Chem120A+Notes+-+Rigid+Rotor

Chem120A+Notes+-+Rigid+Rotor - Rotational Energy Levels 2D...

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Unformatted text preview: Rotational Energy Levels 2D Rotation r O x y The moment of inertia 2 2 2 2 d E I d - = h and the periodic boundary condition ( ) ( 2 ) = + Solution: 2 I mr = Schrodinger Equation 2 2 ( ) e , 0,1,2, 2 in n n E n I - = = = h L Wavefunction Energy Rotational excitation energy decreases as the moment of inertia increases. Estimation of Rotation Excitation Energy 2 2 2 2 2 ( 1) 2 2 1 ( ) 2 n n E I I n I + = - = + h h h Excitation Energy Given that the reduced mass of a diatomic molecule is typically around 10-25 kg, and that the bond length is around 1 , the excitation energy for n = 0 to 1 can be calculated as 2 34 2 24 25 10 2 10 8 2 10 (1.05 10 J s) 5 10 J 2 2 10 kg (10 m) = 5 10 Hz 2 2 3.14 3 10 = 4 10 m 5 10 E I E c ----- = = = = h h Rotational transitions of diatomic molecules occur in the microwave region, and Rotational excited states are thermally accessible at room temperature. (vibrations are in the time scale of ps) Rigid Rotor in 3D Space A particle rotates on a sphere surface around the origin point. Two degrees of freedom are the polar angle and the azimuthal angle . y x z O r Solution: 2 2 2 2 1 1 sin ( , ) ( , ) 2 sin sin Y EY I - + = h Schrodinger Equation Energy 2 ( 1) 2 l l l E I + = - h 1/2 | | (2 1)( | |)!...
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This note was uploaded on 09/28/2009 for the course CHEM 120A taught by Professor Whaley during the Spring '07 term at University of California, Berkeley.

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Chem120A+Notes+-+Rigid+Rotor - Rotational Energy Levels 2D...

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