1.5.15: Chemical Potentials- Solute- Concentration and Molality Scales
- Page ID
- 373396
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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 a given solution we can express the chemical potential of solute \(j\), \(\mu_{\mathrm{j}}(\mathrm{aq})\) in an aqueous solution at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\left(\approx \mathrm{p}^{0}\right)\) using two equations. Therefore, at fixed \(\mathrm{T}\) and \(\mathrm{p}\),
\[\begin{aligned}
&\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)= \\
&\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{c}-\text { scale })+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{c}_{\mathrm{j}} \, \mathrm{y}_{\mathrm{j}} / \mathrm{c}_{\mathrm{r}}\right)
\end{aligned} \nonumber \]
Therefore,
\[\ln \left(\mathrm{y}_{\mathrm{j}}\right)=\ln \left(\gamma_{\mathrm{j}}\right)+\ln \left(\mathrm{m}_{\mathrm{j}} \, \mathrm{c}_{\mathrm{r}} / \mathrm{m}^{0} \, \mathrm{c}_{\mathrm{j}}\right) +(1 / \mathrm{R} \, \mathrm{T}) \,\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})-\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{c}-\text { scale })\right] \nonumber \]
In the latter two equations the composition variables \(\mathrm{m}_{j}\) and \(\mathrm{c}_{j}\) are expressed in the units ‘\(\mathrm{mol kg}^{-1}\)’ and ‘\(\mathrm{mol dm}^{-3}\)’ respectively [1]. The ratio ‘\(\mathrm{c}_{\mathrm{j}} / \mathrm{m}_{\mathrm{j}}\)’ equals the density expressed in the unit ‘\(\mathrm{kg dm}^{-3}\)’. For dilute solutions, \(\mathrm{c}_{\mathrm{j}} / \mathrm{m}_{\mathrm{j}}=\rho_{1}^{*}(\ell)\), the density of the pure solvent.
\[\text { Also, } \mathrm{c}_{\mathrm{r}} / \mathrm{m}^{0}=\left[\mathrm{mol} \mathrm{d \textrm {dm } ^ { - 3 }}\right] /\left[\mathrm{mol} \mathrm{kg}^{-1}\right]=\left[\mathrm{kg} \mathrm{dm}^{-3}\right] \nonumber \]
For dilute aqueous solutions at ambient pressure and \(298.2 \mathrm{~K}\) [2,3],
\[\ln \left(\mathrm{m}_{\mathrm{j}} \, \mathrm{c}_{\mathrm{r}} / \mathrm{m}^{0} \, \mathrm{c}_{\mathrm{j}}\right)=-\ln (0.997) \nonumber \]
With reference to equation (b), with increasing dilution, \(\mathrm{y}_{\mathrm{j}} \rightarrow 1, \gamma_{\mathrm{j}} \rightarrow 1,\left(\mathrm{~m}_{\mathrm{j}} \, \mathrm{c}_{\mathrm{r}} / \mathrm{m}^{0} \, \mathrm{c}_{\mathrm{j}}\right) \rightarrow \mathrm{c}_{\mathrm{r}} / \mathrm{m}^{0} \, \rho_{1}^{*}(\ell)\) Hence,
\[\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{c}-\mathrm{scale})-\mu_{\mathrm{j}}^{0}(\mathrm{aq})=\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{c}_{\mathrm{r}} / \mathrm{m}^{0} \, \rho_{1}^{*}(\ell)\right] \nonumber \]
We combine equations (b) and (e).
\[\ln \left(\mathrm{y}_{\mathrm{j}}\right)=\ln \left(\gamma_{\mathrm{j}}\right)+\ln \left(\mathrm{m}_{\mathrm{j}} \, \mathrm{c}_{\mathrm{r}} / \mathrm{m}^{0} \, \mathrm{c}_{\mathrm{j}}\right)-\ln \left[\mathrm{c}_{\mathrm{r}} / \mathrm{m}^{0} \, \rho_{1}^{*}(\ell)\right] \nonumber \]
\[\ln \left(\mathrm{y}_{\mathrm{j}}\right)=\ln \left(\gamma_{\mathrm{j}}\right)+\ln \left(\mathrm{m}_{\mathrm{j}} \, \mathrm{c}_{\mathrm{r}} / \mathrm{m}^{0} \, \mathrm{c}_{\mathrm{j}}\right)-\ln \left[\mathrm{c}_{\mathrm{r}} / \mathrm{m}^{0} \, \rho_{1}^{*}(\ell)\right] \nonumber \]
Footnotes
[1] A given solution is prepared by adding \(\mathrm{n}_{j}\) moles of solute \(j\) to \(\mathrm{w}_{1} \mathrm{~kg}\) of solvent.
Molality of solute \(\mathrm{j} / \mathrm{mol} \mathrm{kg}{ }^{-1}=\left(\mathrm{n}_{\mathrm{j}} / \mathrm{w}_{1}\right)\)
Total mass of solution/kg \(=w_{1}+n_{j} \, M_{j}\) where molar mass of solute/kg \(\mathrm{mol}^{-1}=\mathrm{M}_{\mathrm{j}}\)
Volume of solution/\(\mathrm{m}^{3} = \mathrm{V}\)
Density of solution \(\rho / \mathrm{kg} \mathrm{m}^{-3}=\left[\frac{\mathrm{w}_{1}+\mathrm{n}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}}}{\mathrm{V}}\right]\)
By convention chemists express the composition of solutions in terms of (i) concentration using the unit ‘\(\mathrm{mol dm}^{-3}\)’ and (ii) molality using the unit, ‘\(\mathrm{mol kg}^{-1}\)’. These composition scales stem from the fact that at \(298.15 \mathrm{~K}\), \(1 \mathrm{~dm}^{3}\) of water has a mass of approx. \(1 \mathrm{~kg}\). So as we swap composition scales a conversion factor is often required .
For dilute solutions \(w_{1}>n_{j} \, M_{j}\) and density of solution \(\rho\) equals the density of the pure solvent (at same temperature and pressure), i.e. density \(\rho=\rho 1(\ell) \mathrm{kg} \mathrm{m} \mathrm{m}^{-3}\)
[2] A typical conversion takes the following form for water at \(298.2 \mathrm{~K}\) and ambient pressure.
\(\begin{aligned}
\text { Density }=0.997 \mathrm{~g} \mathrm{~cm}^{-3} &=0.997\left(10^{-3} \mathrm{~kg}\right)\left(10^{-2} \mathrm{~m}^{-3}\right.\\
&=0.997 \mathrm{X} \mathrm{} 10^{3} \mathrm{~kg} \mathrm{~m}^{-3} \\
=& 997 \mathrm{~kg} \mathrm{~m}^{-3}=0.997 \mathrm{~kg} \mathrm{\textrm {dm } ^ { - 3 }}
\end{aligned}\)
\(\text { Then } \frac{\mathrm{c}_{\mathrm{j}} / \mathrm{mol} \mathrm{dm}^{-3}}{\mathrm{~m}_{\mathrm{j}} / \mathrm{mol} \mathrm{kg}^{-1}}=\frac{\mathrm{n}_{\mathrm{j}} / \mathrm{mol}}{\mathrm{V} / \mathrm{dm}^{3}} \, \frac{\mathrm{w}_{1} / \mathrm{kg}}{\mathrm{n}_{\mathrm{j}} / \mathrm{mol}}=\frac{\mathrm{w}_{1} / \mathrm{kg}}{\mathrm{V} / \mathrm{dm}^{3}}=\rho / \mathrm{kg} \mathrm{dm}^{-3}\)
[3] \(\begin{aligned}
\ln \left(\mathrm{m}_{\mathrm{j}} \, \mathrm{c}_{\mathrm{r}} / \mathrm{m}^{0} \, \mathrm{c}_{\mathrm{j}}\right) &=\ln \left[\left(\mathrm{c}_{\mathrm{r}} / \mathrm{m}^{0}\right) /\left(\mathrm{c}_{\mathrm{j}} / \mathrm{m}_{\mathrm{j}}\right)\right] \\
=& \ln \left[\left(\mathrm{kg} \mathrm{d \textrm {m } ^ { - 3 } ) / \rho ]}=-\ln \left(\rho / \mathrm{kg} \mathrm{d \textrm {dm } ^ { - 3 } )}\right.\right.\right.
\end{aligned}\)