2. THERMODYNAMICS OF LIVING SYSTEMS
65
Or
fc' = κ^ —
. · · = κψκ*
(137)
h
CACB
h
where K* is the constant for the equilibrium,
A + B^±Mt
existing between the reactants and the activated complex. The constant
K* is exactly analogous to any other equilibrium constant. By appropriate concern for activity coefficients it may be related to AF*, Aff*,
and AS
1 , the standard free energy, heat constant, and entropy changes,
respectively, accompanying the formation of the activated state from
the reactants, by means of the familiar thermodynamic relationships.
In this manner, Eq. 137 can be written
h' = κ^. β -Δ**/ΑΓ
(138)
h
where
K = e -AFt/RT
— e -AHt/RT
. e ASt/R
Thus:
k' = K ^ · eAHt /
RT
- e
Ast '
R
(139)
h
The constant, K*, is the ratio of the partition functions of the activated complex to that for the reactants. The partition functions of any
atom or molecule, including the activated complex, have the value
/= y e -B where Ei is the energy of the state of the molecule and the summation
is over all states. As the number of molecules of a species is always
proportional to its partition function, we have an invaluable means of
expressing equilibrium constants as ratios of the appropriate partition
functions. Since the energy of a molecule can be written as the sum of
energies for its degrees of freedom and this energy occurs in the exponent of the partition function, it follows that the over-all partition
function can be written as a product of partition functions for all the
degrees of freedom.
The partition function for three translational degrees of freedom is
/- - '"
Va|F
(140,