This Is Not Melting

If auto-ignition temperature is the temperature when the vapor molecules/atoms just achieve boom velocity, instantaneously as their vibration energy increases with temperature, then we have a good correspondence of temperature of the vapor given its molecular/atomic mass, and kinetic energy that was previously used to define temperature.  As in

$PV=nRT$

we have instead,

$T=density*f(v_{boom})$

where $T$ is temperature and $f(v_{boom})$ is some function of $v_{boom}$.  Consider,

$KE\equiv density*f(v_{boom})$

so,

$KE=\cfrac{1}{2}*density*v_{boom}^2$

this would be kinetic energy per unit volume, but,

$v_{boom}=3.4354*density*\cfrac{1}{particle\,count\,per\,type}$

so,

$KE=\cfrac{3.4354^2}{2}*density^3*\cfrac{1}{(particle\,count\,per\,type)^2}$

And in the case of basic elements,

$KE=density^3*\cfrac{5.901}{(atomic\,\,number)^2}$

where the density (cubed) of the elements changes with $T$.  A plot of $T$ vs $v_{boom}$ of various density will give $f(v_{boom})$ or the function f(x), from which we may then infer $KE=g(T)$ where $g(T)$ is a function of $T$.  Temperature $T$ here then, as intended is a measure of the vibrational kinetic energy within the element.

This seemingly is a definition of temperature without the explicit involvement of temperature particles, derived from the vibrational kinetic energy of the atoms in an elements.  A sentinel point in this definition is $T_{boom}$, the temperature at which $v_{boom}$ is achieved.

This is not melting, but it would be interesting if $T_{boom}$ occurs before the melting point.

Or is it?  Melting?  $v_{boom}$ changes density and could account for the energy released during phase change.  There is a $v_{boom}$ from solid to liquid, a different $v_{boom}$ from liquid to gas and yet another value of $v_{boom}$ for the gas as it disintegrate.  Different $v_{boom}$ values for different densities of the element/compound.

Is $v_{boom}$ responsible for phase change?  Phase change would then be nuclear.  Boiling water for tea would then be nuclear.