I'm reading over a paper by R. Feres and G. Yablonsky titled Knudsen's Cosine Law and Random Billiards, and I can't get around how they don't show directly how Knudsen's Cosine Law was derived. I'm including both that paper and a paper by Knudsen himself titled The Cosine Law in Kinetic Theory of Gasses. I would really appreciate it if someone could provide a step-by-step explanation of how Knudsen obtained his cosine law.

Knudsen says, in his paper, that "when considering a surface element of an area $dS$ of a solid body located in a gas mass at rest, the number of collisions $n'$ per second between the gas molecule and the surface element will have a mean value of $n'=\frac{1}{4}N\bar{c}dS$ where $N$ denotes the number of gas molecules in one cubic centimeter, while $\bar{c}$ is the mean velocity of the gas molecules. the knock-on molecules distributed uniformly over each azimuth strike the surface element, and the number of molecules coming from the solid $d\omega'$ which makes the angle $x'$ with the normal of elements of area, must be $\frac{1}{4\pi}N\bar{c}cos{x'}d\omega dS$ or $\frac{1}{n}n'cos{x'}d\omega'$" (Knudsen 2).

On page 2 of Feres' paper, there is a picture of a solid angle,$d\omega$, created by the angle $\theta$ and the normal vector, which provides some clarity behind the jargon, but it is still fairly unclear to me. It would be much appreciated if someone could explain the concept of Knudsen's Cosine Law more coherently. I am especially interested in a more in-depth look of how it was derived.

The Cosine Law in Kinetic Theory of Gasses by M. Knudsen

Knudsen's Cosine Law and Random Billiards by R. Feres and G. Yablonsky


You can think of it this way:

Let's say you have a big gas-filled chamber in the form of a cube. The gas shall be in thermal equilibrium and no collision between the particles are occurring (very large mean free path).

If you look at a surface element $dA$ of your chamberwalls, the number of particles coming from a certain solid angle $d\Omega$ with a velocity between $v$ and $v+dv$ is

$n dv$ = $n_0 f(v) dv \frac{d\Omega}{4 \pi}$

n$_0$ being the gas density, $f(v)$ the Maxwellian distribution function.

If you want to know how many particles arrive at $dA$ per second, you just take a look at the volume

$dV = v dt \cos\theta dA$

and combine the two equations.

Here $\theta$ shall be defined as in the paper by Feres. So, the surface $dA$ looks smaller for particles that originated at a large $\theta$.

Using the reduced speed $x=v/v_w$, with $v_w=\sqrt{2kT/m}$ ($v_w$ is the most probable velocity coming from the Maxwellian distribution) your distribution goes like $x^3 exp(-x^2)$.

Integration over all velocities leaves you with the cosine law.

But now you are gonna say: sure thing, but I wanna know how many particles leave a surface and not how many arrive at a surface!

If we drill a hole into our chamber at the place where our $dA$ has been and claim that everything stays in thermal equilibrium due to only very few particles leaving the chamber, we will detect a spatial distribution in a cosine-form outside of our source.

In order to use the cosine-law in the context of, for example, thermal evaporation, we have to "claim" thermal equilibrium, which results in the cosine-law being correct only for very low evaporation rates.


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