1.12.9: Expansions- Apparent Molar Isobaric- Composition Dependence
For many aqueous solutions at ambient temperature and pressure the dependence of apparent molar isobaric expansions for solute \(j\) \(\phi\left(E_{p j}\right)\) on molality \(\mathrm{m}_{j}\) is accounted for using an equation having the following general form. [The reason for choosing the molality scale is that \(\mathrm{m}_{j}\) is independent of \(\mathrm{T}\) and \(\mathrm{p}\) whereas concentration \(\mathrm{c}_{j}\) is not.
\[\phi\left(E_{p j}\right)=a_{1}+a_{2} \,\left(m_{j} / m^{0}\right)+a_{3} \,\left(m_{j} / m^{0}\right)^{2} \ldots . . \nonumber \]
At low solute molalities the linear term is dominant. Granted therefore that equation (a) accounts for the observed pattern, we need to explore the analysis a little further. There are advantages in linking \(\phi\left(E_{p j}\right)\) and the partial molar property \(\mathrm{E}_{p j}(\mathrm{aq})\).
For an aqueous solution prepared using \(1 \mathrm{~kg}\) of water and \(\mathrm{m}_{j}\) moles of solute \(j\) at fixed \(\mathrm{T}\) and \(\mathrm{p}\),
\[\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \nonumber \]
Hence,
\[\mathrm{E}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{E}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right) \nonumber \]
\[\left(\frac{\partial \mathrm{E}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)}{\partial \mathrm{m}_{\mathrm{j}}}\right)=\mathrm{m}_{\mathrm{j}} \,\left(\frac{\partial \phi\left(\mathrm{E}_{\mathrm{pj}}\right)}{\partial \mathrm{m}_{\mathrm{j}}}\right)+\phi\left(\mathrm{E}_{\mathrm{pj}}\right) \nonumber \]
But
\[\left(\frac{\partial \mathrm{E}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right.}{\partial \mathrm{m}_{\mathrm{j}}}\right)=\mathrm{E}_{\mathrm{pj}}(\mathrm{aq}) \nonumber \]
\[\mathrm{E}_{\mathrm{pj}}(\mathrm{aq})=\mathrm{m}_{\mathrm{j}} \,\left(\frac{\partial \phi\left(\mathrm{E}_{\mathrm{pj}}\right)}{\partial \mathrm{m}_{\mathrm{j}}}\right)+\phi\left(\mathrm{E}_{\mathrm{pj}}\right) \nonumber \]
Hence the partial molar isobaric expansions for solute \(j\) can be calculated using the apparent molar isobaric expansions and its dependence on molality. Further if equation (a) accounts for the dependence of \(\phi\left(\mathrm{E}_{\mathrm{pj}}\right)\) on \(\mathrm{m}_{j}\), then
\[\mathrm{E}_{\mathrm{pj}}=\mathrm{a}_{1}+2 \, \mathrm{a}_{2} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+3 \, \mathrm{a}_{3} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2} \ldots \ldots \nonumber \]
Therefore, using equations (a) and (g),
\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi\left(\mathrm{E}_{\mathrm{pj}}\right)=\phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty}=\mathrm{E}_{\mathrm{pj}}^{\infty}(\mathrm{aq}) \nonumber \]
In the next stage of the analysis we develop an argument starting with an equation for the chemical potential of solute \(j\) in solution.
\[\mu_{j}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)+\int_{\mathrm{p}^{0}}^{p} \mathrm{~V}_{\mathrm{j}}^{\infty}(\mathrm{aq} ; \mathrm{T}) \, \mathrm{dp} \nonumber \]
Then with \(\mathrm{V}_{\mathrm{j}}(\mathrm{aq})=\left(\frac{\partial \mu_{\mathrm{j}}(\mathrm{aq})}{\partial \mathrm{p}}\right)_{\mathrm{T}}\),
\[\mathrm{V}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{p}\right]_{\mathrm{T}} \nonumber \]
With \(\mathrm{E}_{\mathrm{pj}}(\mathrm{aq})=\left(\frac{\partial \mathrm{V}_{\mathrm{j}}(\mathrm{aq})}{\partial \mathrm{T}}\right)_{\mathrm{p}}\),
\[\mathrm{E}_{\mathrm{pj}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mathrm{E}_{\mathrm{pj}}^{\infty}(\mathrm{aq})+\mathrm{R} \,\left(\frac{\partial \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right)_{\mathrm{T}}+\mathrm{R} \, \mathrm{T} \,\left(\frac{\partial^{2} \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{p} \, \partial \mathrm{T}}\right) \nonumber \]
In the case of dilute solutions we might assert that \(\ln \left(\gamma_{\mathrm{j}}\right)\) is a linear function of molality \(\mathrm{m}_{j}\). Thus[1],
\[\ln \left(\gamma_{\mathrm{j}}\right)=\mathrm{S}_{\gamma} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) \nonumber \]
By definition [2], \(\mathrm{S}_{\mathrm{V}}=\left(\frac{\partial \mathrm{S}_{\gamma}}{\partial \mathrm{p}}\right)_{\mathrm{T}}\) and[3] \(\mathrm{S}_{\mathrm{Ep}}=\mathrm{S}_{\mathrm{V}}+\mathrm{T} \,\left(\frac{\partial \mathrm{S}_{\mathrm{V}}}{\partial \mathrm{T}}\right)_{\mathrm{p}}\) Then[4]
\[\mathrm{E}_{\mathrm{pj}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mathrm{E}_{\mathrm{pj}}^{\infty}(\mathrm{aq})+\mathrm{R} \, \mathrm{S}_{\mathrm{Ep}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) \nonumber \]
Thus we identify the basis of the parameter a2 in equation (a).
Footnotes
[1] \(\ln \left(\gamma_{\mathrm{j}}\right)=[1] \,\left[\mathrm{mol} \mathrm{kg}^{-1}\right] \,\left[\mathrm{mol} \mathrm{kg}{ }^{-1}\right]^{-1}\)
[2] \(\mathrm{S}_{\mathrm{V}}=[1] /\left[\mathrm{N} \mathrm{m}^{-2}\right]=\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1}\)
[3] \(\mathrm{S}_{\mathrm{Ep}}=\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1}+[\mathrm{K}] \,\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1} \,\left[\mathrm{K}^{-1}=\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1}\right.\)
[4]
\[\begin{aligned}
\mathrm{E}_{\mathrm{pj}}(\mathrm{aq}) &=\mathrm{E}_{\mathrm{pj}}^{\infty}(\mathrm{aq})+\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1} \,\left[\mathrm{mol} \mathrm{kg}^{-1}\right] \,\left[\mathrm{mol} \mathrm{kg}^{-1}\right]^{-1} \\
&=\mathrm{E}_{\mathrm{pj}}^{\infty}(\mathrm{aq})+\left[\mathrm{N} \mathrm{m} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1} \\
&=\mathrm{E}_{\mathrm{pj}}^{\infty}(\mathrm{aq})+\left[\mathrm{m}^{3} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right]
\end{aligned} \nonumber \]