Page 75 - Thermodynamics of Biochemical Reactions

70 Chapter 4 Thermodynamics of Biochemical Reactions at Specified pH

4.6 GIBBS-DUHEM EQUATION, DEGREES OF
FREEDOM, AND THE CRITERION FOR
EQUILIBRIUM AT SPECIFIED pH

In discussing one-phase systems in terms of species, the number D of natural
variables was found to be N, + 2 (where the intensive variables are T and P) and
the number F of independent intensive variables was found to be N, + 1 (Section
3.4). When the pH is specified and the acid dissociations are at equilibrium, a
system is described in terms of N’ reactants (sums of species), and the number D‘
of natural variables is N’ + 3 (where the intensive variables are 7; P, and pH), as
indicated by equation 4.1-18. The number N’ of reactants may be significantly less
than the number N, of species, so that fewer variables are required to describe the
state of the system. When the pH is used as an independent variable, the
Gibbs-Duhem equation for the system is
NI
0 = -S’dT+ VdP - nidp: + ln(lO)n,(H)RTdpH (4.6-1)
i=l
which indicates that there are N’ + 3 intensive variables. However, only N‘ + 2 of
them are independent for a one-phase system. This is in agreement with the phase
rule in which the apparent number of independent intensive variables is given by
F’ = N’ - p + 3, where the 3 refers to ?: P, and pH.
The preceding paragraph applies to a system in which there are no biochemi-
cal reactions. Now we consider systems with reactions that are at equilibrium
(Alberty, 1992d). For a chemical reaction system, we saw (Section 3.4) that
D = C + 2 and F = C + 1 for a one-phase system. For a biochemical reaction
system at equilibrium, we need the fundamental equation written in terms of
apparent components to show how many natural variables there are. When the
reaction conditions C vip; = 0 for the biochemical reactions in the system are used
to eliminate one pi for each independent reaction from equation 4.1-18, the
following fundamental equation for G‘ in terms of apparent components is
obtained:

dG’ = -S‘dT+ VdP + C’ pidnLi + RTln(lO)n,(H)dpH (4.6-2)
i=l
where nhi is the amount of apparent component i and C‘ is the number of apparent
components at specified pH. This equation shows that when biochemical reac-
tions are at equilibrium the apparent number D‘ of natural variables is D‘ = C‘
+ 3. Since hydrogen is not included in the C’ apparent components, C’ = C - 1,
where C is the number of components before the pH was specified. Thus the
number of natural variables is the same for a system whether it is considered to
be made up of species or reactants. In other words, making the Legendre transform
has not changed the number of natural variables; it has simply changed an extensive
variable into an intensive variable. The number of apparent components is given
by C’ = N‘ - R’, where N‘ is the number of pseudoisomer groups and R’ is the
number of independent biochemical equations. This can be compared with a
chemical system that contains N, species and whose number of components is
given by C = N, - R, where N, is the number of species and R is the number of
independent chemical reactions.
The Gibbs-Duhem equation that corresponds with equation 4.6-2 is
C’
0 = -S’dT+ VdP - .bid& + RTln(lO)n,(H)dpH (4.6-3)
i= 1
which indicates that when enzyme-catalyzed reactions are at equilibrium, the
number of apparent independent intensive variables is F‘ = C‘ + 2. This is in
agreement with the phase rule written as F‘ = C‘ - p + 3. For a one-phase
system, F‘ = C‘ + 2.