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‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ PHYS 420 Thermal Physics Spring 2016, Problem Set 7 ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ Reading material: Gould and Tobochnik, Chapter 4.2 4.5 Problem 1 (5 points) Problem 4.14, Estimation of the number of states (page 199). Problem 2 (5 points) Problem 4.17, Form of entropy (page 205). Problem 3 (6 points) In class, we derived ܥ ܰ݇ for ideal gas. For real mono atomic gas (like He), the measured ܥ is indeed close to this value (given that the density of the gas is low). For di atomic gas (like H 2 ), ܥ is close to ܰ݇ at room temperature. The value of ܥ becomes even larger at very high temperature. a) Please explain these observations (explain in words, not formulas). b) We have shown that for a mono atomic ideal gas of N atoms, Ω∝ܧ యಿ . Show that for di atomic gas, Ω should be proportional to ܧ ఱಿ in order to have ܥ ܰ݇ . Problem 4 (9 points) Gas mixture We mix a mono atomic gas (total molecule # N1) with a di atomic gas (total molecule # N2). We approximate both as ideal gases, and we can treat each gas as a separate system despite the intermingling of the two gases in space. The internal energy of gas 1 and gas 2 is ܧ and ܧ . The total energy of the two gases ܧൌܧ ൅ܧ is fixed; but the two gases can exchange energy with each other. a) Calculate the probability for gas 1 to have energy ܧ . (Hint: start with the following expression ܲሺܧ ሻൌ Ω ሺܧ ሻΩ ሺܧ Ωሺܧሻ where Ω ሺܧ is the # of microstates for gas 1 at energy ܧ ሺܧ is the # of microstates for gas 2 at energy ܧ ൌܧെܧ , and Ω is the total # of microstates of the mixed system regardless of how ܧ is partitioned into ܧ and ܧ . Think about how to explain this formula.) b) For this mixed system, which energy ܧ has the largest probability to occur?

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