J . Chem. Soc., Faraday Trans. I, 1984,80, 3359-3363 Chemical Equilibria and Related Isochoric Functions BY MICHAEL J. BLANDAMER,* JOHN BURGESS AND BARBARA CLARK Department of Chemistry, The University, Leicester LE1 7RH AND JOHN M. W. SCOTT Department of Chemistry, Memorial University of Newfoundland, St. John’s, Newfoundland, Canada Received 16th March, 1984 The meaning is examined of the term ‘isochoric’ when used in examination of the dependence of equilibrium parameters on temperature and pressure. A number of terms which identify the extrinsic nature of this term are defined. The question of a definition of the standard and reference states is considered. For the most part, the chemical literature describes properties of liquids and solutions at constant temperature and pressure.Certainly equilibrium and rate constants for chemical reaction in solution are usually documented for isobaric- isothermal conditions. In the next stage of an analysis the corresponding derivatives of these constants are examined with respect to temperature at constant pressure and with respect to pressure at constant temperature. Nevertheless many claims are made for derivatives calculated at constant volume,l i.e. isochoric constraint. For the most part these claims are advanced in connection with thermodynamic activation parameters based on the formalism of transition-state theory2 and calculated from the rate constants for chemical reactions in solution. For example, in discussing kinetic data for the solvolysis of t-butyl chloride in methyl alcohol, Hills and Viana3 argue that the isobaric activation parameters are more complicated than the isochoric set.This view is echoed by Whalley and coworkers in their analysis of the dependence of activation parameters on solvent composition for reactions in binary aqueous For example, Gay and Whalley6 conclude that constant-volume activation parameters are more appropriate for fundamental discussion. A marked difference between trends in isochoric and isobaric parameters as a function of solvent composition is borne out by kinetic data reported recently by Holterman and Engbert~.~ These claims for the usefulness of isochoric activation parameters have attracted our interest over the years despite the fact that we have continued to concentrate attention on isobaric-isothermal activation parameters.Part of our hesitancy over isochoric parameters can be understood by attempts to answer two questions : (i) what volume is held constant and (ii) what are the reference (or standard) states for solutes (reactants and transition state) referred to in derived parameters such as the isochoric thermodynamic energy and isochoric heat capacity of activation? Various answers are to be found in the literature. For example, Whalley states8 that the isochoric condition is ‘the volume of an equilibrium mixture of initial and transition state’. [n the form applied to kinetic data for reactions in solution, transition-state theory2 uses the language of equilibrium thermodynamics for chemical equilibria in solutions. The answer to the questions posed above might develop therefore from a consideration 33593360 CHEMICAL EQUILIBRIA AND ISOCHORIC FUNCTIONS of the treatments using classical equilibrium thermodynamics. The literature does not, however, offer the complete answer.Indeed various puzzling statements are encountered. For example, Lown et al.,g in an analysis of the dependence of acid dissociation constants in water on temperature and pressure, write ' At constant volume, the volume of ionisation becomes more negative with increasing temperature '. In the following analysis we examine the definition of isochoric functions from the standpoint of chemical equilibria in solutions. For example, we show that the statement quoted above can be understood if it is reworded to read 'If the pressure is changed such that the molar volume of the solvent remains constant with increasing temperature, the volume of ionisation of acetic acid in water, A, P, decreases with increasing temperature'.The aim of the present paper is to provide a framework for the analysis of kinetic data. CHEMICAL EQUILIBRIUM Consider a closed system at fixed temperature and pressure. Within this system there exists a chemical equilibrium between solutes (j = 2 to j = i) in solvent 1. At chemical equilibrium, where the Gibbs function G is a minimum, the chemical composition is related to the standard equilibrium constant K* (T) by In Ke(T) = In Q(T;p)+jp pe q d p RT where Q( T;p) = ng' = 2 ; j = i) (mi"" yFQ/m*)vj ( 2 ) A,Vm(T;p) = X U = 2 ; j = i)viVy(T;p) (3) and vj is the stoichiometry.The standard equilibrium constant is defined in terms of the chemical potentials of the substances involved in the chemical equilibrium in their corresponding solution standard states at temperature T and standard pressure p*. For a given system the composition quotient Q(T;p) can be defined by the two independent intensive variables T and p . The quotient Q is related to the chemical potentials of solutes in reference states; i.e. the ideal solution where mi = 1, yi = 1 at temperature T and pressure p . The dependence of Q on T and p describes a surface in the In Q-T-p domain: In Q = lnQ[T;p]. (4) At a given temperature 6 and pressure z there exists Q(0; n). Two orthogonal gradients describe the dependence of Q on Tat fixed pressure and on pressure at fixed temperature.MOLAR VOLUME OF SOLVENT The molar volume of the pure solvent c can be defined by the two independent variables T and p : v = V[T;pl. ( 5 ) In other words at a given temperature 0 and pressure z there exists a characteristic molar volume for the solvent, c(6; z).M. J. BLANDAMER, J. BURGESS, B. CLARK AND J. M. W. SCOTT 336 1 COMPARISON The nub of the argument presented here draws together the two sets of data described by eqn (4) and (5). Consider a given temperature 8 and a given pressure n together with the two quantities Q(8; n) and q(O;n). Consider a change in temperature from 8 to 8+A8. According to eqn ( 5 ) there exists a pressure n+Anl where ( 6 ) v(8; n) = v(8+A8; n+Anl) and the pressure increment, An1, is characteristic of pure liquid 1.According to eqn (4) there exists a corresponding equilibrium quotient Q(O+ A8; n + An1). In other words the two quotients Q(8; n) and Q(O+ A8; n + Anl) characterise chemical equilibria in solutions which are individually at minima in Gibbs functions G under conditions where the molar volume of the solvent is the same. Hence the isochoric condition is given by eqn (6) rather than the volumes of the solutions containing the chemical equilibria. For any function of state X which is a function of both T and p, the total differential can be expressed as dX= ( 3 p d T + ( 3 d p . (7) Here we set X = In Q and use the ‘chain rule’ to expand (ap/aT),: and its inverse. condition can be calculated from the properties of the pure solvent: The gradient of the tangent in In Q-7‘-p domain conforming to the isochoric where a,* and ic? are the thermal-expansion coefficient and isothermal compressibility of the solvent at temperature T and pressure p.Another isochoric condition can be formulated with reference to the solvent. Again we start with the two quantities Q(9;n) and q ( 8 ; n ) at temperature 8 and pressure n. With reference to the molar-volume data, consider a change in pressure from n to n+An. According to eqn (4) there exists a temperature O+A8,, characteristic of the solvent, where q ( 8 ; n ) = q(O+A8,;n+An). (9) This isochoric condition leads to the identification of two equilibrium quotients Q(8; n) and Q(O+AO,; n+ An). The gradient of the tangent in the In Q-T-p domain conforming to the isochoric condition [eqn (9)] is given by RELATED PARAMETERS The analysis outlined above can be repeated with reference to the thermodynamic parameters characterising the chemical equilibrium at temperature T and pressure p.Thus the dependence of Ar Vm on T and p can be written in the following form: Ar Ifrn = Ar Vrn[T;p].3362 CHEMICAL EQUILIBRIA AND ISOCHORIC FUNCTIONS The isochoric condition given in eqn (7) can be incorporated in an analysis of the dependence described by eqn (1 1). This in turn leads to the analogue of eqn (8): The derivative calculated by eqn (12) describes the dependence of A, Vm on temperature at constant c. The isochoric condition applies therefore to the solvent and not to either the system under examination or some ideal solution containing unit molalities of the solutes.DERIVED PARAMETERS The van't Hoff equations express the isobaric dependence of Q on temperature in terms of the enthalpy parameter A,Hm and the isothermal dependence of Q on pressure in terms of the volume quantity, A, Vm [eqn (5)]. In analogous fashion we may use the isochoric quantities calculated in eqn (8) and (10) to define two further terms : and Dimensional analysis shows that Ary is expressed in J mol-l and A4 in m3 mol-l, i.e. a volume per mole. DISCUSSION In the introduction we posed two questions concerning isochoric parameters. The analysis presented here and comparison with equations quoted in the references cited in the introduction show that the term 'isochoric' can be understood in terms of variations in T and p which preserve constant molar volume of the pure solvent.In these terms the meaning is unambiguous and leads to a simple connection between the extrinsic isochoric condition and intrinsic isothermal and isobaric dependences of equilibrium parameters. In other words, isochoric conditions are extrinsic to the equilibrium characterised by the quotient Q and standard equilibrium constant Ke( T>. The term isochoric is not used in the present context to signify a thermodynamic change at constant volume, e.g. where the Helmholtz function is the isochoric- isothermal thermodynamic potential function. Under the latter circumstances the condition isochoric is intrinsic to the chemical processes under consideration. By extrinsic we signify above that the isochoric condition refers to another system, i.e.the solvent rather than the solution. This distinction is important because the chemical equilibrium [eqn (l)] describes a system at a minimum in G and the reference states for the solutes are isobaric-isothermal. These comments raise the question as to the significance to be attached to Ary and A4 [eqn (13) and (14)]. The quantity A4 does not seem to have been calculated from experimental data. In contrast Aw is often written as A U e and its temperature dependence as the isochoric heat capacity of reaction. We caution against such practice. The point can be supported as follows. Consider a plot of Arc$, the isobaric heat capacity of reaction, against temperature. With increasing temperature, so the molar volume of the solvent changes (at fixed p ) . Consider the corresponding plot of the isochoric heat-capacity term against temperature. Each point on the plot is calculated for a local isochoric condition for a characteristic change in pressure [cf.M.J. BLANDAMER, J. BURGESS, B. CLARK AND J. M. W. SCOTT 3363 Anl in eqn (7)]. Hence across the plot neither the volume of the solvent nor the local pressure increments are constant. Similar features can be identified in isochoric parameters derived from kinetic data. This point will be explored in another paper. We thank the S.E.R.C. for a grant to B.C. E. Whalley, Adv. Phys. Org. Chem., 1964, 2, 93. S. Glasstone, K. J. Laidler and H. Eyring, Theory of Rate Processes (McGraw-Hill, New York, 1941). G. J. Hills and C. A. Viana, in Hydrogen-bonded Solvent Systems, ed. A. K. Covington and P. Jones (Taylor and Francis, London, 1968), p. 261. B. T. Baliga and E. Whalley, J. Phys. Chem., 1967,71, 1166. B. T. Baliga, R. J. Withey, D. Poulton and E. Whalley, Trans. Furaduy Soc., 1965, 61, 517. D. L. Gay and E. Whalley, J. Phys. Chem., 1968,72, 4145. E. Whalley, Ber. Bunsenges. Phys. Chem., 1966, 70, 958. ' H. A. J. Holterman and J. B. F. N. Engberts, J. Am. Chem. SOC., 1982, 104, 6382. 9 D. A. Lown, H. R. Thirsk and Lord Wynne-Jones, Trans. Furaduy SOC., 1970,66, 51. (PAPER 4/426)