# Do Relativistic Gasses of Identical Particles Obey the Ideal Gas Law?

I am trying to determine the equation of state and see if $$PV = nRT$$ is satisfied. For an ultra-relativistic gas of identical particles in a volume $$V$$ the energy (I am assume there is no potential) is $$E = c\sum_{i = 1}^Np_i$$ where $$p_i = \sqrt{p_{xi}^2 + p_{yi}^2 + p_{zi}^2}$$. Hence I estimate the number of microstates with energy $$E$$ is

$$\Omega(E) = \bigg(\frac{1}{\Delta p}\bigg)^{3N} \int_{0}^{E/c}\Pi\bigg(4\pi p_i^2 dp_i\bigg) = \bigg(\frac{E}{c\Delta p} \bigg)^{3N} \frac{(8\pi)^{N}}{(3N)!}$$

I used the fact that $$\int_{0}^{E/c}\Pi p_i^2dp_i = \frac{2^N}{(3N)!}\bigg(\frac{E}{c} \bigg)^{3N}$$

Now using the equation $$\frac{1}{T} = \frac{\partial S}{\partial E}$$ I obtain $$E = 3Nk_BT$$. However I am not sure how to find an equation for pressure and volume. Any help will be appreciated

• There should be an $V^N$ in you expression of $\Omega$. Commented Mar 18, 2021 at 16:33

Hint: You have to use $$dE=TdS-PdV+\mu dN$$ $$\frac{P}{T}= \left. \frac{\partial S}{\partial V}\right|_{N,E}$$ Where $$S=k_B\ln\Omega$$

• Slightly confused about how (dS/dE at const N,V) = P/V? Should have been 1/T... Commented Mar 18, 2021 at 12:18
• Ah! Sorry, it should be $\partial S/\partial V$. I should confirm it from the book. It's now corrected. Commented Mar 18, 2021 at 12:47
• The first equality implies that entropy $S$ can be measured in the same units as pressure $P$. How is that possible? In addition, it is impossible to differentiate $S$ with respect to $V$, provided $V = const$.
– Gec
Commented Mar 18, 2021 at 15:30
• It's should be $E$, I have corrected it! Commented Mar 18, 2021 at 16:12
• Could be P/T = (dS/dV, at const T), since then TdS-pdV=dU=0, for constant T. Commented Mar 18, 2021 at 16:16

The calculation is easier in the canonical ensemble. Use $$P=-{\partial F\over\partial V}$$ where the free energy is given by $$F=-k_BT\ln{\cal Z}$$ Note that for free particles, the partition function factorizes $${\cal Z}=z^N$$

• Is the no. of particles varying, for it to be in cannonical ensemble? If so, it should become a Grand-Cannonical ensemble. Commented Mar 18, 2021 at 16:32