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:
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.
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 .
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 = .
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 = .