Download Thermodynamics and Statistical Mechanics - Exam 2 | PHYS 4230 and more Exams Physics in PDF only on Docsity! Physics 4230 Examination # 2 8 April 2005 In each problem, be sure to give the reasoning for your answer and define any variables you create. If you use a general formula, state that formula clearly before manipulating it. 1. [5 pts] Two systems, A and B, are placed in thermal contact with each other, but are otherwise isolated. Initially TA > TB. Make no assumptions about the dependence of S on U for either system. Determine whether each statement is true or false, including your reasoning. For example, you might cite a relevant thermodynamic law, or give a simple counterexample. A. A and B will always eventually come to the same temperature T . B. Heat will always flow from system A to system B. C. The entropy of each system will always increase. D. The energy of the total system will be minimized. E. The entropy of the total system will be maximized. 2. [3 pts] Two experiments are performed. In the first, an ideal gas of N atoms at temperature Ti in volume Vi is allowed to expand to a final volume Vf in an adiabatic, quasi-static process. In the second experiment, which has the same initial conditions, the gas undergoes free expansion to the same final volume. Which experiment caused the greatest change in entropy? Why? 3. [3 pts] Consider a system with entropy S which is a function of U , V , and N . Derive a formula for the chemical potential for the case of constant energy and entropy. 4. [3 pts] The dependence of entropy S on energy U for each of two identical systems A and B is shown in the figure below; the state of each system is indicated by the labeled dot. Which system has the higher temperature? Give your reasoning. S U A B 1 5. [8 pts] 100 g of ice at -200◦C is placed in a thermally insulated pool with 1,000,000,000 g of water at 25◦C. Find the total entropy change of the ice plus pool system after it reaches thermal equilibrium. You may assume all processes are at constant volume. Data: cice = 2.1 J/(g◦C), cwater = 4.2 J/(g ◦C),lice−water = 335 J/g . 6. In a paramagnet with spin-1 dipoles, each dipole can take on 3 states: −1, 0, and +1, relative to the direction of the magnetic field, with energies −µB, 0, and +µB, respectively. Consider a spin-1 dipole in thermal equilibrium with a reservoir at temperature T . (a) [5 pts] Derive an expression for the partition function for this dipole. (b) [4 pts] Give expressions for the probabilities of the dipole being in each state. (c) [4 pts] Determine and interpret the low and high temperature limits for your answers to part (b). 2
