Page 80 - Thermodynamics of Biochemical Reactions

4.10 Calculation of Standard Transformed Gibbs Energies of Species 75

Thus the fundamental equation for the transformed Gibbs energy can be written
as
"
dC' = -SdT+ VdP + c pidni
i= 1
- n,(H) ((-il) +) dT- RTln(l0)dpH

P,(n:},pH
Nr
= -SdT+ VdP + 1 pidni - n,(H)dp(H+)
i= 1
where the term in parentheses is the total differential of the chemical potential of
hydrogen ions (see equation 4.1-10).
The summation in equation 4.10-3 can be written in terms of species exclusive
of the hydrogen ion because when species in a pseudoisomer group are in
equilibrium at a specified pH, these species have the same transformed chemical
potential.

Ns-1
dG' = -SdT+ VdP + C &dnj - n,(H)dp(H+) (4.10-4)
j= 1
where N, is the number of different species.
Now the inverse Legendre transform given in equation 4.10-1 is needed. The
differential of the Gibbs energy is given by
dG = dG' + n,(H)dp(H+) + p(H+)dn,(H) (4.10-5)
Substituting equation 4.10-4 into this equation yields

N,- 1
dG = -SdT+ VdP + c pldnj + p(H+)dn,(H) (4.10-6)
j= 1
The amount of the hydrogen component n,(H) in the system is given by equation
4.1-2, and so equation 4.10-6 can be written as
N,
dG = -SdT+ VdP + c (pi + NH(j)p(H+)}dnj (4.10-7)
j= 1
The term in braces is the chemical potential of ion j:

pj = + NH(j)p(H+) (4.10-8)
and so equation 4.1-5 is obtained as expected.
If the apparent equilibrium constant K' for an enzyme-catalyzed reaction has
been determined at 298.15K and AfG'O values can be calculated at the experimen-
tal pH and ionic strength using known functions of pH and ionic strength for all
the reactants but one, the A,G" of that reactant under the experimental condi-
tions can be calculated using equation 4.4-2. So far functions of pH and ionic
strength that yield A,G" are have been published for 131 reactants at 298.15 K
(Alberty, 200 1 f).
When the reactant of interest consists of a single species, Af Go(I = 0) for this
species at 298.1 5 K can be calculated using equation 4.10-8 in the following form
(see equation 4.4-10):
A,GY(I = 0) = AfG'O(pH,I) - N,(j)RTln(lO)pH

+ 2.91482(z; - NH(j))I1I2 (4.10-9)
1 + 1.6Z1'2

A program calcGeflsp has been written to produce output in the form of equation
3.8-1 for a reactant made up of one species. It is given in the package Basic-
BiochemData2.