Essentials of Computational Chemistry

528 15 ADIABATIC REACTION DYNAMICS
which shows that the slope of such a plot should be −H o,‡ /R and the intercept is a
function of S o,‡ /R. With these quantities in hand, the activation free energy may be easily
computed for any temperature within the range of the data points and compared directly to a
theoretical computation of this quantity (extrapolation outside the range of the data points can
be dangerous because enthalpy and entropy are themselves both dependent on temperature, so
it represents an approximation to assume their constancy over a given measurement range).
An alternative analysis having a long history, however, is to simply plot ln k vs. 1/T,this
procedure being motivated by the empirically derived Arrhenius expression
k = Ae −Ea/RT
(15.30)
where A is the so-called pre-exponential factor and Ea is the Arrhenius activation energy.
Rearranging Eq. (15.30) readily illustrates that a plot of ln k vs. 1/T will have slope −Ea/R
and intercept ln A. Simple algebra allows us to express the relationships between the Arrhenius
quantities and the thermodynamic quantities as
and
Ea = H o,‡ + RT (15.31)
A = kBT
h e(1+So,‡ /R)
(15.32)
Because these two different conventions exist (as well as other conventions, e.g., one based
on collision theory, that will not be discussed here), when the term ‘activation energy’ is used
without qualification, it is critical for accurate comparisons that it be established whether
this term refers to an Arrhenius activation energy, a TST activation free energy, a difference
in stationary-point potential energies, a difference in zero-point-including stationary-point
potential energies, etc. The term ‘barrier’ is also often used ambiguously, and care should
be taken to establish its meaning in a given situation.
One point of interest deriving from the equations of TST (and Arrhenius theory) is that
the upper limit for the 298 K rate constant of a unimolecular reaction that takes place with
zero activation energy (of whatever sort) is roughly 10 13 sec −1 . This is, in some sense, a
conceptually obvious result since that is on the order of a molecular vibrational frequency,
which is thought of as the ‘mechanism’ by which a transition state goes to its products.
15.3.1.2 Kinetic isotope effects
As noted in Chapter 10, the zero-point energy, and the translational, rotational, and vibrational
partition functions all depend on the isotopic masses of the atoms. Thus, so too does
the rate constant for a given reaction. A difference in rates observed for two different isotopically
substituted reactants is referred to as a kinetic isotope effect (KIE), usually expressed
as a ratio of rates. Isotope effects are divided into two classes: primary isotope effects refer
to situations where the isotopic substitution involves one of the two atoms involved in a