PHYS-4420 THERMODYNAMICS & STATISTICAL MECHANICS
FINAL EXAMINATION
SPRING 2000
Thursday, May 4, 2000
Your grade will be sent to you by e-mail by 5:00 p.m., Friday, May 5, 2000
NAME: _____SOLUTIONS_____________________
There are four pages to this examination. Check to see that you have them all.
To receive credit for a problem, you must show your work, or explain how you arrived at your
answer.
1. (15%) In the card game Bridge, it is important to know as much as possible about the
distribution of cards in your opponents hands. The game is played by four players formed
into two teams of two each. An example that is often given in books on the subject is the
following. Suppose you know that your opponents team has only four cards of the spade suit
in their possession, divided between their two hands. Then the most probable distribution of
those four cards between the two hands is the one that has three cards in one hand and only
one in the other. The even split, of two cards in each hand, is less probable.
Calculate the probabilities of finding the cards split 3 and 1, and also the probability of
finding them split 2 and 2, to verify the books’ assertions.
Calculate the probability that one opponents hand will contain 0, 1, 2, or 3 spades.
N!
P ( n) 
pnq N n
( N  n)!n!
0
4
4!  1   1 
1
P(0) 
    
4!0!  2   2  16
1
4!  1 
P(1) 
 
(4  1)!1!  2 
3
4 1
1

  
 2  16 4
2
2
4!  1   1 
6 3
P(2) 

    
(4  2)!2!  2   2  16 8
4!  1 
P(3) 
 
(4  3)!3!  2 
3
1
4 1
1

  
 2  16 4
4
0
4!  1   1 
1
P(4) 
    
(4  4)!4!  2   2  16
To find the probability of a 3-1 split, add the probability that a player has 3 spades to the
probability that the same player has 1 spade. Either results in a 3-1 split.
1
3
P(3,1)  P(1)  P(3)  , while P (2,2)  P (2)  , so the books are correct.
2
8
1
2. (10%) A student who drinks coffee with cream in it, gets a cup of coffee, which is very hot,
and a small container of cream which is much colder. The student wants the coffee to be as
hot as possible when it is drunk. The student has to carry the coffee to another room before
drinking it, a trip that could take two or three minutes. In order to have the coffee as hot as
possible when it is drunk, which of the following strategies should the student follow?
A. Carry the hot coffee and cool cream to the other room before mixing, and then add the
cream to the coffee.
B. Add the cream to the coffee at once, and carry the mixture of coffee and cream to the
other room.
Circle your choice, and briefly explain why you made that choice. (Hint: Assume that
radiation is the primary mechanism of cooling.)
By adding the cream at once, the temperature is lowered at the start, and the amount of heat
radiated will be less than it would have been if the coffee had been left hot.
3. (20%) For this problem, you will consider a paramagnetic salt with a fixed number of atoms
(i.e. N is constant, so dN = 0 throughout). When a magnetic field, B, is applied to the salt, it
becomes polarized, and develops a magnetic moment, M. When the field is changed, the
magnetic moment changes, and the magnetic energy changes. The work done when the field
is changed can be shown to be,
dW = MdB.
a) The Helmholtz function is defined as F = E – TS. For a paramagnetic salt, where the only
work done is magnetic, show that,
dF = – MdB – SdT.
(Hint: What form does the First Law assume with only magnetic work?)
dF = dE – TdS – SdT, and the first law is dE = TdS – dW = TdS – MdB. Then,
dF = TdS – MdB – TdS – SdT = – MdB – SdT
b) Use the result of part (a) to show that,
F 
  M .
B T
F 
F 
 dB 
 dT .
B T
T  B
A comparison of the multiplier of dB in this equation with that of part (a) shows that,
F 
  M .
B T
Fince F is a function of B and T, we can write dF 
2
4. (35%) The paramagnetic salt of question 3, consists of N atomic magnets, each of which can
take on only two energies in a magnetic field, –B and +B, where  is the magnetic moment
of the atom.
a) Calculate , the partition function for a single magnetic moment in the magnetic field.
1
(Hint:    e i , where  
.)
kT
i
 = eB + e-B
b) Calculate Z, the partition function for the system of N distinguishable magnetic moments.
Z = N =(eB + e-B) N
c) Use the result of part (b) to find the Helmholtz function for the magnetic salt.
F = – kT ln Z = – kT ln (eB + e-B) N = – NkT ln (eB + e-B)
F 
   M , which you (hopefully) derived in the last problem to get an expression
B T
for M, the magnetic moment of the salt.
d) Use
M 


F 



 NkT ln( e B  e  B ) T  NkT ln( e B  e  B ) 
 
B T
B
B
T
M  NkT
(e B  e  B )
e B  e  B
 B 
M  N tanh 

 kT 
 N
(e B  e  B )
 N tanh( B)
e B  e  B
3
5. (20%) A new satellite that uses a Carnot engine for power, is being designed. For a high
temperature reservoir, it will use a nuclear source at a fixed temperature, T1. Its low
temperature reservoir will be a set of cooling fins that radiate heat into space. The
temperature of the fins depends on the rate at which they radiate energy, and that is
proportional to the fourth power of their temperature, T2. They will maintain a constant
temperature when the rate at which the Carnot engine delivers heat to the fins is equal to the
rate at which the fins radiate heat to space:
dQ2
 AT24
dt
where A is a constant determined by the design of the fins. For this system, the Carnot engine
will have a maximum power output for a specific value of T2 that is between absolute zero
and T1 .
a) Explain briefly why the power output should be a maximum somewhere in this
temperature range.
dW dQ1 dQ2


.)
(Hint: Power is the rate at which work is done, and
dt
dt
dt
If T1 and T2 are close together, T2 will be high and able to radiate heat rapidly, so it is
possible to run the engine at a high rate, but the engine is not efficient. If T2 is lowered,
the engine becomes more efficient, but heat can not be radiated as rapidly, so the engine
has to be run at a slower rate. At some temperature a compromise is found, and a
maximum power output results.
b) Find the value of T2 for which the power output is a maximum. Express your answer in
terms of T1.
dW dQ1 dQ2 dQ1
Q T
P



 AT24 . In addition, we know that 1  1 , so
dt
dt
dt
dt
Q2 T2
T
dQ1 T1 dQ2 T1
Q1  1 Q2 , and

 AT24  AT1T23
T2
dt
T2 dt
T2
dP
 0.
P  AT1T23  AT24  A(T1T23  T24 ) . To find the maximum, solve
dT2
dP
 A(3T1T22  4T23 )  0, (3T1  4T2 )T22  0 . This has two solutions, but T2 = 0 is not
dT2
3
T2  T1 .
possible. Instead,
4
c) What is the efficiency of the Carnot engine when operating at the value of T2 found in
part (b), i.e. when the power output is a maximum.
3
T1
T2
3 1
  1   1  4  1    25%
T1
T1
4 4
4