Lectures on Gas Theory, Vol. 2

[Equation 3] § 3. Number of collisions. § 4. Extension of the molecules.

thus from the volume $Ω dh$ $Ω$ represents the surface area and $dh$ the height, forming the volume of the cylinder and one sees easily that exactly those molecules of the highlighted kind strike the plane $DE$ during the time $dt$, whose centers were located in the oblique cylinder $γ$ at the beginning of the moment in time $dt$.

§ 4. Consideration of the extension of the molecules in the collision number.

In order to find the number $dz$ of these latter molecules, we first determine quite generally the probability that, given a certain fixed position of the remaining molecules, the center of a specific given molecule lies within the cylinder $γ$. The given molecule cannot have a distance from the center of any of the remaining $n - 1$ molecules that is smaller than $σ$ $σ$ refers to the molecular diameter; molecules cannot overlap closer than this distance. We therefore find the space available for the center of our molecule throughout the entire vessel, given the positions of the remaining molecules, in the following way: We construct around the center of each of the $n - 1$ other molecules a sphere of radius $σ$, which we shall call the covering sphere original: "Deckungssphäre"; this is the "excluded volume" around a molecule where no other molecular center can enter of this molecule. Its volume is eight times the volume of the molecule itself, conceived as an elastic sphere. We subtract the total volume $4 π (n - 1) σ^3 / 3$ of all these $n - 1$ covering spheres from the total volume $V$ of the gas, whereby $n$ can also be written for $n - 1$, since $n$ is a very large number.

To find $dz$ now, we compare this space $V - 4 π n σ^3 / 3$, which is available for the center of the specific given molecule in the entire vessel, with the space available for it in the cylinder $γ$. We find the latter if we again subtract from the total volume $Ω dh$ of the cylinder $γ$ the volume of those parts of it which lie within the covering sphere of any of the $n - 1$ remaining molecules. The covering spheres of these $n - 1$ molecules will obviously be distributed on average uniformly throughout the entire volume $V$ of the vessel containing the gas, with the exception of the parts in the vessel very close to the wall. If, therefore, the cylinder $γ$ were located anywhere in the middle of the interior of the vessel, then that part $A$ of the total volume $4 π n σ^3 / 3$