1.7.2: Compressibilities (Isothermal) and Chemical Potentials- Liquids
The (equilibrium) isothermal compressibility of a closed system containing a condensed phase is given by equation (a).
\[\kappa_{\mathrm{T}}=-\frac{1}{\mathrm{~V}} \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{T}} \nonumber \]
\[\text { Or, } \quad \kappa_{\mathrm{T}}=-\left(\frac{\partial \ln (\mathrm{V})}{\partial \mathrm{p}}\right)_{\mathrm{T}} \nonumber \]
Here we assume that over a range of pressures of interest here , \(\kappa_{\mathrm{T}}\) is independent of pressure.
\[\text { Hence at fixed temperature, } \int_{p=0}^{p} d \ln (V)=-K_{T} \, \int_{p=0}^{p} d p \nonumber \]
We define a property \(V(p=0)\), the volume of the system under consideration extrapolated to zero pressure at fixed temperature.
\[\text { Therefore } \ln [\mathrm{V}(\mathrm{p}) / \mathrm{V}(\mathrm{p}=0)]=-\kappa_{\mathrm{T}} \, \mathrm{p} \nonumber \]
\[\text { Or, } \mathrm{V}(\mathrm{T}, \mathrm{p})=\mathrm{V}(\mathrm{T}, \mathrm{p}=0) \, \exp \left(-\mathrm{K}_{\mathrm{T}} \, \mathrm{p}\right) \nonumber \]
For systems at ordinary pressures, \(\kappa_{\mathrm{T}} \, \mathrm{P}<<1\).
\[\text { Hence [1] } \mathrm{V}(\mathrm{T}, \mathrm{p})=\mathrm{V}(\mathrm{T}, \mathrm{p}=0) \,\left[1-\kappa_{\mathrm{T}} \, \mathrm{p}\right] \nonumber \]
For example, in the case of a pure liquid , chemical substance 1 [e.g. water]
\[\mathrm{V}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})=\mathrm{V}_{1}^{*}(\ell ; \mathrm{T}, \mathrm{p}=0) \,\left[1-\kappa_{\mathrm{T} 1}^{*}(\ell) \, \mathrm{p}\right] \nonumber \]
\[\text { But for water }(\ell),\left\lfloor\partial \mu_{1}^{*}(\ell) / \partial \mathrm{p}\right\rfloor=\mathrm{V}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p}) \nonumber \]
\[\text { Hence } \quad \left[\frac{\partial \mu_{1}^{*}(\ell)}{\partial \mathrm{p}}\right]=\mathrm{V}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p}=0) \,\left[1-\kappa_{\mathrm{T1}}^{*}(\ell) \, \mathrm{p}\right] \nonumber \]
Or, following integration between limits ‘\(\mathrm{p}=0\)’ and \(\mathrm{p}\),
\[\begin{aligned}
&\mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})=\mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p}=0) \\
&\quad+\mathrm{p} \, \mathrm{V}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p}=0) \,\left[1-(1 / 2) \, \kappa_{\mathrm{T} 1}^{*}(\ell) \, \mathrm{p}\right]
\end{aligned} \nonumber \]
The latter equation relates the chemical potential of a liquid at pressure p to the isothermal compressibility of the liquid [2].
Footnote
[1] With \(\exp (x)=1+x+\left(x^{2} / 2 !\right)+\left(x^{3} / 3 !\right)+\ldots \ldots\) At small \(\mathrm{x}\), \(\exp (x) \approx 1+x\)
[2] I. Prigogine and R. Defay, Chemical Thermodynamics, transl D. H. Everett, Longmans Green, London, 1953.