The nucleation rate is expressed in the Arrhenius form as given in Equation 19–39 ,
 | (19–39) |
where
and 
are the chemical symbols for the k -th gas reactant
and the stoichiometric coefficient of the k -th gas reactant in the
nucleation reaction, respectively.
Because the nucleation process affects only the number density of class
particle, the change of the number density of each discrete particle class
"bin" in the size distribution, can be expressed as
 | (19–40) |
and
 | (19–41) |
Using the definition of particle size moments given by Equation 19–2
, the change in size moment
due to nucleation can be obtained as
 | (19–42) |
Depending on the sections specified by the user, this size may or may not coincide with the representative size. The nucleated particle is then split into the adjacent sections such that total number density and mass are conserved. For example, consider the input as in Figure 19.1: Dispersed Phase panel to input parameters for sections. . If the nucleation reaction creates a particle with 20 monomers in it, then that particle is split as (20-32)/(16-32) = 0.75 particles in the 5th section that has a representative particle with 16 monomers, and 0.25 particles in the 6th section that has representative a particle with 32 monomers. The total particle number density and mass are thus conserved.
In the case of soot nucleation, the soot particles are assumed to be formed by coalescence of two poly-aromatic hydrocarbon (PAH) precursors. Therefore, the nucleation rate is equal to the collision rate of the two precursors. In the free-molecular regime, the PAH collision rate takes the form
 | (19–43) |
The free-molecule collision frequency is given as
 | (19–44) |
where 
is the collision efficiency and 
is the reduced mass of the collision pair.
If the two precursors are the same PAH species, that is,
i = j = 
and 
= 
= 
, the collision rate becomes
 | (19–45) |
Consequently, the nucleation rate can be expressed as
 | (19–46) |
The Arrhenius rate parameters of the nucleation rate can be obtained by comparing Equation 19–46 against the Arrhenius form:
 | (19–47) |
 | (19–48) |
 | (19–49) |