1.15.5: Heat Capacities: Isochoric: Liquid Mixtures: Ideal
For an ideal binary liquid mixture the molar isobaric heat capacity is given by the mole fraction weighted sum of the isobaric heat capacities of the pure liquid components.
\[\mathrm{C}_{\mathrm{pm}}(\operatorname{mix} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{C}_{\mathrm{p} 2}^{*}(\ell) \nonumber \]
Both \(\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)\) and \(\mathrm{C}_{\mathrm{p} 2}^{*}(\ell)\) can be measured so that \(\mathrm{C}_{\mathrm{pm}}(\operatorname{mix} ; \mathrm{id})\) can be calculated for a given mixture as a function of mole fraction composition. Further
\[\Delta_{\operatorname{mix}} \mathrm{C}_{\mathrm{p}}(\mathrm{id})=0 \nonumber \]
The isochoric heat capacity of the corresponding ideal mixture is related to the isobaric heat capacity using equation (c) [1].
\[\mathrm{C}_{\mathrm{V}_{\mathrm{m}}}(\text { mix } ; \mathrm{id})=\mathrm{C}_{\mathrm{pm}}(\text { mix } ; \mathrm{id})-\frac{\mathrm{T} \,\left[\mathrm{E}_{\mathrm{pm}}(\text { mix} ; \mathrm{id})\right]^{2}}{\mathrm{~K}_{\mathrm{Tm}}(\text { mix } ; \mathrm{id})} \nonumber \]
Equations (a) and (c) provide an equation for \(\mathrm{C}_{\mathrm{Vm}}(\operatorname{mix} ; \mathrm{id})\) in terms of the isochoric heat capacities of the pure liquid components.
\[\begin{aligned}
&\mathrm{C}_{\mathrm{V}_{\mathrm{m}}}(\operatorname{mix} ; \mathrm{id})=\\
&\mathrm{x}_{1} \,\left[\mathrm{C}_{\mathrm{V} 1}^{*}(\ell)+\frac{\mathrm{T} \,\left[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)\right]^{2}}{\mathrm{~K}_{\mathrm{T} 1}^{*}(\ell)}\right]+\mathrm{x}_{2} \,\left[\mathrm{C}_{\mathrm{V} 2}^{*}(\ell)+\frac{\mathrm{T} \,\left[\mathrm{E}_{\mathrm{p} 2}^{*}(\ell)\right]^{2}}{\mathrm{~K}_{\mathrm{T} 2}^{*}(\ell)}\right]\\
&-\frac{\mathrm{T} \,\left[\mathrm{E}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})\right]^{2}}{\mathrm{~K}_{\mathrm{Tm}}(\mathrm{mix} ; \mathrm{id})}
\end{aligned} \nonumber \]
In terms of forming an ideal binary liquid mixture from two pure components,
\[\begin{gathered}
\Delta_{\operatorname{mix}} \mathrm{C}_{\mathrm{V}_{\mathrm{m}}}(\mathrm{id})=\mathrm{x}_{1} \,\left[\frac{\mathrm{T} \,\left[\mathrm{E}_{\mathrm{pl}}^{*}(\ell)\right]^{2}}{\mathrm{~K}_{\mathrm{T} 1}^{*}(\ell)}\right]+\mathrm{x}_{2} \,\left[\frac{\mathrm{T} \,\left[\mathrm{E}_{\mathrm{p} 2}^{*}(\ell)\right]^{2}}{\mathrm{~K}_{\mathrm{T} 2}^{*}(\ell)}\right] \\
-\frac{\mathrm{T} \,\left[\mathrm{E}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})\right]^{2}}{\mathrm{~K}_{\mathrm{Tm}}(\mathrm{mix} ; \mathrm{id})}
\end{gathered} \nonumber \]
The equations become more complicated as we switch conditions from the intensive variables, \(\mathrm{T}\) and \(\mathrm{p}\), to extensive variables such as entropy and volume. The equations become even more complicated when we turn to a description of real mixtures.
Footnote
[1] Consider a closed system subjected to a change in temperature, the system remaining at equilibrium where the affinity for spontaneous change is zero. Then
\[\mathrm{C}_{\mathrm{p}}(\mathrm{A}=0)=\left(\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \mathrm{A}=0} \quad \text { and } \quad \mathrm{C}_{\mathrm{V}}(\mathrm{A}=0)=\left(\frac{\partial \mathrm{U}}{\partial \mathrm{T}}\right)_{\mathrm{V}, \mathrm{A}=0} \nonumber \]
In the following we drop the condition ‘\(\mathrm{A}=0\)’ and take it as implicit in the following analysis. [A similar set of equations can be written for the condition ‘at fixed ξ’.] Then \(\mathrm{C}_{\mathrm{p}}-\mathrm{C}_{\mathrm{V}}=\left(\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right)_{\mathrm{p}}-\left(\frac{\partial \mathrm{U}}{\partial \mathrm{T}}\right)_{\mathrm{V}}\) but by definition, \(\mathrm{H}=\mathrm{U}+\mathrm{p} \, \mathrm{V}\) Then
\[C_{p}-C_{V}=\left(\frac{\partial H}{\partial T}\right)_{p}-\left(\frac{\partial H}{\partial T}\right)_{v}+V \,\left(\frac{\partial p}{\partial T}\right)_{v} \nonumber \]
Using a calculus operation, \(\left(\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right)_{\mathrm{v}}=\left(\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right)_{\mathrm{p}}+\left(\frac{\partial \mathrm{H}}{\partial \mathrm{p}}\right)_{\mathrm{T}} \,\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{v}}\) Then,
\[\mathrm{C}_{\mathrm{p}}-\mathrm{C}_{\mathrm{V}}=\left[\mathrm{V}-\left(\frac{\partial \mathrm{H}}{\partial \mathrm{p}}\right)_{\mathrm{T}}\right] \,\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{V}} \nonumber \]
By definition \(\mathrm{H}=\mathrm{G}+\mathrm{T} \, \mathrm{S}\); then
\[\left(\frac{\partial \mathrm{H}}{\partial \mathrm{p}}\right)_{\mathrm{T}}=\left(\frac{\partial \mathrm{G}}{\partial \mathrm{p}}\right)_{\mathrm{T}}+\mathrm{T} \,\left(\frac{\partial \mathrm{S}}{\partial \mathrm{p}}\right)_{\mathrm{T}} \nonumber \]
A Maxwell equation requires that \(\left(\frac{\partial \mathrm{S}}{\partial \mathrm{p}}\right)_{\mathrm{T}}=-\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}}\) Then, \(\left(\frac{\partial \mathrm{H}}{\partial \mathrm{p}}\right)_{\mathrm{T}}=\mathrm{V}-\mathrm{T} \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}}\) Hence,
\[\mathrm{C}_{\mathrm{p}}-\mathrm{C}_{\mathrm{V}}=\mathrm{T} \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}} \,\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{V}} \nonumber \]
A calculus operation requires that \(\left(\frac{\partial p}{\partial T}\right)_{V} \,\left(\frac{\partial T}{\partial V}\right)_{p} \,\left(\frac{\partial V}{\partial p}\right)_{T}=-1\) Then
\[C_{p}-C_{V}=-T \,\left[\left(\frac{\partial V}{\partial T}\right)_{p}\right]^{2} \,\left[\left(\frac{\partial V}{\partial p}\right)_{T}\right]^{-1} \nonumber \]