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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