Physics 607 Final Exam

Transcription

1 Physics 607 Final Exam Please be well-organized, and show all significant steps clearly in all problems. You are graded on your work, so please do not just write down answers with no explanation! Do all your work on the blank sheets provided, writing your name clearly. (You may keep this exam.) The variables have their usual meanings: E = energy, S = entropy, V = volume, N = number of particles, T = temperature, P = pressure, µ = chemical potential, B = applied magnetic field, C V = heat capacity at constant volume,, C P = heat capacity at constant pressure, F = Helmholtz free energy, G = Gibbs free energy, k = Boltzmann constant. Also, represents an average. e x dx = π, x e x dx = 1 Γ 1 = π, Γ 1 π, ( ) = 1, Γ( z +1) = zγ( z) x n e x dx = 1 0 Γ n + 1 1

2 1. A -dimensional plane of atoms is approximately isolated from the underlying substrate. Since we are concerned with long wavelengths below, we can assume that ε = ω = vp and that there are polarizations per momentum p!". (One is transverse and one longitudinal, but we neglect the difference in wave velocities.) Here ω is the angular frequency and v is the wave velocity, so ε is a phonon energy. Recall that the heat capacity of a single harmonic oscillator is k!ω kt e!ω /kt ( e!ω /kt 1). For a -dimensional system, also recall that the volume in momentum space per state is h / A, where A is the area of the system. This means that the number of states in dp at p is π pdp h / A. (a) (5) Calculate the density of states ρ ε ρ ( ε )dε = number of states in dε at ε. ( ) for the vibrational modes (having polarizations), with (b) (15) At low temperatures, calculate the heat capacity of the system and show that it is proportional to T n, where you will determine n.

3 . An ideal gas of diatomic molecules is enclosed in a large box of volume V at temperature T. The Hamiltonian for the internal degrees of freedom is H = J I + p m + 1 K x where these terms correspond to rotational kinetic energy, vibrational kinetic energy, and vibrational potential energy. Here J is the angular momentum, I is the moment of inertia, m is the reduced mass, p is the momentum associated with vibrations, K is the force constant, and x is the vibrational displacement. (a) (3) Recall that the rotational energies are then j( j +1)!, with j +1 values of m I j, and write down the rotational partition function z rot for a single molecule as a sum over j = 0,1,... (b) (7) Assuming that T is high, show (clearly) how the sum of part (a) becomes an integral over j, and evaluate this integral to obtain the high-temperature form of z rot. (c) (5) Recalling that the rotational energy per molecule is given by ε rot = kt T ln z rot, calculate ε rot as a function of T. (d) (3) Now assume that the temperature is so high that even vibrations can be treated classically (but still in the harmonic approximation). Calculate the classical partition function for vibrations, z vib, recalling that it is an integral over the -dimensional phase space for a single coordinate. (e) (7) Calculate x temperature T., the average value for the square of the vibrational displacement, as a function of the 3

4 3. We wish to determine how the temperature and pressure vary with height in the Earth s atmosphere in a simple but somewhat realistic model: We assume that T and P are determined by adiabatic convection processes within the atmosphere, and that the air molecules can be treated as point particles which all have the same mass m. Recall that the partition function for a classical ideal gas of N indistinguishable particles is given by an integral ( ) N. over phase space with 6 N degrees of freedom, and that the volume of a single state in phase space is h 3 /V Also recall that F = kt ln Z and F = E TS. (As stated at the top of the first page, we use the usual notation with, e.g., F = Helmholtz free energy.) (a) (3) Write down the expression for the partition function of the molecules described in the first two paragraphs above. (b) (3) Evaluate this partition function, obtaining Z in terms of N, v = V N (volume per particle), and the h thermal de Broglie wavelength λ = ( πmkt ). 1/ (c) (3) Write df in terms of dt, dv, and dn, and then calculate the entropy per particle s = S / N. (d) (3) Note that s depends only on a particular quantity involving T and v, so this quantity is constant during an adiabatic process. What is this quantity? (e) (3) Using the ideal gas equation of state, replace v by P in the result of part (d), to obtain P T = T 0 P 0 where you will determine n 1 (which is not necessarily an integer). Here T 0 and P 0 are the temperature and pressure at z = 0. (f) (3) Show a rectangular volume of air between z and z + dz, with cross-sectional area A, and use it to obtain the change in pressure dp as a factor (involving m, the acceleration of gravity g, and the number density n = 1/ v ) multiplying the change in height dz. (g) (3) Using the ideal gas equation of state, and the result from part (e), rewrite dp / dz as dp dz = factor P n where you will determine n (which is not necessarily an integer) and the factor involves m, g, k, T 0, and P 0. (h) (10) Integrate the equation in part (g) to obtain P( z) -- i.e., the pressure P as a function of the height z in this model of the Earth s atmosphere. (i) (3) Sketch a graph of P( z) from z = 0 up to the point where the gravitational potential energy mgz is equal to the thermal energy kt 0 at z = 0. (j) (3) Obtain T ( z) -- i.e., the temperature T as a function of the height z. n 1 4

5 4. (18) In treating a system with the renormalization-group approach, you ultimately find that the renormalization of the partition function is given by ζ ( K ' ) = ζ ( K ) ln f ( K ) where ln Z ( K ) = N lnζ ( K ). Here f ( K ) is a smoothly varying function with no relevant singularities. You have calculated the nontrivial fixed point K C (where K C ' = K C, K ' and K being the coupling constants after and before decimation ). You have also calculated the constant C in the Taylor series expansion K ' = K C + C K K C ( ), C dk ' dk. K=KC α Now you wish to determine the critical exponent α for which C V K K C, with CV d ζ very near the dk critical point. Calculate the (very simple) result for α in terms of C and other constants. Merry Christmas and Happy Holidays! 5