Page 31 - Thermodynamics of Biochemical Reactions

2.4 Gibbs-Duhem Equation and the Phase Rule 25

and the mixed partial derivatives are equal

= (E)) (2.3-2)
x ) : (
the function f is said to be exact, and the integral off is independent of path. If
the mixed partial derivatives are not equal, the function ,f is said to be inexact,
and the integral of the function is dependent on the path. In Section 2.2 it was
noted that heat q and work w are inexact differentials because they depend on the
path of integration. However, all state functions are exact differentials since they
are independent of path of integration. Since the internal energy has an exact
differential, equation 2.3-2 applies to its fundamental equation. In ther-
modynamics relations like equation 2.3-2 are referred to as Maxwell equations.
For equation 2.2-2, the Maxwell equations are

(2.3-3)

(2.3-4)

(2.3-5)

(2.3 - 6)

Equations 2.2-10 to 2.2-12 and equations 2.3-3 to 2.3-6 show that the
thermodynamic properties of a system are interrelated in complicated, and
sometimes unexpected, ways. The next section shows that the intensive variables
for a thermodynamic system are not independent of each other.

W 2.4 GIBBS-DUHEM EQUATION AND THE PHASE RULE

The differential of the integrated form (equation 2.2-14) of the fundamental
equation for the internal energy is
dU = TdS + PdT- PdV- VdP + N, pldn, + N, n,dpl (2.4-1)

I= 1 i= 1
Subtracting the fundamental equation for U (equation 2.2-8) yields
N,
SdT- VdP+ nidpi=O (2.4-2)
i=l
which is referred to as the Gibbs-Duhem equation. This equation is important
because it shows that the N, + 2 intensive properties for a homogeneous system
without chemical reactions are not independent. But N, + 1 of them are indepen-
dent. Note that this is in agreement with the experimental observation of Section
2.1 that the intensive state of a one-phase system can be specified by stating
N, + 1 intensive variables.
The Gibbs-Duhem equation is the basis for the phase rule of Gibbs.
According to the phase rule, the number of degrees of freedom F (independent
intensive variables) for a system involving only PV work, but no chemical
reactions, is given by

F = N, - p + 2 (2.4-3)
where p is the number of phases. This equation is derived in a more general form
later in Chapter 8 on phase equilibria. Thus for a one-phase system without