Considering the conversion to molar heat capacity, C_m

The method appearing here seems to be neglecting that N is the number of atoms, not the number of atoms in a mole.

If we have found that C = 3Nk, to find the molar heat capacity, we would divide by the number of moles, n

Thus, C_m = 3Nk/n.

However, N may be rewritten. If n is the number of moles present, then n*L will be the number of atoms (if L is Avogadro’s number, 6.022 x 10²³ atoms/mol).

So then, C_m = 3(nL)k/n

However, the n cancels out and we have

C_m = 3 L k

Considering that k is J/molecule, we can then see that k * L = J/mol and we could see that it is indeed R.

Finally then, C_m = 3R

A similar analysis can be done with Einstein’s formula in the later section of this web page.