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Chapter 3: Rate Laws

Transition State Theory

Derivation of Vibrational Partition Function q_v

To show

(A12)

Again we solve the wave equation for two molecules undergoing oscillation about an equilibrium position x = 0. The potential energy is shown below as a function of the displacement from the equilibrium position x = 0¹⁷

The uncertainty principle says that we cannot know exactly where the particle is located. Therefore zero frequency of vibration in the ground state, i.e. ν = 0 is not an option¹⁸. When ν₀ is the frequency of vibration, the ground state energy is

(V1)

Harmonic oscillator ¹⁹

Spring Force

potential energy from equilibrium position x = 0

the solution is of the form for t=0 then x=0

where

The potential energy is

(V2)

We now want to show

(V3)

We now solve the wave equation:

(V4)

to find the allowable energies, ε.

Let

,

, where

,

, and

With these changes of variables Eqn. (A15) becomes

(V5)

The solutions to this equation²⁰ will go to infinity unless

ε = 2ν +1

ν = 0, 1, 2, 3 . . .

[c = speed of light]²¹

λ = wave length

(V6)

Measuring energy relative to the zero point vibration frequency, (i.e., ν = 0) gives

Substituting for

in the partition function summation

This is the result we have been looking for!

(V7)

For

, we can make the approximation

(V8)

For m multiple frequencies of vibration

Order of Magnitude and Representative Values

For H₂O we have three vibrational frequencies with corresponding wave numbers, .

and

¹⁷P. W. Atkins, Physical Chemistry, 5th ed. (New York: Freeman, 1994), p. 402.

¹⁸ P. W. Atkins, Physical Chemistry, 5th ed. (New York: Freeman, 1994), pp. 22, 402.

¹⁹P. W. Atkins, Physical Chemistry, 5th ed. (New York: Freeman, 1994), p. 402.

²⁰P. W. Atkins, Physical Chemistry, 5th ed. (New York: Freeman, 1994), p. 22, Appendix 8.

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