# Relativistic Molecules

## Main Question or Discussion Point

Suppose the molecules of an ideal gas move with a speed comparable to the speed of light. I am trying to adapt the kinetic theory to express the pressure of the gas in terms of m and the relativistic energy, but each time I try to derive the expression, I get: P = (1/3)(N/V)(v/c)√(E2 - m2c2).

N/V - number density
v - velocity
c - speed of light
E - relativistic energy
m - rest mass

There shouldn't be the term for velocity. Idk what I am doing wrong.

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Orodruin
Staff Emeritus
Homework Helper
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What is v/c for an ultrarelativistic particle?

Wouldn't it be: pc/E? But in my derivation, I already inputted momentum, so I'm still a bit confused.

I tried this derivation again, I'll show my work this time:

Classical Ideal Gas: P = (1/3)(N/V)mv2 = (2/3) * (1/2)mv2 * (N/V) = (2/3)K(N/V) (assuming only 3 degrees of freedom)
P - pressure
N - # of particles
V - volume
m - mass
v - velocity
K - kinetic energy

Krel = (ϒ - 1)mc2
where
c - speed of light
ϒ - Lorentz factor

So then: P = (2/3)(N/V)(ϒ - 1)mc2

Tell me if I am just crazy, relativity isn't my strongest area.