Transition State Theory

Derivation of Vibrational Partition Function q_v

� To show

�� (Derive)�(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 = 0 (A5p402)

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�� 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. u = 0 is not an option (A5p402 and pA22). When v_o is the frequency of vibration, the ground state energy is

�� (V1)

�� Harmonic oscillator (A5p402)

�� Spring Force �potential energy from equilibrium position x = 0

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

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�� where

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�� The potential energy is

�� (V2)

�� We now want to show

�� (V3)

�� We now solve the wave equation:

�� (V4)

�� to find the allowable energies, .

�� Let , , where , , and

�� With these changes of variables Eqn. (A15) becomes

�� (V5)

�� The solutions to this equation (A5 pA22, i.e., Appendix 8)will go to infinity unless

�� = 2+1

�� = 0, 1, 2, 3 . . .

�� [c = speed of light]

��

�� = wave length

�� (V6)

�� Measuring energy relative to the zero point vibration frequency, i.e., = 0

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�� Substituting for �in the partition function summation

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This is the result we have been looking for!

��

�� (V7)

�� For , we can make the approximation

�� (V8)

�� For m multiple frequencies of vibration

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�� Order of Magnitude and Representative Values

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

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�� and

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