Effusing gas onto the interior of an evacuated sphere

[Toby_phys]

Homework Statement

A gas effuses into a vacuum though a small hole of area $$A$$. Show that if the particles effused into an evacuated sphere and the particles condensed where they collided that there would be a uniform coating. (7.6 of Blundell and Blundell)

Homework Equations

Angular probability distribution. The probability a particle is traveling between $$\theta$$ and $$\theta+d\theta$$ to the normal of the small hole.
$$
cos\theta sin\theta d\theta
$$

Surface area of a ring defined on the surface of a sphere with radius of unity where the angles go from $$\theta$$ to $$\theta+d\theta$$

$$
2\pi sin\theta d\theta
$$

The Attempt at a Solution

Surely it would just be amount of particles over area which is proportional to $$cos\theta$$. This is clearly not uniform

 

[haruspex]

There are a few things wrong with your 2π sin(θ)dθ formula.
First, you have no hypotenuse length in there, and it will not be constant.
Second, you are not considering the oblique and varying angle between the beam and the surface.
Third, different dθ wedges will contain different fractions of the beam.