**Problem : **

Calculate the pressure of a Fermi gas in its ground state.

Remember that
*p* = -
.
We recall that
*U*
_{gs} =
*N*
. Now we need only to
calculate the derviative. Don't forget that
is a function
of the volume. The simplified result is:

**Problem : **

Check that the energy of the ground state of a Fermi gas is correct by calculating the chemical potential from it.

Recall that
*μ* =
.
We take the appropriate derivative, remembering that
is a
function of
*N*
, and find that
*μ* =
. This shouldn't surprise
us; we defined the Fermi energy to be exactly the chemical potential at a
temperature of zero, which is the approximate requirement for the ground
state to be occupied.

**Problem : **

A long series of calculations can be used to derive the entropy of the
Fermi gas, and the result is
*σ* =
*Π*
^{2}
*N*
. From this, calculate the heat capacity at constant
volume.

Remember that
*C*
_{V} = *τ*
.
The algebra is simple, and yields
*C*
_{V} =
*Π*
^{2}
*N*
.

**Problem : **

It turns out that the energy of a Bose gas is given by:
*U* = *Aτ*
^{
}
where
*A*
is a constant that depends only on the volume. From this,
calculate the heat capacity at constant volume.

Using the equation
*C*
_{V} =
,
which comes from the more primitive definition of the heat capacity via the
thermodynamic identity, we find
*C*
_{V} =
.

**Problem : **

Using the knowledge that the entropy goes to zero as the temperature goes to zero, calculate the entropy from the heat capacity.

Remember that
*C*
_{V} = *τ*
. We
solve for
*σ*
, performing the integration from 0 to
*τ*
, and
setting the arbitrary constant equal to 0 in order that the conditions
at
*τ* = 0
are met, and get:
*σ* =
.