Page 145 - Introduction to chemical reaction engineering and kinetics

6.3 Energy in Molecules 127

any arbitrary amount of vibrational energy, but must have one of a relatively small num-
ber of discrete quantities of vibrational energy. This is also true for rotational energy,
although many more states are available. For translational energy, there are usually
so many allowed translational energy states that a continuous distribution is assumed.
Extra energy can also be stored in the electrons, by promoting an electron from an oc-
cupied orbital to an unoccupied orbital. This changes the bonding interactions and can
be thought of as an entirely separate potential energy surface at higher energy. These
energy states are not usually encountered in thermal reactions, but are an important
part of photochemistry and high-energy processes which involve charged species.

6.3.2.2 Distribution of Molecular Energy
In a group of molecules in thermal equilibrium at temperature T, the distribution of
energy among the various modes of energy and among the molecules is given by the
Boltzmann distribution, which states that the probability of finding a molecule within
a narrow energy range around E is proportional to the number of states in that energy
range times the “Boltzmann factor,” e-E’k~T:

P(E) = g(e)e-E’kBT (6.34)

where k, is the Boltzmann constant:

kB = R/N,, = 1.381 x 1O-23J K-’ (6.3-6)

and g(e), the number of states in the energy range E to E + de, is known as the “density
of states” function. This function is derived from quantum mechanical arguments, al-
though when many levels are accessible at the energy (temperature) of the system, clas-
sical (Newtonian) mechanics can also give satisfactory results. This result arises from
the concept that energy is distributed randomly among all the types of motion, subject
to the constraint that the total energy and the number of molecules are conserved. This
relationship gives the probability that any molecule has energy above a certain quantity
(like a barrier height), and allows one to derive the distribution of molecular velocities
in a gas. The randomization of energy is accomplished by energy exchange in encoun-
ters with other molecules in the system. Therefore, each molecule spends some time in
high-energy states, and some time with little energy. The energy distribution over time
of an individual molecule is equal to the instantaneous distribution over the molecules
in the system. We can use molar energy (E) in 6.3-5 to replace molecular energy (E), if
R is substituted for k,.

6.3.2.3 Distribution of Molecular Translational Energy and Velocity in a Gas
In an ideal gas, molecules spend most of the time isolated from the other molecules
in the system and therefore have well defined velocities. In a liquid, the molecules are
in a constant state of collision. The derivation of the translational energy distribution
from equation 6.3-5 (which requires obtaining g(e)) gives the distribution (expressed
as dN/N, the fraction of molecules with energy between E and E + de):

(6.3-7)
which is Boltzmann’s law of the distribution of energy (Moelwyn-Hughes, 1957, p. 37).
The analogous velocity distribution in terms of molecular velocity, u = (2~lrn)~‘~, where
m is the mass per molecule, is: