Chapter 3: Rate Laws
Transition State Theory
Derivation of Transitional Partition Function q'T
To show
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(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
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(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 | ||
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(T3) | |
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(T4) | |
| where | ||
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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
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(T5) |
Substituting (T5) into (T4) we see that only certain energy states are allowed
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(T6) | |
For particle in a a 3-D box of sides a, b, and c |
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| Back to one dimension | ||
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| Therefore relative to the lowest energy level n = 1, the energy is | ||
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(T7) | |
| Then | ||
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(T8) | |
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(T9) |