PHYS-4420 THERMODYNAMICS & STATISTICAL MECHANICS
Quiz 2
SPRING 2002
Friday, April 19, 2002
NAME: ____________________________________________
To receive credit for a problem, you must show your work, or explain how you arrived at your
answer.
1. (30%) Enthalpy is defined as H = E + pV, where E is internal energy, p is pressure, and V is
volume.
a) Show that: dH = TdS + Vdp + dN
b) Use the equation given in part a) to show that: T 
c) Use the equation given in part a) to show that:
(This is one of the Maxwell relations.)
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H 

 S  p , N
T
V
 
 p S , N  S


 p, N
2. (25%) Consider a collection of 3N identical, distinguishable harmonic oscillators, all of
frequency . The energies that one these oscillators can take on (measured relative to the
ground state) are n = nh. Where n is an integer that can take on values from 0 to .
a) Find the partition function , for one of these oscillators.

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(Hint:  x n 
for x < 1)
n 0
1 x
 = _______________
b) Find the partition function Z, for the collection of oscillators. (Since the oscillators are
distinguishable, no N! is needed in the denominator.)
Z = _______________
c) Find the Helmholtz function F, for this collection of oscillators.
(Hint: F = – kT ln Z)
F = _______________
d) Find the entropy S, for this collection of oscillators. Hint: S  
F 
 . This will not
T V , N
give a particularly neat expression.
S = __________________________________
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e) Show that the expression for the entropy that you obtained in part d) goes to zero when T
goes to zero, in agreement with the third law of thermodynamics.
3. (15%) The number density of the gas, mainly hydrogen, that fills interstellar space is about
one molecule per cubic centimeter ( = 1 ×106 m-3). The diameter of the molecules is about
d = 1 ×10-10 m, and the temperature of interstellar space is about 10K.
a) Find the mean free path of the molecules in interstellar space.
l = _____________________
units
b) Find the average speed of the molecules in interstellar space. (The mass of molecular
hydrogen is m = 2 amu, and 1 amu = 1.66 ×10-27 kg.)
v = _____________________
units
c) Find the average time between collisions for molecules in interstellar space. Express your
answer in centuries. (1 year = 3.16 ×107 s, and 1 century = 100 years.)
t = _____________________
units
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4. (30%) One mole of an ideal monatomic gas traverses the cycle shown in the figure. Process
12 takes place at constant volume, process 23 is adiabatic, and process 31 takes place
at constant pressure. In the work that follows, express all answers in terms of the gas constant
R. (Hint: for one mole of an ideal monatomic gas, the internal energy is E  32 RT .)
a) Compute the heat added, and the work done for the process 12.
Q12 = _____________________
units
W12 = _____________________
units
b) Compute the heat added, and the work done for the process 23.
Q23 = _____________________
units
W23 = _____________________
units
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c) Compute the heat added, and the work done for the process 31.
Q31 = _____________________
units
W31 = _____________________
units
d) Compute the net heat added, and the net work done for the entire cycle.
Qnet = _____________________
units
Wnet = _____________________
units
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