PLENTIFUL ENERGY

9.3 Kinetics of the Reactions
Chemical reactions between molecules result from collisions, where the more
energetic molecules undergo reactions. Molecular kinetic energies cover a wide
range, increasing with temperature. The fraction of molecules with kinetic energies
sufficient to cause a reaction is given by a Maxwell-Boltzmann distribution.
Differing slightly from the bell-shaped ―normal distribution‖ seen in all kinds of
phenomena, the Maxwell-Boltzmann distribution has a lop-sided bell shape with a
high energy ―tail‖ trailing out from the maximum. The distribution of energies
broadens as temperature is increased and the maximum moves to higher energies.
The fraction of molecules with energies sufficient to cause reactions is
proportional to an exponential of the form exp((E f -E b )/RT), where (E f -E b ) is the
difference between the free energies of forward and back reactions and RT is the
energy corresponding to temperature T. This expression allows us to actually
calculate the balance between the forward and backward reactions. In addition to
the principal reaction in the ―forward‖ direction, there is always some ―back‖
reaction; it may be small or very small if the free energy differences are substantial,
but always there is some. Note that the smaller the exponential, the smaller the
difference between the two free energies, and the closer the rates of forward and
back reactions, and thus the back reaction fraction gets larger. In our case, because
the back reaction—Pu metal forming from PuCl 3 in the presence of UCl 3 —is what
we want, the smaller (E f -E b ) can be made (again in our case by use of a liquid
cadmium cathode), the easier it is to deposit plutonium metal.
9.4 The Power of Equilibria
The PuCl 3 /UCl 3 ratio in the electrolyte must be high enough to give a back
reaction sufficient for a useful rate of Pu metal deposition. The PuCl 3 /UCl 3 ratios
necessary for useful depositions can be calculated from equilibrium considerations.
In fact, what goes where, and how much, and in what form is extremely important,
and it is possible to calculate it from the simple principles of equilibrium in
the rates of the reactions of each of the elements. The principles of equilibrium are
simple, very general and very powerful. They state that in any reaction the forward
rate must eventually equal the backward rate. Eventually the amount of product will
build up enough that the backward rate—and there will always be a backward
rate—equals the forward rate of product formation. The backward rate—the basic
rate at which the product of a reaction dissociates into the original components of
the reaction—may be small; it may in fact be microscopically small, and present
only due to highly unlikely statistical fluctuations. But equality comes when the
concentrations of reactants have so decreased and the amount of product has so
increased that the two rates overall are equal. In those reactions where the backward
rate is extremely small, present only because of statistical fluctuations, reactions go
193