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21s_562ln08

# 21s_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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MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF CHEMISTRY Physical Chemistry 5.62 EINSTEIN AND DEBYE SOLIDS 1. Einstein model The basic assumption is 3 N harmonic vibrational modes all of the same frequency, ν E . The ther- modynamic functions are given in terms of the reduced parameter, x h ν E / kT . Thermodynamic function Molar value for Einstein solid ( E E 0 ) / 3 RT x ( e x 1) 1 ( A E 0 ) / 3 RT n (1 e x ) S / 3 R x ( e x 1) 1 n (1 e x ) C V / 3 R x 2 e x ( e x 1) 2 2. Debye Model In the Debye model there are 3 N harmonic modes, whose distribution in frequencies is given by ρ ( ν ): ρ ( ν ) = 9 N ν 2 ν max 3 for ν < ν max ρ ( ν ) = 0 for ν > ν max where c is the velocity of sound in the solid, V is the volume, and ν max = c (3 N / 4 π V ) 1 / 3 . The thermody- namic functions for a Debye solid are expressed in terms of the dimensionless parameter y = θ D / T , where the Debye temperature, θ D , is θ D = h ν max / k The thermodynamic functions are often written in terms of the Debye function, D ( y ): D ( y ) = 3 y 3 ± y u 3 ( e u 1) 1 du 0 where u = h ν/ kT . Thermodynamic function Molar value for Debye solid ( E E 0 ) / 3 RT 3 y 3 ± y u 3 ( e u 1) 1 du = D ( y ) 0 ( A E 0 ) / 3 RT 3 y 3 ± y u 2 n (1 e u ) du = n (1 e y ) 1 D ( y ) 0 3 S / 3 R ( E A ) / 3 RT = 3 4 D ( y ) n (1 e y ) C V / 3 R 3 y 3 u 4 e u ( e u 1) 2 du = 4 D ( y ) 3 y ( e y 1) 1
5.62 Lecture # 13 Supplement Page 2 Einstein Functions x ( e x 1) x ln (1 x x ) ( e x x 2 e 1) x 2 ln (1 e x ) ( e x x 1) 0.05 3.99584 0.99979 3.02063 0.97521 0.10 3.30300 0.99917 2.35217 0.95083 0.15 2.89806 0.99813 1.97118 0.92687 0.20 2.61110 0.99667 1.70777 0.90333 0.25 2.38889 0.99481 1.50869 0.88020 0.30 2.20771 0.99253 1.35023 0.85749 0.35 2.05491 0.98985 1.21972 0.83519 0.40 1.92293 0.98677 1.10963 0.81330 0.45 1.80690 0.98329 1.01508 0.79182 0.50 1.70350 0.97942 0.93275 0.77075 0.55 1.61035 0.97517 0.86026 0.75008 0.60 1.52569 0.97053 0.79587 0.72982 0.65 1.44820 0.96552 0.73824 0.70996 0.70 1.37684 0.96015 0.68634 0.69050 0.75 1.31079 0.95441 0.63935 0.67144 0.80 1.24939 0.94833 0.59662 0.65277 0.85 1.19209 0.94191 0.55759 0.63450 0.90 1.13844 0.93515 0.52184 0.61661 0.95 1.08807 0.92807 0.48897 0.59910 1.00 1.04065 0.92067 0.45868 0.58198 1.05 0.99592 0.91298 0.43069 0.56523 1.10 0.95363 0.90499 0.40477 0.54886 1.15 0.91358 0.89671 0.38073 0.53285 1.20 0.87560 0.88817 0.35838 0.51722 1.25 0.83952 0.87937 0.33758 0.50194 1.30 0.80520 0.87031 0.31818 0.48702 1.35 0.77253 0.86102 0.30008 0.47245 1.40 0.74139 0.85151 0.28315 0.45824 1.45 0.71168 0.84178 0.26732 0.44436 1.50 0.68331 0.83185

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