1.18.4: Liquid Mixtures: Binary: Less Common Properties
For an ideal liquid mixture containing i-liquid components four important molar properties are related to the corresponding properties of the pure liquid components using the following equations.
\[\mathrm{C}_{\mathrm{Vm}_{\mathrm{m}}}(\operatorname{mix} ; \mathrm{id})=\sum_{\mathrm{i}} \mathrm{x}_{\mathrm{i}} \,\left\{1-\left[\frac{\mathrm{E}_{\mathrm{pi}}^{*}(\ell)}{\mathrm{C}_{\mathrm{V}_{\mathrm{i}}}^{*}(\ell)}\right] \,\left[\beta_{\mathrm{v}}(\operatorname{mix} ; \mathrm{id})-\beta_{\mathrm{V}_{\mathrm{i}}}^{*}(\ell)\right] \, \mathrm{C}_{\mathrm{V}_{\mathrm{i}}}^{*}(\ell)\right\} \nonumber \]
\[\begin{aligned}
&\mathrm{E}_{\mathrm{Sm}}(\operatorname{mix} ; \mathrm{id}) \\
&=\sum_{\mathrm{i}} \mathrm{x}_{\mathrm{i}} \,\left\{1-\left[\frac{\mathrm{C}_{\mathrm{pi}}^{*}(\ell)}{\mathrm{E}_{\mathrm{Si}}^{*}(\ell)}\right] \,\left[\left[\beta_{\mathrm{v}}(\mathrm{mix} ; \mathrm{id})\right]^{-1}-\left[\beta_{\mathrm{vi}_{\mathrm{i}}}^{*}(\ell)\right]^{-1}\right] \, \mathrm{E}_{\mathrm{Si}}^{*}(\ell)\right\}
\end{aligned} \nonumber \]
\[\mathrm{E}_{\mathrm{Sm}}(\operatorname{mix} ; \mathrm{id})=\sum_{\mathrm{i}} \mathrm{x}_{\mathrm{i}} \,\left\{1-\left[\frac{\mathrm{K}_{\mathrm{pi}}^{*}(\ell)}{\mathrm{E}_{\mathrm{Si}}^{*}(\ell)}\right] \,\left[\beta_{\mathrm{s}}(\operatorname{mix} ; \mathrm{id})-\beta_{\mathrm{Si}}^{*}(\ell)\right] \, \mathrm{E}_{\mathrm{Si}}^{*}(\ell)\right\} \nonumber \]
\[\begin{aligned}
&\mathrm{K}_{\mathrm{Sm}}(\text { mix; id }) \\
&\quad=\sum_{\mathrm{i}} \mathrm{x}_{\mathrm{i}} \,\left\{1-\left[\frac{\mathrm{E}_{\mathrm{pl}}^{*}(\ell)}{\mathrm{K}_{\mathrm{Si}}^{*}(\ell)}\right] \,\left[\left[\beta_{\mathrm{s}}(\text { mix; id) }]^{-1}-\left[\beta_{\mathrm{si}}^{*}(\ell)\right]^{-1}\right] \, \mathrm{K}_{\mathrm{Si}}^{*}(\ell)\right\}\right.
\end{aligned} \nonumber \]
With reference to these four equations, interesting features emerge. If \(\mathrm{V}_{1}^{*}(\ell)\) and \(\mathrm{V}_{2}^{*}(\ell)\) for the two components of a binary liquid mixture having ideal thermodynamic properties are linearly related at different temperatures and pressures then at fixed liquid mixture composition,
\[\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{v}_{1}^{*}(1)}=\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{v}_{2}^{*}(1)}=\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{v}_{(\operatorname{mix} ; \mathrm{d})}} \nonumber \]
Or,
\[\beta_{\mathrm{v}_{1}}^{*}(\ell)=\beta_{\mathrm{v}_{2}}^{*}(\ell)=\beta_{\mathrm{v}}(\mathrm{mix} ; \mathrm{id}) \nonumber \]
Under these conditions the two properties described in equations (a) and (b) are given by the mole fraction weighted sums of the properties of the pure liquids. The internal pressure pint is given by \(\left[\mathrm{T} \, \beta_{\mathrm{V}}-1\right]\). Hence the same condition holds with respect to the two properties defined by equations (a) and (b) if the internal pressures are equal. In practice liquids have different internal pressures. However this difference is often small for chemically similar liquids.
An interesting feature emerges if the molar entropies of the two liquids are linearly related over a range of temperatures and pressures. Thus,
\[\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{s}_{1}^{*}(\theta)}=\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{s}_{2}^{*}(\theta)}=\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{S}(\text { mid; } ; \mathrm{d})} \nonumber \]
Or,
\[\beta_{\mathrm{s} 1}^{*}(\ell)=\beta_{\mathrm{S} 2}^{*}(\ell)=\beta_{\mathrm{s}}(\operatorname{mix} ; \mathrm{id}) \nonumber \]
Therefore a liquid mixture where the components have identical isentropic thermal pressure coefficients, \(\mathrm{K}_{\mathrm{Sm}}(\mathrm{mix} ; \mathrm{id})\) and \(\mathrm{E}_{\mathrm{Sm}}(\mathrm{mix} ; \mathrm{id})\) are given by the mole fraction weighted sums of the properties of the pure components [1].
In the case of an ideal binary liquid mixture the following three equations relate the isochoric heat capacities, isentropic compressions and isentropic expansions to the properties of the component pure liquids.
\[\begin{aligned}
&\mathrm{C}_{\mathrm{Vm}}(\operatorname{mix} ; \mathrm{id}) \\
&=\mathrm{x}_{1} \, \mathrm{C}_{\mathrm{v} 1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{C}_{\mathrm{V} 2}^{*}(\ell) \\
&+\mathrm{T} \,\left\{\left[\frac{\mathrm{x}_{1} \,\left[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)\right]^{2}}{\mathrm{~K}_{\mathrm{T} 1}^{*}(\ell)}\right]+\left[\frac{\mathrm{x}_{2} \,\left[\mathrm{E}_{\mathrm{p} 2}^{*}(\ell)\right]^{2}}{\mathrm{~K}_{\mathrm{T} 2}^{*}(\ell)}\right]-\left[\frac{\left[\mathrm{x}_{1} \, \mathrm{E}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{E}_{\mathrm{p} 2}^{*}(\ell)\right]^{2}}{\mathrm{x}_{1} \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{K}_{\mathrm{T} 2}^{*}(\ell)}\right]\right\}
\end{aligned} \nonumber \]
\[\begin{aligned}
&\mathrm{K}_{\mathrm{Sm}}(\operatorname{mix} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{K}_{\mathrm{S} 1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{K}_{\mathrm{S} 2}^{*}(\ell) \\
&+\mathrm{T} \,\left\{\left[\frac{\mathrm{x}_{1} \,\left[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{p} 1}^{*}(1 \ell)}\right]+\left[\frac{\mathrm{x}_{2} \,\left[\mathrm{E}_{\mathrm{p} 2}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{p} 2}^{*}(\ell)}\right]-\left[\frac{\left[\mathrm{x}_{1} \, \mathrm{E}_{\mathrm{pl}}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{E}_{\mathrm{p} 2}^{*}(\ell)\right]^{2}}{\mathrm{x}_{1} \, \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{C}_{\mathrm{p} 2}^{*}(\ell)}\right]\right\}
\end{aligned} \nonumber \]
\[\begin{aligned}
&\mathrm{E}_{\mathrm{Sm}}(\operatorname{mix} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{E}_{\mathrm{Sl}}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{E}_{\mathrm{S} 2}^{*}(\ell)\\
&+\mathrm{T} \,\left\{\begin{array}{l}
{\left[\frac{\mathrm{x}_{1} \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell) \, \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}{\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)}\right]+\left[\frac{\mathrm{x}_{2} \, \mathrm{K}_{\mathrm{T} 2}^{*}(\ell) \, \mathrm{C}_{\mathrm{p} 2}^{*}(\ell)}{\mathrm{E}_{\mathrm{p} 2}^{*}(\ell)}\right]} \\
-\left[\frac{\left[\mathrm{x}_{1} \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{K}_{\mathrm{T} 2}^{*}(\ell)\right] \,\left[\mathrm{x}_{1} \, \mathrm{C}_{\mathrm{pl} 1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{C}_{\mathrm{p} 2}^{*}(\ell)\right]}{\mathrm{x}_{1} \, \mathrm{E}_{\mathrm{pl}}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{E}_{\mathrm{p} 2}^{*}(\ell)}\right]
\end{array}\right\}
\end{aligned} \nonumber \]
Inspection shows that in each case the condition for simple additivity requires that the sum inside the brackets {……} vanishes. In the case of \(\mathrm{C}_{\mathrm{Vm}}(\operatorname{mix} ; \mathrm{id})\) a sufficient condition ( and most probably also necessary) is that \(\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)=\mathrm{E}_{\mathrm{p} 2}^{*}(\ell)\left[=\mathrm{E}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})\right]\) and that \(\mathrm{K}_{\mathrm{T} 1}^{*}(\ell)=\mathrm{K}_{\mathrm{T} 2}^{*}(\ell)\left[=\mathrm{K}_{\mathrm{Tm}}(\mathrm{mix} ; \mathrm{id})\right]\) at a given \(\mathrm{T}\) and \(\mathrm{p}\). In the case of \(\mathrm{K}_{\mathrm{Sm}}(\mathrm{mix} ; \mathrm{id})\) the required conditions are that \(\mathrm{E}_{\mathrm{pl}}^{*}(\ell)=\mathrm{E}_{\mathrm{p} 2}^{*}(\ell)\left[=\mathrm{E}_{\mathrm{pm}}(\text { mix; } \mathrm{id})\right]\) and that \(\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)=\mathrm{C}_{\mathrm{p} 2}^{*}(\ell)\left[=\mathrm{C}_{\mathrm{pm}}(\operatorname{mix} ; \mathrm{id})\right]\) at a given \(\mathrm{T}\) and \(\mathrm{p}\). But since \(\mathrm{E}_{\mathrm{p}}=-(\partial \mathrm{S} / \partial \mathrm{p})_{\mathrm{T}}\) and \(\mathrm{C}_{\mathrm{p}}=\mathrm{T} \,(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{p}}\), the condition can be restated as follows. Although molar entropies of liquid 1 and 2 may differ, they should have identical isobaric dependences on temperature and isothermal dependence on pressure. In the case of \(\mathrm{E}_{\mathrm{Sm}}(\mathrm{mix} ; \mathrm{id})\) the three conditions are that \(\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)=\mathrm{E}_{\mathrm{p} 2}^{*}(\ell)\left[=\mathrm{E}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})\right]\), \(\mathrm{K}_{\mathrm{T} 1}^{*}(\ell)=\mathrm{K}_{\mathrm{T} 2}^{*}(\ell)\left[=\mathrm{K}_{\mathrm{Tm}}(\mathrm{mix} ; \mathrm{id})\right]\), and \(\mathrm{C}_{\mathrm{pl}}^{*}(\ell)=\mathrm{C}_{\mathrm{p} 2}^{*}(\ell)\left[=\mathrm{C}_{\mathrm{pm}}(\operatorname{mix} ; \mathrm{id})\right]\).
If we extend the foregoing analysis to the variables isentropic compressibilities \(\kappa_{\mathrm{S}}\) and isentropic expansibilities \(\alpha_{\mathrm{S}}\) we find that because \(\kappa_{\mathrm{S}}(\operatorname{mix} ; \mathrm{id})=\mathrm{K}_{\mathrm{Sm}}(\operatorname{mix} ; \mathrm{id}) / \mathrm{V}_{\mathrm{m}}(\operatorname{mix} ; \mathrm{id})\), the condition described above for \(\mathrm{K}_{\mathrm{Sm}}(\text { mix; } \mathrm{id})\) requires that \(\kappa_{\mathrm{S}}(\operatorname{mix} ; \mathrm{id})\) is given by the volume weighted sum of \(\kappa_{\mathrm{sl}}^{*}(\ell)\) and \(\kappa_{S 2}^{*}(\ell)\). Similarly we find that the three conditions described above in the context of \(\mathrm{E}_{\mathrm{Sm}}(\mathrm{mix} ; \mathrm{id})\) is necessary in order that \(\alpha_{S}(\operatorname{mix} ; \mathrm{id})\) is given by the volume weighted sum of \(\alpha_{\mathrm{s} 1}^{*}(\ell)\) and \(\alpha_{\mathrm{s} 2}^{*}(\ell)\).
The conditions described above are expressed in thermodynamic terms but we note that in no case can the properties of real pure liquids comply with these conditions. Nevertheless they provide useful pointers in the task of understanding the properties of real liquid mixtures. Even for a mixture prepared using \(\mathrm{H}_{2}\mathrm{O}(\ell)\) and \(\mathrm{D}_{2}\mathrm{O}(\ell)\), \(\mathrm{C}_{\mathrm{Vm}}(\operatorname{mix} ; \mathrm{id})\) would depart from mole fraction additivity. Only for mixtures of ideal gases would the condition hold for \(\mathrm{C}_{\mathrm{Vm}}(\operatorname{mix} ; \mathrm{id})\). Indeed for a monatomic gas the energy is entirely translational and \(\mathrm{C}_{\mathrm{pm}}=(5 / 2) \ldot \mathrm{R}\).
Then \(\left(\partial \mathrm{S}_{\mathrm{m}} / \partial \mathrm{T}\right)_{\mathrm{p}}=\mathrm{C}_{\mathrm{pm}} / \mathrm{T}=(5 / 2) \, \mathrm{R} / \mathrm{T}\) for both pure gases and the mixture, a consequence of the Sackur-Tetrode equation for the molar entropy of ideal gases.
Footnote
[1] G. Douheret, M. I. Davis, J. C. R. Reis and M. J. Blandamer, Chem. Phys. Chem. Phys.,2001, 2 ,148.