1.8.13: Enthalpies- Liquid Mixtures
- Page ID
- 375302
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)For an ideal binary liquid mixture the Gibbs energy at temperature T is given by equation (a).
\[\mathrm{G}(\operatorname{mix} ; \mathrm{id})=\mathrm{n}_{1} \,\left[\mu_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)\right]+\mathrm{n}_{2} \,\left[\mu_{2}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{2}\right)\right] \nonumber \]
From the Gibbs-Helmholtz equation,
\[\mathrm{H}(\operatorname{mix} ; \mathrm{id})=\mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\ell)+\mathrm{n}_{2} \, \mathrm{H}_{2}^{*}(\ell) \nonumber \]
Hence for an ideal binary liquid mixture,
\[\mathrm{H}_{1}(\operatorname{mix} ; \mathrm{id})=\mathrm{H}_{1}^{*}(\ell) \text { and } \mathrm{H}_{2}(\operatorname{mix} ; \mathrm{id})=\mathrm{H}_{2}^{*}(\ell) \nonumber \]
The molar enthalpy of a real binary liquid mixture is given by equation (d).
\[\mathrm{H}_{\mathrm{m}}=\mathrm{x}_{1} \, \mathrm{H}_{1}(\operatorname{mix})+\mathrm{x}_{2} \, \mathrm{H}_{2}(\operatorname{mix}) \nonumber \]
Therefore the molar enthalpy of mixing for a real binary liquid mixture is given by equation (e).
\[\Delta_{\text {mix }} H_{m}=x_{1} \,\left[H_{1}(\operatorname{mix})-H_{1}^{*}(\ell)\right]+x_{2} \,\left[H_{2}(\operatorname{mix})-H_{2}^{*}(\ell]\right. \nonumber \]
Significantly equations (b) and (e) show that the molar enthalpy of mixing of an ideal binary liquid mixture, \(\Delta_{\text {mix }} H_{m}(\mathrm{id})\) is zero. The latter condition offers an important point of reference for isobaric calorimetry. [1] If we discover that the mixing of two liquids (at constant pressure) is not zero, the measured molar heat of mixing [\(=\Delta_{\text {mix }} \mathrm{H}_{\mathrm{m}}\)] is an immediate indicator of the extent to which the properties of a given mixture are not ideal.
Nevertheless it is important to set down a link between the measured enthalpies of mixing with the activity coefficients of two liquid components. To this end we start with the equation for the chemical potentials of liquid component 1 in a liquid mixture at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) (which is close to ambient); equation (f).
\[\mu_{1}(\mathrm{mix})=\mu_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right) \nonumber \]
where
\[\operatorname{limit}\left(x_{1} \rightarrow 1\right) f_{1}=1 \text { at all } T \text { and } p \text {. } \nonumber \]
The Gibbs - Helmholtz equation yields an equation for the partial molar enthalpy of component 1 in the liquid mixture. Thus
\[\mathrm{H}_{1}(\operatorname{mix})=\mathrm{H}_{1}^{*}(\ell)-\mathrm{R} \, \mathrm{T}^{2} \,\left[\partial \ln \left(\mathrm{f}_{1}\right) / \partial \mathrm{T}\right]_{\mathrm{p}} \nonumber \]
Similarly,
\[\mathrm{H}_{2}(\operatorname{mix})=\mathrm{H}_{2}^{*}(\ell)-\mathrm{R} \, \mathrm{T}^{2} \,\left[\partial \ln \left(\mathrm{f}_{2}\right) / \partial \mathrm{T}\right]_{\mathrm{p}} \nonumber \]
Hence,
\[\begin{aligned}
&\mathrm{H}_{\mathrm{m}}(\operatorname{mix})= \\
&\quad \mathrm{H}_{\mathrm{m}}(\operatorname{mix} ; \mathrm{id})-\mathrm{R} \, \mathrm{T}^{2} \,\left\{\mathrm{x}_{1} \,\left[\partial \ln \left(\mathrm{f}_{1}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}+\mathrm{x}_{2} \,\left[\partial \ln \left(\mathrm{f}_{2}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}\right\}
\end{aligned} \nonumber \]
We also obtain equations for the excess molar enthalpies of the two components (at defined \(\mathrm{T}\) and \(\mathrm{p}\)).
\[\mathrm{H}_{1}^{\mathrm{E}}(\mathrm{mix})=-\mathrm{R} \, \mathrm{T}^{2} \,\left[\partial \ln \left(\mathrm{f}_{1}\right) / \partial \mathrm{T}\right]_{\mathrm{p}} \nonumber \]
and
\[\mathrm{H}_{2}^{\mathrm{E}}(\mathrm{mix})=-\mathrm{R} \, \mathrm{T}^{2} \,\left[\partial \ln \left(\mathrm{f}_{2}\right) / \partial \mathrm{T}\right]_{\mathrm{p}} \nonumber \]
The excess molar enthalpy,
\[\mathrm{H}_{\mathrm{m}}^{\mathrm{E}}(\text { mix })=-\mathrm{R} \, \mathrm{T}^{2} \,\left\{\mathrm{x}_{1} \,\left[\partial \ln \left(\mathrm{f}_{1}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}+\mathrm{x}_{2} \,\left[\partial \ln \left(\mathrm{f}_{2}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}\right\} \nonumber \]
At fixed pressure, the differential dependence of \(\mathrm{H}_{\mathrm{m}}^{\mathrm{E}}(\operatorname{mix})\) on temperature yields the corresponding excess isobaric heat capacity of mixing.
Footnotes
[1] J. B. Ott and C. J. Wormald, Experimental Thermodynamics, IUPAC Chemical Data Series, No. 39, ed. K. N. Marsh and P. A. G. O’Hara, Blackwell, Oxford, 1994, chapter 8.