PHGN341: Thermal Physics
Exam I - February 17, 2012
NAME:
1. It was recently reported that one of the crew modules of the International Space Station was losing air pressure at the rate of about 1.0% per day. Suppose the module has a total volume of 100 m 3 and is kept at a temperature of 298 K and a pressure of 10 5 Pa.
a. (10) What is the rate that air molecules are leaking from the module, dN/dt ?
b. (10) Consider one air molecule with an average mass of m ' 4 .
78 × 10
− 26 kg which elastically bounces from one section of the wall which is normal to the x-axis. Starting from the equipartition theorem, estimate the average change in the x-component of momentum of the molecule if it elastically bounces off the wall?
c. (10) A pressure of 10
5
Pa will produce a net force of F = 10
5
N on some arbitrary one square meter section of the module wall. What is the rate, r , that air molecules are bouncing off this one square meter of wall to produce this average force?
d. (10) Suppose the leak in the module is caused by a small hole that is known to be in this one square meter section of wall. Estimate the area of the hole. Explain your reasoning.
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2. Consider a system of N spin1 non-interacting paramagnetic atoms in an external magnetic field,
~
. The spin of each
2 atom can be either spin-up (along the magnetic field) or spin-down (against the magnetic field). The energy of the i th atom is: U i
µ
B
= − µ
B
B for spin-up and U is the Bohr magneton. If there are i
N
↑
= + µ
B
B for spin-down, where B is the magnitude of the magnetic field and spin-up atoms and N
↓ spin-down atoms, the total energy is therefore:
U = − µ
B
B ( N
↑
− N
↓
) .
a. (10) What is the entropy as a function of the total energy , U ? Dropping the relatively small square-root factors, the approximate multiplicity for the two state system is
Ω( N, N
↑
, N
↓
) '
N
↑
N
− N
↑ N
N
↓
− N
↓
.
b. (10) What is the temperature as a function of the total energy?
c. (10) From your answer in part (b) find the total energy as a function of temperature.
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3. Recall that when two systems are in thermal equilibrium, the composite multiplicity is maximal with respect to variation of the energy of either one of the systems. Suppressing the volume and particle number dependence of the multiplicity, the entropy of two systems in thermal contact is:
S
AB
( U, U
A
) = k log Ω
AB
( U, U
A
) = k [log Ω
A
( U
A
) + log Ω
B
( U
B
= U − U
A
)] , where U = U
A
+ U
B is the total energy (fixed).
a. (10) Show that when the two systems are in thermal equilibrium, the entropies satisfy the condition:
∂S
A
∂U
A
=
∂S
B
∂U
B
.
b. (10) The Sackur-Tetrode expression for the multiplicity of
Ω( N
A
, V
A
, U
A
) =
"
V
A
N
A
N
4 πmU
A
3
A h 2 gas atoms of mass, m , confined to volume, V
A
3 #
N
A
N
A
2 e
5
2
, is where U
A is the energy and h is Planck’s constant. Use the entropic definition of temperature to find the internal energy as a function of temperature.
c. (10) Use the entropic definition of pressure to find the pressure as a function of number, volume, and internal energy.
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