10_562ln08

# 10_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 #10: Quantum vs. Classical. q trans . Equipartition. Internal Degrees of Freedom. GOAL: Calculate average translational energy via quantum and classical descriptions and compare results. QUANTUM DESCRIPTION — Calculate — average kinetic energy in x-direction x ε ε x = ε x P ( ε x ) d ε x 0 kT P ( ε x ) = ( π kT ) 1/2 ε x 1/2 e −ε x 1 1/2 1/2 e x kT d ε x = ε x π kT 0 ε x 1 1/2 kT ( π kT ) 1/2 = 1 kT ε x = π kT 2 2 This is not an accident. It is our first glimpse of “equipartition” of energy, 1 2 kT into each independent degree of freedom. Integral table for two useful integrals x 1/2 e ax dx = π 1/2 2 a 3/2 ( dimension: length 3/2 ) 0 0 x e ax dx = ( π a ) 1/2 ( dimension: length 1/2 ) Since the translational energies in each dimension are uncorrelated, average total energy is the sum of the average energy in each direction. Also, as a consequence of the separability of the Hamiltonian with respect to x, y, z coordinates, total E is sum of E x + E y + E z . ε = 1 2 kT + 1 2 kT + 1 2 kT = 2 3 kT Each degree of freedom contributes 1 2 kT to total energy. Agrees with result from ensemble average ε + ε + ε x y z
5.62 Spring 2008 Lecture #10, Page 2 lnQ trans 3 ε = E = kT 2 T = P α E α = 2 kT N,V α CLASSICAL MECHANICAL DESCRIPTION — calculate ε State of a molecule is described by p momentum, q position. The energy of ~ ~ N–molecule system is ε (p 3N , q 3N ), which is a continuous function of 6N variables. ~ ~ q trans e −ε i kT let's assume this by analogy e (p ~ ,q ~ )/kT dp dq = ~ ~ i quantum classical — integral over 3 momentum and 3 position coordinates for each particle = p x 2 + p y 2 + p z 2 ( ) q trans,cl = ∫∫∫ dq x dq y dq z ∫∫∫ dp x dp y dp z e p 2 / 2mkT = V dp x e p x 2 /2mkT dp y e p y 2 /2mkT dp z e p z 2 /2mkT = V −∞ dp x e p x 2 /2mkT ( ) 3/2 3 = V 2 π mkT [q trans,cl.

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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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10_562ln08 - MIT OpenCourseWare http:/ocw.mit.edu 5.62...

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