1.7.21: Compressions- Isothermal- Binary Aqueous Mixtures
- Page ID
- 374453
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)A given binary aqueous mixture is prepared using \(\mathrm{n}_{1}\) moles of water (\(\ell\)) and \(\mathrm{n}_{2}\) moles of liquid 2. The volume of the mixture, \(\mathrm{V}(\mathrm{mix})\) is given by equation (a) (at fixed temperature and pressure).
\[\mathrm{V}(\operatorname{mix})=\mathrm{n}_{1} \, \mathrm{V}_{1}(\operatorname{mix})+\mathrm{n}_{2} \, \mathrm{V}_{2}(\mathrm{mix}) \nonumber \]
The mixture is perturbed by a change in pressure at fixed temperature along an equilibrium pathway where the affinity for spontaneous change remains at zero.
\[\left(\frac{\partial \mathrm{V}(\mathrm{mix})}{\partial \mathrm{p}}\right)_{\mathrm{T}}=\mathrm{n}_{1} \,\left(\frac{\partial \mathrm{V}_{1}(\mathrm{mix})}{\partial \mathrm{p}}\right)_{\mathrm{T}}+\mathrm{n}_{2} \,\left(\frac{\partial \mathrm{V}_{2}(\mathrm{mix})}{\partial \mathrm{p}}\right)_{\mathrm{T}} \nonumber \]
The isothermal compression of the mixture \([\partial \mathrm{V}(\mathrm{mix}) / \partial \mathrm{p}]_{\mathrm{T}}\) is an extensive property. The partial differentials \(\left[\partial V_{1}(\operatorname{mix}) / \partial p\right]_{T}\) and \(\left[\partial V_{2}(\operatorname{mix}) / \partial \mathrm{p}\right]_{\mathrm{T}}\) are intensive properties. There is merit in defining an intensive molar compression using equation (c).
\[\mathrm{K}_{\mathrm{Tm}}=\frac{\mathrm{K}_{\mathrm{T}}(\operatorname{mix})}{\left(\mathrm{n}_{1}+\mathrm{n}_{2}\right)}=-\left(\mathrm{n}_{1}+\mathrm{n}_{2}\right)^{-1} \,[\partial \mathrm{V}(\operatorname{mix}) / \partial \mathrm{p}]_{\mathrm{T}} \nonumber \]
By definition,
\[\mathrm{K}_{\mathrm{T} 1}(\operatorname{mix})=-\left[\partial \mathrm{V}_{1}(\operatorname{mix}) / \partial \mathrm{p}\right]_{\mathrm{T}} \nonumber \]
And
\[\mathrm{K}_{\mathrm{T} 2}(\operatorname{mix})=-\left[\partial \mathrm{V}_{2}(\operatorname{mix}) / \partial \mathrm{p}\right]_{\mathrm{T}} \nonumber \]
Hence
\[\mathrm{K}_{\mathrm{Tm}}(\operatorname{mix})=\mathrm{x}_{1} \, \mathrm{K}_{\mathrm{T} 1}(\operatorname{mix})+\mathrm{x}_{2} \, \mathrm{K}_{\mathrm{T} 2}(\operatorname{mix}) \nonumber \]
For an ideal binary mixture,
\[\mathrm{V}(\operatorname{mix} ; \mathrm{id})=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{n}_{2} \, \mathrm{V}_{2}^{*}(\ell) \nonumber \]
\(\mathrm{V}_{1}^{*}(\ell)\) and \(\mathrm{V}_{2}^{*}(\ell)\) are the molar volumes of the pure liquids at the same \(\mathrm{T}\) and \(\mathrm{p}\). Therefore, following the argument outlined above,
\[\mathrm{K}_{\mathrm{Tm}}(\mathrm{id})=\mathrm{x}_{1} \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{K}_{\mathrm{T} 2}^{*}(\ell) \nonumber \]
By definition,
\[\mathrm{K}_{\mathrm{T}}^{\mathrm{E}}(\operatorname{mix})=\mathrm{K}_{\mathrm{Tm}}(\operatorname{mix})-\mathrm{K}_{\mathrm{Tm}}(\operatorname{mix} ; \mathrm{id}) \nonumber \]
Hence the excess molar compression is given by equation (j).
\[\mathrm{K}_{\mathrm{Tm}}^{\mathrm{E}}(\operatorname{mix})=\mathrm{x}_{1} \,\left[\mathrm{K}_{\mathrm{T} 1}(\operatorname{mix})-\mathrm{K}_{\mathrm{T} 1}^{*}(\ell)\right]+\mathrm{x}_{2} \,\left[\mathrm{K}_{\mathrm{T} 2}(\operatorname{mix})-\mathrm{K}_{\mathrm{T} 2}^{*}(\ell)\right] \nonumber \]