Page 127 - Mechanism and Theory in Organic Chemistry

Each energy level has contributions from translational, rotational, vibrational,
and electronic substates,

If the state multiplicity gi is the product of the multiplicities of the substates, each
term in the partition function sum can be written in terms of these energies as in
Equation A1.18,

There is a term like Equation A1.18 for each energy level, and there is an energy
level for every combination of each eil, eir, eiv, eie with every other, so the whole
partition function is a multiple sum over all combinations,

where the f's are separated partition functions for the different kinds of motion.
We shall need to know how to evaluate these separated partition functions.
The translational energy levels can be derived from the quantum mechanical
solution for a particle in a box; they are so closely spaced that the partition
function can be evaluated in closed form by integration, and has the value

for each dimension, where M is the mass, h is Planck's constant, and a is the
length of the box. For three dimensi~ns,~

where V is the volume.
The rotational partition function, found in a similar way, ise

where I,, I,, I, are the moments of inertia about three mutually perpendicular

d This value is, strictly speaking, correct only for the gas phase. See, for example, Moore, Physical
Chemistry, p. 627.
Moore, Physical Chemistry, p. 630; Wiberg, Physical Organic Chemistry, p. 221.