ORGANIC REACTWHY IN MIXED AQUEOUS SOLVENTS

A LINK BEIWEEN -CS AND THERMODYNAMICS

ORGANIC REACTIVITY IN MIXED AQUEOUS SOLVENTS

A LINK BETWEEN KINETICS AND THERMODYNAMICS

RI JKSUNIVERSITEIT GRONINGEN

ORGANIC REACTIVITY IN MIXED AQUEOUS SOLVENTS

A LINK BETWEEN KINETICS AND THERMODYNAMICS

PROEFSCHRIFT

ter verlu-ijging van het doctoraat in de Wiskunde en Natuurwetenschappen aan de Rijksuniversiteit Groningen

op gezag van de Rector Magnificus Dr.S.K.Kuipers in het openbaar te verdedigen op

vrijdag 29 november 1991 des namiddags te 4.00 uur

door Wilfried Blokzijl

geboren op 30 maart 1964 te 's-Gravenhage

Promotores: Prof.Dr. J.B.F.N. Engberts Prof.Dr. M. J. Blandamer

Aan mijn ouders

VOORWOORD

Water heeft door de eeuwen heen niet alleen componisten, dichters, schrijvers en schilders maar ook wetenschappers weten te inspireren. Terwijl voor velen de uiterlijke schoonheid van water, in a1 zijn verschijningsvormen, een belangrijke bron van inspiratie was, raakten wetenschappers ook gefascineerd door de moleculaire eigenschappen van de "matrix en de moeder van het leven" (A. Szent-Gyorgy). Ook ik ben de afgelopen jaren ge'infecteerd door het "watervirus". Het complexe gedrag van een ogenschijnlijk zo eenvoudige en alledaagse verbinding als water vormde voor rnij een continue uitdaging die, naar ik hoop, duidelijk wordt bij het lezen van dit proef- schrift.

De afgelopen vier jaar waarin dit onderzoek is uitgevoerd zijn voor rnij bijzonder prettige jaren geweest. Dit is voor een belangrijk deel te danken aan de goede sfeer op het organisch chemisch laboratorium. Vooral de bewoners en de exbewoners van het "waterlab" hebben altijd bijgedragen tot een goed wetenschappelijk werkklimaat en de zo broodnodige gezelligheid.

Mijn grote waardering en dank gaat uit naar Jan Engberts, die binnen zijn werkgroep een sfeer heeft weten te creeren waardoor het mogelijk was op een creatieve manier te "stoeien" met de chemie. De vrijheid en het vertrouwen die hij mij heeft gegeven zijn voor rnij altijd heel belangrijk geweest. Met plezier denk ik terug aan de vele discussies en gesprekken die ik, op het lab of bij andere gelegenheden, met hem heb gevoerd. Deze discussies en gesprekken gingen, vanzelfsprekend, vaak over de chemie, maar ook over de filosofische kanten van het leven. Although the contacts with Mike Blandamer were, due to geographical limitations, less frequent, they were at least as intense. I will not easily forget our two-days sessions. Within these two days, his enthousiasm always convinced me that chemistry is fun. The pile of letters and computer output that crossed the North Sea as well as the frequent telephone calls made doing research time and again "super". I want to express my heartfelt thanks to both Jan and Mike for many discussions during the realisation of the preprint.

Mijn dank gaat ook uit naar de leden van de leescommissie, Pr0f.Dr.H.J.C. Berendsen, Prof.Dr.A.M.van Leusen en Prof.Dr.G.Somsen voor hun vlotte correctie van de preprint en hun waardevolle suggesties.

Bij dit onderzoek was de technische hulp van Marten de Rapper, wiens blik soms a1 voldoende was om weigerachtige UV-apparatuur tot de orde te roepen, onmisbaar. Anno Wagenaar en Wim Kruizinga wil ik bedanken voor hun geduld toen ik steeds maar weer in H,O, en niet in D,O mijn kinetische NMR-experimenten wilde uitvoe- ren. Marinus Suijkerbuijk was altijd weer bereid rnij met raad en daad bij te staan bij de experimenten met en rond het GC-apparaat. Bij de experimenten met de gasfase- osmometer, die helaas niet altijd de venvachte resultaten gaven, kon ik altijd weer rekenen op de hulp van Jan Ebels. Klaas Hovius wil ik bedanken voor zijn syntheti- sche hulp. During three months, Francesca Bortolozzo brought some italian atmos- phere into our laboratory. She did "of course" an important part of the synthetic and kinetic work, described in Chapter 7. Jan Willem Wijnen haalde (alle) onvermijdelijke typefouten uit de preprint. Tenslotte wil ik Willem Kuil bedanken voor het tekenen van de talloze grafieken en figuren. Hier kan nog steeds geen computer tegenop.

CONTENTS

Chapter 1 Solvent effects on organic reactions in aqueous mixtures . An introduction

.................................. 1.1 Solvent effects in chemistry 1 1.1.1 Some words about the historical background ............ 1

. . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Water; menstruum universale? 1 1.2 A survey of approaches to the analysis of solvent effects ............ 2

1.2.1 Correlations with physical properties of the solvent . . . . . . . . 4 1.2.2 Correlations with emperical and semi-emperical solvent

parameters ..................................... 4 1.2.3 Analysis of solvent effects using solubility and transfer

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . parameters 6 1.2.4 Theoretical treatments of solute-solvent interactions in

relation to solvent effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Interactions and reactivity in water and in mixed

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . aqueous solvents 8 1.4 Hydrophobic effects; definitions and the state of the art . . . . . . . . . . . . 8 1.5 The need for a quantitative description of solvent effects in mixed

(aqueous) solvents . Incentives for this study . . . . . . . . . . . . . . . . . . . . . . 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Aims of this study 15

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Survey of the contents 15

Chapter 2 A quantitative analysis of solvent effects in mixed solvents . Development of a theoretical model

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction 19 2.2 Solvent effects in dilute mixed solvents . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2.1 Kiiietics of a reaction; transition state theory . . . . . . . . . . . . 21 2.2.2 Chemical potential and activity coefficients of solutes in

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . dilute mixed solvents 22 2.2.3 Dependence of excess properties on composition . . . . . . . . . 23 2.2.4 Solute-solute interactions; additivity approach . . . . . . . . . . . . 27

2.3 Modifications of the general theoretical model for quantitative analysis of solvent effects in mixed solvents . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Solvolysis reactions 29 2.3.2 Chemical equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Bimolecular reactions 31 2.3.4 Solvent effects on partial molar enthalpies and entropies

of activation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.4 Solvent effects in mixed binary solvents . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.4.1 Kinetics of reactions; transition state theory . . . . . . . . . . . . . 35 2.4.2 Dependences of the standard chemical potential of solutes on

the composition in mixed binary solvents . . . . . . . . . . . . . . . 35 2.5 A comparison of the approaches used to develop a theoretical model for

the analysis of solvent effects in mixed solvents . . . . . . . . . . . . . . . . . . . 39

Chapter 3 Quantitative analysis of solvent effects in highly aqueous media . Application of SWAG procedures and a critical appraisal of the additivity principle

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2 Solute-solute interactions in aqueous solution . . . . . . . . . . . . . . . . . . . . 42 3.3 Medium effects of monohydric and polyhydric alcohols on the hydrolysis

reaction of l.benzoyl.3.phenyl.l,2, 44riazole ...................... 47 3.4 Towards a better understanding of medium effects in dilute aquous

solution of monohydric and polyhydric alcohols ................... 56 ....................................... 3.5 Concluding remarks 60

3.6 Experimental section ....................................... 60

Chapter 4 A survey of solvent effects on some organic processes in dilute aqueous solution

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.2 Solvent effects on the hydrolysis of p-methoxyphenyl dichloroacetate

in aqueous solutions containing urea and alkyl-substituted ureas . . . . . . . 64 4.3 Solvent effects on a bimolecular Diels-Alder reaction . . . . . . . . . . . . . . . 72 4.4 Solvent effects on an intramolecular Diels-Alder reaction . . . . . . . . . . . . 75 4.5 Solvent effects on the keto-en01 equilibrium . . . . . . . . . . . . . . . . . . . . . 76 4.6 Experimental section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Chapter 5 Alkyl substituent effects on the neutral hydrolysis of l.acyl.3.alkyl.1,2, 4.triazoles in highly aqueous media . The importance of solvation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction 83 5.2 Alkyl substituent effects . Current views . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.3 Solvent effects and substituent effects on the hydrolysis of

l~acyl.(3~substituted).1,2, 4.triazoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.4 A link between alkyl substituent effects and solvent effects . . . . . . . . . . 94 5.5 The molecular origin of alkyl substituent effects . . . . . . . . . . . . . . . . . . . 96 5.6 Solvophobic acceleration and substituent effects . . . . . . . . . . . . . . . . . . 97 5.7 Experimental section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

Chapter 6 Diels-Alder reactions in mixed aqueous media . Enforced hydrophobic interaction

6.1 Introduction ............................................ 101 6.2 Diels-Alder reactions in water; mechanistic considerations and

recent developments ...................................... 103 6.3 Kinetic studies of Diels-Alder reactions in mixed aqueous solvents;

a theoretical approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.4 Properties of alcohol water mixtures .......................... 111 6.5 Diels-Alder reactions in water and in mixed aqueous solvents;

...................................... experimental results 111 6.6 A quantitative analysis of solvent effects on Diels-Alder reactions

in mixtures of water and monohydric alcohols . . . . . . . . . . . . . . . . . . . 123 6.7 Hydrophobic effects on Diels-Alder reactions in water . . . . . . . . . . . . . 130 6.8 Hydrophobic effects on Diels-Alder reactions in

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . mixed aqueous solvents 133 6.9 Experimentalpart . . . . . . . . . . . . ............................ 134

Chapter 7 A quantitative analysis of solvent effects on intramolecular Diels-Alder reactions in mixed aqueous solvents

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 7.2 Intramolecular Diels-Alder reactions; an o v e ~ e w . . . . . . . . . . . . . . . . 140 7.3 Intramolecular Diels-Alder reactions of furan derivatives . . . . . . . . . . . 143 7.4 Synthesis and intramolecular Diels-Alder reactions of N-furfuryl-N-

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . alkyl maleamic acids 145 7.5 Solvent effects on the intramolecular Diels-Alder reactions of

. . . . . . . . . . . . . . . . . . . . . . . . . . . . N-furfuryl-N-alkyl maleamic acids 146 7.6 A quantitative analysis of solvent effects on intramolecular

Diels-Alder reactions in mixed aqueous solvents . . . . . . . . . . . . . . . . . 150 7.7 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 Experimental section 154

Chapter 8 Epilogue . A link between organic chemistry in aqueous solutions and hydrophobic effects

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction 159 8.2 A quantitative approach for the analysis of solvent effects in mixed

solvents . Merits and shortcomings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 8.3 Hydrophobic effects . Hydrophobic interactions

and hydrophobic hydration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 8.4 Organic reactivity in mixed aqueous solvents . . . . . . . . . . . . . . . . . . . . 167

References 169

Summary 179

Samenvatting 183

I . Introduction Solvent gects on organic reactions in aqueous mirtwes

CHAPTER 1

Solvent Effects on Organic Reactions in Aqueous Mixtures. An Introduction

2.1 Solvent effects in chemistry

2.1.1 Some words about the hirtorical background

Reactivity of molecules and ions in solution is largely dictated by the solvent. Compa- rison of rate constants in the gas-phase and in solution shows that differences of 10" are not uncommon1. In 1862, Berthelot and PCan de Saint-Gilles were the first to notice the considerable influence of the reaction medium on the rates of homogene- ous chemical reactions2. In 1890, Menschutkin performed the first detailed study of the reaction of trialkylamines with haloalkanes in twenty-three different solvents and stated that "solvents are by no means inert in chemical reactions and can greatly influence the course of themN3. Since then many papers have appeared in which more or less dramatic solvent effects are reported on a large variety of chemical processes. Changing the solvent can, in extreme cases4, lead to rate variations in the order of lo9. Not only chemical reactions, but also chemical equilibria are sensitive to the solvent. This was shown by ClaisenS and wislicenus6, who were, in 1896, among the first authors to draw attention to the considerable solvent effect on keto-en01 tautomeric equilibria. The fact that the solvent can seriously affect spectroscopic properties of molecules in solution was demonstrated by ~ u n d t ' in 1878. During the past century, solvent effects have also been reported on a number of other chemical phenomena, such as aggregation8, complexation9, ionisationlo, conformation" and isomerisation12. Solvent effects on chemical reactivity have received close attention. A long-lived goal of chemists at large has been to establish methods and to provide tools to understand and predict solvent effects on chemical reactions. Ultimately, this knowledge should enable chemists to choose in a rational way a suitable solvent for a particular chemical transformation.

1.1.2 W a r ; menstruum univer~ale?'~

According to ancient Greek philosophy every solution was called "water" and all liquids, able to dissolve compounds were classified as "divine water". Water was in fact the first liquid to be considered a solvent. Many publications and textbooks claim that water is in every aspect a unique solvent and liquid. Water certainly is the most extensively studied liquid. Properties, models and theories have been discussed in detai114-16. Although living organisms depend in a unique way on water as a solvent for biochemical transformations, synthetic organic chemists do not particularly like water

1. Introduction Solvent effects on organic reactions in aqueous mixtures

Table 1-1: Summary of organic reactions performed in homogeneous and heterogene- ous aqueous media.

Reaction References

Intermolecular Diels-Alder reactions

Intramolecular Diels-Alder reactions

Claisen rearrangements 18f,g

Aldol reactions of silyl-en01 ethers 19c

Benzoin condensation 17e

Reduction of alkyl halides with tinhydride 17f

Allylation of carbonyl compounds using zinc 22a,b, 29

as a solvent for chemical reactions. This "hydrophobic" attitude stems largely from the fact that water is far from a "Menstruum Universale" for organic compounds. Moreover, water is often highly reactive towards many organic reagents. Nevertheless, the past decade has witnessed a remarkable reappraisal of water as a solvent for organic reaction^'^-^^. This change in attitude is partly due to the pioneering work of res slow"", who, in 1980, reported intriguing solvent effects of water on notoriously solvent-insensitive Diels-Alder reactions. This discovery inspired others to search for other organic reactions that could benefit likewise from water as a solvent. In addition, the need for solvents that can satisfy the high requirements of current environmental legislation, makes water an interesting choice as a solvent for organic reactions. In Table 1-1 a collection of organic reactions is surnrnarised that are not traditionally performed in water, but were found to benefit from the presence of water as a solvent. The molecular origin of these remarkable and sometimes even spectacular solvent effects in aqueous solution remains unclear.

1.2 A survey of approaches to the analysh of solvent effect^^'-^^

Solvent effects on chemical processes are usually studied in comparison to reactivity in a reference solvent. In some cases comparison is made with gas-phase reactivity. Generally, analysis of solvent effects has been based on an equilibrium solvation model. However, for some organic reactions the reaction rates can be very high and examples are known for which non-equilibrium or dynamic solvation models have to be used to account for reactivity in solution. Consequently, sophisticated theoretical models have been developed. In the past, quantitative and qualitative approaches for the analysis of solvent effects have been developed almost simultaneously.

1. Introduction Solvent effects on organic reactiom in aqueous mixtures

Qualitative descriptions of solvent effects. Almost all qualitative treatments of solvent effects are based on the simple solvation model, developed by Hughes and Ingold in 1935, for explaining solvent effects on substitution and elimination reactionsB. The model considers mainly the change of electrical interactions between solvent and reacting species during the activation process. Solvents are thus classified according to their ability to solvate ions and molecules. A serious shortcoming of this approach is the fact that the solvent is considered as a continuum without defined structure and that specific solvent-solute interactions are completely neglected. In addition, changes in the structure of solvents as a result of changes in solvation during the activation process are neglected. Also entropic contributions to solvent effects are not incorpora- ted into this model. Especially in highly structured solvents such as water, entropic contributions to solvent effects can be significant. Notwithstanding the weak points of these qualitative models, these approaches are simple and are readily applied; they are still very popular in practical chemistry.

Quantitative descriptions of solvent effects. The starting point of most quantitative approaches for the interpretation of solvent effects is based on transition state theory (TST). A general quantitative description of solvent effects is given in Equation 1-1,

in which

A pi,= pic(S) - P ~ H ) and

where ApIsO and ApACO are the transfer standard chemical potentials of the initial state and the activated complex, respectively, for transfer from the reference solvent R to solvent S. The rate constants, found in solvents S and R are k and k,, respectively. The success of any quantitative method in describing solvent effects is determined by the accuracy with which the Gibbs energies of solvation can be calculated. A detailed knowledge of solvent-solute interactions is essential for this goal. The wide range of possible solute-solvent interactions, the actual structure of the solvent, the characteris- tics of the reacting molecules and the activated complex make enormous demands on the theory. (Semi)quantitative approaches can be subdivided into four major catego- ries:

(i) Correlations with physical properties of the solvent. (ii) Correlations with empirical solvent parameters. (iii) Analysis incorporating solubility andfor transfer parameters. (iv) Theoretical treatment of solute-solvent interactions.

In separate sections, these approaches will be briefly outlined and critically discussed.

1. Introduction Solvent pffects on organic reactions in aqueous rnirtures

1.2.1 Correlations with physical properties of the solvent

Solvents can be classified according to many physical properties. Some of these have been used for the correlation of solvent effects. Frequently, the justification for these correlations is found in some theoretical model. Most popular are correlations with the relative permittivity, E, of the solvent. Solvent effects are often related to functions like [(~;l)/(&,+l)] or I/&,. The theoretical basis for these correlations has been given by Eyring, Kirkwood, Laidler and ~andskroene?"'. Another well-known method, developed by Scatchard and Hildebrand, is based on the energy of evaporati- on of a solvent per unit of volume, the cohesive pressure or cohesive energy density, c or aH2 384. Related to this approach are correlations with the internal pressure of the solvent42, -rr. These methods are mainly used to analyse solvent effects on reactions between neutral molecules. Koppel and palm4' emphasised the importance of the polarisability of the solvent, which can be expressed by a function of the refractive index of the solvent, [(n2-1)]/[p2+l)]. In this light, also correlations with the polarisa- bility parameter P are known4 744. Finally, correlations with the viscosity of the solvent have been used to explain solvent effects on diffusion-controlled reactions45746.

Correlations of solvent effects with these solvent parameters are often surprisingly good. The inherent weakness of the method is that the parameters measure macro- scopic properties. Specific solute-solvent interactions that occur on a microscopic scale are completely neglected. The structural changes of the solvent, accompanying the activation process, are also neglected. Correlations with physical properties of the solvent are mainly associated with enthalpic contributions to the overall solvent effect, which makes the method less suitable for reactions in highly structured solvents such as water.

1.2.2 Correlations with empirical and semi-empirical solvent parameters

The ability of a solvent to solvate molecules or ions, sometimes rather ambiguously described as the solvent polarity, is difficult to express in terms of physical solvent parameters. This problem forced chemists to search for empirical or semi-empirical scales of solvent polarity. The list of solvent polarity scales is long and lengthens every yea?'. The scales are based on linear solvent energy relationships (LSER) and use solvent effects on properly selected model processes. In the literature, correlations involving solvent polarity parameters are frequently encountered. Some polarity scales have found a wider application in the analysis of solvent effects and will be described briefly below.

Most popular polarity scales are based on spectroscopic properties. Spectroscopic data describing solvent effects on UVIVIS transitions of a large number of solvato- chromic dyes resulted in a long list of solvatochromic polarity parameters4'. One of the oldest parameters, introduced by ~ o s o w e r ~ ' . ~ ~ in 1958, the Z-value, is still frequently used. Dimroth and ~eichhardt~ '~ ' reported probably the best known and most frequently used solvent polarity parameter, the q 3 0 ) value. The standard probe molecule is a pyridinium-N-phenoxide betaine dye (I), which has a T-T' absorption band with intramolecular charge-transfer characteristics.

I. Introduction Solvent effects on organic reactions in aqueous miaures

This parameter can be determined in many solvents and is very solvent sensitive. Another series of parameters, based on spectroscopic transitions was introduced by Kamlet, Taft and Abraham and is related to specific properties of the s o l ~ e n t ~ ~ . ~ ~ . The a-scale measures the hydrogen bond donor acidity of the solvent, the p-scale the hydrogen bond acceptor basicity of the solvent and the T*-scale the polarisability or dipolarity of the solvent. The a-scale and the p-scale are based on a solvatochromic comparison method, using solvatochromic shifts of 6nitroaniline and N,N-diethyl-4- nitroaniline. Similarly, the T-scale is based on electronic T-T* transitions of seven nitroaromatics. The Acceptor Number (AN), introduced by Gutmanns4, is based on the relative 3 1 ~ - ~ ~ ~ chemical shift values of triethylphosphane oxide related to those of the 1:l adduct Et,PO-SbC15. The scale classifies the solvent according to Lewis acidity or the Electron Pair Acceptor property of the solvent.

A few solvent polarity scales have been related to chemical equilibria. The Donor Number (DN) measures the Lewis basicity of the solvent'0JS~s6. SbCls was used as a reference compound. The parameter is defined by the enthalpy associated with the adduct formation between antimony pentachloride and Electron Pair Donor solvents.

The oldest polarity scales find their origin in kinetic measurements. A famous example is the ionising power, as defined by Winstein and ~runwald '~ and expressed in the solvent Y-scale. This scale has been developed on the basis of the rate constant for the SN1 solvolysis of t-butylchloride in different solvents. Later, this approach was modified by winstein*', ~ e n t l e g ~ ' ~ ' , and others6' in order to extend the procedure to correlations of solvent effects on reactions involving borderline or even pure S,2 mechanisms.

Finally, a large number of polarity scales are based upon solubility data6', transfer parameters6z63, partition coefficient^^^>^' and chromatographic These parameters quantify the phobicity or philicity of a selected compound for the solvent. In physical organic applications the Hansch value^^*^^ as well as the S,-value, recently introduced by brah ham^', are frequently encountered. Both parameters measure the solvophobicity of apolar molecules for different solvents. Some empirical parameters based on partition coefficients, solubility, and Gibbs energies of transfer are mainly used in industrial and engineering applications.

I . Introduction Solvent effects on organic reactions in aqueous mixtures

The inherent weakness of these methods is that the solvent polarity scale is based on a selected process and is therefore not universal. Obviously, only satisfactory correlations can be expected for solvent effects on processes closely related to those used to define the polarity scale. In addition, it is not always clear which of the many types of solute-solvent interactions is expressed by a certain parameter. By careful selection of the model process or by sophisticated comparison methods5z5396&70, some polarity scales have been given a relatively well-defined physical meaning. These methods are based on the fact that some classes of compounds interact with the solvent via a predominant and well-defined mode of interaction. A detailed interpreta- tion of solvent effects in terms of specific interactions based on polarity scales is, however, extremely difficult. The choice of a suitable solvent polarity scale to correlate solvent effects on a new process is often simply pragmatic. A practical limitation is that solvent polarity parameters often cannot be measured in all solvents.

It has become clear that no single solvent parameter can account for the complex nature of solvent effects on different chemical processes. To circumvent this problem, multiparameter approaches have been advocated. Koppel and palm7', and later Kamlet, Taft and brah ham" and others71772 developed sophisticated multiparameter equations, incorporating two to four empirical, semi-empirical or physical solvent parameters. Selection of a set of independent solvent parameters that incorporates all possible contributions to the overall solute-solvent interaction is rather arbitrary. Every single parameter measures a combination of distinct contributions to the overall solute-solvent interaction for which both the identity and magnitude are not properly defined. An interpretation of the results of a linear regression analysis is difficult and often highly speculative. From a more practical point of view, in particular for predictive applications, the use of linear regression analysis is seriously limited by the fact that for a proper calculation many datapoints are required. Finally, the theoretical background of LSER's and its application as a tool to analyse or even predict kinetic parameters for new chemical transformations, has been seriously criticised7'. Discussi- on of the fundamental aspects of LSER's is interesting and often philo~ophical~~.

1.2.3 Analysh of solvent effects using solubility and transfer parameters

Measurement of transfer parameters and solubilities to analyse solvent effects is directly linked to the general Equation 1-1. Many techniques are available to measure the standard chemical potentials and partial molar enthalpies and entropies of transfer of compounds from one solvent to the ~ t h e r ~ ' - ~ ~ . Transfer parameters can be used (i) to develop a solvent polarity scale or (ii) to determine the change in standard Gibbs energy, enthalpy and/or entropy for the transfer of reactants for a particular process. In the latter case, solvent effects can sometimes be analysed in terms of initial state and transition state effects7'. In fact, measurement of transfer parameters is the only way in which solvent effects can be analysed in terms of initial and transition state effects. In some cases, initial state and transition state effects have been subsequently analysed in terms of solvent polarity scales72.

In recent and interesting fundamental studies of reactivity in the gas phase, reactants, present in the gas phase, are stepwise "solvated" by a. limited and accurately

1. Introduction Solvent effects on organic reactions in aqueous mirtures

known number of solvent molecules. The reactivity of these molecules in clusters is monitored as a function of the number of solvent molecules present, and shows that even a small number of solvent molecules can bring about a large change in reactivi- ty77-79. Even with a limited number of solvent molecules, reactivity in the gas-phase starts to resemble the reactivity in the condensed phase.

1.2.4 Theoretical treatments of solute-solvent interactions in rehation to solvent effeca

The past fifteen years have shown an impressive number of sophisticated theoretical studies concerned with reactivity in the condensed phase. However, an unambiguous and universal theory of the liquid state still does not exist. This fact strongly hampers the theoretical treatment of reactivity in solutions. Early theoretical approaches were mainly based on models of Kirkwood and O n ~ a g e P ~ ' ~ .

The first simulations of reactivity in solution appeared in the late sixties. The accessibility of large computers and suitable quantum mechanical methods resulted in a whole new area of theoretical studies of solvent effects. The first ab initio calculati- ons of solvent effects were based on reactive com ounds, "solvated by a few solvent molecules; the so-called super-molecule approachr2. These ab initio approaches are still in the early stages of developments3. Other methods use quantum mechanical approaches to calculate the reaction trajectories in the gas-phase, which are subse- quently transfered into a box of solvent molecules84986. Molecular dynamics is used for allowing reacting systems to equilibrate with the solvent along the total reaction path. Until recently only rather simple and elementary processes, like the substitution reaction of chloromethane with chloride ion, have been studied in detail. Strikingly, reactivity in water has received most attention.

Many theoretical studies are focused on non-equilibrium solvation. Conventional transition state theory is shown to be unsuitable for the treatment of solvent effects in these casess4. Among the topics which can be studied using modern computers are (i) the time-scale of the reaction dynamics, (ii) the extent and importance of energy flow from solvent to reactants and (iii) the involvement of the solvent dynamics in the activation process. Recent calculation^^^ of the solvent effect on a S,Zprocess in water show that water undergoes a substantial reorganisation well before the change in the charge distribution of the reactants. This reorganisation appears to be crucial for the overall reaction.

The development of theoretical models to calculate solvent effects is, however, still in its infancy. Most methods do not take account of mechanistic changes that might be induced by the solvent. The chemical processes, treated by theoretical models, are necessarily still simple and elementary. Consequently, the results do not yet attract the interest of experimental chemists. Unfortunately, the complexity of the methods still creates a considerable gap between experimentalists and theoreticians. It is striking that theoretical approaches of solvent effects do not appear in recent reviews and textbooks on solvent effects in chemist$"32972. Apparently, reading theoretical papers gives experimentalists, dealing with solvent effects, a genuine feeling of dissatisfaction, which makes them turn to their familiar solvent polarity scales.

1. Introduction. Solvent effects on organic reactions in aqueous mi-s

1.3 Interactions and reactivity in water and in m h d aqueous solvents

Organic reactivity in water and in mixed aqueous solvents is determined by interacti- ons of water and cosolvents or cosolutes* with the reactant(s) and the activated com- p l e ~ ~ ~ ~ ~ ~ . Water-solute interactions reflect the fact that water molecules are small, moderately polarisable and able to form a highly structured hydrogen bonded network. Induced dipole-induced dipole interactions of water with solutes are small, but the dipole moment of water does enable significant dipole-induced dipole and dipole-dipole interactions with solutes. Obviously, an important contribution to the overall solute-solvent interactions in water is hydrogen bonding. Finally, the interacti- on of water with charged solutes is very strong.

Hydrophobicity and hydrophobic hydration play an important role in the solvation of reactants and activated complex in water and in mixed aqueous solvents. Hydro- phobic effects are characterised by intriguing thermodynamic properties and are the result of a combination of water-solute and water-water interactions. Traditionally, hydrophobicity and hydrophobic hydration are considered to be a consequence of the preference of water for interaction with other water molecules over interaction with hydrophobic (i.e. apolar) solutes. In a study of solvent effects on reactions in aqueous reaction media, a discussion of hydrophobicity and hydrophobic hydration is essential. Recent views on these hydrophobic effects will be discussed in more detail in Section 1.4.

Interactions of reactant(s) and activated complex with cosolvent or cosolute molecules in aqueous solutions involve all forms of dipolar interactions mentioned above, but are strongly mediated by water. The thermodynamics of these interactions strongly depend on the concentration of the solute molecules. In addition, hydropho- bic interactions play a significant role in intermolecular interactions between solutes in aqueous media. The interactions between hydrophobic compounds or hydrophobic moieties in aqueous media is thermodynamically very complex, and some recent ideas about hydrophobic interactions will be also outlined in Section 1.4.

1.4 Hydrophobic effects; &finitions and the state of the ad9-''

Hydrophobic effects; definition. Disagreements exist in the literature concerning the definition of "hydrophobicity", "hydrophobic effects", "hydrophobic hydration" and "hydrophobic interactions". The terms are strongly .related and some are practically synonymous. Many different definitions for these terms can be encountered in the literatures98. In order to prevent unnecessary ambiguities, a definition of these hydrophobic terms will be given below.

The term "hydrophobic effects" describes all phenomena related to the dissolution of non-polar solutes in aqueous media and is, as such, a quite general term. The term "hydrophobicity" is ambiguous and many different interpretations are found in the

The definition of organic, irreactive compounds in aqueous media as cosolutes or cosolvents is ambiguous. Following thermodynamic formalism, the term "cosolute" is more appropriate in dilute aqueous media, whereas in binary aqueous mixtures the term "cosolvent" is prefered.

1. Introduction. Solvent effects on organic reactions in aqueous r n h r e s

literature. The extent of "hydrophobicity" can be expressed in an experimentally accessible parameter measuring the solubility of apolar substances in waterg9. This parameter involves (a) the breaking of the solute-solute interactions and (b) refilling of the vacancy in the apolar medium, (c) creation of a cavity in the aqueous medium, (d) onset of solute-water interactions and (e) rearrangement of the water molecules surrounding the solute. Alternatively, "hydrophobicity" can be expressed in terms of a transfer process of an apolar compound from an apolar solvent to water. In this case, process (a) is replaced by breaking of the solute-solvent interactions.

The poor interactions between water and apolar solutes make the small but significant solubility of completely apolar compounds in water rather unexpected. "Hydrophobic hydration" has been suggested to account for this "high" solubility of apolar compounds in water. Privalov and Gi111009'03J08~'09~111 used the term "hydration effect" to describe this phenomenon. "Hydrophobic hydration" can be defined as the combination of process (c), (d) and (e).

The term "hydrophobic interaction" seems to express the discomfort of chemists in dealing with non-covalent interactions between apolar molecules in aqueous solution that appear to be predominantly entropic in origin. It is necessary to make a clear distinction between "bulk hydrophobic interactions" and "pairwise hydrophobic interaction^"'^^^^^^. Unfortunately, this distinction is seldomly recognised in discussions about hydrophobic interactions. "Bulk hydrophobic interaction" describes the tendency of apolar molecules or moieties to form solvent unseparated clusters and can be conveniently described as the reverse of process (a)-(e). "Pairwise hydrophobic interaction" refers to the potential of average force between two hydrophobic solutes in water, expressed in G(R), the gradient of which determines the force, necessary to bring the two solutes S from an infinite distance to a distance R, as expressed in Equation 1-292.

In this equation, Uss(R) is the solute-solute interaction potential, or the work required to bring about the same process in vacuum. G~'(R) is the contribution of water to the process in aqueous solution. This term is difficult to quantify and sometimes refered to as hydrophobic interactiong2. The contribution of water to the process of "bulk hydrophobic interactions" is even more difficult to establish experimentally. The terms "hydrophobic hydration" and "hydrophobic interactions" are poorly defined in the literature and the lack of a proper definition has resulted in discussions with a strongly semantic f l a v o ~ r ' ~ ~ ~ ' ~ ~ .

Hydrophobic effects; the state of the art. The classical description of hydrophobic hydration, as put forward by Frank and Evanslo4 in 1945 is still very popular. This "iceberg model" was later quantified by Nemethy and ~cheraga"' and Frank and wedo6. In 1959, hydrophobic interactions were introduced in a famous paper by ~auzrnann~~' . These four papers set the stage for later studies on hydrophobic effects. Figure 1-1 shows the almost exponential increase of papers on hydrophobic effects during the past 25 years. It is impossible to review in this thesis all theories and ideas

I . Introduction. Solvent fleets on organic reactions in aqueous m a r e s

about hydrophobic effects which have been presented in the literature during the last decades. Emphasis will therefore be put on some novel and interesting developments. Recently, the study of hydrophobic effects began to prompt a reconsideration of elder t h e ~ r i e s ' ~ ~ ~ ' ~ ~ ~ ~ ~ ' . An important reason for this development is the accessibility of large computers and sophisticated models which have allowed detailed theoretical studies of hydrophobic effects. Also experimental and theoretical studies of protein folding gave a new impulse to the discussion about hydrophobic effects. "New views on hydropho- bic effects" have been put forward that seriously contrast with classical descriptions112 l19. In this introducto~y chapter as well as in Chapter 8 new views on hydrophobic effects will be compared to the classical theories on hydrophobic effects.

Hydrophobic effects are characterised by a number of remarkable features. The most important feature of hydrophobicity, as expressed in the chemical potential of transfer of a hydrophobic, apolar compound from an apolar phase to water, is the unusual and large heat capacity ~hange"~*"~.

number of t i t l e s

year

Figure 1-1: Number of titles of research papers, containing the word "hydrophobic" as a function of the year.

I . Introduction Solvent effects on organic reactions in aqueous mixtures

Consequently, partial molar enthalpies and entropies of transfer are highly tempera- ture dependent.

The thermodynamic parameters, measuring the process of both pairwise and bulk hydrophobic interactions, are also shown to be strongly temperature dependenta. Privalov et a1.'00~103~108911' reported methods to determine the thermodynamic quantities of hydrophobic hydration, which are very difficult to measure experimentally, and showed that hydrophobic hydration is characterised by a similar temperature depen- dence of enthalpy and entropy terms.

The classical model of hydrophobic effects is mainly based on phenomena observed near room temperature. The hydration shell, surrounding apolar molecules or moieties, is found to be highly structured, minimising the interactions between the apolar molecule or moiety and water and optimising water-water interactions. Detailed t h e o r e t i ~ a 1 ~ " ~ ~ " ~ ~ and computer simulation studies'22-'" confirmed that the structure of the hydrogen bonded network of water is significantly altered in the first solvation shell surrounding apolar molecules or moieties. The transfer of apolar molecules from an apolar solvent to water is, at temperatures near 25"C, characteri- sed by a positive standard Gibbs energy of transfer, a slightly negative enthalpy and a large negative entropy of transfer*. For a broader temperature range, the thermo- dynamic parameters of transfer exhibit very characteristic values at two important temperatures. At a temperature TH, which is near room temperature, the enthalpy of transfer is nearly zero, whereas the entropy is strongly negative. At a temperature Ts, which is near 160C, the entropy of transfer is zero, and the enthalpy is highly positi- ve"'. Apparently, the structure forming capacity of water is lost at these high temperatures. The strong changes of AH0 and TASO as a function of temperature are shown in Figure 1-2. The standard Gibbs energy of transfer, as indicated in Figure 1-2, can be described by Equation 1-386~100~'01~'M~10808111.

1 A GwO = AH* - A Cp[(Ts- ;T) + T

In this equation, AH*, given by AC,(Ts-T,), corresponds to the enthalpy of transfer of an apolar compound from the apolar liquid phase to water at T, and appears to resemble the enthalpy of evaporation of the liquid, apolar solute. The second term on the right, which is always negative, contributes favourably to the standard Gibbs energy of transfer, and is strongly temperature dependent. This term can be identified as due to hydrophobic hydration. Hydrophobic hydration apparently leads to a decrease in the Gibbs energy of transfer. Both the enthalpy and entropy of hydropho- bic hydration are strongly temperature dependent, but appear to be strongly compen- sating, leading to a small, but favourable overall Gibbs energy of transfer. At 160C it is definitely the unfavourable enthalpy of transfer AH*, which causes the large and positive Gibbs energy of transfer of the apolar solutes from an apolar phase to water. This unfavourable term has to be attributed to the disruption of London dispersion interactions between the apolar solute molecules in the pure apolar, liquid state. At room temperature, however, the loss of entropy determines the unfavourable Gibbs energy of transfer. The loss of entropy, associated with the hydrophobic hydration of the apolar molecules, is also accompanied by a similar gain of enthalpy. As a conse-

1. Introduction Solvent effects on organic reactions in aqueous rniriwes

the height of the bamer. Recently, it has been shown that long-range attractive forces are present between large apolar surfaces148. The molecular origin of these hydration forces is also unclear.

1.5 The need for a quantitative description of solvent egects in mixed (aqueous) solve&. Incenlives fir this study

In Section 1.2 it has been shown that simple quantitative models to analyse solvent effects in pure solvents which are based on empirical solvent polarity parameters, lack general applicability. Moreover, the molecular basis of solvent effects remains unclear because of the undefined physical significance of these polarity scales. Macroscopic solvent parameters, derived from physical properties of the solvent give an unrealistic picture of the reaction medium as a continuum without specific structure. Specific interactions between solvent and reacting species, as well as the importance of the structure of the solvent, are neglected. More sophisticated, theoretical approaches to analyse solvent effects in pure solvents are either inaccessible or difficult to use for practical problems.

Many chemical reactions are performed in mixtures of solvents. Particularly solvent mixtures of water and an organic cosolvent are very popular. Quantitative descriptions of solvent effects in mixtures of solvents are even more complicated than those in the pure solvents. It is, for example, an impossible task to determine solvent polarity parameters in every mixture of solvents. Very often therefore, the assumption has been made that these solvent parameters depend linearly on the composition of the mixture. Obviously, the occurence and consequences of preferential solvation are not taken into account. The even more pronounced complexity of specific solute- solvent interactions is also fully neglected.

Patterns of organic reactivity in mixed aqueous solutions are particularly interes- ting. Kinetic data for reactions in water and in mixed aqueous solvents are often intriguing and their interpretation is a real challenge. The study of organic reactivity in water and in aqueous solutions has become more interesting since water has been shown to induce high reaction rate constants as well as high selectivities for a number of organic reactions, both homogeneous and heterogeneous (see Table 1-1). The low solubility of organic reagents can be overcome by addition of organic cosolvents. The consequences of addition of cosolvents for reactivity and selectivity of reactions in water are unknown. A quantitative study of solvent effects on a series of organic reactions in water and in mixed aqueous solvents, based on a general theory for quantitative analysis of solvent effects in mixed solvents, would offer insight into the molecular basis of rate effects on organic reactions in aqueous media. This knowledge should enable organic chemists to select organic reactions, which can benefit from mixed aqueous solvents with a suitable composition with respect to solubility, reactivi- ty and selectivity, in a rational way. However, it is remarkable that no valid quantitati- ve model exists for the analysis of solvent effects on reactions in mixed solvents.

I . Introduction Solvent effects on manic r e a c h in aquew~s mirtwes

1.6 A h of this st&

The general aim of this study was to develop and test a new, simple, but general theoretical model for the quantitative analysis of solvent effects in mixed solvents. An important objective was to draw together transition state, theory and thermodynamic formalism to describe thermodynamic properties of solutions. Following this approach, three important demands, made with respect to a novel theoretical model for the analysis of solvent effects in mixed solvents, can be met.

(i) Kinetic medium effects in mixed solvents can be expressed in terms of thermo- dynamic parameters.

(ii) These thermodynamic parameters can be determined by experimental techni- ques other than kinetic measurements.

(iii) The theoretical model enables a quantitative analysis of observed solvent effects leading to further insight into the solute-solvent interactions which govern solvent effects in mixed solvents.

A major incentive for the development of a new theoretical model for the quantitative analysis of solvent effects on reactions in mixed solvents has been its possible application to study solvent effects on reactions in mixed aqueous media. In fact, this application would involve a rigorous test for the new approach. The second aim of this study was, therefore, to critically appraise the developed theory by studying the kinetics of, first, simple first-order processes and, later, of bimolecular reactions and equilibria, in mixed aqueous solvents. The general strategy involved systematic variation of the nature of the mixed aqueous solvent (i) by varying of the composition of the mixed solvent and (ii) by varying the structure of the cosolvent(s). In addition, different reaction types have been studied and the structure of the reacting molecules has also been changed by changing substituents. The consequences for a quantitative analysis of the observed solvent effects, based on the novel theoretical model, have been examined. Concomittantly, the variation of the structure of the reactants provides an opportunity for a detailed quantitative study of the contribution of solvation effects to the overall substituent effects of alkyl groups on chemical reactivity in mixed aqueous solvents. A challenging goal was found in the application of the developed theory to elucidate the role of water, and eventually of the apolar cosol- vents, in the spectacular rate enhancements of some organic reactions in aqueous media.

1.7 Survey of the contents of thk thesis

Chapter 1 contains a general introduction in the field of qualitative and quantitative methods for the analysis of solvent effects. After a brief historical overview, emphasis is placed on recent methods to analyse solvent effects. In this context conventional methods are critically discussed. The second part of the chapter is devoted to a description of water and mixed aqueous solvents as reaction media. Attention is focussed on intermolecular interactions in aqueous solvents. Hydrophobic effects are

1. Introduction Solvent effects on organic reactions in aqueous mirrures

clearly defined and recent theories on hydrophobic effects are discussed. Finally, the incentives and aims of the study are summarised.

A general theoretical model for the analysis of solvent effects in mixed solvents is reported in Chapter 2. In this chapter, kinetic theory and thermodynamic formalism to describe thermodynamic properties of solutions are drawn together. This leads to theoretical expressions which relate solvent effects in mixed solvents to the composi- tion of the solvent. The general model is elaborated by using two alternative thermo- dynamic descriptions of the reaction medium, dependent on the composition of the medium. Both methods are discussed in detail and are critically compared. Theoretical expressions are derived for solvent effects on a simple unimolecular reaction in mixed solvents. The expressions are modified in order to analyse solvent effects on solvolysis reactions, bimolecular processes and chemical equilibria. Furthermore, the quantitati- ve treatment for the analysis of solvent effects on Gibbs energies of activation is extended to a quantitative analysis of enthalpies and entropies of activation in mixed solvents. A general feature of all theoretical expressions derived is that they describe the dependence of reactivity on the composition of the reaction medium in terms of interactions of the (co)solvent(s) with the reactants and activated complex, respective- ly. These expressions form the basis of the second part of the thesis, and will be used frequently.

In Chapter 3, solvent effects are described on the neutral hydrolysis of l-benzoyl- 3-phenyl-1,2,4-triazole in mixed aqueous solvents that contain monohydric and polyhy- dric alcohols. In the introduction, the vast amount of literature on solute-solute interactions in dilute aqueous solutions is briefly reviewed and some important features are discussed. The new theoretical model, as developed in Chapter 2, is critically tested by analysing the solvent effects of 24 different mono- and polyhydric alcohols on the rates of hydrolysis. The theoretical expression was found to describe the experimental data very well. Particular attention has been paid to the applicability of group additivity approaches for the analysis of solute-solute interactions. It was observed that addivity schemes are only valid under strict conditions and deviation from additivity is discussed in detail. In addition, the effect of urea on solvent effects in mixed aqueous solvents was investigated. Urea reduces the solvent effects of apolar cosolvents, whereas urea itself does not induce a significant solvent effect at all. Solvent effects are expressed in terms of Gibbs energy parameters, describing solute- solute interactions in aqueous solutions, and the results are compared with literature data for the analysis of solute-solute interaction in aqueous media. Emphasis has been placed on the participation of the cosolvent in the solvation shell of the reactant and the activated complex, accompanied by hydration shell overlap.

In the first part of Chapter 4, solvent effects on the Gibbs energy, the isobaric enthalpy and entropy of activation of the neutral hydrolysis of p-methoxyphenyl dichloroacetate in mixed aqueous solvents containing urea and alkyl-substituted ureas, are quantitatively analysed. The theoretica1 model, developed in Chapter 2, is shown to describe the dependence of the activation parameters on the composition of the reaction medium. However, higher-order enthalpic and entropic interaction terms are more important for the description of solute-solute interactions in dilute aqueous solutions than higher order Gibbs energetic interaction terms. The origin of this phenomenon is discussed in detail in terms of hydrophobic effects. In the second part of Chapter 4, the theory is successfully applied in the quantitative analysis of solvent

1. Introduction Solvent EfJects on organic r e a c h in aqueous m h s

effects on the Diels-Alder reaction of methyl vinyl ketone with cyclopentadiene, the keto-en01 equilibrium of 2,epentanedione and the intramolecular Diels-Alder reaction of N-alkyl-N-furfurylmaleamic acid in mixed aqueous solvents.

A thorough study of the medium effect of ethanol and 1-propanol on the neutral hydrolysis of 18 different l-acyl-3-alkyl-1,~4-triazoles is reported in Chapter 5. The dependence of the solvent effect on the alkyl groups in the substrate was examined in detail. The solvent effects depend critically on the substituent. This implies that the substituent effects of alkyl groups are significantly affected by the composition of the solvent. In the introduction conventional quantitative methods for analysing, under- standing and predicting alkyl substituent effects in terms of substituent constants are outlined and discussed. The results show that substituent effects of all@ groups on reactions in aqueous media are strongly governed by solvation effects. It is argued that efforts to describe steric, polar and inductive effects of alkyl substituents either include a substantial contribution of solvation effects or are completely useless. The contribu- tion of solvation to the substituent effect of alkyl groups on reaction rates of reactions in aqueous media is explained in terms of "hydrophobic acceleration".

The main theme of Chapters 6 and 7 is the analysis of solvent effects on Diels- Alder reactions in mixed aqueous solvents, containing monohydric alcohols across the whole mole fraction range. The applicability is tested of the theoretical expression, derived in Chapter 2, for a quantitative anaIysis of solvent effects in binary solvents.

Chapter 6 contains a quantitative study of solvent effects on the rate constants and isobaric activation parameters for bimolecular Diels-Alder reactions of cyclopent- adiene with alkyl vinyl ketones as well as with 5-substituted-1,4-naphthoquinones in mixed aqueous solvents. In addition, standard Gibbs energies of transfer of the reactants, activated complex and products of the Diels-Alder reaction of alkyl vinyl ketones and cyclopentadiene from 1-propanol to aqueous solutions of 1-propanol have been determined and analysed.

Chapter 7 describes the synthesis and the intramolecular Diels-Alder reaction of four N-alkyl-N-furfurylmaleamic acids. The intramolecular Diels-Alder process undergoes a spectacular rate increase in aqueous reaction media. The solvent effects appear to be very similar to solvent effects on the bimolecular process. Emphasis is placed on solvent effects on the stereochemistry of the intramolecular cyclisation.

The solvent effects, reported in Chapters 6 and 7, are satisfactorily described by the derived theoretical expressions. The spectacular rate effects of water on Diels- Alder reactions are largely preserved in highly aqueous reaction media. Preferential solvation of the reactants appears to diminish the rate effect of water in the presence of higher concentrations of cosolvent molecules. The spectacular rate accelerations are ascribed to the apparent decrease of the hydrophobic surface of the apolar reactants during the activation process. This "hydrophobic acceleration" is a result of "enforced hydrophobic interaction" of the reactants during the activation process. The possibility of induction of a more polar activated complex in highly aqueous media is tentatively suggested.

Chapter 8 evaluates the applicability, merits and shortcomings of the proposed theoretical treatment of solvent effects in mixed aqueous solvents. In this thesis, the dependence of solvent effects on reactivity of apolar organic reactants in water on the concentration of cosolvents, has been mainly explained in terms of bulk and pairwise hydrophobic interactions. Based on the quantitative analyses of these solvent effects a

1. Introduction Solvent effects on organic reactions in aqueous mixtwes

novel model is introduced for the description of hydrophobicity and hydrophobic interactions. This model accounts for the most characteristic properties of hydropho- bic effects. Finally, the use of water as a solvent for organic reactions is critically discussed. It is shown that organic reactions can benefit from water as a solvent due to the fact that reactive, hydrophobic species in water and highly aqueous media tend to minimise their hydrophobic exposure to water of the hydrophobic groups during the activation process. Reaction types are suggested which might be accelerated in aqueous solutions.

A major part of the work described in this thesis either has already been publis- hed or will be published in the near f u t ~ r e ' ~ ~ - ~ ~ ~ .

2. A quantitative anaIysis of solvent L'IJeca in mired solvents. Development of a theoretical model

CHAPTER 2

A Quantitative Analysis of Solvent Effects in Mixed Solvents, Development of a Theoretical Model

2.1 Introduction

In this chapter the kinetics of reactions in mixed solvents and thermodynamic formalism for the description of thermodynamic properties of a solution are drawn together. Each separate theme is well established in the chemical literature, and will be briefly introduced. According to the new theoretical model, the combination of kinetic and thermodynamic theories leads to a quantitative description of solvent effects in mixed solvents in terms of the composition of the reaction medium and thermodynamic parameters having well defined physical significan~e'~'~'~~. Mixed aqueous solvents can be classified according to their composition:

(i) Dilute mixed solvents, in which one solvent is present in small mole fractions and is considered as a cosolute;

(ii) Mixed solvents, prepared from comparable amounts of the two components. Dependent on the choice of the reference solvent, one of the solvents is labeled as a cosolvent.

The exact demarcation between these classes of mixed solvents might be somewhat artificial and depends on the characteristics of the reference solvent as well as of the cosolvent(s) or cosolute(s). An important incentive for defining two classes of mixed solvents stems from the fact that reactants andlor activated complex can be preferen- tially solvated by the cosolvent. Two thermodynamic descriptions of the reaction mixture are needed to develop a theoretical model for a quantitative analysis of solvent effects on reactions in mixed solvents. The basic thoughts underlying the model are similar. In the next part of this introduction a brief outline will be given of the general procedures and strategies that have been followed.

Transition state theory constitutes the general basis for the kinetic analysis of solvent effects. Solvent effects in mixed solvents are caused by the interactions of the cosolvent(s)* with the reactant(s) in the initial state and with the activated complex in the transition state. These interactions result in changes of the chemical potential of the reactants as well as of the activated complex which affect the Gibbs energy of activation because reactant(s) and activated complex are solvated to a different extent. In Scheme 2-1, the cosolvent C interacts favourably with the reactant (IS), whereas the interactions between activated complex (AC) and cosolvent are slightly

In this chapter, the term "cosolvent* is generally used and the distinction between "cosolute" and "cosolvent" is not made.

2 A quantitative ana3,sis of solvent effects in mired solvents. Development of a theoretical model

unfavourable. The resulting increase in the Gibbs energy of activation is manifested by an effective decrease in reaction rate constant.

Scheme 2-1

The next step is to set down a thermodynamic description of the chemical potential of reactants and activated complex in mixed solvents. Two approaches have been followed. The first approach is based on the theory on dilute solutions of McMillan and ~ a ~ e r ~ ~ ~ J ~ ~ and is selected for the development of a general theoretical expressi- on that describes solvent effects in dilute mixed solvent^^^'^'^^. The total Gibbs energy of the reaction mixture, including reactants, activated complex, products and cosol- vent(s) is determined by the chemical potentials of all components in the reaction mixture. The chemical potential of the reactant(s) and the activated complex can be described effectively by a standard chemical potential, the concentration and the activity coefficients of the reactant(s) and activated complex in the solution. Changes of the chemical potential of reactant and activated complex are a consequence of changes in the activity coefficient of these species. Changes in the activity coefficients of solutes in a mixed solvent are caused by interactions of the solutes with the added cosolvents and are defined in the non-ideal part of the chemical potential. The non- ideal part of the chemical potential of the solutes contributes to the excess Gibbs energy of the total solution. The excess Gibbs energy can be conveniently expressed in a molality expansion, using virial coefficients. The physical significance of these virial coefficients is given either by the theory of McMillan and Mayer or by lattice models. In dilute solutions of non-electrolytes the second virial coefficient appears to be the most predominant contribution to the overall excess Gibbs energy. The second order virial coefficient is identified as the pairwise Gibbs energy interaction parameter and is subsequently evaluated in terms of the Savage and Wood Additivity of ~ r o u p s " ~ ~ (SWAG) and the Excess Group Additivity (EGA) appro ache^'^".

Molality expansions to describe the excess Gibbs energy of the total reaction

2 A quantitative analysis of solvent eflects in mired solvents. Development of a theoretical model

medium are not suited for solutions that contain large amounts of cosolvent(s). In these binary solvents, an alternative approach is followed. The changes of the chemical potential of the reactant and the activated complex are interpreted in terms of changes in the standard chemical potential of reactant and activated complex. The standard chemical potential of a solute in solution can be quantified using the theory developed by Kirkwood and and later modified by all"^, Ben-Naim1601161 and ~ e w r n a n ' ~ ~ . ~ ~ ~ . This theory expresses intermolecular interactions in solution in terms of inte a1 functions that describe the distribution of cosolvents around solute

164,lg molecules . The general procedures outlined above will be followed leading to theoretical

expressions for solvent effects on unimolecular first-order reactions. In this chapter, these general procedures will be modified in order to broaden the field of application to solvent effects on chemical equiIibria, bimolecular reactions and solvolysis reactions in mixed solvents. This approach will be extended to the analysis of solvent effects on partial molar enthalpies and entropies of activation.

In the last section of this chapter, the two different methods used to develop the theoretical model are compared and critically appraised.

2.2 Solvent effects in dilute mixed solvents

2.2.1 Kinetics of a reaction; transition state theory

A theoretical model will be developed for a quantitative analysis of solvent effects in dilute mixed solvents by considering a simple unimolecular first-order reaction. This reaction involves the conversion of reactant X to products through an activated complex AC'.

X * ACc - products A link between a thermodynamic description of the reaction medium, containing reactants X, activated complex AC', cosolvent(s) C and, eventually, products P, and kinetic data is forged using transition state theory. Recognising the important assump- tions and inherent weaknesses of this theory, the reactant in the initial state is assumed to be in equilibrium with the activated complex in the transition state166. Therefore

In terms of a standard equilibrium constant of activation, 'K', the molality of AC' can be expressed in terms of the molality of reactant X and the activity coefficients of reactant and activated complex.

2. A quantitative anabsis of solvent efects in mired solvents. Development of a theoretical model

In an ideal solution, if the total molality of the solutes is given by m,, in the limit- (q+O)y,=y,=l. In this case, neither the reactants nor the activated complex do interact with surrounding solutes. For dilute solutions, it can be assumed that w m - .=cJc,, where c is expressed in concentration units, mol dm-3. Therefore, in dilute solutions, according to the law of mass action, the rate constant for this reaction, k, is given by

Following transition state theory, in ideal solutions k can be represented by

In non-ideal solutions, m,+O, reactants and activated complex interact with the cosolvent(s) and k becomes

Hence

Equation 2-6 is now a key relationship which relates kinetic data obtained for reactions in the reference solvent, with kinetic data in non-ideal, mixed solvents containing m, mol kg-' of cosolvent(s) via the activity coefficients of the reactant X and the activated complex AC'.

2.2.2 Chemical potential and activity coemients of solutes in dilute mined solvents

In order to evaluate Equation 2-6 further, the activities of reactants and activated complex must be carefully defined. In the treatment described below the reaction medium contains 1 kg of solvent, m, mole of reactant, m, mole of activated complex, m, mole of product(s) and m, mole of cosolvent(s). The chemical potential of the solutes X as well as AC* can be given by Equation 2-7.

The factor RTlny, measures the deviation of the chemical potential from ideal behaviour. m, is, by definition, 1 mol kg-' and p,' is the standard chemical potential of the solute, i.e. that of a hypothetically ideal solution containing a solute concentrati-

2. A quantitative anabsir of solvent effects in mixed solvents. Development of a theoretical model

on of 1 mol kg-'. The chemical potential of the solvent, p,, is defined as in Equation 2-8

where p,*(l) is the chemical potential of the pure solvent and 4 is the practical osmotic coefficient for which limit(m,+O) +=I. M, is the molar mass of the solvent and m, the molality of the solute S.

The total Gibbs energy of the reaction medium, G(sln), in which m,=O, is given by Equation 2-9.

The total Gibbs energy of the reaction medium contains an ideal part G(s1n;id) and a non-ideal part, ~~(sln;nonid), which is defined as the excess Gibbs energy for which

GE(sln;nonid)= RTm,(l-4) + RTmJny, + R%,lny, + Rlh,,hy, 2-10

The activity coefficients of reactants and activated complex are closely related to the excess Gibbs energy of the total reaction mixture according to

This can be shown by taking the derivative of Equation 2-10 with respect to q and by using the Gibbs-Duhem relation for the total reaction mixture

According to the Gibbs-Duhem relation (Equation 2-12) changes in the chemical potential of the solvent affect the chemical potential of the reactants and the activated complex in the reaction mixture. The equations become somewhat more elaborate if m, moles of cosolvent(s) C are present in the mixture. In dilute mixed solvents the cosolvent(s) can be considered thermodynamically as cosolute(s). Therefore, the total Gibbs energy of the reaction medium as well as the Gibbs-Duhem equation can be simply extended with the chemical potential p, of the cosolvent(s). The activity coefficients of the reactants as well as of the activated complex can still be expressed in terms of the excess Gibbs energy of the reaction medium, as shown in Equation 2- 11.

2.2.3 Dependence of excess pmperties on wmposirion

The excess Gibbs energy of the total reaction mixture, as shown in Equation 2-11, can

2 A quantitative anaQsis of solvent effects in mixed solvents. Development of a theoretical model

only be obtained from experimental data and is characteristic for a given solution. Thermodynamic excess properties of solutions are frequently expressed in expansion series of concentration, density or activity. These virial expansions find a theoretical basis in statistical thermodynamics, as shown by, for example, McMillan and ~ a ~ e r ' ~ ~ and Wood et a ~ . ' ~ ~ . According to the theory of McMillan-Mayer (MM-theory), the osmotic pressure, II, can be expressed in a density expansion:143

Virial expansions of the form given in Equation 2-13 are often modified and widely used to represent all kinds of thermodynamic data of solutions. Particularly well- known are the molality expansions to express the practical osmotic coefficient of a solution,

Also the activity of the solvent, y,, can be expressed in terms of the mole fraction of the solute, x,, as shown in Equation 2-15'~~.

A molality expansion for the excess Gibbs energy of a solution containing only solute-j can be derived from the MM-expansion series, as shown in Equation 2 - 1 6 ' ~ ~ ~ .

Here, gjj and g... are the pairwise and triplet Gibbs energy interaction parameters and P

mj is the molality of solute-j. The virial coefficients can be obtained from experimental data, and are sometimes refered to as the Lewis-Randall coefficients16'j. Physical chemists have tried to develop methods to describe the magnitude and sign of these virial coefficients using a theoretical model. In this way, the virial coefficients have been ascribed some physical significance. The magnitude and the sign of the virial coefficients are related to factors such as (i) solute size, (ii) solute-solvent association and (iii) solute-solute interaction^'^^. Concentration expansions of excess thermody- namic properties of solutions have been frequently applied to estimate solute-solute interactions in solutions145. In particular solute-solute interactions in aqueous solution have received much a t t e n t i ~ n ' ~ ' * ' ~ ~ ~ ' ~ ~ . These models have been based on (i) lattice or quasi-lattice theories, (ii) distribution theories, (iii) equilibrium theories and (iv) statistical thermodynamics. These approaches have clearly shown that a oversimplified molecular picture of the virial coefficients can be misleading.

Kauzmann et al.143developed different approaches for calculating the second and third virial coefficient, based on lattice theories as well as on the MM-theory. In a famous paper143 lattice theories were used to show that solvent-solute interactions decrease the magnitude of the virial coefficients. Moreover, the magnitude of the virial coefficients appears to be clearly related to the size of the solute. The use of

2 A quantitative unabsis of solvent effects in mired solvents. Development of a theoretical malel

Ni,Nj as well as Nj,Ni) and the species in the subscripts indicate all particles involved in the cluster integral, shown in Equation 2-18. In case reactants X, activated complex AC (symbol z), product(s) P and cosolvent(s) C are present, Equation 2-19 yields a summation over all the possible interactions between the solutes present in pairs, triplets and higher order.

As shown in Equation 2-11, Iny, is the derivative of the excess Gibbs energy with respect to the molality of x. In the absence of cosolvent(s), the activity of reactants is given by

In the presence of cosolvent(s) this relation is rather elaborate and contains a large number of cross interaction terms. This complicated expression can also be derived for the activated complex and can, together with Equation 2-20, be incorparated into Equation 2-6. A number of important assumptions and simplifications can be made. If the experimental conditions are such that the concentration of reactants, activated complex and therefore products is very low, the medium can, in the absence of cosolvents, be considered ideal. This means that pairwise and higher-order interactions of the reactant, activated complex or product involving other molecules of reactant, activated complex or product can be safely neglected. Moreover, the almost negligible contribution of these terms to the overall excess Gibbs energy in the pure solvent will effectively cancel similar contributions of these interactions in the presence of cosolvent. Combination of Equation 2-20 with Equation 2-6 yields after simplification,

This important equation relates rate constants for reaction in pure solvents to those in solvents that contain m, moles of a cosolvent C in terms of the composition of the mixed solvent and Gibbs energy interaction parameters, describing pairwise and one higher-order interactions between cosolvent C and reactant and activated complex respectively.

In dilute reaction media, where the molality of the cosolvent is also low, the dependence of the excess Gibbs energy as given in Equation 2-20, is effectively described by the pairwise terms. A similar result emerges where the theory uses site- site pair correlation functions98r145. In the presence of small amounts of cosolvent(s), Equation 2-21 can be further simplified to

Inspection of Equation 2-22 shows that ln[k(mc)/k(mc=O;id)] should be a linear

2 A quantitative anabsis of solvent effects in mired solvents. Development of a theoretical model

function of m,. The slope of the linear plot yields the difference in pairwise Gibbs energy interaction parameters, [&-&,I.

It is interesting to note that when the molality of reactants is not low, reactant and products can act as cosolvents for the reaction and thereby affect the rate of the reaction.

2.2.4 Solute-solute interactions; additivity apprwches

The pairwise, but also higher-order Gibbs energy interaction parameters are determi- ned by the nature of the solvent, but more profoundly by the nature of the solute. Solute-solute interactions reflect the interactions between the functional groups, which form the solute molecule and are able to interact with their surroundings. ~ a n g m u i r ' ~ ~ noted that simple additivity principles might be used to reasonably predict pairwise energies or enthalpies of interactions. His "principle of surface action" states that "it is reasonable to assume that the field of force about any particular group or radical in a large organic molecule is characteristic of the group and, as a first approximation, is independent of the nature of the rest of the molecule". Since then, additivity approa- ches are regularly encountered in the literature where they are applied to account for various thermodynamic proper tie^'^^"^^-'^^.

In 1976, Savage and Wood introduced a concept of additivity of groups based on empirical equation^"^^. This Savage-Wood Additivity of Groups (SWAG) model is based on three major assumptions:

(i) Each functional group-i in solute P interacts with each group-j in solute Q. (ii) Each group-i-group-j interaction makes a characteristic contribution to the

solute-P-solute-Q interaction parameter. (iii) Each group interaction parameter is independent of all other functional groups

and their relative position or stereochemistry in solute-P or solute-Q.

These assumptions lead to the following expression, relating pairwise Gibbs energy group interaction parameters to pairwise Gibbs energy solute interaction parameters.

Here n: and njQ are the number of groups-i and groups-j in solute-P and solute-Q respectively. Gij is the pairwise Gibbs energy group interaction parameter.

More recently, Bloemendal and oms sen^'^' developed a similar additivity model for enthalpies in which they take account of the fact that when solute and solvent are identical, gPQ is zero. This model is called the Excess Group Additivity (EGA) approach. The pairwise Gibbs energy solute interaction parameter is related to pairwise Gibbs energy group interactions by

2. A quantitative analysis of solvent effects in mixed solvents. Development of a theoretical model

where n v and njv denote, respectively, the number of groups-i and groups-j in the solvent molecules.

Group additivity approaches are generally considered to be first ap roximations and, in their use, several questionable assumptions are made1"16911.d However, attempts have been made to provide empirical additivity schemes a firm theoretical basis. The pairwise Gibbs energy interaction parameter is connected to the MM- second virial coefficient by the integral function of [exp(-W,&T)], as shown in Equations 2-17 and 2-18, in which W,,,. is the potential of mean force between the pairs of molecules. ~ i l l e ~ l ~ j and woodlS6% submitted that, if WpQ is expressed in terms of additive group contributions, Wij must be very small compared to the thermal energy, kT, thereby allowing expansion of the exponential factor into a linear sum of interaction terms.

For some extreme cases, this assumption is not valid. In order for Equation 2-25 to be valid, nearest neighbour modifications, steric hindrance and specific solvation effects must be absent.

~ i l le~ '@J has pointed to another important theoretical concern in the additivity of Gibbs energy group interaction parameters. Inspection of Equation 2-17 shows that the pairwise interaction term contains a considerable contribution from the partial molar volume of the solute. Partial molar volumes of non-electrolytes in solution are approximately linear functions of the number and types of groups present in the molecule179. The empirical equation, formulating the additivity of group interaction shows a quadratic dependence on the number of the groups present-(see Equation 2- 23 and 2-24)156b.

Recently, Marcus and Bloemendal used a quasi-lattice model to provide the empirical SWAG and EGA schemes with a theoretical Recently, Wood and ~ h o m p s o n ' ~ ~ developed a more sophisticated model, based on site-site interacti- ons. The model is too complex to be suited for a thermodynamic description of dilute mixed reaction media.

Notwithstanding serious criticism, the simple procedure of predicting signs and magnitudes of solute-solute interactions based on additivity schemes is very useful. Since 1976, thermodynamic data for solutions have been reported by Wood et a1.lS6, Lille et a].'@, Somsen et a1.lS7, Barone et a1.181*182, Borghesani et and others- 169p1d16711s111851. These studies have provided interaction parameters and a better understanding of the nature of solute-solute interactions in amidic, aqueous and alcoholic solvents. In a pragmatic sense, a breakdown of the applicability of additivity schemes may be used to signal atypical effects. Some interesting data, reported in the literature for solute-solute interactions in aqueous solvents, will be discussed in Chapter 3.

The Equations 2-23 and 2-24 can be both incorporated into Equation 2-22. Therefore, solvent effects on the kinetics of a simple first-order process in dilute mixed solvents can be described by

2 A quantitative anahsir of solvent Meets in mired solvents, Development of a theoretical model

In dilute mixed solvents, the plot of ln[k/k,,] versus the molality of the cosolvent(s) is predicted to be linear, and the slope of this plot is given by a sum of pairwise Gibbs energy group interaction parameters, which can, in principle, be retrieved from the literature.

2.3 Modification of the general theoretical model for quantitative analysk of solvent efleds in mired solvents

2.3.1 SolvolysB reactions

For many reactions in condensed media, the solvent is also a reactant. The case is considered in which reactant X is solvolysed to products through an activated complex AC*, after attack by N solvent molecules S.

X + N.S * AC* - products The chemical potentials of reactant X and of the solvent are important. The chemical potential of the solvent is also affected by the presence of solutes or cosolvent(s) (see Equation 2-8). In terms of transition state theory, the standard Gibbs energy of activation, A'G', is given by

Here, q=q+m,+m,+m,. By following a procedure similar to that used in section 2.2.1 for a unimolecular first-order process, it can be shown that

2 A quantitative anabsir of solvent ef/ects in mired solvents. Datelopmenr of a theoretical model

Therefore, by analogy to Equation 2-6, the rate of solvolysis of compound X in the solvent S in ideal solutions is related to the rate of solvolysis in the presence of mc moles of cosolvent(s) in terms of the molality of the cosolvents in the reaction mixture, the activity coefficients of reactant and activated complex and the practical osmotic coefficient of the solvent.

Equation 2-31 can be combined with Equation 2-20 to yield

which provides a quantitative description of solvent effects on the rate of solvolysis reactions in dilute mixed solvents if the reactants are present in small amounts. The pairwise Gibbs energy interaction term can be evaluated by using Equations 2-23 and 2-24. The term Ng5M,mc accounts for the solvent effect on the reactivity of the solvent. Generally this contribution is small, but will depend largely on the molar mass of the solvent. In practice, the osmotic coefficient will not deviate markedly from 1 after addition of small amounts of cosolvent(s). It is also possible to express the osmotic coefficient in a molality expansion (cf. Equation 2-14). As shown by Cassel and

if q+m,+m,am, the osmotic coefficient can be expressed in terms of pairwise and higher-order interactions between the cosolvent molecules in the reaction medium.

Experimentally, the osmotic coefficient can be determined from the excess Gibbs energy of mixing of the mixed solvent, according to

2.3.2 Chemical equilibria

Also equilibrium constants characterising chemical equilibria between compound A and compound B depent on the solvent. The equilibrium condition results in the following equation;

2 A quantitative ana&sis of solvent effects in mixed solvents. Development of a theoretical model

In the absence of cosolvents C and in an ideal reaction medium where compounds A and B are only present in trace amounts, y,= y,=l. Addition of cosolvent(s) C affects the activity coefficient of the compounds A and B. The standard equilibrium constant KO(mc) cannot be affected by the presence of the cosolvent C. Since y, and y, depend on m, the equilibrium quotient Q7 which is defined as m$m, does depend on m,. Therefore Q(m,) is related to Q(m,=O;id) by

Combination of Equation 2-36 with Equation 2-20 relates the solvent effect of cosolvent C on the equilibrium quotient Q to the composition of the medium in terms of pairwise Gibbs energy interaction parameters involving interactions between A and C and B and C, respectively. Prerequisite is that compounds A and B are present in small concentrations.

Subsequently, the interaction term between brackets can be evaluated, by using Equation 2-23 or 2-24, in terms of pairwise group interaction parameters.

2.3.3 Bimolecular processes

The analysis described so far finds application in understanding the kinetics of a simple first-order or pseudo-first-order process in mixed solvents. The theoretical model for solvent effects on a birnolecular process is complicated. A bimolecular reaction is represented as involving the conversion of reactants A and B to products through an activated complex AC*.

A + B * AC* - products According to transition state theory

The molality of the activated complex can be represented by

2 A quantitative anabsis of solvent effects in mired solvents. DeveIopment of a theoretical model

where 'K' represents the equilibrium constant for the activation process, here defined in a molality scale; m,= 1 mol kg*'. According to the law of mass action

where k2 is the second-order rate constant, expressed in dm3 mol-' s-'. According to transition state theory, the second-order rate constant can be expressed as

The conversion from molality scale to concentration scale by simple substitution of the concentration scale by the molality scale in Equation 2-39, is impossible. Therefore the molality scale must be first converted into a concentration scale in which W, is the mass of the solvent, and V is the volume of the total reaction mixture