# A maximum temperature

1. Mar 28, 2006

### fehlzunden

While eating at a taco bar a friend and I were discussing various happenings of the day. We both came to the fact that our English teacher told us that there was no such thing as absolute zero when it came to temperature. We continued talking about it and we agreed that temperature was measured by the average kinetic energy of a system as we had been taught in science class. Knowing that temperature is directly correlated with the speed of the molecules in a system it dawned on us that there should also be a maximum temperature. This being when the velocity of the particles in a system hit the speed of light. Seeing as neither of us is any good with thermodynamics we were hoping some one could shed some light on this subject.

2. Mar 28, 2006

### Staff: Mentor

I think you have it backwards: there is an absolute zero and there is not an absolute maximum temperature.

The speed of light is indeed a maximum speed, but kinetic energy does not have a limit, and the equation you learn in high school physics for kinetic energy is only a low-speed approximation. Welcome to the wonderful world of Einstein's Relativity...

3. Mar 28, 2006

### GOD__AM

How can there be an absolute zero if everywhere we look we measure the temp of the CMB to be above absolute zero? Even if we could cool atoms to absolute zero when we try to measure it we will still see the CMB no?

4. Mar 28, 2006

### vanesch

Staff Emeritus
That's about correct: the temperature can (classically) be seen as the average kinetic energy of the particles in the system. There is indeed an absolute zero temperature, which is around -273.16 C.

The error here is the relationship between velocity and kinetic energy. While there is a clear relationship at low (non-relativistic) velocities, at ultra-relativistic states of motion, the speed is essentially c (a tiny bit less) but the energy can still increase indefinitely (the tiny bit then becomes a tiny bit smaller). In the limit of v = c the kinetic energy becomes infinite (and so the temperature that goes with it).

Must be the thermodynamic equivalence of Zeno's paradox :-)

5. Mar 29, 2006

### rbj

dunno if i would trust the competence in physics of your English teacher anymore than i would trust the competence of a physicist in the works of Shakespeare.

there is an absolute zero (where there is no kinetic energy in any particles), but humans have not been able to create an environment where the temp was precisely 0K.

there may be an absolute maximum temperature that would be in the ballpark of the so-called "Planck Temperature" and it's some awful amount. can't remember but it might be around 1035K.

6. Mar 29, 2006

### Staff: Mentor

Just because we are unable to achieve it, that doesn't mean it doesn't exist.

I suspect you are trying to argue something philosophical here, but it just isn't relevant. Even if it only exists as a boundary on calculations, it still exists.

Last edited: Mar 29, 2006
7. Mar 30, 2006

### vanesch

Staff Emeritus
In fact, although reaching absolute zero is not possible, going BELOW the CMB temperature is possible and is being done. The CMB temperature is of the order of the boiling point of He-4 ; ~ 4 K. He-3, for instance, boils at 1.3 K, and with more sophisticated techniques, one can go down much more.

8. Mar 30, 2006

### Pengwuino

I thought absolute 0 was impossible to reach because that would mean we could measure a particle's position with absolute certainty which violates HUP

9. Mar 30, 2006

### vanesch

Staff Emeritus
No, this has actually nothing to do with it. A quantum system in its ground state would have temperature 0, even though the kinetic energy expectation value is not 0.

This can be seen in the canonical ensemble: http://en.wikipedia.org/wiki/Statistical_mechanics

If the system is in its ground state, then P_1 = 1 and all other P_j are 0. As such, this corresponds to $$\beta = \infty$$ or T = 0.

For instance, look at the F-D statistics http://en.wikipedia.org/wiki/Fermi-Dirac_statistics

for T = 0, we have a step function, which corresponds essentially to the occupation of the ground state of the entire system (not the ground state of each individual component!).

Exercise: what is the expectation value of the KE of an electron in a hydrogen atom, and to what "temperature" would that correspond if we had the relationship KE = 3/2 k T ?

We see that the average kinetic energy equals (minus) the binding energy,
13.6 eV, which corresponds to about 105000K...

cheers,
patrick.

Last edited by a moderator: May 2, 2017
10. Mar 30, 2006

### fehlzunden

Thanks

Thank you for you comments on the subject. My friend and I both agree that our English teacher is incorrect on the absolute zero thing which I failed to make clear in the previous post. Mostly I was curious if anyone had come up with some cute way of finding a maximum temperature sorda like Schwarzschild and the black hole thing. Based on your comments I would guess not and that there probably is not one based on my ideas of kenetic energy. Thanks again.