Rigid Rotation 23
To show
(R1) | ||||
where | ||||
(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 24 | ||||
(R4) | ||||
Classical Energy of a rigid rotator is | ||||
(R5) | ||||
where w is the angular velocity (rod/s) and I is the moment of inertia 25 | ||||
(R6) | ||||
where mi is the mass located and distance ri from the center of mass. | ||||
Quantum mechanics solutions to the wave equation gives two quantum numbers, and m | ||||
(R7)26 | ||||
Let J ≡ | ||||
For a linear rigid rotator | ||||
(R8) | ||||
Where B is the rotation constant: | ||||
(R2)27 | ||||
with | ||||
The rotational partition function is | ||||
(R9)28 | ||||
Replacing the by an integral from 0 to ∞ integrating, we obtain the rotational partition function qR for a linear molecule29 | ||||
|
(R10) | |||
where Sy is the symmetry number which is the number of different but equivalent arrangements that can be made by rotating the molecules. | ||||
where | ||||
Sy = symmetry number. 30 For a hetronuclear molecule σ= 1 and for a homonuclear diatomic molecule or a symmetrical linear molecule, e.g., H2, then σ= 2. | ||||
Order of Magnitude and Representative Values | ||||
23P. W. Atkins, Physical Chemistry, 5th ed. (New York: Freeman, 1994), pp. 409, 413, 557, A24.
24P. W. Atkins, Physical Chemistry, 5th ed. (New York: Freeman, 1994), p. 410.
25P. W. Atkins, Physical Chemistry, 5th ed. (New York: Freeman, 1994), p. 555.
26P. W. Atkins, Physical Chemistry, 5th ed. (New York: Freeman, 1994), pp. 408, 413.
27P. W. Atkins, Physical Chemistry, 5th ed. (New York: Freeman, 1994), p. 557.
28P. W. Atkins, Physical Chemistry, 5th ed. (New York: Freeman, 1994), pp. 414, 563, 671.
29 P. W. Atkins, Physical Chemistry, 5th ed. (New York: Freeman, 1994), p. 694.
30For discussion of s, see K. J. Laidler, Chemical Kinetics, 3rd ed. (New York: Harper Collins, 1987), p.99.
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