1.14.52: Partial Molar Properties: Definitions
A given liquid mixture is prepared using \(\mathrm{n}_{1}\) moles of liquid 1 and \(\mathrm{n}_{2}\) moles of liquid 2. If the thermodynamic properties of the liquid mixture are ideal the volume of the mixture is given by the sum of products of amounts and molar volumes (at the same \(\mathrm{T}\) and \(\mathrm{p}\)); equation (a).
\[\mathrm{V}(\operatorname{mix} ; \mathrm{id})=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{n}_{2} \, \mathrm{V}_{2}^{*}(\ell) \nonumber \]
If the thermodynamic properties of the mixture are not ideal, the volume of the (real) mixture is given by equation (b).
\[\mathrm{V}(\operatorname{mix})=\mathrm{n}_{1} \, \mathrm{V}_{1}(\operatorname{mix})+\mathrm{n}_{2} \, \mathrm{V}_{2}(\operatorname{mix}) \nonumber \]
\(\mathrm{V}_{1}(\operatorname{mix})\) and \(\mathrm{V}_{1}(\operatorname{mix})\) are the partial molar volumes of chemical substances 1 and 2 defined by equations (c) and (d).
\[V_{1}(m i x)=\left(\frac{\partial V}{\partial n_{1}}\right)_{T, p, n(2)} \nonumber \]
\[V_{2}(\operatorname{mix})=\left(\frac{\partial V}{\partial n_{2}}\right)_{T, p, n(1)} \nonumber \]
The similarities between equations (a) and (b) are obvious and indicate an important method for describing the extensive properties of a given system. This was the aim of G. N. Lewis [1] who sought equations of the form shown in equation (b). In general terms, we identify an extensive property \(\mathrm{X}\) of a given system such that the variable can be written in the general form shown in equation (e).
\[\mathrm{X}=\mathrm{n}_{1} \, \mathrm{X}_{1}+\mathrm{n}_{2} \, \mathrm{X}_{2} \nonumber \]
where
\[\mathrm{X}_{1}=\left(\frac{\partial \mathrm{X}}{\partial \mathrm{n}_{1}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(2)} \nonumber \]
\[X_{2}(\operatorname{mix})=\left(\frac{\partial X}{\partial n_{2}}\right)_{T_{, p, n(1)}} \nonumber \]
Other than the composition variables, the conditions on the partial differentials in equations (f) and (g) are intensive properties;
- mechanical variable, pressure, and
- thermal variable, temperature.
Partial molar properties can also be defined for different pairs of intensive thermal and non-thermal variables, other than \(\mathrm{T}\) and \(\mathrm{p}\) [2]. The concept of a partial property was extended to intensive properties such as isothermal and isentropic compressibilities [3].
A further distinction between Lewisian and non-Lewisian partial molar properties has been proposed [2,4].
Footnotes
[1] G. N. Lewis, Proc. Am. Acad. Arts Sci.,1907, 43 ,259.
[2] J. C. R. Reis, J. Chem. Soc Faraday Trans.,2,1982, 78 ,1575.
[3] J. C. R. Reis, J. Chem. Soc Faraday Trans.,1998, 94 ,2385.
[4] J. C. R. Reis, M. J. Blandamer, M. I. Davis and G. Douheret, Phys. Chem.Chem.Phys.,2001, 3 ,1465.