Page 182 - Thermodynamics of Biochemical Reactions

182 Chapter 11 Use of Semigrand Partition Functions

of hydrogen atoms in the species. This sum of exponential terms for a
pseudoisomer group may contain many terms and is really a Laplace transform
(Greiner, Neise, and Stocker, 1995).
Substituting equation 11.2-1 in 11.1-5 yields
G'= -N:,,KTln{exp( -[jpl)exp( - N,, In( 10)pH)

+exp(-BLL2)exp(- NH2 In(l0)pH))
= - N:,,kT ln{exp( - Bpi) + exp( - [jp;)) = N~so~i:so ( I 1.2-3)

The transformed chemical potential for the pseudoisomer group (i.e., A- plus HA)
is given by

piso = - kTln{exp( -Bpi) + exp( -[~,LL'~)] (1 1.2-4)
which can also be written as

Pi,, = P;y0 + kTlnIIA1 (1 I .2-5)
where [A] = [A ] + [HA]. Equation 11.2-4 leads to
p,,, = -kTln(exp( -fill;') + exp(-Bp;'))
I0
(1 1.2-6)
This can be demonstrated by substituting p', = pi' + kTln[HA] and
,H; = 1: + kTln[A-] in equation 11.2-4 and using
Ti = exp [ Piso kT - Pi ''1 ( 1 1.2-7)
r0

where rL is the mole fraction of species i.
The fundamental equation of thermodynamics for G' for a dilute aqueous
solution containing a weak acid HA, and its basic form A- when these species
are at equilibrium at a specified pH is (see 4.3-1)

dG' = -S'dT + VdP + &,dN~s,, + N,(H)kTln(lO)dpH (11.2-8)
where N,(H) is the number of atoms of hydrogen in the system. p:so is the
transformed chemical potential of the pseudoisomer group, and the number of
molecules of the pseudoisomer group is given by NI,, = N + N,. Equation 1 1.2-8
can be integrated at constant values of the intensive variables 7: P, nd pH to
obtain

G' = Hisopiso (1 1.2-9)

This is in agreement with statistical mechanical equation 11.2-3.
Equation 11.2-8 indicates that the other thermodynamic properties of the
system can be obtained from

(11.2-11)

(1 1.2-13)