- A box is separated by a partition that divides its volume in the ratio
3:1. The larger portion of the box contains 1000 molecules of Ne gas, and the smaller,
100 molecules of He gas. A small hole is punctured in the partition, and the
system is allowed to settle down and attain an equilibrium state.
- Find the mean number of molecules of each type on either side of the partition.
- What is the probability of finding 1000 molecules of Ne gas in the larger partition, and 100 molecules of He gas in the smaller (in other words, the same distribution as in the initial system)?

- Consider an isolated system consisting of a large number,
, of very weakly-interacting localized particles of spin
. Each particle has a magnetic moment,
, that can point either parallel or antiparallel to an applied magnetic
field,
. The energy,
, of the system is then
, where
is the number of spins aligned parallel to
, and
the number
of antiparallel spins.
- Consider the energy range between
and
, where
is very
small compared to
, but is microscopically large, so that
.
Show that the total number of states lying in this energy range is
- Write down an expression for
. Simplify this expression
using Stirling's approximation (i.e.
) to obtain

- Use the definition
(where
) to show that the energy of the system can be written
- Under what circumstances is the temperature negative?
- Suppose that two otherwise identical spin systems with equal and opposite temperatures are brought into thermal contact. What is the final temperature of the overall system?

- Consider the energy range between
and
, where
is very
small compared to
, but is microscopically large, so that
.
Show that the total number of states lying in this energy range is
- A glass bulb contains air at room temperature and at a pressure of 1 atmosphere.
It is placed in a far larger
chamber filled with helium gas at 1 atmosphere and at room
temperature. A few months later, the experimenter happens to read in a journal that
the particular glass of which the bulb is made is quite permeable to helium,
but not to any other gas. Assuming that equilibrium has been reached by this time,
what gas pressure will the experimenter measure inside the bulb when he/she goes
back to check?
- The heat absorbed by a mole of ideal gas in a quasi-static process in
which the temperature,
, changes by
, and the volume,
, by
, is
given by
- A solid contains
magnetic atoms having spin
. At sufficiently
high temperatures, each spin is completely randomly oriented. In other words, it is equally
likely to be in either one of two possible states. But at sufficiently low
temperature, the interactions between the magnetic atoms causes them to
exhibit ferromagnetism, with the result that their spins become oriented in
the same direction. A very crude approximation suggests that
the spin-dependent contribution,
, to the heat capacity of the solid has an
approximate temperature dependence given by