# Effect of Particle Mass on Thermal Conductivity

Fouriers law of thermal conductivity is $$\vec{q} = -k\nabla T$$

where $q$ is the heat flux, $k$ is the thermal conductivity.

Mass does not seem to appear in the equation. So I'm wondering what if keeping all else constant, but if we change the particle mass (i.e. increase it) in a solid, how would $k$ change?

It would be nice if someone could refer me to a book/article too.

• If you keep EVERYTHING ELSE CONSTANT except for mass, then k doesn't change. Only mass changes per your assumption. – James Jan 7 '16 at 12:53

The expression for thermal conductivity is given by $$K = 1/3 Cvl$$

where $C$ is the heat capacity per unit volume, $v$ is the average velocity of the thermal conducting particles and $l$ is the mean free path of the thermal conducting particles. In a solid, the main contribution of thermal conductivity comes from phonons rather than electrons. Before discussing other aspects, it is better to go to an easier example, that of gases.

Keeping all parameters like temperature fixed, the average velocity of gases come out to be

$$v = c\sqrt{\dfrac{RT}{M}}$$ where $c$ is a constant. In this case the thermal conductivity depends on the mass, and that is the reason that helium and hydrogen have very high thermal conductivity.

In case of solids however this picture gets complicated as the major contribution comes from phonons. So factors such as mean free path and average velocity of phonons are involved. Naively speaking, the average group velocity of phonons follow the same dependence on mass as that of gases, so the thermal conductivity should also. But lattice spacing (a measure of mean path length) as well as effects such as Umklapp processes and geometrical imperfections play a much larger role here. In general mass alone is not the only deciding factor in the thermal conductivity but there are other factors.

The thermal conductivity contains the cross-section for particle collisions. The cross-section itself contains the mass of the particles that are carrying the thermal energy (as well as the mass of the "thing" the particle is colliding with). Sometimes one of these masses can be ignored.

The exact dependence of the cross-section on the mass of the particles depends on the regime you are considering (density, temperature, type of particles involved). But it's in there.