The Maxwell speed distribution is
pspeed(v)=4
! m
2kT
3/2
v2ev2m/2kT .a) Determine the most likely speed (the speed at which the probability is greatest).
b) Show that the mean speed is
v =
#8kT
m .
By what factor does this differ from the most likely speed?c) As the temperature increases, what happens to the most likely speed and the mean
speed?
d) Determine the average kinetic energy for a single molecule using the Maxwell speed distribution and use the result to obtain an expression for the energy of an ideal monoatomic
gas consisting of N such molecules.e) Consider nitrogen molecules (N2). Using the same axes, plot the Maxwell speed distribution for these molecules at 300 K and also 600 K. Indicate any differences in the
graphs.a) The distributions of speed for
two gases, whose particles are
the same, are provided. Describe as accurately as possible
how the temperature of gas A
is related to the temperature
of gas B.0 200 400 600 800 1000 1200 1400
0
0.1
0.2
0.3
0.4
v in m/s
Gas A
Gas Bb) The distributions of speed for
two gases, whose temperatures are the same, are provided. Describe as accurately
as possible how the mass of a
molecule of gas A is related
to the mass of a molecule of
gas B.
0 200 400 600 800 1000 1200 1400
0
0.1
0.2
0.3
v in m/s
Gas A
Gas BThe escape velocity of any particle from a location near Earths surface is about 11 km/s
and any particle with a speed larger than this will leave Earth. This notion can be used to
understand Earths atmosphere.a) Show that the probability with which a particle has a speed greater than vmin is
4
$
vmin/
2kT /m
u2eu2
du.
Hint: Use the subsitution u = v/%2kT /m.b) Determine the probability with which a helium atom will have speed larger than Earths
escape velocity if the temperature is 300 K.c) Determine the probability with which a nitrogen molecule will have speed larger than
Earths escape velocity if the temperature is 300 K. By what factor is the helium molecule
more likely to be able to escape Earth? Use the results to describe how the Earth has
the atmosphere that it has.The integrals in parts b) and c) must be evaluated numerically. Wolfram Alpha and similar
programs can do this.In classical physics a solid can be modeled as a collection of distinguishable three dimensional
harmonic oscillators, each with the same frequency, . Any single oscillator has energy
E = p2
2m
+
1
2
m2r2
where r is the three dimensional position vector, p is the three dimensional momentum
vector and m is the mass of the oscillator. Suppose that the solid consists of N identical but
distinguishable oscillators.a) Determine the partition function for the system of oscillators, assuming that it is in
contact with an environment at temperature T.
2
b) Show that the energy of the system is
E = 3NkT
and determine the heat capacity of the system. This will yield the law of Dulong and
Petit.c) At 298 K and 1 atm, the heat capacity of copper is 385.1 J/kg K, the heat capacity of
aluminum is 897.1 J/kg K, and the heat capacity of iron is 449.3 J/kg K. How do these
compare to the heat capacity, when converted to the correct units, as predicted by the
law of Dulong and Petit?a) A system has two energy levels and contains three Bosons. Provide occupancy diagrams
for all possible states of the system.
b) A Fermi-Dirac system has four energy levels. Provide occupancy diagrams for all possible
states of the system.Consider a system in which there is one energy level, with energy #. Suppose that the particles
in the system are bosons, # > 0, and the chemical potential is = 0.
a) Determine the mean occupancy number, the mean energy and the heat capacity of the
system as T 0.b) Determine the mean occupancy number, the mean energy and the heat capacity of the
system as T .
c) Determine an expression for the temperature at which the mean energy is E = #.Consider a single system state with energy, #.
a) Determine n in the limits as T . Does the result depend on the chemical potential?
What does this imply about the likelihood of a Fermion occupying the state as the
temperature becomes very large?b) Now consider the limit as T 0. There are two cases to consider. Determine n in
the limits as T 0 if # > and separately if # < . What does this imply about the
likelihood of a Fermion occupying the state as the temperature becomes very small?c) Now suppose that T is fixed and consider the mean occupation number as a function of
energy. Determine n for # = .
The next exercises aim to plot the Fermi-Dirac distribution as a function of # for various
temperatures.d) Show that for Fermions
n = 1
e(!!1) + 1
where ## := #/ is a rescaled energy variable and := /kT contains information about
temperature. Suppose that is fixed. Plot, using the same axes, n versus ## in the
range 0 ! ## ! 2, for the following temperatures: T = 0.01/k, (lower temperature),
T = 0.2/k,, T = 1/k, T = 10/k (even higher temperature).e) These graphs should reveal the behavior of Fermions as temperature changes. Suppose
that there are many energy states in the system, each with distinct energies. At very
low temperatures how will the states be populated? Which are very likely to contain a
particle, which are very unlikely to contain a particle? At very high temperatures, how
will the states be populated?
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