1.14.5: Extent of Reaction
- Page ID
- 392427
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)For chemists, chemical reaction is the key thermodynamic process. By definition chemical reaction produces a change in composition of a closed system. The extent of chemical reaction is measured by a quantity \(\mathrm{d}\xi\), where the chemical composition is described by the symbol \(\xi\). An example makes the point.
An aqueous solution is prepared at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) contains solute \(\mathrm{X}\). The latter undergoes spontaneous chemical reaction to form chemical substance \(\mathrm{Y}\).
Thus
| \(\mathrm{X}(\mathrm{aq})\) | \(\rightarrow\) | \(\mathrm{Y}(\mathrm{aq})\) | |
| At \(t = 0\) | \(\mathrm{n}_{\mathrm{X}}^{0}\) | \(0 \mathrm{~mol}\) | |
| At time \(t\), | \(\mathrm{n}_{\mathrm{X}}^{0}-\xi\) | \(\xi \mathrm{~mol}\) | |
| Rate of reaction | \(= \mathrm{d}\xi / \mathrm{dt}\) |
[Time is a legitimate thermodynamic property.]
A key concept states that spontaneous chemical reaction is driven by the affinity for spontaneous change, \(\mathrm{A}\). Then by definition equilibrium corresponds to the state where \(\mathrm{A} = 0\), and \(\mathrm{d}\xi / \mathrm{dt} = 0\).
General Terms
For a system containing \(\mathrm{i}\)-chemical substances, the chemical potential of chemical substance \(\mathrm{j}\) is given by equation (a).
\[\mu_{\mathrm{j}}=\left(\frac{\partial \mathrm{G}}{\partial n_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})} \nonumber \]
Then,
\[\mathrm{dG}=-\mathrm{S} \, \mathrm{dT}+\mathrm{V} \, \mathrm{dp}+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mu_{\mathrm{j}} \, \mathrm{dn} \mathrm{j}_{\mathrm{j}} \nonumber \]
But,
\[\mathrm{dG}=-\mathrm{S} \, \mathrm{dT}+\mathrm{V} \, \mathrm{dp}-\mathrm{A} \, \mathrm{d} \xi \nonumber \]
By comparison,
\[A \, d \xi=-\sum_{j=1}^{j=i} \mu_{j} \, d n_{j} \nonumber \]
But \(\mathrm{dn}_{\mathrm{j}}=\mathrm{v}_{\mathrm{j}} \, \mathrm{d} \xi\) where \(\mathrm{ν}_{\mathrm{j}}\) is positive for products and negative for reactants. Hence,
\[A=-\sum_{j=1}^{j=i} v_{j} \, \mu_{j} \nonumber \]
This remarkable equation relates the affinity for chemical reaction \(A\) with the chemical potentials of the chemical substances involved in the chemical reaction. Moreover at equilibrium, \(A\) is zero. Hence,
\[\sum_{j=1}^{j=i} v_{j} \, \mu_{j}^{e q}=0 \nonumber \]
We have a condition describing chemical equilibrium in terms of the chemical potentials of reactants and products at equilibrium.