Vapour-Liquid-Liquid Equlibria

For a mixture of components, the vapour phase fraction, $y_i$, is linked to the liquid phase fraction, $x_i$ by the non-ideal Raoult’s law, $$y_iP=x_i\gamma_ip_i^o$$ where $\gamma_i$ is the liquid-phase activity coefficient which can be given by many models. If the system is non-ideal enough, then there is also a liquid-liquid phase separation, in this case this equilibrium for this is given by, $$x_i^1\gamma_i^1= x_i^2\gamma_i^2$$ In the case of regular solution theory the liquid-phase activity coefficients can be calculated, for a two-component system, from the solubility parameters of the components as, $$\ln\gamma_1=\frac{v_1}{RT}\left(\frac{x_2v_2}{x_1v_1+x_2v_2}\right)^2(\delta_1-\delta_2)^2$$ $$\ln\gamma_2=\frac{v_2}{RT}\left(\frac{x_1v_1}{x_1v_1+x_2v_2}\right)^2(\delta_1-\delta_2)^2$$ where $\delta_i$ is the solubility parameter and $v_i$ is the molar volume. Although the prediction from regular solution theory is very useful, as only the pure component solubility parameters are needed (as opposed to experimentally fitted values for the mixture), it lacks in the fact that it always predicts activity coefficients higher than unity. This means that the properties for mixtures can only be predicted if they are non-ideal in the fact that when they mix they have positive excess free energy of mixing (i.e. they dislike mixing with each other).

The graph below shows the $x$-$y$ diagram (left) and the $T$-$x$-$y$ diagram (right) for a non-ideal system. The effect of the difference in the solubility parameters on the vapour-liquid-liquid can be explored (when $\delta_1=\delta_2$ the mixture is ideal).