1.22.9: Volume: Liquid Mixtures
- Page ID
- 397793
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)A given binary liquid mixture is prepared using \(\mathrm{n}_{1}\) moles of liquid 1 and \(\mathrm{n}_{2}\) moles of liquid 2 at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\). The term ‘mixture’ usually means that a homogeneous single liquid phase is spontaneously formed on mixing characterized by a minimum in Gibbs energy \(\mathrm{G}\) where the molecular organization is characterized by \(\xi^{\mathrm{eq}}\). \(\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}\) measures the extent to which \(\mathrm{G} /\left(\mathrm{n}_{1}+\mathrm{n}_{2}\right)\) differs from this ratio in the event that the mixing is, in a thermodynamic sense, ideal.
A given binary liquid mixture is displaced to a neighboring state by a change in pressure at constant temperature. The overall composition remains at \(\left(\mathrm{n}_{1} + \mathrm{n}_{2}\right)\) but the organization changes to a new value for \(\xi^{\mathrm{eq}}\) where ‘\(\mathrm{A} = 0\)’. The differential dependence of \(\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}\) on pressure at constant temperature \(\mathrm{T}\) is the excess molar volume \(\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}\).
\[\mathrm{V}(\operatorname{mix})=\left(\frac{\partial \mathrm{G}(\operatorname{mix})}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}=0} \nonumber \]
\[\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}=\left(\frac{\partial \mathrm{G}_{\mathrm{m}}^{\mathrm{E}}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}=0} \nonumber \]
For the molar volume,
\[\mathrm{V}_{\mathrm{m}}=\left(\frac{\partial \mathrm{G}_{\mathrm{m}}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}=0} \nonumber \]
The quantities, \(\mathrm{V}(\mathrm{mix})\), \(\mathrm{V}_{\mathrm{m}}\) and \(\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}\) are interesting because they can be determined [1] whereas the same cannot be said for \(\mathrm{G}(\mathrm{mix})\) and \(\mathrm{G}_{\mathrm{m}}\) although \(\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}\) can be obtained m from vapor pressures of mixtures and pure components.
\[\mathrm{V}_{\mathrm{m}}=\mathrm{V}(\mathrm{mix}) /\left(\mathrm{n}_{1}+\mathrm{n}_{2}\right) \nonumber \]
Density,
\[\rho(\operatorname{mix})=\left(\mathrm{n}_{1} \, \mathrm{M}_{1}+\mathrm{n}_{2} \, \mathrm{M}_{2}\right) / \mathrm{V}(\operatorname{mix}) \nonumber \]
\(\mathrm{M}_{1}\) and \(\mathrm{M}_{2} are the molar masses of the two liquid components. By measuring \(\rho(\mathrm{mix})\) as a function of mixture composition, we form a plot of molar volume Vm as a function of mole fraction composition. The plot has two limits;
\[\operatorname{limit}\left(\mathrm{x}_{1} \rightarrow 1\right) \mathrm{V}_{\mathrm{m}}=\mathrm{V}_{1}^{*}(\ell) \nonumber \]
\[\operatorname{limit}\left(\mathrm{x}_{2} \rightarrow 1\right) \mathrm{V}_{\mathrm{m}}=\mathrm{V}_{2}^{*}(\ell) \nonumber \]
If the thermodynamic properties of the binary liquid mixture are ideal (i.e. \(\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}=0\)),
\[\mathrm{V}_{\mathrm{m}}(\mathrm{id})=\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell) \nonumber \]
Or,
\[\mathrm{V}_{\mathrm{m}}(\mathrm{id})=\left(1-\mathrm{x}_{2}\right) \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell) \nonumber \]
Hence,
\[\mathrm{V}_{\mathrm{m}}(\mathrm{id})=\mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \,\left[\mathrm{V}_{2}^{*}(\ell)-\mathrm{V}_{1}^{*}(\ell)\right] \nonumber \]
The latter is an equation for a straight line. The molar volume of a real binary liquid mixture is usually less than \(\mathrm{V}_{m}(\mathrm{id})\). For a real binary liquid mixture,
\[\mathrm{V}_{\mathrm{m}}(\operatorname{mix})=\mathrm{x}_{1} \, \mathrm{V}_{1}(\operatorname{mix})+\mathrm{x}_{2} \, \mathrm{V}_{2}(\mathrm{mix}) \nonumber \]
The difference between the molar volume of real and ideal binary liquid mixture is the excess molar volume \(\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}\).
\[\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}=\mathrm{x}_{1} \,\left[\mathrm{V}_{1}(\operatorname{mix})-\mathrm{V}_{1}^{*}(\ell)\right]+\mathrm{x}_{2} \,\left[\mathrm{V}_{2}(\operatorname{mix})-\mathrm{V}_{2}^{*}(\ell)\right] \nonumber \]
Or,
\[V_{m}^{E}=x_{1} \, V_{1}^{E}(\operatorname{mix})+x_{2} \, V_{2}^{E}(\operatorname{mix} \nonumber \]
A given mixture, mole fraction \(\mathrm{x}_{2}\) at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\), is perturbed by addition of \(\delta \mathrm{n}_{2}\) moles of chemical substance 2. The system can be perturbed either at constant organization \(\xi\) or constant affinity \(\mathrm{A}\). Here we are concerned with \(\left(\frac{\partial \mathrm{V}}{\partial \mathrm{n}_{2}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(1), \mathrm{A}=0}\), the (equilibrium) partial molar volume of substance 2 in the mixture, \(\mathrm{V}_{2}(\mathrm{mix})\).The condition '\(\mathrm{A} = 0\)' implies that there is a change in organization \(\xi\) in order to hold the system in the equilibrium state. A similar argument is formulated for the (equilibrium) partial molar volume \(\mathrm{V}_{1}(\mathrm{mix})\). Moreover according to the Gibbs-Duhem equation (at constant temperature and pressure),
\[\mathrm{n}_{1} \, d \mathrm{~V}_{1}(\operatorname{mix})+\mathrm{n}_{2} \, d \mathrm{~V}_{2}(\operatorname{mix})=0 \nonumber \]
Further,
\[\mathrm{V}(\operatorname{mix})=\mathrm{n}_{1} \, \mathrm{V}_{1}(\operatorname{mix})+\mathrm{n}_{2} \, \mathrm{V}_{2}(\operatorname{mix}) \nonumber \]
The property \(\mathrm{V}(\mathrm{mix})\) is directly determined from the density \(\rho(\mathrm{mix})\).
\[\mathrm{V}(\operatorname{mix})=\left(\mathrm{n}_{1} \, \mathrm{M}_{1}+\mathrm{n}_{2} \, \mathrm{M}_{2}\right) / \rho(\operatorname{mix}) \nonumber \]
The important point is that the thermodynamic extensive property \(\mathrm{V}(\mathrm{mix})\) is directly determined by experiment whereas we cannot for example measure the enthalpy \(\mathrm{H}(\mathrm{mix})\). The excess molar volume is given by equation (l).
\[\begin{aligned}
\frac{\mathrm{dV}_{\mathrm{m}}^{\mathrm{E}}}{d \mathrm{x}_{1}}=& {\left[\mathrm{V}_{1}(\mathrm{mix})-\mathrm{V}_{1}^{*}(\ell)\right]+\mathrm{x}_{1} \, \frac{\mathrm{dV} \mathrm{V}_{1}(\mathrm{mix})}{\mathrm{dx}_{1}} } \\
&-\left[\mathrm{V}_{2}(\mathrm{mix})-\mathrm{V}_{2}^{*}(\ell)\right]+\mathrm{x}_{2} \, \frac{\mathrm{dV}_{2}(\mathrm{mix})}{\mathrm{dx}_{1}}
\end{aligned} \nonumber \]
Using the Gibbs -Duhem equation,
\[\frac{\mathrm{dV}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{dx}_{1}}=\left[\mathrm{V}_{1}(\mathrm{mix})-\mathrm{V}_{1}^{*}(\ell)\right]-\left[\mathrm{V}_{2}(\mathrm{mix})-\mathrm{V}_{2}^{*}(\ell)\right] \nonumber \]
Or,
\[\left[\mathrm{V}_{1}(\mathrm{mix})-\mathrm{V}_{1}^{*}(\ell)\right]=\frac{\mathrm{dV}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{dx}_{1}}+\left[\mathrm{V}_{2}(\mathrm{mix})-\mathrm{V}_{2}^{*}(\ell)\right] \nonumber \]
From equation (l),
\[\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}=\mathrm{x}_{1} \, \frac{\mathrm{dV}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{dx}_{1}}+\mathrm{x}_{1} \,\left[\mathrm{V}_{2}(\operatorname{mix})-\mathrm{V}_{2}^{*}(\ell)\right]+\mathrm{x}_{2} \,\left[\mathrm{V}_{2}(\operatorname{mix})-\mathrm{V}_{2}^{*}(\ell)\right] \nonumber \]
Hence,
\[\left[\mathrm{V}_{2}(\mathrm{mix})-\mathrm{V}_{2}^{*}(\ell)\right]=\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}-\mathrm{x}_{1} \, \frac{\mathrm{dV}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{dx}_{1}} \nonumber \]
The derivation leading up to equation (s) is the 'Method of Tangents'. Moreover at the mole fraction composition where \(\frac{\mathrm{dV}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{dx}_{1}}\) is zero, \(\left[\mathrm{V}_{2}(\operatorname{mix})-\mathrm{V}_{2}^{*}(\ell)\right]\) equals \(\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}\).
For liquid component 1 the chemical potential in the liquid mixture is related to the mole fraction composition (at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\)).
\[\mu_{1}(\operatorname{mix}, \mathrm{T}, \mathrm{p})=\mu_{1}^{*}(\ell, \mathrm{T})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right)+\int_{\mathrm{p}^{0}}^{\mathrm{p}} \mathrm{V}_{1}^{*}(\ell ; \mathrm{T}) \, \mathrm{dp} \nonumber \]
At temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\),
\[\mathrm{V}_{1}(\operatorname{mix})=\mathrm{V}_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\mathrm{f}_{1}\right) / \partial \mathrm{p}\right]_{\mathrm{T}} \nonumber \]
But
\[\mathrm{V}_{1}^{\mathrm{E}}(\operatorname{mix})=\mathrm{V}_{1}(\operatorname{mix})-\mathrm{V}_{1}^{*}(\ell) \nonumber \]
Hence,
\[\mathrm{V}_{1}^{\mathrm{E}}(\operatorname{mix})=\mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\mathrm{f}_{1}\right) / \partial \mathrm{p}\right]_{\mathrm{T}} \nonumber \]
Then,
\[\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}(\mathrm{mix})=\mathrm{R} \, \mathrm{T} \,\left\{\mathrm{x}_{1} \,\left[\partial \ln \left(\mathrm{f}_{1}\right) / \partial \mathrm{p}\right]_{\mathrm{T}}+\mathrm{x}_{2} \,\left[\partial \ln \left(\mathrm{f}_{2}\right) / \partial \mathrm{p}\right]_{\mathrm{T}}\right\} \nonumber \]
Footnote
[1] R. Battino, Chem.Rev.,1971,71,5.
[2]
- \(\mathrm{C}_{\mathrm{m}}\mathrm{H}_{2\mathrm{m}+2}\) as a solute in \(\mathrm{C}_{\mathrm{n}}\mathrm{H}_{2\mathrm{n}+2}\) solvent; H. Itsuki, S. Terasowa, K. Shinora and H. Ikezawa, J Chem. Thermodyn.,19897,19,555.
- 60[Fullerene} in apolar solvents; P. Ruella, A. Farina-Cuendet and U.W. Kesselring, J. Chem. Soc. Chem. Commun.,1995,1161.