Multi-component mixture phase equilibrium criteria

Question

For a pure species, the equlibrium between liquid phase and vapour phase is given by the equality of molar Gibbs energy in both phase:

$$\underline{G}^l=\underline{G}^v$$

Where $\underline{G}$ with an underline represents molar Gibbs energy.

In a multicomponent system, the equilibrium criterion is the equality of the partial molar Gibbs energy (aka chemical potential) for each component between each phases:

$$\overline{G}^l_i=\overline{G}^v_i$$

Now, because of the relation between molar quantities and partial molar quantities, $\underline{G}=\Sigma x_i\overline{G}_i$, the second criterion contains the first by setting $x_i=1$.

My question is : is the first criterion still valid between phases? If yes this would seem to imply that $x_i^l=x_i^v$, which is clearly wrong.

Answer 1

Your first criterion $ \underline{G}^l=\underline{G}^v $ does not apply for a mixture, though possibly not for the reasons you think.

With a pure material "a mole" has an unambiguous meaning, but for a mixture you have to choose what you're taking to be a mole. You are taking a weighted average of the components present, but unless the composition of the liquid and vapour phases are the same (in general they won't be) "a mole" of the liquid is not the same as "a mole" of the vapour. That's why your equation doesn't apply.

The only conditions under which "a mole" would be the same for the liquid and vapour phases is if the compositions of the liquid and vapour phases were the same, i.e. $x_i^l=x_i^v$. That's why you've ended up concluding that your criterion could only apply if $x_i^l=x_i^v$, and as you say that's usually wrong.