# Kinetic theory of gases - Maximum speed

1. Jul 5, 2011

### mafra

If the aleatory speed of a gas rises with pressure and temperature and there is not an apparent limit for these two, what prevents the particles to reach the speed of light?

thank you

2. Jul 5, 2011

### Staff: Mentor

Have you ever heard of Relativity? It explains that as you increase the speed of an object it requires more and more energy to increase it further. For example, the Large Hadron Collider accelerates protons up to beyond 99% the speed of light. The energy required to get them from 80% to 99%+ is MORE than it took to get them to 80% in the first place.

Have a look at relativity on wikipedia or do a search for google. Also, I read an interesting book called "The Anime Guide To Relativity" I believe. It explains the basics pretty well. It's a good read if you enjoy anime or you don't like very technical books.

3. Jul 5, 2011

### mafra

okay, so
it is saying that, higher the speed of the particles, higher the temperature (or in the other way, idk)

so if it is impossible to particles to reach the speed of light, it is impossible to temperatures get beyond some another value (c².Mm/3.R), or you can't apply this relation to higher speeds?

4. Jul 5, 2011

### Staff: Mentor

That equation will not work when you get to speeds that are a significant fraction of the speed of light. You must use relativistic equations at that point, otherwise your numbers become more and more inaccurate the higher you go.
Edit: Just FYI, I don't know the correct formula to use in this situation. I just know that at very high speeds you must use relativistic formulas instead of classical ones.

5. Jul 5, 2011

### Redbelly98

Staff Emeritus
That formula is based on the assumption that the kinetic energy of a gas molecule is $\frac{1}{2} m v^2$, which is a nonrelativistic approximation. That approximation does not hold at relativistic speeds, and hence the expression $v_{rms} = \sqrt{3RT / M_m}$ is not valid.

6. Jul 5, 2011

### mafra

thank you, people

that answered my question

7. Jul 5, 2011

### ZealScience

First of all, speed of light can't be reached by any means because of energy required is infinity

In addition there is certain limit to the temperature. I've heard there is some "highest temperature that could be reached". But I couldn't remember

And at that enormous velocity it would be no gas any more. Have you seen any gas in stars?

What about using relativistic mass? I think that Tayloring γ factor would give terms like that

8. Jul 5, 2011

### Staff: Mentor

ZealScience, are you saying that there is no gas in stars? If so, that is incorrect. Also, you cannot use relativistic mass for that formula. In fact, i think relativistic mass already gives you the kinetic energy, so there's no need to get the correct answer and try to use it in an inaccurate equation.

9. Jul 6, 2011

### ZealScience

Which star is made of gas? It must have extremely low temperature comparing to normal stars.

I think taylor the (γ-1)mc^2, there would be a term like that. And if you plug in some moderately high speed, you can still find it a good approximation.

10. Jul 6, 2011

### D H

Staff Emeritus
The correct answer is that a relativistic Maxwell-Boltzman gas follows the Maxwell-Juttner distribution, and no, you cannot do a simple gamma correction. The problem is that the Maxwell-Boltzmann distribution has a long tail (excess kurtosis =~ 0.108). The high-speed particles in that long tail are subject to greater relativistic effects than are the particles near the mean.

However, relativistic effects typically are of no concern with regard to thermodynamics. The temperature has to be incredibly high, and the particles incredibly small, for relativistic effects to come into play. For a cloud of electrons, you need to worry about relativistic effects if the temperature is on the order of 1010 K or higher. The last time that such conditions existed was about a second after the big bang.

11. Jul 6, 2011

### Staff: Mentor

Quote from wikipedia:

12. Jul 6, 2011

### cjl

Most stars have surface temperatures of below 10,000 K, which is the temperature required for nearly complete ionization of hydrogen (approximately), so most stars have surfaces that are mostly gas (below around 7000K, there isn't very much ionization at all, and they are nearly 100% gas). Only the very hot stars are close to completely ionized.