Finding the equation of state and internal energy of a given gas.

[quantumkiko]

Consider a gas contained in volume V at temperature T. The gas is composed of N distinguishable particles at zero rest mass, so that the energy E and momentum p are related by E = pc. The number of single-particle energy states in the range p to p + dp is $$4\pi Vp^2 dp$$. Find the equation of state and the internal energy of the gas and compare with an ordinary (ideal?) gas.

In how I understand the problem, you must integrate $$4\pi Vp^2 dp$$ and $$E = pc$$ to get the partition function $$Z(E)$$ then use the entropy $$S = k_b ln Z$$ to get the equation of state given by

$$\left(\frac{\partial S}{\partial V}\right)_{N, E} = \frac{P}{T}$$

and the internal energy by,

$$\left(\frac{\partial}{\partial E}\right)_{N, V} = \frac{1}{T}.$$

Did I understand the problem correctly?

 

[Mapes]

You mean

$$\left(\frac{\partial S}{\partial E}\right)_{N, V} = \frac{1}{T}$$

for the second equation? Looks good.

 

[quantumkiko]

Yes you're right. Sorry for the mistake. And what I meant was you must integrate $$4\pi Vp^2 dp$$ and use E = pc to express the integral in terms of $$E$$ instead of $$p$$. Thank you, I hope I got it correct then. What would be the bounds of the integral?