Page 32 - Radiochemistry and nuclear chemistry

Nuclei, Isotopes and Isotope Separation 21

compounds have different isotopes X and X*, we may have an isotope exchange according
to

AX + BX* = AX* + BX (2.15)

The equilibrium constant k in the reaction is given by

AG ~ = -RT In k = -RT In{([AX*][BX])/([AX][BX*])} (2.16)

where AG ~ is the (Gibb's) free energy and R is the universal gas constant. For values of
fundamental constants see Appendix III.

2.6.1. The partition function

It has been shown that k deviates slightly from 1, giving rise to the isotopic effects
observed in nature for the light elements. This deviation can be calculated by methods of
statistical thermodynamics. Only the main features of this theory are given here. The
equilibrium constant k can be written

k = (FAx* Fsx)/(F FBx*) (2.17)

where F is the grand partition function, which for each molecule includes all possible
energy states of the molecule and the fraction of molecules in each energy state under given
external condition. The grand partition function is defined by

F -- ftr frot fvib fel fnsp (2.18)

where each term fj refers to a particular energy form: translation, rotation, vibration,
electron movement, and nuclear spin. The two latter will have no influences on the
chemical isotope effect, and can therefore be omitted. It can be shown that each separate
partition function fj can be described by the expression

~,..e -/~''~/I'T (2.19)

where Eji is the particular energy state i for the molecule's energy mode j; e.g. for j =
vibration', there may be 20 different vibrational states (i.e. the maximum/-value is 20)
populated, k is the Boltzmann constant; (2.19) is closely related to the Boltzmann
distribution law (see next section).
The term gj, i is called the degeneracy and corrects for the fact that the same energy state
in the molecule may be reached in several different ways. The summation has to be made
over all energy states i.