Page 40 - Radiochemistry and nuclear chemistry

Nuclei, Isotopes and Isotope Separation 29

Because of the law of conservation of energy

EIp -- Eki n
and

hv = kT (2.41)

This development assumes that the vibrational energy is completely converted to fragment
translational energy. This assumption is not always valid for polyatomic fragments, in
which internal excitation may occur. Introducing (2.41) into (2.38) and equaling (2.37) and
(2.38) yields

k[A][BC] = kTh-I[ABC#] (2.42)

It is assumed that ABC # is in dynamic equilibrium with the reactants A and BC. Thus

[ABC#]/([A][BC]) = k # (2.43)

According to (2.17)

k" = FABC"/( 'A FBc ) (2.44)

which with (2.42) yields

k = kT FABC./(h F A Fac ) (2.45)

This expression must be multiplied by a factor r, which is the probability that the complex
will dissociate into products instead of back into the reactants as assumed in (2.43). The
factor g is called the transmission coefficient. The final rate expression thus becomes:

k = g k TFABc#/(h F A FBC ) (2.46)

As is shown in w the grand partition functions F i can be calculated from theory and
spectroscopic data; because these functions are mass dependent k is mass dependent. In
calculating the Fi's, all modes of energy must be included as well as the population of the
different energy states.
For isotopes of the lighter elements, the activation energy term makes the main
contribution to the reaction rate isotope effect, while for the heavier elements the vibrational
frequency causing the decomposition into the products plays the larger role. Because the
energy states usually are more separated for the isotopic molecules of the products and
reactants than for the transition state, isotope effects are usually larger in reaction kinetics
than in equilibria.
Studies of kinetic isotope effects are of considerable theoretical interest, particularly in
organic chemistry. The practical applications are still meager, but this will not necessarily
be so in the future. An example is the decrease in metabolic rate for 13C compounds, which
has led to the suggestion of its use for treatment of certain diseases, as e.g. porphyria.