# What to use as reference state for free energy curve of a gas?

My goal is to determine the temperature at which xenon would sublimate (if it didn't turn into liquid first). To do this, I set the free energy of solid Xenon equal to the free energy of Xenon gas, $$F_s = F_g$$ where

$$F_s(T) = E_s(T) - T S_s(T)$$ $$F_l(T) = E_l(T) - T S_l(T)$$

or equivalently I could check to see where the two curves intersect. I am expecting a plot that looks similar to:

where the intersection would be the sublimation temperature. To generate the curves I use

$$F_s(T) = 3 R T - 3 R T(1 + ln(\frac{k_b T}{\hbar \omega})$$ which is a standard result from the canonical partition function, and

$$F_g(T) = \frac{3}{2} R T - RT(\frac{5}{2} + ln[\frac{V}{N} (\frac{m k_b T}{2 \pi \hbar^2})^\frac{3}{2}])$$

where the entropy term comes from the sackur-tetrode equation for an ideal gas. Using this, I get a plot that looks like:

which means the two free energy curves will never be equal. I think this is a referencing issue -- both are referenced to be zero at T = 1. It seems like the free energy curve of the gas curve must be offset by some appropriate reference value, but I can't figure out what that value should be. Any help would be greatly appreciated. Textbooks about phase diagrams (i.e Porter and Easterling) state that the free energy curves should be referenced to some standard state, but if both the solid and the gas are referenced to a standard state then both will be offset by the same amount and I'm back to the same problem.