Exam2ME501F2009

# Exam2ME501F2009 - 1 Name_ ME 501 Exam #2 2 December 2009...

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1 Name___________________________ 2 December 2009 Prof. Lucht 1. POINT DISTRIBUTION Choose two (2) of problems 1, 2, and 3: Problem #1 50 points _________________ Problem #2 50 points _________________ Problem #3 50 points _________________ You are required to do two of the problems. Please indicate the problems you have chosen. 2. EXAM INSTRUCTIONS Write your name on each sheet. This exam is closed book and closed notes. Seven 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 #2

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2 ME 501 Exam #2 2 Dec 2009 Name___________________________ 1. (50 points) The system shown below has available energy levels of 0, k B , 2 k B , and 3 k B units, where = 100 K. The degeneracy of each of the four levels is given by g j = 10,000 + 10,000 j . The thermodynamic assembly has 1000 particles (N = 1000) and the temperature of the assembly is 200 K. For this dilute assembly, the population distribution for the most probable macrostate is given by the Boltzmann distribution law,      exp / exp / exp / jj B B m p B j gk T T NN N N Z T    (a) Using the Boltzmann distribution law, calculate the most probable macrostate {N 0mp , N 1mp , N 2mp , N 3mp }. Round the populations to the nearest integer. (b) For corrected Maxwell-Boltzmann statistics, the number of microstates in a particular macrostate {N j } is given by WN g N CMB j j j N j j ns e j ! Use the Stirling approximation (ln ! ln ) N N N N to show ln ln N N g N CMB j j j j j ej  (c) What is the entropy (J/K) of the assembly? (d) What is the energy (J) of the assembly? (e) Calculate the number of microstates associated with two macrostates that are very similar to the most probable macrostate. Macrostate A is given by {N 0mp , N 1mp +5, N 2mp -10, N 3m +5}. Macrostate B is given by {N 0mp , N 1mp -5, N 2mp +10, N 3m -5}. Comment on the results.
3 ME 501 Exam #2 2 Dec 2009 Name___________________________ 2. Diatomic hydrogen gas is contained in a rigid pressure vessel with   10 3 . m . The initial pressure is 1 kPa and the temperature is 50 K. (a) Calculate the amount of heat transfer required to raise the temperature of the gas from 50 K to 100 K. Assume that the H 2 is an ideal gas (translational mode fully excited), a rigid rotator, and a harmonic oscillator (see equation sheets). For H 2 , odd-J rotational levels have a nuclear spin statistical weighting factor (NSSW) of 3, and even-J rotational levels have an NSSW of 1. Do not assume that / rot rot ZT  . Instead use the combined rotational- nuclear partition function:  ,, exp J rot nuc rot J J J B Z NSSW g kT     Consider rotational levels with 04 J . Recall that B PN k T   . You may the tables on the next page to be useful in organizing your work.

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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.

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Exam2ME501F2009 - 1 Name_ ME 501 Exam #2 2 December 2009...

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