we can appreciate the heat resistance of some materials. Such materials have a lower gradient, \(\cfrac{d(r_e)}{d\,T}\) just before the kink point on its \(r_e\) vs \(T\) curve. From the post "Gap, Gap, Gap and Quack",
\(\cfrac{d(r_e)}{dt}=\cfrac{d(r_e)}{dT}\cfrac{dT}{dx}\)
where \(x\) is along \(r_e\) and \(\cfrac{dT}{dx}\) is the temperature gradient around the nucleus.
A corresponding high value for velocity, \(\cfrac{d(r_e)}{dt}\) occurs (assuming, \(\cfrac{dT}{dx}\) to be the same). So much so that higher heat input is need to make up for the disparity in kinetic energy before temperature, \(T\) can increase further.
\(\cfrac{d(r_e)}{dt}|_L>\cfrac{d(r_e)}{dt}|_H\)
\((\cfrac{d(r_e)}{dt}|_L)^2>(\cfrac{d(r_e)}{dt}|_H)^2\)
and so,
\((\cfrac{dd(r_e)}{dt}|_K)^2-(\cfrac{d(r_e)}{dt}|_H)^2>(\cfrac{d(r_e)}{dt}|_K)^2-(\cfrac{d(r_e)}{dt}|_L)^2\)
where \(\cfrac{d(r_e)}{dt}|_K\) is at a common point beyond the kink in the graph.
\(\Delta v^2_{BG\,H}>\Delta v^2_{BG\,L}\)
Since, \(E_{BG}=\cfrac{1}{2}m_e\Delta v^2_{BG} \)
\(E_{BG\,H}>E_{BG\,L}\)
The gradient beyond the kink is due to the square root in the expression for \(r_e(T)\). It is the same high value for all material. The temperature \(T\), where the kink occurs in the graph is however, different from material to material.
With this in mind, Heat Capacity of a material is now more complicated.
