- Let
- Let
- Show that

(See Exercise 2.) Thus, deduce that - Consider the integral
- Consider a gas consisting of identical non-interacting particles. The quantum states of
a single particle are labeled by the index
. Let the energy of a particle in state
be
.
Let
be the number of particles in quantum state
. The partition function of the gas is thus
- Demonstrate that
- Show that
- Hence, deduce that

- Demonstrate that
- Use the results of the previous exercise to show that:
- For identical, non-interacting, particles distributed according to the Maxwell-Boltzmann distribution,
- For photons,
- For identical, non-interacting, massive particles distributed according to the Bose-Einstein distribution,
- For identical, non-interacting, massive particles distributed according to the Fermi-Dirac distribution,

- For identical, non-interacting, particles distributed according to the Maxwell-Boltzmann distribution,
- Consider a non-relativistic free particle of mass
in a cubical container of edge-length
, and volume
.
- Show that the energy of a general quantum state,
, can be written
- Demonstrate that the mean pressure of a gas of weakly-interacting non-relativistic particles is

- Show that the energy of a general quantum state,
, can be written
- As an electron moves in a molecule, there exists, at any instance in time, a separation of positive and negative charges within the molecule.The molecule therefore possesses a time-varying electric dipole moment,
. Assuming that the molecule
is located at the origin, its instantaneous dipole moment generates an electric field
- Consider a van der Waals gas whose equation of state is

Here, , where is the molecular mass, is known as the*quantum concentration*, and is the particle concentration at which the de Broglie wavelength is equal to the mean inter-particle spacing. Obviously, the previous expression are only valid when . - Show that the energy density of radiation inside a radiation-filled cavity whose walls are
held at absolute temperature
is
- Apply the thermodynamic relation
to a photon gas.
Here, we can write
, where
is the mean energy density of
the radiation (which is independent of the volume). The radiation pressure is
.
- Considering
as a function of
and
, show that

- Demonstrate that the mathematical identity
leads to a differential equation for
that can be integrated to give
- Hence, deduce that

- Considering
as a function of
and
, show that
- Show that the total number of photons in a radiation-filled cavity of volume
, whose walls are
held at temperature
, is

respectively. Note that the entropy per photon is a constant, independent of the temperature. - Electromagnetic radiation at temperature
fills a cavity of volume
. If thecavity is thermally insulated, and expands quasi-statically, show that
- The partition function for a photon gas is
- Using the standard results

show that

- The partition function for a photon gas is
- Show that the power per unit area radiated by a black-body of temperature
peaks atangular frequency
, where
, and
is the solution of the transcendental equation
- A black (non-reflective) plane at temperature
is parallel to
a black plane at temperature
. The net energy flux density in vacuum
between the two planes is
, where
is the Stefan-Boltzmann constant. A third black plane is inserted
between the other two, and is allowed to come to a steady-state temperature
. Find
in terms of
and
, and show that the net energy
flux is cut in half because of the presence of this plane. This is the principle
of the
*heat shield*, and is widely used to reduce radiant heat transfer. - Consider the conduction electrons in silver, which is monovalent (one free electron per atom), has a mass density
, and an atomic weight of 107. Show that the
Fermi energy (at
) is
*Fermi temperature*--is - Demonstrate that the mean pressure of the conduction electrons in a metal can be written
- Show that the contribution of the conduction electrons in a metal to the isothermal compressibility is
- Justify Equation (8.197).
- A system of
bosons of mass
and zero spin is in a container of volume
, at an absolute temperature
. The number ofparticles is