1) What is the pressure of a gas of free bosons in the limit of vanishing temperature, $T \rightarrow 0$?
2)Argue that for $T \rightarrow 0$ an ideal Fermi gas will have non-vanishing pressure $p_0 > 0$.
We will now use this fact to study a system of two ideal Fermi gases in three dimensions.
A free sliding piston separates two compartments labeled 1 and 2 with volumes $V_1$ and $V_2$ respectively. An ideal Fermi gas with $N_1$ particles with spin 1/2 is placed in compartment 1 and an ideal Fermi gas with $N_2$ particles with spin 3/2 is placed in compartment 2.
As you notice, this problem has already been solved, but I do not understand the vast majority of it.
1) is OK.
2) I do not know how they got equation 15. It is stated to be a continuous approximation but not idea how to even start.
Actually, Griffiths has an interesting section in which treats the fermion distribution:
I know that the Fermi Dirac distribution (which is the one which interests us, since we are working with fermions) has a pretty well-known behavior as $T \rightarrow 0$ (please see figure 5.8 in Griffiths).
But I still do not know how to get it.
From here on I simply got lost. I mean, I have studied the free particle in QM but they go on with the density of states from EQ 16... I do not grasp it.
May you please shed some light on the provided solution (from EQ 15)?
Now I am stuck at EQ 19.