1.2.3: Affinity for Spontaneous Chemical Reaction- Phase Equilibria
- Page ID
- 352501
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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 comprises two phases, I and II, both phases comprising i-chemical substances. We consider the transfer of one mole of chemical substance \(j\) from phase I to phase II. The affinity for the transfer is given by equation (a).
\[\mathrm{A}_{\mathrm{j}}=\mu_{\mathrm{j}}(\mathrm{I})-\mu_{\mathrm{j}}(\mathrm{II}) \nonumber \]
If \(\mu_{j}(\mathrm{II})<\mu_{\mathrm{j}}(\mathrm{I})\), \(\mathrm{A}_{j}\) is positive and the process is spontaneous. If the system is at fixed \(\mathrm{T}\) and pressure, the gradient of Gibbs energy is negative [1].
\[\mathrm{A}_{\mathrm{j}}=-(\partial \mathrm{G} / \partial \xi)_{\mathrm{T}, \mathrm{p}}=\mu_{j}(\mathrm{I})-\mu_{\mathrm{j}}(\mathrm{II}) \nonumber \]
We suppose the mole fraction of substance \(–j\) in phases I and II are \(x_{j}(\mathrm{I})\) and \(x_{j}(\mathrm{II})\). We express the chemical potentials as functions of the mole fraction compositions of the two phases.
\[\begin{aligned}
A_{j}=\mu_{j}^{*}(I)+& R \, T \, \ln \left[x_{j}(I) \, f_{j}(I)\right] \\
&-\mu_{j}^{*}(I I)-R \, T \, \ln \left[x_{j}(I I) \, f_{j}(I I)\right]
\end{aligned} \nonumber \]
Here \(\mathrm{f}_{j}(\mathrm{I})\) and \(\mathrm{f}_{j}(\mathrm{II})\) are rational activity coefficients of substance \(j\) in phases \(\mathrm{I}\) and \(\mathrm{II}\) respectively.
\[\text { At all } T \text { and p, both } \operatorname{limit}\left(x_{j}(I) \rightarrow 1\right) f_{j}(I)=1 \nonumber \]
\[\text { and } \operatorname{limit}\left(\mathrm{x}_{\mathrm{j}} \text { (II) } \rightarrow 1\right) \mathrm{f}_{\mathrm{j}}(\text { II })=1 \nonumber \]
\[\text { By definition } \mu_{j}^{*}(\mathrm{II})-\mu_{\mathrm{j}}^{*}(\mathrm{I})=-\mathrm{R} \, \mathrm{T} \, \ln [\mathrm{K}(\mathrm{T}, \mathrm{p})] \nonumber \]
\(\mathrm{K}_{j}(\mathrm{T}, p)\) is a measure of the difference in reference chemical potentials of substance \(j\) in phases \(\mathrm{I}\) and \(\mathrm{II}\). If the two phases are in equilibrium, there is no affinity for substance \(j\) to pass spontaneously between the two phases. At equilibrium, \(\mathrm{A}_{j}\) is zero. Hence from equations (c) and (f), for the non-equilibrium state,
\[A_{j}=R \, T \, \ln \left[K_{j}(T, p)\right]+R \, T \, \ln \left[\frac{x_{j}(I) \, f_{j}(I)}{x_{j}(I I) \, f_{j}(I I)}\right] \nonumber \]
\[\frac{A_{j}}{T}=R \, \ln \left[K_{j}(T, p)\right]+R \, \ln \left[\frac{x_{j}(I) \, f_{j}(I)}{x_{j}(I I) \, f_{j}(I I)}\right] \nonumber \]
Equation (g) yields the affinity for chemical substance \(j\) to pass between the phases in a non-equilibrium state. In applications of equation (h), we describe the dependence of \(\left(\mathrm{A}_{j} / \mathrm{T}\right)\) on temperature, pressure and composition of the two phases. In other words we require the general differential of equation (h) which is written in the following form.
\[\begin{aligned}
d\left(\frac{A_{j}}{T}\right)=R \,\left(\frac{\partial \ln K_{j}(T, p)}{\partial T}\right) \, d T \\
&+R \,\left(\frac{\partial \ln K_{j}(T, p)}{\partial p}\right) \, d p+R \, d \ln \left[\frac{x_{j}(I) \, f_{j}(I)}{x_{j}(I I) \, f_{j}(I I)}\right]
\end{aligned} \nonumber \]
For the transfer process described by \(\mathrm{K}_{j}(\mathrm{T}, p)\) we obtain equation (j) where \(\Delta_{\text {trans }} \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{~T}, \mathrm{p})\) and \(\Delta_{\text {trans }} V_{j}^{0}(T, p)\) are the standard enthalpy and volume for transfer for chemical substance \(j\).
\[\begin{aligned}
\mathrm{d}\left(\frac{\mathrm{A}_{\mathrm{j}}}{\mathrm{T}}\right)=\left(\frac{\Delta_{\text {trans }} \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{~T}, \mathrm{p})}{\mathrm{T}^{2}}\right) \, \mathrm{dT} \\
&-\left(\frac{\Delta_{\text {trans }} \mathrm{V}_{\mathrm{j}}^{0}(\mathrm{~T}, \mathrm{p})}{\mathrm{T}}\right) \, \mathrm{dp}+\mathrm{R} \, \mathrm{d} \ln \left[\frac{\mathrm{x}_{\mathrm{j}}(\mathrm{I}) \, \mathrm{f}_{\mathrm{j}}(\mathrm{I})}{\mathrm{x}_{\mathrm{j}}(\mathrm{II}) \, \mathrm{f}_{\mathrm{j}}(\mathrm{II})}\right]
\end{aligned} \nonumber \]
\(\Delta_{\text {trans }} \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{~T}, \mathrm{p})\) and \(\Delta_{\text {trans }} \mathrm{V}_{\mathrm{j}}^{0}(\mathrm{~T}, \mathrm{p})\) are properties of pure chemical substance \(j\); i.e. are not dependent on the composition of phases \(\mathrm{I}\) and \(\mathrm{II}\).
Footnotes
[1] Consider the freezing of water;
\[\text { water }(\lambda) \rightarrow \text { water }(\mathrm{s}) \nonumber \]
For this process, \(v\left(\mathrm{H}_{2} \mathrm{O} ; \lambda\right)=-1 ; v\left(\mathrm{H}_{2} \mathrm{O} ; \mathrm{s}\right)=1\)
The general rule is — Positive for Products. The affinity for spontaneous change,
\[A=-\left(\frac{\partial G}{\partial \xi}\right)_{T, p}=-\sum_{j=1}^{j=i} v_{j} \, \mu_{j}=\mu^{*}\left(H_{2} \mathrm{O} ; \lambda\right)-\mu^{*}\left(H_{2} \mathrm{O} ; \mathrm{s}\right) \nonumber \]
At equilibrium (at fixed \(\mathrm{T}\) and \(p\)), \(\mathrm{A} = 0\).
\[\text { Then, } \mu^{*}\left(\mathrm{H}_{2} \mathrm{O} ; \lambda\right)=\mu^{*}\left(\mathrm{H}_{2} \mathrm{O} ; \mathrm{s}\right) \nonumber \]