Page 64 - Thermodynamics of Biochemical Reactions

4.1 Fundamental Equation for a Biochemical Reaction System at Specified pH 59

p(H’) is the specified chemical potential of the hydrogen ion that corresponds
with the experimental pH and ionic strength. It is necessary to use the amount
n,(H) of the hydrogen component in this equation because it is the conjugate
variable to p(H+) (see Section 2.7). The transformed Gibbs energy G’ plays the
same role that the Gibbs energy G does when the pH is not specified. The
introduction of G’ leads to a transformed enthalpy H’ and a transformed entropy
S’ for a reaction system at specified pH. Note that all of these transformed
thermodynamic properties are functions of the ionic strength as well as 7; P, and
pH. Transformed thermodynamic properties had previously been used in connec-
tion with petroleum thermodynamics where partial pressures of molecular hydro-
gen, ethylene, and acetylene can be specified as independent variables (Alberty
and Oppenheim, 1988, 1989, 1992, 1993a, b; Alberty, 1991~).
The amount n,(H) of the hydrogen component in a system is given by the
sum of the amounts of hydrogen atoms in various species in the reaction system.

(4.1-2)
j= 1
In this equation NH(j) is the number of hydrogen atoms in species j, and N, is
the number of different species in the system. The index number for species is
represented byj so that the index number introduced later for reactants (sums of
species) can be i. Substituting equation 4.1-2 and G = Xnjpj (equation 2.5-12)
into the Legendre transform (equation 4.1-1) yields

Ns NS NS Ns- I
G‘= 2 ?Ijp,j- C NH(J’)p(H+)nj= 2 nj{pj-NH(J’)p(H+)) = c njp;
j= 1 j= 1 j= 1 j= 1
(4.1-3)
where the transformed chemical potential pi of species j is given by
p‘.=p.-N H (’ J)p(H+) (4.1-4)
J
Note that the transformed chemical potential of the hydrogen ion is equal to zero
so that there is one less term in the last summation. Equation 4.1-3 shows that
the transformed Gibbs energy G‘ of a system is additive in the transformed
chemical potentials pi of N, - 1 species, just like the Gibbs energy G is additive
in the chemical potentials pj of N, species (see equation 2.5-12). In making the
Legendre transform, the chemical potential of one species (H+) has been changed
from a dependent variable to an independent variable. The roles of n,(H) and
p(H+) in the fundamental equation are interchanged as shown in the next
paragraph.
The derivation of the fundamental equation for the transformed Gibbs energy
G’ starts with the fundamental equation 2.5-5 for the Gibbs energy written in
terms of species:
NS
dG = -SdT+ VdP + c ,ujdnj (4.1-5)
j= 1
In order to obtain the fundamental equation for dG‘, it is first necessary to get
the contribution for the hydrogen component into a separate term. This can be
done by using equation 4.1-4 to eliminate pj from equation 4.1-5:

N,- 1 NS
dG= -SdT+ VdP + C pidnj + 2 NH(j)p(H+)dnj (4.1-6)
j= 1 j= 1
There is one less term in the first summation because p’(H+) = 0, as is evident
from equation 4.1-4. Equation 4.1-2 shows that dn,(H) = XN,(,j)dni, and so
equation 4.1-6 can be written
N,- 1
dG = -Sd7+ VdP + C p;dnj + p(H+)dn,(H) (4.1-7)
j= 1