Heat Resistance, A Kinky Issue

From our understanding of the energy requirements of increasing temperature  $T$ and decreasing electron orbital radius,

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.