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University of Illinois at Chicago
Department of Physics
Thermodynamics and Statistical Physics
Qualifying Exam
January 6, 2012
9:00am-12:00pm
Full credit can be achieved from completely correct answers to
4 questions. If the student attempts all 5 questions, all of the answers
will be graded, and the top 4 scores will be counted towards the exams
total score.
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Mathematical Formulae
Notation:
1
=
kB T
z
2 ( )
erf (z) = dx exp x2 erf is known as the error function
0
2 ( )
erfc (z) = dx exp x2 erfc is known as the complimentary error function
z
Integrals:
dx ln x = x ln x x
dx
= ln x
x
( ) 1 ( )
dx exp ax = 2
erfc ab
2 a
b a
1 exp (a2 )
dx erfc (x) = + a erfc(a)
n
0
( )
1/2
dx x exp(x) = erf n n exp (n)
2
n
0
3 1
dx x3/2 exp(x) = erf( n) n exp(n)(3 + 2n)
0 4 2
Expansions:
1
= 1 + x + x2 + x3 + . . . for x < 1
1x
x2 x3
exp(x) = 1 + x + + + ...
2! [ 3! ]
( 2) 1
erfc(x) = exp x + ... for x
x
sinh(x) = x + ... for x 0
cosh(x) = 1 + ... for x 0
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1. Consider a system consisting of N non-interacting particles each with isospin I = 3/2. The
energies of the states with dierent Iz are given by
E(Iz = 3/2) = E1 ; E(Iz = 1/2) = E2
E(Iz = 1/2) = E3 ; E(Iz = 3/2) = E3
with E1 < E2 < E3 and 12 = E2 E1 23 = E3 E2 .
a) Without using the partition function, give the value of the total energy, E, at temperatures
T = 0, 12 T 23 , and 23 T . Provide a justication for your results. Sketch E as
a function of temperature.
b) What is the occupation of the Iz -states for temperature T Without using the partition
function, give a value of the specic heat for temperature T . Provide a justication for
your results.
c) Without using the partition function, give the value of the average isospin per particle, Iz ,
at temperatures T = 0, 12 T 23 , and 23 T . Provide a justication for your results.
Sketch Iz as a function of temperature.
d) Using the partition function, compute the average isospin per particle, Iz , in the limit
T . How does you result related to those in part c)?
2. Consider an ideal gas of N0 non-interacting spin-less particles each with kinetic energy
m
= v2
2
that is contained in a box. The temperature of the gas is T0 , and the particles are uniformly
distributed throughout the box.
Compute the total energy E0 = E of the N0 particles in the box. Next, one instantaneously
removes all particles from the gas that possess a kinetic energy larger than nkB T (n is an
arbitrary real, positive number). How many particles remain in terms of N0 ? What is the new
total energy, Enew in terms of E0 ? After the remaining particles have returned to equilibrium,
what is the new temperature, Tnew of the gas in terms of T0 ?
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3. Suppose one mole of an ideal gas is subjected to the cyclic process shown below (with temper-
ature V1 , V2 and V3 in states 1, 2 and 3, respectively)
p
1
p1
p2=p3
3 2
V1=V3 V2 V
1 2 is an isothermal expansion.
2 3 is an adiabatic expansion.
3 1 is an isochoric heating step.
All steps are reversible
a) What is the change in internal energy, U , for the entire cyclic process 1 2 3 1.
b) Use the First Law of Thermodynamics to calculate U , Q, and W for the process 1 2.
c) Use the First Law of Thermodynamics to calculate U , Q, and W for the process 2 3.
d) Use the First Law of Thermodynamics to calculate U , Q, and W for the process 3 1.
e) Is the total work done in a cycle positive or negative? What is the eciency, , of this cycle?
In which limit does one obtain = 1.
4. Consider a system consisting of M non-interacting molecules at temperature T . Each of these
molecules possesses vibrations with energies
( )
1
E n = ~ 0 n + where n = 0, 1, 2, 3, ..., N0
2
Let us rst consider the case N0 =
a) Using the partition function, compute the total energy, E, of the system for temperature
T 0 and T . Explain your results. At what temperature occurs the crossover from the
T 0 to the T behavior of E?
b) Compute n for T . What is the physical interpretation of n? What is the relation
of n to the partition function and to E?
c) Consider next the case where N0 is a nite, integer number (i.e., N0 < ). What is now the
form of E for temperature T
d) Compute the specic heat, CV , of the system in the limit T for the two cases N0 =
and N0 < . Explain the dierence in CV between these two cases.
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5. Consider a monoatomic ideal gas.
a) Compute the entropy of an ideal gas as a function of T and V for constant particle number
N starting from
dU = T dS pdV
b) Compute the chemical potential of the ideal gas as a function of p and T starting from the
Gibbs-Duhem relation
SdT V dp + N d = 0
