Page 34 - Radiochemistry and nuclear chemistry

Nuclei, Isotopes and Isotope Separation 23

Thus for a given temperature there is a corresponding average particle velocity. However,
the individual particles are found to move around with slightly different velocities. J. C.
Maxwell has calculated the velocity distribution of the particles in a gas. For the simplest
possible system it is a Boltzmann distribution. In a system of n o particles the number of
particles n E that have kinetic energy > E is given by

O0
n e = n o (2/4"Tr)(kT) -3/2 ~ 4"E e -ea'rdE (2.25)

In Fig. 2.4 nE/n o is plotted as a function of E for three different T's. For the line at 290
K we have marked the energies kT and 3kT/2. While 3kT/2 (or rather 3RT/2) corresponds
to the thermodynamic average translational energy, kT corresponds to the most probable
kinetic energy: the area under the curve is divided in two equal halves by the kT line.
In chemistry, the thermodynamic energy (2.21) must be used, while in nuclear reactions
the most probable energy E' must be used, where

E'= ~/~m(v') 2 = kT (2.26)

v' is the most probable velocity.
Although the difference between E and E' is not large (e. g. at 17 ~ C E = 0.037 eV and
E'= 0.025 eV), it increases with temperature. The most probable velocity is the deciding
factor whether a nuclear reaction takes place or not. Using (2.25) we can calculate the
fraction of particles at temperature T having a higher energy than kT: E> 2kT, 26 %, >
5kT; 1.86 %, and > 10kT, 0.029 %. This high-energy tail is of importance for chemical
reaction yields because it supplies the molecules with energies in excess of the activation
energy. It is also of importance for nuclear reactions and in particular for thermonuclear
processes.

2.6.3. The partial partition functions

So far there has been no clue to why isotopic molecules such as H20 and D20, or H35C1
and H37C1, behave chemically in slightly different manner. This explanation comes from
a study of the energy term Ej, which contains the atomic masses for all three molecular
movements: translation, rotation, and vibration.
(a) Translational energy. The translational energy, as used in chemical thermodynamics
involves molecular movements in all directions of space. The energy is given by the
expression (2.21). A more rigorous treatment leads to the expression

ftr = (27rMkT)3/2VM h-3 (2.27)

for the translational partition function, where V M is the molar volume and h the Planck
constant. Notice that no quantum numbers appear in (2.27); the reason is that they are not
known because of the very small AE's of such changes.