5_60_lecture25

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MIT OpenCourseWare http://ocw.mit.edu 5.60 Thermodynamics & Kinetics Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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5.60 Spring 2008 Lecture #25 page 1 PARTITION FUNCTIONS The partition functions play a central role in statistical mechanics All the thermodynamic functions can be calculated from them! Start with average system energy U = <E> Substitute β ≡ 1/kT E i U = = pE = 1 Ee β E i i i i i Q Use the following result: Q = e β E = − Ee β E i i i ∂β V,N ∂β i V,N i Then 1 β E i 1 Q lnQ lnQ T E = Ee = − = − = − i ( ) Q i Q ∂β V,N ∂β V,N T V,N ∂β V,N ∂β 1 1 = = − 2 ( T ) T ( kT ) kT lnQ U kT 2 = ( T ) V,N Q(N,T,V) leads to natural connection with A(N,T,V) A = U TS U T A using dA = − pdV SdT = + + μ dN ⎝ ∂ T V,N ⎛ ∂ ( A T ) = 1 A A = 1 A U 1 A = − U T V,N T T V,N T 2 T T V,N T 2 T T V,N T 2 2 ⎛ ∂ ( A T ) 2 ⎛ ∂ lnQ U = − T T
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