12_562ln08

12_562ln08 - MIT OpenCourseWare http:/ocw.mit.edu 5.62...

This preview shows pages 1–4. Sign up to view the full content.

MIT OpenCourseWare http://ocw.mit.edu 5.62 Physical Chemistry II Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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

View Full Document
5.62 Lecture #12: Rotational Partition Function. Equipartition Readings: Hill, pp. 153-159; Maczek, pp. 47-53 Metiu, pp. 131-142 motion where n is the number of atoms in the molecule. For a diatomic or a linear polyatomic molecule: 3 TRANSLATIONAL degrees of freedom 2 ROTATIONAL degrees of freedom 3n–5 VIBRATIONAL degrees of freedom 3n TOTAL degrees of freedom DEGREES OF FREEDOM A molecule with n atoms has 3n "degrees of freedom" or 3n coordinates to describe its position and therefore has 3n ways of incorporating energy due to nuclear For a diatomic molecule 3n – 5 = 1 vibrational degree of freedom MOLECULAR ROTATIONAL PARTITION FUNCTION — q rot — DIATOMIC ε rot (J) = J(J + 1) hcB e for J = 0,1,2, g J = 2J + 1 q rot = g( ε ) e ε /kT = (2J + 1) exp[–hcB e J(J+1)/kT] J = 0 allowed rotational energies Question: How do you do the summation? Two cases (Low-T limit case [next Lecture]) Case 1: ε rot /kT 1 or hcB e J(J + 1)/kT 1 [More precisely, we want (E(J + 1) – E(J)) kT at E(J) kT.) rotational states are closely spaced in energy compared to kT — since energy spacings are so close together, can consider
5.62 Spring 2008 Lecture 12, Page 2 ε rot as continuous and use Euler-MacLaurin Summation Formula (draw a picture!) this case is the classical or high-temperature limit. n 1 J = m f J ( ) = m n f(J)dJ + 2 [ f(m) + f(n) ]+ residue so: q rot = 0 (2J + 1) exp[–hcB e J(J+1)/kT] dJ + 2 1 [1 + 0] + J = 0 J = substitute ω = J(J+1) thus d ω = (2J+1)dJ q rot = 0 exp [ hcB e ω / kT ] d ω + 2 1 +… kT e hcB e ω kT 1 = hcB e 0 + 2 = 0 kT 1 hcB e + 2 = kT 1 kT usually can ignore the 1 q rot hcB e + 2 hcB e 2 What happens for a 1 state where J min = 2 rather than 0?

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 11/27/2011 for the course CHEM 5.43 taught by Professor Timothyf.jamison during the Spring '07 term at MIT.

Page1 / 8

12_562ln08 - MIT OpenCourseWare http:/ocw.mit.edu 5.62...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online