1.14.15: Degree of Dissociation
- Page ID
- 374477
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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 aqueous solution is prepared using \(\mathrm{n}_{1}^{0}\) moles of water and \(\mathrm{n}_{\mathrm{A}}^{0}\) moles of a weak acid \(\mathrm{HA}\). The composition of the solution at equilibrium (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)) is described as follows.
| \(\mathrm{HA}(\mathrm{aq})\) | \(\ce{<=>}\) | \(\mathrm{H}^{+}(\mathrm{aq}) +\) | \(A^{\prime}(\mathrm{aq})\) | |
|---|---|---|---|---|
| At \(t=0\) | \(n_{A}^{0}\) | \(0\) | \(0 \mathrm{~mol}\) | |
| At equilibirium, | \(n_{A}^{0}-\xi^{\mathrm{eq}}\) | \(\xi^{\mathrm{eq}}\) | \(\xi^{\mathrm{eq}}\mathrm{~mol}\) | |
| or, | \(\mathrm{n}_{\mathrm{A}}^{0} \,\left[1-\frac{\xi^{e q}}{\mathrm{n}_{\mathrm{A}}^{0}}\right]\) | \(\xi^{\mathrm{eq}}\) | \(\xi^{\mathrm{eq}} \mathrm{~mol}\) | |
| or. | \(\mathrm{n}_{\mathrm{A}}^{0} \,(1-\alpha)\) | \(\alpha \, \mathrm{n}_{\mathrm{A}}^{0}\) | \(\alpha \, \mathrm{n}_{\mathrm{A}}^{0}\mathrm{~mol}\) |
If the volume of the system is \(1 \mathrm{~dm}^{3}\) then, \(\mathrm{c}_{\mathrm{A}}^{0} \,(1-\alpha) \quad \alpha \, \mathrm{c}_{\mathrm{A}}^{0} \quad \alpha \, \mathrm{c}_{\mathrm{A}}^{0} \quad \mathrm{~mol} \mathrm{dm}^{-3}\)
By definition, the degree of dissociation, \(\alpha=\xi^{\mathrm{eq}} / \mathrm{n}_{\mathrm{A}}^{0}\); \(\alpha\) is an intensive variable describing the ‘degree’ of dissociation. If the total volume of the solution is \(\mathrm{V}\), the concentration \(\mathrm{c}_{\mathrm{A}}^{0}=\mathrm{n}_{\mathrm{A}}^{0} / \mathrm{V}\). If the thermodynamic properties of the solution are ideal, the composition of the solution can be described by an equilibrium acid dissociation constant \(\mathrm{K}_{\mathrm{A}}\).
\[\mathrm{K}_{\mathrm{A}}=\alpha^{2} \, \mathrm{c}_{\mathrm{A}}^{0} /(1-\alpha) \nonumber \]
If
\[1-\alpha \cong 1, \alpha^{2}=\mathrm{K}_{\mathrm{A}} / \mathrm{c}_{\mathrm{A}}^{0} \nonumber \]
If the acid is dibasic, the analysis is a little more complicated.
| \(\mathrm{H}_{2}\mathrm{A}\) | \(\ce{<=>}\) | \(\mathrm{H}^{+} +\) | \(\mathrm{HA}^{-}\) | |
|---|---|---|---|---|
| At \(t=0\), | \(\mathrm{n}_{\mathrm{A}}^{0}\) | \(0\) | \(0 \mathrm{~mol}\) | |
| At equilibrium, | \(\mathrm{n}_{\mathrm{A}}^{0}-\xi_{1}\) | \(\xi_{1}+\xi_{2}\) | \(\xi_{1}-\xi_{2}\mathrm{~mol}\) | |
| Or, | \(n_{A}^{0} \,\left[1-\frac{\zeta_{1}}{n_{A}^{0}}\right]\) | \(\mathrm{n}_{\mathrm{A}}^{0} \,\left[\frac{\xi_{1}}{\mathrm{n}_{\mathrm{A}}^{0}}+\frac{\xi_{2}}{\mathrm{n}_{\mathrm{A}}^{0}}\right]\) | \(\mathrm{n}_{\mathrm{A}}^{0} \,\left[\frac{\xi_{1}}{\mathrm{n}_{\mathrm{A}}^{0}}-\frac{\xi_{2}}{\mathrm{n}_{\mathrm{A}}^{0}}\right]\mathrm{~mol}\) |
By definition \(\mathrm{c}_{\mathrm{A}}^{0}=\mathrm{n}_{\mathrm{A}}^{0} / \mathrm{V}\) where \(\mathrm{V}\) is the volume of solution expressed in \(\mathrm{dm}^{3}\). Also by definition \(\alpha_{1}=\xi_{1} / \mathrm{n}_{\mathrm{A}}^{0}\) and \(\alpha_{2}=\xi_{2} / \mathrm{n}_{\mathrm{A}}^{0}\)
Hence from equation (d)
| \(\mathrm{H}_{2}\mathrm{A}\) | \(\rightleftarrows\) | \(\mathrm{H}^{+} +\) | \mathrm{HA}^{-}\) | |
|---|---|---|---|---|
| At equilibrium, | \(\left(\mathrm{n}_{\mathrm{A}}^{0} / \mathrm{V}\right) \,\left[1-\alpha_{1}\right]\) | \(\left(\mathrm{n}_{\mathrm{A}}^{0} / \mathrm{V}\right) \,\left[\alpha_{1}+\alpha_{2}\right]\) | \(\left(\mathrm{n}_{\mathrm{A}}^{0} / \mathrm{V}\right) \,\left[\alpha_{1}-\alpha_{2}\right]\mathrm{~mol}\) | |
| Or, | \(\mathrm{c}_{\mathrm{A}}^{0} \,\left[1-\alpha_{1}\right]\) | \(\mathrm{c}_{\mathrm{A}}^{0} \,\left[\alpha_{1}+\alpha_{2}\right]\) |
\(\mathrm{c}_{\mathrm{A}}^{0} \,\left[\alpha_{1}-\alpha_{2}\right]\mathrm{~mol}\) |
| \(\mathrm{HA}^{-}\) | \(\rightleftarrows\) | \(\mathrm{H}^{+} +\) | \mathrm{A}^{-2}\) | |
| At \(t=0\), | \(0\) | \(0\) | \(0 \mathrm{~mol}\) | |
| Also At equilibrium, | \(\xi_{1}-\xi_{2}\) | \(\xi_{1}+\xi_{2}\) | \(\xi_{2}\mathrm{~mol}\) | |
| Or, | \(\mathrm{n}_{\mathrm{A}}^{0} \,\left[\frac{\xi_{1}}{\mathrm{n}_{\mathrm{A}}^{0}}-\frac{\xi_{2}}{\mathrm{n}_{\mathrm{A}}^{0}}\right]\) | \(\mathrm{n}_{\mathrm{A}}^{0} \,\left[\frac{\xi_{1}}{\mathrm{n}_{\mathrm{A}}^{0}}+\frac{\xi_{2}}{\mathrm{n}_{\mathrm{A}}^{0}}\right]\) | \(n_{A}^{0} \,\left[\frac{\zeta_{2}}{n_{A}^{0}}\right]\mathrm{~mol}\) | |
| \(c_{\mathrm{A}}^{0} \,\left[\alpha_{1}-\alpha_{2}\right]\) | \(\mathrm{c}_{\mathrm{A}}^{0} \,\left[\alpha_{1}+\alpha_{2}\right]\) | \(\mathrm{c}_{\mathrm{A}}^{0} \, \alpha_{2}\) |
Total amount of \(\mathrm{H} in the system
\[=2 \,\left(\mathrm{n}_{\mathrm{A}}^{0}-\xi_{1}\right)+\xi_{1}+\xi_{2}+\xi_{1}-\xi_{2}=2 \, \mathrm{n}_{\mathrm{A}}^{0} \nonumber \]
Total amount of \(\mathrm{A}\) in the system
\[=\mathrm{n}_{\mathrm{A}}^{0}-\xi_{1}+\xi_{1}-\xi_{2}+\xi_{2}=\mathrm{n}_{\mathrm{A}}^{0} \nonumber \]
If the thermodynamic properties of the solution are ideal,
\[\mathrm{K}_{1}=\mathrm{c}_{\mathrm{A}}^{0} \,\left[\alpha_{1}+\alpha_{2}\right] \,\left[\alpha_{1}-\alpha_{2}\right] /\left[1-\alpha_{1}\right] \nonumber \]
If
\[\mathrm{K}_{2}=0, \alpha_{2}=0, \mathrm{~K}_{1}=\mathrm{c}_{\mathrm{A}}^{0} \, \alpha_{1}^{2} /\left(1-\alpha_{1}\right) \nonumber \]
But
\[\mathrm{K}_{2}=\left(\alpha_{1}+\alpha_{2}\right) \, \alpha_{2} \, \mathrm{c}_{\mathrm{A}}^{0} /\left(\alpha_{1}-\alpha_{2}\right) \nonumber \]