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# Lecture09A - Partition Functions for Molecules(18.3-18.8...

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Partition Functions for Molecules (18.3-18.8) Moving beyond atoms to molecules, we have two more independent types of degrees of freedom: vibration and rotation. trans rot vib elec ε ε ε ε ε = + + + ( ) and accordingly: q V T q q q q = and accordingly: q V,T trans rot vib elec ( ) ( ) , N q V T and, as always: Q N,V,T ! N = Fi id di i l l First, consider a diatomic molecule: ( ) ( ) 3/2 1 2 2 B m m k T V T V π + 2 , trans q h = Where the total mass of the molecule (m 1 +m 2 ) is substituted for m in our monatomic solution. Lecture 9 1 For the electronic energy contribution for a diatomic, we need to decide what will be our zero-energy reference point. Relative to their constituent atoms diatomics have electronic energy Relative to their constituent atoms, diatomics have electronic energy. We will define separated atoms at R= in their ground electronic state as our zero for energy. 2 / / 1 2 : ... e B e B D k T k T elec e e Thus q g e g e ε = + + Fig. 18.2 Lecture 9 2 Now we can consider vibrations . 1 0 1 2 f H O h Note: in your text, the v’s (quantum numbers) look a lot like v v v 0,1,2... for a H.O. 2 hv ε = + = the v ’s (frequencies). Be careful! ( ) ( ) v v 1/ 2 / 2 v v 0 v 0 v 0 hv hv hv vib q T e e e e β βε β β + = = = = = = 2 1 for 1: 1 ... n x x x x < = + + + = See the back cover of your text 0 for 1 n x = for common expansions. / 2 hv e β ( ) since 1, 1 hv vib hv e q T e β β < = Wow! Simple. S i i l h i i ( h i ?) lik i lif Statistical mechanicians (mechanics?) like to simplify by using single terms (like β =1/k B T), so they define: vib hv k Θ = B vib This relates the vibrational frequency, , to a vibrational temperature, . v Θ / 2 ib T −Θ 3 ( ) / 1 vib vib vib T e q T e Θ −Θ = 2 vib ln As we noted before: E vib B d q Nk T dT = / 2 1 b vib vib B T Nk Θ Θ Θ = + vib e 2 / ib T C −Θ Θ As I showed in lecture 6: ( ) , , 2 / 1 vib vib V vib vib V vib T e C R n T e Θ −Θ = = , 0 0 V vib as T C , V vib as T C R → ∞ 1 0 368 e ( ) ( ) , 2 2 1 0.368 , 0.921 0.632 1 vib V vib at T C R R R e = Θ = = = That is, at the vibrational temperature, the heat capacity is 92 1% of the high temperature limit 92.1% of the high temperature limit. Lecture 9 4

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