Andrei B. Koudriavtsev . Reginald F. Jameson· Wolfgang ...978-3-642-56770-4/1.pdf · I Sextus Empiricus, 'Adversus mathematicos' IX, 363 2 The works of Democritus are mainly known

Andrei B. Koudriavtsev . Reginald F. Jameson· Wolfgang Linert

The Law of Mass Action

Springer-Verlag Berlin Heidelberg GmbH

Engineering ONLINE lIBRARY

http://www.springer.de/engine/

Andrei B. Koudriavtsev Renigald F. J ameson Wolfgang Linert

The Law of Mass Action

With 80 Figures

, Springer

Dr. ANDREI KOUDRIAVTSEV Analytical Centre, D. Mendeleev University of Chemical Technology of Russia Miusskaya sq. 9, 125047 Moscow,

Russia e-mail: [email protected]

Dr. REGINALD F. JAMESON Department of Chemistry, The University of Dundee DDl 4HN Dundee, Scotland, U.K.

e-mail: R][email protected]

Prof. Dr. Dipl.-Ing. Ing. WOLFGANG LJNERT Inst. for Inorganic Chemistry, Technical University Vienna Getreidemarkt 9/153, 1060 Vienna, Austria e-mail: [email protected]

ISBN 978-3-642-62494-0 ISBN 978-3-642-56770-4 (eBook) DOI 10.1007/978-3-642-56770-4

Library of Congress Cataloging-in-Publicat ion Data

Koudriavtsev, A.B. The law of mass act ion / Andrei B. Koudriavtsev, Reginald F. Jameson, Wolfgang Linert. p. cm. Inc1udes bibliographical references and index. ISBN 978-3-642-62494-0 1. Chemical equilibrium. 2. Statistical thermodynamics. 1. Jameson, Reginald F., 1931 - II. Linert, W. III. Title QD503 .K83 2001 541.3'92--dc21 200lO17048

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Preface

'Why are atoms so small?' asks 'naive physicist' in Erwin Schrodinger's book 'What is Life? The Physical Aspect of the Living Cell'. 'The question is wrong' answers the author, 'the actual problem is why we are built of such an enormous number of these particles'.

The idea that everything is built of atoms is quite an old one. It seems that Democritus himself borrowed it from some obscure Phoenician source l . The arguments for the existence of small indivisible units of matter were quite simple. According to Lucretius2 observable matter would disappear by 'wear and tear' (the world exists for a sufficiently long, if not infinitely long time) unless there are some units which cannot be further split into parts.

However, in the middle of the 19th century any reference to the atomic structure of matter was considered among European physicists as a sign of extremely bad taste and provinciality. The hypothesis of the ancient Greeks (for Lucretius had translated Epicurean philosophy into Latin hexameters) was at that time seen as bringing nothing positive to exact science. The properties of gaseous, liquid and solid bodies, as well as the behaviour of heat and energy, were successfully described by the rapidly developing science of thermodynamics.

Despite this attitude of physicists, chemists of the same period employed the concept of atoms in the description of the transformation of matter by chemical reactions. Not only were the dependencies of properties of compounds on their atomic composition and arrangements of atoms within a molecule established but also all known chemical elements were systematised in Mendeleev's Periodic Table. This allowed for the prediction of the existence of yet undiscovered types of atoms and of the properties of their compounds. Even the number of atoms in a given amount of a substance of known atomic (elemental) composition had at that time been correctly estimated.

However, when describing quantitatively the ability of a compound to take part in a chemical reaction, chemists have had to apply the rules of formal thermodynamics. For such practical purposes thermodynamics supplied excellent tools which effectively substitute the relationships between molecules by the relationships between thermodynamic functions (free energy, entropy, etc.), their

I Sextus Empiricus, 'Adversus mathematicos' IX, 363 2 The works of Democritus are mainly known from commentaries of his opponents. The only

coherent exposition of the ancient atomistic theory available is the poem of Lucretius "De rerum natura".

vi Preface

partial derivatives, temperature, volume, pressure and the numbers of moles. Among these parameters the latter alone has some reference to the molecular (atom istic) concepts of chem istry.

One of these rules, the law of mass action, which defines the relationship between the numbers of moles of products and reactants in chemical equilibrium, is extremely important for chemists. Historically it was first established from experimental observations on the rates of chemical reactions employing the dynamic interpretation of chemical equilibrium. Therefore it is very often considered as an empirical rule and/or as a consequence of the kinetic law ofmass action. In fact, the equilibrium law of mass action follows from the fundamental laws of thermodynamics and thus is an empirical rule to the same extent as is the first law of thermodynamics.

The law of mass action in its widely known form can be formulated only for reactions in ideal gases and ideal crystals although, in practise, real gases at low density and crystalline solids at low temperatures approximate to these ideal states. For non-ideal systems approximating equilibria in any other real aggregate state there are two possible ways of establishing a relationship between the amounts of reactants and products. The first is to derive a specific form of the law of mass action for a given non-ideal system. Another, more successful approach was suggested by Lewis [1], in which the universal ideal form of the law of mass action is retained but, instead of concentrations, functions of these called activities are to be used. The parameters of these functions (activity coefficients) once established for one reaction were found to be applicable in the description of a series of similar reactions. Hence, by making use of a limited number of reference equilibria it was possible to obtain the empirical data necessary for the calculation of activities for a very large number of practically important reactions.

An inquisitive 'naive chemist' would probably ask why a proper law of mass action for a non-ideal system could not be derived? The problem, in fact, is not in the derivation but in the practical applicability of the equations so obtained. The mathematics of the ideal law of mass action is very simple, activities are (or assumed to be) linear or polynomial functions of concentrations. This formalism was very attractive to chemists who, at the beginning of this century, were not accustomed to logarithmic functions. In fact, by the time activities were generally adopted by chemists all the basic theoretical relationships had already been developed within the molecular theory of solutions. However, a large amount of data on activities was obtained before this theory assumed a form allowing either practical calculations or an easy interpretation.

The molecular theory of solutions has thus been developing without any competition with the empirical approach of Lewis in the field of practical calculations. Furthermore, the molecular theory of solutions has been largely used to explain the physical significance of activity coefficients. In other words, instead of producing its own formalism this theory was, regrettably, employed to explain the meaning of parameters arising from empirical relationships!

Preface Vll

The molecular theory of solutions had quite early succeeded in a semI­quantitative description of many phenomena that are outside the scope of an empirical approach. Phase transitions, for example, play an important role in chemical processes, determining their apparent irreversibility when one of the products 'drops out' or escapes into a gaseous or a separate liquid phase. This leads sometimes to a misinterpretation of processes involving a phase transition. For example, the substitution of fatty acids in the form of their salts by carbonic acid (when saturating aqueous solutions of their sodium salts with CO2) is often interpreted as evidence of the stronger acidic properties of carbonic acid. It is completely forgotten that free fatty acids form a separate phase at lower temperatures (although they can be miscible with water at higher temperatures). A theory taking into account molecular interactions predicts such phenomena and the corresponding formalism can be incorporated into the law of mass action.

In biochemistry it is considered as a great wonder that small variation of the energy of hydration (of about 1 kcal mole-I) compared to the absolute magnitude of ca. 100 kcal mole-I can switch biological processes in membranes. However, according to the molecular theory of solutions, this should not be surprising because phase separation in liquid solutions occurs when the difference in energies between hetero- and homo-molecular interactions is above 2RT.

It is all too often believed that the activity coefficient, like a deus ex machina, will always allow for the application of a pseudo-ideal form of the law of mass action or of the van't Hoff and Arrhenius equations. A colleague of one of the authors complained once about the loss of large amounts of reagent in a pilot reactor when the synthesis of ethyleneglycol diacetate from ethylene oxide and acetic anhydride proceeded in a jump, bearing all the signs of a branched chain reaction. The real origin of this misfortune could well have been a problem of scaling (a tub full of hot water cools down more slowly than does a glass full!). But it is also possible that the 'jump' is connected with critical phenomena originating in molecular interactions (this case will be repeatedly considered in this book).

The molecular theory of solutions is based on a statistical mechanical description of collections of large numbers of molecules. Unfortunately the principles of statistical mechanics are all too often omitted from textbook discussions of reactions in solutions. Although they cover practical methods of calculation of equilibrium and rate constants for simple gas-phase reactions 'employing spectroscopic data' the fundamentals of these methods are usually neglected. That statistical mechanical methods can successfully be applied to reactions in non-ideal condensed systems is openly disbelieved (not, it must be admitted, entirely without ground). Until recently there were no examples of a chemical equilibrium in the condensed state better described by the formalism of the molecular theory of solution than by the use of empirical activity coefficients.

It was in 1972 that a spin crossover equilibrium in the solid state was successfully described by a law of mass action derived for a strongly non-ideal system [2]. This equilibrium presents unique possibilities for the checking of

viii Preface

theoretical models of processes in the condensed phase exhibiting, as it does, a large number of the effects of molecular interactions yet, at the same time, remaining a homogeneous system. In fact, spin crossover apparently plays an important role in oxidation processes when it is necessary to convert a paramagnetic dioxygen species into the singlet state. In this book various descriptions of ideal and non-ideal systems will be illustrated by reference to examples of such spin crossover systems.

The statistical mechanical approach to the description of matter gives an answer to the question of the 'naive physicist' mentioned above: the bodies of human beings must necessarily consist of very large numbers of particles in order to avoid fluctuations. This phenomenon (completely neglected by formal thermodynamics) also gives rise to some doubts concerning the applicability of the law of mass action to small systems such as bacteria (i. e. a breakdown of the law of mass action is to be expected). Indeed, the law of mass action applied to the water ionisation equilibrium in vesicles containing 107 molecules of water would apparently result (at pH = 7) in the number of ions per vesicle being either one or zero. Calculation of the mean concentration of ions over an ensemble of such vesicles then results in values considerably different from those predicted by the law of mass action. Whether or not this discrepancy has any connection with fluctuations becomes clear when we consider the origin of the law of mass action from a statistical thermodynamics point of view.

The aim of the present book is to give the reader a consecutive and coherent chain of reasoning leading to the formulation of the law of mass action for ideal and non-ideal systems accompanied by the necessary complementary materials.

The first two Chapters and the Appendix are an aid to those who are not sufficiently well acquainted with statistical mechanics. Chapter 1 begins ab ovo with the ideas relating entropy to probability and a derivation of the Maxwell -Boltzmann distribution law. One of the main tasks of statistical mechanics: the calculation of mean and most probable values of mechanical parameters is explained using several examples.

Chapter 2 explains the Gibbs approach to statistical thermodynamics and outlines the derivation of the relationship between the thermodynamic properties and partition functions of canonical distribution.

Chapter 3 concerns the derivation of the law of mass action for an ideal gas and an ideal crystal. Various forms of this law employing molar concentrations, mole fractions and the numbers of molecules as basic variables are derived and their applicability is discussed. The origin of standard free energy, enthalpy and entropy of reaction is explained and referred to the corresponding terms in the statistical mechanical expression for equilibrium constants. Specific problems of the liquid state are then discussed and a model of an 'ideal liquid' is suggested in order to derive two types of the law of mass action applicable to liquid mixtures and to clarify the choice of a proper variable (concentration) in the law of mass action. The case of the so-called 'breakdown' of the law of mass action is critically analysed.

Preface IX

Chapter 4 introduces improvements into the description of chemical equilibria in liquids. These improvements are based on accounting for a finite volume of molecules (i.e. repulsive interactions). The law of mass action is derived for mono-molecular (spin crossover) and bimolecular (formation of weak molecular complexes) equilibria. The concept of free volume and corresponding theories of the liquid state are then discussed. The law of mass action is derived employing the concept of free volume and additivity of the energy of attractive interactions. Singularities predicted by this law of mass action are discussed.

Chapter 5 gives a brief survey of the description of molecular interactions using a Lennard-Jones binary potential in its various forms. The Lennard-Jones and Devonshire model of the liquid state is also outlined.

Chapter 6 considers the problem of molecular interactions in gases using the Virial Theorem as well as a statistical mechanical treatment of a gas with weak binary interactions. The equation of state of a gas with weak binary interactions is compared with the van der Waals equation and the laws of mass action for equilibria in such systems are analysed. The conditions leading to simplification of the law of mass action and its reduction to an ideal form are discussed.

Chapter 7 concerns reactions in the solid state. An original method of calculation of free energy in a solid system of interacting molecules (suggested by ABK) is described and applied to the description of spin crossover equilibria in the solid state. This formalism (reducible under simplifying conditions to that of the theory of regular solutions) is then used to modify the Bragg - Williams approximation of ordered systems, which provides for a quantitative description of the two-step spin crossover.

Chapter 8 analyses (employing formalisms described in preceding Chapters) the physical backgrounds of empirical correlations including Linear Free Energy Relationships (or LFER's), IsoEquilibrium Relationships (IER) and IsoKinetic Relationships (IKR).

Appendices include the elements of classical mechanics in Hamilton's form, some explanations of the basics of statistical mechanics and complementary materials to other Chapters.

The writing of this book would not have been possible without the fact that the Authors were able to meet regularly in the Institut fUr Anorganische Chemie der Technischen Universitat Wien due to appointment of ABK and RFJ as visiting professors in the Institute.

August 2000 A. B. Koudriavtsev R. F. Jameson

W. Liner!

x Preface

References.

1 G.N. Lewis (1901) Z. Phys. Chern. 38: 205; (1907) Proc. Am. Acad. 43: 259. 2 Slichter CP, Drickarner HG (1972) J. Chern. Phys. 56: 2142

Contents

1 Maxwell - Boltzmann Statistics

I. I Thermodynamics and probability. The Boltzmann - Planck theorem ............ I 1.1. I The Boltzmann - Planck theorem ........................... 5

1.2 The Maxwell - Boltzmann distribution law ........................... 7 1.2. I Continuous Maxwell - Boltzmann distribution ................. 14

1.3 Calculation of most probable and mean values ......................... 17 1.4 Indistinguishable molecules. The Gibbs' paradox ....................... 2 I 1.5 Phase volume and the number of quantum states ........................ 24 1.6 Quantum statistics ........................................... 26

1.6. I Bose - Einstein statistics ................................ 27 1.6.2 Fermi - Dirac statistics ................................. 29 1.6.3 Comparison of the three types of statistics ..................... 29 1.6.4 Degenerate ideal gas. . .................................. 31 1.6.5 Applications of Bose - Einstein statistics:

black-body radiation ................................... 34 1.6.6 Applications of Bose - Einstein statistics:

heat capacity of solids .................................. 35

2 Ensembles, Partition Functions, and Thermodynamic Functions

2.1 Gibbs' approach, or how to avoid molecular interactions .................. 43 2.2 The process of equilibration and increasing entropy ..................... 49 2.3 Microcanonical distribution ..................................... 51 2.4. Canonical distribution ......................................... 52 2.5 The probability of a macros tate ................................... 54 2.6 Thermodynamic functions derived from a canonical distribution ............. 55 2.7 Some molecular partition functions ................................ 57

2.7.1 Degeneracy ......................................... 58 2.7.2 Translational motion ................................... 59 2.7.3 Free rotation ........................................ 61 2.7.4 Vibrational motion: linear harmonic oscillator .................. 62 2.7.5 Total partition function of an ideal system ..................... 63

2.8 Fluctuations ............................................... 64 2.9 Conclusions ............................................... 69

xii Contents

3 The Law of Mass Action for Ideal Systems

3.1 The law of mass action, its origin and formal thermodynamic derivation ........ 71 3.2 Statistical formulae for free energy ................................. 77 3.3 Statistical formulae for ideal systems ............................... 79 3.4 The law of mass action for ideal gases ............................... 81

3.4.1 Conversion to molar concentrations ......................... 83 3.4.2 Conversion to mole fractions .............................. 84 3.4.3 Standard states and standard free energies of reaction .............. 85

3.5 The law of mass action for an ideal crystal. Spin crossover equilibria ........... 89 3.6 Liquids ................................................. 95

3.6.1 The law of mass action for an 'ideal liquid' ..................... 97 3.7 'Breakdown' of the law of mass action ............................. 99 3.8 Conclusions ............................................... 105

4 Reactions in Imperfect Condensed Systems. Free Volume

4.1 Additive volume: a semi-empirical model of repulsive interactions .......... 107 4.1.1 Binary equilibrium in a liquid with repulsive interactions ............. 108 4.1.1 Non-isomolar equilibrium in a liquid with repulsive interactions ........ 111

4.2. Lattice theories of the liquid state ................................ 117 4.3 The Lennard-Jones and Devonshire model .......................... 119 4.4 Chemical equilibria in Lennard-Jones and Devonshire liquids .............. 122 4.5 The non-ideal law of mass action, activities, and standard states ............. 129 4.6 Kinetic law of mass action ..................................... 135 4.7 Conclusions ............................................... 143

5 Molecular Interactions

5.1 Introduction .............................................. 145 5.2 Empirical binary potentials ..................................... 147 5.3 Taking into account nearest, next nearest, and longer range

interactions in the condensed phase ............................... 151 5.4 Frequency of vibrations ....................................... 155 5.5 The shape of the potential well in a cell .............................. 157 5.6 Free volume of a Lennard-Jones and Devonshire liquid ................... 160 5.7 Experimental determination of parameters of the

Lennard-Jones potential ...................................... 164 5.7.1 Compressibility: the Born - Lande method ................... 165 5.7.2 Acoustical measurements: the B.B. Kudryavtsev method .......... 166 5.7.3 Viscosity of gases: the Rayleigh - Chapman method ............ 170

5.8 Conclusions .............................................. 171

Contents xiii

6 Imperfect Gases.

6.1 Introduction. The Virial Theorem ................................ 173 6.2 The Rayleigh equation ........................................ 176

6.2.1 Virial coefficients: the Lennard-lones method for the determination of the parameters of a binary potential ............... 177

6.2.2 Free energy derived from the Rayleigh equation of state ............. 179 6.3 A gas with weak binary interactions:

a statistical thennodynamics approach ............................. 180 6.4 Van der Waals equation of state .................................. 185 6.5 Chemical equilibria in imperfect gases ............................. 188

6.5.1 Isomolar equilibria in imperfect gases ......................... 189 6.5.2 A non-isomolar reaction in an imperfect gas ..................... 192 6.5.3 Separate conditions of ideal behaviour for attractive

and repulsive molecular interactions .......................... 195 6.5.4 Associative equilibria in the gaseous phase ...................... 196 6.5.5 Molecular interaction via a chemical reaction .................... 198

6.6 Conelusions ............................................... 200

7 Reactions in Imperfect Condensed Systems. Lattice Energy

7. I Exchange energy ........................................... 203 7.2 Non-ideality as a result of dependence of the partition function

on the nature of the surroundings ................................. 205 7.3 Exchange free energy ......................................... 208 7.4 Phase separations in binary mixtures ............................... 213 7.5 The law of mass action for an imperfect mixture in the condensed state ........ 216 7.6 The regular solution model of steep spin crossover ..................... 219 7.7 Heat capacity changes in spin crossover ............................ 223 7.8 Negative exchange energy. Ordering. The Bragg - Williams approximation .... 226 7.9. Description of ordering taking into accounttriple interactions .............. 232 7.10 Chemical equilibrium in ordered systems. Two-step spin crossover .......... 234 7.11 Diluted systems ............................................ 240 7.12 Conclusions ............................................... 246

8 Chemical Correlations

8.1 Studies of variations of chemical reactivity .......................... 249 8.1.1 Molecular parameters governing variations of chemical reactivity ....... 250 8.1.2. Solvent effects ......................................... 252 8.1.3. Kinetic studies ......................................... 254 S.I.4 Multidimensionality of variations. Reference reactions .............. 257

8.2 Linear free energy relationship. Modification of reactants ................. 261 8.3 Linear free energy relationship. Variation of solvent .................... 267 8.4 Isoequilibrium and isokinetic relationships ........................... 270

S.4.1 Statistical-mechanical model of the IER in ideal systems ............. 273 8.4.2 The I ER in gas-phase reactions ............................. 276

XIV Contents

8.4.3 Isokinetic relationships ................................... 278 8.4.4 Non-ideality as a source of an IER ........................... 282 8.4.5 IER and exchange energy ................................. 288

8.5 Conclusions ............................................... 293

9 Concluding Remarks ....................................... 295

10 Appendices

10.1 Lagrange equations and Hamilton (canonical) equations ................. 303 10.2

10.3 10.4 10.5 10.6

107 10.8

Phase space ............................................... 309 10.2.1 The phase space of a harmonic oscillator ....................... 310 10.2.2 The phase space of an ideal gas ............................. 311 Derivation of the canonical distribution ............................ 313 Free volume associated with vibrations ......................... 314 Rotational contribution to the equilibrium constant of the ionisation of water ... 316 Forms of the law of mass action employing the r function approximation of the factorial .......................... 317 Derivation of the van der Waals equation of state ...................... 318 Exchange energy ........................................... 319

10.9 Activity coefficients derived from the non-ideality resulting from triple interactions ................................. 319

10.10 The law of mass action for a binary equilibrium in a system with non-additive volume and lattice energy ................................. 320

10.11 Physico-chemical constants and units of energy ...................... 322

11 Index ............................................... 323