Physics Exam: Problems on Thermodynamics and Statistical Mechanics, Exams of Thermal Physics

Download Physics Exam: Problems on Thermodynamics and Statistical Mechanics and more Exams Thermal Physics in PDF only on Docsity! 1 PHYSICS 461 NAME ______________ Sample Exam #2 All work must be shown in order to receive full credit. Work must be legible and comprehensible with answers clearly indicated. Equation sheet is on the last page. PROBLEM POINTS SCORE 1 10 2 35 3 25 4 10 5 20 TOTAL 100 2 1. (10 points) Prove that, for any system in equilibrium with a reservoir at temperature T, the average value of E2 is E2 = 1 Z ∂2Z ∂β 2 , where β=1/kT. 2. (35 points) Consider a paramagnetic system of spins with magnetic moment µ which can point in three possible directions, all in one plane, as shown in the figure. (a) (5 points) Show that the energy of a single spin can have one of two values, -µB and µB/2, in the presence of a magnetic field in the z-direction, r B = Bẑ . (b) (10 points) Assume there is no interaction between the spins, find the partition function for a system of N spins when an external magnetic field is applied along z. (c) (10 points Calculate the average magnetic moment (or magnetization) per spin of this paramagnet when an external magnetic field is applied along z. (d) (10 points) Calculate the average energy per spin of this paramagnet when an external magnetic field is applied along z. 3. (25 points) Consider the following formula for a partition function: Z = V N (2πmkT )5 N /2 (a) (10 points) Calculate the equation of state (pressure in terms of volume and temperature) from this partition function. (b) (10 points) Calculate the heat capacity (at constant volume) from this partition function. (c) (5 points) From the results in (a) and (b) identify the system described by this partition function. x z 2π/3 2π/3