22_562ln08

22_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 Spring 2008 Lecture # 22 Page 1 Einstein and Debye Solids Reading: Hill, pp. 98-105, 490-495 The Einstein (quantum) model (all vibrational modes have the same frequency) gave much better agreement with experiment than the Dulong-Petit (classical) model (equipartition). But as T decreases, the Einstein model C V decreases too fast relative to the experimentally observed (approximately T +3 dependence) behavior of C V . Perhaps it would be more realistic to allow the vibrational frequencies to follow a plausible, computationally convenient, but non-constant probability distribution, ρ ( ν ) Debye Treatment Debye derived an improved model for the thermodynamic properties of solids by assuming that the distribution of normal mode frequencies is equivalent to that for sound waves. C V = k i = 1 3 N ! 6 " h # i kT $ % ( ) 2 e ! h # i kT 1 ! e ! h # i kT ( ) 2 C V Debye = k 0 # max * d #+ ( # ) h # kT $ % ( ) 2 e ! h # kT 1 ! e ! h # kT ( ) 2 density of vibrational frequencies For sound waves traveling in a three-dimensional solid ! " ( ) # " 2 [see Nonlecture derivation, below]. One way of seeing this is that ! = c 2 " k ~ = c 2 " k x 2 + k y 2 + k z 2 ( ) 1/2 , where k ~ is the wave vector such that the wave is described by e i k ~ r ~ and ! = 2 " | ! k | . The density of states with a given ν is the number of ways of choosing k ~ with the corresponding magnitude, k ~ . Just as the degeneracy for a given speed state is proportional to c 2 in the kinetic k ~ theory of gases (as you'll see later in 5.62), the number of ways of picking with magnitude k = | k ~ | is proportional to k 2 . revised 3/21/08 9:01 AM
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5.62 Spring 2008 Lecture # 22 Page 2 So ρ ( ν ) k 2 ∝ ν 2 This figure is supposed to show a spherical shell of radius ! k
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22_562ln08 - MIT OpenCourseWare http/ocw.mit.edu 5.62...

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