Page 49 - Thermodynamics of Biochemical Reactions

3.4 Gibbs-Duhem Equation and the Phase Rule at Chemical Equilibrium 43

fewer independent variables because of the constraints due to the chemical
reactions.
The criterion of spontaneous change and equilibrium for a nonreaction
system is dG d 0 at constant 7; P, and {n,}, but the criterion for a system
involving chemical reactions is dG < 0 at constant 7; P, and {n,,}. Therefore, to
calculate the composition of a reaction system at equilibrium, it is necessary to
specify the amounts of components. This can be done by specifying the initial
composition because the initial reactants obviously contain all the components,
but this is more information than necessary, as we will see in the chapter on
matrices.

3.4 GIBBS-DUHEM EQUATION AND THE PHASE RULE
AT CHEMICAL EQUILIBRIUM

The Gibbs-Duhem equation and the phase rule were discussed briefly in Section
2.4, but now we want to extend those considerations to systems at chemical
equilibrium. The degrees of freedom in a gaseous reaction system at chemical
equilibrium was discussed by Alberty (1993b). The Gibbs-Duhem equation for a
one-phase reaction system at chemical equilibrium is obtained by using the
complete Legendre transform U' = U + PI/ - TS - Cn,,p, to interchange the
roles of amounts of components and the chemical potentials of components. Thus
the Gibbs-Duhem equation corresponding with equation 3.3-4 is

C
0 = -SdT + VdP - 1 n,,dp, (3.4-1)
i= 1
This shows that there are C + 2 intensive variables for a chemical reaction system
at equilibrium, but only C + 1 of them are independent because of this relation
between them; in other words, for a one-phase system at chemical equilibrium the
number F of degrees of freedom is given by F = C + 1. Since this is a one-phase
system, it is evident that the phase rule for the reaction system is

F=C-p+2 (3.4-2)
The number of natural variables is given by

D=F+p=C-p+2+p=C+2 (3.4-3)

Since equations 3.4-2 and 3.4-3 have been introduced in a rather indirect way,
more general derivations are given as follows: The composition of a phase in a
system involving chemical reactions can be specified by stating C - 1 mole
fractions, and the composition of p phases can be specified by stating p(C - 1)
mole fractions. If T and P are independent intensive variables, the number of
independent intensive variables is equal to p(C - 1) + 2. The number of relation-
ships between the chemical potentials of a single component between phases is
p - 1. Since there are C components, there are C(p - 1) equilibrium relationships.
The difference F between the number of independent intensive variables and the
number of relationships is given by F = p(C - 1) + 2 - C(p - 1) = C - p + 2. In
order to describe the extensive state of the system, it is necessary to specify in
addition the amounts of the p phases, and so D = F + p = C + 2. When special
constraints are involved, the number s of these special constraints must be
included in the phase rule to give F = C - p + 2 - s. An example of a special
constraint would be taking the amounts of two reactants in the ratio of their
stoichiometric numbers in a reaction. Further independent work terms in the
fundamental equation increase D and F. The numbers F and D are unique, but
the choices of independent intensive variables and independent extensive variables
are not.