1.10.5: Gibbs Energies- Raoult's Law

We consider a closed system containing a (homogeneous) mixture of two volatile liquids. The closed system is connected to a pressure measuring device which records that at temperature \(\mathrm{T}\) the pressure inside the closed system is \(\mathrm{p}(\text{tot})\). The composition of the liquid mixture is assayed; the mole fractions of the two components of the liquid are \(\mathrm{x}_{1}\) and \(\mathrm{x}_{2}\) (where \(\mathrm{x}_{2} = 1 – \mathrm{x}_{1}\)). Thus the system contains two components so that in terms of the Phase Rule, \(\mathrm{C} = 2\). There are two phases, vapour and liquid, so \(\mathrm{P}\) equals \(2\). Thus in terms of the Rule, \(\mathrm{P} + \mathrm{~F} = \mathrm{~C} + 2\), we have fixed the composition and the temperature using up the two degrees of freedom. Hence the pressure \(\mathrm{p}(\text{tot})\) is fixed.

The foundation of thermodynamics is experiment. So, in considering the properties of water in dilute aqueous solutions, we take account of the observation that the equilibrium vapour pressure \(\mathrm{p}_{1}^{\mathrm{eq}}(\mathrm{aq})\) of water in equilibrium with water in an aqueous solution (at fixed temperature) is approximately a linear function of the mole fraction of water in the solution; equation (a).

\[\mathrm{p}_{1}^{\mathrm{eq}}(\mathrm{aq}) \cong \mathrm{p}_{1}^{*}(\ell) \, \mathrm{x}_{1}\]

Thus as mole fraction \(\mathrm{x}_{1}\) approaches unity (the composition of the solution approaches pure water where \(\mathrm{x}_{1}\) is unity), the equilibrium vapour pressure \(\mathrm{p}_{1}^{\mathrm{eq}}(\mathrm{aq})\) approaches the vapour pressure of pure liquid water \(\mathrm{p}_{1}^{*}(\ell)\) at the same temperature. At this stage we introduce the concept of an ideal solution. We assert that for an ideal solution the approximation (a) is an equation. Thus

\[\mathrm{p}_{1}^{\mathrm{eq}}(\mathrm{aq} ; \mathrm{id})=\mathrm{p}_{1}^{*}(\ell) \, \mathrm{x}_{1}\]

In other words, \(\mathrm{p}_{1}^{\mathrm{eq}}(\mathrm{aq} ; \mathrm{id})\) is a linear function of mole fraction composition of the solution [1]. We have linked the (equilibrium) vapour pressure of the solvent to the composition of the solution. Returning to the results of experiments, we invariably find that as real solution becomes more dilute (i.e. as \(\mathrm{x}_{1}\) approaches unity) \(\mathrm{p}_{1}^{\mathrm{eq}}(\mathrm{aq})\) for real solutions approaches \(\mathrm{p}_{1}^{\mathrm{eq}}(\mathrm{aq} ; \mathrm{id})\) for the corresponding ideal solution at the same temperature. Therefore, we rewrite equation (b) as an equation for real solutions by introducing a new quantity called the (rational) activity coefficient, \(\mathrm{f}_{1}\). Then,

\[\mathrm{p}_{1}^{\mathrm{eq}}(\mathrm{aq}) \cong \mathrm{p}_{1}^{*}(\ell) \, \mathrm{x}_{1} \, \mathrm{f}_{1}\]

where, by definition,

\[\operatorname{limit}\left(\mathrm{x}_{1} \rightarrow 0\right) \mathrm{f}_{1}=1.0\]

Although equations (c) and (d) have simple forms, rational activity coefficients carry a heavy load in terms of information. Thus for a given solution \(\mathrm{f}_{1}\) describes the extent to which interactions involving solvent water in the real solution differ from those in the corresponding ideal solution. The challenge of expressing this information in molecular terms is formidable.

Footnotes

[1] Note that, \(\mathrm{p}_{1}^{\mathrm{eq}}(\mathrm{aq} ; \mathrm{id})-\mathrm{p}_{1}^{*}(\ell)<0\). Thus adding a solute lowers the vapour pressure of the solvent. However the total vapour pressure of a binary liquid mixture can be either increased or decreased by adding a small amount of solute, the change being characteristic of the solute; G. Bertrand and C. Treiner, J. Solution Chem.,1984,13, 43.