Page 106 - Thermodynamics of Biochemical Reactions

5.5 Matrix Forms of the Fundamental Equations for Biochemical Reaction Systems 101

Equation 5.4-12 indicates that the corresponding Gibbs-Duhem equation for
a system of chemical reactions is

-SdT+ VdP - (dp,)n, = 0 ( 5.4- 1 4)
Because of this relation between the C + 2 intensive variables, the number of
intensive degrees of freedom is F = C + 1.

5.5 MATRIX FORMS OF THE FUNDAMENTAL
EQUATIONS FOR BIOCHEMICAL REACTION
SYSTEMS

For a biochemical reaction system at specified pH, equations 5.4-1 and 5.4-2
become

n' = nb + v'k' (5.5-1)
dn' = v'dc (5.5-2)
Therefore equation 4.2-3 can be written in matrix form:

dG' = - S'dT+ VdP + N' p;dn: + RTln(l0) n,(H)dpH (5.5-3)
i= 1
= - S'dT+ VdP + p'dn' + RT ln(10) n,(H)dpH
= - S'dT+ VdP + p'v'dk' + RTln(l0) n,(H)dpH
The primes on the amounts are needed to indicate that they are amounts of
reactants, which are sums of species that are pseudoisomers at specified pH. The
primes on the stoichiometric number matrices and extents of reaction column
matrices are needed to indicate that these matrices are for biochemical reactions
written in terms reactants (sums of species). The primes are needed on the
transformed chemical potentials to distinguish them from chemical potentials of
species.
The biochemical analogues of equations 5.4-11, 5.4-12, and 5.4-13 are

C'
dG' = - S'dT+ VdP + 1 pLidnl,, + RTln(10) n,(H)dpH (5.5-4)
i= 1
= - S'dT+ VdP + pl,dnL + RTln(l0) n,(H)dpH
= - S'dT-t VdP + piA'dn' + RTln(l0) n,(H)dpH
The prime on the amount of a component indicates that these are the components
other than the hydrogen component. The corresponding Gibbs-Duhem equation
is
-S'dT+ VdP - (dpL)nL + RTln(10) n,(H)dpH = 0 (5.5-5)

Since the thermodynamics of a biochemical reaction system is considered at
specific pH, we need to consider equation 5.5-4 in the form
(dG'),, = -S'dT+ VdP + pl,dnL (5.5-6)

and equation 5.5-5 in the form
-S'dT+ VdP - (dpi)nL = 0 (5.5 -7)
These equations look like equations 5.4-13 and 5.4-14, where C' components play
the role of C components in equations 5.4-13 and 5.4-14.
The number D' of natural variables for a system and the number F'
of intensive degrees of freedom for a one-phase system at equilibrium were
discussed in Section 4.6, but now we can discuss these numbers in a more general
way. Table 5.1 gives these numbers for three descriptions of a one-phase reaction