3
$\begingroup$

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?

$\endgroup$
3
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.