1.2.9: Affinity for Spontaneous Chemical Reaction - Isochoric Condition and Controversy

According to the First and Second Laws of thermodynamics, the change in Helmholtz energy \(\mathrm{dF}\) accompanying chemical reaction, change in volume and change in temperature is given by Equation \ref{a}.

\[\mathrm{dF}=-\mathrm{S} \, \mathrm{dT}-\mathrm{p} \, \mathrm{dV}-\mathrm{A} \, \mathrm{d} \xi \label{a}\]

where,

\[\mathrm{A} \, \mathrm{d} \xi \geq 0\]

At constant \(\mathrm{T}\) and \(\mathrm{V}\):

\[\mathrm{dF}=-\mathrm{A} \, \mathrm{d} \xi ; \mathrm{A} \, \mathrm{d} \xi \geq 0\]

If we monitor the change in composition of a closed system held at constant temperature and constant volume, equilibrium corresponds to a minimum in Helmholtz energy. In practice chemists concerned with spontaneous chemical reaction in solutions held at constant temperature, do not constrain the system to a constant volume. Rather they constrain the system to constant temperature and pressure. Therefore the following equation is the key.

\[\mathrm{dG}=-\mathrm{S} \, \mathrm{dT}+\mathrm{V} \, \mathrm{dp}-\mathrm{A} \, \mathrm{d} \xi\]

\[\text { At constant } \mathrm{T} \text { and } \mathrm{p}, \mathrm{dG}=-\mathrm{A} \, \mathrm{d} \xi ; \quad \mathrm{A} \, \mathrm{d} \xi \geq 0\]

The rate at which chemical reaction drives the system to a minimum in Gibbs energy is described by the Law of Mass Action. Presumably a similar law would describe the approach to equilibrium of a system held at constant \(\mathrm{T}\) and \(\mathrm{V}\), the system moving spontaneously towards a lower Helmholtz energy. We do not explore this point. Nevertheless the isochoric condition is often invoked and we examine how this comes about.

(a) Chemical Kinetics

A key problem revolves around the role of the solvent in determining activation parameters; e.g. standard Gibbs energy of activation and standard enthalpy of activation. An extensive literature examines the role of solvents. But densities of solvents are a function of temperature and pressure. Then in attempting to understand the factors controlling kinetic activation parameters there is the problem that intermolecular distances (e.g. solvent-solvent and solvent-solute) are a function of temperature. In 1935 Evans and Polanyi [1] suggested that isochoric activation parameters for chemical reaction in aqueous solution might be more mechanistically informative than conventional isobaric activation parameters; i.e. \([\partial \ln (\mathrm{k}) / \partial \mathrm{T}]_{\mathrm{V}}\) rather than \([\partial \ln (\mathrm{k}) / \partial \mathrm{T}]_{\mathrm{p}}\) where \(\mathrm{k}\) is the rate constant for spontaneous chemical reaction [2,3].

With reference to chemical reactions in dilute aqueous solution the isochoric standard internal energy of activation \(\Delta^{\neq} \mathrm{U}_{\mathrm{V}}^{0}\) is related to the isobaric standard enthalpy of activation \(\Delta^{\neq} \mathrm{H}_{\mathrm{p}}^{0}\) at temperature \(\mathrm{T}\) and the standard volume of activation \(\Delta^{\neq} V^{0}\) using equation (f) where \(\alpha_{p 1}^{*}\) and \(\kappa_{\mathrm{T} 1}^{*}\) are respectively the isobaric expansibilities and isothermal compressibilities of water.

\[\Delta^{\neq} \mathrm{U}_{\mathrm{V}}^{0}=\Delta^{\neq} \mathrm{H}_{\mathrm{p}}^{0}-\mathrm{T} \,\left[\alpha_{\mathrm{p} 1}^{*} / \kappa_{\mathrm{T} 1}^{*}\right] \, \Delta^{\neq} \mathrm{V}^{0}\]

Baliga and Whalley [4] noted that the dependence on solvent mixture composition of \(\Delta^{z} \mathrm{U}_{\mathrm{V}}^{0}\) is less complicated than that for \(\Delta^{\neq} \mathrm{H}_{\mathrm{p}}^{0}\) for solvolysis of benzyl chloride in ethanol + water mixtures at \(298.15 \mathrm{~K}\). A similar pattern was reported by Baliga and Whalley [5] for the hydrolysis of 2-chloro-2-methyl propane in the same mixture at \(273.15 \mathrm{~K}\). [6]

The proposal concerning isochoric activation parameters sparked enormous interest and debate.[7-13] Much of the debate centred on the isochoric condition and the answer to the question ‘what volume is held constant?’ With reference to equation (f), \(\alpha_{p l}^{*}\) and \(\kappa_{\mathrm{T1}}^{*}\) depend on temperature. Then the volume identified by subscript \(\mathrm{V}\) on \(\Delta^{\neq} \mathrm{U}_{\mathrm{V}}^{0}\) is dependent on temperature. Further the molar volumes of binary liquid mixtures depend on \(\mathrm{T}\) and \(p\). In other words the isochoric condition is not global across the given data set. Haak et al. noted [14] that either side of the TMD of water there are pairs of temperatures where the molar volumes of water at ambient pressure are equal. Hence rates of reaction for chemical reactions in dilute aqueous solution at such temperatures would yield pairs of isochoric rate constants. Kinetic data for spontaneous hydrolysis of 1–benzoyl–1,2,4 triazole in aqueous solution at closely spaced temperatures close to the TMD of water reveal no unique features associated with the isochoric condition.[15]

The isochoric condition nevertheless remains interesting. There is however an important point to note. For the most part kinetic experiments investigate the rates of chemical reactions in solution at constant \(\mathrm{T}\) and \(p\). In other words spontaneous change is driven by the decrease in Gibbs energy. The latter is the operating thermodynamic potential function, both \(\mathrm{T}\) and \(p\) being held constant; the rate constant is an isobaric-isothermal property. Further the dependence of rate constant is often monitored on temperature at constant pressure, recognising that the volume of the system changes as the temperature is altered.

In principle spontaneous chemical reaction could be monitored, holding the system at constant \(\mathrm{T}\) and volume \(\mathrm{V}\). Here the direction of spontaneous chemical reaction would be towards a minimum in Helmholtz energy, \(\mathrm{F}\). The dependence of the isochoric – isothermal rate constant on temperature could again in principle be measured. The technological challenge would be immense because one might expect enormous pressures to be required to hold the volume constant when the temperature was increased.

Similar concerns over the definition of isochoric emerge in the context of the dependence on \(\mathrm{T}\) and \(p\) of acid dissociation constants in aqueous solution of ethanoic acid [16].

(a) Electrical Mobilities

The migration of ions through a salt solution under the influence of an applied electric potential can be envisaged as a rate process, analogous to the rate of chemical reaction. In most studies the applied electric field is weak such that the ions are only marginally displaced from their ‘equilibrium’ positions. The derived property is the molar conductivity [17] characterising a given salt in solution at defined \(\mathrm{T}\) and \(p\).

\[\text { Then, } \quad \ln (\Lambda)=\ln (\Lambda[\mathrm{T}, \mathrm{p}])\]

The isobaric dependence of \(\ln (\Lambda)\) on temperature yields the isobaric energy of activation \(\mathrm{E}_{\mathrm{p}}[18] \quad\left[=\mathrm{R} \,\left\{\partial \ln \left(\Lambda^{0}\right) / \partial(1 / \mathrm{T})\right\}_{\mathrm{p}}\right]\). An extensive literature [19] describes the isochoric energies of activation defined by equation (h).

\[\mathrm{E}_{\mathrm{V}}=-\mathrm{R} \, \mathrm{T}^{2}\left\{\partial \ln \left(\Lambda^{0}\right) / \partial\right\}_{\mathrm{V}}\]

The property \(\mathrm{E}_{\mathrm{V}}\) has attracted considerable attention and debate [18-,21]. Nevertheless the same question (i.e. which volume is held constant?) can be asked. The debate emerges partly form the observation that \(\mathrm{T}\) and \(p\) are intensive variables whereas \(\mathrm{V}\) is an extensive variable.

Footnotes

[1] M. G. Evans and M. Polanyi, Trans. Faraday Soc., 1935, 31, 875.

[2] M. G. Evans and M. Polanyi, Trans. Faraday Soc.,1937,33,448.

[3] D. M. Hewitt and A. Wasserman, J. Chem. Soc.,1940,735.

[4] B. T. Baliga and E. Whalley, J. Phys. Chem., 1967, 71, 1166.

[5] B. T. Baliga and E. Whalley, Can. J. Chem., 1970, 48, 528.

[6] See also

1. D. L. Gay and E. Whalley, J. Phys. Chem.,1968,72,4145.
2. E. Whalley, Adv. Phys. Org. Chem.,1964,2,93.
3. E. Whalley, Ber. Bunsenges Phys. Chem.,1966,70,958.
4. G. Kohnstam, Prog. React. Kinet., 1970,15,335.
5. E. A. Moelwyn-Hughes, The Chemical Statics and Kinetics of Solutions, Academic Press, London, 1971.
6. B.T. Baliga, R. J. Withey, D. Poulton and E. Whalley, Trans. Faraday Soc.,1965,61,517.
7. 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.
8. E. Whalley, Ann. Rev. Phys.Chem.,1967,18,205.

[7] E. Whalley, J. Chem. Soc. Faraday Trans.,1987,83,2901.

[8] L. M. P. C. Albuquerque and J. C. R. Reis, J. Chem. Soc., Faraday Trans. 1, 1989, 85, 207; 1991, 87,1553.

[9] H. A. J. Holterman and J. B. F. N. Engberts, J. Am. Chem. Soc., 1982, 104, 6382.

[10] P. G. Wright, J. Chem. Soc., Faraday Trans. 1, 1986, 82,2557.

[11] M. J. Blandamer, J. Burgess, B. Clarke, R. E. Robertson and J. M. W. Scott. J. Chem. Soc., Faraday Trans. 1, 1985, 81, 11.

[12] M. J. Blandamer, J. Burgess, B. Clarke, H. J. Cowles, I. M. Horn, J. F. B. N. Engberts, S. A. Galema and C. D. Hubbard, J. Chem. Soc. Faraday Trans. 1, 1989, 85, 3733.

[13] M. J. Blandamer, J. Burgess, B. Clarke and J. M. W. Scott, J. Chem. Soc., Faraday Trans. 1, 1984, 80, 3359.

[14] J. R. Haak, J. B. F. N. Engberts and M. J. Blandamer, J. Am. Chem. Soc., 1985, 107, 6031.

[15] M. J. Blandamer, N. J. Buurma, J. B. F. N. Engberts and J. C. R. Reis, Org. Biomol. Chem.,2003, 1,720.

[16] D. A. Lown, H. R. Thirsk and Lord Wynne-Jones, Trans. Faraday Soc., 1970, 66, 51.

[17] \(\Lambda=\left[\mathrm{S} \mathrm{} \mathrm{m}^{2} \mathrm{~mol}^{-1}\right]\)

[18] S. B. Brummer and G. J. Hills, Trans. Faraday Soc.,1961,57,1816,1823.

[19] [18] A. F. M. Barton, Rev. Pure Appl. Chem.,1971,21,49.

[20] S. B. Brummer, J. Chem. Phys.,1965,42,1636.

[21]

1. G. J. Hills, P. J. Ovenden and D. R.Whitehouse, Disc. Faraday Soc., 1965, 39, 207.
2. G. J. Hills, in Chemical Physics of Ionic Solutions, edited by B. E. Conway and R. G. Barradas, Wiley, New York, 1966, p. 521.
3. W. A. Adams and K. J. Laidler, Can. J.Chem.,1968,46,2005.
4. F. Barreira and G. J. Hills, Trans. Faraday Soc.,1968,64,1359.