Appendix A
Uniformity of Intensive Potentials as a Criterion of Phase Equilibrium

As discussed in Sec. 2.3, we use the function U to show that the temperature and pressure and the chemical potential of each species must be uniform throughout a closed, heterogeneous system at equilibrium internally with respect to heat transfer, boundary displacement, and mass transfer across phase boundaries. Since we identify equilibrium processes (variations) with reversible processes, the criterion for equilibrium in a closed system is that U is a minimum, and that any variation in U at constant total entropy and total volume vanishes; i.e.,

(A-1)

dU_s,v = 0

An expression for the total differential dU can be written by summing over all the phases, the extension of Eq. (2-20) to a multiphase system:

(A-2)

where α is a phase index, taking values 1 to π, and i is a component index, taking values 1 to m.

On expansion, Eq. (A-2) becomes

(A-3)

The individual variations dS⁽¹⁾, etc., are subject to the constraints of constant total entropy, constant total volume, and constant total moles of each species (chemical reaction excluded). These may be written as:

(A-4)

(A-5)

(A-6)

There are thus π(m + 2) independent variables in Eq. (A-3), and there are m + 2 constraints. The expression for dU may be written in terms of m + 2 fewer independent variables by using the constraining equations to eliminate, for example, dS⁽¹⁾, dV⁽¹⁾ and the m dn_i⁽¹⁾. The result is an expression for dU in terms of (π - l)(m + 2) truly independent variables; i.e., all the variations expressed as dS^(α) etc., are then truly independent, because the constraints have been used to eliminate certain variables. The resulting expression, if we eliminate dS⁽¹⁾, dV⁽¹⁾, and all dn_i⁽¹⁾ as indicated above, is

(A-7)

All variations dS⁽²⁾, dV⁽²⁾, dn₁⁽²⁾, dn₂⁽²⁾, etc., are truly independent. Therefore, at equilibrium in the closed system where dU = 0, it follows that¹

¹ F. B. Hildebrand, 1976, Methods of Applied Mathematics, 2” Ed., (Englewood Cliffs: Prentice-Hall).

(A-8)

Hence T⁽²⁾– T⁽¹⁾ = 0, or

(A-9)

Similarly,

(A-10)

and

(A-11)

Equations (A-9), (A-10), and (A-11) tell us that at internal equilibrium with respect to the three processes (heat transfer, boundary displacement, and mass transfer), temperature, pressure, and chemical potential of each component are uniform throughout the entire heterogeneous, closed system. This uniformity is expressed by Eqs. (2-25), (2-26), and (2-27).

Although chemical reactions have been excluded from consideration in this section, it can be shown that Eq. (2-27) is not altered by the presence of such reactions. For any component i at equilibrium, the chemical potential of i is the same in all phases, regardless of whether or not component i can participate in a chemical reaction in any (or all) of these phases. This is true provided only that all such chemical reactions are also at equilibrium.

However, the existence of chemical reactions does affect the phase rule given by Eq. (2-32). In that equation, m is the number of distinct chemical components only in the absence of chemical reactions. If chemical reactions are considered, then m is the number of independent components, i.e., the number of chemically distinct components minus the number of chemical equilibria interrelating these components.