Finding $c$ with Dulong-Petit law on non-molecular substance

Question

The Wikipedia article states the Dulong-Petit law as follows:

$$c=\frac{3R}M$$

$c$ is specific heat capacity $[\mathrm{\frac J{kg\; K}}]$ and $R$ the gas constant. $M$ is the molar mass in [g/mole], and "grams per mole" makes us think of a number of moles of a substance, as were it a molecular material.

In crystal structures there is no such thing as a "mole" of the substance. An example could be $\ce{CoSb3}$. $\ce{CoSb3}$ is not a molecule, but the simplest formula unit of the endlessly repeating crystal. So which $M$ to plug into the Dulong-Petit formula is not obvious here.

A researcher gave me the following version of the Dulong-Petit formula for this case (which I have used successfully):

$$c=\frac {3R}{M}n$$

• The $M$ still means molar mass [g/mol], treating the formula unit as were it a molecule.

• The $n$ is the number of atoms in the formula unit. So here, $n=4$. Simple.

I do not see how this version is equivalent with the former version. How does "treating the formula unit as was it a molecule" balance out with "multiplying with the number of atoms"? How is

$$\underbrace{\frac{3R}M}_{molecular}\Leftrightarrow \underbrace{\frac{3R}Mn}_{crystalline}$$

equivalent?

Answer 1

It may be useful to start with the specific molar heat capacity $c'$ (per mole, instead of per kilogram), which is somewhat more fundamental. Here $c' = 3R$.

This is true even for your crystalline solid, though we have to decide what "moles" refers to in this case. From a physical perspective, the heat capacity is a result of vibrational motion in three dimensions, and each atom will contribute to this motion. Our "moles" should therefore be "moles of atoms"---for the purposes of this calculation, we consider the atom to be the fundamental unit.

The molar mass should then be $\text{mass of solid/moles of atoms}$, which can be interpreted as a weighted molar mass. You can see that your formula $M/n$ calculates exactly this: the mass per mole of a formula unit, divided by the number of atoms in a formula unit.

Hence $$c = \frac{3R}{M/n} = \frac{3R}{M}n.$$