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## Coexistence in phase diagram

In a nutshell, I am trying to see how to derive the conditions of coexistence between a solid and a liquid from a phase diagram.

The situation is as follows:
Consider a mixture of hard spheres of diameter σ. The potential energy
for a hard sphere system is given by
[itex]\beta U(r) = 0 (r > \sigma)
[/itex]
[itex]
∞ (r ≤ \sigma)
[/itex]
The packing fraction (η) of the system is the amount of space occupied
by the particles.
The equation of state for the hard sphere fluid is approximately
[itex]
\frac{P_{liq}V}{Nk_{B}T}= \frac{1+ \eta + \eta^2 - \eta^3}{ (1 - \eta)^3 }
[/itex]

Another similar equation is given for $P_{sol}$, the pressure in the solid state.

By integration I managed to calculated the free energy as a function of the packing density, using given boundary conditions. This resulted in the following diagram:

Here the free energy is plotted against the packing density. The red line corresponds to the solid phase and the blue line to the liquid phase.

In general I would think one could calculate the minima of the free energy and then draw a common tangent line between them, but in this case there doesn't seem to be any minima. Also, it appears that the free energy of the solid phase is lower than that of the liquid phase even for low densities. Is this even possible? Or should I conclude that my plots are incorrect?

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Recognitions:      Homework  Help     Science  Advisor  Are you sure those equations are meaningful in the full [0,1] range? Especially: How do you get a packing density η > 0.741? (I think it is not an accident that the spike of the solid line is there).  There are materials which do not have a liquid phase at a specific temperature, independent of the packing density.

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