Lecture 29 : Temperature Dependence of Reaction Rates
Other thermodynamic functions from the partition function we have already seen that and that the average energy . This average energy is not the absolute value (since the thermodynamic functions U, S, A and G have no absolute value, their value has to be measured with respect to a standard or a reference value. Let the standard value of U be U(0). Then U is given by
(29.51)
We next obtain entropy. The standard procedure is to use the Boltzmann formula S = kBlnW where W is the number of ways of achieving a given configuration or a given distribution of energies of molecules. We will use a shorter path to s by noting that the formula for entropy of mixing, S = can be generalized by replacing xis by P is, the probabilities of the states of a distribution.
(29.52)
Using
(29.53)
(29.54)
(29.55)
(29.56)
(29.57)
Where Q, the partition function for N molecules is written as qN for distinguishable molecules. For indistinguishable molecules,
Rewriting (29.57), we get the Helmholtz free energy.
or
(29.58)
Just as U is measured relative to U(0), A is measured relative to A(0). As T 0, U(0) = A(0).
To obtain the thermodynamic functions, enthalpy (H) and the Gibbs free energy (G), we need to express pressure in terms of the partition function. From thermodynamics, we know that . Using this in eq (29.58), we get
and that the average energy 
. This average energy is not the absolute value (since the thermodynamic functions U, S, A and G have no absolute value, their value has to be measured with respect to a standard or a reference value. Let the standard value of U be U(0). Then U is given by
can be generalized by replacing xis by P is, the probabilities of the states of a distribution.
0, U(0) = A(0).
. Using this in eq (29.58), we get
or 
or
