Lecture 12 - Rotational partition function q rotational(Heteronuclear 29 29 29 29 βε β = = = 1 4 442 4 4 43 sum over levels 1 2 1 2 1 rot J BJ J

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Unformatted text preview: Rotational partition function q rotational (Heteronuclear) ( 29 ( 29 ( 29 ( 29 βε β- +- = = + = + 1 4 442 4 4 43 sum over levels 1 , 2 1 2 1 rot J BJ J J J q V T J e J e J J J rotational states are described by J, M . for each J value there are 2J+1 possible M values. To sum over states, we would need all the M states ( 29 ( 29 ( 29 θ φ θ φ = + M M J J Y , 1 Y , rotation J J B H θ π = = 2 r 2 we can define a rotational T, 8 h B k k I ( 29 ( 29 ( 29 β- + = + 1 , 2 1 rot BJ J q V T J e dJ o to do the , we consider the change in the differential ( 29 ( 29 ( 29 ( 29 + 1 + 1 1 d d d J J dJ d J J dJ ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 + + = + + + 1 1 2 1 1 1 d d d J J J dJ d J J d J J ( 29 ( 29 ( 29 β- + = + 1 , 2 1 rot BJ J q V T e J dJ ( 29 ( 29 ( 29 θ θ- + = - 1 1 , r rot r J J T q V T e T ( 29 θ = rot r T q T ( 29 π = 2 2 8 rot T q k T h I θ θ β = = r r for 300 1 J varies continuously, because 2 consecutive levels are very cl ose T K T ( 29 ( 29 ( 29- + = + 1 1 1 B kT J J e d J J ( 29 ( 29 ( 29 θ- + = + 1 , 2 1 r rot T J J J q V T J e θ : r when , we have to add all levels without integrating T ( 29 θ θ θ--- = + + + + L 2 6 12 , 1 3 5 7 r r r rot T T T q V T e e e the more terms we add the better; sum until J=3 is usually enough to within 1% or use the Euler-MacLaurin summation formula ( 29 π θ = = rot r 2 8 for most cases, and we can use q kT T T h I ( 29 θ θ θ θ = + + + + L 2 3 1 1 4 , 1 3 15 315 rot r r r r T q V T T T T I = 2 rot ln E N q kT T π 1 = 1 2 2 ln 8 k T T k h T I...
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This note was uploaded on 05/29/2011 for the course CHM 6461 taught by Professor Bowers during the Spring '08 term at University of Florida.

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Lecture 12 - Rotational partition function q rotational(Heteronuclear 29 29 29 29 βε β = = = 1 4 442 4 4 43 sum over levels 1 2 1 2 1 rot J BJ J

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