Deriving Relativistic Pressure of Ideal Gas: Why Am I Getting v/c?

[Jeremy Wittkopp]

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.

 

[Orodruin]

What is v/c for an ultrarelativistic particle?

 

[Jeremy Wittkopp]

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

 

[Jeremy Wittkopp]

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

K_rel = (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.

 

[sardar]

Firstly, it is important to note that the kinetic theory of gases is based on classical mechanics and does not take into account the effects of relativity. Therefore, when trying to adapt the theory to express the pressure of an ideal gas in terms of relativistic energy, there may be some discrepancies.

One possible explanation for the inclusion of the term for velocity (v/c) in the derived expression is that the velocity of the gas molecules is being treated as a relativistic quantity. In classical mechanics, the kinetic energy of a particle is given by 1/2mv^2, where m is the rest mass and v is the velocity. However, in relativity, the kinetic energy is given by the expression E = √(p^2c^2 + m^2c^4), where p is the momentum of the particle. This expression takes into account the effects of special relativity, including the increase in mass as the velocity approaches the speed of light.

Therefore, in order to incorporate the effects of relativity into the kinetic theory of gases, the velocity term (v/c) may have been included in the derived expression for pressure. However, this may not be a completely accurate representation as the kinetic theory does not fully account for the relativistic effects on gas molecules.

In conclusion, the inclusion of the term (v/c) in the derived expression for pressure may be an attempt to incorporate the effects of relativity into the kinetic theory of gases. However, it is important to keep in mind that the theory is based on classical mechanics and may not fully account for the relativistic behavior of gas molecules. Further research and development may be needed in order to accurately describe the pressure of an ideal gas in terms of relativistic energy.