Consider three systems, each with a single particle at temperature 105 K. System A has two
states, one with energy 0 eV and the other with 10 eV. System B has three states, one with
energy 0 eV and the other two each with 10 eV. System C has four states, two each with
energy 0 eV and the other two each with 10 eV. Let ZA be the partition function for system
A, etc,. . . .a) Which of the following is true? Explain your answer.
i) ZA = ZB = ZC.
ii) ZA = ZB != ZC.
iii) ZA = ZC != ZB.
iv) ZB = ZC != ZA.
v) None of the partition functions are the same.Now suppose that system A has a single particle and two states, one with energy 0 eV and
the other with 10 eV. System B has two distinguishable particles and each could be in one of
the two states of system A.
b) Is ZA = ZB in this case? Explain your answer.The molecule CN is often found in interstellar molecular clouds. This molecule has many
states associated with rotational motion. Observations indicate that about 10% of all such
molecules are in any single one of the first three excited states, each of which has the same
energy, 4.7 104 eV, above the ground state.The remaining 70% of the molecules are in
the ground state. Assuming that the molecules are in thermal equilibrium with a heat bath,
determine the temperature of the heat bath.This addresses a famous issue in cosmology and is discussed in detail in P. Thaddeus, Annual
Review of Astronomy and Astrophysics, vol. 10, p. 305 (1972).Consider a system consisting of a single particle that could be in one of two possible states.
The energies of the states are ! and !.
a) Determine an expression for the partition function of the system.
b) Determine an expression for the mean energy of the system.c) Determine an expression for the heat capacity of the system.
d) Suppose that the energies of each state are each changed by adding the same constant
energy, E0. Show that this will change the partition function and the mean energy of
the system according to E E + E0. Show that it will not change the probabilities
with which the system will be in either state.Consider an ensemble of N identical distinguishable quantum harmonic oscillators. The
states of a single oscillator are labeled by n = 0, 1, 2, and these have energy
E = !
!
n +
1
2
.These are all in equilibrium with a bath at temperature T.
a) Use the canonical ensemble formalism (partition function, etc., . . . ) to show that the
mean energy for the ensemble is
E = N!
!1
2 +
e!
1 e!
= N!
!1
2 +
1
e! 1
b) Determine the specific heat capacity (heat capacity per particle) of the ensemble.
c) Determine the mean energy and heat capacity in the high temperature limit kT % !.
d) Show that the mean energy in the low temperature limit (kT & !) is
E = N!
!1
2 + e!/kT
.e) Determine the specific heat capacity in the low temperature limit and verify that C 0
as T 0. This behavior of the heat capacity is also required in all cases by the third law
of thermodynamics.Suppose that a system has states, labeled s = 1, 2, 3,, and that these all have distinct
energies that satisfy E1 < E2 < E3 < . We would like to consider whether it is possible
that the lowest energy state is occupied with certainty and none of the other states are
occupied with certainty. To do this we require that the probabilities satisfy
p2
p1
= p3
p1
= p4
p1
= = 0. (1)a) Suppose that Eq. (1) is true. Find the temperature of the system.
b) Show that if T 0 then Eq. (1) is true.
c) Determine the entropy of the system as T 0.The result lim
T0
S = (correct answer to part c)) is the third law of thermodynamics. The
consequences of this for heat capacities is discussed in section 2.20 of the text.
d) Consider the thermal expansion coefficient at constant pressure of any system
!V
T
P
Use one of the Maxwell relations to relate this to a derivative of entropy and then use the
third law to show that the thermal expansion coefficient must approach zero as T 0.The semi-classical particle in a box model of a gas of distinguishable particles results in
the Helmholtz free energy
F = NkT #
ln V +
3
2
ln !mkT
2!2
$
where m is the mass of a gas molecule.a) Determine the entropy of this gas.
b) Determine the chemical potential of this gas.c) Consider two ideal gases that are initially isolated. Gas A consists of molecules with a
smaller mass, gas B of molecules with a larger mass. Initially each has the same volume,
number of particles and temperature. The two gases are allowed to interact and can
exchange particles and energy. In which direction will particles flow as the gases reach
equilibrium? Explain your answer.7 Mean values of position and momentum for a particle in one dimension
Consider a particle that can move in one dimension. Let x and p denote the position and
momentum of the particle respectively. Suppose that the energy of the particle is
E = p2
2m
+ U(x)
where U(x) is any potential energy.a) Show that the partition function takes the form
Z = ZxZp
where
Zx := x
%
eU(x)dx
and
Zp := p
%
ep2/2mdp
where x and p are constants that are independent of temperature.The constants x and p are irrelevant for the thermodynamics that follows and can both be
set equal to 1.
b) The probability density for the position of the particle, regardless of its momentum is
p(x) = 1
Z
%
eEdp.
Show that
p(x) = 1
Zx
eU(x).c) The mean value of the position, regardless of momentum, is
x := %
x p(x)dx.
Show that
x = 1
Zx
%
x eU(x)dx.
d) Evaluate x for a one dimensional classical harmonic oscillator.e) Consider a particle that is trapped in a vertical region 0 < y ! and which is subject
to potential u(y) = mgy where m > 0. Sketch the potential and indicate its minimum.If a particle were released from rest at any y > 0, where would you expect to eventually
find it? Now suppose that the particle is in contact with a heat bath at temperature T.Determine an expression for y. How does this compare to your previous expectation?
Comment on this as T 0 and T .
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