Thermodynamics & Statistical Mechanics Exam - Spring 2001, Exams of Physics

Download Thermodynamics & Statistical Mechanics Exam - Spring 2001 and more Exams Physics in PDF only on Docsity! 1 PHYS-4420 THERMODYNAMICS & STATISTICAL MECHANICS SPRING 2001 FINAL EXAMINATION Tuesday, May 1, 2001 Your grade will be sent to you by e-mail by midnight, Wednesday, May 2, 2001 NAME: _______________________________________ e-mail address:___________________ There are seven pages to this examination. Check to see that you have them all. To receive credit for a problem, you must show your work, or explain how you arrived at your answer. 1. (20%) Consider a collection of N identical, distinguishable harmonic oscillators, all of frequency . The energies that one these oscillators can take on (measured relative to the ground state) are n = nh. Where n is an integer that can take on values from 0 to . a) (5%) Find the partition function , for one of these oscillators. (Hint: x x n n    0 1 1 for x < 1)  = _______________ b) (5%) Find the partition function Z, for the collection of oscillators. (Since the oscillators are distinguishable, no N! is needed in the denominator.) Z = _______________ c) (5%) Find the Helmholtz function F, for this collection of oscillators. (Hint: F = – kT ln Z) F = _______________ 2 d) (5%) Find the entropy S, for this collection of oscillators. Hint: NVT F S ,       . This will not give a particularly neat expression. However, you can check your result as follows: as T goes to zero, S goes to zero, and as T gets very large, S goes to             h kT NkS ln1 . S = __________________________________ 5 g) (6%) Calculate the work done by the gas and the heat added to the gas as the system goes from d to a. Express your answers in terms of p0 and V0. (Hint: These are both negative.) Wda = __________________ Qda = __________________ h) (3%) Calculate the net work done by the system as it goes around the complete cycle. Express your answer in terms of p0 and V0. W = __________________ i) (3%) Calculate the net heat that goes into the system as it goes around the complete cycle. Express your answer in terms of p0 and V0. Q = __________________ j) (3%) Calculate the efficiency of this engine. (Hint: This should be a pure number. If you can not get rid of constants, be sure you explain what you are trying to calculate.)  = __________________ k) (3%) What would be the efficiency of a Carnot engine that could be operated between the highest and the lowest temperatures that this system reaches. (Hint: This should be a pure number. If you can not get rid of constants, be sure you explain what you are trying to calculate.) Carnot = __________________ 6 l) (2%) Based on your answers to (j) and (k), which would be the better engine? Which would be the better heat pump? (Circle your choices.) Better engine: Carnot cycle Cycle of this problem Better heat pump: Carnot cycle Cycle of this problem 3. (30%) N molecules of an ideal diatomic gas are confined to a vessel of volume V that is maintained at a constant temperature, T. At the temperature T the vibrational degrees of freedom of the molecules are not active. (Each molecule has five degrees of freedom.) a) (4%) What is the internal energy of this gas? Express your answer in terms of N, T, k, and pure numbers. E = ________________ b) (9%) For this gas, find the heat capacities, CV and Cp, and  = Cp/CV. Express your answers in terms of N, k, and pure numbers. (For an ideal gas Cp – CV = Nk) CV = ___________ Cp = ___________  = ___________ c) (4%) The gas is irradiated with ultra-violet light, and one third (1/3) of the N molecules dissociate, each into two atoms. What is the internal energy of this mixed gas? Remember, T is held constant. (The vessel is in contact with a heat reservoir.) Express your answer in terms of N, T, k, and pure numbers. E = ________________ 7 d) (9%) For this mixed gas, find the heat capacities, CV and Cp, and  = Cp/CV. Express your answers in terms of N, k, and pure numbers. CV = ___________ Cp = ___________  = ___________ e) (4%) Determine the ratio of the pressure of the gas after the irradiation to that before the irradiation. Your answer should be a pure number. pafter/pbefore = _____________