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 which impinges on the wall with x-component of velocity, vx, and we consider what happens in a small time interval &Deltat. In the time &Deltat the molecule will travel a distance vx&Deltat in the x-direction. (Let vx be positive so that the molecule is traveling in the +x direction.) The molecule will hit the wall in the time &Deltat if it lies within a distance vx&Deltat of the wall. This distance multiplied by the area, A, creates a small volume Avx&Deltat. If the number density of the gas is N/V then this small volume contains
(1)molecules. The number of these molecules which have velocity, vx, is
(2)Then the total number of collisions is obtained by summing this number of collisions over all positive vx (summation in this case, of course, means we integrate over vx from 0 to ∞),
(3)In Equation 3 we have used the definition of the average of the absolute value of vx,
(4)and the fact that is an even function of vx, so that,
(5a, b, c)So, to find the number of collisions per unit time per unit area, we divide the number of collisions in Equation 3 by the area and by the time interval, &Deltat.
(6)We will show later that
(8)The bottom line is that
(9a, b)We will refer to the expression in Equation 9b as zwall. This expression is the one we were after. It will be of use later when we talk about Knudsen cells and Knudsen flow. It will also be useful in our later discussion of the kinetic molecular theory of transport properties: diffusion, heat conductivity, and viscosity.
We will have to leave our equations in the form of Equation
9b because we do not yet have the form of the velocity probability distribution
function, f, in Equations 2, and 3. As soon as we learn the form
of f we will be able to write equations such as Equation 9b in terms
of temperature, molecular masses, and so on.
From here you can:
Copyright 2004, W. R. Salzman
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