1.4.9: Chemical Equilibria- Dependence on Pressure at Fixed Temperature

A key quantity in the description of a chemical equilibrium is the equilibrium constant. In the majority of cases the symbol used is \(\mathrm{K}^{0}\) indicating with the superscript ‘0’ a standard property. This symbol is used because, again in the majority of cases an equilibrium constant refers to a system at ambient pressure which is close to the standard pressure; i.e. \(10^{5} \mathrm{~Pa}\). In reporting \(\mathrm{K}^{0}\) therefore the temperature is stated but by definition \(\mathrm{K}^{0}\) is not dependent on pressure. However the equilibrium composition of a closed system generally depends on pressure at fixed temperature \(\mathrm{T}\). This problem over symbols and nomenclature is resolved as follows [1-7].

An aqueous solution contains \(i\)-chemical substances, solutes, in chemical equilibrium. For a given solute–\(j\) the dependence of chemical potential \(\mu_{j}(a q ; T ; p)\) on molality \(\mathrm{m}_{j}\) is given by equation (a).

\[\begin{aligned}
&\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})= \\
&\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)+\int_{\mathrm{p}^{\circ}}^{\mathrm{p}} \mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq}) \, \mathrm{dp}
\end{aligned}\]

We define a reference chemical potential for solute-\(j\) \(\mu_{j}^{*}\) at temperature \(\mathrm{T}\) and pressure \(p\) using equation (b).

\[\mu_{\mathrm{j}}^{\#}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T})+\int_{\mathrm{p}^{0}}^{\mathrm{p}} \mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq}) \, \mathrm{dp}\]

Combination of equations (a) and (b) yields equation (c).

\[\mu_{j}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{\mathrm{j}}^{\#}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\]

XK

Here \(\mu_{j}^{H}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\) is the chemical potential of solute-j in an ideal solution (i.e. \(\gamma_{j}=1\)) having unit molality (i.e. \(\mathrm{m}_{\mathrm{j}}=1 \mathrm{~mol} \mathrm{~kg}{ }^{-1}\)) at specified \(\mathrm{T}\) and \(p\). At equilibrium at pressure \(p\) and temperature \(\mathrm{T}\),

\[\sum_{j=1}^{j=i} v_{j} \, \mu_{j}^{\mathrm{eq}}(\mathrm{aq} ; T ; p)=0\]

By definition,

\[\Delta_{\mathrm{r}} \mathrm{G}^{*}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\sum \mathrm{v}_{\mathrm{j}} \, \mu_{\mathrm{j}}^{*}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=-\mathrm{R} \, \mathrm{T} \, \ln \mathrm{K}^{*}(\mathrm{~T}, \mathrm{p})\]

\[\text { and } \mathrm{K}^{\#}(\mathrm{~T}, \mathrm{p})=\prod_{\mathrm{j}=1}^{\mathrm{j}=1}\left[\left(\mathrm{~m}_{\mathrm{j}}^{\mathrm{eq}} / \mathrm{m}^{0}\right) \, \gamma_{j}^{e q}\right]^{v(j)}\]

The differential dependence on pressure [8,9] of \(\mathrm{K}^{\#}(\mathrm{~T}, \mathrm{p})\) yields the limiting volume of reaction, \(\Delta_{\mathrm{r}} \mathrm{V}^{\infty}\).

\[\Delta_{r} V^{\infty}=\sum_{j=1}^{j=i} V_{j} \, V_{j}^{\infty}(a q ; T ; p)=0\]

\[\text { and [c.f. V } \left.=[\partial \mathrm{G} / \partial \mathrm{p}]_{\mathrm{T}}\right] \mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\left[\partial \mu_{\mathrm{j}}^{*}(\mathrm{aq}) / \partial \mathrm{p}\right]_{\mathrm{T}}\]

\[\text { Hence at pressure } p,\left(\frac{\partial \Delta_{\mathrm{r}} \mathrm{G}^{*}(\mathrm{~T})}{\partial \mathrm{p}}\right)_{\mathrm{T}}=\Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\mathrm{T}, \mathrm{p})\]

\[\text { or, }\left(\frac{\partial \ln \mathrm{K}^{\prime \prime}(\mathrm{T})}{\partial \mathrm{p}}\right)_{\mathrm{T}}=-\frac{\Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\mathrm{T}, \mathrm{p})}{\mathrm{R} \, \mathrm{T}}\]

The negative sign in equation (j) means that if \(\ln \mathrm{K}^{\#}\) for a given chemical equilibrium increases with increases in pressure then \(\Delta_{\mathrm{r}} \mathrm{V}^{\infty}\) is negative. But thermodynamics does not define how a given equilibrium constant depends on pressure. This dependence must be measured. Moreover we cannot assume that the limiting volume of reaction \(\Delta_{\mathrm{r}} \mathrm{V}^{\infty}\) is independent of pressure. This dependence is described by the limiting isothermal compressions of reaction, \(\Delta_{\mathrm{r}} \mathrm{K}_{\mathrm{T}}^{\infty}\).

\[\Delta_{\mathrm{r}} \mathrm{K}_{\mathrm{T}}^{\infty}=-\left[\frac{\mathrm{d} \Delta_{\mathrm{r}} \mathrm{V}^{\infty}}{\mathrm{dp}}\right]_{\mathrm{T}}\]

Indeed we cannot assume that \(\mathrm{K}_{\mathrm{T}}^{\infty}\) is independent of pressure but in most cases the precision of the data is insufficient to obtain a meaningful estimate of this dependence. Hence we are often justified in assuming that \(\Delta_{\mathrm{r}} \mathrm{V}^{\infty}\) is a linear function of pressure about a reference pressure \(\pi\), the latter usually chosen as ambient pressure. [10]

\[\Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\mathrm{p})=\Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\pi)-\Delta_{\mathrm{r}} \mathrm{K}_{\mathrm{T}}^{\infty}(\mathrm{p}-\pi)\]

Hence, [11]

\[\ln \left(\mathrm{K}^{\#}(\mathrm{p})\right)=\ln \left(\mathrm{K}^{\#}(\pi)\right)-(\mathrm{R} \, \mathrm{T})^{-1} \, \Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\pi) \,(\mathrm{p}-\pi) -(2 \, R \, T)^{-1} \, \Delta_{r} K_{T}^{\infty} \,\left((p-\pi)^{2}\right)\]

Thus, \(\ln \mathrm{K}^{\#}(\mathrm{p})\) is a quadratic in \((\mathrm{p}-\pi)\).

Alternatively we may express the dependence of \(\ln \left[\mathrm{K}^{\#}(\mathrm{p}) / \mathrm{K}^{\#}(\pi)\right]\) on pressure using the following equation. [12,13]

\[\begin{aligned}
&\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{K}^{\#}(\mathrm{p}) / \mathrm{K}^{\#}(\pi)\right]= \\
&\quad-\Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\pi) \,(\mathrm{p}-\pi)-0.5 \, \Delta_{\mathrm{r}} \mathrm{K}_{\mathrm{T}}^{\infty} \,(\mathrm{p}-\pi)^{2}
\end{aligned}\]

This equation shows how \(\left[\mathrm{K}^{\#}(\mathrm{p}) / \mathrm{K}^{\#}(\pi)\right]\) may be calculated from estimates of \(\Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\pi)\) obtained from independently obtained estimates of partial molar volumes and partial molar isothermal compressions of the chemical substances involved in the chemical equilibrium; e.g. acid dissociation of boric acid. [14,15]

Another approach expresses the ratio \(\left[\mathrm{K}^{\#}(\mathrm{p}) / \mathrm{K}^{\#}(\pi)\right]\) as a function of solvent density at pressure \(p\), \(\rho(p)\) together with density \(\rho(\pi)\) at pressure \(\pi\) and a parameter \(\beta\) using equation (o) [16].

\[\ln \left[K^{\#}(\mathrm{p}) / \mathrm{K}^{\#}(\pi)\right]=(\beta-1) \, \ln [\rho(\pi) / \rho(\mathrm{p})]\]

This approach is closely linked to numerical analysis based on equation (p). [17]

\[\ln \left[K^{\#}(p) / K^{\#}(\pi)\right]=-\left[\Delta_{r} V^{\infty}(\pi) / R \, T\right] \,[p /(1+b \, p)]\]

A rather different approach for chemical equilibria between solutes in aqueous solutions refers to equation (q). \(\mathrm{A}\) and \(\mathrm{B}\) are constants independent of pressure but dependent on temperature; these constants describe the dependence of the molar volume of water on pressure at fixed temperature; Tait’s isotherm [2,18-21].

\[\mathrm{V}_{1}^{*}(\mathrm{p})=\mathrm{V}_{1}^{*}(\pi) \,\left[1-\mathrm{A} \, \ln \left(\frac{\mathrm{B}+\mathrm{p}}{\mathrm{B}+\pi}\right)\right]\]

There are a few case where the experimental data warrant consideration of the dependence on pressure of \(\Delta_{r} K^{\infty}\). Under these circumstances the Owen-Brinkley equation has the following form [2].

\[\begin{aligned}
\mathrm{R} \, \mathrm{T} \, \ln \left[\frac{\mathrm{K}^{\#}(\mathrm{p})}{\mathrm{K}^{\#}(\pi)}\right]=-\Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\pi) \,(\mathrm{p}-\pi) \\
&+\Delta_{\mathrm{r}} \mathrm{K}_{\mathrm{T}}^{\infty}(\pi) \,\left[(\mathrm{B}+\pi) \,(\mathrm{p}-\pi)-(\mathrm{B}+\pi)^{2} \, \ln \left(\frac{\mathrm{B}+\mathrm{p}}{\mathrm{B}+\pi}\right)\right]
\end{aligned}\]

Footnotes

[1] M. J. Blandamer, Chemical Equilibria in Solution, Ellis Horwood PTR, Prentice Hall, New York,1992.

[2] B. B. Owen and S. R. Brinkley, Chem. Rev.,1941,29,401.

[3] S. W. Benson and J. A. Person, J. Amer. Chem. Soc.,1962,84,152.

[4] S. D. Hamann, J. Solution Chem.,1982,11,63; and references therein.

[5] N. A. North, J.Phys.Chem.,1973,77,931.

[6] B. S. El’yanov and E. M. Vasylvitskaya, Rev. Phys. Chem. Jpn, 1980, 50, 169; and references therein.

[7] There is a strong link between this subject and analysis of the dependence of rate constants for chemical reactions on pressure at fixed temperature;

1. R. van Eldik and H. Kelm, Rev. Phys. Chem. Jpn,1980,50,145.
2. C. A. N. Viana and J. C. R. Reis, Pure Appl. Chem.,1996,68,1541.
3. E. Whalley, Adv. Phys.Org. Chem.,1964,2,93.
4. W. J. leNoble, J. Chem. Educ.,1967,44,729.

[8] By definition the standard equilibrium constant \({\mathrm{K}}_{\mathrm{m}}}^{0}\) describes the case where at temperature \(\mathrm{T}\), the pressure is the standard pressure.

[9] \(\frac{\mathrm{d} \ln \mathrm{K}^{*}}{\mathrm{dp}}=\frac{\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]}{\left[\mathrm{J} \mathrm{mol}^{-1} \mathrm{~K}^{-1}\right] \,[\mathrm{K}]}=\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1}\)

[10] \(\Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\mathrm{p})=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]+\left(\frac{\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]}{\left[\mathrm{N} \mathrm{m}^{-2}\right]}\right) \,\left[\mathrm{N} \mathrm{m}^{-2}\right]\)

[11] \(\ln \left(\mathrm{K}^{\#}(\mathrm{p})\right)=[1]+\frac{1}{\left[\mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right]} \,\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right] \,\left[\mathrm{N} \mathrm{m}^{-2}\right] +\frac{1}{[1] \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}]} \,\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} \mathrm{~N}^{-1} \mathrm{~m}^{2}\right] \,\left[\mathrm{N} \mathrm{m}^{-2}\right]^{2}\)

[12] D. A. Lown, H. R. Thirsk and Lord Wynne-Jones, Trans. Faraday Soc., 1968, 64, 2073.

[13] A. J. Read {J. Solution Chem.,1982, 11, 649;1988, 17, 213} uses a simpler form of the equation which has the general form, \(y=m . x+c\). Thus, \(\left[\frac{\mathrm{R} \, \mathrm{T}}{\mathrm{p}-\pi}\right] \, \ln \left[\frac{\mathrm{K}^{\#}(\mathrm{p})}{\mathrm{K}^{\#}(\pi)}\right]=-\Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\pi)-0.5 \,(\mathrm{p}-\pi) \, \Delta_{\mathrm{r}} \mathrm{K}_{\mathrm{T}}^{\infty}\)

[14] G. K. Ward and F. J. Millero, J. Solution Chem.,1974,3,417.

[15] See also pH and pOH; Y. Kitamura and T. Itoh, J. Solution Chem., 1987, 16, 715.

[16] W. L. Marshall and R. E. Mesmer, J. Solution Chem., 1984, 13, 383; and references therein.

[17] B. S. El’yanov and S. D. Hamann, Aust. J. Chem.,1975,28,945.

[18] R. E. Gibson, J.Am.Chem.Soc.,1934,56,4.

[19] S. D. Hamann and F. E. Smith, Aust. J. Chem.,1971,24,2431.

[20] G. A. Neece and D. R. Squire, J. Phys. Chem.,1968,72,128.

[21] K. E. Weale, Chemical Reactions at High Pressure, Spon, London,1967; and references therein.