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 |
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Back to one dimension | ||
Therefore relative to the lowest energy level n = 1, the energy is | ||
(T7) | ||
Then | ||
(T8) | ||
(T9) |
Return to Transition State Theory