1.14.35: Internal Pressure: Liquid Mixtures: Excess Property
The thermodynamic equation of state takes the form shown in equation (a).
\[\left(\frac{\partial \mathrm{U}}{\partial \mathrm{V}}\right)_{\mathrm{T}}=\mathrm{T} \,\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{V}}-\mathrm{p} \nonumber \]
The partial differential \((\partial \mathrm{U} / \partial \mathrm{V})_{\mathrm{T}}\) is the internal pressure, \(\pi_{\mathrm{int}}\) (with units, \(\mathrm{N m}^{-2}\)). A calculus operation relates three interesting partial derivatives in the context of \(\mathrm{p}-\mathrm{V}-\mathrm{T}\) properties; equation (b).
\[\left(\frac{\partial p}{\partial T}\right)_{V}=-\left(\frac{\partial V}{\partial T}\right)_{p} \,\left(\frac{\partial p}{\partial V}\right)_{T} \nonumber \]
For a given liquid at defined \(\mathrm{T}\) and \(\mathrm{p}\), the isobaric (equilibrium) thermal expansion, \(\mathrm{E}_{\mathrm{p}}\) equals \((\partial \mathrm{V} / \partial \mathrm{T})_{\mathrm{p}}\). The isothermal (equilibrium) compression \(\mathrm{K}_{\mathrm{T}}\) is defined by \(-(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{T}}\). According to equations (a) and (b), \(\pi_{\mathrm{int}}\) is given by equation (c).
\[\pi_{\mathrm{int}}=\left(\mathrm{T} \, \mathrm{E}_{\mathrm{p}} / \mathrm{K}_{\mathrm{T}}\right)-\mathrm{p} \nonumber \]
For the purpose of the analysis described here, equation (c) describes the equilibrium molar properties of a given binary liquid mixture at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\). The internal pressure is a non-Gibbsian property of a liquid. Nevertheless it is interesting to compare internal pressures of real and the corresponding ideal binary liquid mixture [1]. In other words we require an equation for the internal pressure of binary liquid mixture \(\pi_{\mathrm{int}}^{\mathrm{id}}\) having thermodynamic properties which are ideal. Marczak[1] uses equation (c) in which the corresponding molar properties of the mixture, mole fraction composition \(\mathrm{x}_{2}\), \(\mathrm{E}_{\mathrm{pm}}^{\mathrm{id}}\left(\mathrm{x}_{2}\right)\) and \(\mathrm{K}_{\mathrm{Tm}}^{\mathrm{id}}\left(\mathrm{x}_{2}\right)\) are given by the mole fraction weighted properties of the pure liquids.
\[\mathrm{E}_{\mathrm{pm}}^{\mathrm{id}}\left(\mathrm{x}_{2}\right)=\sum_{\mathrm{i}=1}^{\mathrm{i}=2} \mathrm{x}_{\mathrm{i}} \, \mathrm{E}_{\mathrm{pi}}^{*}(\ell) \nonumber \]
\[\mathrm{K}_{\mathrm{Tm}}^{\mathrm{id}}\left(\mathrm{x}_{2}\right)=\sum_{\mathrm{i}=1}^{\mathrm{i}=2} \mathrm{x}_{\mathrm{i}} \, \mathrm{K}_{\mathrm{Ti}}^{*}(\ell) \nonumber \]
Equations (d) and (e) can be generalised to multi-component liquid mixtures. From equation (c) for a binary liquid mixture having thermodynamic properties which are ideal, the internal pressure \(\pi_{\mathrm{int}}^{\mathrm{id}}\left(\mathrm{x}_{2}\right)\) is given by equation (f).
\[\pi_{\mathrm{int}}^{\mathrm{id}}\left(\mathrm{x}_{2}\right)=\mathrm{T} \, \frac{\mathrm{E}_{\mathrm{pm}}^{\mathrm{id}}\left(\mathrm{x}_{2}\right)}{\mathrm{K}_{\mathrm{Tm}}^{\mathrm{id}}\left(\mathrm{x}_{2}\right)}-\mathrm{p} \nonumber \]
Or using equations (d) and (e),
\[\pi_{\mathrm{int}}^{\mathrm{id}}\left(\mathrm{x}_{2}\right)=\frac{\sum_{\mathrm{i}=1}^{\mathrm{i}=2} \mathrm{~T} \, \mathrm{x}_{\mathrm{i}} \, \mathrm{E}_{\mathrm{pi}}^{*}(\ell)}{\sum_{\mathrm{i}=1}^{\mathrm{i}=2} \mathrm{x}_{\mathrm{i}} \, \mathrm{K}_{\mathrm{Ti}}^{*}(\ell)}-\mathrm{p} \nonumber \]
Equation (g) is re-written to establish \(\pi_{\mathrm{int,i}}^{*}(\ell)\) as a term on the r.h.s. of the latter equation for \(\pi_{\mathrm{int}}^{\mathrm{id}}\left(\mathrm{x}_{2}\right)\).
\[\pi_{\mathrm{int}}^{\mathrm{id}}\left(\mathrm{x}_{2}\right)=-\mathrm{p}+\frac{\sum_{\mathrm{i}=1}^{\mathrm{i}=2} \mathrm{~T} \, \mathrm{x}_{\mathrm{i}} \, \mathrm{E}_{\mathrm{pi}}^{*}(\ell) \, \mathrm{K}_{\mathrm{Ti}}^{*}(\ell) / \mathrm{K}_{\mathrm{Ti}}^{*}(\ell)}{\sum_{\mathrm{i}=1}^{\mathrm{i}=2} \mathrm{x}_{\mathrm{i}} \, \mathrm{K}_{\mathrm{Ti}}^{*}(\ell)} \nonumber \]
But according to equation (c) , for the pure liquid–\(\mathrm{i}\),
\[\pi_{\mathrm{im}, \mathrm{i}}^{*}(\ell)+\mathrm{p}=\mathrm{T} \, \mathrm{E}_{\mathrm{pi}}^{*}(\ell) / \mathrm{K}_{\mathrm{Ti}}^{*}(\ell) \nonumber \]
Hence from equation (h),
\[\pi_{\mathrm{int}}^{\mathrm{id}}\left(\mathrm{x}_{2}\right)=-\mathrm{p}+\frac{\sum_{\mathrm{i}=1}^{\mathrm{i}=2} \mathrm{x}_{\mathrm{i}} \,\left[\pi_{\mathrm{int,i}}^{*}(\ell)+\mathrm{p}\right] \, \mathrm{K}_{\mathrm{T}_{1}}^{*}(\ell)}{\sum_{\mathrm{i}=1}^{\mathrm{i}=2} \mathrm{x}_{\mathrm{i}} \, \mathrm{K}_{\mathrm{Ti}^{*}}(\ell)} \nonumber \]
In other words,
\[\begin{aligned}
\pi_{\mathrm{idt}}^{\mathrm{id}}\left(\mathrm{x}_{2}\right)=&-\mathrm{p}+\frac{\mathrm{x}_{1} \, \pi_{\mathrm{int1} 1}^{*}(\ell) \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)}{\sum_{\mathrm{i}=1}^{\mathrm{i}=2} \mathrm{x}_{\mathrm{i}} \, \mathrm{K}_{\mathrm{Ti}}^{*}(\ell)}+\frac{\mathrm{x}_{1} \, \mathrm{p} \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)}{\sum_{\mathrm{i}=2}^{*} \mathrm{x}_{\mathrm{i}} \, \mathrm{K}_{\mathrm{Ti}}^{*}(\ell)} \\
&+\frac{\mathrm{x}_{2} \, \pi_{\mathrm{int}, 2}^{*}(\ell) \, \mathrm{K}_{\mathrm{T} 2}^{*}(\ell)}{\sum_{\mathrm{i}=1}^{\mathrm{i}=2} \mathrm{x}_{\mathrm{i}} \, \mathrm{K}_{\mathrm{Ti}}^{*}(\ell)}+\frac{\mathrm{x}_{2} \, \mathrm{p} \, \mathrm{K}_{\mathrm{T} 2}^{*}(\ell)}{\sum_{\mathrm{i}=1}^{\mathrm{i}=2} \mathrm{x}_{\mathrm{i}} \, \mathrm{K}_{\mathrm{Ti}}^{*}(\ell)}
\end{aligned} \nonumber \]
Hence,
\[\pi_{\mathrm{int}}^{\mathrm{id}}\left(\mathrm{x}_{2}\right)=\frac{\sum_{\mathrm{i}=1}^{\mathrm{i}=2} \mathrm{x}_{\mathrm{i}} \, \pi_{\mathrm{int}, \mathrm{i}}^{*}(\ell) \, \mathrm{K}_{\mathrm{Ti}}^{*}(\ell)}{\sum_{\mathrm{i}=1}^{\mathrm{i}=2} \mathrm{x}_{\mathrm{i}} \, \mathrm{K}_{\mathrm{Ti}}^{*}(\ell)} \nonumber \]
By definition [1], for liquid component \(\mathrm{k}\),
\[\Psi_{\mathrm{k}}=\frac{\mathrm{x}_{\mathrm{k}} \, \mathrm{K}_{\mathrm{Tk}}^{*}(\ell)}{\sum \mathrm{x}_{\mathrm{i}} \, \mathrm{K}_{\mathrm{Ti}}^{*}(\ell)} \nonumber \]
In other words,
\[\pi_{\mathrm{int}}^{\mathrm{id}}\left(\mathrm{x}_{2}\right)=\sum_{\mathrm{i}=1}^{\mathrm{i}=2} \psi_{\mathrm{i}} \, \pi_{\mathrm{int,i}}^{*}(\ell) \nonumber \]
The corresponding excess internal pressure at mole fraction \(\mathrm{x}_{2}\),
\[\pi_{\mathrm{int}}^{\mathrm{E}}\left(\mathrm{x}_{2}\right) \nonumber \]
is defined by equation (o).
\[\pi_{\mathrm{int}}^{\mathrm{E}}\left(\mathrm{x}_{2}\right)=\pi_{\mathrm{int}}\left(\mathrm{x}_{2}\right)-\sum_{\mathrm{i}=1}^{\mathrm{i}=2} \psi_{\mathrm{i}} \, \pi_{\mathrm{int}, \mathrm{i}}^{*}(\ell) \nonumber \]
Marczak [1] reports \(\pi_{\text {int }}^{E}\left(X_{2}\right)\) as a function of mole fraction \(\mathrm{x}_{2}\) for two binary liquid mixtures at \(298.15 \mathrm{~K}\);
- methanol + propan-1-ol, and
- tribromomethane + n-octane.
Footnotes
[1] W. Marczak, Phys. Chem. Chem. Phys.2002, 4 ,1889.