NAME:
PHGN341: Thermal Physics
Final Exam
May 6, 2009
For each problem, show all work, explain your reasoning, and state any assumptions.
1. (25) (a) Consider the entropy to be a function of the energy (U ), volume (V ), and number of particles (N ), i.e.
S(U, V, N ). Expand the differential, dS, in partial derivatives of the independent variables, {U, V, N } and solve for dU .
Then using the thermodynamic relation: dU = T dS − P dV + µdN , find {T, P, µ} in terms of the partial derivatives of
the entropy.
(b) From the Sakur-Tetrode expression for the multiplicity of an ideal monotonic gas (found on the equation sheet),
derive an expression for the temperature.
(c) From the Sakur-Tetrode expression for the multiplicity of an ideal monotonic gas (found on the equation sheet),
derive an expression for the pressure.
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2. (25) Consider a spin-1 paramagnetic atom that can have three possible values for its z-component of angular momentum:
m = +1, 0, −1. The atom is in a uniform magnetic field which means its energy depends on its z-component of angular
momentum, i.e. E = −m , where is some energy scale that depends on the magnetic field and the magnetic moment
of the atom. The atom is in thermal contact with a reservoir at temperature, T .
a. Calculate the single atom partition function.
b. Find the average energy of one atom as a function of temperature.
c. For this atom in a 1 T magnetic field, ' 6.0 × 10−5 eV. What is the probability that the atom is in its lowest
energy state (m = +1) at a temperature of 4 K?
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3. (20) When a common star burns up its nuclear fuel, it cools and contracts until it becomes a white dwarf which is
prevented from collapsing further by the degenerate (T = 0) electron pressure.
(a) Assume that the nuclei making up the star have an equal number of protons and neutrons. Let mN the average
mass a nucleon and let me be the mass of an electron (mN >> me ). Derive an expression for the Fermi momentum,
pF = h̄kF , of the electrons for a white dwarf star with radius, R? , and mass, M? . (Since the star is electrically neutral,
the number of electrons, Ne , equals the number of protons.) Give your answer in terms of R? and M? and physical
contants.
3GM 2
(b) The gravitational potential energy of a white dwarf star of uniform density is UG = − 5R?? . Treating only the
kinetic energy of the degenerate electrons, the total energy of the star is Etotal = 35 Ne EF + UG , where EF is the
nonrelativistic Fermi energy (p2F /(2me )). From the total energy, find an expression for the equilibrium radius of the
white dwarf star (i.e. the radius which minimizes the total energy).
(c) Evaluate the equilibrium radius for a white dwarf of one solar mass, M? = 2.0 × 1030 kg.
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4. (30) Consider a thin sheet of material of area A with N atoms which can be approximated as a two-dimensional
membrane with an area density of nA = N/A. Let cs be the speed of sound for the membrane. Treat only one phonon
d2 k
polarization; so the density of states is given by A (2π)
2 , where A is the area of the membrane.
(a) Find an expression for the 2-d cut-off wave number, kmax , that gives the correct area density, nA .
(b) Find an expression for the 2-d Debye temperature.
(c) Write down the integral expression for the 2-d phonon energy at temperature, T (do not evaluate it).
(d) Find an expression for the phonon contribution to the heat capacity in the high temperature limit.
(e) Find an expression for the phonon contribution to the heat capacity in the low temperature limit.
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(f) Now lets do the numbers. Suppose the membrane is graphine. The area density of graphine is nA = 42.2 atoms/nm2
and its speed of sound is cs = 12, 450 m/s. Evaluate your expressions to find the following values in the indicated units:
i) kmax (in nm−1 )
ii) Debye temperature (in K).
iii) Phonon contribution to the heat capacity per atom (in eV/K) in the high temperature limit at T=5000 K.
iii) Phonon contribution to the heat capacity per atom (in eV/K) in the low temperature limit at T= 5 K.
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