1.4.8: Chemical Equilibrium Constants- Dependence on Temperature at Fixed Pressure
A given set of data reports the dependence on temperature (at fixed pressure \(p\), which is close to the standard pressure \(p^{0}\)) of \(\mathrm{K}^{0}\) for a given chemical equilibrium.[1 - 3]
\[\Delta_{\mathrm{r}} \mathrm{G}^{0}=-\mathrm{R} \mathrm{T} \ln \mathrm{K}^{0}=\Delta \mathrm{H}^{0}-\mathrm{T} \Delta_{\mathrm{r}} \mathrm{S}^{0} \nonumber \]
If we confine our attention to systems where the chemical equilibria involve solutes in dilute solution in a given solvent, we can replace \(\Delta_{\mathrm{r}} \mathrm{H}^{0}\) in this equation with the limiting enthalpy of reaction, \(\Delta_{\mathrm{r}} \mathrm{H}^{\infty}\). According to the Gibbs - Helmholtz Equation , at fixed pressure,
\[ \dfrac{ \mathrm{d}\left[\Delta_{\mathrm{r}} \mathrm{G}^{0} / \mathrm{T}\right] }{ \mathrm{dT}} =- \dfrac{ \Delta_{\mathrm{r}} \mathrm{H}^{\infty} }{ \mathrm{T}^{2}} \nonumber \]
Hence
\[\dfrac{ \mathrm{d} \ln \left(\mathrm{K}^{0}\right) }{\mathrm{dT}} = \dfrac{ \Delta_{\mathrm{r}} \mathrm{H}^{\infty} }{ \mathrm{R} \mathrm{T}^{2}} \nonumber \]
or, [4]
\[ \dfrac{ \mathrm{d} \ln \mathrm{K}^{0} }{\mathrm{dT}^{-1}} =- \dfrac{\Delta_{\mathrm{r}} \mathrm{H}^{\infty} }{ \mathrm{R} } \nonumber \]
The latter two equations are equivalent forms of the van ’t Hoff Equation expressing the dependence of \(\mathrm{K}^{0}\) on temperature. This equation does not predict how equilibrium constants depend on temperature. For example the van’t Hoff equation does not require that \(\ln \left(\mathrm{K}^{0}\right)\) is a linear function of \(\mathrm{T}-1\). In fact for simple carboxylic acids, the plots of \(\ln (\text {acid dissociation constant})\) against temperature show maxima. For example, \(\ln \left(\mathrm{K}^{0}\right)\) for the acid dissociation constant of ethanoic acid in aqueous solution at ambient pressure increases with increase in temperature, passes through a maximum near \(295 \mathrm{~K}\) and then decreases. [5-7] At the temperature where \(\mathrm{K}^{0}\) is a maximum, the limiting enthalpy of dissociation is zero. This pattern is possibly surprising at first sight but can be understood in terms of a balance between the standard enthalpy of heterolytic fission of the \(\mathrm{O}-\mathrm{H}\) group in the carboxylic acid group and the standard enthalpies of hydration of the resulting hydrogen and carboxylate ions.
Thus the dependence of \(\mathrm{K}^{0}\) on temperature can be obtained experimentally, the dependence being unique for each system [8]. Nevertheless these equations signal how the dependence forms the basis for determining limiting enthalpies of reaction. The analysis also recognises that \(\Delta_{\mathrm{r}} \mathrm{H}^{0}\) is likely to depend on temperature. There is merit in expressing the dependence of \(\mathrm{K}^{0}\) on temperature about a reference temperature \(\theta\), chosen near the middle of the experimental temperature range [2,3,9]. Over the experimental temperature range straddling \(\theta\), we express the dependence of \(\mathrm{K}^{0}\) on temperature using the integrated form of equation (c).
\[\ln \left[\mathrm{K}^{0}(\mathrm{~T})\right]=\ln \left[\mathrm{K}^{0}(\theta)\right]+\int_{\theta}^{\mathrm{T}}\left[\frac{\Delta_{\mathrm{r}} \mathrm{H}^{\omega}}{\mathrm{RT}^{2}}\right] \mathrm{dT} \nonumber \]
By definition, the limiting isobaric heat capacity of reaction \(\Delta_{\mathrm{r}} C_{\mathrm{p}}^{\infty}\) is given by equation (f).
\[\Delta_{\mathrm{r}} \mathrm{C}_{\mathrm{p}}^{\infty}=\left( \dfrac{ \mathrm{d} \Delta_{\mathrm{r}} \mathrm{H}^{\infty} }{ \mathrm{dT}}\right)_{\mathrm{p}} \nonumber \]
The analysis becomes complicated because we recognise that \(\Delta_{\mathrm{r}} C_{\mathrm{p}}^{\infty}\) depends on temperature. [9] In fact only in rare instances are experimental results sufficiently precise to warrant taking such a dependence into account. [10] A reasonable assumption is that \(\Delta_{r} C_{p}^{\infty}\) is independent of temperature such that \(\Delta_{\mathrm{r}} \mathrm{H}^{\infty}\) is a linear function of temperature over the experimental temperature range.[11]
\[\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{T})=\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\theta)+\Delta_{\mathrm{r}} \mathrm{C}_{\mathrm{p}}^{\infty}(\mathrm{T}-\theta) \nonumber \]
Hence,
\[\begin{aligned}
&\ln \left[\mathrm{K}^{0}(\mathrm{~T})\right]= \\
&\ln \left[\mathrm{K}^{0}(\theta)\right]+\frac{1}{\mathrm{R}} \int_{\theta}^{\mathrm{T}}\left[\frac{\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\theta)}{\mathrm{T}^{2}}+\Delta_{\mathrm{r}} \mathrm{C}_{\mathrm{p}}^{\infty} \left(\frac{1}{\mathrm{~T}}-\frac{\theta}{\mathrm{T}^{2}}\right)\right] \mathrm{dT}
\end{aligned} \nonumber \]
Hence,
\[\begin{aligned}
&\ln \left[\mathrm{K}^{0}(\mathrm{~T})\right]= \\
&\ln \left[\mathrm{K}^{0}(\theta)\right]+\frac{\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\theta)}{\mathrm{R}} \left[\frac{1}{\theta}-\frac{1}{\mathrm{~T}}\right]+\frac{\Delta_{\mathrm{r}} \mathrm{C}_{\mathrm{p}}^{\infty}}{\mathrm{R}} \left[\frac{\theta}{\mathrm{T}}-1+\ln \left(\frac{\mathrm{T}}{\theta}\right)\right]
\end{aligned} \nonumber \]
Numerical analysis uses linear least squares procedures with reference to the dependence of \(\ln K^{0}(T)\) on temperature about the reference temperature \(\theta\) in order to obtain estimates of \(\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\theta)\) and \(\Delta_{r} C_{p}^{\infty}\). The coupling of estimates of derived parameters is minimal if θ is chosen near the centre of the measured temperature range. [2,3] Granted that the analysis yields \(\Delta_{\mathrm{r}} \mathrm{H}^{\infty}\) at a given temperature and pressure, combination with the corresponding \(\Delta_{\mathrm{r}} \mathrm{G}^{0}\) yields the entropy term, \(\Delta_{\mathrm{r}} \mathrm{S}^{0}\).
Other methods of data analysis in this context use (a) orthogonal polynomials, [12] and (b) sigma plots. [13]
An extensive literature describes the thermodynamics of acid dissociation in alcohol + water mixtures. In these solvent systems the standard enthalpies and other thermodynamic parameters pass through extrema as the mole fraction composition of the solvent is changed. [14- 18]
Perlmutter-Hayman examines the related problem of the dependence on temperature of activation energies [19].
Enthalpies of dissociation for weak acids in aqueous solution can be obtained calorimetrically. [20]
Footnotes
[1] R. W. Ramette, J. Chem. Educ.,1977, 54 ,280
[2] M. J. Blandamer, J. Burgess, R. E. Robertson and J. M. W. Scott, Chem. Rev., 1982, 82 ,259.
[3] M. J. Blandamer, Chemical Equilibria in Solution, Ellis Horwood PTR Prentice Hall, New York,1992.
[4] \(\mathrm{d} \ln \mathrm{K}^{0} / \mathrm{dT}^{-1}=\left[\mathrm{J} \mathrm{mol}^{-1}\right] /\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right]=[\mathrm{K}]\)
[5] H. S. Harned and N. D. Embree, J. Am. Chem. Soc.,1934, 56 ,1050.
[6] H. S. Harned and R. W. Ehlers, J.Am.Chem.Soc.,1932, 54 ,1350.
[7] See also ethanoic acid in D2O; M. Paabo, R. G. Bates and R. A. Robinson, J. Phys. Chem., 1966, 70 ,2073; and references therein.
[8] Substituted benzoic acids(aq); L. E. Strong, C. L. Brummel and P. Lindower, J. Solution Chem., 1987, 16 , 105; and references therein.
[9] E. C. W. Clarke and D. N. Glew, Trans. Faraday Soc.,1966, 62 ,539.
[10] H. F. Halliwell and L. E. Strong, J. Phys. Chem.,1985, 89 ,4137.
[11] \(\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\theta)=\left[\mathrm{J} \mathrm{mol}^{-1}\right]+\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] [\mathrm{K}]\)
[12] D. J. G. Ives and P. D. Marsden, J. Chem. Soc.,1965,649 and 2798.
[13] D. J. G. Ives, P. G. N. Moseley, J. Chem. Soc. Faraday Trans.1, 1976, 72 ,1132.
[14] Anilinium ions in EtOH+water mixtures;W. van der Poel, Bull. Soc. Chim. Belges., 1971, 80 ,401; and references therein.
[15] Enthalpies of transfer for carboxylic acids in water+ 2-methyl propan-2-ol mixtures; L. Avedikian, J. Juillard and J.-P. Morel, Thermochim. Acta, 1973, 6 ,283.
[16] Benzoic acid in DMSO + water mixtures; F. Rodante, F. Rallo and P. Fiordiponti, Thermochim. Acta, 1974, 9 ,269.
[17] Tris in water + methanol mixtures; C. A. Vega, R. A. Butler, B. Perez and C. Torres, J. Chem. Eng. Data, 1985, 30 ,376.
[18] F. J. Millero, C-h. Wu and L. G. Hepler, J. Phys. Chem., 1969, 73 ,2453.
[19] B. Perlmutter-Hayman, Prog. Inorg. Chem.,1976, 20 ,229.
[20] F. Rodante, G. Ceccaroni and F. Fantauzzi, Thermochim. Acta,1983, 70 ,91.