1.10.8: Gibbs Energies- Solutions- Solute-Solute Interactions
In real solutions, solute molecules are not infinitely far apart. With increase in solute concentration, the mean separation of solute molecules decreases [1]. Deviations in the properties of real solutions of neutral solutes from ideal can be understood in term of contact, overlap and interaction between cospheres of solvent surrounding solute molecules [2]. Two limiting cases can be identified. In one case overlap occurs between cospheres for which the organisation of solvent molecules are compatible, leading to attractive interaction between two solute molecules; a stabilising effect. In the opposite case the organisation of solvent in the cospheres is incompatible leading to repulsion between the solute molecules; i.e a destabilising effect. These ideas can be formulated quantitatively leading to an understanding of the factors controlling the properties of solutes in aqueous solution.
A given solution is prepared using \(1 \mathrm{~kg}\) of water(\(\ell\)) and \(\mathrm{m}_{j}\) moles of solute \(j\). The chemical potential of the solvent water is related to \(\mathrm{m}_{j}\) using equation (a).
\[\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\ell)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1} \nonumber \]
Here \(\mu_{1}^{*}(\ell)\) is the chemical potential of solvent water at the same \(\mathrm{T}\) and \(\mathrm{p}\); \(\phi\) is the practical osmotic coefficient which is unity for a solution having thermodynamic properties which are ideal; \(\mathrm{M}_{1}\) is the molar mass of water. The difference \(\left[\mu_{1}(\mathrm{aq})-\mu_{1}^{*}(\ell)\right]\) equals \(\left[-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}\right]\). Hence for an ideal solution where \(\phi\) is unity addition of a solute lowers the chemical potential of the solvent; i.e. stabilises the solvent.
The chemical potential of the solute \(\mu_{\mathrm{j}}(\mathrm{aq})\) is related to the molality \(\mathrm{m}_{j}\) using equation (b).
\[\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right) \nonumber \]
We rewrite equation (b) in an extended form.
\[\mu_{j}(a q)=\mu_{j}^{0}(a q)+R \, T \, \ln \left(m_{j} / m^{0}\right)+R \, T \, \ln \left(\gamma_{j}\right) \nonumber \]
Or,
\[\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) \nonumber \]
Hence \(\gamma_{j}\) measures the extent to which the chemical potential of solute \(j\) in the real solution differs from that in an ideal solution. If \(\gamma_{j} > 1\) [and hence \(\ln \left(\gamma_{j}\right)>0\)] \(\mu_{j}(\mathrm{aq})<\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id})\), solute-solute interactions destabilise the solute. If \(\gamma_{\mathrm{j}}<1\) [and hence \(\ln \left(\gamma_{\mathrm{j}}\right)<0\)] \(\mu_{\mathrm{j}}(\mathrm{aq})>\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id})\), and so these interaction stabilise the solute. [\(\mathrm{NB} \gamma_{j}\) cannot be negative.]
The chemical potentials of solute and solvent are linked by the Gibbs – Duhem equation which for aqueous solutions (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)) containing \(1 \mathrm{~kg}\) of water takes the following form.
\[\left(1 / M_{1}\right) \, d \mu_{1}(a q)+m_{j} \, d \mu_{j}(a q)=0 \nonumber \]
Then,
\[\begin{aligned}
\left(1 / \mathrm{M}_{1}\right) \, \mathrm{d}\left[\mu_{1}^{*}(\ell)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}\right] \\
&+\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\right]=0
\end{aligned} \nonumber \]
Or,
\[-\mathrm{d}\left[\phi \, \mathrm{m}_{\mathrm{j}}\right]+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)=0 \nonumber \]
The latter equation links changes in the osmotic coefficient \(\phi\) and \(\gamma_{j}\). In other words, a perturbation which affects the solvent feeds back on to the properties the solute. This is Gibbs-Duhem communication. In the present context equation (g) explains why interaction between cospheres feeds back to the properties of the solute. Consequently the Gibbs-Duhem equation is used to switch between equations describing \(\phi\) and \(\gamma_{j}\). Thus from equation (g),[3]
\[\mathrm{d}\left[\mathrm{m}_{\mathrm{j}} \,(1-\phi)\right]+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right)=0 \nonumber \]
Equation (h) can be written in two forms depending which direction we wish to proceed. Thus from equation (h)[4],
\[(1-\phi)=-\left(1 / m_{j}\right) \, \int_{m(j)=0}^{m(j)} m_{j} \, d \ln \left(\gamma_{j}\right) \nonumber \]
In other words we have an equation for \(\phi\) and in terms of \(\gamma_{j}\). Alternatively [5] we can express \(\gamma_{j}\) in terms of \(\phi\) and its dependence on \(\mathrm{m}_{j}\).
\[\left.\int_{m(j)=0}^{m(j)} d \ln \left(\gamma_{j}\right)=\int_{m(j)=0}^{m(j)} d(1-\phi)+\int\left[(1-\phi) / m_{j}\right)\right] \, d m_{j} \nonumber \]
Then
\[\ln \left(\gamma_{\mathrm{j}}\right)=(\phi-1)+\int_{\mathrm{m}(\mathrm{j})=0}^{\mathrm{m}(\mathrm{j})}(\phi-1) \, \mathrm{d} \ln \left(\mathrm{m}_{\mathrm{j}}\right) \nonumber \]
Hence \(\gamma_{j}\) can be calculated from knowing \(\phi\) and its dependence on dependence on \(\mathrm{m}_{j}\).
Footnotes
[1] R. A. Robinson and R. H. Stokes, Electrolyte Solutions, Butterworths, London, 2nd. edn.,1959, chapter 1.
[2] J. J. Kozak, W. S. Knight and W. Kauzmann, J. Chem. Physics,1968, 48 ,675.
[3] \(\begin{aligned}
&-\phi \, d m_{j}-m_{j} \, d \phi+m_{j} \, d m_{j} / m_{j}+m_{j} \, \ln \left(\gamma_{j}\right)=0\\
&\text { Or, } \mathrm{dm}_{\mathrm{j}}-\phi \, d \mathrm{~m}_{\mathrm{j}}-\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \phi+\mathrm{m}_{\mathrm{j}} \, \ln \left(\gamma_{\mathrm{j}}\right)=0
\end{aligned}\)
[4] \(\begin{array}{r}
\int_{\mathrm{m}(\mathrm{j} j=0}^{\mathrm{m}(\mathrm{j})} \mathrm{d}\left[\mathrm{m}_{\mathrm{j}} \,(1-\phi)\right]=-\int_{\mathrm{m}(\mathrm{j})=0}^{\mathrm{m}(\mathrm{j})} \mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right) \\
\mathrm{m}_{\mathrm{j}} \,(1-\phi)=-\int_{\mathrm{m}(\mathrm{j})=0}^{\mathrm{m}(\mathrm{j})} \mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right)
\end{array}\)
[5] From equation (h), \(\mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right)=\left(1 / \mathrm{m}_{\mathrm{j}}\right) \, \mathrm{d}\left[\mathrm{m}_{\mathrm{j}} \,(\phi-1)\right]\)
Or,
\[\left.\int_{m(j)=0}^{m(j)} d \ln \left(\gamma_{j}\right)=\int_{m(j)=0}^{m(j)} d(\phi-1)+\int\left[(\phi-1) / m_{j}\right)\right] \, d m_{j} \nonumber \]