1.10.31: Gibbs Energies- Liquid Mixtures- Ideal
The molar Gibbs energy of mixing for an ideal binary liquid mixture is given by equation (a);
\[\Delta_{\text {mix }} \mathrm{G}_{\mathrm{m}}(\mathrm{id})=\mathrm{R} \, \mathrm{T} \,\left[\mathrm{x}_{1} \, \ln \mathrm{x}_{1}+\mathrm{x}_{2} \, \ln \mathrm{x}_{2}\right] \nonumber \]
\[\Delta_{\text {mix }} \mathrm{G}_{\mathrm{m}}(\mathrm{id}) / \mathrm{R} \, \mathrm{T}=\mathrm{x}_{1} \, \ln \mathrm{x}_{1}+\mathrm{x}_{2} \, \ln \mathrm{x}_{2} \nonumber \]
In fact the molar Gibbs energy of mixing for an ideal binary mixture is negative across the complete composition range. According to Gibbs - Helmholtz equation, the molar enthalpy of mixing for an ideal binary mixture is given by equation (c).
\[\Delta_{\operatorname{mix}} \mathrm{H}_{\mathrm{m}}(\mathrm{id})=\frac{\mathrm{d}}{\mathrm{d}\left(\mathrm{T}^{-1}\right)}\left[\frac{\Delta_{\operatorname{mix}} \mathrm{G}(\mathrm{id})}{\mathrm{T}}\right]_{\mathrm{p}} \nonumber \]
But mole fractions are not dependent on temperature. Hence,
\[\Delta_{\text {mix }} \mathrm{H}_{\mathrm{m}}(\mathrm{id})=0 \nonumber \]
This important result offers a point of reference. At fixed pressure, the mixing of two liquids to form an ideal binary liquid mixture is athermal. Hence a recorded heat of mixing is a direct measure of the extent to which the properties of a given mixture differ from those defined as ideal. But,
\[\Delta_{\text {mix }} \mathrm{G}_{\mathrm{m}}(\mathrm{id})=\Delta_{\mathrm{mix}} \mathrm{H}_{\mathrm{m}}(\mathrm{id})-\mathrm{T} \, \Delta_{\text {mix }} \mathrm{S}_{\mathrm{m}}(\mathrm{id}) \nonumber \]
For an ideal binary liquid mixture the partial molar entropies of the two liquid components are given by the following equations.
\[\mathrm{S}_{1}(\operatorname{mix} ; \mathrm{id})=\mathrm{S}_{1}^{*}(\ell)-\mathrm{R} \, \ln \left(\mathrm{x}_{1}\right) \nonumber \]
\[S_{2}(\operatorname{mix} ; \mathrm{id})=\mathrm{S}_{2}^{*}(\ell)-\mathrm{R} \, \ln \left(\mathrm{x}_{2}\right) \nonumber \]
\[\mathrm{S}_{\mathrm{m}}(\operatorname{mix} ; \mathrm{id})=\mathrm{x}_{1} \,\left[\mathrm{S}_{1}^{*}(\ell)-\mathrm{R} \, \ln \left(\mathrm{x}_{1}\right)\right]+\mathrm{x}_{2} \,\left[\mathrm{S}_{2}^{*}(\ell)-\mathrm{R} \, \ln \left(\mathrm{x}_{2}\right)\right] \nonumber \]
From equation (h),
\[\Delta_{\operatorname{mix}} \mathrm{S}_{\mathrm{m}}(\mathrm{id})=-\mathrm{R} \,\left[\mathrm{x}_{1} \, \ln \mathrm{x}_{1}+\mathrm{x}_{1} \, \ln \mathrm{x}_{2}\right] \nonumber \]
or,
\[\mathrm{T} \, \Delta_{\text {mix }} S_{m}(\text { id })=-R \, T\left[x_{1} \, \ln x_{1}+x_{2} \, \ln x_{2}\right] \nonumber \]
But across the complete mole fraction range \(\left[x_{1} \, \ln x_{1}+x_{2} \, \ln x_{2}\right] \leq 0\). Over the same range,
\[\mathrm{T} \, \Delta_{\operatorname{mix}} \mathrm{S}_{\mathrm{m}}(\mathrm{id})>0 \nonumber \]
Thus the sign and magnitude of \(\Delta_{\text{mix}}\mathrm{G}_{\mathrm{m}}(\text{id})\) and (with opposite sign) \(\mathrm{T} \, \Delta_{\mathrm{mix}} \mathrm{S}_{\mathrm{m}}(\mathrm{id})\) are defined. A further consequence of equation (d) is that the corresponding isobaric heat capacity variable, \(\Delta_{\mathrm{mix}} \mathrm{C}_{\mathrm{pm}}(\mathrm{id})\) is zero across the whole mole fraction range. Using equation (a), the molar volume of mixing is given by equation (l). Thus,
\[\Delta_{\text {mix }} V_{m}(\mathrm{id})=\frac{\partial}{\partial p}\left[\Delta_{\text {mix }} G_{m}(\mathrm{id})\right]_{p} \nonumber \]
Hence for a binary liquid mixture having ideal properties, across the complete mole fraction range, \(\Delta_{\operatorname{mix}} \mathrm{V}_{\mathrm{m}}(\mathrm{id})=0\). The latter condition requires that the volume of a liquid mixture equals the sum of the volumes of the two liquid components used to prepare the mixture at fixed temperature and pressure. For such a mixture,
\[\mathrm{V}_{\text {mix }}(\mathrm{id})=\mathrm{V}_{2}^{*}(\ell)+\mathrm{x}_{1} \,\left[\mathrm{V}_{1}^{*}(\ell)-\mathrm{V}_{2}^{*}(\ell)\right] \nonumber \]
This simple pattern is not observed. In fact the molar volume of a real binary mixture is usually less than \(\mathrm{V}_{\text{mix} (\text{id})\).
With the benefit of hindsight, we distinguish between Gibbsian and non-Gibbsian on the one hand and between first and second law (thermodynamic) variables on the other hand. Variables \(\mathrm{H}\), \(\mathrm{V}\) and \(\mathrm{C}_{\mathrm{p}}\) are Gibbsian first law variables such that the molar property of an ideal binary liquid mixture is given by the mole fraction weighted sum of the properties of the pure liquids. However Gibbsian second law properties (e.g. entropies and Gibbs energies) require combinatorial terms arising from the irreversible entropy of mixing.
These simple rules do not apply in the case of molar non-Gibbsian properties (e.g. isentropic compressions and isochoric heat capacities) of ideal mixtures.