Temperature of a phase transition A solid can exist in two phases, with energies
$$U_{1}(S,V)=\frac{S^2}{a_1}+b_{1}V(V-2V_{0})$$
$$U_{2}(S,V)=\frac{S^2}{a_2}+b_{2}V(V-2V_{0})$$
where $a_{1},a_{2},b_{1},b_{2},V_{0}$ are positive constants, with $a_{1}>a_{2}$ and $b_{1}<b_2$.
Consider a phase transition between the two phases at pressure $P = 0$. 
At what temperature, $T_0$, does this occur?
I calculated the enthalpy, Helmholtz free energy, and Gibbs free energy for each side, I'm just having trouble figuring out what I need to do with them in order to calculate the temperature a phase transition would occur.  Does someone know what I need to do with those values in order to get $T_{0}$?
 A: Given a phase transition $\phi$ between two phase $\alpha$ and $\beta$:
$$ X_{(\alpha)} \rightleftharpoons X_{(\beta)}$$
When such transformation occurs at a constant Temperature $T_\phi$ there is an equilibrium between the two phases, thus you can write the following equality:
$$ \Delta_\mathrm{\phi}G = \Delta_\mathrm{\phi}H - T_\phi\Delta_\mathrm{\phi}S = 0 $$
You must then determine transition quantities $\Delta_\mathrm{\phi}H$ and $\Delta_\mathrm{\phi}S$ and isolate $T_\phi$, to find out the transition temperature.
Hint: the operator $\Delta_\phi\{\}$ is the Lewis operator, defined as follow:
$$\Delta_\phi X = \left(\frac{\partial X}{\partial \xi} \right)_{p,T} = \sum\limits_i \nu_i x_i$$
Where $\xi$ is the molecular coordinate of the transition, $\nu_i$ the stoichiometric coefficient of the i-th physic-chemical substance and $x_i$ the molar quantity (intensive) related to $X_i$ (extensive, eg. $H_\alpha$).
Furthermore, it is always useful to determine $\Delta_\phi V$ to determine whether boundary work is produced or absorbed. In case of solid, this should be small but not null.
You will find detailed explaination of this in: Atkins, Physical Chemistry, Chapter 4: Physcial Transformation of pure substance.
