1.5.15: Chemical Potentials- Solute- Concentration and Molality Scales
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] \(\begin{aligned}
\ln \left(\mathrm{m}_{\mathrm{j}} \, \mathrm{c}_{\mathrm{r}} / \mathrm{m}^{0} \, \mathrm{c}_{\mathrm{j}}\right) &=\ln \left[\left(\mathrm{c}_{\mathrm{r}} / \mathrm{m}^{0}\right) /\left(\mathrm{c}_{\mathrm{j}} / \mathrm{m}_{\mathrm{j}}\right)\right] \\
=& \ln \left[\left(\mathrm{kg} \mathrm{d \textrm {m } ^ { - 3 } ) / \rho ]}=-\ln \left(\rho / \mathrm{kg} \mathrm{d \textrm {dm } ^ { - 3 } )}\right.\right.\right.
\end{aligned}\)