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PHYS427_Fa08_HE02 B_OpenBook

# PHYS427_Fa08_HE02 B_OpenBook - Physics 427 – Thermal...

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Unformatted text preview: Physics 427 – Thermal Physics November 19, 2008 2nd Midterm Exam ‐‐‐ Part B: Problems ‐‐‐ OPEN BOOK [Use the 8‐page exam book; 80 points total] 1. Consider a gas of N spinless Bose particles. Each of the particles has mass m and all are enclosed in a volume V at temperature T. [7 points each part; 28 points total] a. Find an expression for the density of available single‐particle states D(ε) as a function of the single–particle energy ε. Assume the system is three‐dimensional. Sketch the results, being careful to label the axes and the origin. b. Write down an expression for the mean occupation number of a single particle state, <ne>, as a function of ε, T and μ(T), where μ is the chemical potential. Consider <ne> for a temperature T above the Bose‐Einstein transition temperature, TBE, such that T>TBE. On the same graph you drew for part a, (i.e., use the same x‐axis) sketch <ne> at temperature T. Show the point where ε = μ . c. Write down an integral expression which implicitly determines μ(T). As the temperature, T, is lowered, how does μ(T) change? It will help to refer to your sketch. d. Now find TBE. You may find this useful: ∞ x ∫ dx e x − 1 0 1/ 2 = 1.306π 1 / 2 2. Consider the P−V diagram below. When the system is taken from state i to state f along path iaf, it is found that the heat absorbed by the system is Q = 50 cal and the work done by the system is W = 20 cal. Along path ibf, Q=36 cal. [7 points each part; 28 points total] P a f i c b V a) b) c) d) What is the work done by the system, W for the path ibf? If W = −13 cal for the path fci, what is Q for this path? Take Ei = 10 cal What is Ef? If Eb = 22 cal, what is Q for the process ib? 3. Iodine is contained in a sealed oven. The iodine is partially in atomic form and partially in molecular form. The molecules can split apart, and the atoms can combine: 2I ‐→ I2. The oven's pressure is held at an original value po, and the temperature is held at a value τ = τo, chosen so that the number of I atoms per unit volume equals the number of I2 molecules per unit volume. Now, holding τ fixed, more iodine is added to the oven, increasing the pressure to a new value, 2po. [8 points each 24 points total]. Let n1 and n1' be the original and the new concentration of the I atoms, respectively. Let n2 and n2' be the original and the new concentration of the I2 molecules, respectively. [8 points each part; 24 points total] a.) Write down an equation in terms of the concentrations that guarantees that the pressure is doubled. b.) Write down an equation in terms of the concentrations that guarantees that the temperature is unchanged. c.) Find the ratio n1'/n1 of the new number to the original number of I atoms per unit volume. ...
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