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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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 εε =+ + + d accordingly: q V T q q q ( ) and accordingly: qV,T trans rot vib elec qq = , N qV T ⎡⎤ ( ) ( ) and, as always: Q N,V,T ! qV N ⎣⎦ = First, consider a diatomic molecule: ( ) 3/2 2 2 mmk T VT V π + ( ) 12 2 , B 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 lectronic energy ntribution for a diatomic, we need to ote e ect o c e e gy co t but o o a d ato c, we eed to decide what will be our zero-energy reference point. elative 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 // : ... eB e B DkT kT elec e e Thus q g e g e + Fig. 18.2 Lecture 9 2 Now we can consider vibrations . 1 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 ⎛⎞ = ⎜⎟ ⎝⎠ the v ’s (frequencies). Be careful! () v v1 /2 v 0 v0 hv hv hv vib qT e e e e β βε ββ ∞∞ −+ −− == = = ∑∑ 2 1 r 1: 1 ... n x x x = ++ + = See the back cover of your text 0 for ... 1 n xx x = <+ + + for common expansions. hv since 1, 1 hv vib hv e eq T e <= Wow! Simple. Statistical mechanicians (mechanics?) like to simplify by using single terms (like β =1/k B T), so they define: vib hv Θ = B k vib This relates the vibrational frequency, , to a vibrational temperature, . v Θ ib −Θ Lecture 9 3 / 1 vib vib T vib T e e Θ = 2 ib ln As we noted before: E vib dq Nk T = vib vib Nk ΘΘ vib B dT / 21 vib B T e Θ 2 As I showed in lecture 6: ( ) / , , 2 / 1 vib vib T Vv ib vib T C e CR nT e Θ Θ , 00 as T C →→ , as T C R →∞ 1 368 , 22 1 0.368 ,0 . 9 2 1 0.632 1 vib V vib e at T C R R R e = = = That is, at the vibrational temperature, the heat capacity is 2 1% of the high temperature limit 92.1% of the high temperature limit. Lecture 9 4
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
We can also use these partition functions to calculate the population distribution as a function of temperature. Since all ideal diatomic gases pg will behave the same, we will consider temperatures in terms of Θ vib . /2 ib T hv e β −Θ () ( ) v1 / v0 11 vib vib hv vib T hv ee qT e −+ = == = −− 1/2 th fractional population of n state. hv n n ib e f q vib 0 Typically we want to consider ground state population ( ) versus f 0 00 excited states ( ). By definition: 1 1 en n e nn ff f = = ∑∑ 0 1 So: 1 hv hv hv hv hv e hv f f e e ββ ⎛⎞ = = ⎜⎟ vib qe ⎝⎠ // 0 Or: 1 vib vib TT e fe f e =− = Lecture 9 5 T/
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 04/01/2012 for the course CHEM 444 taught by Professor Gruebele,m during the Spring '08 term at University of Illinois, Urbana Champaign.

Page1 / 4

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online