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).
\[\begin{aligned}
&\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})= \\
&\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)+\int_{\mathrm{p}^{\circ}}^{\mathrm{p}} \mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq}) \, \mathrm{dp}
\end{aligned} \nonumber \]
We define a reference chemical potential for solute-\(j\) \(\mu_{j}^{*}\) at temperature \(\mathrm{T}\) and pressure \(p\) using equation (b).
\[\mu_{\mathrm{j}}^{\#}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T})+\int_{\mathrm{p}^{0}}^{\mathrm{p}} \mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq}) \, \mathrm{dp} \nonumber \]
Combination of equations (a) and (b) yields equation (c).
\[\mu_{j}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{\mathrm{j}}^{\#}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right) \nonumber \]
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 \(\mathrm{T}\),
\[\sum_{j=1}^{j=i} v_{j} \, \mu_{j}^{\mathrm{eq}}(\mathrm{aq} ; T ; p)=0 \nonumber \]
By definition,
\[\Delta_{\mathrm{r}} \mathrm{G}^{*}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\sum \mathrm{v}_{\mathrm{j}} \, \mu_{\mathrm{j}}^{*}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=-\mathrm{R} \, \mathrm{T} \, \ln \mathrm{K}^{*}(\mathrm{~T}, \mathrm{p}) \nonumber \]
\[\text { and } \mathrm{K}^{\#}(\mathrm{~T}, \mathrm{p})=\prod_{\mathrm{j}=1}^{\mathrm{j}=1}\left[\left(\mathrm{~m}_{\mathrm{j}}^{\mathrm{eq}} / \mathrm{m}^{0}\right) \, \gamma_{j}^{e q}\right]^{v(j)} \nonumber \]
The differential dependence on pressure [8,9] of \(\mathrm{K}^{\#}(\mathrm{~T}, \mathrm{p})\) yields the limiting volume of reaction, \(\Delta_{\mathrm{r}} \mathrm{V}^{\infty}\).
\[\Delta_{r} V^{\infty}=\sum_{j=1}^{j=i} V_{j} \, V_{j}^{\infty}(a q ; T ; p)=0 \nonumber \]
\[\text { and [c.f. V } \left.=[\partial \mathrm{G} / \partial \mathrm{p}]_{\mathrm{T}}\right] \mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\left[\partial \mu_{\mathrm{j}}^{*}(\mathrm{aq}) / \partial \mathrm{p}\right]_{\mathrm{T}} \nonumber \]
\[\text { Hence at pressure } p,\left(\frac{\partial \Delta_{\mathrm{r}} \mathrm{G}^{*}(\mathrm{~T})}{\partial \mathrm{p}}\right)_{\mathrm{T}}=\Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\mathrm{T}, \mathrm{p}) \nonumber \]
\[\text { or, }\left(\frac{\partial \ln \mathrm{K}^{\prime \prime}(\mathrm{T})}{\partial \mathrm{p}}\right)_{\mathrm{T}}=-\frac{\Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\mathrm{T}, \mathrm{p})}{\mathrm{R} \, \mathrm{T}} \nonumber \]
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, \(\Delta_{\mathrm{r}} \mathrm{K}_{\mathrm{T}}^{\infty}\).
\[\Delta_{\mathrm{r}} \mathrm{K}_{\mathrm{T}}^{\infty}=-\left[\frac{\mathrm{d} \Delta_{\mathrm{r}} \mathrm{V}^{\infty}}{\mathrm{dp}}\right]_{\mathrm{T}} \nonumber \]
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]
\[\Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\mathrm{p})=\Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\pi)-\Delta_{\mathrm{r}} \mathrm{K}_{\mathrm{T}}^{\infty}(\mathrm{p}-\pi) \nonumber \]
Hence, [11]
\[\ln \left(\mathrm{K}^{\#}(\mathrm{p})\right)=\ln \left(\mathrm{K}^{\#}(\pi)\right)-(\mathrm{R} \, \mathrm{T})^{-1} \, \Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\pi) \,(\mathrm{p}-\pi) -(2 \, R \, T)^{-1} \, \Delta_{r} K_{T}^{\infty} \,\left((p-\pi)^{2}\right) \nonumber \]
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]
\[\begin{aligned}
&\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{K}^{\#}(\mathrm{p}) / \mathrm{K}^{\#}(\pi)\right]= \\
&\quad-\Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\pi) \,(\mathrm{p}-\pi)-0.5 \, \Delta_{\mathrm{r}} \mathrm{K}_{\mathrm{T}}^{\infty} \,(\mathrm{p}-\pi)^{2}
\end{aligned} \nonumber \]
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].
\[\ln \left[K^{\#}(\mathrm{p}) / \mathrm{K}^{\#}(\pi)\right]=(\beta-1) \, \ln [\rho(\pi) / \rho(\mathrm{p})] \nonumber \]
This approach is closely linked to numerical analysis based on equation (p). [17]
\[\ln \left[K^{\#}(p) / K^{\#}(\pi)\right]=-\left[\Delta_{r} V^{\infty}(\pi) / R \, T\right] \,[p /(1+b \, p)] \nonumber \]
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].
\[\mathrm{V}_{1}^{*}(\mathrm{p})=\mathrm{V}_{1}^{*}(\pi) \,\left[1-\mathrm{A} \, \ln \left(\frac{\mathrm{B}+\mathrm{p}}{\mathrm{B}+\pi}\right)\right] \nonumber \]
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].
\[\begin{aligned}
\mathrm{R} \, \mathrm{T} \, \ln \left[\frac{\mathrm{K}^{\#}(\mathrm{p})}{\mathrm{K}^{\#}(\pi)}\right]=-\Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\pi) \,(\mathrm{p}-\pi) \\
&+\Delta_{\mathrm{r}} \mathrm{K}_{\mathrm{T}}^{\infty}(\pi) \,\left[(\mathrm{B}+\pi) \,(\mathrm{p}-\pi)-(\mathrm{B}+\pi)^{2} \, \ln \left(\frac{\mathrm{B}+\mathrm{p}}{\mathrm{B}+\pi}\right)\right]
\end{aligned} \nonumber \]
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;
- R. van Eldik and H. Kelm, Rev. Phys. Chem. Jpn,1980, 50 ,145.
- C. A. N. Viana and J. C. R. Reis, Pure Appl. Chem.,1996, 68 ,1541.
- E. Whalley, Adv. Phys.Org. Chem.,1964, 2 ,93.
- 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.