1.1.11: Activity Coefficients
In descriptions of the properties of the components of liquid mixtures, rational activity coefficients are given the symbols \(\mathrm{f}_{1}, \mathrm{f}_{2}, \mathrm{f}_{3} \ldots\). With reference to solvents their thermodynamic properties are described by rational activity coefficients \(\mathrm{f}_{1}, \mathrm{f}_{2}, \mathrm{f}_{3} \ldots\) and (practical) osmotic coefficient, \(\phi\). The properties of solutes in solutions are described using activity coefficients which are linked to the descriptions of the composition of solutions: molality scale, \(\gamma_{j}\); concentration scale \(\gamma_{j}\); mole fraction scale \(\mathrm{f}_{\mathrm{j}}^{*}\).
These coefficients are intimately related to the concept of activity . Their significance is clarified by equations relating chemical potentials to the composition of a given system; e.g.
- components in liquid mixtures,
- solvents in solutions and
- solutes in solutions.
In all cases they summarise the extent to which the thermodynamic properties of liquid mixtures and solutions are not ideal. The challenge for chemists is to understand in terms of molecular properties why the thermodynamic properties of real systems are not ideal. It has to be admitted that activity coefficients have a ‘bad press’ as far as most chemists are concerned. All too often their importance is ignored. But ‘learn to love activity coefficients! Perhaps the importance of activity coefficients can be understood in the following terms.
The chemical potential of urea as a solute in aqueous solutions, \(\mu_{j}(\mathrm{aq})\) at temperature \(\mathrm{T}\) and pressure \(p\) ( \(\approx\) the standard pressure \(p^{0}\)) is related to the molality of urea \(\mathrm{m}_{j}\) using equation (a).
\[\begin{aligned}
&\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T}, \mathrm{p})=\\
&\mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{T}, \mathrm{p}, \mathrm{m}_{\mathrm{j}}=1 \mathrm{~mol} \mathrm{~kg} \mathrm{~kg}^{-1}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\gamma_{\mathrm{j}}\right)
\end{aligned} \nonumber \]
The term \(\mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{T}, \mathrm{p}, \mathrm{m}_{\mathrm{j}}=1 \mathrm{~mol} \mathrm{\textrm {kg } ^ { - 1 } )}\right.\) describes the chemical potential of solute, urea in an aqueous solution having unit molality where there are no urea - urea (i.e. solute – solute) interactions. Each urea molecule in these terms is unaware of the presence of other urea molecules in the aqueous solution. In a solution having ideal thermodynamic properties \(\mu_{j}(\mathrm{aq} ; \mathrm{T}, \mathrm{p} ; \mathrm{id})\) depends on the molality of solute mj. Thus the osmotic pressure of this solution is a function of the molality of urea. For such an ideal solution there are no urea-urea interactions although there are important urea-water interactions, the hydration of urea. But in a real solution there are also solute-solute interactions. Each solute molecule ‘knows’ there are other solute molecules in the solution. Indeed the extent to which \(\mu_{j}(\mathrm{aq} ; \mathrm{T}, \mathrm{p})\) differs from \(\mu_{j}(\mathrm{aq} ; \mathrm{T}, \mathrm{p} ; \mathrm{id})\) is a function of the molality of the solute, \(\mathrm{m}_{j}\).