Computational Chemistry

change in activation free energy of 5 kJ mol#^1 can alter the rate constant or halflife
by a factor of about 10:

DGz¼100 kJ mol#^1 ; kr¼ 1 : 9 ' 10 #^5 s#^1 ; t 1 = 2 ¼ 4 ' 104 s

DGz¼105 kJ mol#^1 ; kr¼ 2 : 5 ' 10 #^6 s#^1 ; t 1 = 2 ¼ 3 ' 105 s

DGz¼110 kJ mol#^1 ; kr¼ 3 : 3 ' 10 #^7 s#^1 ; t 1 = 2 ¼ 3 ' 106 s

Comparing our calculations with the experimental facts:

CH 3 NC!CH 3 CN reaction

The experimental Arrhenius activation energy and rate constant for the gas
phase isomerization of methyl isocyanide have been reported (at the lowest pressure
used) thus:Ea¼36.27 kcal mol#^1 , i.e. 151.8 kJ mol#^1 , and logA¼10.46, i.e.A¼
2.88' 1010 s#^1 [ 220 ]. We want to compare our calculated activation free energy with
an experimental value, let us calculateDG{fromEaandA. From the Arrhenius
equation Eq. (5.174) and the Eyring equation Eq. (5.197) it follows that

DGz¼#RTlnðA=ðkBT=hÞÞþEa ð 5 : 199 Þ

Using the values of the constants given above for Eq. (5.197), we find

DGz¼# 2 :478 lnðA=ð 6 : 22 ' 1012 ÞÞþEa ð 5 : 200 Þ

with energies in kJ mol#^1 as usual. Using this equation andEaandAfrom [ 220 ], the
experimentally-derivedDG{is 165.1 kJ mol#^1. This is in good agreement with the
calculated values of 158–169 kJ mol#^1 in Table5.11.

CH 2 ¼CHOH!CH 3 CHO Reaction

The reported halflife of ethenol (vinyl alcohol) in the gas phase at room temper-
ature is ca. 30 min [ 221 ], far shorter than our calculated 10^28 –10^29 s. However, the
30 min halflife is very likely that for a protonation/deprotonation isomerization
catalyzed by the walls of the vessel, rather than for the concerted hydrogen
migration (Fig.5.30) considered here. Indeed, the relatedethynolhas been detected
in planetary atmospheres and interstellar space [ 222 ], showing that that molecule,
in isolation, is long-lived. Even under the more confined conditions of the lab,
ethenol can be studied in the gas phase [ 221 , 223 ] and in solution [ 224 ]. All three
methods predict very long halflives for the uncatalyzed reaction.
Cyclopropylidene!allene reaction
Cyclopropylidene has apparently never been isolated [ 225 ], so its halflife
is likely to be short even well below room temperature. Employing a variety of
methods, Bettinger et al. obtained a barrier for its rearrangement to allene of about
4 kcal mol#^1 , i. e ca. 17 kJ mol#^1 [ 226 ], not far from our values of 18–24 kJ mol#^1.

328 5 Ab initio Calculations