**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* = .

Take a Study Break!