PHYS427_Fa08_HE01 B_OpenBook - Physics 427 – Thermal...

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Unformatted text preview: Physics 427 – Thermal Physics October 22, 2008 1st Midterm Exam ‐‐‐ Part B: Problems ‐‐‐ OPEN BOOK (Use the 8‐page exam book) Here are a couple of extra integrals that you may need: ∞ 1. ∫ dx 0 2 1 x e − ax = 2a 1⎛π ⎞ ∫ dx e − ax 2 = ⎜ ⎟ 2⎝a⎠ 0 ∞ 1 2 A popular area of research is “quantum dots”, in which small (nanoscale size) conducting islands are constructed on an insulating substrate. It is shown that a quantum dot can be fabricated such that it will behave like two‐dimensional quantum‐mechanical harmonic oscillator. It may be useful to recall that the energy spectrum of a 2‐D harmonic oscillator is: Emn = ħω (1+m+n). [6 points each] a) Calculate the free energy of a quantum dot. b) Calculate the entropy of the quantum dot c) Sketch the entropy of a quantum dot from T=0 to T>>ħ ω. d) Sketch the heat capacity of a quantum dot from T=0 to T>>ħ ω. An ideal, monatomic gas is allowed to expand reversibly from a volume Vo to three times that volume, 3Vo. [10 points each] a.) If no heat is allowed to enter or leave the gas during the expansion, show that pV5/3 is constant. (Note: starting with pVγ=constant is not enough of a proof.) b.) If the gas contains N atoms and its temperature is held constant at a value o during the expansion, find the change in thermal energy. c.) If the gas contains N atoms and its temperature is held constant at a value o during the expansion, find the change in the gas's entropy. The solar constant, which is the radiant flux from the sun received at the surface of the earth, is about 0.1 W/cm2. Assuming the sun is a black body, what is the temperature of the sun? (The radius of the sun is rS = 7.0×105 km and the distance between the earth and the sun is rSE = 1.5×108 km. [10 points] The three lowest energy levels of a certain molecule are E1 = 0, E2 = ε, E3 = 10 ε. [5 points each] a) Show that at sufficiently low temperatures (state how low), only levels E1 and E2 are populated. b) Find the average energy E of the molecule at temperature T. c) Find the contributions of these levels to the specific heat, Cv, and d) Sketch Cv as a function of T. 3. 4. 5. ...
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This note was uploaded on 12/06/2009 for the course PHYS 427 taught by Professor Flynn during the Fall '08 term at University of Illinois, Urbana Champaign.

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