ProbSet7_solutions(2)

ProbSet7_solutions(2) - Chem C130/MCB C100A Fall 2010...

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Hand in the five problems marked with a * on Thursday, October 21. ANSWER KEY 1. What is the difference between an energy distribution and a microstate? Explain using simple diagrams. 2.* Consider a system of three identical and independent molecules, with energy levels that have values 0,1,2,3, …(arbitrary units of energy). Consider two states of the system, A and B, shown below. States A and B have equal energy (6 units). State A has one molecule in level 4 and two in level 1. State B has 1 molecule each in levels 1, 2 and 3. The multiplicity, W, of each state is given by N! / (n 1 ! x n 2 ! x n 3 ! …x n j ! x….), where N is the total number of molecules and n j is the number of molecules in state j. (i) Calculate the relative probability of observing the molecules in configuration A over configuration B. (ii) What is the change in entropy in going from state A to state B (assume k b =1). (iii) Assume that the 3 molecules in the system have access to only the 5 energy levels shown above. When the total energy of all 3 molecules is increased to Chem C130/MCB C100A. Fall 2010. Page 1 Problem Set 7 UC Berkeley
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the maximum possible value, explain whether the entropy increases or decreases with respect to state B. (iv) Next, assume that the system has an infinite number of possible energy levels of increasing energy. In this case, when the total energy of the 3 molecules is increased does the entropy always increase, or does it sometimes increase and sometimes decrease? Explain your answer. 3. A more accurate version of Stirling’s approximation is given below: ln N ! = N ln N N + ln( 2 π N ) Estimate, by trial and error, how big N has to be for the error introduced by neglecting the last term to be less than 1% of the value of ln N! Chem C130/MCB C100A. Fall 2010. Page 2 Problem Set 7 UC Berkeley
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4. * A reversible process takes place within a system containing 2 moles of a molecule, with the absorption of 10 kJ mol -1 of heat from the surroundings at 300K. What is the ratio of the multiplicities of the initial and final states of the system? 5. Consider a chamber that is divided into two equal right and left halves by a semi- permeable partition that allows A-type molecules to pass through, but blocks the passage of B-type molecules. There are 10 6 type B molecules on the left side of the partition. Consider the two situations described below: (i) There are 330,000 type A molecules on the left side of the partition, and 670,000 type A molecules on the right side of the partition. (ii) There are 320,000 type A molecules on the left side of the partition and 680,000 type A molecules on the right side of the partition. (A) Which situation is more likely, case (i) or case (ii)? What are their relative multiplicities? Use the probabilistic definition of the entropy to calculate the relative multiplicities.
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