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Physics 831: Statistical Mechanics

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Physics 831: Statistical Mechanics Physics 831: Statistical Mechanics Russell Bloomer1 University of Virginia Note: There is no guarantee that these are correct, and they should not be copied 1email: [email protected] Contents 1 Problem Set 1 1 1.1 Kittel 8.1: Heat pump ........................................ 1 1.2 Kittel 8.6: Room air conditioner .................................. 1 1.3 Kittel 8.9: Cooling of nonmetallic solid to T = 0 ......................... 2 1.4 Sterling Heat Engine ......................................... 2 1.5 Unavailability for work ........................................ 3 1.6 Gibbs Free Energy .......................................... 3 2 Problem Set 2 5 2.1 Spin model .............................................. 5 2.2 Paramagnetism of a system of N localized spin-1/2 particles ................... 6 2.3 Kittel 2.3: Quantum harmonic oscillator .............................. 7 2.4 Review Thermal mechanics ..................................... 7 2.5 Review Thermal mechanics ..................................... 8 3 Problem Set 3 9 3.1 Kittel 3.2: Magnetic susceptibility ................................. 9 3.2 Kittel 3.3: Free energy of a harmonic oscillator .......................... 10 3.3 Kittel 3.4: Energy fluctuations ................................... 10 3.4 Kittel 3.10: Elasticity of polymers ................................. 11 3.5 Ising spin chain ............................................ 12 4 Problem Set 4 13 4.1 Application of equal partition theorem ............................... 13 4.2 Thermal Equilibrium of the Sun and Earth ............................ 14 4.3 Kittel 4.3: Average temperature of the interior of the Sun .................... 14 4.4 Kittel 4.6: Pressure of thermal radiation .............................. 15 4.5 Kittel 4.7: Free energy of a photon gas ............................... 15 4.6 Kittel 4.18: Isentropic expansion of photon gas .......................... 16 5 Problem Set 5 17 5.1 5.1: Kittel 4.14: Heat capacity of liquid 4He at low temperature ................... 17 5.2 5.2: Kittel 4.15: Angular distribution of radiant energy flux ...................... 17 5.3 5.3: Qualifying exam file problem ................................... 17 5.4 5.4: Helmholtz free energy for the Debye model ............................ 17 5.5 5.5: Kittel 5.3: Potential energy of gas in gravitational field ...................... 17 i 6 Problem Set 6 19 6.1 6.1: Kittel 5.4: Active transport .................................... 19 6.2 6.2: Kittel 5.7: States of positive and negative ionization ....................... 19 6.3 6.3: Kittel 5.10: Concentration fluctuation .............................. 20 6.4 6.4: Kittel 6.3: Distribution function for double occupancy statistics ................. 20 6.5 6.5: Kittel 6.7: Relation of pressure and energy density ........................ 21 6.6 6.6: Kittel 6.9: Gas of atoms with internal degree of freedom ..................... 21 7 Problem Set 7 23 7.1 7.1: Pressure in types of gases ..................................... 23 7.2 7.2: Kittel 5.13: Isentropic expansion ................................. 23 7.3 7.3: Kittel 6.8: Time for a large fluctuation .............................. 23 7.4 7.4: Kittel 6.10: Isentropic relations of ideal gas ............................ 23 7.5 7.5: Kittel 6.12: Ideal gas in two dimensions .............................. 23 7.6 7.6: Kittel 7.4: Chemical potential versus temperature ........................ 23 7.7 7.7: The absorbtion of gas onto a surface ............................... 23 8 Problem Set 8 25 8.1 8.1: Mixing of two distinct atoms ................................... 25 8.2 8.2: Kittel 7.3: Pressure and entropy of degenerate Fermi gas ..................... 25 8.3 8.3: Kittel 7.6: Mass-radius relationship for white dwarfs ....................... 26 8.4 8.4: Kittel 7.10: Relativistic white dwarfs stars ............................ 27 8.5 8.5: Electrons in the air off a conductor ................................ 27 9 Special Problem Set (9) 29 9.1 Problem 1: Properties of “Photon Gas” ................................ 29 9.2 Problem 2: Engine Cycle ........................................ 29 9.3 Problem 3: Vibrational Modes of a Molecule ............................. 30 9.4 Problem 4: Relativistic Massless Bosons ................................ 31 9.5 Problem 5: Kittel & Kroemer 7.9 and more .............................. 32 10 Problem Set 10 33 10.1 10.1: Collisions with a wall for a Fermi Gas .............................. 33 10.2 10.2: Free energy and pressure of a Boson gas ............................. 33 10.3 10.3: Discontinuity in the slope of the heat capacity of a Bose gas .................. 34 10.4 10.4: Maximum work extracted from an ideal gas ........................... 34 11 Problem Set 11 37 11.1 11.1:A review problem ......................................... 37 11.2 11.2: Dissociation of water ....................................... 37 11.3 11.3: Practice with the Jacobian .................................... 38 11.4 11.4: More practice .......................................... 39 11.5 11.5: van der Waals Gas ........................................ 39 12 Problem Set 12 41 12.1 12.1: Maxwell Relations ........................................ 41 12.2 12.2: Equilibrium conditions ...................................... 41 12.3 12.3: Fluctuation in number of a Fermi gas .............................. 42 12.4 12.4: Fluctuation in volume, pressure, entropy and temperature ................... 42 12.5 12.5: Kittel 10.5: Gas-solid equilibrium ................................ 42 ii 13 Problem Set 13 45 13.1 13.1: Superconduction and Heat Capacity ............................... 45 13.2 13.2: Kittel 10.8: First order crystal transformation .......................... 45 13.3 13.3: Kittel 11.2: Mixing energy in 3He −4 He and P b − Sn mixtures ................ 46 13.4 13.4: Kittel 11.4: Solidification range of a binary alloy ........................ 46 13.5 13.5: Kittel 11.5: Alloying of gold into silicon ............................. 47 iii Chapter 1 Problem Set 1 1.1 Kittel 8.1: Heat pump (a) Show that for a reversible heat pump the energy required per unit of heat delivered inside the building is given by the Carnot efficiency W τh − τl = ηC = (1.1) Qh τh What happens if the heat pump is not reversible? Ql Qh For a reversible system: σh = σl. From the definition of entropy σl = ; σh = . Then τl τh W Qh − Ql σhτh − σlτl ⇒ τh − τl = = σh = σl = ηC X Qh Qh σhτh τh (b) Assume that the electricity consumed by a reversible heat pump must itself be generated by a Carnot engine operating between the temperatures τhh and τl. What is the ratio Qhh/Qh, of the beat consumed at τhh, to the heat delivered at τh? Give numerical values for Thh = 600 K; Th = 300 K; Tl = 270 K. For a Carnot engine W τhh − τl τhh − τl = ⇒ W = Qhh Qhh τhh τhh The ratio is then τ Q τ W/(τ − τ ) Q (τ − τ )/τ Q 1 − l hh = hh hh l ⇒ hh = h l h ⇒ hh = τh τl Qh τhW/(τh − τl) Qh (τhh − τl)/τhh Qh 1 − τhh For τhh = 600 K, τh = 300 K and τl = 270 K, 270 1 − 300 270 = .18 X 1 − 600 (c) Draw an energy-entropy flow diagram for the combination heat engine-heat pump, similar to Figures 8.1, 8.2 and 8.4, but involving no external work at all, only energy and entropy flows at three temperatures. 1.2 Kittel 8.6: Room air conditioner A room air conditioner operates as a Carnot cycle refrigerator between an outside temperature Th and a room at a lower temperature Tl. The room gains heat from the outdoors at a rate A(Th − Tl); this heat is removed by the air conditioner. The power supplied to the cooling unit is P . (a) Show that the steady state temperature of the room is 1 2 21/2 Tl = (Th + P/2A) − (Th + P/2A) − Th (1.2) dQl The rate is dt = A(Th − Tl), so the power is dW 1 dQ (T − T )2 P = = = A h l dt γ dt Tl Solving for Tl P q T 2 − (2T − P/A) T + T 2 = 0 ⇒ T = T + − (T + P/2A)2 − T 2 l h l h l h 2A h h X (b) If the outdoors is at 37oC and the room is maintained at 17oC by a cooling power 2 kW, find the heat loss coefficient A of the room in WK−1. s 1000 1000 2 40000 (1000)2 620000 (1000)2 290 = 310 + − 310 + − 3102 ⇒ 400 − + = 3102 + + − 3102 ⇒ A = 1450 W/K A A A A2 A A2 X 1.3 Kittel 8.9: Cooling of nonmetallic solid to T = 0 Wa saw in Chapter 4 that the heat capacity of nonmetallic solids at sufficiently low temperatures is proportional to T 3, as C = aT 3. Assume it were possible to cool a piece of such a solid to T = 0 by means of a reversible refrigerator that uses the solid specimen as its (varying!) low-temperature reservoir, and for which the high-temperature reservoir has a fixed temperature Th equal to the initial temperature Ti of the solid. Find an expression for the electrical energy required. Th−Tl 3 Rate of change in work is dW = d(Qh − Ql) = γdQl = dQl. We are given C = aT Tl 3 dQl = −dTl ⇒ dQl = −aTl dTl Therefore the work is a 3 dW = − (Th − Tl) Tl dTl Tl For cooling Th → 0 Z Z 0 2 3 a 0 a 4 0 a 4 W = −a ThTl − Tl dTl ⇒ W = − ThT + T ⇒ W = Th X 3 Th 4 Th 12 Th 1.4 Sterling Heat Engine The operation of a certain type of engine involves applying two isothermal steps and two isovolumetric steps per cycle to one mole of diatomic gas. The largest and smallest volumes shown are VL and VS , respectively. You do not need to consider the vibrational degree of freedom of the molecules. (a) Find the efficiency of the engine in terms of Th, Tc, VL and VS . From 1 → 2 the process is isothermal so pV = constant, so the work is Z Z VL cdV VL VL Q12 = pdV ⇒ Q12 = = c ln V ⇒ Q12 = nRTh ln V VS V VS S 5 VL The heat along 2 → 3: Q23 = R (Tc − Th). For 3 → 4: Q34 = nRTc ln . Finally, for 4 → 1: Q41 = 2 VS 5 2 R (Th − Tc).
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