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Temperature is the proportional measure of kinetic energy of the random motion of the constituent micro particles in a system as per wikipedia.

Now I understand that O2 molecules are randomly moving in air at normal temperatures.

I could not find if we could define the speed of these molecules at a certain temperature.

Obviously these particles with rest mass cannot move as fast as light. So they must cause a max limit to temperature.

I wanted to know if there was a limit to the temperature of air (gases) due to this speed limit of their constituent gas molecules? Is this temperature the absolute max (like absolute min 0 K)? Have we ever seen anything close to this in the universe?

Question:

  1. What is the speed of the random motion of O2 molecules in air at normal temperature (20Celsius)

  2. Is there a function or equation that would describe the connection directly between this speed of the O2 molecules and the temperature of the air they are in? Does this speed (must be smaller then c) give a limit to the absolute max temperature?

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  • $\begingroup$ About 500 m/s or 1000 mph. This has a relation to the speed of sound. At much higher temperatures, molecules would break up. So there is an upper limit for the speed of the molecules of oxygen, but no upper limit for the temperature conceptually. There is of course a practical limit, such as what happens in the core of stars and such. Higher than that would be early in the Big Bang, but it gets tricky, because the Fermi temperature of the Big Bang was high, but the thermodynamic temperature of the Big Bang likely started from absolute zero (lowest possible entropy). $\endgroup$ – safesphere Jul 20 '18 at 19:51
  • $\begingroup$ thank you. what I don't understand is, if temperature is related to speed like T=v^2*M/3R, then if v<c then T has to be smaller then some limit no? $\endgroup$ – Árpád Szendrei Jul 21 '18 at 0:57
  • $\begingroup$ No, this is a non-relativistic formula. See the answer of my2cts. There is no upper limit. As the speed approaches the speed of light, the temperature extends to infinity. However, again, at a high enough temperature molecules of oxygen will break down. At a higher yet temperature, atoms will break down. And so on. So for oxygen per se there is indeed an upper limit. $\endgroup$ – safesphere Jul 21 '18 at 2:03
  • $\begingroup$ "As the speed approaches the speed of light, the temperature extends to infinity." Is there a formula for that? $\endgroup$ – Árpád Szendrei Jul 21 '18 at 2:27
  • $\begingroup$ Yes, the last line in the answer of my2cts, where $\gamma=\dfrac{1}{\sqrt{1-\dfrac{v^2}{c^2}}}$ $\endgroup$ – safesphere Jul 21 '18 at 3:06
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"I could not find if we could define the speed of these molecules at a certain temperature."

Here is the non-relativistic answer. Temperature is related to the energy per degree of freedom by $E=\frac{1}{2}k_BT$. $k_B$ is Boltzmann's constant. The speed of an oxygen molecule represents 3 degrees of freedom, one for each dimension or direction. Thus the total kinetic energy is $E_k=\frac{3}{2}k_BT = \frac{1}{2} m(v-\left<v\right>)^2$ from which you find that $v-v\left<v\right>=\sqrt{3k_BT/m}$. At 20'C this gives 477 m/s. It is important to consider the deviation from average $v-\left<v\right>$ because the average velocity does not affect the temperature.

Your question triggered me to check out the relativistic generalization. The definition of T now becomes $3 k_B T = \left< ( v − \left<v\right> ) . ( p − \left<p\right> ) \right>$ according to this site http://numericana.com/answer/heat.htm#temperature. If the average speed is zero, in the rest frame of the gas, $3 k_B T = \left< m\gamma v^2 \right>$. Clearly there is no upper limit to T, as $\gamma$ is unbounded.

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  • $\begingroup$ thank you, can you please tell me why you are saying that there is no upper limit to T? if the particles move faster, then temperature is faster as I understand. particles can't go fast as light. So T should have an upper limit? $\endgroup$ – Árpád Szendrei Jul 21 '18 at 0:52
  • $\begingroup$ I meant as particles move faster, T is higher. But v<c. So T<limit? $\endgroup$ – Árpád Szendrei Jul 21 '18 at 1:04
  • $\begingroup$ As I said, $\gamma=\frac{1}{\sqrt(1-v^2)}$ is unbounded and so is T. $\endgroup$ – my2cts Jul 21 '18 at 14:43
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The quantity you're looking for is the rms (root-mean-square) speed. The relationship is given by $$v_{rms} = \sqrt{\frac{3RT}{M}}$$ where R is the gas constant, T is the temperature and M is the molar mass.

  1. Using the equation above and the values $$R = 8.3145 \frac{kg\ m^2}{s^2\ K\ mol},\ \ \ T = 293.15\ K,\ \ \ M = 3.2\times10^{-2}\ kg$$ you can plug in and evaluate for the quantity you're looking for. Note that the molar mass is 32 g/mol because oxygen is a diatomic molecule.

  2. The equation above is a description of a statistical relationship between temperature and speed. As I understand, there is a theoretical maximum possible temperature according to the laws of physics as we know them, I think posited by Max Planck.

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  • $\begingroup$ thank you. if you make the equation for T=v^2*M/3R. If v<c then T has to be smaller then a certain limit no? $\endgroup$ – Árpád Szendrei Jul 21 '18 at 0:55
  • $\begingroup$ That makes sense within the scope of the physics used here and I believe is the basis for Planck's hypothesis (I could be wrong about that; could be a fun thing to research), but it is very possible that we simply don't have the whole picture yet. For that reason this concept of "absolute hot" is not universally accepted. $\endgroup$ – Lian Jul 21 '18 at 3:11

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