1.14.64: Solutions: Solute and Solvent
A given solution (at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\), where the latter is close to the standard pressure) is prepared using \(1 \mathrm{~kg}\) of water(\(\ell\)) and \(\mathrm{m}_{j}\) moles of a simple solute. The Gibbs energy \(\mathrm{G}\left(\mathrm{w}_{1}=1 \mathrm{~kg}\right)\) is given by equation (a).
\[\begin{aligned}
\mathrm{G}\left(\mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \,\left[\mu_{1}^{*}(\ell)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right] \\
&+\mathrm{m}_{\mathrm{j}} \,\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\right]
\end{aligned} \nonumber \]
In the event that the thermodynamic properties of the solution are ideal,
\[\begin{aligned}
\mathrm{G}\left(\mathrm{w}_{1}=1 \mathrm{~kg} ; \mathrm{id}\right)=&\left(1 / \mathrm{M}_{1}\right) \,\left[\mu_{1}^{*}(\ell)-\mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right] \\
&+\mathrm{m}_{\mathrm{j}} \,\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\right]
\end{aligned} \nonumber \]
The excess Gibbs energy \(\mathrm{G}^{\mathrm{E}}\) for the solution prepared using \(1 \mathrm{~kg}\) of water(\(\ell\)) is given by equation (c).
\[\mathrm{G}^{\mathrm{E}}=\mathrm{G}\left(\mathrm{w}_{1}=1 \mathrm{~kg}\right)-\mathrm{G}\left(\mathrm{w}_{1}=1 \mathrm{~kg} ; \mathrm{id}\right) \nonumber \]
Therefore [1]
\[\mathrm{G}^{\mathrm{E}}=\mathrm{R} \, \mathrm{T} \, \mathrm{m}_{\mathrm{j}} \,\left[1-\phi+\ln \left(\gamma_{\mathrm{j}}\right)\right] \nonumber \]
Hence at fixed \(\mathrm{T}\) and \(\mathrm{p}\), the dependence of \(\mathrm{G}^{\mathrm{E}}\) on \(\mathrm{m}_{j}\) is given by equation (e).
\[(1 / \mathrm{R} \, \mathrm{T}) \, \mathrm{dG}^{\mathrm{E}} / \mathrm{dm}_{\mathrm{j}}=1-\phi+\ln \left(\gamma_{\mathrm{j}}\right)-\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \phi / \mathrm{dm}_{\mathrm{j}}+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right) / \mathrm{dm} \mathrm{j}_{\mathrm{j}} \nonumber \]
According to the Gibbs-Duhem equation, the chemicals potentials of solvent \(\mu_{1}(\mathrm{aq})\) and solute \(\mu_{j}(\mathrm{aq})\) are linked. At fixed \(\mathrm{T}\) and \(\mathrm{p}\),
\[\mathrm{n}_{1} \, \mathrm{d} \mu_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{d} \mu_{\mathrm{j}}(\mathrm{aq})=0 \nonumber \]
Then for a solution prepared using \(1 \mathrm{~kg}\) of water(\(\ell\)),
\[\left(1 / M_{1}\right) \, d \mu_{1}(a q)+m_{j} \, d \mu_{j}(a q)=0 \nonumber \]
In terms of the impact of adding \(\mathrm{dm}_{j}\) moles of solute,
\[\left(1 / M_{1}\right) \, d \mu_{1}(a q) / d m_{j}+m_{j} \, d \mu_{j}(a q) / d m_{j}=0 \nonumber \]
The Gibbs-Duhem relation describes moderation of the effects of added \(\mathrm{dm}_{j}\) moles of solute \(j\) on the changes in chemical potentials of solute and solute. We use the equation which relates these chemical potentials to the composition of the solution. For simple solutes (e.g. urea) at ambient pressure, equation (g) takes the following form.
\[\begin{aligned}
&{\left[1 / \mathrm{M}_{1}\right] \, \mathrm{d}\left[\mu_{1}^{*}(\ell)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right]} \\
&\quad+\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right]=0\right.
\end{aligned \nonumber \]
Hence,
\[\mathrm{d}\left[-\phi \, \mathrm{m}_{\mathrm{j}}\right]+\mathrm{m}_{\mathrm{j}} \,\left[\mathrm{d} \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right]=0\right. \nonumber \]
The simple differential equation (j) is important in developing links between the thermodynamic properties of solutions, solvent and solute. The integrated form of this equation is important. From equation (j),
\[\mathrm{d}\left[-\phi \, \mathrm{m}_{\mathrm{j}}\right]+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\mathrm{m}_{\mathrm{j}}\right)+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right)=0 \nonumber \]
Therefore,
\[-\phi \, d m_{j}-m_{j} \, d \phi+d m_{j}+m_{j} \, d \ln \left(\gamma_{j}\right)=0 \nonumber \]
Or,
\[\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right)=(\phi-1) \, \mathrm{dm} \mathrm{m}_{\mathrm{j}}+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \phi \nonumber \]
Or, with a slight re-arrangement,
\[d \ln \left(\gamma_{\mathrm{j}}\right)=\mathrm{d} \phi+(\phi-1) \, \mathrm{d} \ln \left(\mathrm{m}_{\mathrm{j}}\right) \nonumber \]
Hence we obtain an equation for \(\ln \left(\gamma_{j}\right)\) in terms of the dependence of \((\phi - 1)\) on molality of solute bearing in mind that \(\ln \left(\gamma_{j}\right)\) equals zero and \(\phi\) equals 1 at \(\mathrm{m}_{j} = 0\).
\[\ln \left(\gamma_{\mathrm{j}}\right)=(\phi-1)+\int_{\mathrm{o}}^{\mathrm{m}_{\mathrm{j}}}(\phi-1) \, \mathrm{d} \ln \left(\mathrm{m}_{\mathrm{j}}\right) \nonumber \]
In another approach we start again with equation (j).
\[\mathrm{d}\left[\phi \, \mathrm{m}_{\mathrm{j}}\right]=\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left[\ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\right]+\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left[\ln \left(\gamma_{\mathrm{j}}\right)\right] \nonumber \]
Or,
\[\mathrm{d}\left[\phi \, \mathrm{m}_{\mathrm{j}}\right]=\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left[\ln \left(\mathrm{m}_{\mathrm{j}}\right)-\ln \left(\mathrm{m}^{0}\right)\right]+\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left[\ln \left(\gamma_{\mathrm{j}}\right)\right] \nonumber \]
Or,
\[\mathrm{d}\left[\phi \, \mathrm{m}_{\mathrm{j}}\right]=\mathrm{dm}_{\mathrm{j}} \,+\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left[\ln \left(\gamma_{\mathrm{j}}\right)\right] \nonumber \]
Following integration from ‘\(\mathrm{m}_{j} =0\)’ to \(\mathrm{m}_{j}\),
\[\phi \, \mathrm{m}_{\mathrm{j}}=\mathrm{m}_{\mathrm{j}}+\int_{0}^{\mathrm{m}_{\mathrm{j}}} \mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right) \nonumber \]
\[\phi=1+\left(1 / \mathrm{m}_{\mathrm{j}}\right) \, \int_{0}^{\mathrm{m}_{\mathrm{j}}} \mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right) \nonumber \]
\[\phi-1=\left(1 / m_{j}\right) \, \int_{0}^{m_{j}} m_{j} \, d \ln \left(\gamma_{j}\right) \nonumber \]
In other words \((\phi - 1)\) is related to the integral of \(m_{j} \, d \ln \left(\gamma_{j}\right)\) between the limits ‘\(\mathrm{m}_{j} = 0\)’ and \(\mathrm{m}_{j}\). Equation (e) can be re-expressed as an equation of \(\ln \left(\gamma_{\mathrm{j}}\right)\).
\[\ln \left(\gamma_{\mathrm{j}}\right)=-(1-\phi)+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \phi / \mathrm{dm}_{\mathrm{j}}-\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right) / \mathrm{dm}_{\mathrm{j}}+(1 / \mathrm{R} \, \mathrm{T}) \, \mathrm{dG}{ }^{\mathrm{E}} / \mathrm{dm} \mathrm{j}_{\mathrm{j}} \nonumber \]
Hence from equation (r),
\[\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \phi+\phi \, \mathrm{dm} \mathrm{j}_{\mathrm{j}}=\mathrm{dm} \mathrm{m}_{\mathrm{j}}+\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left[\ln \left(\gamma_{\mathrm{j}}\right)\right] \nonumber \]
Or,
\[\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \phi / \mathrm{dm} \mathrm{m}_{\mathrm{j}}+\phi-1-\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right) / \mathrm{dm} \mathrm{m}_{\mathrm{j}}=0 \nonumber \]
Or,
\[-(1-\phi)+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \phi / \mathrm{dm}_{\mathrm{j}}-\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right) / \mathrm{dm} \mathrm{m}_{\mathrm{j}}=0 \nonumber \]
Then with reference to equation (v), [2]
\[\ln \left(\gamma_{\mathrm{j}}\right)=(1 / \mathrm{R} \, \mathrm{T}) \, \mathrm{dG}^{\mathrm{E}} / \mathrm{dm}_{\mathrm{j}} \nonumber \]
Combination of equations (z) and (d) yields an equation for \((1 - \phi)\) in terms of \(\mathrm{G}^{\mathrm{E}}\). Thus
\[\mathrm{G}^{\mathrm{E}} / \mathrm{R} \, \mathrm{T}=\mathrm{m}_{\mathrm{j}} \,(1-\phi)+\left(\mathrm{m}_{\mathrm{j}} / \mathrm{R} \, \mathrm{T}\right) \, \mathrm{dG}^{\mathrm{E}} / \mathrm{dm}_{\mathrm{j}} \nonumber \]
Or [3],
\[(1-\phi)=(1 / \mathrm{R} \, \mathrm{T}) \,\left[\mathrm{G}^{\mathrm{E}} / \mathrm{m}_{\mathrm{j}}-\mathrm{dG}^{\mathrm{E}} / \mathrm{dm}_{\mathrm{j}}\right] \nonumber \]
Or,
\[(1-\phi)=-\left(\mathrm{m}_{\mathrm{j}} / \mathrm{R} \, \mathrm{T}\right) \,\left\{\mathrm{d}\left[\mathrm{G}^{\mathrm{E}} / \mathrm{m}_{\mathrm{j}}\right] / \mathrm{dm}_{\mathrm{j}}\right\} \nonumber \]
Footnotes
[1]
\[\mathrm{G}^{\mathrm{E}}=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}] \,\left[\mathrm{mol} \mathrm{kg}{ }^{-1}\right] \,[1]=\left[\mathrm{J} \mathrm{kg}^{-1}\right] \nonumber \]
[2]
\[\ln \left(\gamma_{\mathrm{j}}\right)=\frac{1}{\left[\mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}]} \, \frac{\left[\mathrm{J} \mathrm{kg}^{-1}\right]}{\left[\mathrm{mol} \mathrm{kg}^{-1}\right]}=[1] \nonumber \]
[3]
\[(1-\phi)=\left[\frac{[1]}{\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}]}\right] \,\left[\frac{\left[\mathrm{J} \mathrm{kg}^{-1}\right]}{\left[\mathrm{mol} \mathrm{kg}^{-1}\right]}\right]=[1] \nonumber \]