NAME:
PHGN341: Thermal Physics
Final Exam - May 13, 2010
1.
[15] A refrigerator maintains an inner temperature of -15 C. The heat rejection coils operate at +50 C. On average,
the insulation of the refrigerator allows 20 Watts of heat to leak in. Estimate the minimum annual cost of running the
refrigerator if electricity costs $0.10 per kilowatt-hour?
2.
[15] Estimate the temperature of the melting point of ice at 200 bar pressure. (Data: Latent heat: L = 0.333 MJ/kg,
upon freezing 1000 cm3 of water (1 kg) expands to 1091 cm3 .)
1
3.
[20]
2
P
1
3
V
Consider an engine cycle that operates in the shape of a right triangle as shown. The engine operates in the clockwise
direction and uses Helium gas as the medium. Analyze this engine by filling out the table below assuming the following
conditions: P1 = P3 = 1.0 bar, V1 = 1.0 m3 , P2 = 4.0 bar, V2 = V3 = 5.0 m3 . From your table find the efficiency of the
triangle engine under these operating conditions. Show all work and use the back if you need more space.
Step
Work
∆U
1→2
2→3
3→1
2
Q
4.
[20] Satellites have measured that on average the earth is absorbing about 0.85 W/m2 more power per area than it is
radiating. The goal of this problem is to estimate an upper bound to the annual increase in temperature by assuming
this excess heat is absorbed entirely in the atmosphere.
(a) How much excess heat must the atmosphere absorb in one year? (DATA: The radius of the earth is RE = 6.4 × 106
m.)
(b) The heat capacity of an ideal diatomic gas near standard temperature and pressure is cv = 5/2N kB . To estimate
the heat capacity of the earth’s atmosphere we need to know N , i.e. how many molecules it contains. Recall the
isothermal number density versus altitude relation: n(z) = n0 e−z/z0 , where z0 = kB T /(mg) where m is the mass of a
gas molecule and g = 9.8 m/s2 is the acceleration of gravity. Calculate z0 (in m). (DATA: m̄ ' 4.8 × 10−26 kg.)
(c) Use the ideal gas law to evaluate n0 , where the sea-level (z = 0) atmospheric pressure is 1 bar, 105 Pa.
(d) The atmosphere is quite thin compared to the radius of the earth, so the volume element in a spherical shell at
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height z (measuring from the earth’s surface) is dV = 4πRE
dz. Integrate the number density times the volume element
to find the total number of gas molecules in the earth’s atmosphere.
(e) Use your results from above to find the heat capacity of the atmosphere (in J/K) and then estimate the temperature
rise of the earth’s atmosphere in one year.
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5.
[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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6.
[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 (i.e. the temperature such that kB TD = Ephonon = h̄cs kmax ).
(c) Write down the integral expression for the 2-d phonon energy at temperature, T (do not evaluate it).
(d) Calculate the phonon contribution to the internal energy in the high temperature limit.
(e) Calculate the phonon contribution to the internal energy in the low temperature limit.
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(f) (Extra credit) 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 (in eV/K) in the high temperature limit evaluated at T=5000 K.
iii) Phonon contribution to the heat capacity (in eV/K) in the low temperature limit evaluated at T= 5 K.
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