Transition State Theory

Derivation of the Rotational Partition Function q_R

�� Rigid Rotation (A5p409, 413, 557 and A24)

�� To show

�� (R1)

�� where

�� = Rotational Constant�� (R2)

�� Consider a particle of mass m rotating about the z axis a distance r from the origin.

�� (R3)

��

�� This time we convert the wave equation to spherical coordinate to obtain (A5p410)

�� (R4)

�� Classical Energy of a rigid rotator is

�� (R5)

�� where w is the angular velocity (rod/s) and I is the

�� Moment of inertia(A5p555)

�� (R6)

�� where m_i is the mass located and distance r_i from the center of mass.

�� Quantum mechanics solutions to the wave equation gives two quantum numbers, �and m.

�� Magnitude of angular momentum =

�� z component of angular momentum = mh

�� (A5p408,p413) (R7)

��

�� Let J � l

�� For a linear rigid rotator

��E = hcB J (J+ 1)�� (R8)

�� Where B is the rotation constant:

�� (A5p557) (R2)

�� with

�� c = speed of light

�� I = moment of inertia about the center of mass

�� The rotational partition function is

�� (A5p414,563,671) (R9)

�� Replacing the by an integral from 0 to � and integrating, we obtain the rotational partition function q_R for a linear molecule (A5p694)

This is the result we have been looking for!

�� (R10)

�� where S_y is the symmetry number which is the number of different but equivalent arrangements that can be made by rotating the molecules.

��

��

��

�� where��

�� S_y = symmetry number. [For discussion of see Laidler 3^rd Ed. p.99.] For a hetronuclear molecule = 1 and for a homonuclear diatomic molecule or a symmetrical linear molecule, e.g., H₂, then = 2.

�� Order of Magnitude and Representative Values

��

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