2008 Qual Tsmproblems

University of Illinois at Chicago
Department of Physics
Thermodynamics & Statistical Mechanics

Qualifying Examination
January 8, 2008
9.00 am 12:00 pm
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 toward the
exams total score.
1
Problem 1: A cylindrical container of length L is separated into two compartments by a thin
piston, originally clamped at a position L/3 from the left end. The left compartment is filled with
1 mole of helium gas at 5 atm of pressure; the right
compartment is filled with argon gas at 1 atm of pressure.
These gases may be considered ideal. The cylinder is L
submerged in 1 liter of water, and the entire system is
initially at the uniform temperature of 25C, and thermally He Ar
5 atm 1 atm
isolated from the surroundings. The heat capacities of the water
cylinder and the piston may be neglected. When the piston
is unclamped, the system ultimately reaches a new L/3
equilibrium situation.
(a) What is the change in the temperature of the water?
(b) How far from the left end of the cylinder will the piston come to rest?
S S
(c) Starting from dS = dV + dT , find the total increase in the entropy of the
V T T V
system.
(d) Now consider a slightly different situation, in which the left side of the cylinder contains 5
moles of real (not ideal) gas, with attractive intermolecular interactions. The right side still
contains 1 mole of an ideal gas. As before, the piston is initially clamped at a position L/3 from
the left end. When the piston is unclamped and released, does the temperature of the water
increase, decrease, or stay the same? Does the internal energy of the gas increase, decrease, or
remain the same? Explain your reasoning.
Problem 2: A copper block is cooled from TB to TA using a Carnot engine operating in reverse
between a reservoir at TC and the copper block. The copper block is then heated back up to TB by
placing it in thermal contact with another reservoir at TB. ( TC > TB > TA )
(a) What is the limiting value of the heat capacity per mole for the copper block at high
temperatures?
(b) Find the total entropy change of the universe in the cyclic process B A B and show that
it is greater than zero?
(c) How much work is done on the system, consisting of the copper block and the Carnot engine?
(d) For the cyclic path B A B , does the system absorb heat from the reservoirs or reject
heat?
2
Problem 3: Two magnetic spin systems with N1 = 500 spins and N2 = 1000 spins are placed in
an external magnetic field H. They are initially thermally isolated from one another and are
prepared with spin excess values of m1 = +20 and m2 = +250, where m = N N . The magnetic
moment of each spin is denoted by m . Assume that one can use the Gaussian approximation for
the number of states of each system with energies between E and E + dE:
2N E2
(N, E ) = exp dE

2N m H
2 2
2N m2 H 2
(a) What is the total energy in each system (in units of m H ) and the initial temperature of each
system (in units of m H / k )?
(b) The two systems are brought into thermal contact. What are the spin excess values m1eq and
m2eq and the final temperature, after they reach equilibrium?
(c) What is S , the total change in the entropy of the combined system (in units of k), and what
is the probability of finding the system in the initial configuration relative to the probability of
finding the system in the equilibrium configuration?
Problem 4: Consider a 2-dimensional polymer,

consisting of N links, each of length l. Each link has
four allowed orientations, pointing in the +x or x
direction, or in the +y or y direction.
(a) Show that the average end-to-end

r r distance squared
for this polymer is given by < L L >=< L2 >= Nl 2
(b) Now consider a force f that extends the length of the

polymer in the x-direction. In this simplified two-
dimensional picture, the three allowed energy levels of
each link are (i) = fl , if the link is oriented in the direction of the applied force, = +fl , if the
link is oriented opposite to the direction of the applied force, and (iii) = 0 , if the link is
oriented perpendicular to the direction of the applied force. Find the average extension < Lx > in
the x-direction as a function of the applied force, and show that, at low forces (fl << kBT), the
polymer behaves like a Hookean spring.
(c) The polymer is stretched quasistatically and isothermally such that the average extension in
the x-direction is 5% of its unperturbed size L = N l ? Assume that the polymer still behaves as
a Hookean spring. What is the change in the Helmholtz free energy of the polymer?
(d) Now consider that the polymer is stretched quasistatically and adiabatically. Will the
temperature of the polymer increase or decrease in this process?
3
Problem 5: Consider a system consisting of impurity atoms in a semiconductor. Suppose that
the impurity atom has one "extra" electron (with two degenerate spin states), compared to the
neighboring atoms (e.g. a phosphorus atom occupying a lattice site in a silicon crystal). The extra
electron is easily removed, contributing to conduction electrons, and leaving behind a positively
charged ion.
(a) What is the probability that a single donor atom is ionized? Express your result in terms of
the ionization energy I, and the chemical potential of the "gas" of ionized electrons.
(b) If every conduction electron comes from an ionized donor, and the conduction electrons
behave like an ideal gas, write down an expression that relates the number of conduction
electrons N c and the number of donor atoms N d , in terms of the volume V of the sample, the
temperature T and fundamental constants.
(c) Show that in the limit of low (kT << I ) and high (kT >> I ) temperatures, the ratio N c / N d has
the expected values.
(d) Write down an expression for the Gibbs free energy of the conduction electrons in terms of
Nc, T and V and show that it is an extensive quantity.
Equations and constants:
kB =1.3811023 J/K; NA = 6.0221023; R = 8.315 J/mol/K; 1 atm = 1.013105 N/m2
Hyperbolic functions
e x e x e x + e x sinh x
sinh x = cosh x = tanh x =
2 2 cosh x
Inequality
ln( y ) y 1
Maxwell's relations
T P T P T V T V
= ; = ; = ; =
V S S V V P S T P S S P P V S T
Ideal gas
VZ
3/2
h2
= k BT ln int ;

v q =
Nv q 2mk BT

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University of Illinois at Chicago

Thermodynamics & Statistical Mechanics

Full credit can be achieved from completely correct answers to 4

(a) What is the change in the temperature of the water?

Problem 4: Consider a 2-dimensional polymer,

(a) Show that the average end-to-end

(b) Now consider a force f that extends the length of the

Equations and constants:

kB =1.3811023 J/K; NA = 6.0221023; R = 8.315 J/mol/K; 1 atm = 1.013105 N/m2

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