# Understanding liquid-liquid phase separation

Physics Asked on January 6, 2021

The above figure shows how the regular-solution free energy might lead to liquid–liquid equilibrium. In the region between the compositions $$x^I_A$$ and $$x^{II}_A$$, $$G$$ increases with $$x_A$$. If the overall concentration lies between these two values, it is always favorable for the
system to spontaneously phase-separate into two regions, one with composition $$x^I _A$$ and
another with composition $$x^{II} _A$$. The common tangent line gives the free energy of the phase-separated state.

Okay, so if I have solution with composition between $$x^I_A$$ and $$x^{II}_A$$, it will phase separate into whichever potential well they are closes to. That makes sense.

The above graph is just for illustration purposes. It is not directly related to the problem I have.

My question is, if I have an expression for the excess Gibbs energy of the Margules kind, given by,
$$G^{ex} = RT(c x_1 + kc x_2)x_1x_2$$
how do I find the region where the solution will spontaneously phase separate?
I know $$G = G^{ideal}+G^{ex}$$, so $$G = mu_1 + mu_2 + RTln x_1 + RTln x_2 + G^{ex}$$.

I believe that solution will most definitely spontaneously phase out when
$$frac{partial ^2 G}{partial x_1^2} < 0$$
I can solve for values of $$c$$ for which the solution separates.

If, however, $$c$$ is dependent on temperature, such as $$c=wT$$, how would I find the critical composition and temperature? Would I still impose
$$frac{partial ^2 G}{partial x_1^2} < 0?$$
But that is only one constraint I have, but I need to solve for $$T_c$$ and $$x_c$$.

How do I go ahead with this problem?