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

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Discussion Overview

The discussion revolves around deriving the relativistic pressure of an ideal gas, particularly focusing on the challenges faced when incorporating relativistic effects into the kinetic theory of gases. Participants explore the implications of relativistic speeds on pressure calculations and the role of various parameters in the derivation.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant expresses confusion over their derived expression for pressure, which includes a term for velocity (v/c), suggesting it should not be present.
  • Another participant questions the significance of the term v/c for ultrarelativistic particles, indicating a potential area of exploration.
  • A different participant proposes an alternative expression involving momentum (pc/E) but admits to confusion regarding its integration into their derivation.
  • One participant shares their derivation steps, starting from the classical ideal gas pressure equation and transitioning to a relativistic form, incorporating the Lorentz factor (ϒ) and relativistic kinetic energy.
  • The same participant expresses uncertainty about their understanding of relativity, indicating a lack of confidence in their derivation process.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus, as there are multiple competing views and expressions for the relativistic pressure of an ideal gas. The discussion remains unresolved with ongoing questions and clarifications.

Contextual Notes

Participants mention various assumptions and parameters, such as the number density (N/V), rest mass (m), and relativistic energy (E), but do not fully resolve the implications of these in the context of their derivations.

Jeremy Wittkopp
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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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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.
 

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