Exam1SolnME501F2011

# Exam1SolnME501F2011 - 1 Name ME 501 Exam#1 19 October 2011...

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1 Name___________________________ 19 October 2011 Prof. Lucht 1. POINT DISTRIBUTION Problem #1 40 points _________________ Problem #2 45 points _________________ Problem #3 15 points _________________ 2. EXAM INSTRUCTIONS Write your name on each sheet. This exam is closed book and closed notes. Four equation sheets are attached. When working the problems, list all assumptions, and begin with the basic equations. If you do not have time to complete evaluation of integrals or of terms numerically, remember that the significant credit on each problem will be given for setting up the problem correctly and/or obtaining the correct analytical solution. ME 501 Exam #1

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2 ME 501 Exam #1 19 Oct 2011 Name___________________________ 1. (40 points) The system shown below has available energy levels of 0, 2, 4, 6, and 8 units. The degeneracy of each of the four levels is 3. The thermodynamic assembly has 6 particles (N = 6) and a total system energy of 14 units (E = 14). (a) For bosons, what are the available macrostates? (There are 8.) 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:    ,, 1! !1 ! jj mBE jBE Ng WW    (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: ! !! j mFD jFD jjj g N (c) What is the energy and entropy of entropy of an assembly of fermions that occupies the energy levels shown with N=15 and T= 0 K? What is the energy and entropy of an assembly of bosons with N=15 and T= 0 K? j 4 0 1 2 3 j 8 0 2 4 6 g j 3 3 3 3 3
3 (a) For bosons the available macrostates are: N=6, E=14 j j g j A B C D E F G H I J 4 8 3 1 1 1 0 0 0 0 0 0 0 3 6 3 1 0 0 2 1 1 1 0 0 0 2 4 3 0 1 0 0 2 1 0 3 2 1 1 2 3 0 1 3 1 0 2 4 1 3 5 0 0 3 4 3 2 3 3 2 1 2 1 0 噺嚇  1! 3! 3! 2! 6! 6 5 3 3 135 1!2! 1!2! 0!2! 4!2! 2 !1 ! 3! 3! 3! 5! 5 4 3 3 3 270 1!2! 1!2! 1!2! 3!2! 2 3! 5! 4! 1!2! 3!2! 2!2! jj A j B C Ng W W W       54 43 3 180 22 4! 3! 5! 4 3 5 4 3 180 1!2! 2 2 180 3! 3! 4! 4! 4 3 4 3 3 3 324 1!2! 1!2! 2 2 3! 6! 3! 6 5 33 1 3 1!2! 1!2! 2 D ED F G W WW W W   22 5 180 180 3! 7! 7 6 36 3 1!2! 5!2! 2 1827 ln 7.510 1.037 10 / = most probable macrostate HD ID J ABCDEFGH I BB W WWWWWWWWW Sk k J K F   噺嚅

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4 (b) for fermions, macrostates A, G, and J are excluded because j j Ng .
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Exam1SolnME501F2011 - 1 Name ME 501 Exam#1 19 October 2011...

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