ME501HW3SolnF2011

# ME501HW3SolnF2011 - 1 ME 501 Homework #3 Due Friday,...

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1 ME 501 Homework #3 Due Friday, October 14, 2011 Prof. Lucht (E-mail address: Lucht@purdue.edu) 1. The system shown below has available energy levels of 0, 2, 4, and 8 units. The degeneracy of each of the four levels is 4. The thermodynamic assembly has 7 particles (N = 7) and a total system energy of 16 units (E = 16). (a) For bosons, what are the available macrostates? How many microstates are associated with each of these macrostates and what is the most probable macrostate? What is the entropy of the system? For Bose-Einstein statistics, the number of microstates for a given macrostate   j N is given by: WW N g Ng mBE jBE jj ,, () ! !( )!   1 1 (b) For fermions, what are the available macrostates? How many microstates are associated with each of these macrostates and what is the most probable macrostate? What is the entropy of the system? For Fermi-Dirac statistics, the number of microstates for a given macrostate   j N is given by: g N mFD jFD j jjj ! ) ! (c) What is the entropy of an assembly of fermions that occupies the energy levels shown with N=16 and E = 56?

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2 (a) For bosons the available macrostates are: N=7, E=16 j j g j A B C D E F G H 3 8 4 2 1 1 1 0 0 0 0 2 4 4 0 2 1 0 4 3 2 1 1 2 4 0 0 2 4 0 2 4 6 0 0 4 5 4 3 2 3 2 1 0  1! 5! 3! 3! 8! 5 4 8 7 6 1 1 560 2!3! 0!3! 0!3! 5!3! 2 3 2 !1 ! 4! 5! 7! 5 4 7 6 5 4 1400 1!3! 2!3! 4!3! 2 3 2 6! 1!3! 1!3! 2!3! 3!3! jj A j B C Ng W W W        54 654 4 4 3200 23 2 1400 7 6 5 6 5 4 700 4!3! 3!3! 3 2 3 2 5! 5! 6 5 4 5 4 5 4 2000 3!3! 2!3! 2!3! 3 2 2 2 1400 9! 9 8 7 4 336 1!3! 6!3! 3 2 DB E F GB H WW W W W  22 10,996 ln 9.305 1.285 10 / = most probable macrostate ABCDEFGH BB WWWWWWWW Sk k J K C    (b) for fermions, macrostates A and H are excluded because j j . ! 4 3 41 1 2 4 1!3! 2!2! 0!4! 4!0! 2 !! 4 4 4 384 1!3! 1!3! 3!1! 2 24 j B j jjj C g W N W
3 23 4! 4 4!0! 3!1! 4 3 4 3 4 144 2!2! 2 2 24 604 ln 6.404 8.841 10 / = most probable macrostate E F GB BCDEFG BB W W WW WWWWWW Sk k J K C         (c) For fermions, for N=16, all quantum states are occupied and 4 j N for all levels. The energy of the assembly is 0+8+16+32 = 60 units. For this macrostate,  !

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## This note was uploaded on 12/26/2011 for the course ME 501 taught by Professor Na during the Fall '10 term at Purdue University-West Lafayette.

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ME501HW3SolnF2011 - 1 ME 501 Homework #3 Due Friday,...

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