1.8.13: Enthalpies- Liquid Mixtures
For an ideal binary liquid mixture the Gibbs energy at temperature T is given by equation (a).
\[\mathrm{G}(\operatorname{mix} ; \mathrm{id})=\mathrm{n}_{1} \,\left[\mu_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)\right]+\mathrm{n}_{2} \,\left[\mu_{2}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{2}\right)\right] \nonumber \]
From the Gibbs-Helmholtz equation,
\[\mathrm{H}(\operatorname{mix} ; \mathrm{id})=\mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\ell)+\mathrm{n}_{2} \, \mathrm{H}_{2}^{*}(\ell) \nonumber \]
Hence for an ideal binary liquid mixture,
\[\mathrm{H}_{1}(\operatorname{mix} ; \mathrm{id})=\mathrm{H}_{1}^{*}(\ell) \text { and } \mathrm{H}_{2}(\operatorname{mix} ; \mathrm{id})=\mathrm{H}_{2}^{*}(\ell) \nonumber \]
The molar enthalpy of a real binary liquid mixture is given by equation (d).
\[\mathrm{H}_{\mathrm{m}}=\mathrm{x}_{1} \, \mathrm{H}_{1}(\operatorname{mix})+\mathrm{x}_{2} \, \mathrm{H}_{2}(\operatorname{mix}) \nonumber \]
Therefore the molar enthalpy of mixing for a real binary liquid mixture is given by equation (e).
\[\Delta_{\text {mix }} H_{m}=x_{1} \,\left[H_{1}(\operatorname{mix})-H_{1}^{*}(\ell)\right]+x_{2} \,\left[H_{2}(\operatorname{mix})-H_{2}^{*}(\ell]\right. \nonumber \]
Significantly equations (b) and (e) show that the molar enthalpy of mixing of an ideal binary liquid mixture, \(\Delta_{\text {mix }} H_{m}(\mathrm{id})\) is zero. The latter condition offers an important point of reference for isobaric calorimetry. [1] If we discover that the mixing of two liquids (at constant pressure) is not zero, the measured molar heat of mixing [\(=\Delta_{\text {mix }} \mathrm{H}_{\mathrm{m}}\)] is an immediate indicator of the extent to which the properties of a given mixture are not ideal.
Nevertheless it is important to set down a link between the measured enthalpies of mixing with the activity coefficients of two liquid components. To this end we start with the equation for the chemical potentials of liquid component 1 in a liquid mixture at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) (which is close to ambient); equation (f).
\[\mu_{1}(\mathrm{mix})=\mu_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right) \nonumber \]
where
\[\operatorname{limit}\left(x_{1} \rightarrow 1\right) f_{1}=1 \text { at all } T \text { and } p \text {. } \nonumber \]
The Gibbs - Helmholtz equation yields an equation for the partial molar enthalpy of component 1 in the liquid mixture. Thus
\[\mathrm{H}_{1}(\operatorname{mix})=\mathrm{H}_{1}^{*}(\ell)-\mathrm{R} \, \mathrm{T}^{2} \,\left[\partial \ln \left(\mathrm{f}_{1}\right) / \partial \mathrm{T}\right]_{\mathrm{p}} \nonumber \]
Similarly,
\[\mathrm{H}_{2}(\operatorname{mix})=\mathrm{H}_{2}^{*}(\ell)-\mathrm{R} \, \mathrm{T}^{2} \,\left[\partial \ln \left(\mathrm{f}_{2}\right) / \partial \mathrm{T}\right]_{\mathrm{p}} \nonumber \]
Hence,
\[\begin{aligned}
&\mathrm{H}_{\mathrm{m}}(\operatorname{mix})= \\
&\quad \mathrm{H}_{\mathrm{m}}(\operatorname{mix} ; \mathrm{id})-\mathrm{R} \, \mathrm{T}^{2} \,\left\{\mathrm{x}_{1} \,\left[\partial \ln \left(\mathrm{f}_{1}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}+\mathrm{x}_{2} \,\left[\partial \ln \left(\mathrm{f}_{2}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}\right\}
\end{aligned} \nonumber \]
We also obtain equations for the excess molar enthalpies of the two components (at defined \(\mathrm{T}\) and \(\mathrm{p}\)).
\[\mathrm{H}_{1}^{\mathrm{E}}(\mathrm{mix})=-\mathrm{R} \, \mathrm{T}^{2} \,\left[\partial \ln \left(\mathrm{f}_{1}\right) / \partial \mathrm{T}\right]_{\mathrm{p}} \nonumber \]
and
\[\mathrm{H}_{2}^{\mathrm{E}}(\mathrm{mix})=-\mathrm{R} \, \mathrm{T}^{2} \,\left[\partial \ln \left(\mathrm{f}_{2}\right) / \partial \mathrm{T}\right]_{\mathrm{p}} \nonumber \]
The excess molar enthalpy,
\[\mathrm{H}_{\mathrm{m}}^{\mathrm{E}}(\text { mix })=-\mathrm{R} \, \mathrm{T}^{2} \,\left\{\mathrm{x}_{1} \,\left[\partial \ln \left(\mathrm{f}_{1}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}+\mathrm{x}_{2} \,\left[\partial \ln \left(\mathrm{f}_{2}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}\right\} \nonumber \]
At fixed pressure, the differential dependence of \(\mathrm{H}_{\mathrm{m}}^{\mathrm{E}}(\operatorname{mix})\) on temperature yields the corresponding excess isobaric heat capacity of mixing.
Footnotes
[1] J. B. Ott and C. J. Wormald, Experimental Thermodynamics, IUPAC Chemical Data Series, No. 39, ed. K. N. Marsh and P. A. G. O’Hara, Blackwell, Oxford, 1994, chapter 8.