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System of two ideal Fermi gases in three dimensions.

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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.

asked Nov 29 in Physics Problems by JD_PM (500 points)
edited 6 days ago by JD_PM
I have been thinking that this question may be too broad. Should I edit it and focus on one point?
This question concerns a topic which I do not understand well. I recommend that you ask John Rennie if he is able to help.

1 Answer

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It is all about knowing that the ground-state pressure of the ideal Fermi gas is:

$$P_o = \frac{2}{5} n \epsilon_f$$

And actually (after some dimensional analysis), I realised that equation 19 is wrong as it should be:

$$p_o = \frac{2}{5}( \frac{3 h^3}{(2s+1)8 \sqrt{2} \pi m^{3/2}})^{2/3} n^{5/3}$$

From that point on, it drags that mistake.

answered 6 days ago by JD_PM (500 points)
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