# Upper limit to the temperature of a body

I wonder if there is an upper limit to the temperature analogous to absolute zero due to the following:

$$K_{\mathrm{avg}} = \frac{3}{2}kT$$

where $K_{\mathrm{avg}}$ is average kinetic energy. Average kinetic energy is due to the average velcoity of the material particles in motion which cannot travel at the speed of light? So is there a temperature at which the molecules would achieve the speed of light on average and therefore cannot reach that temperature?

I know this must be wrong but I want to know what’s wrong with it because I have never heard of an upper limit to temperature in principle.

That particular line of thought is wrong (as you suspect) because there is no upper limit to kinetic energy. ${1 \over 2} m v^2$ is only the low velocity approximation, the full form is $(\gamma-1) m c^2$ . As you give more energy to a particle travelling near the speed of light (as happens in the LHC) the KE increase manifests itself in an increase in $\gamma$ even though there is very little increase in $v$.
There was once an idea that there could be an ultimate maximum temperature, before we knew about quarks and thought that protons, neutrons, Sigmas, Deltas, pions and all that zoo were truly 'elementary'. There seemed to be more and more of these states with increasing energy. If the spectrum of such states got sufficiently dense sufficiently fast, then there would be a limiting temperature ($T_H$, the Hagedorn temperature): as you approached it then as you put energy into a system it would produce more and heavier particles, rather than increasing their individual energies so you could never exceed $T_H$. Nowadays we see it as a temperature (around 6 trillion degrees) at which ordinary elementary particles become the quark-gluon plasma.