PHYS-4420 THERMODYNAMICS & STATISTICAL MECHANICS
Quiz 2
SPRING 2003
Friday, April 18, 2003
NAME: ____________________________________________
To receive credit for a problem, you must show your work, or explain how you arrived at your
answer.
1. (15%) The Helmholtz Free Energy is defined as F = E – TS, where E is internal energy, T is
F
 , where
temperature, and S is entropy. For a system where dW = pdV, show that S  
 T V , N
N is the number of particles in the system.
2. (40%) One mole of an ideal monatomic gas traverses the idealized Diesel cycle shown in the
figure. Process 12 is adiabatic, and process 23 takes place at constant pressure. Process
34 is also adiabatic, and process 41 takes place at constant volume. (Hint: for one mole
of an ideal monatomic gas, the internal energy is E  32 RT .)
The temperatures of the gas at points 1, 2, 3, and 4 are:
T1 = 300 K,
T2 = 1600 K,
T3 = 1800 K,
1
T4 = 365 K
a) (10%) Find V1/V2, the ratio of the volume of the gas at point 1 to the volume at point 2.
(This is the compression ratio for the Diesel cycle.)
V1/V2 = _____________
b) (10%) Find the magnitude of the heat added to the gas in the process 23. Express the
answer in terms of the gas constant R.
Q1 = _____________
c) (5%) Find the magnitude of the work done by the gas in the process 23. Express the
answer in terms of the gas constant R.
W23 = _____________
d) (5%) Find the magnitude of the heat removed from gas in the process 41. Express the
answer in terms of the gas constant R.
Q2 = _____________
e) (5%) How much work is done by the gas in one complete cycle? Express the answer in
terms of the gas constant R.
W = _____________
2
f) (5%) Find the efficiency of a heat engine operating with this cycle.
 = _____________
3. (15%) A thin walled vessel of volume V contains N particles of gas which slowly leak out
through a small hole of area A. No particles enter the volume through the hole. (You can
imagine the vessel is in outer space.) Find the time required for for the number of particles in
the vessel to decrease to N/2. Assume that the temperature within the vessel remains
constant. Express your answer in terms of A, V, and v .
t1/2 = ____________________
3
4. (30%) Consider a solid consisting of N atoms, each of which has a magnetic dipole moment.
When a magnetic field is applied to the solid, each dipole either points parallel to the
magnetic field or antiparallel to the magnetic field (nothing else is allowed). The interaction
energy of a dipole that is parallel to the field is – H, and one that is antiparallel to the field
is H, where  is the magnitude of a dipole moment, and H is the magnetic field strength.
The system is contact with a heat reservoir which is at temperature of T.
a.) (5%) What is the partition function  for a single dipole? (Note that it has two energy
levels.)
 = ____________________
b.) (5%) What is the partition function for the entire system of N dipoles? (Note that the N
dipoles are distinguishable because they are in particular positions.) You can give your
answer in terms of  from part (a).
Z = ____________________
c.) (5%) What is the average energy of the N-dipole system at temperature T?
E = __________________
d) (5%)What is the energy of this N-dipole system in the limit of T0? There is no need to
do a calculation in this case. Simply present a well-reasoned argument if you prefer.
ET  0 = __________________
4
e) (5%) If the magnetic field is turned off, while the solid is at a temperature close to T = 0,
the energy of the spin system will change. Based on your previous answers, calculate the
change of the energy. Be careful, signs are important. A positive answer means the
energy increases, and a negative answer means the energy decreases.
E = __________________
f) (5%) As stated earlier, the spin system is in contact with a heat reservoir. If the heat
reservoir is finite, its temperature can change if heat is added or removed. Suppose it is
finite. Based on your work so far, what will happen to the temperature of the reservoir
when the magnetic field is turned off. (Circle the correct answer.)
THE TEMPEATURE WILL INCREASE
THE TEMPERATURE WILL DECREASE
THE TEMPERATURE WILL REMAIN THE SAME
5