12_562ln08

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

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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 .
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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
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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?
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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.

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

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