In: Physics
Consider N massless non-interacting spin-s fermions in a
three-dimensional box of volume V .
(a) Find the Fermi energy EF as a function of N, V , and
s.
(b) For zero temperature, find the pressure in terms of N, V , and
EF .
(c) Plot the occupation of states as a function of the energy at a
temperature of T =EF /(10k). Your graph can be a sketch
by hand. However, the effect of the finite temperature should be
indicated clearly.
(d) Suppose that you start with a gas of these particles in a box
of volume V near T = 0.
Now suppose that a valve is opened such that the gas can undergo a
free expansion, allowing it to come to equilibrium at a new volume
that is twice as large as the initial volume. Assuming no work or
heat transfer occurs during the expansion, does the temperature of
the gas go up, down, or stay the same? Explain your reasoning.