Reference: Daniel V. Schroeder, *An Introduction to Thermal Physics*, (Addison-Wesley, 2000) – Problem 1.64.

We can estimate the thermal conductivity of a gas such as helium using the approximate formula

where is the average molecular velocity, which we can approximate by the rms speed, which is

The mean free path is based on the idea that the mean path length is equal to the length of a cylinder of radius equal to the molecule’s diameter and volume equal to the average volume per molecule , so that

where is the radius of the molecule. The heat capacity is

where is the number of degrees of freedom of the molecule.

At room temperature and pressure we can work out for helium by looking up a few properties using Google. The effective radius of a helium atom seems to depend a lot on which web page you find. It seems that the van der Waals radius is defined as half the distance between two nuclei when two non-bonded atoms are at their closest possible approach. The most commonly quoted value for helium is

This gives a mean free path of

The mass of a helium-4 atom is about 4 atomic mass units or

The rms velocity at is therefore

Since helium is monatomic, it has only 3 degrees of freedom so and

Putting all this together gives an estimate of :

This is only about half the measured value of around 0.142. Using a radius of around gives a better result, and this value is given on a few web sites so who knows? In any case, we’d expect for helium to be higher than air, since the lower mass of the molecule (it being a single atom) gives it a higher speed so it will transport energy faster.

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