> I consider fugacity to be the same as an activity coefficient, observed deviation of expected internal chemical potential from an ideal solution or gas.
If I am remembering correctly, the assumption that fugacity is equal in the three phases allows you to assume that the phase changes are solely due to increased internal kinetic energy, rather than vibratory, rotational, or electronic excitation. This lets you solve for the work of the system let's you assume equilibrium and makes the math easier to avoid accounting for disproportionation of the phase states, I believe