Page 69 - Thermodynamics of Biochemical Reactions

64 Chapter 4 Thermodynamics of Biochemical Reactions at Specified pH

4.2-l), and [' is the apparent extent of the biochemical reaction. It is necessary to
put the primes on these quantities to differentiate them from the stoichiometric
numbers vj and extents of reaction ( of the underlying chemical reactions written
in terms of species. If there is a single biochemical reaction, substituting
dni = vidj" in equation 4.1-18 yields

dG' = -S'dT+ VdP + (,z1 yip;) d(' + RTln(lO)n,(H)dpH (4.2-3)

so that
A,G' = (g) " (4.2-4)

T,p,pH = c v;p;
i= 1
where A,G' is referred to as the transformed reaction Gibbs energy. At chemical
equilibrium, ArG' is equal to zero so that

(4.2-5)
i=l
This is the equilibrium condition. In Chapter 3 we saw that the corresponding
condition for a chemical reaction is equation 3.1-6. Note that equation 4.2-5 has
the same form as biochemical equation 4.2-1.
The expression for the transformed chemical potential of a reactant is given
by
pi = pio + RTln[Bi] (4.2-6)
where pio is the standard value (that is the value for an ideal 1 M solution) and
Bi represents the ith reactant (pseudoisomer group). This looks reasonable in
relation to pi = py + RTln[Bj] for the chemical potential of species j, but we will
consider equation 4.2-6 in greater detail in Section 4.4. Substituting equation 4.2-6
in equation 4.2-4 yields
Arc' = c vip:' + RT~v;ln[B,] (4.2-7)

At equilibrium, A,G' = 0, and so
Cvi$ = -RT,?l,v:ln[B,]
= -RTCln([BilVi)

= -RTlnTIIB,IYL = -RTlnK' (4.2- 8)
where lI is the product sign. The concentrations in equation 4.2-8 are equilibrium
concentrations, but it is conventional to omit the subscripts "eq" in writing
expressions for equilibrium constants. The apparent equilibrium constant is given
by
K' = n[Bi]"' (4.2-9)
This confirms that K' is written in terms of concentrations of pseudoisomer
groups and that there are no terms for hydrogen ions. When dilute aqueous
solutions are considered, the convention is that [H,O] is omitted, but the
contribution for p"(H,O) in equation 4.2-8 is included.
When equation 4.2-4 is substituted in equation 4.2-3, the following fundamen-
tal equation is obtained for the transformed Gibbs energy:

dG' = -S'dT+ VdP + A,G'd(' + RTln(lO)n,(H)dpH (4.2-10)
This form of the fundamental equation has two Maxwell equations of special
interest. The first Maxwell equation is

(4.2-1 1)