1.7.22: Compressions- Isothermal- Liquid Mixtures Binary- Compressibilities
The isothermal compressibility of a given binary liquid mixture having ideal thermodynamic properties is related to the isothermal compressions of the liquid components using equation (a) [1].
\[\mathrm{K}_{\mathrm{T}}(\operatorname{mix} ; \mathrm{id})=\frac{\mathrm{K}_{\mathrm{T} 1}^{*}(\ell)+\mathrm{x}_{2} \,\left[\mathrm{K}_{\mathrm{T} 2}^{*}(\ell)-\mathrm{K}_{\mathrm{T} 1}^{*}(\ell)\right]}{\mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \,\left[\mathrm{V}_{2}^{*}(\ell)-\mathrm{V}_{1}^{*}(\ell)\right]} \nonumber \]
The excess compression for a given binary liquid mixture is defined by equation (b).
\[\mathrm{K}_{\mathrm{Tm}}^{\mathrm{E}}=\mathrm{K}_{\mathrm{Tm}}(\mathrm{mix})-\mathrm{K}_{\mathrm{Tn}}(\mathrm{mix} ; \mathrm{id}) \nonumber \]
Or,
\[\mathrm{K}_{\mathrm{Tm}}^{\mathrm{E}}=\mathrm{K}_{\mathrm{Tm}}(\mathrm{mix})-\left[\mathrm{x}_{1} \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{K}_{\mathrm{T} 2}^{*}(\ell)\right] \nonumber \]
The isothermal compressibilities of ideal and real binary liquid mixtures are defined by equations (d) and (e) respectively.
\[\kappa_{\mathrm{T}}(\operatorname{mix} ; \mathrm{id})=-\frac{1}{\mathrm{~V}(\operatorname{mix} ; \mathrm{id})} \,\left(\frac{\partial \mathrm{V}(\mathrm{mix} ; \mathrm{id})}{\partial \mathrm{p}}\right)_{\mathrm{T}} \nonumber \]
\[\kappa_{\mathrm{T}}(\operatorname{mix})=-\frac{1}{\mathrm{~V}(\operatorname{mix})} \,\left(\frac{\partial \mathrm{V}(\operatorname{mix})}{\partial \mathrm{p}}\right)_{\mathrm{T}} \nonumber \]
For a given binary liquid mixture we can define an excess compressibility using equation (f).
\[\kappa_{\mathrm{T}}^{\mathrm{E}}=\kappa_{\mathrm{T}}(\operatorname{mix})-\kappa_{\mathrm{T}}(\operatorname{mix} ; \mathrm{id}) \nonumber \]
Then
\[\begin{aligned}
&\kappa_{\mathrm{T}}^{\mathrm{E}}(\mathrm{mix})=-\frac{1}{\mathrm{~V}(\mathrm{mix})} \,\left(\frac{\partial \mathrm{V}(\mathrm{mix})}{\partial \mathrm{p}}\right)_{\mathrm{T}} \\
&+\frac{1}{\left[\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell)\right]} \,\left(\frac{\partial\left[\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell)\right]}{\partial \mathrm{p}}\right)_{\mathrm{T}}
\end{aligned} \nonumber \]
A similar equation was used by Moelwyn-Hughes and Thorpe [3]. They introduced the concept of a compressibility of the excess volume.
\[\Delta \kappa_{\mathrm{T}}(\operatorname{mix})=-\frac{1}{\Delta \mathrm{V}(\operatorname{mix})} \,\left(\frac{\partial \Delta \mathrm{V}(\mathrm{mix})}{\partial \mathrm{p}}\right)_{\mathrm{T}} \nonumber \]
In publications by Prigogine and by Moelwyn-Hughes and Thorpe the analysis was taken a step further to facilitate analysis of experimental results. However approximations were made in both treatments. An exact formulation was given by Missen [4] in terms of volume fractions of both components in the corresponding having ideal thermodynamic properties, \(\phi_{1}(\text { mix;id })\) and \(\phi_{2}(\text { mix;id })\). Hence,
\[\kappa_{\mathrm{T}}^{\mathrm{E}}(\operatorname{mix})=-\frac{1}{\mathrm{~V}_{\mathrm{m}}(\operatorname{mix})} \,\left[\left(\frac{\partial \mathrm{V}_{\mathrm{m}}^{\mathrm{E}}}{\partial \mathrm{p}}\right)_{\mathrm{T}}+\mathrm{V}_{\mathrm{m}}^{\mathrm{E}} \, \kappa_{\mathrm{T}}(\operatorname{mix} ; \mathrm{id})\right] \nonumber \]
A partial compressibility was defined by Moelwyn-Hughes [5]. For liquid 1 in a binary liquid mixture at defined \(\mathrm{T}\) and \(\mathrm{p}\), the partial compressibility is defined by equation (j).
\[\kappa_{T_{1}}(\operatorname{mix})=-\frac{1}{V_{1}(\operatorname{mix})} \,\left(\frac{\partial V_{1}(\operatorname{mix})}{\partial p}\right)_{T} \nonumber \]
Similarly for component 2,
\[\kappa_{\mathrm{T} 2}(\operatorname{mix})=-\frac{1}{\mathrm{~V}_{2}(\operatorname{mix})} \,\left(\frac{\partial \mathrm{V}_{2}(\operatorname{mix})}{\partial \mathrm{p}}\right)_{\mathrm{T}} \nonumber \]
The excess compressibility of a given binary liquid mixture \(\mathrm{K}_{\mathrm{T}}^{\mathrm{E}}(\operatorname{mix})\) was defined in equation (f). Hence,
\[\begin{aligned}
\kappa_{\mathrm{T}}^{\mathrm{E}}(\operatorname{mix})=& \phi_{1}(\operatorname{mix}) \, \kappa_{\mathrm{T} 1}^{\mathrm{E}}(\operatorname{mix})+\phi_{2}(\operatorname{mix}) \, \kappa_{\mathrm{T} 2}^{\mathrm{E}}(\operatorname{mix}) \\
&+\left[\phi_{1}(\operatorname{mix})-\phi_{1}(\operatorname{mix} ; \mathrm{id})\right] \, \kappa_{\mathrm{T} 1}^{*}(\ell)+\left[\phi_{2}(\operatorname{mix})-\phi_{2}(\operatorname{mix} ; \mathrm{id})\right] \, \kappa_{\mathrm{T} 2}^{*}(\ell)
\end{aligned} \nonumber \]
It may be noted that ‘true’ partial properties can also be defined for the isothermal compressibility [6]. Then the properties introduced in equations (j) and (k) would be termed specific partial isothermal compressions [6].
It is also possible to formulate a set of equations incorporating rational activity coefficients for the two components of the binary liquid mixture. We start with the equation for the partial molar volume of component 1.
\[V_{1}(\operatorname{mix})=V_{1}^{*}(\ell)+R \, T \,\left(\frac{\partial \ln \left(f_{1}\right)}{\partial p}\right)_{T} \nonumber \]
\[K_{T 1}(\operatorname{mix})=K_{T 1}^{*}(\ell)-R \, T \,\left(\frac{\partial^{2} \ln \left(f_{1}\right)}{\partial p^{2}}\right)_{T} \nonumber \]
Similarly
\[\mathrm{K}_{\mathrm{T} 2}(\operatorname{mix})=\mathrm{K}_{\mathrm{T} 2}^{*}(\ell)-\mathrm{R} \, \mathrm{T} \,\left(\frac{\partial^{2} \ln \left(\mathrm{f}_{2}\right)}{\partial \mathrm{p}^{2}}\right)_{\mathrm{T}} \nonumber \]
Therefore
\[\mathrm{K}_{\mathrm{Tm}}(\operatorname{mix})=\mathrm{K}_{\mathrm{Tm}}(\operatorname{mix} ; \mathrm{id})-\mathrm{R} \, \mathrm{T} \,\left[\mathrm{x}_{1} \,\left(\frac{\partial^{2} \ln \left(\mathrm{f}_{1}\right)}{\partial \mathrm{p}^{2}}\right)_{\mathrm{T}}+\mathrm{x}_{2} \,\left(\frac{\partial^{2} \ln \left(\mathrm{f}_{2}\right)}{\partial \mathrm{p}^{2}}\right)_{\mathrm{T}}\right] \nonumber \]
The two liquid components are characterised by their molar excess properties.
\[\mathrm{K}_{\mathrm{T} 1}^{\mathrm{E}}(\operatorname{mix})=-\mathrm{R} \, \mathrm{T} \,\left(\frac{\partial^{2} \ln \left(\mathrm{f}_{1}\right)}{\partial \mathrm{p}^{2}}\right)_{\mathrm{T}} \nonumber \]
and
\[\mathrm{K}_{\mathrm{T} 2}^{\mathrm{E}}(\operatorname{mix})=-\mathrm{R} \, \mathrm{T} \,\left(\frac{\partial^{2} \ln \left(\mathrm{f}_{2}\right)}{\partial \mathrm{p}^{2}}\right)_{\mathrm{T}} \nonumber \]
Therefore
\[\mathrm{K}_{\mathrm{Tm}}^{\mathrm{E}}(\mathrm{mix})=-\mathrm{R} \, \mathrm{T} \,\left[\mathrm{x}_{1} \,\left(\frac{\partial^{2} \ln \left(\mathrm{f}_{1}\right)}{\partial \mathrm{p}^{2}}\right)_{\mathrm{T}}+\mathrm{x}_{2} \,\left(\frac{\partial^{2} \ln \left(\mathrm{f}_{2}\right)}{\partial \mathrm{p}^{2}}\right)_{\mathrm{T}}\right] \nonumber \]
Also
\[\mathrm{K}_{\mathrm{T} 1}^{\mathrm{E}}=-\left(\frac{\partial \mathrm{V}_{1}^{\mathrm{E}}}{\partial \mathrm{p}}\right)_{\mathrm{T}} \text { and } \mathrm{K}_{\mathrm{T} 2}^{\mathrm{E}}=-\left(\frac{\partial \mathrm{V}_{2}^{\mathrm{E}}}{\partial \mathrm{p}}\right)_{\mathrm{T}} \nonumber \]
In other words
\[\mathrm{K}_{\mathrm{Tm}}^{\mathrm{E}}=-\left(\frac{\partial \mathrm{V}_{\mathrm{m}}^{\mathrm{E}}}{\partial \mathrm{p}}\right)_{\mathrm{T}} \nonumber \]
Isothermal compressions of liquid mixtures can be directly measured [7]. Hamann and Smith [8] report measurements using binary liquid mixtures at \(303 \mathrm{~K}\) and two pressures. Hamann and Smith define excess isothermal molar compressions \(\mathrm{K}_{\mathrm{T}}^{\mathrm{E}}(\phi)\) in terms of volume fraction weighted isothermal compressions of the pure liquids. The volume fractions are defined as follows.
\[\phi_{1}=\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell) /\left[\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell)\right] \nonumber \]
\[\phi_{2}=x_{2} \, V_{2}^{*}(\ell) /\left[x_{1} \, V_{1}^{*}(\ell)+x_{2} \, V_{2}^{*}(\ell)\right] \nonumber \]
Then
\[\mathrm{K}_{\mathrm{T}}^{\mathrm{E}}(\phi)=\mathrm{K}_{\mathrm{Tm}}(\mathrm{mix})-\left[\phi_{1} \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)+\phi_{2} \, \mathrm{K}_{\mathrm{T} 2}^{*}(\ell)\right] \nonumber \]
For most binary aqueous mixtures \(\mathrm{K}_{\mathrm{T}}^{\mathrm{E}}(\phi)\) is negative, plots of \(\mathrm{K}_{\mathrm{T}}^{\mathrm{E}}(\phi)\) against \(\phi_{2}\) being smooth curves. The minima in aqueous mixtures containing \(\mathrm{THF}\) and propanone the minima are near 0.4 and 0.6 respectively [0].
Footnotes
[1] For a binary liquid mixture having ideal thermodynamic properties,
\[\mathrm{V}_{\mathrm{m}}(\operatorname{mix} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell) \nonumber \]
Then
\[\mathrm{K}_{\mathrm{Tm}}(\operatorname{mix} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{K}_{\mathrm{T} 2}^{*}(\ell) \nonumber \]
But
\[\kappa_{\mathrm{T}}(\operatorname{mix} ; \mathrm{id})=\frac{\mathrm{K}_{\mathrm{Tm}}(\operatorname{mix} ; \mathrm{id})}{\mathrm{V}_{\mathrm{m}}(\operatorname{mix} ; \mathrm{id})} \nonumber \]
Then,
\[\kappa_{\mathrm{T}}(\operatorname{mix} ; \mathrm{id})=\frac{\mathrm{K}_{\mathrm{T} 1}^{*}(\ell)+\mathrm{x}_{2} \,\left[\mathrm{K}_{\mathrm{T} 2}^{*}(\ell)-\mathrm{K}_{\mathrm{T} 1}^{*}(\ell)\right]}{\mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \,\left[\mathrm{V}_{2}^{*}(\ell)-\mathrm{V}_{1}^{*}(\ell)\right]} \nonumber \]
[2] I. Prigogine, The Molecular Theory of Solutions, North Holland, Amsterdam, 1957, p.18.
[3] E. A. Moelwyn-Hughes and P. L. Thorpe, Proc. R. Soc. London, Ser. A,1964, 278A , 574.
[4] R. W. Missen, Ind. Eng. Chem. Fundam., 1969, 8 ,81.
[5] E. A. Moelwyn-Hughes, Physical Chemistry, Pergamon, London, 2nd. Edn., 1965, .817
[6] J. C. R. Reis, J. Chem. Soc. Faraday Trans.,1998, 94 ,2385.
[7] J. E. Stutchbury, Aust. J. Chem.,1971, 24 ,2431.
[8] S. D. Hamann and F. Smith, Aust. J. Chem.,1971, 24 ,2431.
[9] For a detailed report on the properties of liquid mixtures see G. M. Schneider, Pure Appl. Chem.,1983, 55 ,479 ; and references therein.