermic. • If one propagation step is endothermic, its standard-enthalpy variation must be smaller than that of the exothermic step.

322

Chapter 10, Chain reactions

If this were not so, equilibrium would be unfavorable and conversion would remain minimal. If both forward steps are exothermic, both reverse steps are endothermic and so will have smaller rate coefficients. In the competition for each of the two chain carriers, the respective forward step, being exothermic, is bound to win out. Thus: • If both propagation steps are highly exothermic, both are irreversible. If one forward step is endothermic, it competes for a chain carrier with the endothermic reverse of the other forward step, but the latter involves the larger freeenergy variation and so is disadvantaged. On the other hand, the exothermic forward step competes for the other chain carrier with the exothermic reverse of the endothermic forward step; both being exothermic, they will have low activation energies and may or may not occur with comparable ease. Accordingly: • If one propagation step is endothermic and the other is exothermic, the endothermic step may be reversible, the exothermic step is not. The likelihood of reversibility of the endothermic step is greater, the smaller the free-energy decrease accompanying the overall reaction. This is because the -AH° values of the two competing exothermic steps will then be more similar, and so will be their activation energies and rates. Lastly, the termination mechanism can be conjectured on the basis of thermochemical data. The propagation step with the larger drop in free energy, or the exothermic step if one is endothermic, is apt to be "faster. "* The chain carrier consumed by this step is depleted while the other accumulates. Since termination normally is controlled by coupling of the more abundant chain carrier: • Termination is controlled by the chain carrier produced by the more highly exothermic step, or by the exothermic step if the other step is endothermic, (However, see the preceding section for exceptions to control by the most abundant chain carrier.) While certainly not valid without exceptions and no substitute for a thorough understanding of the reaction at hand, these rules can serve well for preliminary orientation and as working hypotheses where their stated premises are valid. 10,5. Transmission of reactivity: indirect initiation, chain transfer So far we have taken for granted that initiation produces a chain carrier and that termination occurs by coupling of the chain carriers. Such behavior is the norm, but there are exceptions, owing to the ability of radicals to transmit their reactivity * "Fast" as used here refers to how soon a reactant is likely to react, not to reaction rate (see Section 4.2). Granted quasi-stationary behavior and long chains, the rates of the two propagation steps are equal (see Section 10.3).

70.5. Indirect initiation, chain transfer

323

to other species. Specifically, initiation may be indirect in that the radical it produces is not a chain carrier, but reacts with another molecule to form a chain carrier. More importantly, the kinetic chain may be broken by reaction of a chain carrier with another molecule, producing a radical that may or may not start a new chain. This is called chain transfer. Indirect initiation. The typical step sequence of indirect initiation is initiation transmission

in

• 2 R-

R- -f S

(10.31)

• X -f- ...

where X is a chain carrier but R- is not, S may be a solvent molecule, and another molecule may or may not be produced in the second step. With R- at trace level, the rates of the two steps are equal, and the slow first is rate-controlling: r.

= 2t.,C init

(10.3)

as for single-step initiation. The fact that initiation may be indirect has no effect on the rate. [If R- is the most abundant radical, it will recombine in addition to transmitting; this recombination does not reduce the chain-carrier population, which then is kept in balance by coupling of R- with the most abundant chain carrier.] Examples of indirect initiation will be encountered later in this chapter in the Rice-Herzfeld mechanisms and hydrocarbon oxidation (see next section). Also, initiation of radical polymerization usually is a two-step process (see Section 11.3). Chain transfer. While indirect initiation remains without effect on the form of the rate equation, chain transfer may profoundly affect kinetics because it may contribute an additional and possible dominant termination mechanism. chain transfer

X+ S

• ...

(10.32)

Here, X is any chain carrier, and S may but need not be a solvent molecule. The product may be, or may include, a radical that starts a new chain. In such cases, chain transfer does not decrease the radical population. If the new chain is of the same kind, the reaction continues at its pace. In the rare instances in which it starts a different kind of chain, two parallel reactions, each with its own termination mechanism, must be considered. Alternatively, chain transfer to a molecule such as carbon tetrachloride can produce radicals of low reactivity, thereby contributing to the termination of the kinetic chain. Species S then acts as a retardant. Assuming for simplicity X+X coupling as the normal termination, the net termination rate in such cases is A-,^„ = -(2k,^Ci

+

fcchxsQCs)

where k^^^xs is the rate coefficient of chain transfer from X to S.

(10-33)

324

Chapter 10. Chain reactions

If chain transfer is the dominant termination mechanisms, the equality of initiation and termination rates according to eqn 10.12 leads to a chain-carrier population that is proportional to the initiator concentration. For example, for chain transfer by X, the radical that is produced by initiation: C

=

Ik

C

'"'^ ^" k C

(10.34)

With eqn 10.9, a radical population proportional to the initiator concentration gives a chain-reaction rate that is first order instead of half order in initiator. However, for chain transfer to outrun coupling, its rate must be high, and species S then acts essentially as an inhibitor (see also Section 10.8). If the rates of termination by X-hX coupling and chain transfer by X are comparable, one finds

q ^

^chXS Q

'

fc,„,c„

1/2

^hxs<-s

(10.35)

^cX

and the chain-reaction rate becomes of order between one half and one in initiator. Equations for the chain-carrier concentrations at the various other possible combinations of initiation, coupling, and chain transfer involving the other chain carrier as well are more lengthy, but the conclusions as to kinetic behavior and reaction order with respect to the initiator are qualitatively the same. Chain transfer is of particular interest in radical polymerization, where it affects not only the polymerization rate, but also the molecular weight of the product (see Section 11.3). 10.6. Reactions with more than two radicals So far, only a very simple type of chain reaction has been considered, that with two chain carriers generated by initiation and with a two-membered propagation cycle. The high reactivity of radicals in general, however, can often lead to a much more complex behavior. Mechanisms with as many as 86 different elementary steps have been proposed as early as 1978 [20], and with the advent of cheap, fast, and easy-to-use computers there is now no lack of conjectured networks of even larger sizes, especially in petroleum processing and combustion engineering. The possible combinations of steps and topologies of networks stagger the imagination and make a comprehensive coverage at this place out of question. Instead, a few examples will be given and one relatively simple specific case will be discussed in detail in order to illustrate principles as well as a way of deriving rate equations.

70.6. Reactions with more than two radicals

325

It would be relatively easy to extend the mathematics in the preceding sections to reactions with single propagation cycles of more than two members (provided there is no chain branching). The forward and reverse reactions of the cycle could simply be expressed with clockwise and counter-clockwise A segment coefficients (see Section 6.3). However, there is litde point in writing such equations because the high reactivity of the chain-carrying radicals in real systems makes linear pathways of any length a rarity. In particular, many chain reactions involve hydrogen or oxygen atoms, and these react with almost any molecule they encounter. The so-called Rice-Herzfeld mechanisms of thermal degradation of organic compounds may serve as a typical example. 10.6.1. Rice-Herzfeld mechanisms: thermal cracking Thermal cracking of organic substances is an important reaction in the petroleum industry and has been extensively studied for over seventy years. At least for simple alkanes, the decay is first order in good approximation and therefore was long believed to occur in a single, unimolecular step [21]. However, in the 1930s, Rice and coworkers [22-24] established the presence of radicals under the conditions of the reaction by means of the Paneth mirror technique [25,26]. This observation led Rice and Herzfeld to propose a chain mechanism [22,27,28]. Extensive later studies proved the essential features of their mechanism to be correct not only for hydrocarbons, but also for many other types of organic substances. The principal features of what has come to be called Rice-Herzfeld mechanisms are [21,29]: initiation: rupture of a weak bond in the reactant molecule, generating two radicals; transmission: abstraction of a hydrogen atom from a reactant molecule by one of the radicals, to yield a product and the first chain carrier; first propagation step: decay of this chain carrier into a product molecule and another, second chain carrier (which may or may not be H-); second propagation step: hydrogen abstraction from another reactant molecule by the second chain carrier to yield a product molecule (H2 if the second carrier is H •) and another specimen of the first chain carrier; termination: coupling of the most abundant radical. As an example, thermal cracking of ethane will be examined here in detail. Example 10.2. Thermal cracking of ethane. Thermal cracking of ethane at temperatures in the range of 800 to 1200 K and ambient or lower pressures yields mainly ethene and hydrogen: C2H6

-•

C2H4 + H2

(10.36)

326

Chapter 10. Chain reactions but small amounts of methane, butane, and propene, are also formed. The reaction is generally held to be essentially first order in ethane [30]: (10.37) — ^app/'cC 'cc

(CC = ethane). Rice and Herzfeld [27] proposed the mechanism initiation transmission:

C^H, — CH3- + C2H6 — •

2CH3CH4 + C2H5 •

^^^^C,H,

propagation:

rate 2k,^^pcc (10.38) ^piPcc •

kpiPn' Pec

(CC- = C2H5-)- Initiation breaks the ethane carbon-carbon bond, which is weaker than the carbon-hydrogen bonds. The most abundant radical under most conditions of interest is C2H5- [30], so that coupling of two of these to butane should be the dominant termination mechanism, probably accompanied to a small extent by disproportionation to ethene and ethane [31,32]: C.H„

termmation

2CA.

(10.39) C2H4 + C2H^

Propene, formed in small amounts at high degrees of conversion, is believed to arise from a side reaction C2H5- + C2H4 —• CjHg + CH3 • in several steps [33]. Other side reactions at high conversion are CH3 • + H2 —• CH4 + H •, H- -f C2H4 —• C2H5 •, H- -hC2H4 - ^ C2H3' + H2, and CH3- + C2H4 —• CH4 + C2H3- [34,35]. These reactions leave the number of radicals unchanged, and so have little effect on the algebraic form of the rate equation for ethane disappearance and the reaction order. Taken at face value, the mechanism 10.38 with termination 10.39 leads to a reaction order of one half in ethane, in seeming contradiction to experimental observation. It is intriguing to trace the efforts at reconciliation. Rice and Herzfeld [27], at a time when still little was known about radicals and chain reactions, had tried to account for the observed first-order behavior by postulating a "mixed" termination C2H5- -H H- —>• C2H6. However, since C2H5outnumbers H • by several orders of magnitude under typical reaction conditions, this assumption proved untenable [30]. Thereupon Kiichler and Theile [36] suggested that initiation is bimolecular in ethane; provided termination occurs without a

70.6. Reactions with more than two radicals

327

collision partner, the overall rate then is first order in ethane. A case can be made that the butane molecule, formed by termination, has enough internal degrees of freedom to carry off the recombination energy without help by a partner even if ethane needs one for dissociation. This explanation had to be abandoned when methane formation, at least initially due exclusively to initiation, was found to be first order in ethane [33,37]. To save the day, Quinn [33] invoked Lindemann's theory of unimolecular decay [38] and applied it to the first propagation step, C2H5 • —• C2H4 -f H •. According to Lindemann, activation by binary collision must precede unimolecular decay and becomes rate-controlling at very low pressure. At start, ethane is the only available collision partner. With ethane in that role: QHs'

+ CjHg

•

(C2H5 • )activated +

Q^g

The overall rate of ethane consumption then is of order one-and-a-half in ethane if the rate of C2H5 • —> C2H4 + H • is controlled by activating collision, and of order one half if controlled by decay of the activated radical. According to Quinn, first-order behavior was observed because the reaction was studied in the "fall-off" range of pressure, that is, where rate control of C2H5 • decay shifts from one step to the other. Indeed, at very low pressures the initial rate varies with (pcV)^^ [31]. Quinn studied initial rates—i.e., in the absence of reaction products—in a limited pressure range of 60 to 230 Torr. His hypothesis can explain the dependence on initial pressure he observed, but not what is normally defined as first-order behavior, namely, a rate proportional to the reactant concentration or partial pressure in the course of the reaction in the presence of products formed. This is because ethene (and, for that matter, almost any other molecule with the possible exception of H2) can also serve as activating collision partner. Indeed, addition of inerts has been found to boost the rate [36]. Since one mole of ethane produces approximately one mole of ethene, the concentration of potential collision partners is/7c=c +Pcc = PQQ and remains essentially unchanged, so that there is no effect on the form of the rate equation and the reaction order (for simplicity, this assumes ethene to be as effective a collision partner as is ethane, and H2 to be ineffective.) Nevertheless, textbooks to this day accept Quinn's explanation, if not Rice and Herzfeld's. First-order behavior (as normally defined) at any pressure can be rationalized if the first propagation step is made reversible. This is not unreasonable because the step in the forward direction is strongly endothermic ( + 159 k J m o r ^ , so its reverse should make itself felt long before the reverse of the overall reaction becomes noticeable. The rate of this reverse step is proportional to a product, so that the retardation it exerts increases with progressing conversion. This translates into a higher apparent reaction order. Quantitatively, the mechanism 10.38 with termination 10.39 and reversible first step gives a rate equation of the form KP^^

(10.40)

1 + *-piPc=c/Vcc (for derivation, see farther below). Since both steps - p i and p2 are exothermic and bimolecular, the ratio of their rate coefficients should not be far from unity (see Sec-

328

Chapter 10. Chain reactions tion 10.4). Setting /:_pi lk^2 — ^-^ one finds a behavior within 1% of first order up to over 40% conversion (see farther below). However, the initial rate now is proportional to the square root of initial pressure, at odds with Quinn's experimental results and therefore possible only at pressures above the "fall-off" range, i.e., where activating collision no longer affects the rate. Also, the acceleration by added inerts remains unexplained. Moreover, step —pi being more exothermic than step p2, the ratio of their rate coefficients is expected to be larger than unity. In and below the fall-off range, if Quinn's hypothesis of an activating collision partner for C2H5 • decay is accepted, a factor ipccY with 0 < n < \ appears in the numerator and the second denominator term of eqn 10.40 and can produce the sought-for pressure dependence of the initial rate; overall first order in ethane now requires A:_pi (pccY^i - 0-5, a ratio that is more believable, but confined to a pressure range around/?cc ^ (^p2/^-pi)^^"So far we have taken for granted that the reaction is conducted at constant volume, as in the kinetic studies by Kiichler [36], Laidler [39], Quinn [40], and Lin [37]. In a plug-flow reactor as used in some other work [30], the gas expands as the mole number doubles when ethane forms ethene and hydrogen. Failure to correct for expansion would let the reaction order seem farther from first (see Section 3.3.4) and so cannot help to explain unexpected first-order behavior. Expansion keeps reducing the concentrations of the collision partners as conversion progresses. As a detailed calculation shows, this can produce an apparent reaction order close to one up to moderate conversion in and below the fall-off range, where activating collisions affect the rate. However, this effect alone cannot explain first-order behavior at constant pressure above the fall-off range, nor at constant volume at any pressure. None of the explanations described here is entirely satisfactory, and no other simple ones come to mind. Reaction behavior appears to be more complex than the original Rice-Herzfeld network 10.38 suggests [34,35]. Derivation of eqn 10.40 and apparent reaction order. Indirect initiation supplies CjHj-, the chain carrier that dominates termination, so that eqn 10.14 applies. The substitutions are Xpi = k^^, Xp2 = ^pzPcc >^-pi = ^-piPc=C' ^-p2 = ^-2PPH2' ^ = C2H5(CC •), and in = C2H6 (CC), and give . 1/2

^nitPcC

^pi^p2/^cc" ^-pi^-piPc^cPu,

^cCC-

^piPcc "^ ^-piPc^c

(9.41)

Since step - p i is strongly exothermic while step - p 2 is endothermic (-159 vs. -1-22 kJmol"^), it can be assumed that k_pi is significantly larger than k_p2- The second denominator term then makes itself felt before the second numerator term does, that is, already at conversions so low that the reverse of the overall reaction is still negligible. Without the second numerator term, eqn 10.41 in one-plus form and with K = ikmi^/k^cc-y'^kpi gives eqn 10.40. To establish the apparent (power-law) reaction order at low conversion, eqn 10.41 without the second numerator term must be integrated. For this purpose, pcc and /?c=c are expressed in terms of the fractional conversion of ethane; at constant volume:

10.6. Reactions with more than two radicals

Pec = Pcc(l - /cc)'

Pc^c

329

Pccfo

so that K(Pcc)"\l 'cc

- fccf"

1 ^ ^b/cc

{K - ^-pi/^p2- 1)

(9.42)

The general relationship between reaction time t and rate — r^ at constant volume is [41,42] PA:\-r;'dA /A=0

Integration with A = ethane and eqn 9.42 for the rate — r^ gives 1/2

(Pec)

/A

J (l-/cc)''^

/.=o

d/c.

2(Pcc)' (l-/cc)"^

With /Tb — —0.5 (i.e., k_pi lk^2 ^ O-^) ^^^ ^i^ne dependence of fractional conversion is within 1% of t = -ln(l -/cc) (constant volume, first order) up to/cc = 0.42. Thermal cracking of ethane is an excellent example of an intricate mechanism that leads to a kinetic behavior obeying a simple, first-order rate law in good approximation over a fairly wide range of conditions. It also serves to show how easily such a deceptively simple rate law is misinterpreted. Moreover, the example illustrates an important general point: A reverse propagation step with rate proportional to a product concentration produces an apparent overall reaction order that is higher than without the reverse step. This can happen even at conditions under which the overall reaction is irreversible.

The cause is the increasing retardation by the reverse step with progressing conversion as the product builds up [43]. This retardation can become effective even if the other propagation step and therefore the overall reaction are irreversible, or at a conversion so low that the reverse overall reaction is still insignificant. The hydrogen-bromide reaction is a simple example (see Example 10.1 in Section 10.3). A comment on deducing mechanistic details of Rice-Herzfeld-type reactions from apparent reaction orders is called for. Usually, a termination mechanism giving the desired result is postulated or, failing that, a collision partner in initiation, termination, or propagation steps is invoked. A formal scheme relating overall reaction orders to such mechanistic features, developed as early as 1948 by Goldfinger

330

Chapter 10. Chain reactions et al. [44], is quoted to this day in some textbooks. However, uncritical application can easily result in misinterpretation. The scheme implies that a collision partner in the initiation step must be another reactant molecule although most other molecules could serve just as well, it glosses over the requirements of thermodynamic consistency and that "mixed" termination is possible only under exceptional circumstances, it does not account for the effects of reverse steps, nor does it address the need for a volume correction in gas reactions at constant pressure and with change in mole number. In fact, an unusual apparent reaction order in an empirical rate equation may very well stem such ignored facets, and the mechanism may be contrary to what one is led to believe when taking the Goldfinger scheme at face value. Ethane cracking is a case in point.

Higher hydrocarbons. Thermal cracking of higher hydrocarbons is believed to occur with Rice-Herzfeld-type mechanisms [45,46]. Of course, with more carbon atoms in the molecule, more radicals of different carbon numbers appear and produce a greater variety of products. As a still relatively simple example, the network of principal steps in cracking of A2-butane is [47,48]:

C.H,,

initiation

2qH3

(10.43) propagation

C.H,

H

QH,,

termination -^ C,H,+ C,H,

There are three propagation cycles, all of which have the butyl radical, C4H9-, in common. Even this network is grossly simplified in that it omits, among other steps, any reactions of products, hydrogen abstraction from alkanes other than butane, and the presence of propyl radicals, which can arise from a step C4H10 —• C3H7- + CH3- and activate their own propagation cycle.

10.6. Reactions with more than two radicals

331

Rates of thermal cracking are first-order in good approximation for propane, butane and still higher hydrocarbons [21]. This is remarkable because chain mechanisms with initiation by break-up of a reactant normally result in reaction orders of one half or one-and-a-half, depending on which radical is consumed by termination. First-order behavior can result from "mixed" termination, which, however, can in most cases be ruled out as dominant mechanism (see Section 10.3). A more probable explanation is a combination of effects: that key hydrocarbon radicals participate in several steps of different molecularities, that some steps are reversible, and that some unimolecular ones require collision partners. As the complexity of the reaction of even as simple a molecule as /2-butane demonstrates, the number of steps increases steeply with carbon number. It becomes almost astronomical for complex mixtures of higher hydrocarbons as encountered in industrial petroleum processing. Here, a more promising approach pioneered chiefly Froment and co-workers [49], is to focus on the reactions not of individual reactants, but of molecular configuration that the reactants have in common. The guiding idea is that the rate coefficients and activation energies are similar for like events—say, hydrogen abstraction by H- from the -CH2~ group of an alkyl chain, or coupling of two alkyl radicals—as long as the immediate vicinity of the reaction site is the same, even if other parts of the molecules differ in size and structure (see also Sections 12.3 and 12.4). For single higher hydrocarbons, conventional step-by-step modeling has become feasible thanks to the extensive data base on relevant rate coefficients and activation energies that is has been compiled over the last few years [50,51]. Other organic compounds. Rice-Herzfeld mechanisms appear to be the rule in thermal degradation of many other types of organic compounds, among them aldehydes [21,43,52-54] and ketones [21,55]. Many of these reactions are approximately first order. Decomposition of acetaldehyde, extensively studied, is of order one-and-ahalf, easily explained with a Rice-Herzfeld mechanism and eqn 10.18 or 10.19 [22,56]. The reaction order is found to increase toward two at high conversion [43,56]. As seen in the example of ethane cracking and the hydrogen-bromide reaction, such a "creeping up" of the reaction order with progressing conversion is a typical symptom of a reverse step in the propagation cycle [43]. 10.6.2. Hydrocarbon oxidation Reactions of organic compounds, especially hydrocarbons, with oxygen in the gas or liquid phase at moderate temperatures (below 150° C) are important both as an industrial process and as a natural decomposition phenomenon that is to be suppressed if possible. They are chain reactions, but differ from thermal cracking in that they usually require initiation. An initiator may have been added intentionally or be present as an impurity or early minor product, possibly a hydroperoxide that had accumulated upon prolonged standing in contact with air.

332

Chapter 10. Chain reactions A typical mechanism of oxidation of a hydrocarbon, RH, is [57-62] initiation:

initiator —• 2 R' • R' • + O2 —• R'OO •

transmission:

propagation:

rate 2 /CinhCin

R'OO- + RH —• R'OOH + R O

(10.44) ^pl^R-P02

^p2^R00 • ^RH

ROOH termination:

2 ROO • —• inactive products

or

2R • —• inactive products

or

ROO • 4- R • —• inactive products

2 ^cROO • ^ R O O •

2 kcR. CR . 2 KR • ROO • Q • Qc

As a rule, the first propagation step is highly exothermic, and the second is endothermic. If so, k^^ is several orders of magnitude larger than kp2' As a result, the concentration of ROO- is much higher than that of R- (see eqn 10.8), even in liquid-phase reactions in which the concentration of RH is much higher than that of dissolved oxygen. Accordingly, coupling of ROO- is the dominant termination mechanism in most cases. At very low partial pressures of oxygen, however, ROO • is no longer as abundant, and the other two possible termination mechanisms may also come into play. The hydroperoxide, ROOH, may decompose into radicals that start new chains to give alcohols, ketones, and acids. Since the decomposition of the hydroperoxide gives rise to additional chain carriers, the reaction can be selfaccelerating and may evolve into an explosion. Being highly exothermic, the first propagation step in the cycle 10.44 can safely be regarded as irreversible. If the hydroperoxide were stable, the second propagation step should be reversible. However, it is unstable and likely to decompose in other ways before it has time to react with R -, which is at very low concentration. Accordingly, the assumption that both steps are irreversible is usually justified. Termination at other than very low oxygen pressures is controlled by ROO •, the chain carrier functioning as reactant in the second propagation step (Y in the general equations in Section 10.3). Thus, if new chains initiated by the decomposition of the hydroperoxide product can be disregarded, the rate of hydrocarbon consumption is described by eqn 10.19. At oxygen pressures so low

70.6. Reactions with more than two radicals

333

that all three termination mechanisms contribute, the rate is given by eqn 10.21. With rjnit = 2/:initCin, Xpi = k^iPo2 (validity of Henry's law is assumed here), and \2 — ^PICRH, these equations give =

(^init IKKOO ' )

^p2 Qn

CRH

^

*

^

and ,1/2.

,

^1/2

^ 1/2

^ c R • (^p2 ^ R H )

^

^cROO • ( ^ p l / ^ 0 , )

"^ ^ c R • ROO • ^ p l ^ p 2 Po,

(10.46)

^RH

for normal and very low oxygen pressures, respectively. Behavior according to eqn 10.45—orders of one half in initiator, one in the organic reactant, and zero in oxygen—is very common. If the oxygen pressure is reduced substantially, a beginning dependence on that pressure and a fall-off of the reaction order with respect to the organic reactant is observed, an indication that R- + R - andR- + R 0 0 - terminations have started to contribute [58]. Termination by R • + R • coupling alone would result in a rate that is first order in oxygen and zero order in hydrocarbon (eqn 10.18), but is unlikely even at quite low oxygen pressures. In all instances, the rate is of order one half in the initiator. The reaction is not "clean." Hydroperoxide decomposition yields aldehyde and ketone. Moreover, at other than quite low conversion, further oxidation leads to scission of carbon-carbon bonds and formation of acids [63]. However, if a borate ester or boroxine is added, secondary alcohol can be obtained in good yield (see Example 5.5 in Section 5.4). Example J0.3. Oxidation of cyclohexane [63-65]. Air oxidation of cyclohexane to a mixture of cyclohexanol and cyclohexanone is an important step in a process for production of adipic acid and caprolactam in du Font's Nylon synthesis. The reaction is carried out in the presence of a small amount of a cobalt salt (typically naphthanate or 2-ethylhexanoate) at 140 to 165° C and moderate pressure (e.g., 10 atm). The primary reaction product is cyclohexyl hydroperoxide: < ^

+0^

• (^^^OOH

(10.47)

which, however, decomposes quickly. The reaction is run at low conversion to achieve reasonable yields of the desired final products, cyclohexanol and cyclohexanone [66]. The formation of the hydroperoxide is believed to proceed with a chain mechanism much like 10.44 (R- being C^U^^-), except that its product acts as initiator. Cobalt controls the conversion of the hydroperoxide to radicals for initiation [67], in all likelihood by a Haber-Weiss redox cycle 10.4 (see Section 10.2).

334

Chapter 10. Chain reactions The first propagation step is highly exothermic, the second is endothermic. Cobalt also promotes the conversion of the hydroperoxide to cyclohexanol and cyclohexanone. Conditions can be adjusted so that this conversion is fairly rapid, and the reverse propagation step C^H^ • + CgHnOOH —• C6H12 + CgHnOO • then remains insignificant. A likely termination is [59-61] " > 0 O' (3"00-

•

( ^ 0 0 - O O H ( ^

< ^ • O2

(10.48)

OOH The fairly harsh conditions required to break the carbon-hydrogen bond in cyclohexane cause various side reactions, and the yield to the desired end products (based on cyclohexane converted) is only about 60 to 70%, even at low conversion. A higher yield could be obtained with added borate ester or boroxine (see Example 5.5 in Section 5.5), but this would require hydrolysis of the resulting cyclohexyl ester and is not practical in a process that calls for a dry product. Initiation may occur in other ways. In autoxidation, the hydrocarbon itself functions as initiator by reacting with oxygen to form a hydroperoxide. If so, CRHPO2 replaces Q^ in the rate equations. The reaction orders then are between the following limits: one-and-a-half in hydrocarbon and one half in oxygen at moderate to high oxygen pressures, and one half in hydrocarbon and one-and-a-half in oxygen at very low oxygen pressures. Also, the reaction may be initiated photochemically, possibly in the presence of a sensitizer [68]. The rate then is proportional to the square root of the light intensity. Because of their great importance in chemical industry, much effort has been devoted to the study of hydrocarbon oxidation, and a large data base of rate coefficients and activation energies of common elementary reaction steps has been compiled [50,51,60,69-71]. 10.7. Reactions with chain branching: the hydrogen-oxygen reaction As mentioned at the outset, certain chain reactions include steps in which more chain carriers are formed than consumed, and this may cause a detonation. In many cases, branching is caused by oxygen, whose atom has two unpaired electrons. In a reaction with chain branching, there is competition between branching and termination. As the chain-carrier population increases after initiation, production and elimination of chain carriers may or may not reach a balance: If termination is unable to keep pace with branching, the chain-carrier population grows exponentially, and a detonation ensues. The essential features of this process will be shown with a specific example, that of the hydrogen-oxygen reaction.

10.7.

Reactions

with chain branching:

Example 10.4. The hydrogen-oxygen

the H2-O2 reaction

335

reaction [72-76]. The hydrogen-oxygen reaction

2 H2 + O2 —• 2 H2O

(10.49)

is one of the most interesting and most thoroughly studied reactions with chain branching. AUhough only two elements—hydrogen and oxygen—are involved, they form a large number of molecular and radical species, and these can undergo many steps that are interconnected in a labyrinthine fashion. An earlier volume of this series [77] starts (under the title Minima minimorum) with a listing of thirty such steps. At low pressure and moderate temperature (say, 400 to 700 K), the propagation cycles can be represented in a simplified fashion by a network of just three interlocking steps:

(10.50)

Two of the three steps involve chain branching: H- + O2 —• O: + OH- and O: + H2 —• OH • + H •. Initiation can be triggered by an electric discharge or occur by dissociation H2H-M—•H+H-+M or reaction H2 + O2 + M - >

OH • + OH • + M

where M is a collision partner or the wall of the reaction vessel. Possible termination mechanisms are adsorption of H • at the vessel wall or reaction of H • with oxygen: H- —• H,dsorbed H- + O2 + M ->

HOO-

rate + M

-K^^uPw

(10.51)

-KU'02PHPO2PM

(10.52)

The hydroperoxy radical, HOO •, is fairly unreactive. It builds up to relatively high concentrations and is then likely to react with itself according to 2 HOO- —• H2O2 + O2

(10.53)

336

Chapter 10. Chain reactions or be deactivated at the vessel wall. Because the gas mixture lacks inert large molecules that could effectively carry off the released energy, coupling of H- is not competitive under usual conditions. The question of greatest interest is whether or not a detonation will ensue once the reaction has been initiated. To answer it, an equation describing the growth rate of the chain-carrier population is sought. Because population growth is considered, the Bodenstein approximation of quasi-stationary states of chain carriers cannot be applied across the board. However, as long as the system is still remote from detonation, an approximate description can be obtained by application of that approximation to OH • and O: only, but not to H •. This is admissible because, in the propagation cycle, H- is consumed only by a highly endothermic step (ca. + 70 kJmol""^) and thus becomes more abundant than OH • and O: by several orders of magnitude, with the result that on an absolute scale the growth rates of the latter two radicals are negligible by comparison with that of H •. The Bodenstein approximations for OH- and O: amount to ^O:

^OH-

-

KIPH'PO,

= KiP^-Po,-^

-

^3P0:PH,

K^PO.PH,-

=

^

K^Pon-Pn, =

^

respectively, and yield Po:

=

KIPH'POJKZPH,

PoH' = %2PH'PO,^K^POPH)IKIPH,

(10-^4)

^^KIPH'POJKIPH,

(10.55)

(Equation 10.54 has been used to replace po, in eqn 10.55.) The growth rate of Hcan now be calculated. Since H • is by far the most abundant chain carrier, its growth rate is representative of that of the total chain-carrier population, D •. Allowing for both termination mechanisms 10.51 and 10.52: ^L'

=

^H- =

^.na^ KiPowPu,-'

K^Po.Pnr

PU'(K2P0,'-

KdsH^- Kw

OPO.PM)

and with eqns 10.54 and 10.55:

For a batch system at constant volume, r^. = dp^Jdt, integrated over time: P^(t) =

jr,.d/

= _ ^ ( l

and eqn 10.56 can be

-exp[(^-5)r])

r=o

where A = '^^p2Po

chain-branching effect

^ = ^adsH +

termination effect

KH.OPOPM

(10.57)

10.7.

Reactions with chain branching: the H2-O2 reaction

337

[Even if the gas is not confined in a closed reaction vessel, the increase in mole number becomes so rapid as detonation is approached that pressure builds up, so that, for the integration of eqn 10.56 in this range, constant volume is a better approximation than constant pressure.] Equation 10.57 shows the behavior of the gas mixture to depend critically on the relative strength of the branching and termination effects. IfB > A, termination can keep pace with branching, and the chain-carrier population approaches a quasistationary level -^

P^^it) =

>nit

^ - ^

^adsH + ^cH • O.Po.Pu "

^KlPo.

The chain-branching effect, A, is seen to counteract the termination effect, B, but does not overcome it. On the other hand, '\i B < A, chain-carrier production outruns elimination by termination, and the population begins to increase exponentially as the exponential in eqn 10.57 becomes dominant. When this starts to happen, the Bodenstein approximations for OH- and O: and eqns 10.56 and 10.57 derived from them lose their validity. Moreover, with the accompanying rise in temperature and pressure, other steps enter the picture to produce a more complex behavior. A result is the anomalous shape of the detonation limit, shown in Figure 10.2. Equation 10.57 provides some clues about the dependence of 10 the detonation limit on conditions. thermal explosion An increase in oxygen pressure promotes branching more than termination, and so favors detonation; an addition of a heavy inert component P 0.1 (M) does the opposite. A large [atm] surface-to-volume ratio makes for a large value of /CadsH^ favoring ter0.01 mination. The propagation step p2, in which an oxygen-oxygen bond is 0.001 broken, is highly endothermic and 400 450 500 550 thus has a high activation energy; r[°C] accordingly, a temperature increase strongly promotes chain branching Figure 10.2 Explosion limit of 2:1 and detonation (above about 1000 mixtur of H2 and O2 as function molar mixture K, the stoichiometric gas mixture is of temperature and pressure (schematic) explosive at any pressure). The example of the hydrogen-oxygen reaction and eqn 10.57 nicely illustrate a critical facet of reactions with chain branching: the competition between the rates of excess chain-carrier production by branching and increased elimination by termination. However, reality is more complex. Systems with chain branching by their nature involve highly aggressive radicals and, therefore, a large number of

338

Chapter 10. Chain reactions

possible steps of different molecularities and very different activation energies. As a result, variations in pressure and temperature, bound to occur when population growth becomes fast, are apt to produce dramatic shifts in control between various mechanisms. To some extent this is, of course, true for all types of reactions, but the potential for exponential self-acceleration caused by chain branching aggravates matters greatly. The procedure of arriving at eqn 10.57 as an approximation for growth of the chain-carrier population is generally applicable: (1) (2)

use of the Bodenstein approximation for the radicals except the most abundant one, in order to obtain an equation for the growth rate of the latter; integration of this equation over time at constant volume.

As the example of the hydrogen-oxygen reaction has shown, this procedure provides clues about the sensitivity of the system and its dependence on conditions. Because of the many simplifications in its derivation, however, it can not be used to predict detonation limits. 10.8. Inhibition and induction periods As mentioned at the outset, chain reactions, relying on radicals as chain carriers, are sensitive toward any substances that can destroy or trap such radicals. The interference with chain propagation can assume two forms. An added substance can reduce the reaction rate to almost nil or bring it to essentially a complete and permanent stop. This is called inhibition. It occurs if the inhibitor catches practically all radicals produced by the initiator. Under different conditions, an added substance or impurity can delay the start of a chain reaction for some period of time, called an induction period, without affecting its later course. Inhibition [78]. An inhibitor is itself being consumed as it traps radicals. To be effective, it must therefore be present in an excess over the initiator. In practice, this limits effective inhibition to chain reactions apt to be set off by small amounts of an initiator other than the bulk reactant. The most common application of inhibition is for protection of sensitive chemicals whose decomposition or polymerization by chain mechanisms may easily be triggered. A typical example is the stabilization of highly reactive monomers such as styrene or methyl methacrylate by hydroquinone, 4-r^rr-butyl-catechol, or TEMPO (a nitrosoxide) [79-82]. Radical polymerization of styrene is easily set off by impurities or the slightest amount of a radical producing agent and, being highly exothermic, can result in a thermal explosion. Another example is the use of antioxidants for protection of polymers against degradation by radicals produced by oxygen from air or ultraviolet radiation from sunlight [83,84].

10.8. Inhibition and induction periods

339

Induction periods. An induction 2 X inhibitor period typically occurs if a radical and Co2^ trap is present at a concentration much lower than that of the initiating substance (reactant or added o initiator). An often cited example is the temporary inhibition of the oxidation of cumene by a cresol derivative [57,85]: With twice as much inhibitor, the induction period is twice as long, and once the inhibitor is used up, the reaction is faster if cobalt is present time (see Figure 10.3). Figure 10.3. Effect of initiation (by AIBN and There are, however, other Co^"*") and inhibition (by 2,6-di-r-butyl-/7phenomena that can lead to a delay cresol) on oxidation of cumene in glacial acetic of a chain reaction. In oxidation acid (adapted from Moore and Pearson [78]). of hydrocarbons, for example, the primary reaction product is a hydroperoxide that, in turn, can act as initiator (see Example 10.3 in Section 10.6.2). With no initiator at start, the reaction begins at a very low rate and then picks up speed as it produces its own initiator. Such behavior is best classified as self-accelerating (see Section 8.9).
77 t-H

40

-

o H

oO

^/
20

=3

_ .oC

V3

^ S ' ^ ^

^y^

(-1

a. n

-^ ^'^'^ ^.^ : _ 100

1 •

T

200

1

1

300

1

1

400

time [s]

Figure 10.4. Pressure versus time in low-temperature oxidation of methane in constant volume batch (methane-to-oxygen mole ratio 2:1; adapted from Hoare [86]).

340

Chapter 10. Chain reactions

Self-acceleration of oxidation of a hydrocarbon caused by radicals arising from the reaction can involve complicated mechanisms. An example is low-temperature oxidation of methane [59,86]. The key radical turns out to be C H j - , produced from methane by initiation and giving rise to other radicals. The propagation mechanism with six interlocking steps and chain branching is such that build-up of CH3- accelerates the rate (see Figure 10.4, preceding page). Yet another possible mechanism that can lead to an induction period is reversible coupling (or other reaction with itself) of the most abundant radical to yield a product of low stability. Initially, none of that product is present. In the early stages of the reaction, its formation can be the dominant and highly effective termination, keeping the reaction at a low rate. With time, however, the metastable product builds up and approaches equilibrium with the radicals from which it is formed. Picture the product as a reservoir into which the reaction drains radicals until it is filled to capacity. By default, another and slower termination mechanism then takes over, and the reaction speeds up accordingly. Summary Chain reactions rely on highly reactive radicals that convert reactants to products while cycling through a step sequence like a DO loop in FORTRAN, in which they are consumed and produced anew. The step sequence, called propagation, is initiated by a reaction or event that generates radicals, and is terminated by reaction of radicals with one another to form one or more stable products or, more rarely, by their deactivation by some other mechanism. With few exceptions, termination is by reaction of the most abundant radical with itself. Such termination is second-order in radicals, and this leads to rate equations with exponents of one half or integer multiples of one half. The steric and frequency factors of reactions of radicals with one another or with small molecules are rather alike, and the activation energies are very low for exothermic reactions and only barely higher than the standard-enthalpy changes A//° for endothermic ones. In simple cases this makes it possible to use thermochemical data to identify which of the many possible steps dominate kinetics. The radicals generated by initiation are not necessarily the chain carriers. Rather, they may transmit their reactivity to other species which then act as chain carriers. If so, initiation is indirect. Similarly, chain carriers may transmit their reactivity to other species, which then may or may not start new chains. This is called chain transfer. Whether initiation is direct or indirect makes litde difference for kinetics. In contrast, chain transfer can have a profound effect. If a new chain of the same kind is started, the reaction continues at its pace, but chain transfer producing an unreactive radical may dominate termination, and the reaction then is first order rather than half order in initiator. If such transfer is fast, the reaction may be retarded or inhibited. Very common in thermal degradation of organic substances, especially hydrocarbons, are so-called Rice-Herzfeld mechanisms. One of the two chain carriers abstracts a hydrogen atom from the reactant to form a product plus the other chain carrier, which then reacts to yield another product plus the first chain carrier.

References

341

Other typical chain reactions include those of hydrogen with halogen in the gas phase and oxidation of organic substances. A special facet of the latter reactions is that an intermediate can act as initiator, and the reaction then is self-accelerating. Certain chain reactions involve steps which generate more radicals than they consume. This is called chain branching. The inherent self-acceleration may or may not outrun termination. If it does, a detonation results. A classical example is the reaction of hydrogen with oxygen. Chain reaction may be inhibited by agents acting as radical traps. They may also exhibit a delayed onset of the reaction. Such an induction period may be caused by a trace inhibitor that must first be destroyed by the radicals, or by reversible formation of a metatstable compound from the most abundant radical. Examples include the hydrogen-bromide reaction, thermal cracking of ethane and /i-butane, oxidation of cyclohexane, and the hydrogen-oxygen reaction.

References General references Gl.

S. W. Benson, The foundations of chemical kinetics, McGraw-Hill, New York, 1960 (updated and corrected reprint, Krieger, Melbourne, 1982, ISBN 0898741947). G2. M. Boudart, Kinetics of chemical processes, Prentice-Hall, Englewood Cliffs, 1968. G3. E. T. Denisov, T. G. Denisova, and T. S. Pokidova, Handbook of free radial initiators, Wiley, New York, 2003, ISBN 0471207535. G4. R. J. Farrauto and C. H. Bartholomew, Fundamentals of industrial catalytic processes. Chapman & Hall, London, 1997, ISBN 0751404063, Chapter 12. G5. G. F. Froment and K. B. Bischoff, Chemical reactor analysis and design, Wiley, New York, 2nd ed., 1990, ISBN 0471510440. G6. A. A. Frost and R. G. Pearson, Kinetics and mechanism: a study of homogeneous chemical reactions, Wiley, New York, 2nd. ed., 1961. G7. B. C. Gates, Catalytic chemistry, Wiley, New York, 1992, ISBN 0471517615. G8. V. N. Kondratiev, in The theory of kinetics. Vol. 2 of Comprehensive chemical kinetics, C. H. Bamford and C. F. H. Tipper, eds., Elsevier, Amsterdam, 1969, ISBN 0444406743, Chapter 2. G9. K. J. Laidler, Chemical kinetics, McGraw-Hill, New York, 3rd ed., 1987, ISBN 0060438622. GIO. J. W. Moore and R. G. Pearson, Kinetics and mechanism: a study of homogeneous chemical reactions, Wiley, New York, 3rd ed., 1981, ISBN 0471035580. G i l . G. W. Parshall and S. D. Ittel, Homogeneous catalysis: the application and chemistry of catalysis by soluble transition metal complexes, Wiley, New York, 1992, ISBN 0471538299.

342 G12. G13. G14. G15. G16. G17. G18. G19.

Chapter 10, Chain reactions M. J. Pilling and P. W. Seakins, Reaction kinetics, Oxford University Press, 1995, ISBN 0198555288. A. Pross, Theoretical and physical principles of organic reactivity, Wiley, New York, 1995, ISBN 0471555991. W. A. Pryor, Free radicals, McGraw-Hill, New York, 1966 (reprinted by Books on Demand, Ann Arbor, publication projected, ISBN 0608099430). F. O. Rice and K. K. Rice, The aliphatic free radicals, Johns Hopkins Press, Baltimore. 1935. N. N. Semenov, Some problems of chemical kinetics and reactivity, Pergamon, New York, Vol. 1, 1958. E. W. R. Steacie, Atomic and free radical reactions, Reinhold, New York, 2nd ed., 1954 (2 volumes). J. I. Steinfeld, J. S. Francisco, and W. L. Hase, Chemical kinetics and dynamics, Prentice-Hall, Englewood Cliffs, 2nd ed., 1999, ISBN 0137371233. G. S. Yablonskii, V. I. Bykov, A. N. Gorban', and V. I. Elokhin, Kinetic models of catalytic reactions, in Comprehensive chemical kinetics, R. E. Compton, ed., Elsevier, Amsterdam, Vol.32, 1991, ISBN 0444888020.

Specific references 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

M. Bodenstein, Z. Physik. Chem., 85 (1913) 329. F. Haber and J. Weiss, Naturwissenschaften, 20 (1932) 948; Proc. Roy. Soc, A 147 (1934) 332. M. Bodenstein and H. Liitkemeyer, Z. Physik. Chem., 114 (1925) 208. G. R. Gavalas, Chem. Eng. Sci., 21 (1966) 133. M. Bodenstein and S. C. Lind, Z. Physik. Chem., 57 (1907) 168. J. A. Christiansen, Kgl. Dansk Videnskab Selskab, Mat-fys, 1 (1919) 14. K. F. Herzfeld, Ann. Physik, 59 (1919) 14. M. Polanyi, Z. Elektrochem., 16 (1920) 49. M. G. Evans and M. Polanyi, Trans, Faraday. Soc, 34 (1938) 3 and 11. Semenov (ref. G16), pp. 27-28. Boudart (ref. G2), Section 8.1. Moore and Pearson (ref. GIO), pp. 199-201. H. S. Johnston and C. Parr, J. Am. Chem. Soc, 85 (1963) 2544. Pross (ref. G13), Chapter 10. E. Rabinowitch and H. L. Lehmann, Trans. Faraday Soc, 31 (1935) 689. Steinfeld et al. (ref. G18, Section 2.3.1. W. Jost and G. Jung, Z. Physik. Chem., B 3 (1929) 83. M. Ritchie, Proc Roy. Soc, A 146 (1934) 828. e.g., see Froment and Bischoff (ref. G5), pp. 36-37. K. M. Sundaram and G. F. Froment, Ind. Eng. Chem. Fundam., 17 (1978) 174.

References 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50.

343

Steacie (ref. G17), Chapter IV (includes copious references to earlier work). F. O. Rice, J. Am. Chem. Soc, 53 (1931) 1959; 55 (1933) 3035. F. O. Rice, W. R. Johnston, and B. L. Evering, /. Am. Chem. Soc, 54 (1932) 3529. Rice and Rice (ref. G15), Chapter III. F. A. Paneth and W. Hofeditz, Chem. Ber., 62 B (1929) 1335. Rice and Rice (ref. G15), Chapter IV. F. O. Rice and K. F. Herzfeld, J. Am. Chem. Soc., 56 (1934) 284. Rice and Rice (ref. G15), Chapter VII. Froment and Bischoff (ref. G4), p.27. Steacie (ref. G17), Section IV-8 (includes copious references to earlier work). K. J. Laidler, and B. W. Wojciechowski, Proc. Roy. Soc, A 260 (1961) 91. Laidler (ref. G9), Section 8.5.3. C. P. Quinn, Proc. Roy. Soc, A 275 (1963) 190. M.J. Pilling, in Modem gas kinetics: theory, experiment and application, M. J. Pilling and I. W. M. Smith, eds., Blackwell, Oxford, 1987, ISBN 0632016159, Chapter C5. Pilling and Seakins (ref. G12), Section 9.4.1. L. Kiichler and H. Theile, Z. Physik. Chem., B 42 (1939) 359. M. C. Lin and M. H. Back, Can. J. Chem., 44 (1966) 505; 45 (1967) 3165. F. A. Lindemann, Trans. Faraday Soc, 17 (1922) 598. K. J. Laidler and B. W. Wojciechowski, Proc Roy. Soc, A 259 (1960-61) 257. J. H. Purnell and C. P. Quinn, J. Chem. Soc, 1961, 4128. C. G. Hill, Jr., An introduction to chemical engineering kinetics and reactor design, Wiley, New York, 1977, ISBN 0471396095, Section 8.5.1. O. Levenspiel, Chemical reaction engineering, Wiley, New York, 3nd ed. 1972, ISBN 047125424X, Chapter 5. M. Letort, J. Chim. Phys., 34 (1937) 206 and 256. P. Goldfinger, M. Letort, and M. Niclause, Contribution a Vetude de la structure moleculaire, Victor Henri Commemorative Volume, Desoer, Liege, 1947-48, p. 283; see Froment and Bischoff (ref. G4), Table 1.4.1, p. 29. D. A. Leathard and J. H. Purnell, Annu. Rev. Phys. Chem., 21 (1970) 197. Benson (ref. Gl), p. Section XIII.9. J. H. Purnell and C. P. Quinn, Proc. Roy. Soc, A 270 (1962), 267. K. J. Laidler, Chemical kinetics, McGraw-Hill, New York, 2nd ed., 1965,, pp. 406-407. G. F. Froment, Fundamental kinetic modeling of complex processes, Chapter 5 in Chemical reactions in complex mixtures, A. V. Sapre and F. J. Krambeck, eds.. Van Nostrand Reinhold, New York, 1990, ISBN 0442007256. Landolt-Bomstein, New Series, Radical reaction rates in liquids, H. Fischer, ed.. Springer, Berlin, Part II Vol. 13 (5 subvolumes), 1983-1985, ISBN 0387126074, 0387132414, 0387117253, 0387121978, 0387136762; Vol. 18 (5 subvolumes), 1994-1997, 3540560548, 3540560556, 3540560564, 3540603573, 3540560572.

344 51.

52. 53. 54. 55. 56. 57. 58. 59. 60.

61.

62.

63. 64. 65. 66. 67. 68. 69.

70.

71. 72. 73. 74. 75.

Chapter 10. Chain reactions A. B. Ross, W. G. Mallard, W. P. Helman, G. V. Buxton, R. E. Huie, and P. Neta, NDRL-NIST Solution kinetics database, Version 3.0, Notre Dame Radition Laboratory, Notre Dame, and National Institute of Standards, Gaithersburg, 1998. W. D. Walters, in A. L. Fries and A. Weisberger, eds.. Techniques of organic chemistry, Interscience, NY, vol. 8, 1953, pp. 283-296. M. Letort, Chim. Ind., 76 (1956) 430. J. G. Calvert and J. T. Gruver, 7. Am. Chem. Soc, 80 (1958) 1313. Rice and Rice (ref. G15), Chapter VIII. Frost and Pearson (ref. G6), pp. 256-258 (includes references to earlier work). G. A. Russell, /. Chem. Educ, 36 (1959) 111. Semenov (ref. G16), p. 399. Kondratiev (ref. G8), Section 3.3.2. E. T. Denisov and T. Denisova, Handbook of antioxidants: bond dissociation energies, rate constants, activation energies, and enthalpies of reactions, CRC Press, Boca Raton, 2nd ed., 2000, ISBN 0849390044, Chapter 2. V. A. Kritsman, G. E. Zaikov, and N. M. Emanuel', Chemical kinetics and chain reactions: historical aspects, English translation Nova Science, Commack, 1995, ISBN 1560721669, Section VII-2 (many references to Russian work). L. Reich and S. S. Stivala, Autoxidation of hydrocarbons andpolyolefins: kinetics and mechanisms, Dekker, New York, 1969; reprint Books on Demand, Ann Arbor (projected), ISBN 0783707754. Gates (ref. G7), p. 67. I. V. Berezin, E. T. Denisov, and N. M. Emanuel', The oxidation of cyclohexane, Pergamon, Oxford, 1966, ISBN 0080113788. Parshall and Ittel (ref. Gil), Section 10.3. J. W. Parshall, 7. Mol. CataL, 4 (1978) 243. C. A. Tolman, J. D. Druliner, F. J. Krusic, M. J. Nappa, W. C. Seidel, I. D. Williams, and S. D. Ittel, J. Mol. CataL, 48 (1988) 129. P. D. Bartlett, ACS Symp. Ser., 69 (1978) 15. D. L.Baulch, C. J. Cobos, R. A. Cox, C. Esser, P. Frank, T. Just, J. A. Kerr, M. J. Pilling, J. Troe, R. W. Walker, and J. Warnatz, J. Phys. Chem. Ref Data, 21 (1992)411. D. L. Baulch, C. J. Cobos, R. A. Cox, P. Frank, G. Hayman, T. Just, J. A. Kerr, T. Murrells, M. J. Pilling, J. Troe, R. W. Walker, and J. Warnatz, /. Phys. Chem. Ref Data, 23 (1994) 847. Handbook of chemistry and physics, D. R. Lide, ed.-in-chief, CRC Press, Boca Raton, 80th ed., 1999-2000, ISBN 0849304806, Section 5, pp. 505-519. Kondratiev (ref. G8), Section 3.2. R. R. Baldwin and R. W. Walker, Essays Chem., 3 (1972) 1. P. Gray and S. K. Scott, Chemical oscillations and instabilities: non-linear chemical kinetics, Oxford University Press, 1990, ISBN 0198558643, Section 15.1. Pilling and Seakins (ref. G12), Section 10.3.1.

References 76. 77. 78. 79. 80. 81. 82.

83. 84. 85. 86.

345

Steinfeld et al. (ref. G18), Section 14.2 Yablonskii et al, (ref. G19), Section 1.1. Moore and Pearson (ref. GIO), pp. 398-401. V. W. Bowry and K. U. Ingold, 7. Am. Chem. Soc, 114 (1992) 4992. G. Odian, Principles of polymerization, Wiley, 3rd ed., 1991, ISBN 0471610208, Section 3.7b. G. Moad and D. H. Solomon, The chemistry of free radical polymerization, Pergamon, 1995, ISBN 0080420788, Section 5.4. S. S. Chen, Styrene, in Kirk-Othmer, encyclopedia of chemical technology, J. I. Kroschwith and M. Howe-Grant, eds., 7th ed. Wiley, New York, Vol. 22, 1997, ISBN 0471526916, p.956 (see p. 973). Yu. A. Shliapnikov, Antioxidative stabilization of polymers, Taylor & Francis, London, 1996, ISBN 0748405771. Denisov and Denisova (ref. 60), Chapter 9. H. S. Blanchard, J. Am, Chem. Soc, 82 (1960) 2014. D. E. Hoare, Combustion of methane, in Low temperature oxidation, W. Jost, ed., Gordon & Breach, New York, 1968, p. 125.

Chapter 11 Polymerization In principle, polymerization is a chemical reaction like any other.* However, in most cases it occurs under circumstances that differ from those of ordinary reactions, with consequences for kinetic behavior. For example, although the reaction mixture may be homogeneous at start, the polymer may precipitate, and monomer diffusion into the newly formed particles may then become an important factor. Even if the reaction mixture does remain homo-geneous, entanglement of polymer chains grown to great length may impede the mobility of the reactive groups at their ends. Depending on conditions, this may slow the reaction down, may accelerate it into a runaway, or may bring it to a standstill short of complete conversion. Such effects are matters of polymer reaction engineering and can be dealt with here only in outline. The present chapter is intended mainly as a guide to show how the principles, methods, and mathematics developed in earlier parts of this book can be used where they are applicable, namely, in homogeneous polymerization at no more than moderate conversion or in dilute solution (several phases may be present, provided the reaction is confined to a homogeneous liquid). Even in such cases, an accuracy and reliability as high as for ordinary homogeneous reactions cannot be expected. As in previous chapters, the presentation remains restricted to kinetic principles and their applications in modeling. No attempt is made to review the intricacies of reactivities of monomers and their molecular causes or their effect on polymer structure and properties. For details of these, the reader is referred to excellent texts on the subject [Gl-GU]. In polymerization kinetics, not only rates are of interest, but also molecular weights and their distributions. To the extent that this can be done without getting embroiled in major complications, such aspects are included. 11.1. Types of polymerization reactions From a point of view of chemistry, a distinction can be made between condensation and addition as mechanisms of polymerization, depending on whether or not a small molecule such as H2O or HBr is cast off when monomers link up [1]. Examples are The term polymerization is meant to include oligomerization (formation of low polymer).

348

condensation polymerization

addition polymerization

Chapter 11.

Polymerization

OH OH J^ H ^^-^N--^ C^] ^ § ^ ^ C s J '"

^

OH OH r-^^N'^'\<^^N-^^ CsX T 9 J '"' + HP

biroi

(H.l)

(11-2)

However, a classification into condensation and addition polymers as originally envisaged by Carothers [2] is no longer appropriate because some polymers can be synthesized by either method. For example, Nylon-6, a polymer with repeating units -NH(CH2)5CO-, can be produced either from 6-aminocaproic acid by condensation polymerization or from caprolactam by addition polymerization. As far as mathematics of reaction kinetics is concerned, there is no fundamental difference between condensation and addition polymerization: Whether or not a small molecule is cast off when two monomer or polymer molecules link up has little impact in this respect. On the other hand, a distinction relevant to kinetics is that between step growth and chain growth as reaction mechanisms. In step-growth polymerization, monomer or already formed polymer is added step by step to the growing polymer at its functional groups in an ordinary chemical reaction, and such link-up is the only reaction that occurs. Chain-growth polymerization differs in that a "reactive center" must first be created at a molecule, and new monomer molecules are then added successively until some other reaction or event brings the process to a halt and produces "dead" polymer that reacts no further: Chain growth must be initiated and usually also involves a termination reaction. It is sustained by a small minority of highly reactive chain carriers that act as an assembly line for churning out dead polymer molecules. The most common type of chain-growth polymerization is radical polymerization. An initiator or a photochemical reaction produces a radical that attaches itself to a monomer molecule, creating a group with odd-electron configuration (reactive center or propagating center) at which monomer molecules are added until two such centers react with one another or, more rarely, a center is deactivated by some other process. This is a mechanism much like that of ordinary chain reactions (see Chapter 10; the term "chain" in chain growth refers to that kind of mechanism, not to the growing molecular chain of repeating units in the polymer.) Unlike ordinary chain reactions, chain-growth polymerization need not involve radicals. The reactive center may instead be a carbanion or carbocation generated by intermolecular transfer of a proton or electron. Depending on the sign of the ionic charge on the chain carriers, the overall reaction is called anionic or cationicpolymerization. As in radical polymerization, initiation is required. How-

U—J

03

0)

> CO

O

'a3

S 2

o

E CI.

o

C/3

o

(4-1

s;

o

O

O

C3

.2

a.

.9

p

o CI.

O

C/5


I'll ^ o t^ S '-a

C/5

> ^

N

.2 (u

•^ "o

'o a.

''3
3 .g 2S

.2

G

'o

a s

•a

^ >>

o ^ •5 (D

O

o -^

c §

11.1. Types of polymerization reactions

2 o

o

%

C/3

o

O

349

350

Chapter 11. Polymerization

ever, termination by reaction of two chain carriers with one another cannot occur because charges of the same sign repel one another. As a result, reactive centers may be left over when all monomer is used up {living polymers). This can be utilized for production of specialty polymers. Coordination polymerization is yet another variation on the same theme. Here, polymerization is initiated by attachment of a monomer molecule to a metal complex. The polymer grows by successive insertion of monomer molecules at the metal. Growth stops when the metal complex detaches itself or the reactive center becomes deactivated by some intended or inadvertent event. Stereo-specific polymers can be produced. 11.2. Step-grov^^th polymerization The earliest polymers of practical use were prepared by step-growth reactions, most notable among them Bakelite, a phenol-formaldehyde copolymer first marketed in 1910 [4]. Its name was long almost synonymous with synthetic plastics and resins, has become generic, and is no longer restricted to phenol-formaldehyde copolymers. Most but not all step-growth polymerizations are condensations. 11.2.1. Functionality Step growth involves reactions of functional groups with one another. For example, the functional groups in polymerization of 6-aminocaproic acid to Nylon-6 [5,6] n H,N--^^^-v^ COOH '

• H - [ N-^^^^^v^ C ] - OH + n-1 H p ^H O^n

(11.3)

are — NH2 and — COOH. The monomer, carrying two groups per molecule, is said to be bifunctional. The functionality may be higher. For example, glycerol with its three hydroxyl groups is trifunctional in condensation polymerization with, say, a dicarboxylic acid or organic dihalide. The functionality may vary with reaction conditions. For example, in basecatalyzed copolymerization of phenol and formaldehyde, both monomers are bifunctional at ambient temperature, but phenol becomes trifunctional if the temperature is raised sufficiently. Copolymerization at ambient temperature can produce essentially linear, liquid, resole-type "prepolymers" of low molecular weight. Upon acidification and heat-curing, methylene and ether crosslinks formed by the now trifunctional phenol units transform the polymer into an insoluble resin [7]. The original Bakelite was such a "thermosetting" product. An additional functionality that comes into play only when the reaction conditions are changed is called latent functionality.

11.2. Step-growth polymerization

OH

351

OH

H^CO heat

HO

^OH

heat-cured resin

resole-type oligomers

11.2.2.

Mechanism and rate

Homopolymerization. In the simplest type of step growth, a bifunctional monomer reacts successively with itself, eventually forming a polymer with a large number of repeating units. The reaction may be an addition, but more commonly is a condensation. Although condensation usually is reversible, its equilibrium is driven toward complete conversion by removal of the small and volatile cast-off molecule: M

M M

M

(11.4) - ^

" ^ Q

where M is the monomer, Q is a cast-off small molecule, and Pj is a polymer molecule with i repeating units. A reaction involving only one kind of monomer, as in 11.4, is called homopolymerization. Polymerization of 6-aminocaproic acid to Nylon-6 (reaction 11.3 on facing page) is an example from industrial practice. Since the functional end groups of the polymer molecules Pj formed are the same as those of the monomer M from which they were formed, one must expect monomer link-up per reactions 11.4 to be accompanied by link-up of polymer molecules with one another. Also, if the cast-off small molecule is not effectively removed, polymer molecules may split up again: link-up:

Pi + Pj

-•

split-up:

PR +

-> P; + Pn,

Q

Pk + Q

(k = i + j )

(11.5)

(k = £ -f m)

(11.6)

(i+j = £+m)

(11.7)

Moreover, a so-called interchange reaction interchange:

Pi + Pi

-> Pf + P .

352

Chapter 11. Polymerization

may scramble polymer fragments of different length. Even if normal condensation is dominant and irreversible, interchange may occur if the temperature is raised (e.g., upon further processing). All steps in reactions 11.4 and 11.5 involve functional groups of the same kinds. One may therefore assume that all of them, except possibly the first linking of two monomers, have approximately the same rate coefficients [8] (see "shortsightedness" of reaction steps, Section 12.3). Moreover, if the reaction is run in dilute solution or stopped at reasonably low conversion, the coefficients may be assumed to remain unchanged as conversion progresses. In practice, these assumptions are usually valid, but not without exceptions [9-11], and must therefore be verified. However, even granted their validity, the wealth of simultaneous rate equations for all the participants makes it quite cumbersome to obtain concentration histories for the monomer and the individual polymer entities. [Analytical solutions can be obtained if condensation is irreversible and polymer-polymer link-up and interchange per reactions 11.5 and 11.7 can be disregarded [12], but are of little more than academic interest.] Easier to come by and just as useful is information about conversion of the functional groups. Each link-up in reactions 11.4 and 11.5 eliminates the two functional end groups that react with one another, and any interchange by reaction 11.7 leaves their number unchanged. Accordingly, the disappearance of functional groups is a bimolecular reaction and so essentially follows second-order kinetics. If the reverse reaction and polymer split-up are insignificant or suppressed, e.g., by removal or elimination of the cast-off small molecule Q as it is formed, the rate is -r^

= 2kC^^

(11.8)

where F stands for the functional groups of both kinds. Of particular interest is the fractional conversion of the functional groups,/p = 1 ~ Cp/Cp^, as a function of reaction time t or reactor space time r: batch:

/p(0 = 1

continuous stirred tank:

/p(7) = 1 -

\ i^iktc; (1 + SkrC^y^'^ - 1 1 4kTC^

(11.9)

(11.10)

Derivation. For liquid-phase batch, where -rp = —dCp/dr, eqn 11.9 is obtained by integration of eqn 11.8 over time; for a continuous stirred tank, eqn 11.10 is obtained from eqn 11.8 and the material balance for the functional groups, —rp= (Cp° — Cp)/T. Equations 11.9 and 11.10 assume the reverse reaction to be negligible or suppressed, the rate coefficient to be independent of conversion, and no significant fluid-density variation to occur upon reaction.

11,2. Step-growth polymerization

353

Copolymerization. Up to this point we have considered polymerization of a single monomer that carries two different functional end groups, those of one kind reacting with those of the other. Many commercial polymers, however, are produced by condensation of two monomers, each of which carries functional groups of the same kind at both ends. An example is Nylon-6,6, a polymer with alternating diamine and dicarbonyl units, made from 1,6-diaminohexane and adipic acid [5]: O

H

-OH

H

In such cases also, all linking steps are reactions of the same groups, so that the rate coefficient can once again be assumed to have approximately the same value for all. However, the two types of functional groups now are on different monomers and therefore are not necessarily present in stoichiometric amounts. For stoichiometric mixtures of monomers with different functionalities, the rate equation 11.8 and eqns 11.9 and 11.10 for fractional conversion remain valid. For nonstoichiometric mixtures, eqn 11.8 must be replaced by (11.11)

^^F,^F„

where F^ and Fg are the two different functional groups. The fractional conversion of FA, the group that is in the minority, is now given by batch:

CSTR:

/p = 1 -

4 = 1

(11.12)

Clexp(Akt)-Cl

J_fMr

AkTC;

IkTC:

(1 + Akrf

111

(11.13)

where A =

Cp° - c;

Derivation. Equation 11.12 is obtained from the integrated form of eqn 11.11 found in standard texts [13,14]: InCCp^Cp^ = IniClIC;) +

(Cl-C;)kt

Equation 11.13 follows from eqn 11.11 and the material balance for FA in the CSTR. More than two monomers may participate in copolymerization, and functionalities may be higher. Nevertheless, eqns 11.8 to 11.10 or eqn 11.11 remain applicable as long as the reverse reaction is negligible or suppressed and the

Chapter 11. Polymerization

354

rate coefficient is independent of conversion. However, the latter assumption becomes questionable as monomers of higher functionalities begin to form crosslinks (see also discussion of gel point farther below). CycUzation. The two functional end groups of a monomer or polymer molecule might react with one another to form a cyclic compound [15,16]. An example is the formation of caprolactam as a by-product in condensation polymerization of 6-aminocaproic acid to Nylon-6 [5]:

H^N-

•COOH

np

(11.14)

Such so-called cycUzation can occur to the almost complete exclusion of polymerization if five- or six-membered rings are formed [17]. Smaller rings are not favored because their bond angles are strained; neither are much larger ones because a functional end group on a long chain is likely to react with an end group of another molecules before it has a chance to come close to that on the other end of its own (see Figure 11.2). Rarely will more than one or perhaps two of the species involved undergo cycUzation to any significant extent.

number of atoms in ring Figure 11.2. Dependence of extent of cycUzation on size of ring formed (from Odian [16]). CycUzation comes about by a reaction of the same kinds of functional groups as the link-ups of monomer or polymer molecules and, like these, consumes two functional groups. CycUzation therefore does not change the form of the rate equation 11.8 or 11.11 for consumption of functional groups. However, the degree of polymerization and the molecular weight are affected (see below).

11.2. Step-growth polymerization

355

11.2.3. Degree of polymerization and molecular weight Number-average degree of polymerization. _Fov step-growth polymerization, the number-average degree of polymerization, A^, is traditionally defined as 7j ^

number of monomer molecules at start number of molecules after polymerization

Q J J^X

As will be seen, it can be related in a relatively simple fashion to the fractional conversion of functional groups [18], provided cyclization is insignificant. In step-growth polymerization of bifunctional monomers, each molecule, whether monomer or polymer, carries two functional groups: There are always half as many molecules as functional groups. Thus, when the number of unreacted groups has decreased to a fraction 1 - / p of the initial, the number of molecules has decreased to that same fraction of its initial. By virtue of its definition 11.15, the number-average degree of polymerization (monomer included in averaging) is the reciprocal of the latter fraction: N = 1/(1-/p) (11-16) In homopolymerization or in copolymerization of stoichiometric mixtures of bifunctional monomers, /p can be replaced by^ means of eqn 11.9 or 11.10. [If monomer is excluded from the mole count, TV = (2 - / F ) / ( 1 - / F ) (see eqn 11.85 in Section 11.5.2). At the usual high degrees of polymerization (/p-» 1), the difference becomes negligible.] Example ILL Control of molecular weight. Assume the end use of a polymer made by step-growth homopolymerization of a bifunctional monomer requires a numberaverage molecular weight MW of about 50,000 and that the molecular weight of the repeating unit is 100. Accordingly, the number-average degree of polymerization should be 50,000/100 = 500. Equation 10.16 with A^ = 500 and solved for/p gives a fractional conversion of functional groups/p = 0.998. There is little margin for deviation because a small variation in conversion results in a large change in molecular weight: /p = 0.987 would give A^ = 333 and MW = 33,333, and such_a polymer may not have the required mechanical strength; /p = 0.999 would give A^ = 1,000 and MW = 100,000, and that product may well be too stiff for processing. An easier way of controlling the molecular weight would be to add 0.2 mole percent of a monofunctional compound that reacts with and deactivates one percent of one of the functional groups, and then drive conversion of the remaining 99 percent of that group essentially to completion. Equation 11.16 for bifunctional monomers is a special case of the more general Carothers equation [18] that is applicable to monomers with any functionalities: N = 2/(2 - n,f,)

(11-17)

356

Chapter 11,

Polymerization

where % is the effective average functionality. In copolymerization of mixtures with stoichiometric amounts of functional groups, it is the average functionality; in copolymerization of nonstoichiometric mixtures, groups that cannot react because they exceed the stoichiometric amount are not counted in the averaging [19]. Example 11.2. Number-average degree of polymerization in step-growth polymerization of nonstoichiometric mixture of monomers. A mixture of two moles of glycerol and five moles of phthalic acid reacts. There are 2x3 = 6 - O H groups from glycerol and 5x2 = 10 -COOH groups from phthalic acid on 2+5 = 7 molecules. Only six of the ten acid groups can react with the six —OH groups, the other four are not counted. The effective average functionality thus is (6+6)/7 = 1.714. According to eqn 11.17, the maximum number-average degree of polymerization that can be obtained, at complete conversion of glycerol, is 2/(2 — 1.714) = 6.99. The Carothers equation becomes invalid if cyclization occurs to a significant extent. Cyclization reduces the number of functional groups, but leaves the number of molecules unchanged. This violates the underlying premise that there are always twice as many functional groups as there are molecules. Gel point. The Carothers equation can also be used to estimate the conversion needed to reach the so-called gel point in condensation polymerization involving monomers with functionalities higher than 2. The gel point is defined as the state of conversion at which gel formation caused by crosslinking begins to become apparent. With the assumption that this occurs when practically all molecules of the limiting monomer have reacted, the requisite fractional conversion of functional groups can be estimated with a rearranged form of the Carothers equation: /p = 2/np - 2/n^N As the number-average degree of polymerization, TV, is driven as high as possible, the second term on the right-hand side becomes negligible, so that

For example, in condensation of an equimolar mixture of glycerol with a trifunctional acid such as citric, «"F = 3 (both monomers are trifunctional), and the gel point is reached at a fractional conversion of groups of 2/3 = 0.667. Gelation actually begins before all molecules of the limiting monomer have reacted. The Carothers equation therefore overestimates the conversion of functional groups needed to reach the gel point. A more rigorous statistical treatment by Flory [20,21] and Stockmayer [22,23] considers the probability that a unit becomes attached to two chains [24]. This approach gives lower values for the fractional conversion at the gel point. Experimental observations suggest that the actual gel point typically falls between the estimates with Carothers' and Flory's equations [20,21,25]. A computer simulation of sol-gel distribution at high conversion has been published [26].

11.2. Step-growth polymerization

357

Molecular weight and molecular-weight distribution. The Carothers equation 11.17, where applicable, provides the number-average degree of polymerization of the reaction mixture (unreacted monomer included in the mole count). Usually, conversion of monomer is driven to a very high degree of completion and cyclization is suppressed. The number-average molecular weight of the polymer can then be obtained in good approximation from the number-average degree of polymerization simply by multiplication with the molecular weight of the structural unit (average weight if two different units alternate). However, the molecular-weight distribution is harder to come by and cannot be predicted with the same degree of accuracy. Here, the utility of mathematical theory is more in showing trends and relative magnitudes of effects than in quantitative predictions or application to design. Since step growth is a sequential reaction, the distribution of products it yields depends on the type of reactor (see Section 5.4). Analytical solutions can be obtained only under grossly simplifying assumptions and, therefore, are of little use in practice. In principle, the simultaneous rate equations for all participants, or at least their ratios, would have to be known and solved under the respective conditions. The complications here are that polymer link-up (reaction 11.5), interchange (reaction 11.7) and, more rarely, cyclization (reactions such as 11.14) may occur. Polymer link-up shifts the distribution to higher molecular weights and broadens it, and interchange and cyclization distort the relationship between fractional conversion and molecular weight. The most useful and most commonly employed simplified approach dates back to Flory [27,28] and is based on the premise of equal reactivity of functional groups and statistical growth. The most important application is to polymerization of bifunctional monomers and can be sketched as follows (Flory's derivation is more elaborate). In homopolymerization or copolymerization of stoichiometric mixtures of two monomers, the probability that two functional groups have reacted to form a link is given by the fractional conversion of groups, /p. A polymer with j -h 1 repeating units contains one more link than one with only j units. Therefore, its existence is less probable by a factor/p than that of the latter. In view of the large number of molecules involved, the ratio of the existence probabilities is also that of the mole fractions, Xj ^ i and Xj. Accordingly: X.,,.

= f,x,

(j > 1)

(11.19)

or, for the mole fraction as an explicit function of fractional conversion:

^j = fr'lifr /

- fj-\i-f,)

(jsi)

(11-20)

n=l

[The sum converges to 1/(1 -/p). Note that the mole count includes the monomer, but not the solvent, cast-off small molecules, or any inerts.] According to this statistical approach:

358

Chapter 11. Polymerization

In step-growth polymerization of bifunctional monomers, the mole fractions of successive polymers (with increasing number of structural units) are in a declining geometrical progression. The factor by which the mole fractions of two successive polymers differ is given by the fractional conversion of the functional groups. A mole-fraction distribution that is a declining geometrical progression is called a Schulz-Flory distribution or most probable distribution and is quite common [29,30]. As later examples will show, it can arise from other mechanisms as well and can therefore not be taken as evidence for step growth. Quantitatively, the weight fraction of polymer with j structural units as a function of fractional conversion of functional groups is given by

^j = W-'li^ir' /

=

0^1)

J/F^-HI-/P)^

(11.21)

n=l

[The sum converges to 1/(1 -/p)^.] This formula is for addition polymerization and requires a small correction for the weight of the cast-off small molecule in condensation polymerization. Although the mole-fraction distributions show a monotonic decline with the number of repeating units in the polymer, the molecular-weight distributions have maxima. This is because, in the low-polymer range, the weight increase with number of repeating units overcompensates the decrease in mole fraction. With progressing conversion, the maximum shifts to higher molecular weights and flattens (see Figure 11.3). 0.04

0.04 /p = 0.90 - \ /p = 0.95

o

0.02

0.02

^ 3

^ W p = 0.98 /P = 0.99 1

0

50

-1

100

—1^

150

number of structural units, j Figure 11.3. Schulz-Flory mole-fraction distribution (left) and corresponding molecular-weight distribution (right) at different degrees of fractional conversion of functional groups (adapted from Flory [27]).

11.3. Radical polymerization

359

Derivation ofeqn 11.21. The weight of polymer with j repeating units is jA^jAfW^, where A^j is the number of moles of that polymer and MW^ is the molecular weight of the repeating unit. The total weight of the mixture (including monomer) is N°AfWM, where A^° is the number of moles of monomer at start. The weight fraction of the polymer with j repeating units is the ratio of these two weights: w. = 1-J

^ = LJ.

(j > 1)

With A^j = NXj by virtue of the definition of mole fractions (A^ = number of moles including monomer), NIN° = 1 - / F , and eqn 11.20 forXj, this gives eqn 11.21. 11.3. Radical polymerization Chain growth differs from step growth in that it involves initiation and usually also termination reactions in addition to actual growth. This makes its kinetic behavior similar to that of chain reactions (see Chapter 10). However, the chain carriers in chain-growth polymerization need not be radicals, as they are in ordinary chain reactions. Instead, they could be anions, cations, or metal-complex adducts. While the general structure of kinetics is similar in all types of chain-growth polymerizations, the details differ depending on the nature of the chain carriers. The most common type of chain-growth polymerization is radical polymerization and will be examined first. 11.3.1. Mechanism and rate * A majority of commercial polymers are produced by radical polymerization. Foremost among these are polystyrene, polyethene (i.e., polyethylene), poly(vinyl chloride), poly (vinyl alcohol), poly (vinyl acetate), and poly (methyl methacrylate). In each of these, polymerization involves an olefinic double bond. However, radical polymerization is not restricted to such monomers. At its simplest, the mechanism of radical polymerization consists of radical production by an initiator (initiation), link-up of the radical with a monomer molecule (often considered part of initiation), addition of further monomer (propagation), and eventual deactivation (termination) of the growing polymer radicals by coupling, also called recombination, that is, by link-up of two radicals with one another. This is much as in ordinary chain reactions (see Section 10.3). Radical polymerization of styrene may serve as an example [31,32]. * For an excellent coverage of chemical and structural effects and their mechanistic implications, see a recent book by Moad and Solomon [G6].

360

Chapter 11. Polymerization Example 11.3. Radical polymerization of styrene. Styrene is a highly reactive monomer. If not stabilized, it polymerizes slowly even without an initiator [33]. Commercial polystyrene is produced with peroxy or azo compounds as initiators. The mechanism of polymerization initiated by 2,2'-azo-to-isobutyronitrile (AIBN) is as follows: N=N

mitiation

link-up with monomer

\ CN

>•

CN

CN

propagation

N.

O

rate = 2A„,q„

^Ink^M^X

^P^MQ]

termination

2^tcQp •

where in is the initiator, X is the radical produced by its decay, M is styrene monomer, and DP • is the total of all styrene-containing radicals (including that with only a single styrene unit). The initiation rate also involves an effectiveness factor/ which reflects the fact that some of the radicals from the initiator may become deactivated by reactions with one another before they manage to initiate a kinetic chain. The factor 2 in the initiation and termination rates appears because two radicals are produced or consumed in the respective reactions. The rate equations of propagation and termination presuppose that the rate coefficients do not change with growing length of the polymer chain. Because of their very low concentrations, quasi-stationary behavior of the radicals can safely be assumed (Bodenstein approximation). Initiation and termination rates then are equal in absolute value: ^/^init^n

=

^KCl

(11.22)

Solved for the radical population Cj^p.; ^EP-

~

C/^init/^tc)

Cin

(11.23)

11.3. Radical polymerization

361

The polymerization rate, which can be identified as the rate of monomer consumption, is given by -^M = ^ , „ . C „ q . + ^PCMQP. (11-24) The first term is the consumption by Hnk-up with the radical from the initiator, the second is the consumption by addition to the growing polymer radicals. Under practical conditions of production of high-molecular weight polymer, the first term is negligible compared with the second (long-chain approximation, see Section 10.3). If this can be assumed, elimination of Qp. by means of eqn 11.23 gives ^p C/^init /^tc )

^ in

(11.25)

^M

making the rate first order in styrene monomer and half order in initiator. [If the first term in eqn 11.24 cannot be disregarded, the rate includes an additional term 2/^init Qn» obtained with the Bodenstein approximation for the initiator radical X •, according to which the initiation and link-up rates can be taken as equal.] Rate behavior of this kind is observed for many other olefinic monomers. As an example, Figure 11.4 shows the rate of methyl methacrylate polymerization also to be first order in monomer and about half order in initiator. However, the mechanism in Example 11.3 is by no means universal. In outline, others involve: Termination by disproportionation. In Example 11.3, coupling of two polymer radicals was assumed to be the only termination mechanism, as is indeed essentially true for polymerization of styrene [34]. However, various other mechanisms may contribute to termination or even dominate it. The most common of these is disproportionation, mainly observed for tertiary and other sterically hindered radicals [35]. An example is methyl methacrylate [34] (see reaction 11.26 below). In disproportionation, two polymer radicals react with one another, transferring a 20

100 o

o X

10

10

0.3

10

1 CM

0.1

0.2

0.3

[M]

Figure 11.4. Rate of methyl methacrylate chain polymerization. Left: rate first order in monomer (redox initiator) [36]; right: rate approximately half order in initiator (benzoyl peroxide) [37].

362

Chapter 11. Polymerization

hydrogen atom to produce a stable saturated polymer molecule and another with a carbon-carbon double bond:

Like coupling, disproportionation as a reaction of two polymer radicals is second order in radicals. A contribution of disproportionation to termination thus does not alter the algebraic form of the rate equation 11.25, but the termination rate coefficient k^^ becomes the sum of two second-order coefficients k^^ and k^^ for coupling and disproportionation, respectively. However, the degree of polymerization, the molecular weight, and the molecular-weight distribution are affected by disproportionation (see Section 11.3.4). Chain transfer. Another mechanism of chain breaking, that is, of stopping the growth of a polymer radical, is chain transfer to monomer, another polymer molecule, the solvent, or some other inadvertently present or intentionally added species.* Chain transfer to monomer in most cases predominantly yields a saturated monomer radical and a polymer molecule with double bond. Hydrogen transfer from the monomer to the polymer radical, leaving the double bond on the monomer radical, can also occur but is energetically disfavored: CH

H,C 1

CH.I CH 11 R

^ ^

^

•CH 11 R

+

1 R

1

1 R

^ -^

CH_

(11.27)

H,C

CH,

1

R

+

i R

Polymer molecules produced by the unsaturated monomer radical carry terminal vinyl groups that can react with other radicals to form radicals with reactive centers along their carbon chains. Further growth then yields branched polymer [39]. * In the literature there is a lack of consensus on terminology regarding "termination." We follow Kennedy and Marechal [38]: Termination irretrievably ends the kinetic chain (e.g., by coupling, disproportionation, or chain transfer to produce an inactive radical); chain breaking ends the growth of the respective polymer radical by whatever mechanism without necessarily terminating the kinetic chain, which, upon chain transfer, may or may not continue on another molecule.

11.3. Radical polymerization

363

Chain transfer, whether to monomer, polymer, solvent, or an added transfer agent, breaks the kinetic chain, but does not per se terminate it. Unless the new radical is unreactive, chain polymerization continues, though on a different molecule. In many instances, the reactivity of the new radical is comparable to that of the old one and re-initiation of the new chain is fast. Monomer consumption then continues at its pace according to eqn 11.25, and only the degree of polymerization and the molecular weight are affected; if chain transfer is also very fast relative to propagation, only low polymer is produced {telomerization). On the other hand, the new radical generated by chain transfer may be unreactive. Chain transfer then decreases the rate of monomer consumption {retardation) or, if transfer is fast relative to propagation, polymerization may stop altogether {inhibition). Chain transfer producing an unreactive radical acts as another mechanism terminating the kinetic chain, in this case by a reaction that is first order rather than second order in polymer radicals (see also Section 10.5). Deactivating chain transfer to monomer is quite common in polymerization of allyl monomers [40-42]. Allyl radicals such as that of allyl acetate are resonance-stabilized, with the result that polymerization rates and molecular weights remain low. Moreover, with chain transfer as the dominant termination mechanism, the termination rate is first order in radicals. This lets the radical population become proportional to the initiator concentration and leads to a polymerization rate that is first order rather half order in initiator and zero order in monomer. Derivation. The Bodenstein approximation of a quasi-stationary radical population allows the absolute values of the initiation and termination rates to be equated. With terminating chain transfer being first order in radicals and initiator:

Solved for Qp.:

This gives a propagation rate that is first order in initiator and zero order in monomer:

The rate of monomer consumption is the sum of the rates of propagation and link-up of the initiator radical with monomer. The latter rate equals the initiation rate (Bodenstein approximation of quasi-stationary behavior of initiator radicals). The monomer consumption rate thus becomes r,, + rp = Ifk. ,(1 + kJkJC. J init^ p trM^ init

D

-J

init ^

D

trM''

in m

and is also first order in initiator and zero order in monomer.

(11.28)

364

Chapter 11. Polymerization

Another possible chain-breaking mechanism is chain transfer to polymer [43,44]. Here, a new reactive center is formed on the polymer chain of the receiving molecule, usually along its chain rather than at either end:

•CH

CH,

CH2

(11.29)

•CH

New growth from such a center produces a branch. Chain transfer may also occur to a carbon atom of the same polymer molecule five, six, or seven positions distant from the original reactive center [45]. This is called backbiting and is regarded as the mechanism of formation of short branches in polyethene polymerization [46,47]. Transfer agents that lead to production of unreactive radicals may be added to limit molecular weight [48-51]. Best suited are agents whose radicals are stabilized by adjacent groups or by resonance. The effectiveness of a transfer agent is characterized by its transfer constant, defined as the ratio of the rate coefficients of chain transfer and propagation: I?.

trM

(11.30)

p

Table 11.1 lists approximate values of transfer constants of some common agents in polymerization of styrene, methyl methacrylate, and vinyl acetate. Table ILL Approximate values of transfer constants of selected transfer agents in polymerization of styrene, methyl methacrylate, and vinyl acetate at 60°C (averaged and rounded values from Eastmond [52]). monomer transfer agent

isopropylbenzene isopropanol chloroform carbon tetrachloride carbon tetrabromide ^-butane thiol

styrene

methyl methacrylate

vinyl acetate

1*10-' 3*10-' 3*10-' 0.01 50 25

2*10-' 6*10-^ 1*10' 1*10-'

0^01 0.004 0.015 40 50

11.3. Radical polymerization

365

Dependence of rate coefficients on polymer chain length. The rate equations in Example 11.3 were derived with the assumption that the rate coefficients do not depend on the degree of polymerization of the polymer radicals and remain constant as more polymer molecules are formed. There are two major exceptions: For most monomers, the propagation rate coefficient, k^, is somewhat higher for the first one or two propagation steps than for later addition to longer polymer radicals [53]. This is of concern only in oligomerization, not if polymerization is carried to high molecular weight, as is the more common practice. Potentially more troublesome is a decline in coefficient values at high conversion. Unless polymerization is carried out in dilute solution, the mixture stiffens and reactive groups have a harder time finding partners to react with. The long-chain polymer radicals become entangled with other polymer chains, and while the small molecules of monomer can still find access to radical groups on the polymer with reasonable ease, the frequency of encounters of such radical groups with one another decreases sharply [54]. Termination then is impeded, causing the reaction to accelerate {Trommsdorff effect or gel effect, see high-concentration curves in Figure 11.5) [54-58]. This calls for care in handling of large amounts of liquid monomers such as vinyl compounds, whose polymerization is strongly exothermic: An accidental initiation may result in an explosive runaway (or, in corporate parlance that knows no disasters, an "unscheduled polymerization"). In the absence of solvent, propagation also may come to a stand-till, short of complete conversion of monomer (see curve for 100%). Moreover, because of the arrested termination, the final polymer may still contain reactive centers.

c o

^

time [min] Figure 11.5. Conversion as a function of time for polymerization of methyl methacrylate at different concentrations in benzene at 50 °C (adapted from Schulz and Harborth [55]).

366

Chapter 11. Polymerization

11.3.2. Photochemical initiation Vinyl compounds absorb ultraviolet light in the range of 200 nm. Irradiation, say, with a mercury lamp produces radicals that can initiate polymerization [59]. The initiation rate is '•ini. = I... - erC^d

(11.31)

where I^^^ is the intensity of absorbed light, 7° is the intensity of incident light, L ~

%^M

- k c^'^ (i>€rd

(11.32)

Polymerization rate equations for other termination mechanisms are readily obtained by replacement of Ifk,^^^ C,^ by 4>hhs or (j>eFCy^ d in the respective equations for conventional initiation. Initiation can also be achieved by irradiation of a reaction mixture to which an absorbing sensitizer has been added. The sensitizer then is the source of radicals and, in eqns 11.31 and 11.32, Qen replaces CM, and C^C^ll^ replaces CM^^ (sen = sensitizer). 11.3.3. Chain length A relatively easily determined quantity indicative of the extent of polymerization is the kinetic chain length, commonly defined as the average number of propagation steps occurring between initiation and termination and thus equalling the ratio of the rates of propagation and termination. However, more useful in connection with molecular weights is what we may call the radical chain length, v, given by the average number of propagation steps between initiation and chain breaking rather than termination, and directly related to the length of the polymer radicals: " = '•p/(-W


("" ^chbr is the total chain-breaking rate from all mechanisms.) Of course, if all chain breaking also terminates the kinetic chain, the kinetic and radical chain lengths are equal. Regardless of the chain-breaking mechanism, the propagation rate is '•p = ^PCMQP.

(11-34)

11.3. Radical polymerization

'iGl

The radical chain length depends on the mechanism of chain breaking. The most common of these is coupling. Here, the chain-breaking (and termination) rate is -'-.. = - ' - . =

24 Q V

(11-35)

With this and eqns 11.33, 11.34, and 11.23 the radical chain length becomes V = k^C^/2K,C,,,

= V2k,(fk,^,k,yC^Q;'''

(11.36)

and is seen to be inversely proportional to the radical population. Accordingly: • any attempt to boost the polymerization rate by an increase in the radical population—e.g., by use of more or a better initiator—comes at a sacrifice in molecular weight [61,62]. This important conclusion becomes plausible if one considers the reaction orders of propagation and chain breaking. Propagation is first-order in radicals, whereas chain breaking is second-order (granted coupling as the chain breaking mechanism, as is most common). As a consequence, an increase in the radical population speeds up chain breaking more than propagation, so that growth is stopped after fewer propagation steps. For chain breaking by disproportionation or any combination of coupling and disproportionation, eqns 11.35 and 11.36 also apply, but with a rate coefficient k^^ or ^tc -f-1^^ instead of ti^ alone. For chain breaking predominantly by chain transfer to monomer, the chainbreaking rate is 'chbr

~

'trM

"

'^trM ^ M ^ E P •

(neglecting the chain-breaking contribution from termination) and the radical chain length becomes . ^ k/K, (11.37) Similarly, for chain-breaking predominantly by chain transfer to solvent or an agent, the chain-breaking rate and radical chain length are — - r

-

k C C

and ^ = %IKs)CJC,

(11.38)

respectively (S = recipient species). The radical chain lengths for chain transfer (eqns 11.37 and 11.38) are independent of the size of the radical population. Accordingly, the conclusion above does not apply. Rather, if chain transfer is the dominant chain-breaking mechanism, an increase in radical population can accelerate polymerization without sacrifice in molecular weight. This is because both propagation and chain breaking then are first order in radicals and so are accelerated equally by an increase in the radical population.

368

Chapter 11. Polymerization

The radical chain lengths resulting from the various chain-breaking mechanisms can be summarized as follows: chain-breaking mechanism

disproportionation chain transfer to monomer chain transfer to S (transfer agent or solvent)

radical chain length 1/2 M ^in - 1 / 2 ^ ,-,-1/2

V2 k^ {fk,^,, k,^ K IKu

CM Q ;

(ATp /A:trs) C ^ / C s

If several mechanisms contribute significantly to termination, their rates are additive. 11.3.4. Degree of polymerization and molecular weight While the radical chain length as the ratio of the propagation and chain-breaking rates refers to conditions existing at the moment, the degree of polymerization and the molecular weight characterize a polymer that may have been produced over a span of time during which the conditions varied. Here, only some idealized situations with conditions held constant can be described in any detail. Number-average degree of polymerization. For step-growth polymerization, The number-average degree of polymerization was defined as the ratio of the numbers of monomer molecules at start and of molecules after polymerization (eqn 11.15), and so included unreacted monomer in the molecule count. More practical for chain-growth polymerization is to exclude the unreacted monomer: r; ^ number of monomer units in total polymer Q ^ ^g) ^°' number of polymer molecules (subscript pol added to indicate restriction to polymer). The number-average degree of polymerization is usually taken to be the radical chain length, or twice that length if termination is by coupling where one polymer molecule is formed from two radicals. This is permissible only for high degrees of polymerization, as becomes apparent from the fact that the minimum degree of polymerization according to the definition 11.39 is 2 (dimer being the lowest possible polymer) whereas the radical chain length as ratio of the propagation and chain-breaking rates can in principle be lower. Many complications can arise. Only two relatively simple cases will be examined here: polymerization with chain breaking by coupling and by chain transfer to an agent that produces an unreactive radical. In chain breaking exclusively by coupling, the total consumption of monomer per unit time is

77.5. Radical polymerization

369

r .. + r init

p

and the n u m b e r of radical chains broken per unit time is chbr

trm

tc

init

T h e n u m b e r of polymer molecules produced per unit time is half as large because two p o l y m e r radicals are consumed when forming o n e polymer molecule. Accordingly, with eqn 11.33 and noting that in this case -r^hbr = ~^trm = ^inu— KT

-r _

P°'

r M

_

'¥iF~

+r

'ink

l.Jj-

'p

2 ( 1 + V)

(11.40)

Vir..

This formula correctly gives the limit of 2 at infinitesimal chain length. In chain breaking exclusively by chain transfer to a deactivating transfer agent, T r , total m o n o m e r consumption per unit time is again the s u m of the initiation and propagation rates. However, of the m o n o m e r radicals M - produced by initiation and link-up, some are lost by chain transfer to the agent; only a fraction P ends u p as polymer, where P is the probability that a radical will add m o n o m e r rather than b e terminated by transfer: p

^

propagation rate propagation rate + transfer rate

^

^p r^ + (-r^^ir)

(11.41)

Accordingly, the number of m o n o m e r molecules converted to polymer per unit time is ~ ^^Wpol

~

-* ''init "^ ''p

and the n u m b e r of polymer radicals terminated p e r unit time, equaling the number of m o n o m e r radicals that added monomer, is ^"'"trTr/pol

~

^''init

This gives a degree of polymerization ^pol

=

(-''M)pol/(-^rTr)pol

=

(^^init ^ ^pV^^init

and after replacement of P with eqn 11.41 and noting that, h e r e , —^chbr = Hnitiv , = 2 + r / r . , pol

p

= 2 + ^

(11.42)

init

H e r e , t o o , the limit at infinitesimal chain length is seen to b e 2 as called for. Other cases are m o r e complex. F o r example, in termination by disproportionation, molecules that have undergone initiation and link-up but h a v e been deactivated by disproportionation before managing to add further m o n o m e r must b e

370

Chapter 11. Polymerization

excluded from the mole count. Also, if the concentrations of monomer and initiator decline during polymerization, as they do in batch reactors, the degree of polymerization of the eventual total product is given by the integral over time or conversion. The exception here is the case of termination by chain transfer to monomer, in which the radical chain length and degree of polymerization are independent of concentrations. Molecular weight. The number-average molecular weight is obtained from the number-average degree of polymerization by multiplication with the molecular weight of the monomer, plus the formula weight of the end group from the initiator (a correction that is significant only at low degrees of polymerization). Molecular-weight distribution. For low conversion or polymerization in dilute solution, approximate simple formulas can be derived. Complications arising at high conversion in bulk polymerization will be outlined later. The probability P that a polymer radical of any length continues to grow is given by p ^

propagation rate propagation rate + chain-breaking rate

(11.43)

where chain transfers to monomer, solvent, or transfer agent are included in the chain-breaking rate. Chain transfer to polymer can be disregarded as unlikely at low conversion. With M = monomer, Tr = transfer agent, and S = solvent, and with propagation rate

^p ^M Q p •

chain-breaking rate

^ITM^MQ:/"

transfer to monomer

+ ^trTrO'rQp*

transfer to transfer agent

"^ ^trS ^ S ^ZP

transfer to solvent

'

^

^ V^tc •*" ^ t d ) ^ E P •

coupling and disproportionation

the probability is 1

P =

^-Ku

[

-2(t?,.^jqp./Q

(11 4

where the t?i = kjk^ are the respective transfer constants (see eqn 11.30). Consider first the case of chain breaking exclusively by disproportionation or (terminating) chain transfer. Here, chain breaking converts polymer radicals into dead polymer molecules without change in the number of repeating units they contain. A polymer radical with j4-l repeating units has resulted from one more propagation step with probability P than a polymer radical with j repeating units, so the existence of the former is less probable by a factor P than that of the latter. In terms of mole fractions instead of existence probabilities:

11.3. Radical polymerization

371

where {x^^^^ is the mole fraction based on total polymer (monomer not counted) of a polymer with i repeating units. The mole fraction of the next larger polymer molecule is smaller by a factor P < \. This is a Schulz-Flory distribution (declining geometrical progression; see Section 11.2.3): •

Radical polymerization with chain breaking exclusively by disproportionation or chain transfer yielding unreactive radicals produces a Schulz-Flory molefraction distribution.

This is as in step-growth polymerization of bifunctional monomers (eqn 11.19). The argument and result are formally the same as in that case, with the probability P taking the place of the fractional conversion/p of functional groups. However, there is one important difference: •

In step-growth polymerization, the geometric factor is the fractional conversion and increases toward unity with progressing conversion. As a result, the distribution shifts toward higher molecular weights and flattens as conversion progresses. In contrast, in radical polymerization, the factor is the probability of the growth reaction and remains constant or decreases with monomer concentration, so that any shift in the distribution is toward lower weights (granted the rate coefficients do not vary with monomer conversion).

This reflects a fundamental difference between step-growth and radical chain-growth kinetics: In step growth, conversion starts with formation of low polymer, and molecular weight increases as conversion progresses and the polymer molecules grow by link-up with monomer or other polymer molecules. In radical chain growth, polymer is produced by a very small family of chain carriers that quickly settle down to a quasi-stationary molecular-weight distribution and function as an assembly line that churns out dead polymer of a matching distribution. The number of polymer molecules increases widi conversion, but their molecular-weight distribution remains essentially constant if monomer and initiator are replenished. The mole and weight fractions as explicit functions of the probability of the growth reaction are (^j)poi = P J - ^ / f ^ P - ^

= PJ-^(l-P)

(]>2)

(11-46)

0^2)

(11-47)

/ n=l

(>^j)po, = j p j - 7 f : ( « + i ) f " - ' / n=i

= ^^':'^^"/^' I

-

r

[The sums converge to 1/(1—P) and (2-P)/(l-P)^, respectively.] Unlike in step growth, the fractions are based on total polymer, to the exclusion of monomer, and are therefore written with the extra subscript pol. This difference comes about

372

Chapter 11. Polymerization

because the probability argument leading to eqn 11.43 is for molecules produced by the chain carriers and so does not apply to the monomer. It also accounts for the difference in algebraic forms between eqns 11.47 and 11.21. For the more frequently encountered case of termination exclusively by coupling, eqn 11.46 indicates what fraction of polymer radicals have j repeating units, and random pairing of the members of a radical population with this distribution must then be considered. According to Flory [28,63] the result is

(w.)^, = i^il-Pfp-^

(UAS)

This is a Poisson-type distribution, narrower than that from eqn 11.47. Any chain transfer that does not terminate the kinetic chain reduces the molecular weight, but does not alter the shape of the distribution, which remains given by eqn 11.48. In practice, several or all chain-breaking mechanisms may contribute. If so, the weight fraction can be taken as the weighted average of those derived for coupling and for chain transfer plus disproportionation (eqns 11.47 and 11.48). High conversion. Molecular weights and molecular-weight distributions at high conversions are less predictable and more dependent on the nature of the monomer. The two principal additional factors that come into play are chain transfer to polymer and the decrease of the rate coefficients for coupling and disproportionation with increasing polymer content, viscosity, and chain entanglement. The more polymer, the greater is the chance of (non-terminating) chain transfer to polymer rather than monomer. Large polymer molecules offer larger targets for chain transfer and so are more likely recipients. This shifts growth to larger polymer and thereby broadens the molecular-weight distribution. The slow-down of termination is most pronounced for large polymer molecules because these are more prone to chain entanglement. This also results in favored growth of large molecules and adds to the broadening of the molecularweight distribution.

11.4. Ionic polymerization The chain carriers in chain-growth polymerization may be anions or cations rather than radicals. Such ionic polymerization shares many features with radical polymerization, but differs in one important respect: Since ions of the same charge sign repel one another, spontaneous binary termination by reaction of two chain carriers with one another cannot occur. In fact, the reaction may run out of monomer with chain carriers still intact. There are also subtle mechanistic differences between anionic and cationic polymerization, which will therefore be examined separately.

11.4, Ionic polymerization

373

11.4.1. Anionic polymerization Compounds capable of forming carbanions stabilized by delocalization of the negative charge can be made to undergo anionic polymerization under appropriate conditions. Typical representatives are compounds with conjugated double bonds, such as styrene and butadiene, or with hetero-atoms that are more electronegative than carbon, among them ethene oxide and caprolactam. Historically important is Buna 5, the first successful commercial synthetic rubber, produced by sodiuminitiated copolymerization of butadiene and styrene [64]. A delicate balance of base strengths and a proper choice of solvent and reaction conditions is essential: A carbanion whose base strength is too high may deactivate itself by deprotonating a pro tic solvent or forming a stable complex with a polar aprotic one; a solvent with too low a dielectric constant will not sufficiently encourage ionization, while one with too high a dielectric constant will deactivate the carbanion. Under suitable conditions, anionic polymerization is faster than radical polymerization and so can be conducted at lower temperatures. The main reasons are fast initiation by an ionic reaction and absence of an effective termination mechanism. However, the sensitivity to impurities is much greater, and choice and control of reaction conditions are more delicate. Water, oxygen, carbon dioxide, and other substances able to react with carbanion chain carriers must be strictly excluded. The key feature distinguishing anionic (and cationic) from radical polymerization is the absence of spontaneous binary termination and has already been mentioned. Unless chain transfer occurs, polymer chains keep growing until all monomer is used up. At that stage, the polymer still carries reactive centers [65] —it is said to be a "living polymer" [66-68]—, and polymerization can be started anew by addition of further monomer. Block copolymers can be synthesized from a living polymer by addition of a different monomer [69,70]. Because of the strong dependence on the nature of the solvent and initiator and the high sensitivity to impurities and reaction conditions, quantitative predictions of rates and molecular weights are difficult to make and less reliable than in radical polymerization. Initiation [68]. The three most common kinds of initiators for anionic polymerization are alkali-metal alky Is, metal amides, and elementary alkali metals. In initiation by an alkali-metal alkyl, the alkyl links up with the monomer to form a carbanion, leaving the metal as a cation to compensate the negative charge. An example is the initiation of styrene polymerization by butyl lithium [71-73]: ""CH

H C Li+

(11.49)

374

Chapter 11. Polymerization

This is a technique especially suited for production of living polymers. In initiation by a metal amide, carried out in liquid ammonia, the amide dissociates into a metal cation and an amide anion, NH2", which then adds to the monomer [74]:

K+

NH;

+

CH X X

' ^A- ^^ w^ HC K+

— ••

(11.50) ^ ^

Chain transfer to the ammonia solvent by deprotonation of the carbanion is common in such systems [74]:

"9" ^ ' +

NH3

"'

9"^

+

N H ; K-

(11.51)

In initiation by alkali metal, an electron is transferred from the metal to the monomer (the metal is usually charged as a solid or colloid). Sodium-initiated butadiene polymerization provides an example [75,76]: Na

+

U^C^''^^^^^^

•

JJ^/^^CH;

Na+

(11.52)

In such systems, dimerization of the carbanion or electron transfer from another alkali metal atom can produce a di-anion that adds monomer at both ends, complicating kinetics [76]. An interesting variation is initiation by a combination of an alkali metal and an aromatic with condensed rings, e.g., naphthalene. The aromatic anion radical transfers an electron to a monomer such as styrene, which then dimerizes and grows at both ends [66,77]. Propagation. Regardless of its degree of polymerization, the carbanion and its metal counter ion Me^ can be expected to exist in a spectrum of different forms, ranging from a covalently bonded species at one extreme to separate ions at the other, with a contact ion pair and a solvent-separated ion pair as intermediates (Winstein spectrum) [78,79]: [PMe] M covalently bonded

• [P-Me^] < contact ion pair

• P" Me"' < solvent-separated ion pair

• P" -h Me^ separate ions

(1153)

11.4. Ionic polymerization

375

(The contact and solvent-separated ion pairs are also called intimate and loose ion pairs, respectively.) Since ionic dissociation-association reactions in general are very fast, quasi-equilibrium of the species can reasonably be assumed. Monomer is inserted at the carbon atom carrying the negative charge. The covalently bonded species tends to be unreactive, and reactivity increases sharply with progressing dissociation (left to right in 11.53). Under most conditions, the free carbanion, P", exists only at a much lower concentration than do the other forms: It is a lapc (least abundant propagating center) constituting only a negligibly small fraction of the total of the species in 11.53 (analogous to a lacs in catalysis, see Sections 8.5.1 and 9.3). However, because of its high reactivity, its contribution to the propagation rate may nevertheless be significant or even dominant. The nature of the solvent strongly affects the equilibria and thus the propagation rate and the form of the rate equation. For example, in a solvent with poor solvating power—e.g., a hydrocarbon—there is less tendency to dissociate, and free ions and a solvent-separated ion pair may not even exist. Granted quasi-equilibrium of the various species in the reactions 11.53 at any number of repeating units in P, the relationships between the concentrations are

where the ^oi ^^e the respective equilibrium constants. If these do not vary as the chains grow, eqns 11.54 apply to the totals of all covalent species, contact ion pairs, etc., regardless of the lengths of their chains. The concentrations can then be expressed in terms of the total population EP~ of potential propagating centers: ^[PMe]

where

~

^j;p-/^»

^[P-Me*]

~ ^01 ^ E P " ' ^ '

EP- = [PMe] + [P-Me1 + P'Me^ + pK ^

I .K,,

.K,,

(11.55) (11.56)

-KJC^^.

(11.57)

All species in the Winstein spectrum 11.53 might contribute to propagation. Leaving all possibilities open, the propagation rate can be written

(11.58)

1

.K,^.K,,.KJC^^^

C

C

where the k^ are the respective rate coefficients of monomer addition. Equation 11.58 covers all bases. Of particular interest in practice are three special and fairly common situations:

376

Chapter 11. Polymerization

Case I:

The free anion is the lapc (least abundant propagating center) but, thanks to its high reactivity, provides a significant rate contribution. If the free anion is the lapc, the last term in the denominator of eqn 11.58 is negligible, so that K = I + KQ^ -h KQ2. Moreover, without an added electrolyte, the concentrations of P~ and Me^ are necessarily equal. In this case, the last of eqns 11.55 gives

{KJ(l^K,,.Kjf" ^ 1 / 2

Cp. = tion rate is '"p p

~

(11.59)

V M ^ ^aa ^- ELi'P - •*• " ^ D b ^^ErP "/- * -^ iva

where

K =

k, + A:,/:^, + k^K,^ 1-^0.-^02

^ '

'

^ "

^03 '

1-^01-^02]

If only the free anion contributes to the rate, the first term in eqn 11.59 is negUgible. Case //:

As in Case I, the free anion is the lapc and provides a significant rate contribution. An inert strong electrolyte with same cation as that produced by the initiator has been added. A strong electrolyte with same cation can be added to keep an overly pesky reaction in check. Without the negligible last term in its denominator, eqn 11.58 gives

% = iK-K/Cu.-)C,,-C^

{k^^ k,KJil.K,,.K,,))

(11.60)

where k^ is defined as in Case I above. A large excess of added electrolyte may suppress the rate contribution from the free anion entirely, and the second term then disappears. Case III:

The free anion is the lapc and does not contribute significantly to the rate. This may be the case in a solvent of low polarity, in which lack of dissociation keeps the free ion at too low a concentration to be effective. If the free anion does not contribute significantly to the rate, the last term in the numerator of eqn 11.58 also becomes negligible, so that:

where k^ is as defined in Case I. As long as the free anion is the lapc, the rate equations are of the algebraic forms of eqns 11.58 to 11.61, regardless of which of the other three propagating centers or combination of these contributes to the rate.

11 A . Ionic polymerization

2>11

Polymerization without termination: living polymers. If care is taken, all termination reactions c a n b e avoided. Polymerization then proceeds until all m o n o m e r is used u p o r the reaction is quenched by addition of a deactivating transfer agent. Under such conditions, the rate of monomer consumption is the s u m of the rates of initiation a n d propagation:

where r^ is given b y eqn 11.58 or one of its simpler special forms. T h e contribution from initiation is negligible except at low degrees of polymerization, a n d m a y require a n effectiveness factor. Moreover, if initiation is fast relative to propagation, as is almost always true, the total population of propagating centers, E P " , is equal to the amount of initiator initially added (multiplied with the effectiveness factor if called for). In the three situations discussed above:

•

•

•

Case I (free anion is lapc, but contributes to the rate): The polymerization rate is first order in monomer and of order between one half and one in initially added initiator. From eqn 11.59:

Accordingly, a plot of -r^lC^{C^^^''^ versus (Q)^^^ is linear with intercept k^ (contribution from free anion) and slope k^ (contribution from ion pair). An example is shown in Figure 11.6, left (next page). Case II (as Case I, but strong electrolyte with conmfion cation has been added): The rate is first order in monomer and of order between zero and minus one in the common cation, possibly approaching zero at very large excess of the added electrolyte. From eqn 11.60:

so that a plot of -ry^lCyiC^^ versus l/CMe+ is linear with intercept k^ (contribution from ion pairs) and slope k^ (contribution from free anion). For an example, see Figure 11.6, right. Case III (free anion is lapc and does not contribute significantly to the rate): The rate is first order in monomer and initially added initiator (see eqn 11.61).

If no quenching agent is added, the number-average degree of polymerization is given by the ratio of the number of monomer molecules at start and number of polymeric carbanion propagating centers, EP~ (also called "living ends"). Granted fast initiation, the latter equals the number of initiator molecules added at start, so that (AOpo, =

C^/Q:

(11-63)

Chapter 11. Polymerization

378

200 12

o c

U=8 h

<_ 0.2

0.4

0.6

0.8

0.1

0.2

Figure 11,6. Rate of anionic polymerization of styrene initiated by sodium naphthalene in 3-methyl tetrahydrofuran at 20°C. Left: linear variation of rate with (Cjn)^^^; right: inverse linear variation of rate with concentration of Na"*" in presence of added sodium tetraphenyl borate. (Data from Schmitt and Schulz [80].) An effectiveness factor may have to be used if not all of the initiator produces carbanion propagating centers. If the carbanion produced by initiation dimerizes to give a di-anion, as in lithium-initiated butadiene polymerization, the degree of polymerization is twice as large as indicated by eqn 11.63. As in other situations, the number-average molecular weight is given by multiplication of the degree of polymerization with the molecular weight of the monomer. If initiation is very fast relative to propagation, as is usually the case, all propagating centers start growing at about the same time and grow at the same rate. If other termination reactions can be avoided, a quenching agent added at the appropriate time can stop growth when a desired molecular weight has been reached. This makes it possible to produce polymer standards of specific, highly uniform molecular weights [81,82]. •

Fast initiation and suppression of termination in living polymerization can lead to a reaction behavior that is fundamentally different from that of radical polymerization. In the latter, the size of the chain-carrier population is dictated by a balance of initiation and termination rates. Chain carriers in quasistationary molecular-weight distribution convert monomer to polymer of a matching distribution. In contrast, in ionic living polymerization, the initiator is used up quickly, producing a population of propagating centers that keep adding monomer to themselves. In radical polymerization, the number of polymer molecules increases with conversion of monomer, but the molecular-weight distribution remains essentially unchanged. In ionic living polymerization, polymer radicals keep growing to ever larger sizes as monomer is consumed, but their number does not change.

11.4. Ionic polymerization

379

Polymerization with termination. Unless care is taken to prevent it, termination may occur by chain transfer to the solvent or impurities. Many different combinations of possibilities exist. For illustration, one example may suffice. Example 11.4. Anionic polymerization of styrene in ammonia [74]. Polymerization of styrene in ammonia can be initiated by potassium amide and is believed to proceed as follows. Initiation consists of styrene

(see also reaction 11.50). KNH2 dissociation is fast and at equilibrium, making the initiation rate '-ini.

= *taicCNH,<^M

= ^U.U'STKNH.QNH.CM/Q.

(11-64)

where K^^^ii is the dissociation constant of KNH2. The free carbanion remains at so low a concentration that it does not contribute significantly to the propagation rate. Accordingly, that rate is given by eqn 11.61 (Case III): ^P

= ^aQp-^M

(11.65)

where k^ is a lumped rate coefficient defined as in eqn 11.59. There is substantial chain transfer to the ammonia solvent and some to water that is likely to be present as a trace impurity. Transfer may occur from the covalent species or either kind of ion pair or a combination of these, but the transfer rates are of the same algebraic form and so can be lumped without loss of accuracy. The rates are r.s = ^,rsQp-Cs (11-66) '-.H.o = W ^ r P - C „ p

(11-67)

respectively. Transfer to ammonia produces a new NH2~ anion (see reaction 11.51) which immediately initiates another chain: The kinetic chain is not terminated and the polymerization rate is not affected. Transfer to water produces an unreactive OH" anion and so terminates the kinetic chain. Equating the initiation rate to the rate of termination by chain transfer to water (Bodenstein approximation for the population of propagation centers) and solving for Qp- one finds C

- (k K

Ik

^C

C Ic

C

With this substitution in eqn 11.65 the propagation rate becomes ''p

=

^QNH^C'M/Q.CHP

\k

=

^b^init^KNH,/^trH,oj

(11.68)

380

Chapter 11. Polymerization The rate of monomer consumption contains an additional contribution from initiation (see eqn 11.62). If the kinetic chains are long, this contribution is negligible, and the rate, then given by eqn 11.69, is first order in potassium amide, second order in styrene monomer, and of orders minus one in potassium ion and water. The number-average degree of polymerization can be obtained from the rates of propagation (eqn 11.65) and chain breaking (sum of eqns 11.66 and 11.67) as in radical polymerization with termination by chain transfer to a transfer agent (see eqn 11.42): (N) , = l-ll±

= 2 ^

^-^

(11.70)

The molecular-weight distribution is a Schulz-Flory distribution as in eqn 11.45. 11.4.2. Cationic polymerization With respect to kinetics and mechanism, cationic and anionic polymerization share the key features that distinguish ionic from radical polymerization: the absence of effective binary termination, and initiation that is fast relative to propagation. There are differences, however. Cationic polymerization tends to be faster and may have to be conducted at temperatures below 0° C. Initiation may be more complex, and carbocations, while less sensitive to impurities than carbanions, are more prone to undergo a variety of chain transfer or rearrangement reactions that break the kinetic chain. As a result, high molecular weights are harder to obtain, and predictions of rates and molecular-weight distributions are less reliable than with anionic polymerization. Because of the analogies to anionic polymerization, the discussion here will be restricted to comments on differences in chemical detail. Cationic polymerization is applied almost exclusively to monomers with olefinic double bonds. Susceptible are double bonds whose carbon atoms carry electron-donating substituents such as alkyl groups. Thus, isobutene with two methyl groups adjacent to the double bond polymerizes readily, propene with only one is sluggish, and ethene with none is inert; a-methyl styrene is more reactive than styrene; vinyl ethers are reactive, but vinyl chloride is not. The most important commercial product is butyl rubber, produced by copolymerization of isobutene with small amounts of isoprene, initiated by AICI3, BF3, or TiCl4 [83]. Initiation [68]. The general mechanism of initiation of cationic polymerization is proton capture by the monomer. Bronsted acids can be used, but the choice is limited: The acid must be strong enough to be an effective proton donor, but its anion must not be so nucleophilic that it would form covalent bonds. Perchloric, sulfuric, phosphoric, and substituted sulfonic acids meet the requirements. More commonly used for production of high molecular-weight polymers are Lewis acids. Examples are BF3, AICI3, ZnCls, TiCl4, and PCI5 among many others.

11.4. Ionic polymerization

381

Initiation by a Lewis acid requires the presence of a proton donor (protogen) such as water, alcohol, or hydrogen halide. Generation of the carbocation is preceded by link-up of the initiator with the protogen, as shown here for polymerization of isoprene induced by BF3 [84]: BF3

+

Ufi

^—•

BF3OH2 - C - [BF3OH]^ " " ^ H .

(11.71)

C«3

or, in general terms: M in + pr

<

•

in-pr

-^^^^—•

P^^ A"

(11.72)

where in is the Lewis-acid initiator, pr is the protogen, P^ is the monomeric carbocation, and A~ is the associated anion.* Usually but not necessarily, the ratecontrolling step in this process is the second, and the first is at quasi-equilibrium. Initiation can also occur spontaneously, either by so-called self-ionization [86,87] as in 2 AlBr3 <

M • AlBr2-^AlBr,- •^^^^=-—• AlBr^M^ AlBr^-

^

or by direct reaction with monomer [88,89]: TiCl4 + M

•

TiCljM^ C r

(11.74)

Polymerization of highly reactive monomers can also be initiated by iodine: Iodine adds to the double bond, and hydrogen iodide is then split off and acts as the protogen [90]. Other possible initiation methods include photochemical initiation [91], ionizing radiation [92], and electrolysis [93]. Termination. As in anionic polymerization, termination by coupling or disproportionation cannot occur, leaving chain transfers as the most likely mechanisms. With regard to kinetics of chain breaking, the various chain-transfer reactions are analogous to those in radical polymerization (see Section 11.3.1) and need not be described again in detail. In cationic polymerization, the most common chain transfer is to monomer and leaves the polymer with a double bond while generating a monomeric chain carrier of the same structure as the original one [94]. Cationic polymerization can be effectively retarded or quenched by addition of transfer agents such as alkyl or aryl amines or phosphines that form stable quaternary cations [95]: * A different notation with the protogen as "initiator" and the Lewis acid as "co-initiator" has been suggested [85], but has so far not been generally accepted.

382

Chapter IL Polymerization

HC-A-

+

NR3

•

HC~NR3A-

R

(1175)

R

Water, alcohols (in combination with KOH), acids, and anhydrides in sufficiently high concentrations act in the same fashion [96]. It is interesting that such agents can function as the needed protogens at low concentrations, but have a retarding or quenching effect at high concentrations. A termination unique to cationic polymerization, suggested [97] but still controversial [98], is by reaction with the counterion: '"\ HCMBF3OH]R

"*\ •

HC—OH

+

BF3

(11.76)

R

Propagation. As in anionic polymerization, a spectrum of dissociation states of the propagating centers can exist [99] (Winstein spectrum [78]; see carbanionic dissociation 11.53 in previous section). Accordingly, the general rate equation 11.58 and its simpler special forms 11.59 to 11.61 apply with obvious adjustments under the respective conditions. In cationic polymerization under practical conditions, dissociation may be weaker and the solvated carbocation may not exist. On the other hand, in some cases the covalently bonded species can contribute to or even dominate the propagation rate. This is referred to aspseudo-cationicpolymerization [99-102]. It does not alter the form of the general rate equation 11.58 for propagation because the reaction orders for propagation are the same for the covalent species as for the contact and solvent-separated ion pairs. However, more complex behavior has also been observed [103,104]. Living polymers. To suppress termination in order to produce living polymers by ionic polymerization, the propagating centers must be stabilized against termination, but left reactive enough for propagation. These conflicting demands are harder to balance in cationic than in anionic polymerization, and special techniques had to be developed [105-107]. Even so, the cationic reactive ends in living polymers begin to decay slowly once all monomer is used up. 11.5. Coordination polymerization Coordination polymerization is another variant of chain-growth polymerization. The kinetic chain starts with addition of monomer to a metal complex, propagation is by successive insertion of monomer at the metal, and termination occurs when the metal complex separates from the polymer. Inasmuch as the complex is restored

11.5. Coordination polymerization

383

upon termination, this is a catalytic process. Nevertheless, the metal complex is often called an initiator rather than a catalyst because of the analogies to other modes of chain-growth polymerization, in some processes also because the catalyst is left in the final, solid polymer product. By far the most important industrial coordination polymerization processes are Ziegler-Natta polymerizations of 1-olefins [108-111], most notably the production of high-density polyethene [112] and stereo-specific olefin polymers and copolymers [109]. These processes employ solid catalysts, and mechanistic details are not yet fully understood. The earliest Ziegler-Natta catalysts were insoluble bimetallic complexes of titanium and aluminum. Other combinations of transition and Group I-III metals have been used. Most of the current processes for production of high-density polyethene in the United States employ chromium complexes bound to silica supports. Soluble Ziegler-Natta catalysts have been prepared, but have so far not found their way into industrial processes. With respect to stereo-specificity they cannot match their solid counterparts. Since the pioneer work of Ziegler [113] and Natta [114,115] in the 1950s, different types of soluble catalysts for coordination polymerization of monomers with olefinic double bonds have been developed. With the possible exception of some soluble Ziegler-Natta catalysts, all appear to be complex hydrides of transition metals, among these nickel, cobalt, and palladium [108]. The most important industrial processes employing homogeneous coordination catalysts are polymerizations of ethene and 1-olefins, in particular Shell's ethene oligomerization that produces straight-chain olefins in very high selectivity (see Example 11.5 farther below). 11.5.1. Mechanism Mechanisms of coordination polymerization differ somewhat, depending on the nature of the metal complex and the monomer. However, the principal features are common to most. Kinetics are best understood for homogeneous reactions. The reaction requires as a first step the attachment of the monomer to the metal of the hydride complex, typically at a free coordinative site as a x-complex of the olefinic double bond. The monomer then inserts itself between the metal and the hydrogen, forming a a-bond with the metal and vacating the coordinative site it had occupied. That site then accepts another monomer molecule, which subsequendy inserts itself between the first monomer molecule and the metal (Cossee mechanism [116,117]; a more complicated pathway has also been suggested [118,119]). This chain of events repeats itself until either a polymer molecule with unsaturated end group splits off after having transferred a hydrogen to the metal and thereby restored the catalyst to its original hydride form, or the chain is intentionally terminated by hydrogenolysis to produce a saturated polymer molecule and restore the hydride catalyst. For ethene polymerization:

384

Chapter 11. Polymerization

LMeH

1-olefin

ethene

LMe

H

y

LMe' /

(11.77)

I I I

H.

LMe'

ethene ethene

LMe'

LMe'

y In homogeneous coordination polymerization, the L are ligands such as cyclooctadiene. The pathway of termination by hydrogenolysis is shown dashed. For the purpose of reaction mathematics and granted quasi-stationary behavior, the two sequential steps of monomer addition and insertion can be consolidated into a single step (see Rule 8.55 in Section 8.6). If this is done, the network can be written

(11.78)

where M is the monomer, cat is the catalyst, the Pj* are the metal-complex alkyls, and the Pj are dead polymer molecules, subscript i denoting the respective number of their monomer units. All growth steps Pj* -f- M —• l?{li ^re chemically the same except for the length of the polymer chain, and so are all termination steps Pj* —• Pj 4- cat (or Pj* -h H2 —• Pj + cat) (i > 2) in which polymer is split off. The respective rate coefficients, k^ and k^^^, can thus be assumed not to vary significantly with chain length i. Note that the addition of the first monomer molecule to the catalyst occurs with a different rate coefficient, to be written ko^.

11,5. Coordination polymerization

385

11.5,2. Molecular-weight distribution and degree of polymerization Granted the invariance of the rate coefficients for propagation, k^, and catalyst splitoff, k^,^, the net rate of formation of an adduct Pj*i with j +1 monomer units is

Given quasi-stationary behavior of the propagating centers, the Bodenstein approximation Tp* = 0 can be used. With eqn 11.79 and solved for Cp* /Cp* this gives

<^p;/^p- = KCJ{k^C^.k,J

(jsl)

(11.80)

Equation 11.80 assumes spontaneous termination; for termination by hydrogenolysis, k^,^ should be replaced by kl^p^^. [The right-hand side of eqn 11.80 also expresses the probability that an adduct will continue to add monomer rather than split off catalyst and produce a dead polymer molecule.] Since all adducts Pj* produce polymer Pj with the same rate coefficient k^,^, the ratio in eqn 11.80 is also that of the concentrations or mole fractions of the respective polymer molecules Pj+i and Pj that are formed:

W^i

= Kcjik^c^.Kj

(j>i)

(11.81)

Moreover, that ratio is independent of the number of monomer units in the polymer molecules, so that eqn 11.81 describes a declining geometrical progression: Homogeneous coordination polymerization under stationary or quasi-stationary conditions produces a Schulz-Flory mole-fraction distribution. (This is not true for some solid polymerization catalysts.) The distribution is as in radical polymerization with termination by disproportionation or terminating chain transfer (eqn 11.45 in Section 11.3.4) and, with increase of the progression factor as conversion increases, in step-growth polymerization of bifunctional monomers (eqn 11.19 in Section 11.2.3). According to eqn 11.81, the progression factor is

g = KCJik^C^.KJ

(11.82)

and remains constant if the monomer concentration is kept constant. Example 11.5. Ethene oligomerization in the Shell Higher Olefin Process. The Shell Higher Olefin Process, abbreviated SHOP, produces predominantly internal straightchain olefins in the Cio to C^g carbon-number range from ethene [108,110,120,121]. Oligomerization of ethene to straight-chain 1-olefins of even carbon numbers is only

Chapter 11,

386

Polymerization

the first step. It is followed by catalyzed isomerization to internal olefins and finally olefin metathesis that breaks olefins at their double bonds and reassembles the fragments in a random fashion.* Olefms of carbon numbers higher or lower than desired are recycled. Only the oligomerization step is of interest here. SHOP oligomerization is a three-phase process. The reactor contains a polar liquid catalyst phase, a nonpolar liquid product phase, and ethene gas. Ethene consumed in the catalyst phase is resupplied from the gas phase, and oligomer produced in the catalyst phase separates out to form the product phase. The cataTable 11.2. Ethene oligomer formation lyst is a nickel hydride complex measured in laboratory batch reactor at with a phosphine carboxylate group 100°C and 80 atm pressure with a SHOPtype catalyst. [121,122] in 1,4-butanediol: product

weight g

mmol

C4H8 QH12

3.74 4.38 4.57 4.47 4.20 3.83 3.42 3.02 16.20

66.79 52.14 40.80 31.93 25.00 19.50 15.23 11.91

C 10^20 ^12^24

C14H28 C16H32 C18H36

higher


mmol ratio 0.781 .783 .783 .783 .780 .781 .782

• Ph Ni 0^"^0^ (Ph = organic phosphine). The mechanism is as in 11.77, with spontaneous termination. Table 11.2 shows a typical Schulz-Flory distribution obtained with this process.

Given the Schulz-Flory distribution, the mole and weight fractions of polymer with j monomer units (based on polymer) as a function of that number j are (11.83)

^)pol

I

n=l

and (^PPOI

= j^^-7E(^-^i)r-^

(j>2)

(11-84)

2-g

respectively (the sums converge as shown). The number-average molecular weight is * The metathesis reaction is still occasionally referred to as disproportionation although it bears no resemblance to disproportionation as a chain-growth termination step. Rather, it is analogous to interchange in step-growth polymerization (reaction 11.7 in Section 11.2.2) except that the reactants are broken apart at their double bonds and the fragments reconnected by double bonds.

11.5. Coordination polymerization

n=l

/

387

2-g 1-g

n=l

(11.85)

or, after replacement oi g with eqn 11.82: p

M

(11.86)

= 2 + ^

2 + kC^lt

(AO,pol

irm

where v is the radical chain length (see Section 11.3.2). At sufficiently high degrees of polymerization {g approaching unity), eqns 11.85 and 11.86 reduce to (AOpo, = 1/(1 - g) = V (11-87) The degree of polymerization is seen to show the same dependence on the geometric-progression factor as in step-growth polymerization of bifiinctional monomers (eqn 11.16) and radical polymerization with chain breaking by disproportionation or terminating chain transfer (eqn 11.42). 11.5.3. Polymerization rate If the condensed network 11.78 is taken at face value, the rate of polymerization (i.e., of monomer consumption) is (11.88) ^Ol^cat^M •*" ^ p Q p - ^ M Vm^P; - r The exact distribution of metal over free catalyst and the propagating centers is usually not known. A rate equation in terms of total metal, EMe, is therefore desired. This turns out to be k C

1+ 1^^ ^ \ t

p

"*"

'^»

M

^01 ^ M

trm

c

c

(11.89)

"^ ^ t r m

The rate is first order in total metal and, depending on conditions, approximately first order in monomer. Note that eqn 11.89 is in terms of liquid-phase monomer concentration; if the monomer must be supplied from a gas phase, as in SHOP, Langmuir-type absorption may produce a tendency toward saturation kinetics. Derivation. To replace the catalyst concentration in eqn 11.88, the Bodenstein approximation is applied to that species: ^trm^EP-

C.a. ^

^01 ^ c a t ^ M

-

^

K..C,,.lk,,C^

(11.90)

According to eqn 11.83, the concentration of the first propagating center, Pi*, is C„

QPVE^""'

= c,p.(i-^)

(11.91)

388

Chapter 11. Polymerization Lastly, the metal balance is ^Me = C „ , . C , ,

(11.92)

Equations 11.90 to 11.92 permit Qat and Cp* to be expressed in terms of Qp*, and then the latter in terms of total metal, QMC- The result is eqn 11.89. 11.6. Chain-growth copolymerization The description of chain-growth kinetics in the preceding sections has focused on polymerization of single monomers. However, the great majority of polymers produced on a large scale are copolymers. In fact, our present-day ability to tailormake polymers of desired mechanical and chemical properties owes a great debt to the progress in the science of copolymerization. To give only two examples: Butyl rubber is a copolymer of isobutene and small amounts of isoprene [83], and Saran is a copolymer of vinyl chloride and vinylidene chloride [123]. In essence, a second monomer, and maybe even a third, is included to modify polymer properties such as elasticity, tensile strength, etc. Kinetic aspects of step-growth copolymerization have been examined in Section 11.2.2. The principal features of chain-growth copolymerization are very different, but are alike for all types of chain growth, that is, for radical, anionic, cationic, and coordination polymerization. Many chain-growth copolymerizations include dienes such as divinyl benzene or divinyl adipate that act as crosslinking agents and lead to gel formation. Polymerization kinetics in such cases are complex and are beyond the scope of a this book. Here, only binary copolymerization of monofunctional monomers will be examined. For an excellent and extensive treatment that includes copolymerization of more than two monomers as well as crosslinking by bifunctional monomers, the reader is refer to Odian's book [124]. The two principal aspects to be considered here are copolymer composition and polymerization rate. The ways of deducing molecular weights and molecularweight distributions are essentially the same as in homopolymerization and will not be reiterated. 11.6.1, Polymer composition: reactivity ratios and copolymer equation A key facet of copolymerization is the possible disparity of reactivities of the monomers. Traditional procedure is to assume, at least as an approximation, that the reactivity of a growing propagating center depends only on the identity of its reactive end unit (i.e., the last monomer added), not on the composition and length of the rest of its chain [125-127] {first-order Markov or terminal model; see also

1L6. Chain-growth copolymerization

389

"shortsightedness" of reaction steps, Section 12.3). Even so and even with only two monomers, four different types of reactions can occur and must be considered: Each of the two monomers can react with an end group of its own kind or with one of the other kind. With two monomers M^ and Mg the possible reactions are: MA

+ .••-MA-

•

-

M * '"B

-

MA

+ .• •

MB

+ .••-MA*

-

MB

+ ...-MB*

-

^ •

...-MAM/

rate AT^A^A^A*

...-MBMA*

^AB^A^B*

...-M^MB*

^BA^B^A*

...-MBMB*

%B^B^B*

where the Mj* are the reactive end units or propagating centers, be they radical, cationic, anionic, or complexed, and the k^ are the respective rate coefficients (first subscript refers to monomer). The rates at which the monomers are consumed are -'"A

~

^ A A ^ A ^ A " •*•

^AB^A^B-

-''B

~

^ B A ^ B ^ A - •*"

^BB ^ B ^ B * '^BB^B^B-

(11.93) (M in subscripts is suppressed for simplicity). To find what composition the polymer will have at a given monomer composition, an equation for the ratio of the monomer consumption rates as a function of the concentrations of the monomers is needed. With eqns 11.93 and the Bodenstein approximation of quasi-stationary behavior of either propagating center, M^* or MB*, one obtains -''A

'"A-

c.iPS. - c,)

- ' • B

''B-

CeiPb^^B^ ^A)

(11.94)

where

= ^\JKk

and

p, s k^Jk^^

(11.95)

are termed reactivity ratios. Equation 11.94, known as the copolymer equation (or copolymerization equation) expresses the ratio in which M^ and Mg units are added to the polymer at given monomer concentrations, that is, the instantaneous copolymer composition. Alternatively, that composition can be expressed as the mole fraction y^ = r^J(rj^* + r^*) of M^ units added (based on added units) as a function of the mole fraction x^ = C^/(C^ + C^) (based on total monomer population): •^A(Pa - 0

+ ^A

^A(Pa + P b - 2 ) + 2XA(1-Pb) + Pb

(11.96)

390

Chapter 11.

Polymerization

Derivation. The ratio of the monomer consumption rates 11.93 is ^AA^A^A* 'B

^AB^A^B-

^AA^A(^A-^^B-)

'^BB^B^B^ B B ^ B ^ B - •'' ^ '^BA'^B^A^BA^B^A-

"^

^AB^A

(11.97)

^BB ^ B "^ ^BA ^ B ^ ^ A ' ^ ^ B ' ^

The Bodenstein approximation for the end units — M^* is ^AB^A^B-

^BA^B^A-

"

^

and gives Cj^./C^,

—

(11.98)

^AB^A^^BA^B

Replacement of C^JC^* in eqn 11.97 with this expression and use of eqns 11.95 for the rate coefficient ratios gives eqn 11.94. That equation in terms of mole fractions and with -r^l{-r^) = y^ly^ = JA/(1->^A) gives Pa^A + -^A-^B

y^ Pa^A

+

2XAXB

+

Pb-^B

Equation 11.96 is obtained from this with x^ = \ - x^.

1.0 0.8 o B c >^ .2 'o 0.6 cd

G

0.4 0.2 C/3

0 1.0

"^ 0.8

^ 0.6

^Cf>. "^OA . "^e

< %

^ ^ \ 0.4 ^ 0.2

0.6 0.4

.0^

^ .^^^


0 0

%.

''^^.

'^er,

Figure 11.7. Instantaneous copolymer composition in radical copolymerization of styrene and 2-vinyl thiophene; mole fraction of styrene in polymer as function of initial mole fraction and fractional conversion of styrene; calculated with reactivity ratios p^ = 0.35 and p^ = 3.10 (from Mayo and Walling [128]).

11.6. Chain-growth copolymerization

391

The copolymer equation 11.94 describes the instantaneous copolymer composition at given S monomer concentrations. An example is shown in Figure 11.7. In batch polymerization, of course, the concentration ratio of .2 o the monomers does not remain constant as conversion progresses, so that the instantaneous copolymer composition varies with time. The prediction of the composition of total polymer at any given J I L. instant then requires integration 0 0.2 0.4 0.6 0.8 1.0 over time [130]. styrene mole fraction in monomer feed Compilations of reactivity ratios for various pairs of monoFigure 11.8. Instantaneous copolymer compomers in radical polymerization sition in cationic, radical, and anionic styrene/ have been provided by Eastmond methyl methacrylate copolymerization initiated [131] and Odian [132]. The re- by SnCl4, benzoyl peroxide, and Na in liquid activity ratios for pairs of given NH3, respectively (from Pepper [129]). monomers can be very different for the different types of chain-growth copolymerization: radical, anionic, cationic, and coordination copolymerization. Although the copolymer equation is valid for each of them, the copolymer composition can depend strongly on the mode of initiation (see Figure 11.8). Several special cases are of interest: Case I:

random copolymers (also called ideal or statistical copolymers)

PaPb = 1

Here, k^^ /k^^ = kj^^ Ik^^, that is, the probability of adding M^ rather than Mg is the same for both kinds of end groups -M^* and -Mg*. The monomer that is preferentially added becomes enriched in the polymer relative to the monomer mixture. Examples of such behavior are radical copolymerizations of styrene and butadiene (PaPb — 1-1)» vinyl chloride and vinyl acetate (PaPb = 0-9), and vinylidene chloride and vinyl chloride (PaPb =1-1) [133]. Mathematically, the dependence of the instantaneous copolymer composition on the monomer composition is analogous to that of liquid-phase on gas-phase compositions in ideal vapor-liquid equilibria. This explains why the term "ideal" was chosen (not to be understood as an optimum or desirable condition).

392

Chapter 11. Polymerization

Case 11:

alternating copolymers

Pa = Pb = 0

In this more interesting case, each monomer adds only to an end unit of the other kind (k^^ = 0, k^^ = 0). In the polymer, units M^ and MB then alternate. Coordination copolymerization of olefins and carbon monoxide, catalyzed by complex hydrides of Pd(II) or Rh(I) in the presence of an alcohol co-solvent to yield polyketo esters, provides an example [134,135]: Olefin and carbon monoxide are added alternatingly, and reaction with alcohol terminates the kinetic chain and restores the catalyst. For ethene as the olefin: nH2C = CH2 + n C O

Case III:

block copolymers

4- ROH

•

H-(CH2CH2CO)„-OR

Pa > I, p^ > 1

In this hypothetical case, the monomers add preferentially to end groups of their own kind (/T^A > ^AB» ^BB > ^BA)- As a result, "blocks" M^ M^ MA ••• ^^d MB MB MB ... of units of the same kind are formed and join to yield a polymer in which they alternate. Such behavior has been reported for some coordination copolymerizations [136,137], but has not been conclusively established. Block copolymers have great importance for many practical applications, but are more conveniently produced from living polymers (see Section 11.4).

11.6.2. Polymerization rate While the copolymer equation is universal in that it applies to all kinds of chaingrowth copolymerization, an equally universal equation for the polymerization rate cannot be arrived at. For assessing the composition of the copolymer, only the ratio of the monomer consumption rates was needed, and that ratio was found to be a unique function of the monomer concentrations and rate coefficients. In contrast, the polymerization rate is composed of the absolute values of the monomer consumption rates, and these depend also on the concentrations of the propagating centers and thereby indirectly on the mechanism and rate of termination. In copolymerization, several different combinations of initiation and termination mechanisms are possible, giving rise to a variety of different polymerization rate equations. Only two cases will be singled out here: radical copolymerization with termination by coupling, and ionic polymerization with termination by chain transfer to a deactivating agent or impurity. For other combinations, the derivation of rate equations follows along the same lines. Radical polymerization. No matter whether the propagating centers are radicals, anionic, cationic, or coordinated, the propagation rate is equal to the sum of the consumption rates of the two monomers, given by eqns 11.93:

11.6. Chain-growth copolymerization

"^

^ A A ^ A ^ A - ^ ^ A B ^ A ^ B * "^ ^BA ^

393

^ A * "^ ^B ^ B ^ B '

In radical polymerization with termination by coupling, there are three possible termination steps: reaction of end groups —M^* with -M^*, of -M^* with -Mg*, and of — Mg* with -Mg*. Each eliminates two reactive end groups. Leaving the possibility open that all steps contribute significantly, the termination rate is -^trm

~

^V^tAAWv*

"^ ^ t A B ^ A ' ^ B *

''" ^ t B B ^ B ' )

*

^

The initiation rate for radical polymerization is r.

= Ifk.C.

init

J

init

in

(11.101) ^

^

(granted one initiator molecule produces two radicals). With the Bodenstein approximations of quasi-stationary behavior of the individual propagating centers and their total, the rate in terms of the monomer concentrations is found to be [138]: r P

=

(11.102) / 2 2 2 2\^'^ \ i^lAA / ^ B A ) ^ A + (^tAB / ^ A B % A ) Q ^ ^ B + ( % B / ^ A B ) ^ B /

where p^ and p^ are the reactivity ratios defined in eqns 11.95 (derivation is given farther below). Equation 11.102 is rather unwieldy. However, it can often be simplified: Binary terminations as reactions of two radicals with one another have low activation energies and large rate coefficients that, with few exceptions, are of the same order of magnitude (monomers giving strongly stabilized radicals are poorly suited for polymerization; see also Section 10.4). As a result, in copolymerization: •

Termination tends to be dominated by the more abundant radical.

This is much as in ordinary chain reactions (see Section 10.3). The ratio of the radicals with -M^* and -Mg* end groups is given by eqn 11.98. Accordingly, termination is likely to occur by coupling of -M^* end groups if /^ABQ » ^BAQ» or of -Mg* end groups if the opposite is true. If termination is by —M^* coupling, the second and third denominator terms in eqn 11.102 can be dropped as unimportant; if it is by -Mg* coupling, the first and second terms can be dropped. Ionic polymerization. In ionic polymerization with termination by deactivating chain transfer, the propagation rate equation is also given by 11.99, but the initiation and termination rates are different. In initiation, each initiator molecule produces only one propagating center: r . = fk..a (11.103) imt

J

init

In

^

^

394

Chapter 11. Polymerization Chain transfer can occur from end groups of either or both kinds:

iK.•C^• - K^.c^-)c,.

-r

(11-104)

where Tr is the transfer agent or impurity. With the Bodenstein approximations as for radical copolymerizatiom, the propagation rate becomes (Pa^A + 2CACB + PbCB^)AnitCin

(11.105)

where p^ and Pt. are the reactivity ratios defined by eqns 11.95. Equations 11.102 and 11.105 state the propagation rates. The polymerization rates include in addition the consumption of monomer in initiation. However, this contribution is negligible except in oligomerization. Derivation of eqns 11.102 and 11.105. For both radical and ionic copolymerization, the Bodenstein approximation for the propagating centers -Mg* yields r^.

^BA^B^A-

^B •

—

^AB^A^B-

"

^

(11.106)

(^BA ^ B ' ^AB ^ A ) ^ A •

Using this to replace Cg* in eqns 11.99 and 11.100 one obtains ^AA^AB^A + 2 A : A B % A ^ A ^ B + % B ^ B A ^ B

(11.107)

C^.

^AB^A

and ^tAA(^AB^A)

"^ ^ t A B ^ A B ^ B A ^ A ^ B + ^tBB(^BAC^B)

C^.2

(11.108)

(^AB ^ A )

respectively. The Bodenstein approximation for the total population of propagating centers amounts to equating the initiation and termination rates. For radical polymerization this gives, with eqn 11.101 and 11.108 and solved for Q*: /^init ^in CA-

=

^AB^A ^IAA^^AB^A)

^

^tAB^AB^BA^A^B

Using this to replace C^* in eqn 11.107 one finds

"^ ^tBB V^BA ^ B )

Summary

r

= P

(^AA^ABVA

"^ 2 A : A B ^ B A ^ A ^ B + % B % A ^ B

395

) (Ainit C^j^^)

/ ^ ^ V1/2 I^IAA^^AB^A) "^ ^tAB^AB^BA^A^B "^ ^ I B B ^ ^ B A ^ B ) /

(11.109) ^

^

Dividing numerator and denominator by /TAB^BA ^^^ replacing the ratios /C^A /^BA ^^^ by the respective reactivity ratios with eqns 11.95 one obtains eqn 11.102. For ionic polymerization, equating the initiation and termination rates (eqns 11.103 and 11.104), replacing Cg* with eqn 11.106, and solving for C^*, one obtains

^BB/^AB

(^trA^AB^A •*" ^trB^BA^fi) ^ T r

Replacement of Q* in eqn 11.107 by this expression, division of numerator and denominator with A:ABA:BA, and introduction of the reactivity ratios yields eqn 11.105.

Summary A distinction can be made between condensation and addition polymerization, depending on whether or not a small molecule such as water or hydrogen halide is cast off when monomers link up. With respect to kinetics, a more relevant distinction is between step growth and chain growth. In step-growth polymerization, molecules link up with one another by reaction of their functional end groups, and that is the only reaction occurring. Molecular weight increases with progressing conversion. In chain-growth polymerization, initiation is required to produce chain carriers or reactive centers that then add monomer molecules successively until some event terminates the kinetic chain or monomer is used up. The number of polymer molecules increases with progressing conversion, the molecular weight as a rule remains constant. Radical, anionic, cationic, and coordination polymerization proceed with chain-growth mechanisms. Step growth is essentially a second-order reaction of the functional groups. The number-average molecular weight is related in a simple fashion to the fractional conversion of functional groups by the Carothers equation. That equation can also be used to estimate the gel point (state of conversion at which crosslinks begin to form) in polymerization of monomers with more than two functional groups per molecule. If the monomers are bifunctional and statistical growth can be assumed, the mole fractions of successive polymer molecules (with increasing number of monomer units) are in a declining geometrical progression. This is called a Schulz-Flory distribution. Radical polymerization requires initiation to produce radicals that link up with monomer molecules to produce reactive centers. Additional monomer molecules are then added successively at these centers. In this way, a small family of polymer radicals acts as an assembly line to produce "dead" polymer. The most common termination mechanisms are reactions of two polymer radicals with one another, either by coupling to yield one larger dead polymer molecule or, more rarely, by disproportionation to convert two

396

Chapter 11.

Polymerization

radicals into a saturated and an unsaturated dead polymer molecule. With either of these terminations, the polymerization rate is first order in monomer and of order one half in initiator. Chain transfer to monomer, polymer, or solvent can also occur. Such chain breaking stops the growth of the polymer radical and may or may not terminate the kinetic chain, which might continue on another molecule. If transfer terminates the kinetic chain, the polymerization rate is first order rather than half order in initiator. In bulk polymerization, the increase in viscosity with conversion reduces the rate coefficients. Termination by coupling or disproportionation, involving two polymeric radicals, is more strongly affected than propagation. This causes self-acceleration (Trommsdorff effect) and, under certain conditions, can result in a runaway. Chain breaking exclusively by disproportionation or chain transfer produces a Schulz-Flory molecular-weight distribution. Chain breaking predominantly by coupling produces a higher degree of polymerization and a narrower, Poisson-type molecular-weight distribution. Compounds capable of forming carbanions stabilized by delocalization of the negative charge can be made to undergo anionic polymerization. A key feature distinguishing anionic (and cationic) from radical polymerization is that binary termination cannot occur because ionic charges of same sign repel one another. Other termination mechanisms can be suppressed, and the polymer then still contains propagating centers when all monomer is used up ("living polymers"). Typical initiators are alkali metals, their alky Is, and metal amides. Rate behavior, reaction orders, and molecular-weight distributions depend on conditions. In particular, the rate can be reduced by addition of a salt with the same cation as that of the initiator. As a rule, initiation is fast compared with propagation, so that the propagating centers start growing at the same time and add monomer at the same rate. This makes it possible to produce polymers of specified molecular weights and narrow molecular-weight distributions by addition of a deactivating agent when polymerization has progressed as far as desired. Cationic polymerization is similar to anionic polymerization in that binary termination by recombination or disproportionation cannot occur. The most common initiators are Bronsted or Lewis acids and iodine. A plethora of possible side reactions make it difficult to attain high molecular weights or prepare living polymers. Also, theoretical predictions of rates, molecular weights, and molecular-weight distributions are in general not reliable. In coordination polymerization, monomer forms an adduct with a transition-metal complex, and further monomer is then successively inserted between metal and carbon. Termination occurs when the metal complex splits off from the polymer or the chain is broken intentionally by hydrogenolysis. Since the initiator is restored to its original form, the process is catalytic. The most important industrial processes are Ziegler-Natta polymerizations of a-olefins and employ solid catalysts. Most catalysts for coordination polymerization are hydride complexes of transition metals. An important example is the Shell Higher Olefin Process (SHOP) for homogeneous oligomerization of ethene with a complex nickel catalyst. The molecular-weight distribution is a Schulz-Flory distribution. The rate is first order in the catalyst metal. In chain-growth copolymerization, the composition of the polymer depends on the concentrations and relative reactivities of the monomers. The relative reactivities can be drastically different in radical, ionic, and coordination polymerization. Three special cases

References

397

are random, alternating, and block copolymerization. In random copolymerization, the preference of adding monomer M^ rather than Mg is the same for polymers with reactive end group M^ as for those with reactive end group MB; the sequence of M^ and Mg units in the polymer then is random. In alternating copolymerization, each monomer adds preferentially to reactive end groups of the other kind; in the product, M^ and MB units then alternate. In rarely observed block copolymerization, each monomer adds preferentially to end groups of its own kind; the product then consists of alternating long "blocks" of monomer units of the same kind. However, a more practical method of producing block copolymers is via living polymers. Rate behavior in chain-growth copolymerization is complex. The presence of reactive end groups of different types and different reactivities makes for a profusion of possible propagation and termination steps. While copolymer composition is essentially dictated by the relative amounts and relative reactivities of the monomers, the rate depends in addition on the population of propagating centers and thereby on the termination mechanism. Examples include control of molecular weight in step-growth polymerization, number-average degree of polymerization in step-growth polymerization of nonstoichiometric monomer mixtures, radical and anionic polymerizations of styrene, and ethene oligomerization to linear 1-olefins in the Shell Higher Olefins Process. References General references Gl.

G2. G3. G4. G5. G6. G7, G8.

G. C. Eastmond, The kinetics offree radical polymerization of vinyl monomers in homogeneous solutions, in Comprehensive chemical kinetics. Vol. 14a, C. H. Bamford and C. F. H. Tipper, eds., Elsevier, Amsterdam, 1967, ISBN 044441486X. P. J. Flory, Principles of polymer chemistry, Cornell University Press, Ithaca, 1953. P. C. Hiemenz, Polymer chemistry: the basic concepts, Dekker, 1984, ISBN 082477082X. C. D. Holland and R. G. Anthony, Fundamentals of chemical reaction engineering. Prentice Hall, Englewood Cliffs, 2nd ed., 1989, ISBN 0133356396, Chapter 10. J. P. Kennedy and E. Marechal, Carbocation polymerization, Wiley, New York, 1982, ISBN 0471017876. G. Moad and D. H. Solomon, The chemistry of free radical polymerization, Pergamon, Oxford, 1995, ISBN 0080420788. M. Morton, Anionic polymerization: principles and practice. Academic Press, New York, 1983, ISBN 0125080808. G. Odian, Principles of polymerization, Wiley, New York, 3rd ed., 1991, ISBN 0471610208.

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Chapter 11.

Polymerization

A. Rudin, The elements of polymer science and engineering, Academic Press, San Diego, 2nd ed., 1999, ISBN 0126016852. S. R. Sandler and W. Karo, Polymer syntheses. Academic Press, Boston, 2nd ed., Vol I-III, 1992-1996, ISBN 0126185115, 0126185123, 0126185131. M. Szwarc, Ionic polymerization fundamentals, Hanser, Munich, 1996, ISBN 3446185062.

Specific references 1. 2. 3. 4. 5. 6. 1.

8. 9. 10. 11. 12. 13. 14.

15. 16. 17. 18. 19. 20. 21.

Flory (ref. G2), Section II-2. W. H. Carothers, J. Am. Chem. Soc, 51 (1929) 2548. F. G. Helfferich and P. E. Savage, Reaction kinetics for the practical engineer, Course #195, AIChE Educational Services, New York, 7th ed., 1999, Section 6.10. L. H. Baekeland, J. Ind. Eng. Chem., 1 (1909) 149; 6 (1913) 506. J. N. Weber, Polyamides, in Kirk-Othmer, Encyclopedia of chemical technology, 4th ed., J. I. Kroschwitz and M. Howe-Grant, eds., Wiley, New York, Vol. 19, 1996, ISBN 0471526886, p. 472. Sandler and Karo (ref. GIO), Vol. II, Chapter 2, Section 2. P. W. Kopf, Phenolic resins, in Kirk-Othmer, Encyclopedia of chemical technology, 4th ed., J. I. Kroschwitz and M. Howe-Grant, eds., Wiley, New York, Vol. 18, 1996, ISBN 471526878, p. 6037.22. Flory (ref. G2), Section III-7. E. G. Lovering and K. J. Laidler, Can. J. Chem., 40 (1962) 31. M. Kronstadt, P. L. Dubin, and J. A. Tyburczy, Macromolecules, 11 (1978) 37. Odian(ref. G8) p. 57. R. W. Missen, C. A. Mims, and B. A. Saville, Introduction to chemical reaction engineering and kinetics, Wiley, New York, 1999, ISBN 0471163392, Section 7.3.2. J. W. Moore and R. G. Pearson, Kinetics and mechanism: a study of homogeneous chemical reactions, Wiley, New York, 3rd ed., 1981, ISBN 0471035580, p. 23. S. M. Walas, Reaction kinetics, in Perry's chemical engineers' handbook, 7th ed., D. W. Green, and J. O. Maloney, eds., McGraw-Hill, New York, 1997, ISBN 0070498415, Table 7.4. Flory (ref. G2), Section III-l. Odian (ref. G8), Section 2-5. M. Stoll, A. Rouve, and G. Stoll-Comte, Helv. Chim. Acta, 17(1934) 1289. W. H. Carothers, Trans. Faraday Soc, 32 (1936) 39. S. H. Pinner, J. Polym. Sci., 21 (1956) 153. P. J. Flory, J. Am. Chem. Soc, 63 (1941) 3083, 3091, and 3096. Flory (ref. G2), Section IX-1.

References 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32.

33. 34. 35.

36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53.

399

W. H. Stockmayer, J. Chem. Phys,, 11 (1943) 45. W. H. Stockmayer, 7. Polym. ScL, 9 (1952) 69; 11 (1953) 424. Hiemenz (ref. G3), Section 5.8. R. H. Kienle and F. E. Petke, J. Am. Chem. Soc, 62 (1940) 1053; 63 (1941) 481. Y.-K. Leung and B. E. Eichinger, in Characterization of highly crosslinked polymers, ACS Symp. Ser., 243 (1984) 21. J. P. Flory, Chem. Rev., 39 (1946) 137. Flory (ref. G2), Section VIII-1. G. V. Schulz, Z. Physik. Chem., B 30 (1935) 379. P. J. Flory, J. Am. Chem. Soc, 62 (1940) 1561. Sandler and Karo (ref. GIO), Vol. I, Chapter 1, Section 2. D. B. Priddy, Styrene plastics, in Kirk-Othmer, Encyclopedia of chemical technology, 4th ed., J. I. Kroschwitz and M. Howe-Grant, eds., Wiley, New York, Vol. 22, 1996, ISBN 0471526916, p. 1034. Moad and Solomon (ref. G6), p. 92. Eastmond (ref. Gl), pp. 64-65, Chapter 1. G. Moad and D. H. Solomon, in Comprehensive polymer science, G. C. Eastmond, A. Ledwith, S. Russo, and P. Sigwalt, eds.. Vol. 3, Pergamon, Oxford, 1989, ISBN 0080325157, p. 147. T. Sugimura and Y. Minoura, J. Polym. Sci., A-1, 4 (1966) 2735. G. V. Schulz and F. Blaschke, Z. Physik. Chem. (Leipzig), B 51 (1942) 75. Kennedy and Marechal (ref. G5), pp. 193-194. Rudin (ref. G9), Section 6.8.2. P. D. Bartlett and R. Altschul, /. Am. Chem. Soc, 67 (1946) 816. Moad and Solomon (ref. G6), Section 5.3.3.4. Sandler and Karo (ref. GIO), Vol. Ill, Chapter 8, Section 2. Moad and Solomon (ref. G6), Section 5.3.4. Odian (ref. G8), Section 3-6d. M. J. Roedel, J. Am. Chem. Soc, 75 (1953) 6110. Odian (ref. G8), pp. 257-258. Rudin (ref. G9), p. 217. C M . Starks, Free radical telomerization. Academic Press, New York, 1974, ISBN 0126636508. T. Corner, Adv. Polym. ScL, 62 (1984) 95. W. Heitz, in Telechelicpolymers: synthesis and applications, E. J. Goethals, ed., CRC Press, Boca Raton, 1989, ISBN 0849367646, p. 61. B. Boutevin, Adv. Polym. ScL, 94 (1990) 69. Eastmond (ref. Gl), Chapter 3. Landolt-Bomstein, New Series, Radical reaction rates in liquids, H. Fischer, ed., Springer, Berlin, Part II, Vol. 13a, 1984, ISBN 0387126074.

400 54. 55. 56. 57. 58.

59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82.

83. 84.

Chapter 11.

Polymerization

G. V. Schulz and G. Harborth, MakromoL Chem., 1 (1948) 106. E. Trommsdorff, H. Kohle, and P. Lagally, MakromoL Chem., 1 (1948) 169. G. V. Schulz, Z. PhysiL Chem. (Frankfurt), 8 (1956) 290. Odian (ref. G8), Section 3-lOa. M. Kucera, Mechanism and kinetics of addition polymerizations, in Comprehensive chemical kinetics. Vol. 31, H. G. Compton, ed., Elsevier, Amsterdam, 1992, ISBN 0444987959, Chapter 3, Section 1.3. Odian (ref. G8), Section 3-4c. Odian (ref. G8), Section 3-5a. Rudin (ref. G9), Section 6.6. Flory (ref. G2), pp. 334-336. E. Tschunkur and W. Bock, Ger. Pat. 532,456, 1929; 570,980, 1933 (to I. G. Farbenindustrie). K. Ziegler, Angew. Chem., 49 (1936) 499. M. Szwarc, M. Levy, and R. Milkovich, J. Am. Chem. Soc, 78 (1956) 2656. M. Szwarc, Carbanions, living polymers, and electron transfer processes, Interscience, New York, 1968, ISBN 0470843055. M. Szwarc and M. van Beylen, Ionic polymerization and living polymers, Chapman & Hall, London, 1993, ISBN 0412036614. D. C. Allport, Block copolymers, Elsevier, Amsterdam, 1991, ISBN 0853345570. Morton (ref. G7), Chapter 9. H. L. Hsieh, J. Polym. ScL, A 3 (1965) 163. K. F. O'Driskoll, E. N. Ricchezza, and J. E. Clark, J. Polym. Sci., A 3 (1965) 3241. Sandler and Karo (ref. GIO), Vol. I, p.36. W. C. E. Higginson and N. S. Wooding, J. Chem. Soc, 1952, 760 and 1178. C. S. Marvel, W. J. Bailey, and G. E. Inskeep, J. Polym. Sci., 1 (1946) 275. N. G. Gaylord and S. S. Dixit, J. Polym. Sci., Macromol. Rev., 8 (1974) 51. M. Szwarc, in Ions and ion pairs in organic reactions. Vol. 2, M. Szwarc, ed., Wiley, New York, 1974, ISBN 0471843083, Chapter 4. S. Winstein, E. Clippinger, A. H. Fainberg, and G. C. Robinson, /. Am. Chem. Soc, 16 {195A) 2591. M. Szwarc, in Ions and ion pairs in organic reactions, M. Szwarc, ed., Wiley, New York, 1972, ISBN 0471843075, Chapter 1. B. J. Schmitt and G. V. Schulz, Eur. Polym. J., 11 (1975) 119. Morton (ref. G7), Section 8.3. Hiemenz (ref. G3), pp. 407-410. E. Kresge and H.-C. Wang, Butyl rubber, in Kirk-Othmer, Encyclopedia of chemical technology, 4th ed., J. I. Kroschwitz and M. Howe-Grant, eds., Wiley, New York, Vol. 18, 1996, ISBN 471526878, p. 603. J. P. Kennedy and R. G. Squires, Polymer, 6 (1965) 579. Kennedy and Marechal (ref. G5), Section 2.2.

References 85. 86. 87. 88. 89. 90.

91. 92.

93. 94. 95. 96. 97. 98. 99. 100. 101. 102.

103. 104. 105. 106. 107. 108.

109.

401

Kennedy and Marechal (ref. G5), Section 2.2. M. Chmelir and M. Marek, Collect. Czech. Chem. Commun., 32 (1967) 3047. A. Gandini and H. Cheradame, Adv. Polym. Sci., 34/35 (1980) 1. P. Sigwalt, Makromol. Chem., 175 (1974) 1017. G. Sauvet, J. P. Vairon, and P. Sigwalt, /. Polym. Sci., Polym. Chem., 16 (1978) 3047. G. Sauvet and P. Sigwalt, Carbocation polymerization: general aspects and initiation, in Comprehensive polymer science. Vol. 3, G. C. Eastmond, A. Ledwith, S. Russo, and P. Sigwalt, eds., Pergamon, Oxford, 1989, ISBN 0080325157, Chapter 39. J. V. Crivello, Annu. Rev. Mater. Sci., 13 (1983) 173. V. T. Stannett, J. Silverman, and J. L. Gamett, Polymerization by high-energy radiation, in Comprehensive polymer science. Vol. 4, G. C. Eastmond, A. Ledwith, S. Russo, and P. Sigwalt, eds., Pergamon, London, 1989, ISBN 0080325157, p.317. O. F. Olaj, Makromol. Chem., Macromol. Symp., 8 (1987) 235. Kennedy and Marechal (ref. G5), Section 4.3. M. Biswas and P. Kamannarayana, J. Polym. Sci., Polym. Chem., 14 (1976) 2071. A. R. Mathieson, in The chemistry of cationic polymerization, P. H. Plesch, ed., Macmillan, New York, 1963, ISBN 0080102891, Chapter 6. G. Heublein, Zum Ablauf ionischer Polymerisationsreaktionen, Akademie Verlag, Berlin, p. 125. Kennedy and Marechal (ref. G5), p. 220. Kennedy and Marechal (ref. G5), Section 3.1. D. C. Pepper and P. J. Reilly, Proc. Roy. Soc, A 291 (1966) 41. P. H. Plesch and A. Gandini, The chemistry of polymerization processes. Monograph No. 20, Society of Chemical Industry, London, 1966. D. J. Dunn, The cationic polymerization of vinyl monomers, in Developments in polymerization. Vol. 1, R. N. Haward, ed., Appl. Sci. Publishers, London, 1979, ISBN 0853348227, Chapter 2. P. H. Plesch, Makromol. Chem., Macromol. Symp., 13/14 (1988) 375 and 393. K. A. Matyjaszewski, Makromol. Chem., Macromol. Symp., 13/14 (1988) 389. R. Faust, A. Fehervari, and J. P. Kennedy, /. Macromol. Sci., Chem., A 18 (1982-83) 1209. J. Puskas, G. Kaszas, J. P. Kennedy, T. Kelen, and F. Tiidos, J. Macromol. Sci., Chem., A 18 (1982-83) 1229 and 1263. M. Sawamoto and J. P. Kennedy, J. Macromol. Sci., Chem., A 18 (1982) 1275. J. P. CoUman, L. S. Hegedus, J. R. Norton, and R. G. Finke, Principles and applications of organotransition metal chemistry. University Science Books, Mill Valley, 2nd ed., 1987, ISBN 0935702512, Chapter 11. Odian (ref. G8), Chapter 8.

402 110.

HI. 112.

113. 114. 115. 116. 117. 118. 119. 120. 121. 122. 123.

124. 125. 126. 127. 128. 129. 130. 131. 132. 133. 134. 135. 136. 137. 138.

Chapter 11.

Polymerization

G. W. Parshall and S. D. Ittel, Homogeneous catalysis: the application and chemistry of catalysis by soluble transition metal complexes, Wiley, New York, 1992, ISBN 0471538299, Chapter 4. Sandler and Karo (ref. GIO), Vol. I, Chapter 1, Section 5. Y. V. Kissin, High density polyethylene, in Kirk-Othmer, Encyclopedia of chemical technology, 4th ed., J. I. Kroschwitz and M. Howe-Grant, eds., Wiley, New York, Vol. 17, 1996, ISBN 047152686X, p. 724. K. Ziegler, E. Holzkamp, H. Breil, and H. Martin, Angew. Chem,, 67 (1955) 541. G. Natta, Chim. Ind. (Milan), 37 (1955) 888; 38 (1956), 124; 39 (1957) 993. G. Natta, /. Polym. ScL, 16 (1955) 143; Modem Plastics, 34 (1956) 169, 261. P. Cossee, /. CataL, 3 (1964) 80. E. J. Arlman and P. Cossee, /. Catal, 3 (1964) 99. K. J. Ivin, J. J. Rooney, C. D. Stewart, M. L. H. Green, and J. Mahtab, /. Chem. Soc, Chem. Commun., 1978, 604. M. L. H. Green, Pure AppL Chem., 100 (1978) 2079. E. R. Freitas and C. R. Gum, Chem. Eng. Prog., 75(1) (1979) 73. M. Peuckert and W. Keim, Organometallics, 2 (1983) 594. W. Keim, F. H. Kowaldt, R. Goddard, and C. Kriiger, Angew. Chem. Int. Ed., 17 (1978) 466. R. A. Wessling, D. B. Gibbs, P. T. DeLassus, B. E. Obi, and B. A. Howell, Vinylidene monomer and polymers, in Kirk-Othmer, Encyclopedia of chemical technology, 4th ed., J. I. Kroschwitz and M. Howe-Grant, eds., Wiley, New York, Vol. 24, 1997, ISBN 0471526932, p. 882. Odian (ref. G8), Section 6-6. T. Alfrey, Jr., and G. Goldfmger, J. Chem. Phys., 12 (1944) 205 and 332; 14 (1946) 115. F. R. Mayo and F. M. Lewis, J. Am. Chem. Soc, 66 (1944) 1594. F. T. Wall, J. Am. Chem. Soc, 66 (1944) 2050. F. R. Mayo and C. Walling, Chem. Rev., 46 (1950) 191. D. C. Pepper, Quart. Rev. (London), 8 (1954) 88. I. Skeist, /. Am. Chem. Soc, 68 (1946) 1781. Eastmond and E. G. Smith (ref. Gl), Appendix to Chapter 4. Odian (ref. G8), Table 6-2. Rudin (ref. G9), pp. 247-248. T. W. Lai and A. Sen, Organometallics, 3 (1984) 866. A. Sen and J. S. Brumbaugh, J. Organomet. Chem., 279 (1985) C5. P. Prabhu, A. Schindler, M. H. Theil, and R. D. Gilbert, J. Polym. Sci., Polym. Lett., 18(1980)389. R. W. Gumbs, S. Penczek, J. Jagur-Grodzinski, and M Szwarc, Macromolecules, 2 (1969) 77. C. Walling, J. Am. Chem. Soc, 71 (1949) 1930.

Chapter 12 Mathematical Modeling Mathematical models play an essential role in process development. How best to construct a kinetic model depends on the process development strategy that has been chosen. The present chapter discusses such strategies and suggests approaches to mathematical modeling suited to them. 12.1. Strategies of process development Evolutionary approach (left-hand tier in Figure 12.1). Until the 1950s, practically all chemical processes were developed with evolutionary methods, and many still are: The chemist, having discovered a new and potentially useful reaction, replicates his bench-scale experiments in a larger vessel. If there is commercial promise, engineers then take over and construct a still larger "semi-technical unit." A small pilot plant might follow, and eventually a fair-sized pilot plant that provides the operating experience on which the design of the full-sized plant can be based. In a nutshell, scale-up is by a number of small steps. In each of these, the vagaries of scale-up are apt to cause some problems or failures that must be remedied by tinkering, but the scale-up factors are small enough that no serious risk is incurred. This approach is reliable and almost always successful. Its disadvantage is that, for a large-scale process, it takes a lot of time and manpower, both chronically in short supply. A knowledge of mechanisms is not required and even serves no useful purpose other than providing guidance to the research chemist or troubleshooting engineer. Empirical approach (center tier in Figure 12.1). About halfway through the twentieth century, competitive pressures in chemical industry increased to the point that a shortening of development time was accorded a high priority. At the same time, computers were becoming more powerful and more readily available and statistics gained in popularity. This combination instigated an empirical approach to process development that relies heavily on statistics. Typically, after a chemist has done his job, engineers make an educated guess at probable optimum design and operating

404

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c

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X W

, C


1) -3 H ex O

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c
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c D OH

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CT3 r^H

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^ g

11

03

c ^

J

^

6 ^ ^ S

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13

c)? ' S ^

'§ 1 'S

1

O

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O

Chapter 12. Mathematical Modeling

13

a ^ 4> PQ

c

.2 i-H

E

Cfl

cn O CT3

c«

CO

OJ

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C3 ^

O PQ

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12.1. Strategies of process development

405

conditions for the future plant and build an integrated demonstration unit —really a miniature pilot plant—that faithfully models the anticipated plant with all its reactors and separation trains, recycle loops, etc., on a laboratory scale. This unit is then used to explore a broad range of operating conditions by varying the operating parameters with greatest economy of effort, guided by theory of statistical design of experiments [2-4]. When sufficient data have been collected, a computer program constructs a multidimensional operating surface and searches on it for an optimum. A pilot plant is then built and operated within a narrower range of conditions in the vicinity of the supposed optimum, in order to fme-tune the design of the full-scale plant. In this approach, the chemistry is essentially a "black box." No effort is made to elucidate mechanisms. Rather, empirical equations are fitted to observed reactor dynamics. This approach is quite workable and may be the best if the process is not complicated and the scale not large. However, with, say, six or seven or more design parameters, hundreds of runs are needed for a complete statistical design. Each run may take two to three days to line the demonstration unit out to a steady state. Development time and consumption of man-time and chemicals then become prohibitive, so that short-cuts must be taken, a risky procedure if the scale-up factor is large. Moreover, information about the chemistry, thermodynamics, quantum mechanics, etc., and observations of transient behavior upon start-up, shut-down, and switches to different operating conditions have no input into the mathematical optimization. Whenever he makes no use of information available for free, the engineer does not work at highest efficiency. All this militates against the empirical approach to development of large and complex processes. Fundamental approach (right-hand tier in Figure 12.1). Concurrent with the empirical approach just described, a quite different philosophy was developed in the 1960s, pioneered chiefly by Mobil Oil in petroleum processing and Shell in industrial chemicals. The basic idea is to establish the true mathematics of the chemical reactions by elucidation of mechanisms in short-duration bench-scale experiments in order to make possible a direct and safe scale-up to the full-sized plant. Demonstration unit and pilot plant are relegated to assessing long-term effects such as catalyst life and corrosion, fine-tuning the design, providing proof of operability, piloting process control, and producing samples for customers, but are freed from the task of scanning wide ranges of operating conditions. The shining goal of direct scale-up from the laboratory bench can rarely be attained. Nevertheless, the approach has proved it can come close to the ideal of process development: to do the best possible job in the shortest possible time with least possible manpower at least possible expense. The fundamental approach is at its best if the scale is large, the process can be expected to have a long life with additional plants being built as market share is conquered, and the chemistry is neither trivial nor overwhelmingly complex, con-

406

Chapter 12. Mathematical Modeling

ditions often given especially in industrial chemicals production. In such situations the fundamental approach requires less equipment, manpower, and chemicals than does the empirical approach. It also has the advantage of providing more insight into details of chemical mechanisms, a knowledge often valuable for research as well as for development of future processes based on the same or a similar chemistry. A drawback is that the approach is also the most demanding on the expertise of the development team. This problem is aggravated at least at the present time by a lack of thorough schooling in chemical kinetics (as distinct from reactor design) in our current chemical engineering curriculum. Moreover, if the scale is small, the chemistry is very complex, timing is more important than efficiency, and the lifetime of the process is uncertain, as is often the case in biotechnology, an effort to establish mechanisms in detail may not pay off. Addressing fundamental kinetics, this book is chiefly intended as an aid to practitioners of the fundamental approach. In part this is because little can be said in generally applicable terms about the other approaches as they largely rely on experience with the specific chemistry at hand and on intuition—not to mention folklore—as well as the well-documented principles of statistical design. Balanced coverage requires, however, tiiat the limitations of the fundamental approach be pointed out and alternatives be mentioned. Pursuit of strategies. The brief outlines above may serve to characterize the possible options in process development. They describe clear-cut, "pure" strategies, but, of course, modifications are possible. For efficient development it is important to decide as early as possible on the merits of the case which tack to take. If the chosen approach is evolutionary or empirical, expected best operating conditions for the eventual plant should dominate the design of the experimental program. In contrast, if the fundamental approach is taken, the clear aim should be to establish networks and mechanisms beyond reasonable doubt with the fewest possible experiments, even if their conditions are far remote from those likely to be employed in the plant. Only when networks and mechanisms are believed to be in hand, should predictions for expected plant conditions be made and verified. An attempt to look at plant conditions while pursuing the fundamental approach is apt to syphon off valuable manpower and time from the principal effort:

Don't mix your strategies

To be sure, this dictum is not absolute. Complementary experiments on an as-timepermits basis might settle pressing questions or give valuable leads. However, the delay of the principal task they almost always entail should be weighed against the expected benefits.

12.2. Effective fundamental modeling

407

12.2. Effective fundamental modeling How best to approach mathematical modeling of kinetics depends on what strategy of process development was chosen. The evolutionary strategy, which essentially relies on tinkering upon stepwise scale-up, has little use for kinetic modeling. For the empirical strategy, empirical equations are sought that can fit observation, a task for which, as a rule, mathematicians are better suited than engineers, and whose details have no proper place in this book. As far as the chemistry of the process is concerned, the most common procedure is to start out with power-law rate equations. Here, fundamental kinetics can offer one piece of advice: If a power-law rate equation requires fractional exponents, one-plus equations with integer exponents should be tried instead. If chain mechanisms or pre-dissociation may be involved, one-plus equations with exponents that are integer multiples of one half should also be tried. This is because, as has been seen, such one-plus equations are more likely to reflect actual mechanisms and so to provide approximate fits over wider ranges of conditions than do power laws. Of course, there is no guarantee that one-plus equations will do better. In any event, whether power law or one-plus, the equations chosen must be fine-tuned and, in the end, will rarely resemble their simple initial forms. Other than this, fundamental kinetics has little, if anything, to offer in an evolutionary or empirical process development. In fact, an attempt to introduce it into what aims to be the best possible empiricism might only cause distracting complications. The balance of the present section will therefore be restricted to fundamental kinetic modeling based on the concepts and procedures developed in the preceding chapters. 12.2.1. Complete fundamental modeling with Bodenstein approximation A prerequisite for fundamental mathematical modeling is that the reaction network or networks have been established. This will be taken for granted here (for network elucidation, see Chapter 7). Software for direct, "brute-force" solution of the rate equations is available [5-10] and can be used if the network consists of only a few elementary steps. In practice, however, effective fundamental modeling usually calls for a reduction in the number of simultaneous rate equations and their coefficients. As Chapter 6 has shown, a systematic application of the Bodenstein approximation to all trace-level intermediates can achieve this, at least unless the network is largely non-simple.

408

Chapter 12. Mathematical Modeling

The approximations of a rate-controlling step, quasi-equilibrium steps, and long chains in chain reactions and the concept of relative abundance of catalyst-containing species in catalysis or of propagating centers in ionic polymerization can often be used for additional simplification (see Sections 4.2, 4.3, 8.5, 9.3, 10.3, and 11.4.1). A procedure suited in many cases consists essentially of the following steps [11]: (1) (2) (3) (4)

(5)

(6) (7) (8) (9)

Identify any non-trace intermediates and steps with two or more molecules of intermediates as reactants; cut the network into piecewise simple portions at such intermediates and steps (Sections 6.5 and 7.3.3); establish all independent stoichiometric constraints and yield-ratio equations (Section 6.4.3) for possible replacement of rate equations; reduce the network or its piecewise simple portions to a form or forms with only pseudo-single, pseudo first-order steps between adjacent nodes and between nodes and adjacent end members (Section 6.4); compile the rate equations in terms of A coefficients (segment coefficients) for all end members of the network or its piecewise simple portions, except those which can be replaced by stoichiometric constraints (Sections 6.3 for pathways and 6.4.2 for networks); use yield-ratio equations to replace rate equations where the former are of simpler form than the latter, establish equations for all A coefficients in terms of\ coefficients (pseudofirst order coefficients, Section 6.2); replace all X coefficients by k coefficients (true coefficients), multiplied by coreactant or co-product concentrations where appropriate (Section 6.2); for single pathways and small networks or network portions, reduce the rate equation or equations to their most convenient one-plus forms with lumped phenomenological coefficients (Section 7.2.1); for large networks or network portions, do so with the equations for the A coefficients.

In networks with loops or multiple catalytic cycles, the use of iS (loop) or V (collective) instead of A coefficients may provide further simplification (see Sections 6.4 and 8.8). A detailed example will illustrate the procedure. Example 12.1. Hydroformylation of cyclohexene with phosphine-substituted cobalt hydrocarbonyl catalyst. The most probable network of cyclohexene hydroformylation catalyzed by a phosphine-substituted cobalt hydrocarbonyl is shown on the facing page. HCo(CO)3Ph (cat) is in equilibrium with the CO-deficient HCo(CO)2Ph (cat') and CO. For greater generality, quasi-equilibrium of these species with the TTcomplex, Xj, is not assumed. Actual hydroformylation olefin —• aldehyde proceeds via a Heck-Breslow pathway (cycle 6.9 that includes the trihydride, X2) but without

409

72.2. Effective fundamental modeling

(ole)

OC Ph HCo OC CO

} "g<^

(cat)

OC Ph HCo. H

(ale) OC Ph HjCo^H OC ^ C - ( > HO ^ - ^ (X,) OC Ph OC

^C-< ) HO ^ - ^ (Xs)

(X.) ' /-^

410

Chapter 12. Mathematical Modeling the tetracarbonyl acyl (Y in 6.9), whose concentration remains insignificant at the lower pressures used with the phosphine-substituted catalyst. Paraffin by-product is presumed to be formed from the trihydride (see Example 7.5, pathway IV), and aldehyde hydrogenation is via pathway 7.28 of Example 7.4. The steps X2 —• paraffin + cat', X4 —• X5, Xg —• aid + cat', and X9 —• ale + cat' are irreversible. Equilibrium with other catalyst species (Example 8.3) is not accounted for. This example is to illustrate procedure, not demonstrate the power of the method, therefore the choice of an olefin with relatively simple kinetics: Cyclohexene exists only as internal d^-olefin, so that olefin isomerization and production of isomeric aldehydes and alcohols need not be considered, and the aldehyde formed carries its -CHO group on a secondary carbon atom, so that aldol condensation remains insignificant (these complications will be included in the next example). Break-up into piecewise simple portions and their reduction. The network 12.1 is "simple" except that aldehyde, an intermediate, builds up to higher than trace concentrations. Thus it can be cut at the aldehyde into two piecewise simple portions which share aldehyde and the ligand-deficient catalyst, cat'. After reduction the portions can be written ole

^4—• X2 — •

aid

I

and

aid —• ale

par with five end members and five A coefficients (ole and aid rather than cat' are written as starting species, for better distinction between the two portions.) Stoichiometric constraints and yield ratio equations. The stoichiometric constraints are conservation of skeletal carbon: conservation of hydrogen: conservation of oxygen:

-''ole

= r

-''H.

=

-'*co

+ r ,^ + r,

par

=

aid

V ^ r

'"ald^ 2/-,,, + r

'aid

(12.2)

ale

(12.3) (12.4)

'ale

and obviate compilation of the rate equations for olefin, H2, and CO. The yield ratio paraffin-to-hydroformylation products is ^ar

^,p

(12.5)

^2,ald

and can be used instead of the rate equation for paraffin. Rate equations in terms of A coefficients. With the rate equations for olefin, paraffin, H2, and CO replaced by the stoichiometric constraints and the yield ratio of paraffin to hydroformylation products, the rate equations for aldehyde and alcohol remain to be established. In terms of A coefficients these are:

12.2. Effective fundamental

modeling

411

-^02^2,31(1

c.

^O.alc ^ a l d

^20 -^ ^2.par "^

K

(first term in equation for r^jd is obtained with eqn 6.20). These equations are in terms of the CO-deficient catalyst, cat'. In view of the equilibrium cat <-> cat' + CO: ^cat'

"^

^czi^c^i'PcO

(X^at = dissociation constant of catalyst) the rates can be written instead: ^02 ^2,aid ^ 2 0 ^ ^2.par ~

(12.6)

A' C Pco

^2.

^O.alc^cat

(12.7)

'Pco

where ^02

=

and

^02^cat

^0,alc

= K...K^.

A coefficients and their one-plus forms. The derivation of equations for the A coefficients can be simplified with rules from Section 7.3.1 by shortening and consolidating the pathways X2 —• aid and aid —• ale: Since the step X4 —• X5 is irreversible, the pathway portion X5 —• aid does not affect rate behavior (Rule 7.12), and the step sequences X3 + CO —• X4 —• X5, aid + cat' —• X7 —• Xg, and Xg + H2 —• X9 —• ale + cat' can be consolidated into single steps X3 4- CO —• X5, aid + cat' —• Xg, and Xg + H2 —• ale -f cat', respectively (Rule 7.24). With these simplifications, the general procedure for simple pathways or segments (Section 6.3) gives \\

^12^cat

(12.8)

^02

1+

\ 2

KP»

^10^21

X12 +

\Q

^2,par

^.par

~

^35 +

^2

^S^g.alc-^cat \ . a l c -^

\o

1 + fc,p„ (12.10)

'^2,par

^23^35/'cO

\ . A', O.alc

(12.9) '^10

^35/^CO ^ ^32

KP^ cyco 1 + ^dPco

^08^8,alc^cat^ald/^H,

K^MPH,

^8,ale.PH, + ^1'80

1 + k,p

(12.11) (12.12)

The phenomenological coefficients k^ to kf in the one-plus forms are related to the coefficients of the individual steps and the catalyst dissociation constant by

412

Chapter 12, Mathematical Modeling

^d ^

^35/^32'

^e ^

KzK,z\c^cJKo'

^f ^

KMC^KO

Model equations. The mathematical model requires eight concentration-independent coefficients: k^ to A:f, ^p^r, and ^21- From these it calculates the five A coefficients with eqns 12.8 to 12.12; from these, the rates of aldehyde and alcohol with eqns 12.6 and 12.7; and finally the rates of olefin, paraffin, H2, and CO with eqns 12.2, 12.5, 12.10, 12.11, 12.3, and 12.4, respectively. Alternatively, eqns 12.8 to 12.12 can be used to replace the A coefficients in eqns 12.6 and 12.7 in order to obtain explicit rate equations for aldehyde and alcohol in terms of the phenomenological coefficients. However, the resulting rate equations are more cumbersome. This example has shown how the procedures developed in earlier chapters can be used effectively for modeling. The reaction system has seventeen participants: olefin, paraffin, aldehyde, alcohol, H2, CO, HCo(CO)3Ph, HCo(CO)2Ph, and nine intermediates. "Brute force" modeling would require one rate equation for each, four of which could be replaced by stoichiometric constraints (in addition to the constraints 12.2 to 12.4, the brute-force model can use that of conservation of cobalt). Such a model would have 22 rate coefficients (arrowheads in network 12.1, not counting those to and from co-reactants and co-products), whose values and activation energies would have to be determined. This has been reduced to two rate equations and nine simple algebraic relationships (stoichiometric constraints, yield ration equations, and equations for the A coefficients) with eight coefficients. Most impressive here is the reduction from thirteen to two rate equations because these may be differential equations. For this example and seven others from this book, Table 12.1 illustrates the reduction of complexity achieved, showing a comparison of the numbers of rate and other equations and their coefficients of reduced and "brute force" models. The latter are understood to consist of the rate equations for all participants except those that can be replaced by stoichiometric constraints, and the constraints used in this fashion. The greatest reductions are where it counts most: in the possibly differential rate equations. Also important is the reduction in the number of coefficients. This is because the problem with brute-force modeling today is not so much the actual calculations, but the experimental work required to obtain values for all the coefficients and their activation energies. The reduced models in Table 12.1 rely on the validity of the Bodenstein approximation for all intermediates except the aldehyde in hydroformylation, but are otherwise free of assumptions. In every case, equations that are as simple or even simpler have long been derived, but only with much more restrictive assumptions, most commonly that of a single rate-controlling step and quasi-equilibrium everywhere else. Of course, such equations should be used in preference if their assumptions can be substantiated.

12.2. Effective fundamental modeling

413

Table 12.1. Reduction of complexity: comparison with "brute force" models. number of reaction

model

rate equations*

stoichiom. constraints

1^

4

reduced

1

aldol condensation (Example 8.2)

brute force

1^

2

reduced

1

aldehyde hydrogenation (Example 7.4)

brute force

9

"oxo" reaction (Example 6.2)

brute force

8

reduced

1

hydrocyanation (Example 8.7)

brute force

7

reduced

1

Wilkinson reaction (Example 8.9)

brute force

6

reduced

1

hydroformylation (Example 11.1)

brute force

13

4

reduced

2

3

nitration of aromatics (Example 6.1)

brute force

reduced

1

yield ratios

1

4

1 4

coefficients

6 3 5 3 5 2

1 1 1 1 1 1

16

1

2 1 12

4

1

4

10

4

1

8 II 22 II 1

8**

* Does not include equations that can be replaced by stoichiometric constraints or yield ratios. ** If quasi-equilibrium ole + cat' —• Xj can be assumed as in network 6.9, k^^ is negligible and only seven coefficients are needed.

12.2.2. Streamlining for large networks Industrial practice often confronts the development engineer with networks that are considerably more complicated than that of cyclohexene hydroformylation in the example above. Additional simplifications may then be desirable or necessary in order to arrive at a model that remains manageable in the highly iterative applications called for in reactor design and optimization and possibly on-line process control. A useful and usually successful way of achieving such streamlining is to place all network nodes at end members or non-trace intermediates, ignoring the fact that some of them may be at trace-level intermediates [11].

Chapter 12. Mathematical Modeling

414

The segments in a streamlined model might not correspond to those in the actual network with trace-level node intermediates. This raises the question what to use as A coefficients. The choice is somewhat arbitrary. A good starting point is to trace, in the actual network, the most direct paths between what are the node and end members in the streamlined network, and to establish the A coefficients for these paths regardless of any branches along them. The following example will illustrate this procedure. Example 12.2. Streamlined network for hydroformylation of n-heptene catalyzed by phosphine-substituted cobalt hydrocarbonyl. In hydroformylation of straight-chain olefins with a phosphine-substituted cobalt hydrocarbonyl catalyst, the model must account for three complications that are absent with cyclohexene: isomerization by migration of the double bond along the hydrocarbon chain, formation of isomeric aldehydes and alcohols, and condensation of the straight-chain aldehyde to "heavy ends" (chiefly an alcohol of twice the carbon number, such as 2-ethylhexanol from propene via n-butanal and a Cg aldol). A streamlined network for w-heptene is:

(Po)

H HC-COH H

(12.13)

2-hexyldecanol

CHjOH Az-octanol

CHO ;2-octanal

1-heptene

(AJ HCO 2-methylheptanal (K,) ^-heptane (Q)

"^

H2COH

-•

2-methylheptanol (P2)

2-heptene H^COH

HCO 2-ethylhexanal

2-ethylhexanol

(K3)

(P3)

3-heptene (A3)

HCO 2-propylpentanal (K,)

(catalyst and co-reactants not shown to avoid clutter).

^

H2COH 2-propylpentanol (P4)

12.2,

Effective fundamenal

modeling

415

The aldehydes arise from the olefin isomers by addition of CO to a carbon atom on either side of the double bond in coupled parallel steps (see Example 5.3 in Section 5.3) and are hydrogenated to the respective alcohol isomers. Fortunately, only the straight-chain aldehyde condenses to a significant extent. In comparison, the detailed network consist of 111 steps. A derivation of the rate equation for 2-propylpentanal (K4) may serve to illustrate the resulting streamlined mathematics. The rate is given by \

= \KC^'^K:,C^.

(12.14)

The shortened and consolidated, direct pathways A3 —• K4 and K4 —• P4 are cat <—^

ole H2 CO cat' -^^^^—• Xj ^ ' ^ X3 —=^*—• X5 ••• CO

H, H2 V V cat <—^ cat' -^^^^^—• Xg -^^'—• ale + cat' CO aid

with A coefficients of the forms AC v ^ «

AAK

=

— 1 + k^^^ + KPCO^PH,

AC J v x i r

'

A^4P4 = '

- ^ Pco(^

(12.15)

^ KPH)

respectively (in both equations, the first denominator term before reduction to one-plus form has been omitted because of equilibrium in the first step of the pathway). A comparison shows that eqn 12.14 with eqns 12.15 gives a much simpler rate equation than does eqn 12.6 with eqns 12.8 to 12.11 for an analogous step sequence. There is no simplification in the rate equation for aldehyde —• alcohol because that pathway has no branch in the detailed network. While the reduced models discussed previously invoke only the Bodenstein approximation for trace-level intermediates, the additional streamlining is apt to introduce some errors, and these are hard to estimate beforehand. A safe way to proceed is to compile both a "research model" based on the detailed network, and a streamlined "process model." Apart from its use in evaluation of bench-scale experiments, the research model can serve to assess, by comparison, the nature, direction, and magnitude of the error of the process model under operating conditions of the plant. If found satisfactory, the process model can then be used for reactor design and optimization, possibly after some fine-tuning. Provided steric considerations and experience strongly suggest its topology, a streamlined model may also come into play as working hypothesis before the results of kinetic studies are in, or as a poor-man's mechanistic model if elucidation of kinetics in detail is not possible. In such applications, the concentration dependence of the A coefficients must be determined empirically.

416

Chapter 12. Mathematical Modeling

12.2.3. Determination of coefficients The discussion up to this point has addressed the compilation of a mathematical model, that is, how best and most concisely to express a network and mechanism in terms of equations. A second and no less important task is to obtain the best numerical values of the model's coefficients. The determination of coefficients in power-law and one-plus rate equations has been described in Sections 3.3, 3.5, and 7.2. Here, the principles will be placed in context. There are two ways of arriving at values of coefficients. The conventional, old-fashioned, graphical procedure seeks plots that give straight lines and determines coefficient values from their slopes and intercepts. Examples have been seen in earlier chapters. Instead, linear- or nonlinear-regression computer software can be used for "optimization," that is, finding the set of coefficient values that fits the experimental data with least error [2,12-16] (see Section 3.5). Both methods have their pros and cons, and each has its place in kinetic modeling. The main shortcoming of the graphical method is that it relies on "eyeballing," a subjective activity, and is therefore unlikely to yield values that provide an optimum fit. It has the advantage of presenting a picture to look at, making it easier to spot any untoward behavior. On the other hand, statistical regression, if properly used, turns out best values. However, unless the program is equipped with a plotting routine, it is largely a black-box affair, and that makes it difficult in complicated cases to discern where these values really come from. For example, a graph may show a fit to be excellent except for one point or a pair of points that are far off, raising a red flag that an experimental error may have occurred and prompting a check; in contrast, a primitive regression program accepts the point or points at face value and has them bias the entire set of values it returns as "best." There are still other pitfalls to beware of. First, primitive regression programs seek the best correlation without screening for systematic error, and so may reject a correct model with slightly larger, but random error. Second, the program may converge onto a false optimum unless started with an approximately correct guess. Third, even if the optimum is statistically correct, it may be the wrong answer because a very different, but only slightly inferior parameter set is physically correct. Lastly, there is the psychological danger of being overly impressed by a good correlation, misinterpreting it as evidence of correctness of the model.* Statistics is a very fine tool, but no substitute for human judgment. * A story, possibly apocryphal and admittedly from old times, tells of a research team at Dow Chemical's Midland facility submitting to their brethren in Applied Mathematics a bogus model and a heap of random numbers alleged to be experimental results, and receiving within a day the reply that statistical optimization of the coefficients gave an excellent correlation, proving the model to be correct. And there is of course no end of jokes about statisticians misapplying the tenets of their profession to situations in their everyday lives.

12.3. "Shortsightedness" of elementary reaction steps

All

In the early stages of model development, the engineer should keep his eyes open for any symptoms that might point to effects overlooked or not yet accounted for. Here, graphs are generally preferable to abstract statistics. Once the equations of a model have been established, statistics software is invaluable for obtaining the objectively best set of coefficients. Even here, however, old-fashioned graphics still has an input in that it can provide a good initial guess for regression, thereby greatly reducing the chance of convergence onto a false optimum. 12.3. "Shortsightedness" of elementary reaction steps A useful principle that often holds and can be applied to ease the work load of development is: The reactivity of a group depends strongly on the local configuration, but litde or not at all on the size and shape of the reactant molecule more than two atoms away.

For example, the reactivities of primary and secondary hydroxy 1 groups of alcohols differ greatly, but neither depends much on the lengths and configurations of the hydrocarbon chains to which they are attached, granted the chains are longer dian three carbon atoms. One might say, the elementary reaction at a group usually is "shortsighted." There are exceptions, especially in structures with steric hindrance or resonance, the most notable being the effect of conjugation in molecules with multiple double bonds or aromatic rings. Therefore, the principle of shortsightedness must be tested in each case before it can be relied upon. What amounts to invoking the shortsightedness principle has become standard practice in polymerization kinetics where, for most purposes, the rate coefficients of step or chain growth and termination are taken to be independent of the lengths of the polymer chains (see Sections 11.2.1, 11.3.1, 11.4.1,and 11.5.1). In fact, the principle is the basis of Carother's approach to step-growth polymerization kinetics leading to his equations 11.6 and 11.7. In other fields of kinetics, the principle is particularly useful in two situations: for pathways with repetitive, consecutive addition reactions such as multiple ethoxylation or paraffin oxidation (see examples in Section 5.5); and for networks with coupled parallel steps, as in olefin reactions such as hydrogenation, hydroformylation, and hydrocyanation that involve double-bond migration in concert with conversion to products [17]. Consider as a prototype the network 12.13 of ^-heptene hydroformylation, keeping in mind that the arrows represent multistep pathways and that the reactions of higher straight-chain olefins involve still more

418

Chapter 12. Mathematical Modeling

parallel pathways of internal olefin isomers to aldehyde isomers and on to alcohol isomers. In such networks, all but one of the aldehyde-to-alcohol conversions involve the reaction of an aldehyde group on a secondary carbon atom, so that all these pathways can be assumed to involve essentially the same rate coefficients of their steps. Only the conversion of the straight-chain aldehyde (Az-octanal to noctanol in network 12.13) must be expected to occur with somewhat different rate coefficients. Likewise, all conversions of the internal olefin isomers to aldehydes and paraffin, except that of the 2-olefin, can be assumed to occur with practically equal rate coefficients. If the network is large, such equalities of coefficients can greatly reduce the number of coefficient values that must be determined experimentally. Apart from its usefulness in the construction of mathematical models, the shortsightedness principle packs notable predictive power. As an example, an olefmic double bond can exist in only six configurations:

internal

terminal

-CIS

internal -trans

terminal at branch

internal at branch

internal between branches

Usually, the reactivity is highest for the terminal position and lowest for the positions at branches. Once the reactivities with regard to a specific reaction have been determined for all six positions, the reaction behavior of all types of monoolefins except those with strained rings or bulky substituents can be predicted with reasonable confidence. The reactivities of three structurally differed hexene isomers in hydroformylation catalyzed by phosphine-substituted cobalt hydrocarbonyls may serve as an example [17]: /

cyclohexene

«-hexene

3,3' dimethyl-1-butene

Cyclohexene exists only as internal cis-oltfin and is moderately reactive. In contrast, Az-hexene, regardless of whether charged as 1- or internal olefin or a mixture of these, is quickly isomerized to a near-equilibrium mixture containing some 5 to 10% of the isomer with terminal double bond, whose reactivity is about two orders of magnitude higher than those with internal ones. Accordingly, n-hexene is more reactive than cyclohexene with only internal double-bond positions. Lastly, neohexene (3,3'-dimethy 1-1-butene) has its double bond locked in the terminal position —no double bond can exist adjacent to a quaternary carbon atom—and so should

12.3. "Shortsightedness" of elementary reaction steps

419

have the highest reactivity if not sterically hindered. (This is an unsubstantiated prediction, as the hydroformylation reactivity of that olefin seems not to have been studied to date.) In heterogeneous catalysis with its steric effects and other complications, the shortsightedness principle cannot be expected to yield quantitative predictions. Nevertheless, in can provide valuable qualitative guidance and insight. Catalytic cracking of paraffins may serve as an example. Barring overriding mass transfer effects, the cracking rate of paraffins of like structure increases with carbon number (see Figure 12.2): The longer the hydrocarbon chain, the greater is the number of carbon-carbon bonds exposed to cracking. If all such bonds have approximately the same reaction probability, this makes for a higher rate. The rates are lower for branched than for straight-chain paraffins, at least in part owing to the shape selectivity of the catalyst and its lower mass-transfer rates for bulkier molecules.

pentanes

0.23

0.01 hexanes

0.71

0.36

0.22

0.09

0.09

heptanes

1.0

YY 0.05

0.52

0.38

0.09

0.17

0.06

T 0.06

Figure 12.2. Relative rates of cracking of paraffins over zeolite ZMS-5 at 35 atm and 340 °C (data from Chen and Garwood [18]).

420

Chapter 12, Mathematical Modeling

Once its validity has been confirmed, the shortsightedness principle can also be used in exploratory evaluations and process development for prediction of reactivities within homologous series. Keep in mind that equal reactivity of a group on molecules of different sizes does not necessarily entail equal overall reaction rates in a practical context. Again taking hydroformylation of straight-chain olefins as an example: Ethene and propene exist only as 1-olefins and so are highly and about equally reactive. With each carbon atom added to the chain, the percentage of 1-olefin in the quickly established near-equilibrium isomer mixture decreases, and so does the overall reactivity. If the catalyst is a phosphine-substituted cobalt hydrocarbonyl, the overall rate at near-equilibrium of isomerization decreases more than a hundred-fold from ethene and propene to «-octadecene. A fundamental model allows the reaction behavior of all olefins of the series to be predicted with reasonable accuracy once the individual rate coefficients for reaction of the terminal and internal double-bond positions have been determined by experiments with only one member of the series. A historical footnote may illustrate this point. In the 1960s, Shell had developed and put into operation hydroformylation processes for n-butanol from propene and for detergent-range alcohols (C^o to Cjg) from higher olefins. A new process for plasticizer-range alcohols (C5 to C9) was then developed, based entirely on the predictions of a fundamental model and only a few confirmatory bench-scale kinetic experiments with one olefin in that range. Development and plant construction were completed in record time and design capacity was reached almost immediately, even though the plant processed some olefins with which no experiments at all had been done. [However, the plant was shut down and cannibalized in short order: The product tanks were full and customers failed to materialize. A brilliant technical success, but a sad economic failure. Such is lifeil

12.4. Lumping and continuous mixtures Many reactions of interest in industrial practice, especially in petroleum processing and hydrocarbon combustion, involve mixtures of so many reactants that individual modeling of each is out of question. Lumping then becomes necessary. Reactants can be lumped into a single or several so-called supercomponents or "lumps." A variant of this approach, potentially useful if the number of reactants of the same kind is very large, is to treat their mixture as a continuum rather than as a collection of discrete components. Alternatively, lumping can be done by reactive configurations regardless of what reactant molecules they are part of. Lumping of reactants [19-21]. The oldest and most primitive, yet useful form of lumping collects all reactants in a single lump. This can be done if all reactants are of the same kind—say, are paraffins—and undergo the same reaction, though with different rate coefficients. As the reaction progresses, the reactant mixture becomes

12.4. Lumping and continuous mixtures

All

richer in its more sluggish components. The reaction rate of the lump therefore decreases more steeply with conversion than do those of the pure individual components. This makes for an apparent reaction order of the lump that is higher than that of its components (see also Section 5.3, Case II, for a two-component mixture). The classical example is Mobil Oil's lumping of gas oil feedstocks for catalytic cracking in the 1950s into a single lump with second-order kinetics, although the order is first for individual components [22,23]. As it turned out later, the secondorder behavior is not coincidental, but has been given a theoretical explanation (see Example 12.3 below). The kinetic behavior of the lump is, of course, only an approximation to that of the ^^^ °^^ •fpp/^ a t o p i c

mixture of its real components. Greater accuracy can be obtained if the reactants are grouped into several lumps rather than a single one [24]. This alone, however, does not address the problem that, in some of the most important applications such as gas oil or naphtha cracking, gasoline ^ coke products identical or similar to initial reactants product dry gas are formed and then, in turn, undergo reaction. An early example of a model that includes a Figure 12.3. Mobil Oil's threereaction of the product is Mobil Oil's three- lump model for gas oil cracking. lump model for cracking of gas oil [25,26] that accounts for formation of gas and deactivating coke from both the feedstock and the desired gasoline product (see Figure 12.3). This was soon replaced by a 10-lump model which also keeps track of substituent groups and distinguishes between heavy and light components [27]. That model has gone through many updates. Continuous mixtures. The staggering complexity of feedstocks in refinery operations and the great economic incentive to optimize the latter has renewed interest in a parallel approach in which the reactant mixture is taken to be an entity with a continuous distribution of rate coefficients [28-30]. The mathematics become quite complicated in the general case of reversible and higher-order reactions, and some aspects are still problematic. However, if the reaction of the individual reactants is first order, as is often true in refinery operations, the approach is well in hand and has proved fruitful and given interesting results. One of these is a better understanding of how the reaction order of a lump depends on the distribution of the rate coefficients of the original reactant mixture, as shown in the following example. Example 12.3. Continuous mixtures of reactants undergoing the same irreversible first-order reaction [29-32]. If the reaction of the components is first order and the initial distribution of the reactants by their rate coefficients is g°{k), the distribution at time t is

422

Chapter 12. Mathematical Modeling gik,t) = g'{k) txp(-kt)

and the "lump" is g(kj)

= \^8(k,t)

=

f^g%k)txp(-kt)

For an initial gamma distribution, used in most of this work, one finds g{Kt)lg\k) = (1 4- kt/a)-^ where a is the parameter of the gamma distribution (a = 1 gives an exponential distribution, a = oo gives an impulse) and k is the average rate coefficient. This corresponds to a rate equation of order 1 + 1/a for the lump:

(gy For an exponential initial distribution (a = 1), the lump reaction thus is second order. Such second-order kinetics are observed in some refinery operations including catalytic cracking and hydrodesulfurization. Others, such as steam deactivation upon coke bum-off, however, occur at reaction orders of 2 to 5, indicating an initial distribution that diverges at A:-*0 [33]. Lumping by reactive configurations. As discussed in the preceding section on "shortsightedness" of chemical reactions, the order and rate of a reaction at a molecular configuration often depends only on the immediate vicinity, not on the molecule's structure or size more than an atom or two away. The first to take advantage of this idea was Carothers, who simplified the mathematics of stepgrowth polymerization by focusing on the conversion of functional groups rather than molecules [34]. More recently, this idea has been used for modeling of reactions of complex mixtures in petroleum processing, an approach pioneered chiefly by Froment and co-workers [35-37]. While the number of reactants, intermediates, and products, their isomers, and their potential reactions is staggering, only a relatively modest number of reactions at elementary molecular configurations are involved. The approach to hydrocarbon cracking taken by the Froment school is to model the actual elementary steps of radicals at the various molecular configurations [38]. These are relatively few: initiation; hydrogen abstraction from a primary, secondary, or tertiary carbon; and radical decomposition by scission of a carboncarbon bond in /3-position to the unpaired electron. Boolean relation matrices are used to reflect the structures of the hydrocarbon reactants by indicating the existence and location of all their carbon-carbon bonds. Computer software generates reaction networks on the basis of known rate coefficients and activation energies at the various positions. Froment states the number of components in naphtha cracking as around 200, that of radicals as 40, and that of elementary radical steps

12.5. Model validation

423

as of the order of 10,000, but the total number of independent rate coefficients as only 120 [38]. Even "only" 120 rate coefficients and their activation energies are far more than were known or could be measured directly when this work was initiated. The majority had to be determined with statistical optimization routines from observed kinetics, a procedure that relies on the postulated reaction networks to be correct and the calculation not yielding a false or physically incorrect optimum. Not surprisingly, here and there a coefficient was later found to be seriously off the mark as better experimental techniques were developed and more information became available. This is an ongoing endeavor being continuously perfected. An important advantage of the method is its giving detailed information about not only reactant behavior, but also product composition. Moreover, while the chemistry—of radical chain reactions in hydrocarbon processing—is a "black box" in both reactant lumping and continuous mixtures, it is the starting point in lumping by reactive configurations. It can be seen that the former methods become ever more complex to meet the escalating demands on accuracy of results and predictions. Quite likely, they will lose out in the long run to the truly fundamental approach based on elementary reaction steps, whose power is bound to grow with an increasing data base and improvements in experimental techniques.

12.5. Model validation The goals of fundamental mathematical modeling are to predict what has not yet been explored and to make a risk-free scale up by large factors possible, ideally even a scale-up from the laboratory bench to the full-sized plant. For this purpose, models cannot be based on plausibility, they must be established beyond reasonable doubt. Plausible models will do as working hypotheses in research, exploratory evaluations, and the early stages of development, but not as a basis for plant design. The young development engineer must learn to distinguish sharply between plausible hypotheses and well-established fact, in his own mind as well as in his presentations to his management and discussions with his peers. It is true that in chemical kinetics one can disprove, but never definitely prove. A theory or model is conclusively disproved by experimental evidence to the contrary, but compatibility with such evidence is no proof for correctness because alternative explanations are always possible and new experiments might support those.* Nevertheless, a very high degree of reliability can certainly be * Tolman [39] quotes Oxford's Peter Atkins as saying, "a proof in a chemical reaction is less like a mathematical proof and more like a proof in a court of law," and adds, "as in the court, the reputation and rhetoric of the lawyer ... may swing opinions regarding correctness of the mechanism."

424

Chapter 12. Mathematical Modeling

attained. Moreover, what matters for a design is only the accuracy (within the required limits) of the model's equations, not the correctness of all assumptions that have led to them. While a design engineer may do fine with just two, his model needs three legs to stand on: A model for scale-up, design, and optimization must: • conform with all experimental evidence, • be explainable by an eminently plausible mechanism, and • have proved its predictive power.

No model should be accepted that does not meet all three of these criteria. To be sure, the first criterion, of conforming with all experimental results, is hardly ever met in practice. Where deviations show up, the design of the experiments, their evaluation, and the reproducibility of the results must be checked. More often than not, the disagreement can be traced to an artifact, an equipment malfunction, a calculation error, an incorrect entry in a lab record book, or some other extraneous source, and can then safely be disregarded. However, if a divergent result is confirmed, the proposed model is inadequate and must be refined or rejected. Regarding the second criterion, the proposed mechanism must withstand scrutiny from all angles such as stereochemistry, thermodynamics, molecular-orbital theory, experience with analogous chemistry, etc. Such matters have been discussed in the context of network elucidation in Section 7.4. The third criterion, a good track record of correct predictions, is vital because any set of experimental observations can be explained in different ways, each of which is apt to give different projections of behavior under other conditions. The most conclusive support for a model comes from experimental confirmation of counterintuitive predictions. If a prediction is rejected as absurd because it runs counter to common sense, and experiments then confirm it, the engineer can finally have reasonable confidence in the reliability of his model. The seasoned fundamental modeler will go to great lengths to seek out such counterintuitive features and take devious delight in seeing them confirmed. A case in point is the (correct) prediction of the fundamental model of hydroformylation with base-stabilized, phosphine-substituted cobalt hydrocarbonyl catalysts that both rate and yield are higher at lower pressure although gas is consumed and volume therefore shrinks. The higher rate appears to contradict the Le Chatelier principle [it does not, because that principle applies to equilibria, not to rates], and the higher yield at lesser effort and lower cost offends the third law of engineering: no free lunch. The higher rate

Summary

425

stems from a better utilization of cobalt as HCo(CO)3Ph catalyst thanks to a higher phosphine-to-CO ratio at lower pressure (see Example 8.3 in Section 8.2.2); the better yield at lower pressure results from the increase in the aldehyde hydrogenation rate (see Example 7.3 in Section 7.2.2) without increase in the competing rate of aldehyde condensation to heavy ends (see Example 8.2.1).

Summary What kind of mathematical modeling is best for process development depends on the development strategy chosen. Evolutionary development by stepwise scale-up by small factors essentially relies on tinkering rather than modeling. Empirical development with statistical methods seeks empirical equations to fit observed reactor dynamics. There is little input for reaction-kinetic modeling except that, for first trials, one-plus equations are preferable to power-law equations with fractional exponents. Fundamental development based on fully elucidated networks and mechanisms, addressed in this book, is best suited for large processes with long life expectancy. The experimental program to support evolutionary or empirical development should be dictated by anticipated plant conditions. The program for a fundamental development should aim for the most efficient way of elucidating networks, even if that calls for experiments under conditions far from those expected for the plant. In a typical industrial process, a key task of fundamental modeling of an established network is to reduce the number of rate equations and coefficients. The issue is not so much one of computation time, but of reduction of the experimental effort needed to obtain values for the coefficients and their temperature dependences. The most powerful tool for this purpose is the Bodenstein approximation. A detailed example describes step by step how a model with a minimum of equations and coefficients can be established. For very large networks, a detailed fundamental model may be too cumbersome for highly iterative use in reactor design or optimization. An option then is to use a streamlined model in whose network all branches are placed at non-trace intermediates. This introduces an error that can be assessed by comparison with the detailed model. Numerical values of the coefficients in a model can be determined graphically or by regression with statistics software. Graphical methods are more likely to show up symptoms that indicate effects not yet accounted for, and are therefore preferable in the early stages of modeling. They can also help to provide good initial guesses for regression by computer. A final best set of coefficients should be obtained by regression. An often helpful principle is that of "shortsightedness" of reaction steps: Usually, the reactivity of a group does not depend on the size and shape of the molecule more than two atoms away. Pathways involving like chemistry, say, conversion of aldehyde groups on secondary carbon atoms of molecules with different chain lengths, can thus be assumed to involve closely similar rate coefficients. In large networks this can significandy reduce the number of coefficient values to be determined. The principle can also provide predicdons of still unknown rate behavior, particularly in homologous series. It is not valid without exceptions, and must therefore be tested before use.

426

Chapter 12, Mathematical Modeling

Most reactions in petroleum processing have so may reactants that their individual modeling is out of question. In such cases, the participants can be lumped into one or more supercomponents (lumps), the mixture can be treated mathematically as a continuum, or kinetics can be based on conversion not of individual molecules, but of reactive configurations that the reactants have in common. Only this last method can account for the actual reaction mechanism. To be acceptable for scale-up and reactor design, a model must meet three criteria: It must conform with all experimental evidence, be explainable by an eminently plausible mechanism, and have established a good track record of correct predictions, preferably counterintuitive ones. The establishment of the model equations of a detailed and a streamlined network and the shortsightedness principle are illustrated with cases of olefin hydroformylation.

References 1. 2.

3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

F. G. Helfferich and P. E. Savage, Reaction kinetics for the practical engineer. Course #195, AIChE Educational Services, New York, 7th ed., 1999, Chapter 1. G. E. P. Box, W. G. Hunter, and J. S. Hunter, Statistics for experimenters: an introduction to design, data analysis, and model building, Wiley, 1978, ISBN 0471093157. C. D. Hendrix, Chem, Tech., 9(3) (1979) 167. M. J. Anderson and P. J. Whitcomb, Chem. Eng. Prog., 92 (12) (1996) 51. F. J. Weigert, Comput. Chem. (Oxford), 11 (1987) 273. R. J. McKinney and F. J. Weigert, GEAR and GIT, Project SERAPHIM disks PC3003A and PC3003B, 1987. ACUCHEM and ACUPLOT programs, NIST Chemical Kinetics Data Center, Gaithersburg, MD, 1988 (for isothermal constant-volume batch only). W. Braun, J. T. Herron, and D. K. Kahaner, Int. J. Chem. Kinet., 20 (1988) 51. CHEMKIN (FORTRAN program), Sandia National Laboratory, Livermore, CA, 1990. R. J. Kee, F. M. Rupley, and J. A. Miller, Sandia Report SAND 889-8009, UC401, 1990. Helfferich and Savage (ref. 1), Section 8.2. N. R. Draper and H. Smith, Applied regression analysis, Wiley, New York, 3rd ed., 1998, ISBN 0471170828. R. D. Cook and S. Weisberg, Applied regression including computing and graphics, Wiley, New York, 1999, ISBN 04713171IX. S. Chatterjee, A. S. Hadi, and B. Price, Regression analysis by example, Wiley, New York, 3rd ed., 2000, ISBN 471319465. B. S. Gottfried, Spreadsheet tools for engineers: EXCEL 2000 Version, McGrawHill, New York, 2000, Solver program. SYSTAT (from Systat, Inc., Evanston IL).

Summary 17. 18. 19. 20. 21.

22. 23. 24. 25. 26.

27. 28. 29. 30. 31. 32. 33.

34. 35. 36. 37. 38.

39.

421

Helfferich and Savage (ref. 1), Chapter 10. N. Y. Chen and W. E. Garwood, /. CataL, 52 (1978) 453. J. Wei and J. Kuo, Ind. Eng. Chem,, 8 (1969) 114. V. W. Weekman, Jr., AIChEMonogr. Ser, 75 (1979) no.ll, 3. K. B. Bischoff, Some current issues in kinetic lumping of discrete mixtures. Chapter 9 in Chemical reactions in complex mixtures, A. V. Sapre and F. J. Krambeck, eds., Van Nostrand Reinhold, New York, 1990, ISBN 0442007256. F. H. Blanding, Ind. Eng. Chem., 45 (1953) 1186. V. W. Weekman, Jr., and D. M. Nace, AIChE /., 16 (1970) 397. P. S. van Damme, G. F. Froment, and W. B. Balthasar, Ind. Eng. Chem., Prod. Res. Dev., 20(1981)366. V. W. Weekman, Jr., Ind. Eng. Chem., Proc. Des. Dev., 7 (1968) 90. A. V. Sapre, Kinetic modeling at Mobil: an historical perspective, Chapter 12, in Chemical reactions in complex mixtures, A. V. Sapre and F. J. Krambeck, eds.. Van Nostrand Reinhold, New York, 1990, ISBN 0442007256. S. M. Jacob, B. Gross, S. E. Voltz, and V. W. Weekman, Jr., AIChE J., 22 (1976)701. T. de Bonder, VAjfinite, Gauthier-Villars, Paris, 1931. R. Aris, Arch. Rati Mech. Anal, 11 (1968) 356. R. Aris, AIChE J., 35 (1989) 1398. T. C. Ho and R. Aris, AIChE J., 33 (1987) 1050. F. J. Krambeck, AIChE J., 34 (1988) 877. F. J. Krambeck, Continuous mixtures in fluid catalytic cracking and extensions, Chapter 3 in Chemical reactions in complex mixtures, A. V. Sapre and F. J. Krambeck, eds.. Van Nostrand Reinhold, New York, 1990, ISBN 0442007256. H. W. Carothers, Trans. Faraday Soc, 32 (936) 39. P. J. Clymans and G. F. Froment, Comput. Chem. Eng., 28 (1984) 899. M. A. Baltanas, K. K. Van Raemsdonck, G. F. Froment, and S. R. Mohedas, Ind. Eng. Chem. Res., 28 (1989) 899. L. P. Hillewaert, J. L. Dierickx, and G. F. Froment, AIChE J., 34 (1988) 17. e.g., see G. F. Froment, Fundamental kinetic modeling of complex processes. Chapter 5 in Chemical reactions in complex mixtures, A. V. Sapre and F. J. Krambeck, eds.. Van Nostrand Reinhold, New York, 1990, ISBN 0442007256. C. A.Tolman and J. W. Faller, in Homogeneous catalysis with metal phosphine complexes, L. H. Pignolet, ed., Plenum, New York, 1983, ISBN 03064121IX, p. 22.

Chapter 13 Unusual Thermal and Mass-Transfer Effects Thermal and mass-transfer effects are matters of reaction engineering and reactor design, not of kinetics as understood here. For standard situations—that is, reactions with positive activation energies and rate equations with reaction orders that are positive or zero for reactants and negative or zero for products—current texts on reaction engineering provide excellent treatments [1-10]. In some instances, however, the special nature of a multistep reaction may result in unusual and quite different behavior. Only such cases will be examined here. 13.1. Anomalous temperature dependence As discussed in Section 2.3, a single reaction step obeys a power-law rate equation whose algebraic form is independent of temperature, with a rate coefficient whose temperature dependence in general is given by the Arrhenius equation k = Atxp(-EJRT)

(1.3)

The activation energy, E^, is positive and in good approximation constant. Accordingly, the logarithm of the rate coefficient varies linearly with reciprocal absolute temperature: dlnfc E^ ^2.2) d(l/T) R and the rate increases with increasing temperature. However, this is not always true. In multistep reactions, the following anomalies are occasionally observed: • The activation energy may be negative, so that the rate decreases with increasing temperature; • the algebraic form of the rate equation may change with a change in temperature; and • although the form of the rate equation remains the same, the apparent activation energy may change to a different value with a change in temperature.

430

Chapter 13, Unusual thermal and mass-transfer effects Another, very rare anomaly is that single reaction steps of radicals at very low temperatures may not obey the Arrhenius equation at all. Normal Arrhenius behavior arises from an activation barrier which the reaction must surmount. In many exothermic reactions of radicals with one another or with small molecules, there is no such barrier to speak of (see Section 10.4), and a hindrance from other effects, say, molecular rotation, may increase as the temperature is raised. This makes the rate decrease in a non-Arrhenius fashion with increasing temperature. Examples include reactions of CN- with O2, ethene, ethyne, and ammonia and of OH- with O:, butenes, and HBr. This interesting anomaly has recently been reviewed by Sims [11]. The discussion here and in other part of this book assumes no such steps are involved.

Although the anomalies are rare, the fact that they may occur makes it risky to extrapolate with the Arrhenius equation into temperature regions not covered by experiments. The anomaly most often encountered is a negative, but approximately temperature-independent activation energy. It will be examined first. 13.1.1. Negative apparent activation energy, lower rate at higher temperature To be specific, the "apparent" activation energy of a reaction is the quantity E^ calculated with eqn 1.3 or 2.2 above from the temperature variation of the apparent rate coefficient in an empirical power-law rate equation that fits experimental results reasonably well. It is obtained without any assumptions as to a mechanism. The rate coefficients of all individual steps increase with temperature (barring participation of anomalous non-Arrhenius steps). Accordingly, a decrease of the reaction rate with increasing temperature can only be produced by reverse steps whose rate coefficients increase more sharply than do those of the forward ones: A decrease in rate with increase in temperature is possible only if the reaction involves at least one reverse step.

The overall reaction, however, need not be reversible. Consider as the simplest example a reaction with pathway h <-^ X -> V

(13.1)

where X remains at trace level. The rate equation, obtained with the Bodenstein approximation for X is k k

k ^ k

13,1. Anomalous temperature dependence

431

so that the apparent rate coefficient is given by k

k

(13.2)

app ^XP"^

^XA

The rate decreases with increasing temperature if the denominator increases by a larger factor than does the numerator. To establish a more restrictive criterion in terms of activation energies of steps, we need a rule for activation energies of combinations of rate coefficients. As can be deduced from eqn 2.2 above, the activation energy (£"4)1*2 of a product k^kj of rate coefficients k^ and ^2 with activation energies {E^^ and (E^i is d\n{k^k^) (£a)l.2 = ~R

d(l/7)

= -/?.

d(i/r)

lnit„ sXo^t'EJRT

\IT Figure 13.1. Arrhenius plot for reaction with negative apparent activation energy (schematic). {Ink, + \nk^) = (EX + (EX

(13.3)

Similarly, for a ratio A:, / ^ of rate coefficients: (^a)./2

= -R

dln{kjk^) d(l/7)

-/?.

d(l/7)

(In/:, - Ink,) = (E^ - (E^

(13.4)

In words:

The activation energy of a product of rate coefficients equals the sum of the activation energies of the coefficients. The activation energy of a ratio of rate coefficients equals the difference of the activation energies of the coefficients.

To return to the reaction with pathway 13.1 and apparent rate coefficient given by eqn 13.2: For the rate to decrease with increasing temperature, at least one of the two denominator terms in eqn 13.2 must increase by a larger factor than does the numerator, that is, have a higher activation energy than the latter. This can only be the second term, the reverse coefficient kxA, because the activation energy (^Jxp of the first term is necessarily lower than that of the numerator, (£a)Ax 4- (EJxp- Accordingly:

432

•

Chapter 13. Unusual thermal and mass-transfer effects

The rate of a reaction with pathway A <--^ X —• P may decrease with increasing temperature if the activation energy of its reverse step is higher than the sum of those of its two forward steps.

This is a necessary condition (in the absence of non-Arrhenius steps), but not a sufficient one: Even if it is met, the rate may increase with increasing temperature because the activation energy of the first denominator term in eqn 13.2 is lower than that of the numerator and may exert the stronger influence. The condition above can be generalized for simple pathways A ^— ... ^ — Xj ^>

Xj,, ^-> ... ^ i ^ P

(arbitrary number of steps, co-reactants and co-products not shown; see Section 6.3 for simple pathways and their rate equations): • The forward rate of a simple reaction may decrease with increasing temperature if the pathway contains at least one intermediate Xj whose enthalpy (—AH°)Q^ of exothermic formation from the reactants is larger than the activation energy (^^Jjj+i of the subsequent step Xj —• Xj+p This ensures that at least one denominator term, the j'th, in the rate equation increases with temperature by a larger factor than does the numerator, and is a necessary condition, but not a sufficient one (again granted absence of nonArrhenius steps). Derivation. Apart from concentrations, the ratio of the numerator and the j'th denominator term is the ratio of the products of the first j +1 forward coefficients and the first j reverse coefficients. The product of the ratios of forward and reverse coefficients of the first j steps equals the equilibrium constant of formation of Xj from the reactants or a power of it. According to the van't Hoff equation d\nK/d(l/T)

= - AHVR

and the Arrhenius equation 2.2, this constant decreases more steeply with increasing temperature than /TJJ+I increases if {-AH\^ is larger than {E^\^+i. For more complex cases, specific conditions are not readily derived beyond the obvious requirement that the denominator in the equation for the apparent rate coefficient must increase with temperature by a larger factor than does the numerator. Negative apparent activation energies, if within a limited temperature range, are most often found in heterogeneous catalysis, where adsorption must precede the reaction on the catalyst surface. Adsorption equilibria are less favorable at higher temperature, and this effect may outweigh that of a higher rate coefficient of the surface reaction if the (negative) adsorption enthalpy is larger than the activation energy of the surface reaction [12] (see also Section 9.4.1).

13.1. Anomalous temperature dependence

433

13.1.2. Change in rate control with change in temperature As the steps of a multistep reaction have different activation energies, a change in temperature may affect their balance and cause a shift in rate control, leading to a rate behavior not in accordance with the Arrhenius equation. Two situations can be distinguished: • shift of rate control to a different step within a pathway, or • shift of rate control to an alternative, parallel pathway. In both instances, the algebraic form of the rate equation and, with it, the reaction orders are apt to change also. However, this is not always so. Shift of rate control within a pathway. As with resistances in sequence or traffic on a one-lane road, rate control within a pathway rests with the slowest step, the "bottleneck" (see Section 4.2). If the step that is slowest at low temperature has a high activation energy, an increase in temperature may let it become faster than another with lower activation energy, which then assumes rate control. More often than not, the two steps have different co-reactants, or one may have a co-reactant and the other not. The algebraic form of the rate equation then changes with the change in rate control, and the experimental determination of the rate equations at low and high temperatures makes it apparent that rate control shifts with temperature. An Arrhenius plot covering both temperature regions cannot be drawn because the rate equations differ; the apparent rate coefficient may even have different dimensions in the two regions. However, since a change in control requires the step that is rate-controlling at low temperature to have the higher activation energy, the low-temperature Arrhenius plot must have the steeper slope. It is possible, of course, that the algebraic form of the rate equation does not change with the change in rate control. This is true if neither step has a co-reactant or if both have the same co-reactant or co-reactants. If so, the resulting Arrhenius plot shows two distinct temperature regimes with constant, but different slopes, connected by a curved transition region where rate control switches from one step to the other, as shown in Figure 13.2, left diagram, next page. As pointed out above, the slope in the low-temperature region (right-hand side in an Arrhenius plot versus reciprocal temperature) is the steeper one [13]. The diagrams in Figure 13.2 also show linear extrapolations of the Arrhenius straight lines into the respective other temperature regions. These represent the behavior that would be observed if rate control would not change with temperature. It can be seen that, in the left diagram, the solid lines are below the dashed ones in both temperature regions; that is, in both regions the observed apparent rate coefficient (solid line) is lower than the extrapolated one (dashed line). Accordingly, in both regions the slower mechanism is realized. A behavior as in Figure 13.2, left diagram, is a certain indication of a shift in rate control, even though no change in the form of the rate equation is observed.

434

Chapter 13. Unusual thermal and mass-transfer effects

\nk

IIT \IT Figure 13.2. Arrhenius plots for reactions with shift in rate control but no change in form of rate equation (schematic). Left: shift to different step within same pathway; right: shift to alternative, parallel pathway. Dashed lines are linear extrapolations into respective other temperature regions (from Helfferich and Savage [14]). To sunmiarize: Rate control in a pathway may switch to a different step upon temperature variation if the step controlling at low temperature has a higher activation energy than at least one other. The apparent activation energy of the pathway then is higher in the low-temperature region.

Behavior of this general type is also occasionally found in systems in which a reactant must be supplied from another phase by mass transfer, e.g., in heterogeneous catalysis or homogeneous hydrogenation. Usually, the activation energy of the reaction is fairly high, but that of mass transfer is only a few kilojoule per mole. Rate control may then shift from reaction at low temperature to mass transfer or a combination of mass transfer and reaction at high temperature. Shift of rate control to alternative pathway. Often, two or more possible pathways from the same reactant or reactants to the same product or products exist side-byside. As with electric resistances in parallel or traffic along parallel roads, the rate then is dominated by flow along the fastest pathway. A dirt track parallel to a freeway contributes little to traffic flow! The rate-controlling step along the fastest pathway may have a low apparent activation energy, so that a temperature increase may cause another pathway to become faster and assume control. Usually, such a shift also produces a change in the algebraic form of the rate equation. As with shift of control within a pathway, a determination of the rate equations in the two temperature regions then makes the shift in control obvious, and an Arrhenius plot

13.1. Anomalous temperature dependence

435

covering both temperature regions cannot be drawn. However, since a shift in control requires the pathway that is rate-controlling at low temperature to have the lower activation energy, the Arrhenius plot for the low-temperature region must have the gentler slope. This is the opposite of what happens with shift in rate control within one and the same pathway (see above). Again, the algebraic form of the rate equation might not change with the shift in control. If it does not, the Arrhenius plot shows two straight-line portions connected by a curve in die temperature range where the shift occurs. This time, however, the slope is steeper in the high-temperature region, as shown in Figure 13.2, right diagram. The diagram also includes linear extrapolations into the respective other temperature regions. In contrast to the behavior upon shift of control within a pathway (left diagram), the observed apparent rate coefficient (solid line) in both regions now is larger than the extrapolated one (dashed line): In both regions, liht faster mechanism is realized. Such behavior is strong evidence for a shift of rate control to another pathway despite the fact that no change in the algebraic form of the rate equation is observed [13]. To summarize: Rate control in a network with parallel pathways may switch from one to another if the pathway controlling at lower temperature has a lower apparent activation energy than the other. The apparent activation energy of the network then is lower in the low-temperature region.

13.1.3. Rate maxima and minima In very rare instances, a shift of rate control with temperature may occur from a mechanism with positive to one with negative apparent activation energy or vice versa. If so, the rate has a maximum or minimum in the temperature range of the shift. A rate maximum is sometimes seen in heterogeneous catalysis [12,15]. If the rate has a maximum, the controlling mechanism in each of the two temperature regions is the slower one; if the rate has a minimum, it is the faster one. In view of the rules established in the previous section, a corollary is: A maximum or minimum in the temperature dependence of the rate indicates a shift of rate control: a maximum, a shift to another step in the same pathway; a minimum, a shift to an alternative, parallel pathway.

436

Chapter 13. Unusual thermal and mass-transfer effects

13.1,4. Activation energies of phenomenological coefficients The phenomenological coefficients in a one-plus rate equation derived for a specific mechanism are composites of individual rate coefficients of steps. Activation energies for the former can be established from their temperature dependence with the Arrhenius equation. With regard to their activation energies, two types of phenomenological coefficients can be distinguished: those consisting entirely of products, ratios, or ratios of products of individual rate coefficients; and those which also involve additive terms. As was shown in Section 13.1.1 and with an addition for ratios of products: • • •

the activation energy of a product of individual rate coefficients is given by the sum of the activation energies of the latter; the activation energy of a ratio of individual rate coefficients is given by the difference of the activation energies of the latter; and the activation energy of a ratio of products of individual rate coefficients is given by the sum of the activation energies of the coefficients in the numerator minus the sum of the activation energies of those in the denominator.

In all these three cases, the activation energy of the phenomenological coefficient as the sum, or difference, or difference of sums of constant activation energies is itself constant. In contrast, if the phenomenological coefficient involves additive terms, the same cannot be said. Here, a constant activation energy results only if all but one of the additive terms remain insignificant, or if all significant terms have the same activation energy. Accordingly

A significant variation of the activation energy of a phenomenological coefficient with temperature indicates that the coefficient involves additive terms of individual rate coefficients of steps.

The reverse is not necessarily true: The additive terms may have very similar activation energies or one of them may dominate, and the activation energy of their sum then remains approximately constant. The curvature of an Arrhenius plot for a phenomenological coefficient can sometimes be used to distinguish between rival models: A model in which a phenomenological coefficient is a product, ratio, or ratio of products of individual rate coefficients of steps can in general not be correct if an Arrhenius plot of that coefficients is distinctly curved. Note, however, that the curvature may arise from a shift of rate control with temperature (see Section 13.1.2).

13.2, Uncommon heat-transfer problems

437

13.2. Uncommon heat-transfer problems In most instances in practice, the rate of and heat evolution by an exothermic chemical reaction are smooth functions of temperature and reactant concentrations, with an apparent reaction order not higher than three and an apparent activation energy rarely above 200 kJ mol'^ Engineers designing reactors tend to take this for granted. Simultaneous or multistep reactions, however, may behave differently. Two quite common situations are those in which the starting material is a mixture of reactants of very different reactivities, or the reactant is a highly reactive isomer, most of which is quickly converted to much less reactive ones as conversion to products progresses. In both cases, the initial very fast conversion of a highly reactive component in the starting material results in a short burst of very high heat release, a fact a reactor designer ignores at his peril. A strong burst of heat release at beginning conversion may result if the starting material is a mixture containing a component that is much more reactive than the others, or is a highly reactive isomer that undergoes isomerization to less reactive ones while conversion to products proceeds.

The first of the two situation above is occasionally encountered in petroleum processing, where multicomponent mixtures of reactants are common. The second may arise in reactions with coupled parallel steps [14] (see Section 5.3). Example 13.1. Hydroformylation of long-chain l-olefins with phosphine-substituted cobalt hydrocarbonyl catalysts. Hydroformylation of long-chain l-olefins with phosphine-substituted cobalt hydrocarbonyl catalysts provides a striking example of coupled parallel steps and the potential of an unconmion heat-transfer problem. The network is of the type 13.5 below, with the Aj as the olefin isomers and the Pj as the isomeric alcohol products (arrows represent multistep pathways; see also Example 5.3, Figure 5.9, and network 5.43 in Section 5.3 and network 7.40 in Section 7.4). A4

/ p,

->

*\

/

F> 2

A^3

^

"

(13.5)

\

/ P

\ P4

The 1-olefin, A^, is about two orders of magnitude more reactive than the other isomers with the double bond in internal positions, but in equilibrium at typical reaction temperatures (150 to 200°C) it constitutes only a few percent of total olefin. Fast double bond migration lets a steady-state olefin isomer distribution be closely

438

Chapter 13. Unusual thermal and mass-transfer ejfects approached before much olefin has been converted. As a result, 1-olefin if 1-olefin is charged to the equilibrium reactor, the initial abundance of isomer mixture this highly reactive isomer causes Q reaction and heat release rates at the inlet of a continuous reactor or at start in a batch reactor to be abnormally high. However, most of the 1-olefin is consumed very quickly by isomerization and 1.0 reaction to products, making the /o. rate (at constant temperature) Figure 13,3. Instantaneous isothermal drop to a fev^ percent of that at heat release Q as function of olefin the inlet or start. Figure 13.3 conversion for hydroformylation of longshows such behavior and, for chain olefins with cobalt phosphinocomparison, that of the same hydrocarbonyl catalysts (schematic). olefin with double bonds in isomerization equilibrium. Interestingly, the initial burst of reaction also aggravates a very serious masstransfer problem, as will be shown in the next section.

13.3. Uncommon mass-transfer problems Mass transfer in combination with even quite "normal" reaction kinetics can produce a wealth of phenomena including multiple steady states, instabilities, and oscillations. An example is the behavior of nonisothermal catalyst particles outlined in Section 9.5.2. Such phenomena are covered in detail in standard texts on reaction engineering, to which the reader is referred. The examination in this section will remain restricted to effects produced by vagaries of multistep or multiple simultaneous reactions. 13.3.1. Mass transfer and reaction rate In heterogeneous catalysis, the reactants must be supplied to the solid catalyst from a contacting fluid by mass transfer (see Section 9.5). Likewise, some reactions kinetically classified as homogeneous because they occur in a single fluid phase, consume reactants that require mass transfer from a gas phase (or occasionally from another liquid phase). Homogeneous hydrogenation is an example. This is a typical problem of reaction engineering and is treated in some detail in almost all modern texts of that field [1,3,4,9,16,17]. Customarily, a power law or LangmuirHinshelwood kinetics is assumed for the rate of the chemical reaction and is then combined with a standard linear-driving force or Fickian diffusion treatment of mass

13.3. Uncommon mass-transfer problems

439

transfer. A mass-transfer limitation lowers the rate, which in some extreme situations can become entirely mass transfer-controlled. Certain types of multistep reactions, however, can produce a totally different and very interesting behavior that may involve instability. In a great majority of cases of practical interest, the reaction orders are positive for reactants and negative or zero for products, as is taken for granted in standard texts. If mass transfer from another phase is not fast enough to deliver as much of a reactant as the reaction consumes, the concentration of that reactant drops somewhat and, as a result, the reaction slows down to some extent. Similarly, in a reversible reaction whose product leaves the phase in which the reaction occurs, slow mass transfer of the product reduces the rate. However, if the reaction is product-promoted (positive reaction order with respect to a product) or reactantinhibited (negative reaction order with respect to a reactant), a mass-transfer limitation may increase rather than decrease the rate. It does so in a product-promoted reaction if mass transfer is slow in removing a product whose reaction order is positive. It does so in a reactant-inhibited reaction if it is slow in supplying a reactant whose order is negative. A lagging of mass transfer increases the concentration difference between the phases that acts as driving force and, thereby, boosts the mass-transfer rate. Also, an increase of conversion rate lowers the concentrations of reactants whose orders are positive (no reaction can have negative orders with respect to all reactants). Both these effects work against a continuing acceleration of the reaction rate. In a product-promoted reaction, lagging removal of the accelerating product merely maintains self-acceleration, which, of course, would not come into play at all if the respective product were removed as soon as it is formed. In a reactant-inhibited reaction, however, the accelerating effect of lagging supply of the respective reactant may win out over the increased mass transfer rate and cause a catastrophic depletion: The phase runs out of reactant or some other process takes over. There is a sharp limit of stability beyond which that happens.

In a product-promoted reaction, fast mass transfer of the accelerating product into another phase slows the rate by depressing self-acceleration. In a reactant-inhibited reaction, slow mass transfer of the inhibiting reactant from another phase accelerates the rate and may cause instability.

The acceleration and chance of instability in a reactant-inhibited reaction are less pronounced if mass transfer of the respective reactant is fast. Thus, the very high mobility of hydrogen makes the effects unlikely to occur in the most common situation of reactant inhibition, that is, in heterogenous hydrogenation (see Example 9.4 in Section 9.3.1).

440

Chapter 13. Unusual thermal and mass-transfer effects Example 13.2. Potential mass transfer-induced instability in olefin hydroformylation [14]. The rate of homogeneous olefin hydroformylation with cobalt hydrocarbonyl catalysts in a liquid phase obeys in good approximation the Martin equation -

-

k C C ^ °'' ^^^ 1 + kj)^Jp

(6.12)

(see Example 6.2 in Section 6.3). The rate is of negative order in CO and positive order in H2, both being reactants that must be supplied by mass transfer from the gas phase. H2 as the much smaller molecule has the much higher mass-transfer coefficient. Accordingly, any mass-transfer limitation makes itself felt with respect to CO first. Because of the negative order with respect to CO, a decrease in CO concentration in the liquid increases the rate. As described above, this acceleration may win out over the counteracting effects of decreases in olefin and H2 concentrations. If so, reactant depletion of the liquid becomes self-accelerating. However, as the liquid becomes depleted of CO, the hydrocarbonyl catalyst becomes unstable and metallic cobalt is plated out. This catalyst loss eventually reins in the accelerating reaction. One might say, a mass-transfer limitation under normal conditions acts as a gentle brake on the reaction, slowing it down at worst to the rate that mass transfer to the reacting phase can sustain, whereas in homogeneous hydroformylation with cobalt hydrocarbonyl catalysts, mass transfer imposes an upper limit on the amount of catalyst the system will tolerate. Assume a small amount of catalyst is used; mass transfer then has no trouble supplying as much CO as the reaction consumes (note conversion is first order in catalyst). However, if now the amount of catalyst and thereby the conversion rate are increased, the point may be reached where mass transfer can no longer keep up with CO consumption. The self-accelerating reaction then depletes the liquid of CO to the extent that the catalyst added beyond the limit of mass-transfer stability decomposes. In "heterogenized" hydroformylation with metal phosphino-hydrocarbonyl complexes anchored on a swollen polymer (see Section 9.6), another interesting effect can be observed. On the safe side of mass-transfer instability, the rate under same conditions of fluid- and gas-phase composition and pressure and with same amount of catalyst complex may be higher in the heterogeneous than in the homogeneous case. This is counterintuitive because the former requires mass transfer (normally retarding) of H2 and CO not only from gas to liquid, but also from liquid into the catalyst beads. Higher rates in the heterogenized system at same amount of catalyst complex have indeed been observed in some cases [18]. Several effects can cause this and, in hydroformylation, mass-transfer induced acceleration is almost certainly one of them. The example of the mass-transfer effect in olefin hydroformylation is interesting in still another respect. If a phosphine-substituted cobalt hydrocarbonyl is used as catalyst in combination with 1-olefin as reactant, a very strong burst of reaction ensues at the reactor inlet or at start of a batch reaction (see Example 12.1). This places extreme demands on the supply of CO from the gas phase by mass transfer and can easily lead to instability and high catalyst losses unless appropriate precautions were taken in the design.

Summary

441

A rigorous mathematical treatment of mass-transfer instability is difficult and requires a detailed knowledge of fluid dynamics in the reactor, information hard to come by. The best way to proceed in process design therefore is to establish approximately where the limit of stability lies, and then keep away from it by a safe margin. No plant operates without excursions from design operating conditions, and the risk that it could stray into the region of instability must be minimized. 13.3.2. Mass transfer and selectivity Mass transfer into or out of a catalyst particle or liquid phase can affect not only the reaction rate, but also the selectivity. Unless mass transfer is fast compared with reaction, faster reactants will outrun their brothers into the phase where the reaction occurs, so that selectivity shifts in favor of the products they produce. A more subtle effect, most noticeable in reversible reactions, is that products slow to leave the scene are at a disadvantage. Such behavior is most prominent in zeolites (see Section 9.7), in which differences in diffusion coefficients can be very large. A striking example of unusual behavior of this kind is cracking of paraffins on zeolite T or erionite, materials with window-and-supercage structure (see Section 9.7). With C22 or C23 Az-paraffm as starting material, the reaction yields almost no products in the C7 to C9 range, but plenty of lighter and heavier ones [19,20]. This happens because the diffusion coefficients are abnormally low in the range around Cg, a size that best matches the dimensions of the supercages. A mathematical analysis that invokes the analogy to Brownian motion of a rod across equally spaced potential barriers can explain the effect [21]. Mass transfer can also affect the selectivity of a single, multistep reaction by altering the reactant ratios. An example is homogeneous liquid-phase hydroformylation with phosphine-substituted cobalt hydrocarbonyl catalysts. Alcohol and paraffin byproduct are formed from olefin, H2, and CO. Mass transfer of CO is slower than that of H2, so that the H2:C0 ratio in the liquid phase shifts in favor of H2. This causes more paraffin to be formed (see Example 7.5 in Section 7.3.2). Summary Apparent activation energies of reactions may be negative, corresponding to a decrease in rate with increase in temperature, if the pathway contains at least one reverse step with an activation energy that is high compared with those of the forward steps. A change in temperature may cause a shift in rate control to another step in the same pathway or to an alternative, parallel pathway. Usually, this entails a change in the rate equation, and so makes it obvious that such a shift occurs. If the rate equation remains the same, the apparent activation energy changes from one value to another over a narrow temperature range, giving an Arrhenius plot with two straight-line portions

442

Chapter 13. Unusual thermal and mass-transfer effects

a curve. Whether the rate equation changes or not, the activation energy is higher in the low-temperature region if rate control shifts within the pathway, and is higher in the hightemperature region if the shift is to another pathway. In the first case, the slower of the two possible mechanisms dominates the rate; in the second case, the faster one. A shift in rate control may cause the rate to have a maximum or minimum at some temperature. A rate maximum indicates a shift of rate control to a another step in the same pathway; a minimum, a shift to an alternative, parallel pathway. Phenomenological coefficients in reduced rate equations are composites of individual rate coefficients of steps. If a coefficient consists entirely of a product, ratio, or ratio of products of individual rate coefficients, its activation energy is essentially temperature-independent. In contrast, if the coefficient involves additive terms, its activation energy varies with temperature unless the terms have similar activation energies or one of them dominates. An initial burst of high heat release may occur if the starting material of an exothermic reaction is a mixture of components of very different reactivities, or is a highly reactive isomer that is quickly isomerized to less reactive ones as conversion to products progresses. In product-promoted reactions in which the accelerating species exits into another phase, or in reactant-inhibited reactions in which the respective reactant enters from another phase, slow mass transfer may boost rather than depress the reaction rate. In reactant-inhibited reactions, slow supply of the inhibiting reactant may cause the system to become unstable: There may be a sharp stability limit beyond which catastrophic selfacceleration occurs until the inhibiting reactant has become exhausted or some other phenomenon has come into play. Heterogenized catalysis of a reactant-inhibited reaction may be faster than the respective homogeneous reaction with same amount of catalyst. Unless mass transfer is fast compared with reaction, it can affect selectivity in reactions in which a reactant enters from, or a product exits into, another phase. Products with greater mobility, or formed from reactants with greater mobility, are favored. If several reactants of a single reaction enter from another phase, their ratios shift in favor of the more mobile ones, possibly causing a corresponding selectivity shift. The anomalous heat- and mass-transfer problems are illustrated with examples from catalytic cracking and olefin hydroformylation with cobalt hydrocarbonyl catalysts. References 1. 2. 3. 4. 5.

G. Astarita, Mass transfer with chemical reaction, Elsevier, Amsterdam, 1967. J. J. Carberry, Chemical and catalytic reaction engineering, McGraw-Hill, New York, 1973, ISBN 0070097909, Sections 3-10 and 4-2 and Chapter 6. A. R. Cooper and G. V. Jeffreys, Chemical kinetics and reactor design, PrenticeHall, Englewood Cliffs, 1971, ISBN 0131286781, Chapter 9. P. V. Danckwerts, Gas-liquid reactions, McGraw-Hill, New York, 1970. H. S. Fogler, Elements of chemical reaction engineering, Prentice-Hall, Englewood Cliffs, 3rd ed., 1999, ISBN 0135317088, Chapters 8 and 10.

References 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

443

G. F. Froment and K. B. Bischoff, Chemical reactor analysis and design, Wiley, 2nd ed., 1990, ISBN 0471510440, Chapters 3, 6, and 7. C. G. Hill, Jr., y4n introduction to chemical engineering kinetics & reactor design, Wiley, New York, 1977, ISBN 0471396095, Chapter 10. O. Levenspiel, Chemical reaction engineering, Wiley, New York, 3nd ed., 1999, ISBN 047125424X, Chapters 6 and 13. O. Levenspiel, The chemical reactor omnibook, O.S.U. Book Stores, Corvallis, OR, 1989, ISBN 0882461648, Section 41. G. D. Ulrich, A guide to chemical engineering reactor design and kinetics, Ulrich Research and Consulting, Lee, 1993, Chapter 5. I. R. Sims, Res. Chem. Kinet., 4 (1997) 121. T. J. Yoon and M. A. Vannice, 7. Catal, 82 (1983) 457. J. A. Christiansen, Adv. Catal, 5 (1953) 311, Section 5.2. F. G. Helfferich and P. E. Savage, Reaction kinetics for the practical engineer. Course #195, AIChE Educational Services, New York, 7th ed., 1999, Chapter 11. R. Badilla-Ohlbaum, H. J. Neuberg, W. F. Graydon, and M. J. Phillips, J. Catal, 47 (1977) 273, Carberry (ref. 2), Chapter 6. Levenspiel (ref. 8), Chapter 13. C. U. Pittman, Jr., and L. R. Smith, J. Am. Chem. Soc, 97 (1975) 1749. N. Y. Chen, S. J. Lucki, and E. B. Mower, J. Catal, 13 (1969) 329. R. L. Gorring, J. Catal, 31 (1973) 13. J. M. Nitsche and J. Wei, AIChE J., 37 (1991) 661.

Chapter 14 Instability, Periodic Reactions, and Chaos In some rare instances. Chemical reactions may be unstable, periodic, or chaotic. The practical engineer must be aware of these phenomena, but will do his best to avoid them. He is therefore not concerned with their rather complex mathematics and vocabulary, only with a comprehension of when and how they might arise, so he can stay clear. This chapter attempts to provide some idea of the danger signs without delving into the finer aspects of quantitative theory. For the latter, the reader is referred to the specialty literature (G1-G5). Instabilities can arise from interplay of quite normal reaction kinetics with other rate processes such as heat transfer or control systems. Such situations are well understood and covered in standard texts. Therefore, emphasis here is on behavior produced by vagaries of multistep kinetics, an aspect given short shrift in today's reaction-engineering education. However, easily understood examples with normal kinetics are included to illustrate basic principles. Hard and fast rules stating when a reaction will be unstable, periodic, or chaotic cannot be given. Even detailed mathematical theories can only say whether a reaction with given network may show such behavior, depending on parameter values and operating conditions. Moreover, mathematics are different for batch and continuous stirred-tank reactors since the former are described by simultaneous differential equations, the latter by simultaneous algebraic equations. The brief account here is by no means all-encompassing. It can only provide a rough outline. 14.1. Instability A "normal" chemical reaction—with reaction orders that are positive for reactants and negative or zero for products—is stable under isothermal conditions as its rate declines smoothly and steadily with progressing conversion. Instability requires feedback, that is, a result of the reaction must affect the rate. Some product or intermediate, heat produced or consumed, or coupling with some other mechanism (e.g., process control) must do so: •

To be unstable, a reaction must be autocatalytic (if single-step), or multistep, or nonisothernial, or coupled with another mechanism.

Moreover, the feedback, or at least part of it, must be destabilizing, that is, it must cause small deviations from the status quo to escalate:

446 •

Chapter 14. Instability, periodic reactions, and chaos Instability requires feedback to provide acceleration that increases with increasing reaction rate, or deceleration that decreases with decreasing reaction rate.

[It can be Argued that every reaction has its built-in feedback in that it consumes its reactants and forms products that may affect its rate. However, this feedback is stabilizing if the reaction orders are positive for reactants and negative or zero for products.] The conditions above are necessary, but not sufficient. For example, an exothermic, nonisothermal reaction is stable unless the temperature is so high that its heat generation (with exponential temperature dependence) exceeds the heat loss to the environment or a cooling coil (with linear temperature dependence). A chain reaction with chain branching is stable unless the radical population is so large that acceleration by chain branching (with exponential dependence on the radical population) outruns termination (with quadratic dependence). Unfortunately, in complex networks it may be difficult to unravel the interplay of steps and identify possibly destabilizing effects. If no self-accelerating steps, step sequences, or cycles are apparent, the reaction is probably stable. However, a few computer runs under relevant conditions may be indicated to make sure. 14.2. Runaway and multiple steady states As the examples of nonisothermal and chain-branching reactions have shown, destabilizing feedback typically entails two phenomena that differ in their dependences on a reaction parameter and contend with one another: heat generation and loss in the nonisothermal reaction, chain branching and termination in the chain reaction. The stationary or quasi-stationary balance of the accelerating and decelerating effects (intersections of curves in Figure 14.1) is unstable if the former varies more strongly with the reaction parameter than does the latter: in the nonisothermal reaction if heat generation increases more steeply with temperature than does heat loss; in the chain reaction if acceleration by chain branching increases more steeply with the size of the radical population than does termination. Instability in a non-flow system—that might be an exothermic batch reaction or a pile of oily rags subject to autoxidation—is apt to escalate into a conflagration or thermal explosion (see Figure 14.1, left diagram). However, instability might be confined to a narrow region close to equilibrium, and the reaction then runs out of reactants before the temperature rise can become catastrophic. Flow systems may also admit multiple steady states and hysteresis. Example 14.1. Multiple steady states and hysteresis in a nonisothermal continuous stirred-tank reactor (CSTR) [1,2]. In a CSTR, the curve for the temperature dependence of heat loss to the cooling coil is linear (loss proportional to temperature difference) while that for heat generation by the reaction is S-shaped (Arrhenius ex-

14.2.

Runaway and multiple steady states

batch

447

1 /

/ '^ ^ /

/ /// / /

heat loss

Q

f unstable

/ heat generation

^ / /

stable T" T

T'

^c

-*c

^c

Figure 14.1. Stable and unstable states. Heat generation and loss curves for batch reactor (left) and CSTR (right); T^ = cooling coil temperature (schematic). ponential dependence at low conversion, but approaching a finite limit as conversion approaches equilibrium; see Figure 14.1, right diagram). Depending on the temperature of the cooling coil (foot point of loss curve) and the heat-transfer effectiveness (steepness of the curve) there may be as many as three intersections of the two curves. Each intersection corresponds to an exact compensation of heat generation and loss, and thus to a steady state. The low- and high-temperature steady states are stable because, at these, the curve for heat loss is steeper than that for heat generation, so that any small excursion to higher or lower temperature is self-correcting (heat loss greater than gain at higher temperature, heat gain greater than loss at lower temperature). In contrast, the intermediate steady state is unstable because, at it, the heat generation curve is the steeper one, so that any small excursion escalates. Which steady state the system approaches at given and constant flow rate and coil temperature depends on prior history. If the reactor operates at a "low" steady state and the cooling coil temperature is allowed to rise, corresponding to a shift of the cooling curve to the right, the steady-state temperature and conversion creep up, but only until the heat-loss curve becomes tangential to the heat generation curve (at coil temperature T^ in Figure 14.1, right). At this point, instability sets in and ignition occurs as behavior switches to the "high" steady state on coil temperature or residence time the current heat-loss curve. Similarly, if the reactor is at a "high" steady state Figure 14.2. Stable and unstable steady and the coil temperature is lowered, states in an exothermic CSTR (schematic). corresponding to a shift of the heat-

448

Chapter 14. Instability, periodic reactions, and chaos

loss curve to the left, the steady-state temperature and conversion drop steadily up to a point of flame-out (also called extinction, at coil temperature T^" in Figure 14.1, right, preceding page), when behavior switches to the "low" steady state on the current heat-loss curve. Instead of the coil temperature, the flow rate can be changed to lengthen or shorten the residence time. This amounts to shifting the heat generation curve left or right. Back and forth variation of the coil temperature or flow rate can produce a hysteresis loop (see Figure 14.2, preceding page). Reactions with "normal" kinetics were chosen for these examples because they are easiest to understand. Similar effects may arise under isothermal conditions if the reaction itself involves two different, contending pathways or steps whose dependences on a reaction parameter correspond to intersecting curves. The interplay of chain branching and termination in a chain reaction is a case in point—even though the reaction can hardly be kept isothermal once it has become unstable. Detonation would result even under hypothetical isothermal conditions! In continuous stirred-tank reactors (CSTRs), complex kinetics may give rise to multiple steady states even in isothermal operation, especially in heterogeneous catalysis. However, to unravel the causes may be difficult. Here, Feinberg's network theory can help [3]. It operates with a deficiency index that is a readily calculated zero or positive integer. The most useful result of the theory is: •

A reaction with deficiency zero in an isothermal CSTR has exactly one stable and no unstable steady states.

Networks with higher deficiencies may or may not admit multiple steady states. The user need not concern himself with details of the calculation of deficiencies: A computer program CHEMICAL REACTION NETWORK TOOLBOX available on the internet [4] can do the job for him. Feinberg's network theory [3-7]. The deficiency of a network is calculated from its number of complexes, n, linkage classes, /, and rank, s. Here, a "complex" is defined as the reactant(s) or product(s) of a step. For example, the complexes of a step A 4- B <—> K are A+B and K. A network consists of one or more "linkage classes," defined as portions consisting of complexes connected with one another by arrows, but not with complexes of another class; For example, a network A •P A ^^^-> K

•Q

with five complexes A, 2A, K, P, and Q has two linkage classes {A,P} and {2A,K,Q}. Lastly, the "rank" of a network is the number of its linearly independent steps [8]. The formula for the deficiency index, 6, is 6 = n — I - s

14,2, Runaway and multiple steady states

449

Before the formula can be used for a CSTR, the network must be augmented by addition of "pseudo-steps" for the in- and outflows. For any species i present in the effluent but not the feed, that step is i—• 0; for any species i present in both the feed and the effluent, that step is i -4—• 0. For example, the "true" network of the reaction and the augmented CSTR network might be "true" network

augmented CSTR network A

A

•K

• P

• K

V.,^^ I ^

•

P

(14.1)

0

The complex 0 contains nothing at all and is a mathematical artifice, not a species. [The pseudo-steps and thus the augmented network are not subject to the rules of stoichiometry and thermodynamics. For example, the CSTR network 14.1 violates microscopic reversibility in that it allows net circular reactions.] The CSTR network 14.1 has four complexes (A, K, P, and 0), only one linkage class, and rank 3. Accordingly, it has a deficiency of zero and will sustain just one stable and no unstable steady states. Beyond establishing unequivocally that deficiency-zero networks cannot sustain multiple steady states, network theory provides information on at least some types of networks of higher deficiencies. For example, in a CSTR, the reaction A A <—• K ^^^^-^ P has deficiency one and, according to theory, can sustain at most one stable steady state. On the other hand, in a CSTR, the catalytic reaction A

C+2 O

(14.2)

B

also of deficiency one, can sustain two stable and one unstable steady states. In a number of situations, a graphical procedure can establish whether multiple steady states can be sustained [5]. For further details, the reader is referred to Feinberg's publications.* * Feinberg's theory [3-7] is more complicated and is not restricted to CSTRs. It subdivides network categories by introducing the concepts of weak reversibility and of strong and strong terminal linkage classes. For example, although the reaction 14.1 itself is irreversible, its CSTR network is "weakly reversible" because, wherever you go in arrowhead direction, there is a path in arrowhead direction back to your starting point (e.g., from P back to K via 0 and A). All CSTR networks of homogeneous reactions are weakly reversible, TOOLBOX output is in this language.

450

Chapter 14. Instability, periodic reactions, and chaos

In some instances, network theory can also help in network discrimination by allowing to check whether a pair of stable steady states observed under identical operating conditions is compatible with a postulated network. To this end, the ''signature'' of the network is established. The signature is a set of equalities and inequalities involving the logarithmic ratios of the concentrations in the two steady states. For example, the CSTR network 14.2 has the signature

^®^^A+/^0'

^'^^^

^^^^^

0^^®+^®'

^®+^®=^P

where

Mi ^ In(qvq-)

(i = A, B, P , ( A ) , ( B ) , 0 )

and where primes and double primes refer to the two steady states. The TOOLBOX program provides signatures of some deficiency-one networks. Even if not all steady-state concentrations have been measured, a proposed network may be seen not to conform and so to be incorrect. [Each signature has a complement with all inequalities reversed; this corresponds simply to an interchange of the two steady states.] Feinberg's network theory is based on a number of premises: • isothermal behavior, • constant fluid density, • uniform catalytic sites, • uniform surface coverage, and • Langmuir equations for adsorption and desorption rates. While constructed to encompass all types of reactors, the theory is mainly oriented toward continuous stirred-tank reactors and steady-state multiplicity. [For batch reactions, the TOOLBOX returns the not very enlightening information that no steady state can be sustained; for the hydrogen-oxygen reaction network 9-50 (deficiency zero in batch) it gives no warning that the system is explosive.] 14.3. Periodic reactions Imagine a moron in a shower. The water is cold, so he keeps turning the knob toward hot. By the time the control takes effect, he has overcompensated and gets scalded. So he starts turning the knob back toward cold, overcompensating again and thereby making the cycle start anew. The moron in the shower teaches us that control with delay can cause a phenomenon to become periodic. Delay is essential for periodic behavior, but here we are concerned only with delay that comes from the reaction mechanism itself.

14.3. Periodic reactions

451

Let us specify what we are looking for: reactions in which concentrations of participants repeatedly alternate having maxima. Successive concentration maxima are normal behavior in reactions with sequential steps (see Section 5.4). However, their patterns do not repeat themselves. To achieve a repetition, the last step of the pathway must restore the original reactant, and in order to sustain a net circular reaction, the cycle must convert external reactants to products (see Section 2.5.2): •

A periodic reaction, like any other, must involve continuous net conversion of reactants to products of lesser free energy.

A catalytic cycle meets the requirement of continuous conversion, but fails to produce alternating concentration maxima. For that, delayed feedback must restore a depleted reactant or intermediate. This cannot be one of the main reactants that drives the reaction and whose consumption can therefore not be reversed: •

A reaction may be periodic if its network contains a cycle that provides for restoration of a depleted reactant or intermediate while being driven by continuous conversion of main reactants to products.

Periodic behavior can result if conversion switches to another pathway and either gives the depleted reactant time to recover, or restores it by forming it as one of its new products. A quick take-over results if the reaction along the new pathway is self-accelerating. However, it then needs a mechanism for removal of its accelerator, so that it will fade out in turn and let the other reaction become dominant again to restart the cycle. Nature provides many examples of such behavior. Example 14.2. Foxes and rabbits. Predator-prey fluctuations are well-understood periodic phenomena in nature as well as the basis of board games such as Struggle or Foxes and Rabbits [9]. In terms of these: Grass grows as available area permits, rabbits feast on grass and multiply when doing so, foxes feast on rabbits and multiply when doing so, and hunters in constant number shoot foxes. At start there is a good bit of grass, some rabbits, and very few foxes. The rabbit population grows in leaps and bounds, but then starts to decline as it depletes the grass supply and becomes decimated by foxes. The fox population, in turn, grows while plenty of rabbits are around, but then starts to decline as prey becomes scarce and hunters take their toll. With both rabbits and foxes again few in numbers, grass recovers, and the cycle starts anew. This periodic "reaction" can be approximated by a step sequence in which G is grass, R is rabbits, F is foxes, H is hunters, and ... is available space: (1)

...

• G

(2)

R + G

• 2R

(3)

F + R

• 2F

(4)

H + F

• H -h ...

452

Chapter 14. Instability, periodic reactions, and chaos This is a typical Lotka-Volterra sequence [10-12], well known in ecology and biochemistry. Steps for death of rabbits and foxes from natural causes can be added. The continuing fluctuations with sharpened maxima arise from the competition of the two "autocatalytic" steps, 2 and 3, whose rates increase rather than decrease with decreasing concentration of one of their reactants, and the free-energy input keeping the cycle going is the growth of grass and its irreversible conversion to dead foxes. Given plenty of grass, step 2 takes off while step 3 is delayed until there are lots of rabbits around. Both step 2 and, later, step 3 fade out when their respective food supply becomes scarce and casualties mount. Meanwhile, decimation of rabbits lets grass grow again, so the cycle can start anew. Without rabbits, grass would take over. Without foxes, rabbits would crowd out the grass. Without hunters, foxes would displace rabbits in a world in which grass starts to flourish again.

The famous Belousov-Zhabotinsky reaction* owes its fluctuations to a similar interplay of reactions along two pathways subject to feedback, but is vastly more complicated (80 possible steps have been listed [18]). The reaction along the second pathway is self-accelerating and takes over when the first begins to deplete a lowconcentration reactant, but this second reaction is curbed by a bimolecular termination step. A subsequent reaction with an organic intermediate then restores the depleted reactant and lets the first reaction become dominant again. The slowness of this step provides the necessary delay. The oscillation with a period of the order of a minute causes the solution to change its color. What makes the reaction even more fascinating is that it can give rise to chemical waves that propagate in space {spatio-temporal patterns) [19-26]. Example 14.3. The Belousov-Zhabotinsky reaction [22,27-29]. The reaction is an oxidation of malonic acid by bromate ion in sulfuric acid, catalyzed by a Ce(III)/Ce(IV) redox couple. Many variations with other organic acids and transitionmetal ions are possible [22] (Belousov used citric acid, and manganese, ruthenium, or iron can replace cerium). The color of the solution alternates between clear [Ce(III)] and pale yellow [Ce(IV)], and more dramatically between red and blue if ferroin is added as indicator. * The reaction was discovered accidentally in 1951 by Boris Belousov, a scientist with military rank of Brigadier and director of a government biochemistry installation in Moscow. Belousov's attempts to publish his discovery in a chemistry journal met with rejection with the remark he would have to prove why established thermodynamic theory was in error [13-15] (shades of Morgenstern, see Preface?). A second attempt years later remained equally fruitless. He shared his notes with colleagues but, in disgust, decided never again to try to publish, and was not swayed even later. Not until 1959 was at least a summary issued [16]. However, Anatol Zhabotinsky, at the time a biophysics graduate student at Moscow State University, was handed Belousov's notes by his professor, studied and slightly modified the reaction, and managed to publish his findings in 1964 in reputed Russian journals [17]. It took four more years for worldwide acclaim to follow. Biochemists, unencumbered by thermodynamic prejudice and familiar with cyclic phenomena in nature, became the first true believers in the West. Belousov died in 1970, just as he started to become famous, and ten years before his and Zhabotinsky's work was honored with a Lenin Prize.

14.3. Periodic reactions

453

In the simplified Field-Koros-Noyes network [27] that reflects the essential features of the reaction, a first pathway starts with a three-step sequence with stoichiometry Br03- + 5 Br- + 6 H^ • 3 Br2 + 3 H2O Br2 then brominates the organic acid. This reaction depletes Br~ and lets a second pathway take over that starts with a self-accelerating two-step sequence that also oxidizes the cerium catalyst and so produces the color change: BrOj- + 2 H^ + 2 Ce(III) + HBrOj

• 2 Ce(IV) + 2 HBr02 + H2O

(HBr02 is an intermediate in the first pathway.) Self-acceleration is limited by a bimolecular termination 2 HBr02 <

• HOBr + Br03- + H^

Ce(IV) then oxidizes the organic acid, freeing Br" and reverting to Ce(III), thereby reversing the color change. As Br~ is restored, the first pathway becomes dominant again to restart the cycle. Various modifications have been suggested [22,30,31]. Stripped to its bare essentials, the reaction can be represented by a five-step "oregonator" [28,32,33]: A+Y • X+P X+Y • 2P A+X • 2X + 2Z 2X • A+P B + Z

•

1/2/Y + ...

(concentrations of major reactants A and B held constant; / , with 0.67 < / < 1, accounts for incomplete yield of Y; lettering is the authors' and does not correspond to usage elsewhere in this book.) This stripped model gives the proper oscillations, but not the correct wave shape and dependence on the bromate concentration. Unstable quasi-stationary states. In the examples shown so far, periodic behavior was caused by two contending reactions. However, a reaction involving a selfaccelerating conversion of one intermediate into another can become periodic all by itself [34]. Imagine a reaction converting A into P via two intermediates X and Y. Normally, intermediates at low concentrations attain quasi-stationary states (see Section 4.4). However, if the reaction of X to Y is self-accelerating, such quasistationary states may be unstable within a certain range of conversion. In this range, the concentrations of X and Y then oscillate. A quasi-stationary state is stable if a small excursion from it is selfcorrecting, but is unstable if the excursion escalates. Specifically, in a system as described here, stability is ensured if the net rates rx and r^ decrease if the concentrations of X and Y increase. If this is true for one of the intermediates, but not for the other, the stabilizing and destabilizing tendencies counteract one another, and the (necessary and sufficient) condition for stability becomes

454

Chapter 14. Instability, periodic reactions, and chaos stable if

(14.3)

(9^x/aCx)cv. + (a^Y/9CY)c,, < 0

No other stable or unstable steady or quasi-stationary state exists and, if the reaction is unstable, the concentrations of X and Y oscillate around their unstable quasistationary states until condition 14.3 is met again. Example 14.4. Test for stability of quasi-stationary states [35]. Let the (multistep) partial reaction X —• Y and its rate equation be X + 2Y

3Y,

rxY —

(14.4)

kxY^xCy

(a trace of Y must be present initially). With single steps Anet rates of A, X, and Y then are ^AxQ\ »

^AX^A ~ % Y ^ X ^ Y >

Xand Y-

''Y - % Y ^ X ^ Y ~ ^ y P ^ Y

• P, the

(14.5)

With the conditions of quasi-stationary behavior, ^x = 0 and ^y = 0, eqns 14.5 give Q

q = Cexp(-^^^r),

"• k

^

k

_

^AX ^ A

(14.6)

C '

'^AX'^XY^A

so that Cj^Cy

—

(14.7)

^YP'^XY

Also, from eqns 14.5 and with eqn 14.7 for CxC^: dr^

dr^ ^XY ^ Y »

xY

V

=

^v

Y

YP

~~

"vw

(14.8)

C„t

so that, with eqn 14.6 for Cy, dr^ =

^Y

^XY C v

^XY(^AX ^ A )

(14.9)

With the stability conditions 14.3 this can be seen to give A^YP

unstable if

C.

<

k

(14.10)

k'

A batch reaction may be stable at first, but start to oscillate when the concentration of the reactant has dipped below this critical limit. In a CSTR, oscillation occurs if the residence time is short enough for condition 14.10 to be met. [The unrealistic "cubic autocatalysis" form 14.4 was chosen here for simplicity's sake (a "quadratic" form X-HY—• 2Y does not produce instability). Gray and Scott [34] discuss this case in much greater detail. They include reverse steps as well as an uncatalyzed, parallel step X—• Y; the latter keeps the mathematics from

14.4,

Complex oscillations and chaos

455

derailing when, at high conversion, the constancy of C^CY becomes untenable, and lets the reaction become stable again once Q has dropped below another limit.] In a similar manner, oscillations may be caused by the interplay of a nonisothermal reaction with its heat release. This can happen in a reaction as simple as A ->

time [s]

K —• P

if the second step is highly exothermic whereas the first is not. If so, conversion K—• P is thermally self-accelerating, but fades out as it depletes K. This lets the reactor cool again and allows K to be replenished by A. The quasi-stationary state may be unstable. As in the previous example, this can be established by testing whether a small excursion escalates. However, the mathematics is more formidable as it now involves activation energies, heat generation, heat transfer, and heat capacity. A formal and very detailed analysis has been given by Gray, Scott, and coworkers [36-38].

time [s]

time [s]

Figure 14.3. Computed concentration and temperature histories for a nonisothermal batch reaction A ^ K -* P with highly exothermic second step (from P. Gray and S. K. Scott [39]).

14.4. Complex oscillations and chaos Even more exotic behavior has been found by those searching for it, but is exceedingly rare: Oscillations may be periodic but contain waves of different amplitudes, or be entirely aperiodic. Only a glimpse of what may happen can be given here. Complex periodic oscillations. For reasons only mathematical analysis can reveal, oscillations can become more complex than the normal pattern of a single wave repeating itself (1^ pattern). In general, the higher the frequency of a wave, the lower is its amplitude. The period may contain one high- and one low-amplitude wave (1^ pattern) or one highand several low-amplitude waves (T patterns). Also, such periods may alternate, or a regular 1^ pattern may alternate with a period of no oscillation at all ("bursting").

456

Chapter 14. Instability, periodic reactions, and chaos

In general, in a CSTR, the amplitude and period of an oscillation decrease if the residence is shortened. However, this decrease is not smooth. Typically, there are some preferred, relatively stable 1^ oscillations at wide residence-time windows, but in between the patterns may be composites as described above. A case in point is the BelousovZhabotinsky reaction [40-42]. Most of the observed wave forms and pattern have been successfully reproduced by computation with a modified oregonator [43]: A 4- Y <

• X + P

X + Y <

• 2P

A + X <

• 2W

C 4- W <

• X -h Z

2X <

• A -f P

Z

•

VifY + C

Chaos [G3,44]. The sequence of complex wave patterns may be aperiodic, that is, without identical repetitions. Such a "chaotic" reaction is unpredictable and seems to obey no rules. However, appearances are deceptive. As has often been pointed out, "chaos" in physical sciences is a misleading term, though too appealing to be changed: The behavior is not random. Chemical chaos is strictly deterministic. Given sharply defined starting conditions, the course the reaction will take can be exactly calculated. Moreover, the variables (concentrations, temperature, etc.) remain restricted to well-defined limits. What makes the behavior unpredictable and seemingly random is that minute differences in conditions—smaller than the error of any physical measurement, smaller than the margins of control of variables, smaller even than the round-off error of a computation—can lead to dramatically different results. This is butterfly effect of meteorology: that the bat of a butterfly's wing over Tokyo can spawn storms to come to New York City. As indicated above, a change in a variable—say, the residence time in a CSTR—can cause period doubling and further complications of the wave pattern. Chaos can be viewed as the ultimate result of such variations piling complications upon one another while ever shortening the span within which a new pattern persists. Two dependent mathematical variables are needed for a network to be capable of periodic behavior. A third is required to permit chaos. It has become quite a sport among the apostles of nonlinear chemical dynamics to invent ever new simple, if not exactly realistic networks that can admit chaos. A classical example, and one of the simplest, is the Hudson-Rossler model [45]. The core of the network is P A + B

• A

Q

• B

• 2B

B

• R

To make this quadratic-autocatalysis network capable of periodic behavior, the "step" B — • R is given a nonlinear rate equation —r^ = k^^C^I{C^-\-K) as might result from saturation kinetics (see Section 8.3.1). To allow chaos, another step producing an otherwise inert species C is added:

For the practical engineer it will rarely matter whether an unstable reaction oscillates or is chaotic. If it does, he may be better served playing with a numerical simulation of the reaction rather than trying his hand at a theoretical analysis.

Summary

457

Summary A reaction can evolve into a runaway or be periodic or chaotic. Such behavior requires feedback: A result of the reaction—a product or intermediate or the heat released—must affect the rate. This feedback must be destabilizing. Thermal feedback is the most common cause and is well documented in standard texts. The focus here is on feedback caused by the vagaries of reaction kinetics. Instability typically arises from the interaction of two phenomena with different dependences on a reaction parameter: In a nonisothermal reaction, the dependence on temperature is exponential for heat generation by the reaction, but linear for heat loss to the cooling coil or environment; in a reaction with chain branching, the dependence on radical population is exponential for acceleration by branching, but quadratic for chain termination. A reaction is unstable if acceleration outruns retardation. This can cause an explosion or, in a CSTR, lead to multiple steady states. Feinberg's network theory can help to assess whether an isothermal reaction admits multiple steady states in a CSTR. A reaction may be periodic if its network provides for restoration of a reactant or intermediate that has been depleted, while conversion of main reactants to products continues. Periodic behavior often results from competition of two or more contending mechanisms. Predator-prey fluctuations in ecology (Lotka-Volterra mechanisms) provide an easily visualized example. The Belousov-Zhabotinsky reaction—catalyzed oxidation of malonic acid by bromate—involves a similar competition between two pathways. Periodic behavior can also result from instability of quasi-stationary states in a reaction with intermediates. A quasi-stationary state is unstable if a small excursion from it escalates. The self-acceleration required for instability can arise from thermal feedback or from an "autocatalytic" step in the reaction pathway. Oscillations may involve periods with more than a single wave, and may be chaotic (with no recurring periods at all). Aperiodic behavior is called chaos, but is not random: The seeming randomness results from the fact that minutes difference in starting conditions can lead to drastically different behavior (butterfly effect of meteorology). Even relatively simple hypothetical networks can produce chaotic behavior. However, while periodic oscillation are possible in networks with only two independent mathematical variables, chaos requires at least three. Examples include multiple steady states in isothermal CSTRs, predator-prey fluctuations, the Belousov-Zhabotinsky reaction, and a test for stability of quasi-stationary states in reactions with a self-accelerating intermediate steps. References General references Gl. G2.

R. J. Field and M. Burger, eds.. Oscillations and traveling waves in chemical systems, Wiley, New York, 1985, ISBN 0471893846. P. Gray and S. K. Scott, Chemical oscillations and instabilities: non-linear chemical kinetics, Oxford University Press, 1990, ISBN 0198558643.

458

G3. G4. G5.

Chapter 14. Instability, periodic reactions, and chaos

S. K. Scott, Chemical chaos, Oxford University Press, 1993, ISBN 0198556586. S. K. Scott, Oscillations, waves, and chaos in chemical kinetics, Oxford University Press, 1994, 0198558449. I. R. Epstein and J. A. Pojman, An introduction to nonlinear chemical dynamics: oscillations, waves, patterns, and chaos, Oxford University Press, Oxford, 1998, ISBN 0195096703.

Specific references 1.

3.

C. G. Hill, h.,An introduction to chemical engineering kinetics & reactor design, Wiley, New York, 1977, ISBN 0471396095, Section 10.6. H. S. Fogler, Elements of chemical reaction engineering, Prentice-Hall, Englewood Cliffs, 3rd ed., 1999, ISBN 0-13-531708-8, Section 8.6. M. Feinberg, Chem. Eng, ScL, 42 (1987) 2229; 43 (1988) 1.

4.

M. Feinberg and P. Ellison, CHEMICAL REACTION NETWORK TOOLBOX. Versions

2.

5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

1.02 and 1.10a, available at http://www.che.eng.ohio-state.edu/-feinberg/CRNT, 1999. P. Schlosser and M. Feinberg, Chem. Eng. Sci., 49 (1994) 1749. P. Ellison, The advanced deficiency algorithm and its application to mechanism discrimination. Thesis, Dept. Chem. Eng., University of Rochester, 1998. P. Ellison and M. Feinberg, J. Mol. Catal., 154 (2000) 155, For independence in multistep networks, e.g., see Hill (ref. 1), Section 2.9. e.g., see M. Eigen and R. Winkler, Das Spiel: Naturgesetze steuem den Zufall, Piper, Munich, 1975, ISBN 3491021514, Section 6.4. A. J. Lotka, J. Am. Chem. Sac, 42 (1920) 1595; Elements of physical biology, Williams & Wilkins, Baltimore, 1925. V. Volterra, Mem. R. Acad. Naz. dei Licei,, Ser. VI, Vol. 2 (1926); Nature, 118 (1926) 558. Epstein and Pojman (ref. G5), Section 1.1. A. T. Winfree, J. Chem. Educ, 61 (1984) 661. S. E. Shnol', Heroes and Villains of Russian Science, Kron Press, Moscow, 1997 (in Russian). A. Pechenkin, Studies in history and philosophy of modem physics, 33 (2002) 269. B. P. Belousov, Sb. Ref Radiats. Med., Medgiz, Moscow, 1959, 145; English translation in Field and Burger (ref. Gl), Appendix. A. M. Zhabotinsky, Proc. Acad. Nauk SSSR, 159 (1964) 392; Biofizika, 9 (1964) 306. L. Gyorgy, T. Turanyi, and R. J. Field, J. Phys. Chem., 94 (1990) 7162. A. N. Zaikin and A. M. Zhabotinsky, Nature, 225 (1970) 225. R. J. Field and A. T. Winfree, J. Chem. Educ, 56 (1979) 754. W. Geiseler, J. Phys. Chem., 86 (1982) 4394.

References 22.

23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45.

459

R. J. Field, Experimental and mechanistic characterization of bromate-ion-driven chemical oscillations and traveling waves in closed systems, in Field and Burger (ref. Gl), Chapter 2. J. J. Tyson, A quantitative account of oscillations, bistability, and traveling waves in the Belousov-Zhabotinsky reaction, in Field and Burger (ref. Gl), Chapter 3. A. M. Zhabotinsky, F. Buchholtz, A. B. Kiyatkin, and I. R. Epstein, J. Phys. Chem,, 97(1993)7578. Gray and Scott (ref. G2), Section 14.5. R. Kapral and K. Showalter, AIP Conf Proc, 675 (2003) no. 1, 265. R. J. Field, E. Koros, and R. M. Noyes, J. Am. Chem. Soc, 94 (1972) 1877. Gray and Scott (ref. 1), Section 14.1. M. J. Pilling and P. W. Seakins, Reaction kinetics, Oxford University Press, 1995, ISBN 0198555288, Section 11.5. R. J. Field and H. D. Forsterling, /. Phys. Chem., 90 (1086) 5400. S.-H. Yau, Mechanistic studies on the cerium-catalyzed Belousov-Zhabotinsky reaction, thesis, University of Marburg, 2001. R. J. Field and R. M. Noyes, J. Chem. Phys., 60 (1974) 1877. W. C. Troy, Mathematical analysis of the oregonator model of the BelousovZhabotinsky reaction, in Field and Burger (ref. Gl), Chapter 4. Gray and Scott (ref. G2), Chapters 2 and 3. Gray and Scott (ref. G2), Section 2.6. P. Gray, S. R. Kay, and S. K. Scott, Proc. Roy. Soc, A 416 (1988), 321. B. F. Gray and M. J. Roberts, Proc. Roy. Soc. A416 (1988) 391. Gray and Scott (ref. G2), Chapters 4 and 5. Gray and Scott (ref. G2), Section 4.3. J. L. Hudson, M. Hart, and D. Marinko, J. Chem. Phys., 71 (1979) 1601. J. Maselko and H. L. Swinney, Phys. Lett., A119 (1987) 403. Gray and Scott (ref. G2), Section 4.3. K. Showalter, R. M. Noyes, and K. Bar-Eli, J. Chem. Phys., 69 (1978) 2514. Gray and Scott (ref. G2), Chapter 13. J. L. Hudson and O. E. Rossler, Chaos in simple three- and four-variable systems, in Modeling of patterns in space and time (Lecture notes in biomathematics, vol. 55), W. Jager and J. D. Murray, eds., Springer, Berlin, 1984; see also Gray and Scott (ref. G2), Section 13.4.

Appendix: Software and Data Bases I: Software for Numerical

Simulation

Acuchem

For reactions in isothermal, constant-volume batch reactor. reaction steps of mechanism Input: rate coefficients criteria for integration output: concentration histories of participants Companion program Acuplot displays results on screen Hardware: IBM-PC and compatibles with math processor NIST Developer W. Braun et al., J. Internat. Chem. Kinetics, Literature: 20 (1988) 22. Available from NISI’, Gaithersburg, MD 20899

CHEMKN

For batch, plug flow, and CSTR. Includes gas-phase isothermal, nonisothermal, and nonisobaric reactions, heterogeneous catalysis, and thermochemical database for calculation of equilibrium constants. Many subprograms for special situations (shock waves, flames, partially stirred reactors, etc.) are available. reaction steps Input: rate coefficients activation energies output: depends on reactor type selected essentially any PC (written in FORTRAN) Hardware: Sandia National Laboratory Developer: R. J. Key et al., Chemkin Collection, Release Literature: 3.7.1, Reaction Design Inc., San Diego, 2003 Available as commercial software package from Reaction Design, 6440 LuskBlvd. SanDiego, CA 94121, www.ReactionDesign.com.

Version 3.7. 2003

CHEMICAL REACTION NETWORK TOOLBOX For multistep isothermal batch and flow reactors (easy to use). reaction steps Input: rate coefficients stop time number of iterations choice of network, differential equations, output: concentration histories Hardware: IBM-compatible PSs (uses WINDOWS) Developed: M. Feinberg, Ohio State University Literature: M. Feinberg, Chem. Eng. Sci., 42 (1987) 2229; 43 (1988) 1 Availableoninternetwww.che.eng.ohio-state.edu/ -feinberg/CRNT

Version l.lOa, 1995

462

II:

Appendix:

Software and data bases

Software for Statistics

SYSTAT

Commercial computer program for regression analysis Hardware: essentailly any PC Systat Inc. Developer: Available from Systat Inc., Evanston IL

EXCEL 2000

Spreadsheet program for regression analysis Hardware: essentailly any PC Literature: B. S. Gottfried, Spreadsheet tools fo engineers, McGraw Hill, New York, 2000, Solver program

See also:

R. D. Cook and S. Weisberg, Applied regression including computing and graphics, Wiley, New York, 1999 S. Chatterjee et al., Regression analysis by example, Wiley, New York, 3rd ed., 2000

III:

Data bases

Reaction kinetics

Reaction rates in Liquids, Landolt-Bornstein, New Series, Part II, Vol 13 (5 subvolumes), 1983-85; Vol 18 (5 sub-volumes), 1994-97 D. L. Baulch et al., J. Phys. Chem. Ref. Data, 21 (1992) 411; 23 (1994) 847 NIST Chemical Kinetics Data Base (1998) Windows Version 2Q98 E. T. Denisov and T. Denisova, Handbook of antioxidants: Bond dissociation energies, rate coeflcients, activation energies, and enthalpies of reactions, CRC Press, Boca Raton, 2nd ed., 2000

Glossary of Symbols Quantities a A Cj © Dj Djk

(normalized) catalyst activity [eqn 9.27] dimensionless pre-exponential factor in Arrhenius equation 1.3 same as rate coefficient concentration of species i or intermediate Xj mol V ^ denominator in Christiansen rate equation [eqn 8.30] depends on network effective diffusion coefficient of species i in catalyst particle l^t~^ denominator in rate eqn for simple pathway or segment j ^^—•k [eqn 6.6] depends on pathway Djj sum of terms of row j4-1 in Christiansen matrix [Section 8.4] depends on catalytic cycle e = 2.71828 (basis of natural logarithms) — Ea molar activation energy kJmol"^ / effectiveness factor of free radicals for initiation [eqn 11.22] — ^ fractional conversion of reactant i [eqn 1.4] — Fj molar flow rate of species i mol t~^ (AGf°)i standard Gibbs free energy of formation of species i kJmol"^ AH° standard enthalpy change of reaction kJmol"^ J, flux of species i [eqn 1.12] mol l~^t~^ k rate coefficient (see subscripts for special rates) depends on rate eqn k^pp apparent depends o n rate eqn k^, A:b, ... phenomenological depends o n rate eqn k^ of disproportionation mo\~^Vt~^ k, of adsorption of species i atm~^ t~^ or mol wt^aT^ r ^ /:_i of desorption of species i r"^ or wt^at V~^ t~^ ^ij of step i —• j depends o n molecularity k^^ mass-transfer coefficient [eqn 1.13] lt~^ k number of forward steps in simple pathway or catalytic cycle — K equilibrium constant (see subscripts for special reactions) depends o n stoichiometry K of overall reaction in heterogeneous catalysis [eqn 9.5] depends o n stoichiometry jfiTcat of dissociation of complex catalyst mol V~ ^ K^ of acid or base dissociation mol V~^ or atm"^ ^i of adsorption of i atm~^ Ky^ of reaction i^^—^j depends on stoichiometry / length / ifjj loop coefficient of reduced loop segment i—>] [eqn 6.15] t~^ m, n reaction orders — MW molecular weight — MW number-average molecular weight (monomer included in mole count) -

464 n

Glossary of symbols number of sites in catalytic reaction [eqn 9.7]

—

Hj stoichiometric n u m b e r of species i — N n u m b e r - a v e r a g e degree of polymerization ( m o n o m e r included in mole count) — (N)poi ( m o n o m e r not included in mole count) -

A^i p, P P P, q^ r To r r^ r, rj, Tj R s s^ S 5i 5i / ti,2 fst r^, ty T y V wtcat Wj

number of moles of species i or Xj mol partial pressure of species i atm total pressure atm probability — polymer with i structural units concentration of species i on catalyst distance from center of catalyst particle / radius of (spherical) catalyst particle / rate (see subscripts for special rates) mol V~^ t~^ or mol wt^aT^ t~^ of catalyst deactivation [eqn 9.28] t~^ net, of formation of species i or X-, mol V~^ t~^ or mol wt^at"^ t~^ forward and reverse formation rates of species i mol V'^ t~^ gas constant kJ mol"^ deg~^ standard deviation — variance — catalyst surface area [eqn 9.24] /^ instantaneous selectivity to product i [eqn 1.9] mol/mol cumulative selectivity to product i [eqn 1.8] mol/mol time t half-time of reaction / Student's / time required to reduce amount of limiting reactant to fraction x ov y t absolute temperature deg K volume, reactor volume V volumetric flow rate V r~^ weight of heterogeneous catalyst weight fraction of polymer with j monomer units (monomer included in mole count) (Wj)po, (monomer not included in mole count) — X fractional distance from equilibrium [eqns 5.3, 5.7] — Xj mole fraction of monomer i in copolymerization monomer mixture — Xj mole fraction of polymer with j monomer units (monomer included in mole count) — (Xj)po, (monomer not included in mole count) — ji yield of product i [eqn 1.5] mol/mol y, mole fraction of unit i in copolymer [eqn 11.96] — l^j yield ratio, instantaneous, of products i and j [eqn 1.7] mol/mol Fjj cumulative, of products i and j [eqn 1.6] mol/mol Are, relative error — t] catalyst effectiveness factor [eqn 9.26] dimensionless

Glossary of

d

Pa. Pb 7

Thiele modulus [eqn 9.26] dimensionless transfer constant of ... [eqn 11.30] pseudo-first order rate coefficient of step j — • k [eqns 6.1, 6.2] t~^ segment coefficient of pathway or linear network segment i—•] [eqns 6.5] r^ radical chain length in chain reactions [eqn 11.33] stoichiometric coefficient of species i (negative for reactants) reactivity ratios in chain copolymerization [eqns 11.95] true variance [eqn 3.36] — = V/y°, reactor space time [eqn 3.3] t

Subscripts c... ch... init

465

symbols

(for rates, equilibria, etc.)

coupling of ... chain transfer to initiation

ISO isomerizaUon p propagation pl,p2,... propagation step 1,2,

rx trm t...

surface reaction termination termination by ..

M mM s

mole per liter millimole per liter second

Superscripts bulk fluid phase initial, at entry

s 00

catalyst surface at equilibrium

Units atm h kJ K

atmosphere hour kilojoule degree Kelvin

L min mol

liter minute mole

Species ale aid A, B, ... AIBN cat, cat' F, FA, FQ he in inh K,L,Ki, ... lacs lapc L macs masi M M

alcohol aldehyde reactants azo-^/^-isobutyronitrile catalyst functional groups in step-growth polymerization heavy ends initiator inhibitor reaction intermediates low-abundance catalyst-containing species low-abundance propagating center in ionic polymerization ligand most abundant catalyst-containing species most abundant surface intermediate in heterogeneous catalysis collision partner (or wall of reaction vessel) monomer in polymerization

466 MA,MB MA*,MB*

Me ole par poi P, Q, ... P~ Pi Ph R• S Tr X, Xj XL Y Ecat U E° DP • UP" O ®

Glossary of symbols monomers in copolymerization end groups in copolymerization metal olefin paraffin catalyst poison reaction products anionic propagation center in anionic polymerization polymer with i monomer units organic phosphine* radical solvent or other molecule in chain reaction transfer agent in chain reaction trace-level intermediate, chain carrier in chain reactions lumped species [Section 8.5.3] chain carrier in chain reaction total catalyst material (including amounts bound as intermediates) [eqn 8.16] total catalytic sites total catalyst material in cycle, excluding external pathways total of all radicals in chain-growth polymerization total of all propagating centers in anionic polymerization catalytic site [network 9.2] catalytic site occupied by species i [network 9.2]

* With apologies to the organic chemist who reads Ph as phenyl, this abbreviation is used because many industrially important phosphines are cyclic and do not correspond to the formula PR3.

Author Index Abel, E. 120 {11) Acrivos, A. 87f, 89 {13) Affrosman, S. 296 {41) Ahn, D. J. 33 {23) Albright, R. L. 295 {31) Alemdaroglu, N. H. 141 (75,76) Alfrey, T., Jr. 388 (725) AUport, D. C. 373 {69) Altman, C. 143 {20)\ 227f {39) Altschul, R. 363 {40) Amundson, N. R. 292 {25) Anderson, M. J. 405 (4) Anthony, R. G. 17, 36 {G6), 33 {29)\ 347, 397 {G4) Aris, R. 87f, 89 {14)\ 292 {22,23)\ All {29,30,31) Arlman, E. J. 383 (777) Arndt, W. ixf Arrhenius, S. 11 (5) Astarita, G. 429, 438 (7) Atkins, P. W. 425f B Back, M. H. 327, 328 {36) Badilla-Ohlbaum, R. 435 (75) Baekeland, L. H. 350 {4) Bailey, W. J. 374 {75) Baldwin, R. R. 335 {73) Baltanas, M. A. 422 {36) Balthasar, W. B. 421 {24) Bar-Eli, K. 456 {43) Barlie, G. C. 298 {55) Barroe, P. E. 295, 296 {37) Bartholomew, C. H. 341 {G4) Bartlett, P. D. 334 {68)\ 363 {40) Bashford, C. L. 50 {22) Bashkirov, A. N. 122 (76) Baulch, D. L. 155 {26,27)\ 196 (75,74); 334 (69,70); 462 Baumgartner, H. J. 218 {20) Becker, E. D. 50 (77) Bell, R. P. 215 (72) Belousov, B. P. 452f (76)

Benson, S. W. 95, 107, 131 (G7), 120 (75), 123 {20)\ 330 {46), 341 {01) Berezin, I. V. 333 {64) Bernasconi, C. F. 50 (76) Beylen, M. van 373 {68) Bhore, N. A. 12 {4)\ 165f, 166 {3) Billig, E. 140 (72); 218 (79) Bischoff, K. B. 12 {4)\ 17, 36 {G4), 20 (9); 128 {26)\ 165f, 166 {3)\ 265 (70); 273, 305 (G5), 290 (75), 292 {18), 299 (67), 302 {68,69)', 321 (79), 325 {29), 341 {G5)', 420 (27); 429 (6) Biswas, M. 381 {95) Blaine, S. 40 (7) Blanchard, H. S. 339 {85) Blanding, F. H. 112 (5); 421 {22) Blaschke, F. 361 {37) Bock, W. 373 {64) Bodenstein, M. 2; 86 (7), 87f (72); 310, 316(7), 311, 316, 320(5), 316(5) Bolton, J. R. 51 {39) Bonchev, D. 133f, 158 (2) Bor, G. 219 {23) Bory, B. H. 118 (70) Boudart, M. 17, 35 (G7); 128 {25)', 230f {42)', 280 (5), 285 {8); 318 (77), 341 {G2) Boutevin, B. 364 (57) van Boven, M. 141 (75) Bowen, J. R. 87f, 89 (75) Bowry, V. W. 338 {79) Box, G. E. P. 65 {48)', 405, 416 (2) Bradley, J. N. 50 (6) Braun, W. 407 {8), 461 Breil, H. 383 (775) Breslow, D. S. 140, 141 (70,77); 204 {42) Brezonik, P. L. 229 {41) Briggs, G. E. 223 {30) Broich, F. 122 (77) Brown, C. K. 201 {30) Brumbaugh, J. S. 392 (755) Brunauer, S. 32 (75), 33 {28) Bryant, D. R. 140 (72); 218 (79)

468

Author index

Buchholtz, F. 452 (24) Burger, M. 445, 457 (Gl) Burk, D. 225 (31) Buxton, G. V. 331, 334(57) Bykov, V. I. 133f, 158 (7); 335 (77), 342 (G79) Caldin, E. F. 48 (2) Calvert, J. G. 331 (54) Calvert, R. B. 205 (47) Carberry, J. J. 95, 131 (G2); 135f (5); 273, 305 (01), 288 (10), 290 (77), 297 (48), 299 (59); 429 (2), 438 (16) Carothers, W. H. 348 (2), 355 (18); All (34) Chance, B. 48 ( ^ Chang, C.-H. 33 (23) Chapman, D. L. 87f (77) Chatterjee, S. 65 (57); 416 (14); 462 Chen, N.-H. 95, 131 (03); 419 (18) Chen, N.-Y. 297 (47,48), 299 (58); 441 (19) Chen, S. S. 338 (82) Cheradame, H. 381 (87) Chern, J.-M. 135, 143, 149, 151 (9); 244 (48), 253f (64) Chmelir 381 (86) Christiansen, J. A. 2; 20 (7); 135f, 143 (3); 111 (36-38); 316 (6); 433, 435 (75) Chu, S-Y. 201 (24) Churchill, S. W. lOf (7); 39, 73 (01) Clark, J. E. 373 (72) Clary, D. 124 (22); 266 (76) Clausius, R. 2 Clippinger, E. 374, 382 (78) Clymans, P. J. 422 (35) Cobos, C. J. 155 (26,27); 196 (13,14); 334(69,70) Coenen, J. W. E. 201 (27) Collman, J. P. 141 (17); 200 (23), 201 (25,57); 217, 269 (Gl), 218 (22), 246 (52,59); 383, 385 (108) Conklin, C. W. 122 (18) Connors, K. A. 39, 73 (02), 50 (17), 65 (57); 95, 131 (04); 214 (77), 269 (02) Cook, R. D. 65 (50); 416 (75); 462

Cooper, A. R. 429, 438 (5) Cordes, E. H. 211 (5) Corner, T. 364 (49) Cossee, P. 383 (116,117) Cotton, F. A. 205 (45) Cottrell, T. L. 199 (79) Coulson, C. A. 201 (34) Cox, R. A. 155 (26,27); 196 (75,74); 334 (69,70) Crank, J. 302 (77) Crivello, J. V. 381 (97) de Croon, M. H. J. M. 201 (27) Csicsery, S. M. 297 (50) D Damkohler, G. 291, 293 (16), 293 (27) van Damme, P. S. 421 (24) Danckwerts, P. V. 429, 438 (4) Daniel, C. 246 (58) Datta, S. C. 266 (73) Davies, C. W. 296 (42,44) Day, J. N. E. 266(73,74) DeCoursey, W. J. 65 (52) Degnan, T. F., Jr. 297 (47) DeLassus, P. T. 388 (725) DeLury, D. B. 120 (75) Deming, L. S. 32 (18) Deming, W. E. 32 (18) Denbigh, K. G. 17, 35 (02), 26 (77), 27 (75,76) Denisov, E. T. 196 (16); 332, 334 (60), 333 (64), 338 (84), 341 (03); 462 Denisova, T. 196 (16); 332, 334 (60), 338 (84), 341 (03); 462 Deyrup, A. J. 214 (7) Dimitrov, V. I. 163 (7) Dierickx, J. L. 422 (37) Dixit, S. S. 374 (76) de Donder, T. 421 (28) Dotzlaw, G. 51 (28) Draper, N. R. 65 (49); 416 (72) Druliner, J. D. 197 (77); 236 (43), 236, 237, 239, 262 (44); 333 (67) Dubin, P. L. 352 (10) Dumez, F. J. 302 (68) Dunn, D. J. 382 (102) Dwyer, F. G. 297 (48)

Author index

Eadie, G. S. 225 (33) Eastmond, G. C. 347, 397 (Gi), 361 (34), 364 (52), 391 (131) Edward, J. T. 213 (6) Eichinger, B. E. 356 (26) Eigen, M. 50 (5,8); 451 (9) Ellison, P. 448, 449f (^,6,7) Elokhin, V. I. 133f, 158 (7); 335 (77), 342 (G19) Emanuer, N. M. 98 (i); 332, 334 (67), 333 (64) Emmett, P. H. 33 (28) Epstein I. R. 445, 458 (G5), 452 (12,24) Escher, M. C. 28 Espenson, J. H. 39, 73 (G5); 95, 131 (G5) Esser, C. 155 (26); 196 (ii); 334 (69) Evans, G. O. 296 (38) Evans, M. G. 318 (9) Evering, B. L. 325 (23) Eyring, H. 20 (4-7) Fainberg, A. H. 374, 382 (78) Falbe, J. 141 (14); 218 (18), 269 (GJ) Faller, J. W. 51 (33,40); 202, 205 (38), 205(46); 423f (59) Farrauto, R. J. 341 (G4) Faust, R. 382 (105) Fehervari, A. 382 (105) Feinberg, M. 448, 449 (3,4,5,7); 461 Feinstein, K. 50 (23) Fekete, L. 219 (23) Fetterly, L. C. 122 (18) Field, R. J. 445, 457 (01), 452 (18,20, 22,27), 453 (27,30,32) Finke, R. G. 141 (17); 200 (23), 201 (25,37); 217, 269 (G7), 218 (22), 246 (52,59); 383, 385 (708) Finn, M. G. 255 (65) Fisher, R. A. 66 (55) Fleming, G. R. 50 (73) Fleming, I. 50 (26) Flory, P. J. 120 (14); 347, 397 (G2), 347 (7), 352 (8), 354 (75), 356 (20,27); 357 (27,28), 358 (27,30), 372 (28,63)

469

Forsterling, H. D. 453 (30) Fogler, H. S. 17, 35 (G3); 39, 73 (G4); 273, 305 (G2), 290 (72), 299 (60); 429 (5); 446 (2) Fowler, R. H. 33 (27) Francisco, J. S. 95, 131 (G9); 265 (69), 270 (G72); 320f (16), 335 (76), 342 (G18) Franckaerts, J. 286 (9) Frank, P. 155 (26,27); 196 (13,14); 334 (69,70) Franklin, J. L. 127f (24) Franses. E. J. 33 (23) Freitas, E. R. 385 (720) Frenklach, M. 124 (22); 266 (76) Freundlich, H. M. F. 32 (79) Frilette, V. J. 298 (52) Frizzell, D. H. 155 (28); 196 (75) Froment, G. F. 17, 36 (G4), 20 (9); 128 (26); 265 (70); 273, 305 (G3), 286 (9), 290 (73), 292 (18), 299 (60), 302 (68, 69); 321 (79), 324 (20), 325 (29), 331 (49), 341 (G5); 421 (2^; 422 (35,36, 37), 423 (38); 429 (6) Frost. A. A. 82 (3); 135f (^; 201 ( 7 ^ , 266 (75) 269 (G^; 331 (56), 341 (G6) Fu, X. Y. 246 (58) Gandini, A. 381 (87), 382 (707) Garcia-Ochoa, F. 169 (6) Garnett, J. L. 381 (92) Garwood, W . E. 297 (48), 299 (58); 419 (i5) Gates, B. C. 210 (3); 217, 269 (G5), 246 (53), 249 (67); 295, 296 (37); 333 (63), 341 (G7) Gavalas, G. R. 312 (4) Gaydon, A. G. 50 (7) Gaylord, N. G. 374 (76) Geiseler, W. 452 (27) George, W. O. 50 (24) Gibbons, D. 51 (29) Gibbs, D. B. 388 (723) Gibbs, J. W. 2 Gijbels, R. 51 (37) Gilbert, R. D. 392 (136)

470

Author index

Gilderson, P. W. 122 {18) Gillespie, R. J. 82 (7) Givens, E. N. 298 (55) Glasstone, S. 20 (6) Goddard, R. 386 (722) Gobel, T. 255 {66) V. Goethe, J. W. ix Goldfarb, I. J. 198 {18) Goldfinger, G. 388 (725) Goldfinger, P. 330 {44) Goodwin, R. D. 303 {72) Gorban', A. N. 133f, 158 (7); 335 {77), 342 (G79) Goring, R. L. 293 {28) Gorring, R. L. 441 {20) Gottfried, B. S. 65 (55); 416 (75); 462 Grasemann, H. 122 (77) Gray, B. F. 455 {37) Gray, P. 335 {74), 445, 457 {G2), 452 {25), 453 {28,34), 454 {35), 455 {36,38,39), 456 {42,44,45) Graydon, W. F. 435 (75) Green, M. L. H. 383 {118,119) Gross, B. 421 {27) Gruver, J. T. 331 {54) Guggenheim, E. A. 33 {22) Gum, C. R. 385 {120) Gumbs, R. W. 392 (7576) Gupte, A. A. 236f Gyorgy, L. 454 (75) H Haag, W. O. 298 {56) Haber, F. 311 (2) Hadi, A. S. 65 (57); 416 {14) Haldane, J. B. S. 223 {30) Hall, T. L. 205 {49) Halpern, J. 201 {28)', 246 {54-56)', 248 (55) Halsey, G. D. 33 {24) Hammes, G. G. 225, 226 {35) Hammett, L. P. 214 {7,10), 269 {G6) Hampson, R. F. 155 {28)', 196 (75) Han, J. 246 {58) Hanes, C. S. 225 {32) Harborth, G. 365 {55) Harris, D. A. 50 {22)

Harris, R. K. 51 {32) Hart, M. 456 {40) Hartley, F. R. 295 {36) Hartridge, H. 48 (5) Hase, W. L. 95, 131 {G9y, 265 {69), 270 (G72); 320f (76), 335 {76), 342 (G78) Hayman, G. 155 {27)', 196 {14)', 334 {70) Heck, R. F. 140, 141 {10,11)', 204 {42) Hegedus, L. S. 141 (77); 200 {23), 201 (25,57); 217, 269 (G7), 218 {22), 246 {52,59)', 383, 385 {108) Heinemann, H. 298 {54) Heitz, W. 364 (50) Helfferich, F. G. 11 (2); 33 {26)', 103, 114 (2), 117 (7), 122 {18)', 135 {6,8,9), 143, 149, 151 (9), 145, 148, 153 {8), 157 {30)', 175-179, 187, 190, 191, 195 (7); 178, 179, 187, 190 (9); 216, 217 (75), 240 {47)', 295 {30,33), 296 {30,43, 45), 297 (46), 303 {70)', 349 (5); 404 (7), 408, 413 (77), 417, 418 (77); 434, 437, 440 {14) Helman, W. P. 331, 334 (57) Hendrix, C D . 405 (5) Henri, V. 221 {29) Herron, J. T. 155 {28)', 196 (75); 407 {8) Herzfeld, K. F. 316 (7), 325, 326 {27) Heublein, G. 382 {97) Hicks, J. S. 293, 294 {27) Hiemenz, P. C. 347, 397 {G3), 356 {24), 378 {82) Higginson, W. C. E. 374, 379 {74) Hill, C. G., Jr. 17, 36 {G5), 26 {12), 27 (77); 39, 73 {G5), 57 {41), 59 {43,45), 60 {46), 53 (57); 95, 131 {G6), 116 (5); 265 {67); 273, 306 {G4), 281 (7), 290 (7^, 292 (79), 299 {62), 303 {73); 329 (47); 429 (7); 446 (7), 448 {8) Hillewaert, L. P. 422 {37) Hinshelwood, C. N. 2; 20 (5); 273 (2) Ho, T. C. 421 (57) Hoare, D. E. 339, 340 {86) Hofeditz, W. 325 {25) van't Hoff, J. H. 22 (70) Hoffman, H. L. 295 {35) Hoffmann, R. 86 (9); 201 {24,32,33,35), 202 {32,35,37)

Author index Hofstee, B. H. J. 225 (34) Holland, C. D. 17, 36 (G6), 33 (29); 347, 397 (G4) Holzkamp, E. 383 (113) Holzworth, J. F. 50 (9) Hoste, J. 51 (37) Hougen, O. A. 273, 306 (G5), 273, 276 (i), 278, 287 (4) Howell, B. A. 388 (123) Hsieh, H. L. 373 (71) Hudson, J. L. 456 (40,45) Hughes, E. D. 82 (7) Huie, R. E. 331, 334 (57) Hunter, J. S. 65 (48); 405, 416 (2) Hunter, W. G. 65 (48); 405, 416 (2) Hurle, I. R. 50 (7) Hurwitz, H. 288 (10) Hwang, Y.-L. 295 (33) I Ingold, C. K. 82 (7,2); 215, 216 (13), 266 (73,74), 269 (G7) Ingold, K. U. 338 (79) Inskeep, G. E. 374 (75) Ittel, S. D. 141 (18), 154 (27); 217, 270 (GIO), 236 (45); 333 (65,67), 341 (G77); 383, 385 (770 Ivin, K. J. 383 (118) Iyer, S. D. 167, 168 (5) J Jackman, L. M. 205 (45) Jacob, S. M. 421 (27) Jacobsen, E. N. 255 (65) Jagur-Grodzinski, J. 392 (137) Jardine, F. H. 246 (49,50) Jeffreys, G. V. 429, 438 (3) Jenkins, R. 51 (36) Jennings, W. 50 (18) Jefabek, K. 296 (39) Jesson, J. P. 237 (46), 246 (57) Johnston, H. S. 318 (75) Johnston, W. R. 325 (23) Jones, A. C. 205 (44); 219 (24) Jost, W. 320 (17) Jung, G. 320 (17) Just, T. 155 (26,27); 196 (13,14); 334 (69,70)

471

K Kaeding, W. W. 298 (55) Kahaner, D. K. 407 (8) Kamannarayana, P. 381 (95) Kapral, R. 452 (26) Karo, W. 347, 398 (GIO), 350 (6), 359 (31), 363 (42), 373 (73), 383 (777) Kastrup, R. V. 50 (27); 220 (26) Kaszas, G. 382 (106) Kay, S. R. 455 (36) Kee, R. J. 407 (10) Keim, W. 385, 386 (727), 386 (722) Kelen, T. 382 (106) Kemball, C. 33 (22) Kennedy, J. P. 347, 397 (G5), 362f (38), 381 (84,94), 381f (85), 382 (98,99,105107) Kerr, J. A. 155 (26,27); 196 (7i,7^; 334 (69,70) Key, R. J. 461 Kienle, R. H. 356 (25) Kiricsi, I. 297 (50) King, E. L. 143 (20); 227f (39) Kissin, Y. V. 383 (772) Kiyatkin, A. B. 452 (24) Klein, M. T. 12 (4); 165f (2,3), 152 (3), 167, 168 (5) Knopf, P. W. 350 (7) Knorre, D. G. 98 (7) Kochavi, D. 225, 226 (35) Kochi, J. K. 205 (48) Kohle. H. 365 (56) Koros, E. 452 (27) Koetitz, K. F. 122 (18) Koga, N. 246 (58) Konaka, R. 205 (50) Kondratiev, V. N. 332, 334, 340 (59), 335 (72), 341 (G8) Koshida, K. 122 (19) Kowaldt, F. H. 386 (126) Krambeck, F. J. 421 (32), 422 (33) Kresge, E. 380, 388 (83) Kritsman, V. A. 332, 334 (67) Kronstadt, M. 352 (10) Kriiger, C. 386 (722) Krusic, F. J. 333 (67) Kucera, M. 366 (59) Kuchler, L. 326, 327, 328 (36)

472

Author index

Kuczynski, G. C. 302 {65) Kuipers, H. J. A. M. 201 {27) Kuo, J. 420 (79) Kurata, N. 122 (79) Lagally, P. 365 {56) Lai, T. W. 392 {1343) Laidler, K. J. 17, 36 {G7), 20 (6); 39, 50, 73 {G6)\ 86 {8)\ 326 (57,52), 327 (57), 328 {39), 330 (^5), 341 {G9)\ 352 (9) Lambie, D. A. 51 {29) Langmuir, I 32 (2^,27), 32, 33 (27); 34 (20); 273 (7) L'Aiinunziata, M. F. 51 {30) Lappert, M. F. 205 {49) Leathart, D. A. 330 {45) Lednor, W. P. 205 {49) Lehmann, H. L. 320 (75) Leonhardt, A. 296 {40) Letort, M. 329 {43), 330 {44), 331 {43,53) Leung, Y.-K. 356 {26) Levason, W. 219 {25) Levenspiel, O 17, 36 (G8); 39, 73 {G7), 59 {42,44)', 95, 100, 131 {G7)\ 269 (G8); 273, 306 (G6), 290 (75), 292 {20), 299 {63); 329 (^2); 429 (8,9), 438 (9,77) Levy, M. 373, 374 {66) Lewis, F. M. 388 (726) Lin, M. C. 327, 328 {37) Lin, S. H. 20 (7) Lin, S. M. 20 (7) Lind, S. C. 316 (5) Lindemann, F. A. 2; 20 (2); 327 {38) Lindner, D. L. 246 {57) Lineweaver, H. 225 (57) Logsdon, J. E. 118 (9) Long, F. A. 211 {4) Lotka, A. J. 452 {10) Levering, E. G. 352 (9) Lucki, S. J. 441 (79) Liitkemeyer, H. 311, 316, 320 (5) Luss, D. 292 {25) Lynn, J. L., Jr. 118 {10)

M Maatman, W. R. 298 (57) Mahler, J. M. 141 (79) Mahtab, J. 383 {118) Mallard, W. G. 155 {28)', 196 (75); 331, 334 (57) Mann, B. E. 51 {32) Marcandalli, B. 50 (9) Marechal, E. 347, 397 {G5), 362f (55), 381f (55), 381 {94), 382 {98,99) Marek, M. 381 {86) Marinko, D. 458 {40) Marko, L 255 {65) Marko, L. 188 {lOy, 219 {23) Martin, A. R. 141 (75) Martin, H. 383 (775) Marvel, C. S. 374 {75) Maselko, J. 456 {41) Mathieson, A. R. 382 {96) Matsen, F. A. 127f {24) Matsumoto, M. 201 {26) Matyjaszewski, K. A. 382 {104) de Mayer, L. 50 {8) Mayo, F. R. 388 (726), 390 {128) McAuliffe, C. A. 219 {25) McCoy, J. J. 218 (27) Mclntyre, P. S. 50 {24) McKinney, R. J. 236, 237, 239, 262 {44)', 407(6) McNair, H. M. 50 (79) McWeeny, R. 201 {34) Meacock, S. C. R. 213 (6) Meakin, P. Z. 246 {57) Melander, L. 203 {40) Menten, M. L. 220-222 {27) Merola, J. S. 50 {27)', 220 {26) Michaelis, L. 2; 220-222 {27) Milkovich, R. 373, 374 {66) Millen, D. J. 82 (7) Miller, J. A. 407 (70) Miller, J. M. 50 (79) Mims, C. A. 265 {72)', 352 (72) Minoura, Y. 361 {36) Missen, R. W. 265 {72)', 352 (72) Mittlefehldt, E. 50 (75)

Author index Moad, G. 338 (87); 347, 359f, 397 {G6), 360 (33), 361 (55), 363 (41), 364 (43), 365 (54) Mohedas, S. R. 422 (36) Moore, J. W. 17, 36 (G9), 20 (8); 85 (6); 95, 100, 131 (GS); 210 (2), 265, 266 (68), 270 (09); 318 (i2), 338, 339 (78), 341 (Gi(?); 353 (13) Morgenstem, C. ix Morton, M. 347, 397 (G7), 373 (70), 378 (81) Morukuma, K. 246 (58) Mosbach, K. 296 (40) Mower, E. B. 298 (57); 441 (79) Muhammadi, N. A. 248 (60) Mullineaux, R. D. 218, 219 (77) Murrells, T. 155 (27); 196 (14)-, 334 (70) Murray, J. P. 296 (41) Myers, P. S. 165f (^ N Nace, D. M. 421 (23) Nagy-Magos, Z. 219 (23) Nappa, M. J. 333 (67) Natta, G. 383 (114,115) Nakatsu, K. 201 (26) Neta, P. 331, 334(57) Neuberg, H. J. 435 (75) Newman, S. F. 122 (78) Niclause, M. 330 (44) van Nisselrooij, P. F. M. T. 201 (27) Nitsche, J. M. 441 (27) Norrish, R. G. W. 50 (10) Norton, J. R. 141 (77); 200 (22), 201 (25,57); 217, 269 (G7), 218 (22), 246 (52,59); 383, 385 (708) Noyes, R. M. 452 (27); 453 (32), 456 (43) Nycander, B. 118, 110(8) O Obi, B. E. 388 (725) O'Connor, C. 246 (57) Odian, G. 338 (80); 347, 397 (G8), 352 (77), 354 (16), 364 (44,46), 365 (58), 366 (60), 367 (67), 383 (109), 388 (72^, 391 (752)

473

O'DriskoU, K. F. 373 (72) Okamoto, T. 246, 248 (55) Olah, G. A. 214 (8) Olaj, O. F. 381 (93) Oldenburg, C. C. 281 (6) Olson, D. H. 298 (56) Oltay, E. 141 (16) Oppenheim, A. K. 87f, 89 (75) Orchin, M. 198 (78); 218 (27) Osborn, J. A. 246 (49,50) Oswald, A. A. 50 (27); 220 (26) Otsuka, S. 201 (26) Owen, B. D. R. 296 (42 ) Paneth, F. A. 325 (25) Parr, C. 318 (75) Parshall, G. W. 141 (78), 154 (27); 217, 270 (G70), 236 (45); 333 (65,66), 341 (G77); 383, 385 (770) Paul, M. A. 211 (4) Pearson, R. G. 17, 36 (G9), 20 (8); 82 (5), 85 (6); 95, 100, 131 (G8); 135f (^; 210 (2), 215 (14), 265, 266 (68), 266 (75), 269 (G4), 270 (G9); 318 (72), 331 (56), 338, 339 (78), 341 (G6,G10); 353 (75), 335 (702), 341 (G6,G10) Pechenkin, A. 452f (75) Penczek, S. 392 (137) Penniger, J. M. L. 141 (15,16) Pepper, D. C. 382 (700), 391 (729) Perrin, C. L. 82 (4) Perry, R. H. 1 (7) Petke, F. E. 356 (25) Peuckert, M. 385, 386 (727) Phillips, M. J. 435 (75) Pilling, M. J. 73 (G8); 87f, 89 (75); 155 (26,27); 196 (13,14); 326 (34,35), 328 (35), 334 (69,70), 335 75), 342 (G72); 452 (29) Pinner S. H. 356 (79) Pittman, C. U., Jr. 296 (38); 440 (78) Plank, C. J. 298 (53) Plesch, P. H. 382 (101,103) Pojman, J. A. 445, 458 (G5), 452 (72) Pokidova, T. S. 341 (G3) Polanyi, M. 203 (39); 316 (8), 318 (9)

474

Author index

Poole, C. F. 50 (20) Poole, C. P., Jr. 51 (38) Poole, S. K. 50 (20) Prabhu, P. 392 (136) Prater, C. D. 127, 128 (23) Pregosin, P. S. 51 (34) Price, B. 65 (51); 416 (14) Priddy, D. B. 359 (32) Prokop, Z. 296 (39) Pross, A. 201, 202 (36); 318 (14), 342 (G13) Pryor, W. A. 342 (G14) Purnell, J. H. 328 (40), 330 (45,^7) Puskas, J. 382 (106) Querol, J. 169 (6) Quinn, C. P. 326, 327 (33), 328 (40), 330 (47) R Rabinowitch, E. 320 (15) van Raemsdonck, K. K. 422 (36) Rase, H. F. 281 (6) Reed, R. I. 82 (1) Reich, L. 332 (62) Reilly, P. J. 382 (100) Rempel, G. L. 248 (60) Revillon, A. 296 (39) Ricchezza, E. N. 373 (72) Rice, F. O. 2; 325 (22-24,26-28), 326 (27), 331 (22,55), 342 (G14) Rice, K. K. 325 (24,26,28), 331 (55), 342 (G15) Rideal, E. K. 33 (22) Ritchie, W. 320 (iS) Roberts, G. W. 292 (24) Roberts, M. J. 455 (37) Robinson, G. C. 374, 382 (78) Roedel, M. J. 364 (45) Rossler, O. E. 456 (45) RoUman, L. D. 299 (57) Romero, A. 169 (6) Rooney, J. J. 383 (118) Rosevaere, W. E. 63 (47) Rosinski, E. R. 298 (53) Ross, A. B. 331, 334(57)

Roughton, F. J. W. 48 (3) Rouve, A. 354 (17) Rudin, A. 347, 398 (G9), 362 (39), 364 (47), 367 (62), 391 (755) Rulliere, C. 50 (75) Rupilius, W. 218 (27) Rupley, F. M. 407 (10) Russell, G. A. 332, 339 (57) Sandler, S. R. 347, 398 (GIO), 350 (6), 359 (57;, 363 (42), 373 (73), 383 (777) Sapre, A. V. 421 (26) Satterfield, C. N. 17, 36 (GIO); 273, 306 (G7), 292 (27,2^, 299 (64) Saunders, W. H., Jr. 203 (40) Sauvet, G. 381 (89,90) Savage, P. E. 11 (2); 40 (7); 65f; 66 (56) 157 (30); 175-179, 187, 175, 176, 180 (7); 216, 217 (75), 240 (47); 349 (5); 404 (7), 408, 413 (77), 417, 418 (17); 434, 437, 440 (14) Saville, B. A. 265 (72); 352 (72) Sawamoto, M. 382 (107) Schindler, A. 392 (136) Schlosser, P. M. 448, 449 (5) Schmidt, L. D. 265 (77) Schmitt, B. J. 378 (80) Schulz, G. V. 358 (29), 361 (37), 365 (55,57), 378 (80) Scott, R. P. W. 50 (27) Scott, S. K. 335 (74), 445, 457 (G2,G3, G4), 452 (25), 453 (28,34), 454 (35), 455 (36,38,39), 456 (42,44,45) Seakins, P. W . 73 (G8); 326, 328 (35}, 335 (73), 342 (G72); 452 (29) Segel, I. H. 221, 222, 211 (28), 229 (40), 241 (62,63), 270 (G77) Seidel, W . C. 236, 237, 239, 262 (44); 333 (67) Semenov, N. N. 2; 199 (20); 318 (10), 332, 333 (58), 342 (G16) Sen, A. 392 (134,135) Shapley, J. R. 205 (47) Sharma, M. M. 295 (32) Sharpless, K. B. 255 (65,66) Shestakov, G. K. 154 (22)

Author index Shliapnikov, Yu. A. 338 (83) Shnor, S. E. 452f (24) Showalter. K. 452 (26), 456 (43) Shulman, R. A. 286 (10) Siddiqui, F. A. 33 (23) Sigwalt, P. 381 (88,89,90) Silverman, J. 381 (92) Sims, I. R. 430 (77) Sinfelt, J. H. 288 (20) Skeist, I. 391 (750) Slaugh, L. H. 205 (44); 218, 219 (77), 219 (24) Smith, B. C. 50 (25); 204 (43) Smith, C. M. 297 (47) Smith, E. G. 391 (757) Smith, H. 65 (49); 416 (72) Smith, L. R. 440 (28) Smith, M. A. 66 (56) Smith, R. M. 51 (57) de Soete, D. 51 (37) Solomon, D. H., 338 (82); 299, 359f, 397 (G6), 360 (33), 361 (35), 363 (42), 364 (43), 365 (54) Squires. R. G. 381 (84) Spooncer, W. W. 205 (44); 219 (24) Stange, G. 50 (9) Stannett, V. T. 381 (92) Starks, C. M. 364 (48) Steacie, E. W. R. 325, 331 (27), 326, 328 (30), 342 (G27) Steinfeld, J. I. 95, 131 (G9); 265 (69), 270 (G72); 320f (76), 335 (76), 342 iG28) Stevens, W. R. 236, 237, 239, 262 (44) Stevenson, S. A. 302 (66) Stewart, C. D. 383 (778) Steward, I. 3 (2) Stiegman, A. E. 201 (29) Stivala, S. S. 332 (62) Stockmayer, W. H. 356 (22,23) Stoll, M. 354 (27) Stoll-Comte, G. 354 (77) Stremple, P. P. 50 (28) Sugimura, T. 361 (36) Sullivan, J. H. 85, 86 (5), 87 (70) Sundaram, K. M. 324 (20)

475

Sundstrom, V. 50 (24) Svendsen, J. S. 255 (65) Swift, T. J. 50 (72) Swinney, H. L. 456 (42) Szabo, A. L. 203 (39) Szabo, P. 205 (23) Szabo, Z. G. 95, 107, 131 (G70), 120 (72), 123 (27) Szwarc, M. 347, 398 (G77), 373 (66-68), 374 (66,77,79), 392 (237) Taylor, H. S. 33 (24) Teller, E. 32 (75), 33 (28) Temkin, M. I. 33 (30), 34 (57) Temkin, O. N. 133f, 158 (2), 154 (22, 23), 155 (23,24) Terabe, S. 205 (50) Theil, M. H. 392 (236) Theile, H. 326, 327, 328 (36) Thiele, E. W. 291 (27) Thiele, H. 51 (35) Thomas, G. G. 296 (44) Tolman, C. A. 51 (40); 200 (22), 202, 205 (38); 236, 237, 239, 262 (44), 237 (46), 246 (57); 333 (67); 423 (39) Tolman R. C. 27 (23,24) Tomlin, A. S. 87f, 89 (75) Treger, Yu. A. 154 (22) Troe, J. 155 (26,27); 196 (75,7^; 334 (69,70) Trommsdorff, E. 365 (56) Trowse, F. W. 48 (2) Troy, W. C. 453 (33) Tschunkur, E. 373 (64) Tudos, F. 382 (206) Turanyi, T. 87f, 89 (25), 454 (28) Tyburczy, J. A. 352 (70) Tyler, D. R. 201 (29) Tyson, J. J. 452 (23) U Ulrich, G. D. 429 (70) Underhill, L. K. 87f (77) Vairon, J. P. 381 (89)

476

Author index

Van Winkle J. L. 175 (8), 194 (ii); 217 (16) Vannice, M. A. 432, 435 (12) Virk, P. S. 165f (2) Volmer, M. 33 (25) Volterra, V. 452 (77) Voltz, S. E. 421 (27) Voorhies, A. 302 (67) W Wai, J. M. S. 255 (65) Walas, S. M. 1 (7); 353 (14) Walker, R. W. 155 (26,27); 196 (13,14); 334 (69JO), 335 (73) Wall, F. T. 388 (127) Walling, C. 390 (128), 393 (138) Walsh, D. E. 299 (57) Walters, W. D. 331 (52) Wang, H.-C. 380, 388 (83) Wang, N.-H. L. 33 (23) Warnatz, J. 155 (26,27); 196 (75,7^; 334 (69,70) Watson, K. M. 165f (4); 273, 306 {G5), 273, 276 (3) Weber, J. N. 350, 353, 354 (5) Weber, U. 51 (35) Weekman, V. W., Jr. 112 (4); 293 (28,29); 420 (20), 421 (23,25,27) Wei, J. 127, 128 (23); 420 (19); 441 (27) Weibull, B. 118, 120(5) Weigert, P. J. 409 (5,6) Weisberg, S. 65 (50); 416 (75); 462 Weiss, J. 311 (2) Weiss, M. D. 51 (28) Weisz, P. B. 293, 294 (27), 297 (49), 298 (52,53), 303 (72) Wenthe, A. M. 211 (5) Wertz, J. E. 51 (39) Wessling, R. A. 388 (123) Westley F. 155 (28); 196 (75)

Westman, A. E. R. 120 (75) Whitcomb, P. J. 405 (4) Wiberg, K. B. 203 (41); 214 (9) Wilhelmy. L. 210 (7) Wilkinson, F. 135 (7) Wilkinson, G. 201 (30); 246 (49-51) Williams, D. H. 50 (26) Williams, I. D. 333 (67) Winfree, A. T. 452f (75), 452 (20) Winkler, R. 451 (9) Winstein, S. 374, 382 (78) Wojciechowski, B.W. 326, 327 (57), 328 (39) Wong, C. S. 246 (54) Wooding, N. S. 374, 379 (74) Woodward, R. B. 201 (32,33,35), 202 (32,35) Wu, M. M. 298 (55) Wynne-Jones, W. F. K. 20 (5)

Yablonskii, G. S. 133f, 155, 158 (7); 335 (77), 342 (G79) Yagupsky, G. 201 (30) Yang K. H. 278, 287 (4) Yates, F. 65 (55) Yau, S.-H. 453 (57) Yoon, T. J. 432, 435 (72) Yoshida, T. 201 (26) Young, J. F. 246 (49,50)

Zaikin, A. N. 454 (79) Zaikov, G. E. 332, 334 (67) Zakhariev, A. 246, 248 (55) Zeigarnik, A. V. 133f, 158 (2) Zhabotinsky, A. T. 452f(77), 452 (19^4). Ziegler, K. 373 (65), 383 (775)

Subject Index ab initio calculations 246 absolute reaction rates, theory of 20 abundance, relative of catalyst-containing species 230-232, 236, 240-243 of catalyst-surface species 280 of chain carriers 313, 322, 332 of propagating centers 375-377 of radicals 323,325,326,338,340 acetal hydrolysis 211-212 acetylene see ethyne acid-base catalysis 214-217 activated complex 20, 298 activated molecules 19-20, 327 activation 251 activation energy 11, 21-23, 291, 292, 317-318, 431, 432, 434 negative 23, 125, 431, 430-433 of phenomenological coefficients 436 activities, thermodynamic 20-21 addition polymerization 347-349 adenosine phosphates 225-226 adipic acid 333, 353 adiponitrile 236-237, 262-264 ADP see adenosine phosphates adsorption 14, 32-34, 273-275 equilibria 14, 32-34 equilibrium constant 284, 285 group, in heterogeneous catalysis 276, 277, 278-279 isotherms 14, 32-34 rates 15, 34, 273 AmN 310-311, 360 alcohols see also hydrogenation, hydroformylation long-chain primary 118 secondary 122-123 alcoholysis 214 aldehydes see aldol condensation, hydroformylation, hydrogenation aldol condensation 157-158, 215-216 alkali metals as initiators 374

alkali-metal alkyls as initiators 373 alkylation, of toluene 298 allowed reactions see WoodwardHoffmann exclusion rules aluminum bromide as initiator 381 aluminum chloride as initiator 380 aluminum complexes in coordination polymerization 383 amines, as transfer agents 381 amino acids 229 6-aminocaproic acid 348, 350, 354 anmionia synthesis 34, 299 analytical support 50-51 anionic polymerization 348-349, 373-380 apparent reaction order 19, 329 aromatics, nitration of see nitration Arrhenius equation 11, 21-23, 284, 431 Arrhenius plot 21 anomalous 431, 434 asymmetric dihydroxylation 255-256 ATP see adenosine phosphates autocatalysis 165, 183, 265-266, 445, 454 see also self-acceleration autocatalytic steps 452 autoxidation, of hydrocarbons 169, 266, 334, 446 2,2'-azo-M5-isobutyronitrile 310-311, 360 B backbiting 364 Bakelite 350 batch reactors 39-43, 47 benzoyl peroxide, as initiator 310, 361 BEBO 318 Belousov-Zhabotinsky reaction 452-453, 456 BET adsorption isotherm 33 bimolecular steps 18 block-copolymers 349, 373, 392 Bodenstein approximation 87-92, 124, 134, 136, 148, 209, 223, 230, 312, 336, 360, 363, 389, 407 bond energies 199, 320

478

Subject index

bond energy-bond order method (BEBO) 318 borate esters 122, 333, 334 boron trifluoride, as initiator 380, 381 boroxines 122, 333, 334 Boudart's entropy criteria 285 Boudart's theorems 280 branches, of networks 8 Briggs-Haldane kinetics 223, 224, 226 bromate ion, as oxidant 452-453 Bronsted acids, as initiators 380 Brunauer-Emmett-Teller (BET) isotherm 33 Buna-S 373 burn-off, of coke 300, 422 butadiene 202, 373, 391 butadiene polymers, hydrogenation of 248 rt-butanal, condensation of 216-217 A2-butane, cracking of 330 butanol, from propene 103-105 butterfly effect 456 butyl lithium, as initiator 373 butyl rubber 380, 388 4-r^rr-butylcatechol, as inhibitor 338 7-butyrolactone 98-99 caprolactam 348, 354, 373 carbanions 348, 373 Carbitol 118 carbocations 348, 380 Carothers equation 355-356, 358, 395 catalysis 8 by acids 210-214 acid-base 214-217 complex 209, 214-220 heterogeneous 2, 4, 127, 165, 230f, 273-303, 315 heterogenized 295-297 homogeneous 209-267, 273 by metal complexes 217-220 single-species 209, 210-214 trace-level 209, 220 catalysts 30-31 deactivation 299-303 decay 252, see also deactivation dual-form, multiple-form 256-258 poisoning 252

catalytic cycles 9, 29-30, 210 with common pathways 259-265 connected 256-258 with external reactions 243-252 general formula for single cycles 227-229 multiple 253-265 catalytic cracking 112, 298, 299, 303 catalytic sites 273, 274, 277 cationic polymerization 348, 349, 380-382 Cellosolve 118 Ce(III)/Ce(IV) redox couple 452-453 chabazite 297, 299 chain branching 310, 334-338, 446, 448 chain breaking 362, 366-8, 370, 371, 381, 387 chain carriers 309,311 chain-growth polymerization 348-349, 383 chain length 366-368 chain reactions 77, 86-87, 89, 158, 309340 chain transfer 322-324, 340, 362-364, 367-374, 379, 381-2, 385, 387, 392 chaotic reactions 2, 178, 445, 456 Chem algorithm 149-152, 154 chi-square statistic 69 Christiansen mathematics 141f, 227-229, 274, 275, 289 Christiansen matrix 227-228 circular reactions 27-30 clockwise rate coefficients 28-30 cobalt hydrocarbonyls 139 phosphine-substituted 103-104, 114-115 157, 175, 190, 195, 200, 218-219 257-258, 295-296, 408-409, 414, 418, 420, 424, 437, 440, 441 coefficients, criteria for 284 co-initiators 38If coke 300, 421 coking 300 collapsed network 260, 262 collective coefficients 147 collision 18-19, 22 collision partner 320, 335 combustion see oxidation competing catalytic reactions 253-256 competing steps 124-125 competitive adsorption 32

Subject index competitive inhibition 249 complex reactions 7 consecutive steps 118 concentration, effect on selectivity 107, 123-124 concurrent steps 101, see also parallel steps condensation reactions 124, 214 see also aldol condensation condensation polymerization 347-349, 350 confidence interval 65-66 consecutive steps 118, see also sequential steps consistency, thermodynamic 26-27, 142, 320f, 330 consistency checks, in chain reactions 321 consistency criteria 25, 26-30 continuous mixtures 421, 422-423 continuous stirred-tank reactors 10, 43-44, 47 multiple steady states in 446-449 conversion 12 fractional 12-13, 14 of functional groups 352-353 effect on selectivity 107-108, 121-124 coordination polymerization 348-349, 382-388 copolymers 388-389 alternating 392 block 349, 373, 392 ideal 391 random 391 statistical 391 copolymer composition 389-392 copolymer equation 389 copolymerization chain-growth 388-395 step-growth 353 co-reactant entries, sequence of 184 counter-clockwise rate coefficients 28-30 coupled parallel steps 109-117, 129 coupling, of radicals 311-314, 320, 322325, 332-333, 340, 359, 361-362, 366368, 370, 381, 393 co-variance 72 cracking, catalytic 298, 299, 303, 419, 421-422, 441

479

cracking, thermal 321, 325-331 crosslinking, of polymers 350, 354, 356, 388 CSTR 10, 43-44, 47; see also reactors cubic autocatalysis 454 curve-crossing approach 318 cyclization 354 cyclohexane, oxidation of 333-334 cyclohexanol, from cyclohexane 333-334 cyclohexanone, from cyclohexane 333-334 cyclohexene 202, 408-410, 418 D databases 155, 196, 331, 462 deactivation, of catalysts 299-303 mechanisms 300 dead polymer 348 decarboxylation, of organic acids 210 decay of catalysts 252, 299-303 unimolecular 18, 19-20, 327 deficiency, of network 448-450 degree of polymerization 355-357, 363, 368-369, 378, 380,385, 387 dehydration, of ethanol 286 dehydrogenation, of methylcyclohexane 288-289 of paraffins 101 delayed control 450 Delplot 165-171 demonstration units 2, 3, 403-405 density variation 13, 44, 59 desorption 34, 273, 274-275 detailed balancing 28 detergents 120 detonation 155, 334 deuterium labeling 197 1,6-diaminohexane 353 differential evaluation method 52 differential reactors 45-46, 47 diffusion coefficient, effective 290 diffusion limitation 296 see also mass-transfer limitation dihydroxylation, asymmetric 255-256 disguised kinetics 59, 292 disproportionation 361-362, 367-368, 370372, 386f

480

Subject index

dissociation 172 see also pre-dissociation upon adsorption 33, 276, 278-279, 281, 283 electrolytic, in ion exchangers 295 distance from equilibrium 96 fractional 96-97, 101-102 distribution coefficient 14, 34 divinyl adipate, as crosslinking agent 388 divinyl benzene, as crosslinking agent 388 double-bond migration 109, 114, 141, 198 see also isomerization drain effect 113, 117 driving-force group, in Hougen-Watson formula 276, 277, 278-279, 281 dual-form catalysts 256-258 Eadie-Hofstee plot 225 effectiveness factor 292, 360, 377, electric conductivity 97 electric field 50 electrolytic dissociation 84 electron spin resonance 51, 205 elementary steps see steps Eley-Rideal mechanism 274 empirical approach 403-405, 407 enantio-selectivity 255-256 enthalpy, standard, of reaction 22, 199, 317-319, 432 enzyme kinetics 220-226 enzyme-mimicking polymers 296 equilibrium requirement 26 see also quasi-equilibrium equilibrium constant 26-27,28-31, 125, 142 erionite 297, 299 error analytical 65 in approximations 80, 81, 85, 89 constant absolute and relative 69-71 correlation 71 propagation 67 systematic 72, 416 zero-time 62 error recognition 60-64 Escher's world 28 ESR 51, 205

essential activation 251 ester hydrolysis 165, 183, 203, 266 esterification 210, 214 ethoxylation 118, 120-121 2-ethylhexanal, hydrogenation of 175-177 ethane, cracking of 325-329 ethanol, dehydrogenation of 286 ethene addition to butadiene 202 dimerization 201-202 hydroformylation 421 oligomerization 383, 385-386 polymerization 364, 383-385 ethene oxide 120, 373 ethoxylation 118, 120-121 ethylene see ethene ethyne dimerization 154-155 Evans-Polanyi procedure 318 evolutionary approach 403-404, 407 experimental methods 39-64 exponents in rate equations 12 fractional 19, 127, 171-172, 315, 425 extent of reaction 14 extinction 448 fast reactions, study of 48-50 "fast" steps 79 faujasite 297 feedback 445-446 Feinberg network theory 448-449 femtosecond techniques 50 ferroin 452 flash photolysis 50, 86 flame-out 448 Flory-Schulz distribution see Schulz-Flory distribution fluid-density, variations of 12-13, 44, 98 see also volume variation flux 15 forbidden reactions see WoodwardHoffmann exclusion rules fouling, of catalysts 299, 305 Foxes and Rabbits game 451-452 fractional conversion 12-13, 14 fractional distance from equilibrium 96-97, 101-102

Subject index fractional exponents, in rate equations 19, 127, 171-172, 315, 324, 425 fractional-life methods 56,114 free energy 28 standard 8, 28-29 Freundlich adsorption isotherm 32 functionality, of monomers 350 fundamental approach 3, 405-406 fundamental modeling 407-412 fundamental reaction kinetics 2, 3 gas chromatography 50, 203 gas-liquid reactions 4, 42, 44, 47, 58 gas oil, cracking of 112, 421 gel effect (Trommsdorff effect) 365 gel point 356 general base catalysis 215-216 Gibbs free energy 28 Gillespie-Ingold mechanism 82, 90, 138 Goldfinger scheme 329-330 global evaluation method 61-63 glucose 225-226 glycol ethers 118 graph theory 158 H Haber-Weiss redox cycle 311, 333 half-time of reaction 56 halogenation 104 Hammett acidity function 214 Hanes plot 225 heat curing 350-351 heat transfer 439-411 in heterogeneous catalysis 283, 290, 293 Heck-Breslow mechanism 140, 141, 188, 408 Henry coefficient 14 Henry's law 14, 140, 236, 333 ^-heptene, hydroformylation of 414, 417 heterogeneous catalysis see catalysis n-hexene 418 n-hexanol, from 1-pentene 114-117 hexaphenylethane 60-62 hexokinase 225 van't Hoff equation 22, 284, 285, 432 homogeneous catalysis see catalysis

481

homogeneous reaction see reactions homogeneous source 110, 112, 129 homo-polymerization 351 Hougen-Watson formula 275-279, 280, 281, 282,283, 289 Hougen-Watson kinetics 273, 274, 280, 281, 283, 286 Hudson-Rossler model 456 hydration, of olefins 259 hydrocarbons see combustion, oxidation hydrocarbonyls see cobalt hydrocarbonyIs, rhodium hydrocarbonyls hydrocyanation 104, 197-198, 217, 236-239, 259, 262-264 hydrodesulfurization 422 hydroformylation 78, 101, 103-105, 114117, 139-142, 157-158, 165, 171, 183, 188, 190-192, 195-196, 200, 203, 217, 218-219, 257-258, 259, 295-296, 408412, 413-415, 437-440 hydrogen ion, as catalyst 210-214, 195 hydrogen-bromide reaction 86, 158, 202, 310, 311, 316-7, 319-321, 329, 331 hydrogen-chloride reaction 86, 202 hydrogen-halide reactions 202 hydrogen-iodide reaction 85-86, 199-200, 202 hydrogen-oxygen reaction 311, 335-337, 253 hydrogenation 104, 217, 253, 273 of aldehydes 175-177,183,187-189, 246, 281 of olefins 246-249 hydrohalogenation 104, 259 hydrolysis 214 of butyronitrile 167 of esters 165, 183, 203, 266 hydroquinone, as inhibitor 338 hydroperoxides 169, 266, 310, 331-334 hydroperoxy radicals 335 hysteresis, in nonisothermal CSTR 446447 I ignition 447 imaginary equivalent pathway 231, 241-242

482

Subject index

induction period 310,339 infrared spectra 50, 204-205, 237 inhibition 249-251, 338-339, 363 initial rates 58, 148, 207, 327 initial yield ratio 153 initiation, of chain reaction 309, 310-311, 359, 373-374, 380-381 indirect 322-323 photochemical 311, 334, 366, 381 instability 445-447 of quasi-stationary states 453-455 instantaneous selectivity 14 see also selectivity instantaneous yield ratio 13 see also yield ratio integral evaluation method 52 interchange reaction 351 intermediates as starting materials 164, 195, 204 trace-level 87-89, 133 intimate ion pairs 375 iodine, as initiator 381 ion exchangers, catalysis by 295 ion pairs 374-375 ionic copolymerization 388, 391, 393-395 ionic polymerization 348, 349, 372-382 irreversible reactions 8 island formation 273 isobutene 380, 388 isomerization 96, 109-117, 198, 274 of xylenes 298 isoprene 380, 381, 388 isotope techniques 51, 203-204 K keto acids 229 ketone, thermal degradation of 331 kinetic chain length 366 kinetic group, in Hougen-Watson formula 276-279 kinetic isotope effect 203 kinetic order see reaction order labeling 51, 203-204 lacs (low-abundance catalyst-containing species) 232

lacs (low-abundance catalyst-surface species) 280 laps (low-abundance propagating center) 375-377 Langmuir adsorption isotherm 32, 171, 282 Langmuir-Hinshelwood kinetics 273, 274275, 280, 283, 300-301, 438 laser techniques 50 latent functionality 350 least-squares fitting 68-71 Le Chatelier principle 219f, 424 Lewis acids 210, 262, 380-381 ligand-deficient catalysts 244-249 limiting reagent 13 Linde-A zeolite 297, 298 Linde-L zeolite 297 Lindemann theory 327 linear networks 133f Lineweaver-Burk plot 225 living ends 377 living polymers 349, 350, 373, 374, 377, 382 long-chain approximation 312, 361, 408 loops, of parallel pathways 8, 27-31, 146147 circular reactions in 27-30 loop coefficients 146 loose ion pairs 375 Lotka-Volterra sequence 452 lumping 420-423 of denominator terms 173 of products 195 ofreactants 112,219-220,420-421 by reactive configurations 422-423 of species in quasi-equilibrium 233-234 lumps 420-422 M M-Forming 298 macs (most abundant catalyst-containing species) 230-231, 236, 240-241 (most abundant catalyst-surface species) 280, 283 malonic acid 455 Markov model 388 Martin equation 141, 191, 258, 440

Subject index masi 230f, 280f mass spectroscopy 51 mass transfer, unusual effects of 438-440 in heterogeneous catalysis 290-295 mass-transfer coefficient 15 mathematical modeling 403-425 matrix for simple pathways 137, 179 Christiansen's 227-228 matrix notation for multistep reactions 25 maximum likelihood 68 maximum model 163 mean value 65, 66 metal amides, as initiators 374 metal complexes, fixed, as catalysts 295-296 metathesis reaction 386 methyl methacrylate 361, 364, 391 a-methyl styrene 380 methyl-r^rr-butyl ether 295 methylcyclohexane dehydrogenation of 288-289 Michaelis constant 223 Michaelis-Menten kinetics 221-224, 288 microscopic reversibility 27-31, 320f mixed termination, of chain reaction 314, 326, 330f model discrimination 284-286 validation 423-425 modeling, mathematical 347-395 molecular-orbital synmietry see Woodward-Hoffmann exclusion rules molecular weight, of polymers 355, 357, 370, 378, 387 distribution 357-358, 370-372, 385-386, highly uniform 378 molecularity 12, 17-18, 199 mole-fraction distribution (of polymers) 357, 358, 386 moron in shower 450 most probable distribution see Schulz-Flory distribution MTBE 295 multiple catalytic cycles 253-265 multiple-form catalysts 256-258 multiple reactions 7 multiple steady states 446-450 multistep reactions 7

483

N naphtha, cracking of 421-422 neohexene 418 networks 3, 8 collapsed 260, 262 first-order, general solution 127-130 linear 133f non-simple 133-134, 155-158, 194-197 piecewise simple 134, 155-158, 195196, 408, 410 simple, 133-134, 145-155, 190-194 streamlined 413-415 network elucidation 163-205 network notation 9 network properties, and kinetic behavior 178-196, 239-243 network reduction 136, 145-147, 410 network theory 448-450 neutron activation 51 nickel, as catalyst 281 nickel complexes, as catalysts 217, 236239, 262-263, 383, 386 nitration of aromatics 82-83,90-91, 138 NMR 50, 51, 205, 237, 239, 246 nodes, of networks 8 non-Arrhenius steps 430 noncompetitive inhibition 249 nonideality 20-21 nonisothermal reactions 445-448, 455 non-simple pathways and networks see networks, pathways nuclear magnetic resonance see NMR number-average degree of polymerization see degree of polymerization numerical work-up 60-64 Nylon-6 348, 350, 351, 354 Nylon-6,6 353 O objective function 68-71 Occam's razor 86, 173 olefins see hydroformylation, isomerization, oligomerization, polymerization oligomerization 347f of ethene 383, 385-388 once-through differential reactors 46, 47

484

Subject index

one-plus rate equations 71, 171-177, 185, 407 Onsager's axiom 27 optimization, statistical 416 order see reaction order oregonator 453, 456 oscillation 295, 452-456 osmium tetroxide, as catalyst 255-256 overall reaction orders 12, 57-58 oxidation 217 of hydrocarbons 101, 331-334, 339 of paraffins 122-123, 169, 259 0X0 catalyst 139, 157, 191 paraffins see also hydrocarbons autoxidation of 169 by-product in hydroformylation 157-158, 190-193, 259 cracking of 298 oxidation of 122-123,331-334 parallel pathways see pathways, parallel parallel steps 101-109 parameter estimation 68-71 partheite 297 pathways 8 elucidation 187-189 forward and reverse 31 non-simple 133-134, 155-157, 194-196 irreversible 179-180 parallel 8, 30-31, 87, 146 piecewise simple 134, 155-157 simple 133-134, 135, 179-197 1 -pentene, hydroformylation of 114-117 4-pentenenitrile, hydrocyanation of 236239 periodic reactions 178, 445, 450-457 peroxides see benzoyl peroxide, hydroperoxides phase equilibrium 14 phenol-formaldehyde copolymers 350-351 phenomenological coefficients 171 activation energies of 436 phosphate transfer by hexokinase 225-226 phosphines as transfer agents 381 ^^^hydrocarbony Is, phosphine-substituted

photochemical initiation 311, 334, 366, 381 photon-echo techniques 50 piecewise simple networks and pathways 134, 155-158 pilot plants 2, 3, 403-405 ping-pong mechanisms 229 point-by-point evaluation 62 poisoning, of catalysts 252, 300, 303 Poisson distribution 372 Polanyi equation 318 polyethene (polyethylene) 359, 383 polymer standards 378 polymerization 2, 118, 217, 347-496 of olefins 383-388 stereospecific 383 poly(methyl methacrylate) 359 polystyrene 359-361 poly(vinyl acetate) 359 poly(vinyl alcohol) 359 poly(vinyl chloride) 359 power-law rate equations 12,51, 170-171 see also rate equations predator-prey fluctuations 451-452 pre-dissociation 125-127, 407 prepolymers 350 pressure, reactors for elevated 41 gas-phase reactions at constant 59 probability 17-18, 68, 356-357, 369-370, 385 see also reaction probability process development 403-406 process model 415 process rates 10-11 product-promoted reactions see reactions product shape selectivity 298 promoters 299 proof, in kinetics 86, 423f propagating center 348, 375-378, 382, 385, 387-389, 392-394, 396-397 propagation 309, 312, 315, 359-360, 363370, 374-379, 382, 392-393 propanal, hydrogenation of 281-282 propene, hydroformylation of 103-105, 414, 420 protogen 381 provenance 165, 170 pseudo-cationic polymerization 382

Subject index pseudo-components 128 pseudo-first order rate coefficients (X coefficients) 133, 134-135 pseudo reaction orders 57-58 pyrolysis, of hydrocarbons 196 see also oxidation quadratic autocatalysis 454, 456 quadrimolecular 20 quasi-equilibrium 82, 110, 125-127, 230, 233-235, 240, 375, 408 quasi-equilibrium approximation 84-86 quasi-stationary states 87-88, 209, 274 see also Bodenstein approximation quenching, of polymerization 378, 381 R radicals 205, 309 radical chain length 366 radical copolymerization 388-393 radical polymerization 348, 349, 359-372 radical traps 309, 341 radioactive decay 88-89 radioactivity 51 rake mechanism 274 random error 68, 72 rank 12, 165-171, 194 rates 10-11 see also reaction rates of change, process rates 10-11 rate coefficients 11, 17 apparent 51 determination of 51-52, 56, 58, 62, 64, 417-420 and equilibrium constant 31,142 ratios of, in chain reactions 317-319 pseudo-first order 133, 134-135 temperature dependence 11, 21-23, 284 rate control see rate-controlling steps change in, with temperature 433-436 rate-controlling steps 78-84, 184, 232-233, 236, 275-277, 284, 408 identification of 285-286 rate equations 11 ideal 19, 20 unabridged, of multistep reactions 23-26 for simple pathways 135-145

485

reactant-inhibited reactions see reactions reactant inhibition 250-251 reactant shape selectivity 297-298 reactions catalytic see catalytic cycles chaotic 456 circular (in loop) 27-31 complex 7 fast 48-51 forbidden see Woodward-Hoffmann exclusion rules gas-liquid 4, 42, 44, 47, 58, 273 gas-phase, at constant pressure 59 homogeneous 7 irreversible 8 multiple 7 multistep 7 nonisothermal 445-448, 455 pathways of forward and reverse 31 periodic 178, 450-456 product-promoted 165, 183, 265-267, 439 reactant-inhibited 165, 183, 267, 439 reverse, pathway of 31 reversible 8, 31, 58, 95-100 simple 133 5"^^fl/^c?networks, pathways simultaneous 7, 112 substrate-inhibited 165, 267 reaction enthalpy see enthalpy of reaction reaction kinetics 1-3, 17 fundamental 3 reaction orders 12, 17-20, 164-165, 173174, 180-183 apparent 19, 51 determination of 51-60 fractional 19, 127, 164, 240, 283 see also fractional exponents and mass transfer 291-292 of lumps 421-422 negative 165, 174, 182-183, 242, 283 overall 12, 57-58 pseudo 57-58 effect on selectivity 123 in simple pathways 180-183 variable 124-125 reaction paths 128-129 reaction probability 79, 135f

486

Subject index

reaction rates absolute, theory of 20 determination of 39-64 initial 58, 148, 286-287 and mass transfer 291-293, 438-441 maxima and minima of 435 of steps, equality in quasi-stationary pathways 89 temperature dependence 11, 21 -23, anomalous 429-432 reactive centers 348, 350, 362, 364, 373 reactivity ratios 389 reactors 39-50 batch 39-43, 47 continuous stirred tank (CSTR) 43-44, 47 differential 45-48 for study of fast reactions 48-50 semi-batch 42 tubular 44-45, 47 reactor type, and selectivity 107, 121-124 recombination 311 see also coupling recycle differential reactors 46, 47 reduction of complexity 77-92, 229-239 refractive index 97 regression 60, 71-72, 416 relative abundance see abundance relaxation techniques 50 research model 415 resole-type prepolymers 350-351 resonance 8f, 363, 364, 417 reversibility 8 of steps in chain reaction 320, 322, 329 microscopic 27-31, 320f reversible reactions 8, 31, 58, 95-100 rhodium complexes, as catalysts 217, 220 Rice-Herzfeld mechanisms 325-331 rotation, of plane of polarized light 97 runaway 446-447 Saran 388 saturation kinetics 222, 288 scale-up 2, 3, 4, 403, 405, 407, 423 Schulz-Flory distribution 358, 371, 380, 385, 386 segments 8 segments coefficients 136, 146, 410-411

selectivity 13-14, 107-109, 121-123 and mass transfer 441 self-acceleration 265-267, 283-284, 310, 332, 338, 340-341, 396, 439-440, 451453, 455, 457 see also autocatalysis semi-technical unit 405 sensitizers 334, 366 sequential steps 118-124, 129 series-parallel steps 124 shape selectivity 297-299 Shell Higher Olefin Process 383, 385-386 shock tube 50 SHOP 383, 385-386 shortsightedness of elementary steps 389, 417-420, 422 silent partners 165, 182, 245, 251 silica fouling, of anion exchangers 299 silver ion, as catalyst 295, 296 simple pathways and networks 133-134 simultaneous reactions 7,112 sintering 299, 302 site, on solid surface 33, 276 see also catalytic sites "slow" and "fast" steps 79 snowflake network 150-152 software 461 spatio-selectivity 298 spatio-temporal patterns 452 specific base catalysis 215-216 spectra 50, 51, 204-205, 237, 248, 258 spin-state labeling 51 stability 445-450 of quasi-stationary states 453-455 standard deviation 65 standard enthalpy of reaction 22 see also enthalpy standard free energy 8, 28-29 statistics 17, 65-73, 403-406, 416-417 steady states, multiple, in CSTR 446-450 steps, elementary 8, 17-20 autocatalytic 445, 452, 454 bimolecular 18 combinations of 95-130 competing 124-125 consecutive 118 consolidation of 186-187, 240 coupled parallel 109-117, 129

Subject index (still steps) irreversible 8, 138, 179-180, 235, 240, parallel 101-109 quadrimolecular 20 quasi-equilibrium 82, 84-87, 233-235 rate-controlling 78-84, 184, 232-233, 236, 275 reverse, in chain propagation 327-329 sequential 118-124, 129 in series 118 series-parallel 124 "slow" and "fast" 79 trimolecular 18-19, 199 unimolecular 17-18 step-growth polymerization 348, 349, 350-359, 371 stereochemistry 197 stereospecific polymers 383 stoichiometric coefficients 14, 18, 24 stoichiometric constraints 77-78, 408, 410, 412-413 stoichiometric numbers 13 stoichiometry 17-19 stopped-flow technique 48-49 Student's distribution 66 styrene 360-361, 364, 373, 374, 379, 380, 391, 397 substrate-inhibited reactions 165, 267 sucrose inversion 210 supercages, in zeolites 297, 441 supercomponents 420 see also lumps surface reaction 274-276, 278-279, 287 synergistic adsorption 32 systematic deviation see error, systematic tagging 51, 203 telomerization 363 Temkin adsorption isotherm 33 temperature dependence of rates see reaction rates TEMPO inhibitor 338 terminal model 388 termination, of chain reaction 309, 311315, 322-324, 359-362, 373, 381, 393 thermal effects 429, 437-438 see also nonisothermal reactions

487

thermochemical data 199 thermodynamic activities 20-21 thermodynamic consistency 26-27, 142 thermodynamics 1, 2, 199 thermosetting polymers 350 Thiele-Damkohler theory 291-293 Thiele modulus 292, 293, 294 titanium tetrachloride, as initiator 380, 381 titanium complexes, in coordination polymerization 383 Tolman's 16- or 18-electron rule 200-201 toluene 288-289, 298 tortuosity effect 290 trace-level intermediates 78, 87-89, 133 transfer agents 364, 369-370, 377, 381 transfer constants 364, 370 transition state 20 transition-state selectivity 298 transmission of reactivity 322-324 see also chain transfer trimolecular steps 18, 19 Trommsdorff effect 365 tubular reactors 44-45, 47 U ultraviolet light, initiation by 311 ultraviolet spectra 204, 237 uncertainty, relative 67 unimolecular steps 17-18 V variance 65, 67, 72 vinyl acetate 364 vinyl chloride 380, 388, 391 vinyl ethers 380 vinyl ethyne 154-155 2-vinyl thiophene 390 vinylidene chloride 388, 391 volatilization 300 volume variation 40, 59 expansion, contraction 293 see also fluid-density variation correction for 59 W Wei-Prater solution for first-order networks 127-130

488

Subject index

Wilkinson's catalyst 246-249 Winstein spectrum 374, 382 Woodward-Hoffmann exclusion rules 86, 201-202 X-ray fluorescence 51 xylene 298 Yang-Watson tabulation 278-279 yield 13 yield ratio 13 equations 153-155, 190-194, 410

zeolites 297 zero time, in batch reaction 41, 42, 47, 62 Ziegler-Natta polymerization 383 zinc chloride, as initiator 380 ZMS-5 zeolite 297, 298 X coefficients 133, 134-135 A coefficients (segment coefficients) 136, 146, 410-411 4n + 2rule 201-202 16- or 18-electron rule 200-201