We want to calculate how many molecules collide with the surface per unit time and per unit surface area. We will use the same technique we used in computing the pressure against a surface except now there is no momentum exchange, we only want to count the collisions.

Once again we consider a portion of the wall of area, A, and a molecule with x-component of velocity, v_x, and think about what happens in a small time interval D t. In the time D t the molecule will travel a distance v_xD t in the x-direction. (Let v_x be positive so that the molecule is traveling in the +x direction.) The molecule will hit the wall in the time D t if it lies within a distance v_xD t of the wall. This distance multiplied by the area, A, creates a small volume Av_xD t. If the number density is N/V then this small volume contains

(1)             

molecules. The number of these molecules which have velocity, v_x, is

(2)              .

Then the total number of collisions is obtained by summing this number of collisions over all positive v_x (summation in this case, of course, means we integrate over v_x from 0 to ¥ ),

(3)             

So,

(4)              .

Recall that |v_x| is defined as,

(5)             

since |v_x| is an even function of v_x.

We will show later that

(6)              ,

where

(7)              .

The bottom line is that

(8ab)             

This expression is the one we were after. It will be of use later when we talk about Knudsen cells and Kundsen flow. It will also be useful in our later discussion of the kinetic molecular theory of transport properties: diffusion, heat conductivity, and viscosity.