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)?

**EDIT:**

Now I am stuck at EQ 19.