Ex2.7 Black holes, white dwarfs and neutron stars-Shapiro, Teukolsky

[Vrbic]

Homework Statement

Show that from (*) that for a nonrelativistic Maxwell-Boltzmann gas,
$n=g\bigg(\frac{nkT}{2\pi\hbar^2}\bigg)^{\frac{3}{2}}e^{\frac{\mu-mc^2}{kT}}$
$P=nkT$
$e=nmc^2+\frac{3}{2}nkT$ [itex][/itex]

Homework Equations

(*): $f(E)=e^{\frac{\mu-E}{kT}}$
$E=\sqrt{p^2c^2+m^2c^4}$
$n=\frac{g}{h^3}\int f(E)d^3p$
(#) $h=2\pi\hbar$

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.

 

[mark726]

Hello there,

I see that you are trying to derive the Maxwell-Boltzmann distribution for a nonrelativistic gas from the given equation (*) and the equations for energy and number density. Your approach seems to be correct, but there are a few mistakes in your calculations.

Firstly, you have assumed that the mean energy of the particles is equal to their rest energy, which is mc^2. This is not true for a nonrelativistic gas, as the mean kinetic energy of the particles is given by E_k=mv^2/2=p^2/2m=3kT/2. This is because the particles in a nonrelativistic gas have a very low velocity compared to the speed of light, so their kinetic energy can be approximated by classical mechanics.

Secondly, when you substitute p=\sqrt{3mkT} in the expression for number density, you have forgotten to take the derivative of the exponential term with respect to p. It should be dp instead of p when you take the derivative.

Lastly, in your final expression for number density, you have made a mistake in the exponent. It should be (3mkT)^{3/2} instead of (3^{1/3}mkT)^{3/2}.

So, the correct expression for number density in terms of the given parameters is:

n=ge^{\frac{\mu-mc^2}{kT}}\bigg(\frac{3mkT}{2^{5/2}\pi^{1/2}\hbar^3}\bigg)^{3/2}

Now, to show that P=nkT, we can use the ideal gas law, which states that PV=nRT, where R is the gas constant. Since we are dealing with a nonrelativistic gas, we can use the classical expression for pressure, which is P=nkT. Therefore, we have:

P=nkT=ge^{\frac{\mu-mc^2}{kT}}\bigg(\frac{3mkT}{2^{5/2}\pi^{1/2}\hbar^3}\bigg)^{3/2}kT

=ge^{\frac{\mu-mc^2}{kT}}\frac{3^{3/2}}{2^{3/2}\pi^{1/2}\hbar^3}(mkT)^{3/