Computational Chemistry

is the calculation ofequilibriumconstants from reaction free energies. The crudest
way to calculate a rate constant is to use the Arrhenius equation [ 140 , 149 ]

k¼Ae#Ea=RT $ð 5 : 184 ¼ 5 : 174 Þ

and to simply approximate the preexponential factorAby that known for a similar
reaction (a typical value for unimolecular reactions is 10^12 –10^15 [ 150 ])and to
approximateEabyDEtotal0K (Eq.5.181and discussion). Theoretically more satisfying
is to representEabyDH{þRT, using the temperature in question, in accordance
with

Ea¼DHzþRT $ð 5 : 185 ¼ 5 : 175 Þ

for a gas-phase unimolecular reaction, and by

Ea¼DHzþ 2 RT ð 5 : 186 Þ

for a gas phase bimolecular reaction [ 151 ]. The main problem with this is that the
preexponentialAvaries by a large factor even for, say, reactions which are formally
unimolecular [ 150 ]:

CH 3 NC!CH 3 CN 3.98' 1013

Cyclopropane!propene 1.58' 1015

C 2 H 6 !2 CH 3 2.51' 1017

so that this method of guessingAby analogy could give a value that was out by a
factor of 10^4 unless one was judicious (or lucky) enough to choose a good model
reaction. Theexponentialfactor is prone to smaller errors, since calculatingDH{to
within 10 kJ mol#^1 is now feasible, and an error of this size corresponds to an error
factor in exp(#DEa) of exp(#10/2.48)¼57 (atT¼298 K). This may seem to be
itself very big, but an easy method of reliably calculating rate constants to within a
factor of 100 might be useful for estimating the stability of unknown substances. In
fact, a very useful and simple rule is that the threshold barrier for the stability of a
substance at room temperature is about 100 kJ mol#^1 ; allow a latitude of about
20 kJ mol#^1 [ 152 ]. This rule has been used frequently in the computational search
for stable nitrogen allotropes [ 153 ].
Note that for unimolecular processes the halflife, an intuitively more meaningful
quantity than the rate constant, is simply

t 1 = 2 ¼

ln 2

kr

¼

0 : 693

kr

i.e. the halflife of a unimolecular reaction is approximately the reciprocal of its rate
constant.

302 5 Ab initio Calculations