We are all familiar with phase transitions where a substance transforms from one stable phase to another at an equilibrium temperature. For example, ice will be in equilibrium with liquid water at 273.15 K and 1 atmosphere pressure, or liquid water will be in equilibrium with water vapor at 373.15 K and one atmosphere. We will now take a look at the thermodynamic principles involved in phase transitions.

I will assume at the beginning that we know nothing about the conditions under which two phases can be in equilibrium. The only things we know are the criteria for equilibrium under certain conditions. That is, we know that

                (1)

                (2)

                (3)

We will do a series of three thought experiments under different sets of conditions and use the above criteria to tell us things about the temperature, pressure, and chemical potential of a system of two phases in equilibrium.

I. A closed isolated system.

Consider a closed isolated system consisting of two phases, phase α in equilibrium with phase β . We will call the temperature of the α phase T_αand the temperature of the β phase T_β.

               (4)

               (5a,b)

               (6a, b)

                    (7)

               (8)

               (9)

If the system is at equilibrium then  There is only one temperature defined and there in no need to distinguish between the temperatures of the two phases.

II. A system at constant volume and temperature

Since we now know that at equilibrium there is only one temperature defined for the two phases let us now remove the system from isolation and place it in a heat bath at temperature, T. We will still hold the volume constant. Under conditions of constant temperature and volume we know that the criterion for equilibrium is that the Helmholtz free energy seeks a minimum. That is,

Our system now looks like,

Let us now transfer a small amount of volume, dV > 0, from phase α to phase β .

We know that dA = − SdT − pdV = − pdV for a constant temperature system. So,

               (10)

               (11)

               (12)

If the system is at equilibrium then  so that there is only one pressure defined for the system.

III. The system at constant pressure and temperature

Since we now know that at equilibrium both phases are at the same temperature and pressure, let us remove the constant volume restriction and place our system in a heat bath at temperature, T, and a pressure bath at pressure, p. (The atmosphere is a good example of a pressure bath at approximately one atmosphere pressure. You can make chambers to hold different pressures if you wish.) Our system looks like,

At constant temperature and pressure we know that

               (13)

               (14)

               (15a, b)

               (16a, b)

               (17)

               (18)

               (19)

If the system is in equilibrium then,

               (20)

Our conclusion, then, is that two phases in equilibrium must have the same temperature, pressure and chemical potential. The above sequence of derivations can easily be extended to include more phases and/or extended to include mixtures, where we would find that the temperature, pressure, and the chemical potential of each component must be the same in every phase. We will use these facts a little later in our derivation of the Gibbs phase rule.

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