Computational Chemistry

energy” to mean Gibbs free energy. A free energy change can be thought of as

an enthalpy change adjusted by a temperature-weighted entropy change:

DG¼DH#TDS $ð 5 : 173 Þ

TheTDSterm is often a minor contributor toDGat room temperature or
below, but will dominate at sufficiently high temperatures. If the entropy change
is positive (increased freedom of motion), this tends to make the free energy
change favorable (negative). Entropy can also be viewed in terms of dispersal of
energy. A change in free energy is the best indicator of the ease, as measured by
rate, or the extent, as measured by completeness, of a chemical reaction. Rate
and completeness are quantitated by the rate constant, and the equilibrium
constant, which can be calculated, respectively, from the free energy of activa-
tion and the free energy of reaction. These two energy differences (transition
state energy minus reactants energy, and products energy minus reactants
energy) can often be calculated quantum-mechanically fairly readily for mole-
cules, and the results can be used to calculate activation and reaction energy,
respectively. Free energies of formation have been tabulated and the values can
be used to calculate free energies of reaction and thus equilibrium constants.
Such tables should be better for such purposes than enthalpy tables, but in fact
are less widely used. This is probably because free energy tends to be harder to
measure than enthalpy, and could not be calculated accurately until fairly
recently, largely because of the problem of calculating accurate vibrational
frequencies. Free energies are readily obtained from experiment when equilib-
rium constants (Eq.5.183) can be accurately measured, and enthalpies can
usually be obtained from combustion measurements.
Symbol: Gibbs free energy is denoted eponymously byG, after Josiah Willard
Gibbs, ca. 1873, who single-handedly created much of chemical thermodynam-
ics. In the older literatureFwas sometimes used.Equation: sinceG¼H#TS,
the free energy of a molecule can be calculated from its enthalpy (above) and
entropy at temperature T; the entropy is calculated by standard statistical
mechanics methods [ 130 ].
6.Helmholtz free energy(or Helmholtz energy) is the work obtainable from a
system at a constant temperature and volume. It is much less used in chemistry
than Gibbs free energy, because most chemical reactions occur at constant
pressure, not constant volume. However, Helmholtz free energy is relevant to
reactions with rapid pressure changes (explosions).
Symbol: Helmholtz free energy is denoted in chemistry byA(GermanArbeit,
work), in physics byF, free energy.Equation:A¼U#TS, whereU¼internal
energy,T¼Temperature,S¼entropy.
7.Arrhenius activation energyis the energy term in an empirical equation that
shows the dependence of the rate constant on temperature (J. H. van’t Hoff,
1884, interpreted by S. Arrhenius, 1889):

k¼Ae#Ea=RT $ð 5 : 174 Þ

5.5 Applications of the Ab initio Method 295