1.22.12: Volume of Reaction: Dependence on Pressure
- Page ID
- 397796
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Consider a chemical equilibrium between two solute \(\mathrm{X}(\mathrm{aq})\) and \(\mathrm{Y}(\mathrm{aq})\) in aqueous solution at fixed \(\mathrm{T}\) and \(\mathrm{p}\). We assume that the thermodynamic properties of the two solutes are ideal. The chemical equilibrium is be expressed as follows.
\[\mathrm{X}(\mathrm{aq}) \Leftrightarrow \mathrm{Y}(\mathrm{aq}) \nonumber \]
The (dimensionless intensive) degree of reaction \(\alpha\) is related to the equilibrium constant \(\mathrm{K}^{0}\) using equation (b) [1].
\[\alpha=\mathrm{K}^{0} /\left(1+\mathrm{K}^{0}\right) \nonumber \]
At fixed temperature,
\[\frac{\mathrm{d} \alpha}{\mathrm{dp}}=\frac{\mathrm{K}^{0}}{\left(1+\mathrm{K}^{0}\right)^{2}} \, \frac{\mathrm{d} \ln \left(\mathrm{K}^{0}\right)}{\mathrm{dp}} \nonumber \]
Or,
\[\frac{\mathrm{d} \alpha}{\mathrm{dp}}=-\frac{\mathrm{K}^{0}}{\left(1+\mathrm{K}^{0}\right)^{2}} \, \frac{\Delta_{\mathrm{r}} \mathrm{V}^{0}(\mathrm{aq})}{\mathrm{R} \, \mathrm{T}} \nonumber \]
\(\Delta_{\mathrm{r}} \mathrm{V}^{0}(\mathrm{aq})\) is the limiting volume of reaction. The (equilibrium) volume of the system at a defined \(\mathrm{T}\) and \(\mathrm{p}\) is given by equation (e).
\[\mathrm{V}(\mathrm{aq})=\mathrm{n}_{\mathrm{x}} \, \mathrm{V}_{\mathrm{x}}^{\infty}(\mathrm{aq})+\mathrm{n}_{\mathrm{Y}} \, \mathrm{V}_{\mathrm{Y}}^{\infty}(\mathrm{aq})+\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell) \nonumber \]
\(\mathrm{V}_{1}^{*}(\ell)\) is the molar volume of solvent, water. If \(\mathrm{n}_{\mathrm{x}}^{0}\) is total amount of solute, (i.e. \(\mathrm{X}\) and \(\mathrm{Y}\)) in the system,
\[\mathrm{V}(\mathrm{aq})=(1-\alpha) \, \mathrm{n}_{\mathrm{X}}^{0} \, \mathrm{V}_{\mathrm{x}}^{\infty}(\mathrm{aq})+\alpha \, \mathrm{n}_{\mathrm{X}}^{0} \, \mathrm{V}_{\mathrm{Y}}^{\infty}(\mathrm{aq})+\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell) \nonumber \]
Or,
\[\mathrm{V}(\mathrm{aq})=\mathrm{n}_{\mathrm{X}}^{0} \, \mathrm{V}_{\mathrm{x}}^{\infty}(\mathrm{aq})+\alpha \, \mathrm{n}_{\mathrm{X}}^{0} \, \Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\mathrm{aq})+\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell) \nonumber \]
\(\Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\mathrm{aq})\) is the limiting volume of reaction. We assume that at temperature \(\mathrm{T}\), the properties \(\mathrm{V}_{\mathrm{x}}^{\infty}(\mathrm{aq}), \Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\mathrm{aq}) \text { and } \mathrm{V}_{1}^{*}(\ell)\) are independent of pressure. Hence using equations (d) and (g) [2,3],
\[\left(\frac{\partial \mathrm{V}(\mathrm{aq})}{\partial \mathrm{p}}\right)_{\mathrm{T}}=-\mathrm{n}_{\mathrm{X}}^{0} \, \frac{\left[\Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\mathrm{aq})\right]^{2}}{\mathrm{R} \, \mathrm{T}} \, \frac{\mathrm{K}^{0}}{\left[1+\mathrm{K}^{0}\right]^{2}} \nonumber \]
We have taken account of the fact that,
\[\frac{\mathrm{d} \ln \left(\mathrm{K}^{0}\right)}{\mathrm{dp}}=-\frac{\Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\mathrm{aq})}{\mathrm{R} \, \mathrm{T}} \nonumber \]
Similarly [2]
\[\left(\frac{\partial \mathrm{V}(\mathrm{aq})}{\partial \mathrm{T}}\right)_{\mathrm{p}}=\mathrm{n}_{\mathrm{X}}^{0} \, \frac{\Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\mathrm{aq}) \, \Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq})}{\mathrm{R} \, \mathrm{T}^{2}} \, \frac{\mathrm{K}^{0}}{\left[1+\mathrm{K}^{0}\right]^{2}} \nonumber \]
Equation (h) shows that irrespective of the sign of \(\Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\mathrm{aq})\), the contribution to \(\left(\frac{\partial V(a q)}{\partial p}\right)_{\mathrm{T}}\) is always negative. No such generalisation emerges with respect to equation (j) [4]. A closely related subject concerns the dependence of rate constants on pressure leading to volumes of activation [5].
Footnotes
[1] From equation (a)
| \(\mathrm{X}(\mathrm{aq})\) | \(\Leftrightarrow\) | \(\mathrm{Y}(\mathrm{aq})\) | ||
| At \(t = 0\), | \(\mathrm{n}_{\mathrm{X}}^{0}\) | 0 | \(\mathrm{mol}\) | |
| At equilib; | \(\mathrm{n}_{\mathrm{X}}^{0} - \xi\) | \(\xi\) | \(\mathrm{mol}\) | |
| In volume \(\mathrm{V}\) | \(\mathrm{n}_{\mathrm{X}}^{0}\) | \(\xi / \mathrm{V}\) | \(\mathrm{mol m}^{-3}\) |
[2]
\[\begin{aligned}
&\mathrm{dV} / \mathrm{dp}=\left[\mathrm{m}^{3}\right] /\left[\mathrm{N} \mathrm{m}^{-2}\right]=\left[\mathrm{m}^{5} \mathrm{~N}^{-1}\right]\\
&=\frac{\left[\mathrm{m}^{6}\right]}{[\mathrm{N} \mathrm{m}]}=\left[\mathrm{m}^{5} \mathrm{~N}^{-1}\right]
\end{aligned} \nonumber \]
[3]
\[\begin{aligned}
&\frac{\mathrm{dV}}{\mathrm{dT}}=\frac{\left[\mathrm{m}^{3}\right]}{[\mathrm{K}]}\\
&\mathrm{n}_{\mathrm{X}}^{0} \, \frac{\Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\mathrm{aq}) \, \Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq})}{\mathrm{R} \, \mathrm{T}^{2}} \, \frac{\mathrm{K}^{0}}{\left[1+\mathrm{K}^{0}\right]^{2}}\\
&=[\mathrm{mol}] \, \frac{\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right] \,\left[\mathrm{J} \mathrm{mol}^{-1}\right]}{\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}]^{2}}=\left[\mathrm{m}^{3} \mathrm{~K}^{-1}\right]
\end{aligned} \nonumber \]
[4] See for example, J. E. Desnoyers, Pure Appl.Chem.,1982, 54,1469.
[5] (i) W. J. leNoble, J. Chem. Educ.,1967,44,729. (ii) B. S. El’yanov and S. D. Hamann, Aust. J Chem.,1975,28,945. (iii) B.S. El’yanov and M. G. Gonikberg, Russian J. Phys. Chem., 1972, 46, 856.