Essentials of Computational Chemistry

15.3 TRANSITION-STATE THEORY 527
but it is sufficiently rare for a unimolecular reaction to have different standard-state volumes
for the activated complex and the reactant that one rarely gives thought to this point). Care
must be taken then such that if the molecular translational partition function is computed
for a volume of, say, 24.5 L (the volume occupied by one mole of an ideal gas at 298 K
and 1 atm pressure), and the rate constant is in, say, molecules cm −3 sec −1 , the appropriate
conversion in standard states is made.
In a very general form, then, we have the canonical expression
k = kBT
h
Q ‡
QR
Q o R
Q ‡,o e−V ‡ /kBT
(15.25)
where R refers generically to either unimolecular or bimolecular reactants, and V ‡ is the
difference in zero-point-including potential energies of the reactants and TS structure. When
working in molar quantities, Eq. (15.25) becomes
k = kBT
h
Q ‡
QR
Q o R
Q ‡,o e−V ‡ /RT
(15.26)
in which case one often absorbs the standard-state partition functions back into the exponential
to write
k = kBT
h e−Go,‡ /RT
(15.27)
where Go,‡ is referred to as the free energy of activation. Note that using Eq. (10.6) we
may also write
k = kBT
h e−H o,‡ /RT S
e o,‡ /R
(15.28)
15.3.1.1 Relation between theory and experiment
Operationally, the theoretical computation of a rate constant using TST typically employs
Eq. (15.26). One locates all necessary stationary points – one TS structure and one or two
minima – and evaluates their energies and their partition functions under the rigid-rotorharmonic-oscillator
approximation. Experiment, on the other hand, measures rate constants
according to the methodologies outlined in Section 15.1, typically with the goal of deriving
such quantities as the free energy of activation. However, the experimental data may be
analyzed in a variety of ways, and it is critically important to ensure experimental/theoretical
comparisons are made under consistent conditions.
One analysis of experimental data involves carrying out rate constant measurements at a
series of temperatures, and then plotting ln(k/T ) against 1/T (a so-called Eyring plot). We
may rearrange Eq. (15.28) to
o,‡
k H So,‡ kB
ln =− + + ln
T RT R h
(15.29)