Page 48 - Thermodynamics of Biochemical Reactions

42 Chapter 3 Chemical Equilibrium in Aqueous Solutions

Chapter 5 on matrices. In any case, the things that are conserved are referred to
as components.
The preceding section was based on the fundamental equation for G in terms
of the extent of reaction, but in order to identify the D natural variables for a
one-reaction system at equilibrium, we need to apply the condition for equilib-
rium Cvipi = 0 (equation 3.1-6) that is due to the reaction. That is done by using
each independent equilibrium condition to eliminate one chemical potential from
equation 2.5-5. This is more easily seen for a simple reaction:

A+B=AB (3.3-1)
At chemical equilibrium, equation 3.1-6 indicates that pA + pB = pAB. Using this
relation to eliminate pAB from the fundamental equation yields

dG = --SdT + VdP + p,(dnA + dn,,) + pB(dnB + dn,,)
= --SdT + VdP + pAdncA + pg dnCB (3.3-2)
where nCi is the amount of component i; ncA = nA + nAB and ncB = ng + nAB. This
form of the fundamental equation for G applies at chemical equilibrium. It is easy
to see that nA + nAB is conserved because every time a molecule of A disappears,
a molecule of AB appears. These two conservation equations are constraints on
the equilibrium composition. The other constraint is K = [nAB/V]/[nA/V][nB/V]
where the amounts are equilibrium values; thus there are three equations and
three unknowns, nA, n,, and nAB. The natural variables for this reaction system
at chemical equilibrium are 7: P, ncA, and ncB, as shown by equation 3.3-2. Note
that the number of natural variables has been decreased by one by the constraint
due to reaction 3.3-1. When chemical reactions are involved in a system, pi and
nCi are conjugate variables (see Table 2.1) as indicated by equation 3.3-2.
Usually statements of problems on chemical equilibrium include the initial
amounts of several species, but this doesn’t really indicate the number of
components. The initial amounts of all species can be used to calculate the initial
amounts of components. The choice of components is arbitrary because pA or p,
could have been eliminated from the fundamental equation at chemical equilib-
rium, rather than pAB. However, the number C of components is unique. Note
that in equation 3.3-2 the components have the chemical potentials of species.
This is an example of the theorems of Beattie and Oppenheim (1979) that “(1) the
chemical potential of a component of a phase is independent of the choice of
components, and (2) the chemical potential of a constituent of a phase when
considered to be a species is equal to its chemical potential when considered to
be a component.” The amount of a component in a species can be negative.
The number C of components in a one-phase system is given by
C=N,-R (3.3-3)
where N, is the number of different species and R is the number of independent
reactions. The source of this equation and answers to questions about the number
of components and the choice of components are clarified by the use of matrices,
as described in Chapter 5. The amounts of components can be calculated from
the amounts of species by use of a matrix multiplication (equation 5.1-27). When
there are no reactions in a system, it is not necessary to distinguish between
species and components.
Now we are in position to discuss a closed reaction system where several
reactions are occuring. Equation 3.3-2 can be generalized to

C
dG = -SdT + VdP + 1 pidnci (3.3-4)
i=l
This form of the fundamental equation for G applies to a system at chemical
equilibrium. Note that the number D of natural variables of G is now C + 2,
rather than N, + 2 as it was for a nonreaction system (see Section 2.5). There are