10.9 Statistics Applied to Biochemical Reactions
491
10.9.1 Molecular Numbers
Suppose there are A kinds of indistinguishable atoms within a macrosystem,
making up M species of molecules. Each of the molecules of type J can exist
in certain energy states J i labeled by the index ‘i’. Let the number of molecules
of type J in the i-th possible energy state of these molecules be N J i , with N J =
i N J i , and the total number of atoms of kind k be n k . If molecule J has n J k bound
atoms of kind k, then the total of these kinds of atoms reads
M
J =1
i
n J k N J i = n k ,
(10.194)
which we intend to hold fixed. There will be A of these atom-conservation
conditions. In addition, if the molecules are involved in a chemical reaction, we
have the stoichiometric balancing condition for each kind of atom:
J
ν J n J k = 0 . ,
(10.195)
a generalization of Eq. (9.210).
10.9.2 The Case of Molecules Acting Independently
We say that molecules in a system act independently if the partition function for the
system can be factorized into a product of partition functions for each species of
molecule.
Near equilibrium, the entropy of such a mixture of independent reactants and
products is
S = k B log W
= k B
J
i
N J i log (g J i /N J i ) + N J
= k B
J
Ni
N Ni (−
k
n J k α k + ββ J i ) + N J
= −k B
J
N J
k
n J k α k + k B β
J
i
N Ni J i + k B N
= k B
J
N J ln (eZ J /N J ) + k B βE .
(10.196)