1.8.5: Enthalpies- Solutions- Equilibrium and Frozen Partial Molar Enthalpies
- Page ID
- 374779
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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 system at fixed \(\mathrm{T}\) and \(\mathrm{p}\) is at thermodynamic equilibrium. The enthalpy of the system is perturbed by adding \(\delta \mathrm{n}_{j}\) moles of chemical substance \(j\). We imagine two possible limiting changes to the system. In one limit the enthalpy of the system changes to a neighbouring state where the extent of chemical reaction remains constant; i.e. at fixed \(\xi\). In another limit the enthalpy of the system changes to a neighbouring state where the affinity for spontaneous change \(\mathrm{A}\) remains constant. The two differential changes in enthalpy are related.
\[\left(\frac{\partial H}{\partial n_{j}}\right)_{A}=\left(\frac{\partial H}{\partial n_{j}}\right)_{\xi}-\left(\frac{\partial \mathrm{A}}{\partial n_{j}}\right)_{\xi} \,\left(\frac{\partial \xi}{\partial A}\right)_{n_{j}} \,\left(\frac{\partial H}{\partial \xi}\right)_{n_{j}} \nonumber \]
We identify the state being perturbed as the equilibrium state where \(\mathrm{A} = 0\) and the composition-organisation is represented by \(\xi^{\mathrm{eq}\). We identify two quantities describing the impact of adding \(\delta \mathrm{n}_{j}\) moles of chemical substance \(j\).
Equilibrium partial molar enthalpy,
\[\mathrm{H}_{\mathrm{j}}(\mathrm{A}=0)=\left(\frac{\partial \mathrm{H}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{A}=0} \nonumber \]
Frozen partial molar enthalpy,
\[\mathrm{H}_{\mathrm{j}}\left(\xi^{\mathrm{eq}}\right)=\left(\frac{\partial \mathrm{H}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \xi^{\mathrm{eq}}} \nonumber \]
Because the triple product term on the r.h.s. of equation (a) is not zero at equilibrium (i.e. at \(\mathrm{A} = \text { zero}\) and \(\xi = \xi^{\mathrm{eq}}\)), then \(\mathrm{H}_{\mathrm{j}}(\mathrm{A}=0)\) is not equal to. By convention, the term ‘ partial molar enthalpy is taken to mean \(\mathrm{H}_{\mathrm{j}}(\mathrm{A}=0)\).