1.7.2: Compressibilities (Isothermal) and Chemical Potentials- Liquids

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Page ID: 373610

Michael J Blandamer & Joao Carlos R Reis
University of Leicester & Faculdade de Ciencias

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The (equilibrium) isothermal compressibility of a closed system containing a condensed phase is given by equation (a).

\[\kappa_{\mathrm{T}}=-\frac{1}{\mathrm{~V}} \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{T}}\]

\[\text { Or, } \quad \kappa_{\mathrm{T}}=-\left(\frac{\partial \ln (\mathrm{V})}{\partial \mathrm{p}}\right)_{\mathrm{T}}\]

Here we assume that over a range of pressures of interest here , \(\kappa_{\mathrm{T}}\) is independent of pressure.

\[\text { Hence at fixed temperature, } \int_{p=0}^{p} d \ln (V)=-K_{T} \, \int_{p=0}^{p} d p\]

We define a property \(V(p=0)\), the volume of the system under consideration extrapolated to zero pressure at fixed temperature.

\[\text { Therefore } \ln [\mathrm{V}(\mathrm{p}) / \mathrm{V}(\mathrm{p}=0)]=-\kappa_{\mathrm{T}} \, \mathrm{p}\]

\[\text { Or, } \mathrm{V}(\mathrm{T}, \mathrm{p})=\mathrm{V}(\mathrm{T}, \mathrm{p}=0) \, \exp \left(-\mathrm{K}_{\mathrm{T}} \, \mathrm{p}\right)\]

For systems at ordinary pressures, \(\kappa_{\mathrm{T}} \, \mathrm{P}<<1\).

\[\text { Hence [1] } \mathrm{V}(\mathrm{T}, \mathrm{p})=\mathrm{V}(\mathrm{T}, \mathrm{p}=0) \,\left[1-\kappa_{\mathrm{T}} \, \mathrm{p}\right]\]

For example, in the case of a pure liquid , chemical substance 1 [e.g. water]

\[\mathrm{V}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})=\mathrm{V}_{1}^{*}(\ell ; \mathrm{T}, \mathrm{p}=0) \,\left[1-\kappa_{\mathrm{T} 1}^{*}(\ell) \, \mathrm{p}\right]\]

\[\text { But for water }(\ell),\left\lfloor\partial \mu_{1}^{*}(\ell) / \partial \mathrm{p}\right\rfloor=\mathrm{V}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})\]

\[\text { Hence } \quad \left[\frac{\partial \mu_{1}^{*}(\ell)}{\partial \mathrm{p}}\right]=\mathrm{V}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p}=0) \,\left[1-\kappa_{\mathrm{T1}}^{*}(\ell) \, \mathrm{p}\right]\]

Or, following integration between limits ‘\(\mathrm{p}=0\)’ and \(\mathrm{p}\),

\[\begin{aligned}
&\mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})=\mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p}=0) \\
&\quad+\mathrm{p} \, \mathrm{V}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p}=0) \,\left[1-(1 / 2) \, \kappa_{\mathrm{T} 1}^{*}(\ell) \, \mathrm{p}\right]
\end{aligned}\]

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.