Physics 607 Final Exam

Size: px

Start display at page:

Download "Physics 607 Final Exam"

Jonas Stanley
6 years ago
Views:

1 Physics 607 Final Exam Please show all significant steps clearly in all problems. 1. Let E,S,V,T, and P be the internal energy, entropy, volume, temperature, and pressure of a system in thermodynamic equilibrium whose number of particles N is fixed. (a) (5) With E taken to be a function of its natural variables S and V, write TdS in terms of de and dv. (b) (10) Now take T and V to be the independent variables in the expression for de, and write TdS in terms of dt and dv. Divide the resulting expression by dt to show that [ C P C V = P + ( ) ]( ) E V. (1) V T T P (c) (5) Use the result of part (b) to obtain C P C V for a classical ideal gas. (d) (5) The equation of state and energy of a van der Waals gas are respectively (P + a v 2 ) (V Nb)=Nk B T (2) E = E (T ) a N 2 (3) V where v V/N. Give a clear and complete interpretation of the terms involving a and b in the above equations. I.e., pretend you are explaining the fundamental origin of these terms to a class of undergraduate students! (e) (5) Use the result of part (b) to obtain C P C V for a van der Waals gas. 1

2 2. Suppose we postulate that the entropy of an ideal gas of fermions or bosons is given by S = [n k log n k ± (1 n k )log(1 n k )] (4) k where the upper sign holds for fermions and the lower sign for bosons. Here n k is the number of particles in the single-particle state labeled by k. (a) (5) Demonstrate with clear arguments that the entropy per particle goes to zero at zero temperature for fermions. (b) (5) Similarly, demonstrate that the entropy per particle goes to zero at zero temperature for bosons. (c) (10) Use the method of Lagrange multipliers to show that the above expression for S leads to the Fermi-Dirac distribution function n k for a system of fermions in thermal equilibrium at temperature T, if the particle number N and the total energy E are fixed. (d) (10) Similarly, use the method of Lagrange multipliers to show that the above expression for S leads to the Bose-Einstein distribution function n k for a system of bosons in thermal equilibrium at temperature T, with the particle number N and total energy E again fixed. 2

3 3. In studying the Benard instability, we started with the Navier-Stokes equations of hydrodynamics, and after a series of approximations (plus Fourier-transforming) obtained the differential equation ( ) d 2 3 dz 2 κ2 V (Z) = Rκ 2 V (Z) (5) for the onset of the instability. Here R gaαd4 ρ 2 0 c V ηk (6) is the Rayleigh number and a (T 0 T d ) /d (7) is the applied temperature gradient.also, Z = z/d and κ = kd are the scaled values of the coordinate perpendicular to the bottom and top planes and the magnitude of the wavevector parallel to these planes. With the velocity required to vanish at the boundaries, it follows that V (Z) =A sin(nπz). (8) (a) (10) Let R c be the smallest value of R at which there is an instability. Determine the value of κ corresponding to this value of R, and show that it represents a wavelength of λ =2 2 d. (9) (b) (10) Calculate R c, and show that its numerical value is approximately 657. (c) (10) Please give a coherent description of the Benard instability. Again, pretend that you are teaching an undergraduate physics class about the qualitative nature of this phenomenon. Explain the various physical quantities in the expression for R, and why the instability should depend on them in the way it does. 3

4 4. Let Z be the canonical partition function for a system in thermodynamic equilibrium at temperature T, and let β =1/k B T. (a) (7) Obtain an expression for the average energy E in terms of log Z/ T. (b) (7) Using the fact that df = SdT PdV + µdn (10) and the standard expression for F in terms of log Z, obtain an expression for the entropy S in terms of log Z and log Z/ T. (c) (7) Combine the results of (a) and (b) to show that S = k B log j exp ( E j /k B T )+ 1 T j E j exp ( E j /k B T ) j exp ( E j /k B T ). (11) (d) (2) Is this result consistent with the third law of thermodynamics? Explain. (e) (7) Using the same approach as in (a), relate the variance σ 2 E in the energy to the heat capacity at constant volume, C V. 4

5 5. [This problem is copied from States of Matter, by David Goodstein of Caltech (available in Dover paperback).] There is an empirical formula relating the Debye temperature Θ D and the melting temperature T m of solids: Θ D = C ( Tm Mv 2/3 ) 1/2 (12) where M is the atomic mass (in amu), v is the volume per atom (in A 3 ), and C is a constant, roughly the same for all solids. (a) (5) Estimate the value of C from whatever information you have in your head about the properties of some particular solid. If you lack some particular datum, make an order of magnitude guess. The formula was first proposed by Lindemann in According to the textbooks, it has no theoretical explanation. In parts (b) and (c) you will provide a theoretical explanation. (b) (10) Recalling that the energy of an oscillator is twice its potential energy, calculate the mean square displacement x 2 of an atom in a Debye solid. (Hint: For the kth mode, you can show that the energy per atom is E k = Mωk x 2 2 k, whereω k is the frequency and x 2 k is the contribution to the mean square displacement of the kth mode. Since the modes are independent, x 2 = k x 2 k. Recall that melting occurs at high temperature.) (c) (10) Assuming a crystal melts when the root mean square displacement is some reasonable fraction (say 0.1) of the interatomic spacing, estimate C in the Lindemann formula. (d) (5) By the same kind of arguments, find the melting temperature of a Debye solid in two dimensions. 5

UNIVERSITY OF SOUTHAMPTON

UNIVERSITY OF SOUTHAMPTON PHYS2024W1 SEMESTER 2 EXAMINATION 2011/12 Quantum Physics of Matter Duration: 120 MINS VERY IMPORTANT NOTE Section A answers MUST BE in a separate blue answer book. If any blue