# Order of phase transition: Which free energy to use?

I am unclear about the adjective "first" or "second" used in phase transitions. Take the liquid-gas transition for example. If we vary the volume of the system at constant T, at some point we will have two phases co-existing. The relevant free energy of the complete system (liquid+gas), the Helmholtz free energy $A$, varies continuously such that its second derivative is zero and first derivative is constant. On the other hand between the phases themselves $A$ is discontinuous, but intensities like T, P etc are same. If one studies the variation along a P-T isotherm (vary pressure of the system at constant temperature), the relevant free energy of the total system $G$ is continuous but non-differentiable at the transition. Exactly what is the criterion for calling this "first" order?

I have checked wikipedia and a few online lecture notes, and which free energy derivatives should one consider is not clear to me.

EDIT: I just realized a similar question has been raised a few days ago. Not sure how it slipped my attention. However, in light of that one, my question can be simplified to which free energy do I choose for deciding order? I can study a phase transition using different thermodynamic potentials, the free energy being the one that corresponds to my independent variables.

In the case of the second order transition it can be described by any potential of your choice. The choice then depends upon the thermodynamic variables you need, rather than upon the transition order. However, there is the so-called, theorem about small perturbations. You may find it in Landau and Lifshitz, Theoretical Physics, V. 5 Statistical Mechanics, §15 "The free energy and thermodynamic potential". This theorem states that a small perturbation of a system makes the equal contribution to the any one of the thermodynamic potentials, independent upon this potential. In other words all the peerturbations are equal to one another. The phase transition free energy falls under the condition of this theorem: it is small with respect to the basis free energy. Therefore, it is all the same, what potential to use.

In the case of the first order transition the situation is basically the same, as soon as we discuss the transition into a new phase of a whole body. So you choose the potential whose variables are more suitable for you.

The important difference only arises in the case when a limited portion of the medium transforms into a new phase, while the rest of the body stays in the old one. It is different, since you have simulutaneously the jump of the solid volume and of the number of particles under the first order transition. So you cannot fix the volume and the number of particles simultaneously. In this case it is illegal to use, say, the free energy, whose variables are temperature, volume and the number of particles. You need instead to use the so-called, omega-potential with the variables temperature, volume and the chemical potential. You may find a discussion of this point in the same Landau and Lifshitz book, in §147 "Effective Hamiltonian". arguments become important in the fluctuation theory of phase transitions of even the second order, since a fluctuation is like a local phase transition with the spatial size equal to the correlation radius.

Here it is important to know, however, that in literature people often do not pay attention to the choice of the potential and unrigorously call any such a potential "the free energy". It is no problem unless one starts to calculate the thermodynamc parameters as the derivatives of such a free energy, where it matters which potential to take.