(T1)
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
V(X)
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)
where
(T4)
We now use the boundary conditions
|
|
= 0, sin 0
= 0, and cos 0=1
therefore B = 0
the wave equation is now
= A sin kx
At x = a = 0
will be zero provided
therefore ka = n
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
(T9)
Return to Transition State Theory
