Can Particles in Extreme Conditions Exhibit Quantum Degeneracy?

[Anabelle37]

Homework Statement

By estimating the De Broglie wavelength and the inter-particle spacing, find whether the following systems of particles are degenerate:
(i) a gas of neutrons in a neutron star with mass density 10^17 kg/m^3 at the temperature of 10^8K
(ii)a gas of oxygen molecules at pressure 10atm(=10^6 pascals) at room temperature

Homework Equations

De Broglie wavelength, l= h/(sqrt(2*pi*m*k*T))
inter-particle spacing: (4/3)*pi*d^3=1/n
h = 6.626x10^-34 Js
k= 1.38x10^-23 J/K

The Attempt at a Solution

for degeneracy d< l

De Broglie wavelength, l= 6.626x10^-34 /(sqrt(2*pi*10^17 *1.38x10^-23*10^8))
= 2.25x10^-35?

Is m in the DeBroglie equation meant to be mass? so if I have mass density how do I found the mass if idon't know the volume of the gas??

Also for the inter-particle spacing what is the number density(n) for each example??

Thanks

 

[Thaakisfox]

The "m" in the de Broglie wavelength formula is the mass of one gas particle. In this case the mass of one neutron.

"n" is the number density i.e. the number of particles in unit volume. This can be expressed with the mass of a neutron and the given mass density of the neutron gas as follows:

consider V to be the volume of the gas, and N to be the total number of particles in the gas. Then: n=N/V
Now if rho is the mass density then the volume of the gas is: V=(m*N)/rho since m is the mass of a single particle, and so m*N is the total mass of the gas. Now write this in the formula for the number density, and you will see that N will cancel, only the m mass of a single neutron and the rho mass density will stay back, which are known.