Now consider a thin layer of gas of dimensions L, L and Δx as shown in Fig. 4.8. The layer contains target molecules represented by the shaded circles. We now imagine that a very large number N of bullet molecules (black dots in the same figure) are projected towards the face of the layer with a random distribution. It is assumed that the thickness of the layer is so small that no target molecule can hide behind one another. Most of the bullet molecules will pass through the layer but some will collide with target molecules.

Fig. 4.8 Collision Cross-Section

The ratio of the number of collision, ΔN to the total number of bullet molecules N, is equal to the ratio of the area presented by the target molecules to the total area presented by the layer:

(4.43)

Target area of a single molecule is the area of a circle of radius d (exclusion radius). Thus,

σ = πd2

(4.44)

where σ is called the microscopic collision cross section of one (equivalent) molecule. The exclusion radius is represented in Fig. 4.9

Fig. 4.9 Exclusion Radius

If there are n target molecules per unit volume, then total target are is nσL²Δx. The total area of the layer is L². Thus,

(4.45)

where nσ is known as the macroscopic collision cross section of equivalent molecules. In any system, the unit of macroscopic collision cross section is a reciprocal length.
Each of the ΔN collisions diverts a molecule from its original path or scatters it out of the beam and thereby decreases the number remaining in the beam. Let us rename ΔN as the decrease in number N.
Thus

ΔN = - NnσΔx

(4.46)

or,

(4.47)

In reality N decreases in stepwise fashion as individual molecules make collisions. But if N is very large we can consider it to be a continuous function of x leading to

(4.48)

Then,

ln N = - nσx + C

(4.49)

where C is a constant. At x = 0, N = N₀ (initial number of molecules before collision) resulting in C = ln N₀.
Thus

N = N0exp(- nσx)

(4.50)

This is known as the survival equation. It represents the number of molecules N, out of an initial number N₀, that has not yet made a collision after travelling a distance x.
Inserting expression for in Eq (4.46), we obtain

ΔN = N0nσexp(- nσx)Δx

(4.51)

where N is the number of molecules making their first collision after having travelled a distance between x and x + Δx.