Page 50 - Thermodynamics of Biochemical Reactions

44 Chapter 3 Chemical Equilibrium in Aqueous Solutions

Equation 3.3-4 shows that for a chemical reaction system, the number F of
intensive degrees of freedom and the number D of extensive degrees of freedom
are given by
F = N, - R -p + 2 (3.4-4)
D = N, - R + 2 (3.4-5)

W 3.5 ISOMER GROUP THERMODYNAMICS

In discussing the thermodynamics of complex reaction systems, it is helpful to
have ways of reducing the complexity so that it is easier to think about the system
and to make calculations. One of these ways is to aggregate isomers and make
thermodynamic calculations with isomer groups, rather than species (Smith and
Missen, 1982; Alberty, 1983a, 1993b). Examples of isomer groups are the butenes
(3 isomers) and pentenes (5 isomers), where the numbers of isomers exclude
cis-trans and stereoisomers. At higher temperatures these isomers are in equilib-
rium with each other, and so thermodynamic calculations can be made with
butenes and pentenes. The reason this can be done is that the distribution of
isomers within an isomer group is independent of the composition of a reaction
system and of the other reactions that occur. The distribution of isomers within
an isomer group depends only on temperature for ideal gases and ideal solutions.
The fundamental equation for G can be used to show that when isomers are
in equilibrium, they have the same chemical potential. Therefore terms for isomers
in the fundamental equation for G can be aggregated so that the amounts dealt
with are amounts of isomer groups, rather than amounts of species. Since the
number of isomers of a reactant can be significant, this can make a significant
reduction in the number of chemical terms in the fundamental equation at
chemical equilibrium.
There are two ways to express the Gibbs energy GI,, of a group of isomers at
chemical equilibrium. The first method simply uses a sum of the terms for the
individual isomers, and the second method utilizes the chemical potential p15, for
the isomer group at equilibrium and the amount nlco of the isomer group as in

(3.5-1)

The number of isomers in an isomer group is represented by N,,,. At chemical
equilibrium, all of the isomers have the same chemical potential, and this chemical
potential is represented by pica. The amount of an isomer group is represented by
n,,, = Cn,. For a group of gaseous isomers at cquilibriurn, the chemical potential
of the isomer group in a mixture of ideal gases is given by

nisop (3.5-2)
n, P"
piso = p;\o + R T In ~
where n, is the total amount of gas in the system, P is the sum of the partial
pressure of the isomers, and Po is the standard state pressure. At equilibrium the
chemical potential of isomer i is given by

ni P
n, Po
pi = ,u: + RTln ~ (3.5-3)
These two equations can be written as

(3.5-4)

(3.5-5)