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5_60_lect26_27

# 5_60_lect26_27 - MIT OpenCourseWare http/ocw.mit.edu 5.60...

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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 #26-27 page 1 STATISTICAL THERMODYNAMICS Calculation of macroscopic thermodynamic results Entropically driven examples: Free expansion of a gas V 2 gas vacuum V 1 gas Lattice model for ideal gas translation: Molecular volume v, Total volume V All molecular positions have equal energy ε trans = 0 All system microstates have equal energy E trans = 0 V 1 Calculate S = k ln Ω in each state V 2 Molecular degeneracy g = V/v System degeneracy Ω = g N /N! = (V/v) N /N! For expansion from volume V 1 to V 2 , Ω () Vv N N ! Δ= S kln Ω− kln Ω = kln 2 = kln 2 2 1 Ω Vv N N ! 1 () 1 Δ= SN k l n V 2 = n R l n V 2 V 1 V 1 V Should look familiar! And Δ= Δ−Δ= H T S 2 G n R
5.60 Spring 2008 Lecture #26-27 page 2 Ideal gas mixture N A V A N B V B N = N A + N B V = V A + V B Assume same initial (p,T) for A & B same (p,T) for mixture Assume equal molecular volumes & lattice cell sizes. Then initially ( Vv ) N A ( V v ) N B B S = kln Ω + kln Ω = kln Ω Ω = kln A 1 A B A B N! N! A B After mixing: Count how many ways to distribute N A molecules of A and N B molecules of B among the (V/v) lattice sites As before, the number of ways to distribute N molecules among (V/v) sites is (V/v) N . To correct for indistinguishability, divide by N A !N B ! So the final state entropy is

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5_60_lect26_27 - MIT OpenCourseWare http/ocw.mit.edu 5.60...

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