Page 181 - Thermodynamics of Biochemical Reactions

11.2 Transformed Gibbs Energy for a System Containing a Weak Acid 181

When a system at specified T and P contains two species and one of them is
in equilibrium with that species in a reservoir through a semipermeable mem-
brane, the transformed Gibbs energy of the system is calculated from the
semigrand partition function r’(T, P, pl, N,) by use of
G’(T P, p,, N,) = -kTInI-’(T, P, p,, N2) (1 1.1-4)
Statistical mechanics is often thought of as a way to predict the ther-
modynamic properties of molecules from their microscopic properties, but statis-
tical mechnics is more than that because it provides a complementary way of
looking at thermodynamics. The transformed Gibbs energy G‘ for a biochemical
reaction system at specified pH is given by

G‘ = - kTln I-’ ( 1 1.1 -5)
The semigrand partition function r’ corresponds with a system of enzyme-
catalyzed reactions in contact with a reservoir of hydrogen ions at a specified pH.
The semigrand partition function can be written for an aqueous solution of a
biochemical reactant at specified pH or a system involving many biochemical
reations. The other thermodynamic properties of the system can be calculated
from r‘.
The further transformed Gibbs energy G” for a biochemical rection system at
specified pH and specified concentrations of certain coenzymes is given by
G“ = - kTln I-’’ ( 1 1.1 -6)
where r” is the corresponding semigrand partition function. All the remaining
thermodynamic properties of the system can be calculated by taking partial
derivatives of G“ or r”.
This is not the place to discuss the partition functions E, Y, A, and in detail,
but it is important to point out that B and Y differ by the factor exp(fiN,p,) and
A and r differ by the same factor. Thus holding the chemical potential of species
1 constant has the effect of introducing an exponential factor that depends on the
number of molecules of species 1 and the chemical potential of species 1.

H 11.2 TRANSFORMED GIBBS ENERGY FOR A SYSTEM
CONTAINING A WEAK ACID AND ITS BASIC
FORM AT A SPECIFIED pH

The forms of semigrand partition functions for biochemical rection systems can
be illustrated, starting with an aqueous solution of a weak acid and its basic form
at a specified pH (Alberty, 2001~). The semigrand partition function r’ for this
system is given by
I-’(T P, PH, KO)

= (exp( -[lp,)exp( - N,, In(l0)pH) + exp( -Pp2)exp(- N,, In(10)pH}N~50

= {exp( - bpi) + exp( - [jpi)}Niso (1 1.2- 1)
where ,LL~ and p2 are the chemical potentials of species 1 and 2 (i.e., A- and HA)
and 11: = p, - N,,p(H+). N,, and N,, are the numbers of hydrogen atoms in
these two species N:so is the number of molecules in the pseudoisomer group,
N, + N,. Equation 11.2-1 has been given in the form of the partition function
that applies at zero ionic strength in order to keep it simpler, but this complica-
tion can be taken into account by using

2.91482(zj - NH(j))I”2
p\ = pj(Z = 0) + N,(j)RTIn(lO)pH - (1 1.2-2)
1 + 1.61”2
Note that the exp( -Bpi) terms for species are weighted according to the number