has been strongly criticized [ 144 ], but in the author’s opinion this viewpoint works
quite well used pictorially at the molecular level and is more useful in interpreting
reactions than is the counterview of dispersal of energy. Suffice it to say that a
spontaneouslydisordered system is more probable than an ordered one, and the
entropy of a system is proportional to the logarithm of its probability [ 145 ].
Intuitively, we see thatDS>0 for a process in which the product or the transition
state is less symmetrical or has more freedom of motion than the reactants is less
ordered. For example, ring-opening reactions, since they relieve constraints on
intramolecular motion, should be accompanied by an increase in entropy. Note
that an increase in entropy favors a process: it increases a rate constant (affects
activation entropy) or an equilibrium constant (affects reaction entropy), while an
increase inenthalpydisfavors a process.
Details on the calculation of entropies is given in [ 130 ]andthebookbyHehre,
Radom, Schleyer and Pople, who also tabulate the errors in calculated entropy for
small molecules composed of elements from H to F [ 146 ]. Errors in calculated
entropies at 300 K are 1.7, 1.3 and 0.8 J mol#^1 K#^1 (0.4, 0.3 and 0.2 cal mol#^1 K#^1 )
at 300 K, for frequency calculations at the HF/3–21G(), 6–31G and MP2/6–31G
levels, respectively. From Eq.5.175this corresponds to an error in free energy at
300 K of 300'(0.8þ0.8) J mol#^1 ¼480 J mol#^1 or 0.5 kJ mol#^1 , for the MP2/
6–31G calculations. This is much smaller than the enthalpy error of ca. 10 kJ
mol#^1 which can be reliably obtained with practical high-accuracy methods (see
below) and shows that in current ab initio work errors in free energies can be
expected to come mainly from the enthalpy. Many programs, e.g. Gaussian and
Spartan, automatically calculate the correction terms to be added toDEtotal0K in
Eq.5.182at the end of a frequency calculation, and print out the 298.15 K enthalpy
or the correction to the 0 K enthalpy. Reaction entropies are needed to calculate free
energies of reaction (from Eq.5.179¼5.173), from which equilibrium constants
[ 147 ] can be calculated:
DGreact¼#RTlnKeq $ð 5 : 183 ÞWhere several species are in equilibrium, the ratios are proportional to their
Boltzmann exponential factors. For example, if the relative free energiesGof A,
B and C are 0, 5.0 and 20.0 kJ mol#^1 (hereGfor species A has been set to zero
and B and C lie 5.0 and 20.0 kJ mol#^1 higher) then
[A] : [B] : [C]¼exp(#0/RT) : exp(#5.0/RT) : exp(#20.0/RT);At room temperatureRT¼2.48 kJ mol#^1 and so at this temperature[A] : [B] : [C]¼1 : 0.133 : 0.000315¼3175 : 422 : 1
Activation entropies are useful because they can give information on the struc-
ture of a transition state (as stated above, a more confined transition state is
signalled by a negative, unfavorable, activation entropy), but the ab initio calcula-
tion ofrateconstants [ 148 ] from activation free energies is not as straightforward as
5.5 Applications of the Ab initio Method 301