Page 128 - Thermodynamics of Biochemical Reactions

7.2 Further Transformed Gibbs Energy at Specified Oxygen Concentration 125

7.2 FURTHER TRANSFORMED GIBBS ENERGY AT
SPECIFIED OXYGEN CONCENTRATION

In order to introduce the chemical potential of molecular oxygen as a natural
variable, the following Legendre transform is used to define a further transformed
Gibbs energy G” (Alberty, 1996b):
G” = G‘ - n:(O2)p’(O2) (7.2-1)

where nh(0,) is the amount of molecular oxygen in the system, free and bound.
Substituting G” = C and G‘ = C p:dn: (equation 7.1-2) yields the following
expression for the further transformed chemical potential p; of reactant i
(i = 0 - 4).

P:’ = p; - N&(OZ)P‘(02) (7.2-2)

where N,(02) is the number of 0, molecules bound by i. Note that p”(0,) = 0.
Using equation 7.2-2 to eliminate p’(T), p‘(T(02)), p‘(T(O,),), p‘(T(0J3), and
F’(T(O~)~) from equation 7.1-1 yields
dG‘ = -S‘dT+ VdP + p”(T)dn’(T) + p”(T(02))dn’(T(02))

+ C1”(T(02>2)dn’(T(oZ>,) + p”(T(02)3)dn‘((T(02)3)
+ p”(T(02),)dn’(T(Oz)4) + p‘(O2)dn;(O2) + RTln(lO)n,(H)dpH
(7.2-3)

Taking the differential of G” in equation 7.2-1 yields

dG” = dG‘ - n:(02)dp’(0,) - p’(Oz)dnL(02) (7.2-4)
Substituting equation 7.2-3 yields

dG“ = -S’dT+ VdP + kt”(T)dlI’(T) + p”(T(O,))dn’(T(O2))
f ~”(T(02>2>dn’(T(02)2) + p”(T(02),>dn’((T(02)3)

+ p”(T(O,),)dn’(T(O,),) - nh(o,>dP‘(oz>
+ RTln(lO)n,(H)dpH (7.2-5)
At specified 7: P, p’(02), and pH,

(dG‘’)TP ic (0,)pH = p”(T)dn’(T) + p”(T(02))dn’(T(02)) + p”(T(02>2)dn’(T(02>2)
f ~”(T(02>3)dn’((T(02>,) + p”(T(02>,>dn’(T(o,>,) (7.2-6)

The four reactions at specified ~’(0~) can be written as

T = T(0,) (7.2-7)

(7.2-8)

(7.2-9)

(7.2- 1 0)

These reactions do not balance oxygen because its chemical potential is specified.
At specified [O,], these five forms of hemoglobin are pseudoisomers, and they
have the same further transformed chemical potential: p”(T) = p”(T(0,)) =