As all the particles crossing the shell are coagulating and each collision results in coagulation, the coagulation rate is simply the collision rate. Thus it can be said that jA is the rate of coagulation per particles per unit time. If we consider all the particles, the rate at which particles are coagulating is given by

(10.19)

which has the characteristic form of a second-order process with a diffusion-controlled rate.

In the above equation, K_r is the rate constant for rapid coagulation given by

(10.20)

Similar to molecular kinetics, we can express the rate coefficient in the form of an Arrhenius expression:

In our case the pre-exponential factor is simply the collision frequency. We have assumed that when particles collide they stick. There is no barrier in place and thus the activation energy is zero. Hence K_r, in this case is independent of temperature.

From Einstein-Stokes equation for the Brownian self-diffusion, diffusion coefficient for spherical particles is given by

(10.21)

Combining equation (10.19) and (10.20) we get