Consider a two-body collision between two similar gas molecules of initial velocity ν1 and ν2. After the collision, let the final velocities be ν3 and ν4. The probability for the occurrence of such a collision will be proportional to the number of molecules per unit volume having these velocities, that is to the product f (ν1) f (ν2). Thus the number of each collisions per unit volume per unit time is c f (ν1) f (ν2) where c is a constant. Similarly, the number of inverse collisions per unit volume per unit time is c f (ν3) f (ν4) where c is also a constant. Since the gas is in equilibrium and the velocity distribution is unchanged by collisions, these two rates must be equal. Further in the centre of mass these two collisions appear to be equivalent so that c' = c. We can then write
where A and α are constants. The negative sign is essential to ensure that no molecule can have infinite energy.
Let N(ν)dν be the number of molecules per unit volume with speeds ν to +dν, irrespective of direction. As the velocity distribution is assumed to be spherically symmetrical, N(ν)dν is equal to the number of velocity vectors whose tips end up in the volume of the shell defined by the radii ν and +dν, so that
where gamma functions have been used for the evaluation of the two integrals. Further,
