Chapter 3: Rate Laws

Transition State Theory

Derivation of Transitional Partition Function q'_T

To show

(T1)

Translational Energy

We solve the Schrödinger equation for the energy of a molecule trapped in an infinite potential well. This situation is called "a particle in a box". For a particle in a box of length a

The potential energy is zero everywhere except at the walls where it is infinite so that the particle cannot escape the box. Inside the box

		(T2)
The box is a square well potential where the potential is zero between x = 0 and x = a but infinite at x = 0 and x = a (A5p392). The solution is to the above equation is
		(T3)
		(T4)
where

We now use the boundary conditions

Ψ = 0, sin 0 = 0, and cos 0=1

∴ B = 0

The wave equation is now

Ψ = A sin kx

At x = a, Ψ = 0

Ψ will be zero provided

∴ ka = nπ

where n is an integer, 1, 2, 3

(T5)

Substituting (T5) into (T4) we see that only certain energy states are allowed

		(T6)
For particle in a a 3-D box of sides a, b, and c

Back to one dimension

Therefore relative to the lowest energy level n = 1, the energy is
		(T7)
Then
		(T8)
		(T9)