1.18.2: Liquid Mixtures: General Equations
- Page ID
- 394366
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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 liquid mixture is prepared using liquid-1 and liquid –2 at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\), the latter being close to the standard pressure. The chemical potentials, \(\mu_{1}\left(\operatorname{mix} ; \mathrm{x}_{1}\right)\) and \(\mu_{2}\left(\operatorname{mix} ; \mathrm{x}_{2}\right)\) are related to the mole fraction composition, \(x_{1}\) and \(x_{2} (= 1 - x_{1})\) using equations (a) and (c) where \(\mu_{1}^{*}(\ell)\) and \(\mu_{2}^{*}(\ell)\) are the chemical potentials of the two pure liquid components at the same \(\mathrm{T}\) and \(\mathrm{p}\);
\[\mu_{1}\left(\operatorname{mix} ; \mathrm{x}_{1}\right)=\mu_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right) \nonumber \]
where
\[\operatorname{limit}\left(x_{1} \rightarrow 1\right) f_{1}=1 \nonumber \]
\[\mu_{2}\left(\operatorname{mix} ; \mathrm{x}_{2}\right)=\mu_{2}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{2} \, \mathrm{f}_{2}\right) \nonumber \]
where
\[\operatorname{limit}\left(\mathrm{x}_{2} \rightarrow 1\right) \mathrm{f}_{2}=1 \nonumber \]
A general equation for activity coefficient \(\mathrm{f}_{1}\) takes the following form [1].
\[\ln \left(f_{1}\right)=\sum_{k=1}^{k=\infty} \alpha_{k} \, x_{s}^{\lambda(k)} \nonumber \]
Equation (e) satisfies the condition,
\[\operatorname{limit}\left(x_{2} \rightarrow 0\right) \ln \left(f_{1}\right)=0 ; f_{1}=1 \nonumber \]
The parameter \(\alpha_{\mathrm{k}}\) is characteristic of the mixture, temperature and pressure. The property \(\lambda_{\mathrm{k}}\) is a real number. In the limit that the liquid mixture is dilute in chemical substance liquid-2, equation (e) simplifies to equation (g).
\[\ln \left(f_{1}\right)=\alpha \, x_{2}^{\lambda} \nonumber \]
In general terms [2],
\[x_{1} \, d \ln \left(f_{1}\right)+x_{2} \, d \ln \left(f_{2}\right)=0 \nonumber \]
We combine equations (e) and (h) with \(\lambda_{k} \geq 2\) [3].
\[\frac{\mathrm{d} \ln \left(\mathrm{f}_{1} / \mathrm{f}_{2}\right)}{\mathrm{dx}_{2}}=\frac{1}{\mathrm{x}_{2}} \, \frac{\mathrm{d} \ln \left(\mathrm{f}_{1}\right)}{\mathrm{dx} \mathrm{x}_{2}}=\sum_{\mathrm{k}=1}^{\mathrm{k}=\infty} \alpha_{\mathrm{k}} \, \lambda_{\mathrm{k}} \, \mathrm{x}_{2}^{\lambda(\mathrm{k})-2} \nonumber \]
Equation (i) is integrated to yield equation (j) where \(\mathrm{I}\) is the constant of integration.
\[\ln \left(f_{2}\right)=\ln \left(f_{1}\right)-\sum_{k=1}^{k=\infty} \frac{\alpha_{k} \, \lambda_{k} \, x_{2}^{\lambda(k)-1}}{\lambda_{k}-1}-I \nonumber \]
Hence [4,5]
\[\begin{aligned}
\ln \left(f_{2}\right) &=\ln \left(f_{1}\right)-\sum_{k=1}^{k=\infty} \frac{\alpha_{k} \, \lambda_{k}}{\lambda_{k}-1} \,\left(x_{2}^{\lambda(k)-1}-1\right)-\sum_{k=1}^{k=\infty} \alpha_{k} \\
&=\ln \left(f_{1}\right)-\sum_{k=1}^{k=\infty} \alpha_{k} \,\left[\frac{\lambda_{k}}{\lambda_{k}-1} \,\left(x_{2}^{\lambda-(k-1)}-1\right)-1\right]
\end{aligned} \nonumber \]
In other words, granted that \(\ln \left(f_{1}\right)\) is known as a function of \(x_{2}\), then \(\ln \left(f_{2}\right)\) can be calculated.
Footnotes
[1] I. Prigogine and R. Defay, Chemical Thermodynamics, transl. D. H. Everett, Longmans Green, London, 1954.
[2] For a binary liquid mixture , the Gibbs-Duhem equation relates activity coefficients \(\mathrm{f}_{1}\) and \(\mathrm{f}_{2}\). Thus,
\[-S \, d T+V \, d p+n_{1} \, d \mu_{1}+n_{2} \, d \mu_{2}=0 \nonumber \]
At fixed \(\mathrm{T}\) and \(\mathrm{p}\), \(\mathrm{n}_{1} \, \mathrm{d} \mu_{1}+\mathrm{n}_{2} \, \mathrm{d} \mu_{2}=0\)
Divide by \(\left(n_{1}+n_{2}\right)\); \(x_{1} \, d \mu_{1}+x_{2} \, d \mu_{2}=0\)
\[\begin{gathered}
\mathrm{x}_{1} \, \mathrm{d}\left[\mu_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right)\right]+\mathrm{x}_{2} \, \mathrm{d}\left[\mu_{2}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{2} \, \mathrm{f}_{2}\right)\right]=0 \\
\mathrm{x}_{1} \, \mathrm{d} \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right)+\mathrm{x}_{2} \, \mathrm{d} \ln \left(\mathrm{x}_{2} \, \mathrm{f}_{2}\right)=0 \\
\mathrm{x}_{1} \, \mathrm{d} \ln \left(\mathrm{x}_{1}\right)+\mathrm{x}_{1} \, \mathrm{d} \ln \left(\mathrm{f}_{1}\right)+\mathrm{x}_{2} \, \mathrm{d} \ln \left(\mathrm{x}_{2}\right)+\mathrm{x}_{2} \, \mathrm{d} \ln \left(\mathrm{f}_{2}\right)=0
\end{gathered} \nonumber \]
But
\[\mathrm{x}_{1} \, \mathrm{d} \ln \left(\mathrm{x}_{1}\right)+\mathrm{x}_{2} \, \mathrm{d} \ln \left(\mathrm{x}_{2}\right)=\left(\mathrm{x}_{1} / \mathrm{x}_{1}\right) \, \mathrm{dx} \mathrm{x}_{1}+\left(\mathrm{x}_{2} / \mathrm{x}_{2}\right) \, \mathrm{dx} \mathrm{x}_{2} \nonumber \]
Also \(\mathrm{x}_{1}+\mathrm{x}_{2}=1\) so that \(\mathrm{dx}_{1}+\mathrm{dx}_{2}=0\)
[3] From equation (h) for a binary liquid mixture at fixed \(\mathrm{T}\) and \(\mathrm{p}\),
\[\begin{array}{r}
\left(1-x_{2}\right) \, \frac{d \ln \left(f_{1}\right)}{d x_{2}}+x_{2} \, \frac{d \ln \left(f_{2}\right)}{d x_{2}}=0 \\
\frac{d \ln \left(f_{1}\right)}{d x_{2}}-x_{2} \, \frac{d \ln \left(f_{1}\right)}{d x_{2}}+x_{2} \, \frac{d \ln \left(f_{2}\right)}{d x_{2}}=0
\end{array} \nonumber \]
We divide by \(x_{2}\) and rearrange the equation.
\[\frac{\mathrm{d} \ln \left(\mathrm{f}_{1}\right)}{\mathrm{dx} \mathrm{x}_{2}}-\frac{\mathrm{d} \ln \left(\mathrm{f}_{2}\right)}{\mathrm{dx}_{2}}=\frac{1}{\mathrm{x}_{2}} \, \frac{\mathrm{d} \ln \left(\mathrm{f}_{1}\right)}{\mathrm{dx} \mathrm{x}_{2}} \nonumber \]
Or,
\[\frac{\mathrm{d} \ln \left(\mathrm{f}_{1} / \mathrm{f}_{2}\right)}{\mathrm{dx} \mathrm{x}_{2}}=\frac{1}{\mathrm{x}_{2}} \, \frac{\mathrm{d} \ln \left(\mathrm{f}_{1}\right)}{\mathrm{dx_{2 }}} \nonumber \]
[4] From equations (e) and (j),
\[\ln \left(f_{2}\right)=\sum_{k=1}^{k=\infty} \alpha_{k} \, x_{2}^{\lambda(k)}-\sum_{k=1}^{k=\infty} \frac{\alpha_{k} \, \lambda_{k} \, x_{2}^{\lambda(k)-1}}{\lambda_{k}-1}-I \nonumber \]
But at \(x_{2} = 1, f_{2} = 1\). Then,
\[0=\sum_{\mathrm{k}=1}^{\mathrm{k}=\infty} \alpha_{\mathrm{k}}-\sum_{\mathrm{k}=1}^{\mathrm{k}=\infty} \frac{\alpha_{\mathrm{k}} \, \lambda_{\mathrm{k}}}{\lambda_{\mathrm{k}}-1}-\mathrm{I} \nonumber \]
[5] J. N. Bronsted and P. Colmart, Z. Phys. Chem.,1934,A168, 381 ( as quoted in reference 1).