Thermodynamics and Statistical Mechanics Exam - Spring 2005, Exams of Physics

Download Thermodynamics and Statistical Mechanics Exam - Spring 2005 and more Exams Physics in PDF only on Docsity! PHYS-4420 THERMODYNAMICS & STATISTICAL MECHANICS SPRING 2005 FINAL EXAMINATION Tuesday, May 10, 2005 NAME: _______________________________________ There are seven pages to this examination. Check to see that you have them all. CREDIT GRADE Problem 1 30% Problem 2 20% Problem 3 30% Problem 4 20% TOTAL 100% To receive credit for a problem, you must show your work, or explain how you arrived at your answer. 1. (30%) The Gibbs function is defined as G = U – TS + PV, where U is internal energy, T is temperature, S is entropy, P is pressure, and V is volume. a) (10%) Show that: dG = – SdT + VdP b) (10%) Use the equation given in part a) to show that: P T G S          c) (10%) Use the equation given in part a) to show that: PT T V P S                 1 (This is one of the Maxwell relations.) 2 chambers holding N0/2 molecules, and the initial state when all the molecules were in one chamber. S =__________________ 5 4. (20%) A new satellite that uses a Carnot engine for power, is being designed. For a high temperature reservoir, it will use a nuclear source at a fixed temperature, T2. Its low temperature reservoir will be a set of cooling fins that radiate heat into space. The temperature of the fins depends on the rate at which they radiate energy, and that is proportional to the fourth power of their temperature, T1. They will maintain a constant temperature when the rate at which the Carnot engine delivers heat to the fins is equal to the rate at which the fins radiate heat to space: 4 1 1 AT dt Qd  where A is a constant determined by the design of the fins. For this system, the Carnot engine will have a maximum power output for a specific value of T1 that is between absolute zero and T2 . a) (5%) Explain briefly why the power output should be a maximum somewhere in this temperature range. (Hint: Power is the rate at which work is done, and .12 dt Qd dt Qd dt dW  ) b) (10%) Find the value of T1 for which the power output is a maximum. Express your answer in terms of T2. 6 c) (5%) What is the efficiency of the Carnot engine when operating at the value of T1 found in part (b), i.e. when the power output is a maximum. 7