# Homework Help: Ex2.7 Black holes, white dwarfs and neutron stars-Shapiro, Teukolsky

1. Nov 18, 2014

### Vrbic

1. The problem statement, all variables and given/known data
Show that from (*) that for a nonrelativistic Maxwell-Boltzmann gas,
$n=g\bigg(\frac{nkT}{2\pi\hbar^2}\bigg)^{\frac{3}{2}}e^{\frac{\mu-mc^2}{kT}}$
$P=nkT$
$e=nmc^2+\frac{3}{2}nkT$ 

2. Relevant equations
(*): $f(E)=e^{\frac{\mu-E}{kT}}$
$E=\sqrt{p^2c^2+m^2c^4}$
$n=\frac{g}{h^3}\int f(E)d^3p$
(#) $h=2\pi\hbar$
3. The attempt at a solution
So I suppose that nonrelativistic mean $E=mc^2$, than $n=\frac{g}{h^3}\int e^{\frac{\mu-mc^2}{kT}}4\pi p^2dp=ge^{\frac{\mu-mc^2}{kT}}\frac{4\pi}{3h^3}p_f^3$. $E_k=mv^2/2=p^2/2m=3kT/2$=>$p=\sqrt{3mkT}$. Employ this in previous $n=ge^{\frac{\mu-mc^2}{kT}}\frac{4\pi}{3h^3}(3mkT)^{3/2}$ due (#) $n=ge^{\frac{\mu-mc^2}{kT}}\bigg(\frac{3^{1/3}mkT}{2^{2/3\pi^{4/3}}\hbar^3}\bigg)^{3/2}$. Do somebody see some mistake?
Thank you very much.

2. Nov 23, 2014