1.7.2: Compressibilities (Isothermal) and Chemical Potentials- Liquids
- Page ID
- 373610
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.