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:
p =
n
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 :
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 = .