Essentials of Computational Chemistry

Ink
15.3 TRANSITION-STATE THEORY 537
A B
1/T
Figure 15.6 Typical changes in rate constants as a function of temperature for light-isotope (above)
and heavy-isotope (below) substituted systems. In the high-temperature regime A, Arrhenius or TST
plots will be essentially linear and yield good estimates of the activation parameters. In the intermediate
region B, such plots may still be linear if the sampled temperature range is too small, but any activation
parameters inferred therefrom will be of little utility. Additionally, KIE values in this region are very
sensitive to the temperature. In the low-temperature regime C, the rate constant is almost entirely a
result of tunneling, and little information about the PES can be gleaned from kinetic analysis
an activation enthalpy of 11.6 kcal mol −1 . However, Sherer and Cramer (2003) found for
the hydrogen-atom-transfer rate-determining step that Eqs. (15.40) and (15.41) predict the
tunneling transmission coefficient κ to drop from 93 to 4 over the experimental temperature
range of 300 to 400 K. When the apparent rate constants were divided by their corresponding
κ values, an Eyring plot of the corrected ‘true’ semiclassical rate constants provided an activation
enthalpy of 19.2 kcal mol −1 . This latter result agreed well with a value of 20.3 kcal
mol −1 computed from DFT, and illustrates the magnitude of the quantitative difference that
may arise when tunneling is ignored in experimental Eyring analyses.
Figure 15.6 forms the basis for a more general discussion of tunneling, reaction rates, and
kinetic isotope effects. The rate constant for an exergonic chemical reaction does not actually
go to zero as the temperature goes to zero. Instead, after the temperature drops sufficiently,
all reactant systems will be in their lowest energy state, that state will have some rate of
tunneling through the barrier, and that rate is the non-zero asymptote that will be approached.
Of course, an analysis that neglects tunneling would interpret a rate that is independent of
temperature as corresponding to an activation enthalpy of zero (see Section 15.3.1.1) which
may be very far from correct.
Moreover, because tunneling is less efficient for heavy isotopes, the transition to tunnelingdominated
kinetics occurs at lower temperatures for heavier isotopes (Figure 15.6, region B),
C