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

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

View Full Document Right Arrow Icon
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.
Background image of page 1

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

View Full DocumentRight Arrow Icon
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 = ++
Background image of page 2
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 =
Background image of page 3

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

View Full DocumentRight Arrow Icon
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.
Background image of page 4
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 .
Background image of page 5

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

View Full DocumentRight Arrow Icon
Image of page 6
This is the end of the preview. Sign up to access the rest of the document.

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 Right Arrow Icon
Ask a homework question - tutors are online