Origin of the 1/4 factor in the molecular flux formula

[raintrek]

The molecular flux (number of collisions per unit area per unit time) in Kinetic theory is given by

$$F=\frac{1}{4} n \bar{c}$$

where c bar is the average molecular speed and n is the density of molcules in the gas phase (molecules/m^3)

I was wondering about the origin of the 1/4 term ... is it significant?

 

[turin]

Such a factor is due to the fact that the molecules are randomly directed. If they were all going in the same direction, then F would depend on the orientation of the plane that is experiencing the collisions, and the maximum value would be $n\bar{c}$. However, since the molecules are randomly directed, on average only half of them will hit the surface, and the half that does hit the surface will do so at various angles, not head on, so this makes the effective velocity that is directed toward the surface less than the speed of the molecules.