## Equilibrium between two phases

The equilibrium criterion of minimum Gibbs free energy (Sect. 1.11) can be applied to any of the phase transitions described in the previous section. At fixed pressure and temperature, let the system contain nI moles of phase I and nII moles of phase II, with molar Gibbs free energies of gI and gII, respectively. The total Gibbs free energy of the two-phase mixture is:

The requirement of equilibrium is that G remain unchanged (at its minimum value) for any variations in the state of the system. Because p and T are fixed, so are the molar Gibbs free energies of the individual phases, gI and gII. The only possible change is the conversion of some of one phase to the other. Since the system is closed, an increment dnI of phase I implies an equal and opposite change dnII = -dnI in phase II. The equilibrium criterion of Eq (1.20a) yields:

This is an expression of chemical equilibrium. It complements the conditions of thermal equilibrium (TI = TII) and mechanical equilibrium (pI = pII). Since the Gibbs free energy is defined by g = h - Ts, another form of Eq (5.2) is:

where the subscript tr means transition from one phase to the other. Ahtt = hn - h and Astr = sn - si are, respectively, the enthalpy and entropy differences of the substance in the two states at the temperature Ttr of the transition. Equation (5.3) provides a relation between the two property changes:

Aht,

'tr Ttr

By common usage, the subscript "sat" is replaced by "vap" for a liquid (denoting vaporization) or "sub" for sublimation of a solid

Equation (5.4a) was used in Sect. 3.5.2 by a different approach for the specific case of water vaporization. It is used in Problem 5.6 as part of a calculation of the absolute entropy of a substance.

By common convention, the phase transition denoted by the generic subscript tr in Eq (5.4) or sat in Equation (5.4a) are written in the form of a simple chemical reaction with the higher-enthalpy phase on the right-hand side. Thus, vaporization of a liquid is written in chemical reaction terminology as: L = g with the liquid (L) as phase I and the vapor (g) as phase II. The equal sign denotes equilibrium between liquid and vapor. With this convention, Ahvap is a positive quantity. Transitions between condensed phases are written as:

And the enthalpy change AhM for melting or Ahtt for crystal-structure changes are positive.

The consequence of the above convention is that the entropy of the transition must also be positive, since temperature cannot be negative. Thus, the entropy of the vapor is always greater than that of the co-existing liquid state, corresponding to the relative degrees of disorder of these two phases. Similarly, AsM for melting and Astr for solid/solid transitions are also positive quantities

The equilibrium condition of Eq (5.3) can be regarded as a balance between the positive enthalpy change of the transition, which tends to favor the more tightly bound state (i.e., the liquid), and the positive entropy change that acts to drive the system towards the state of greater disorder (i.e., the vapor). These competing tendencies just balance at a unique temperature, Ttr, where the Gibbs free energies of the two phases are equal.

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