Lecture 9 - Partition Functions for Molecules(18.3-18.8...

This preview shows pages 1–6. Sign up to view the full content.

Lecture 9 1 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 =+ + + () and accordingly: q V,T trans rot vib elec qq q q = ( ) , and, as always: Q N,V,T ! N qVT N = First, consider a diatomic molecule: ( ) 3/2 12 2 2 , B trans mm k T qV T V h π ⎡⎤ + = ⎢⎥ Where the total mass of the molecule (m 1 +m 2 ) is substituted for m in our monatomic solution.

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

View Full Document
Lecture 9 2 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. Fig. 18.2 We will define separated atoms at R= in their ground electronic state as our zero for energy. 2 // 12 : ... eB e B DkT kT elec e e Thus q g e g e ε = ++
Lecture 9 3 Now we can consider vibrations . v 1 v v 0,1,2. .. for a H.O. 2 hv ε ⎛⎞ =+ = ⎜⎟ ⎝⎠ Note: in your text, the v’s (quantum numbers) look a lot like the v ’s (frequencies). Be careful! () v v1 /2 v v0 hv hv hv vib qT e e e e β βε ββ ∞∞ −+ −− == = = ∑∑ 2 0 1 for 1: 1 ... 1 n n x xx x x = <= + + + = See the back cover of your text for common expansions. since 1, 1 hv 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 B hv k Θ = vib his relates the vibrational frequency, , to a vibrational temperature, . v Θ / 1 vib vib T vib T e e −Θ =

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

View Full Document
Lecture 9 4 2 vib ln s we noted before: E vib B dq Nk T dT = / 21 vib vib vib B T Nk e Θ Θ Θ =+ As I showed in lecture 6: () 2 / , , 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 , 22 1 0.368 , 0.921 0.632 1 vib V vib e at T C R R R e = = = That is, at the vibrational temperature, the heat capacity is 92.1% of the high temperature limit.
Lecture 9 5 We can also use these partition functions to calculate the population distribution as a function of temperature. Since all ideal diatomic gases will behave the same, we will consider temperatures in terms of Θ vib .

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

View Full Document
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 / 16

Lecture 9 - Partition Functions for Molecules(18.3-18.8...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online