When a new particle is created, its surface can be covered by elements or fractional structures from the precursors. The initial surface coverage of new particles is given by the stoichiometric coefficients of surface species products in the nucleation reaction. Determination of the initial surface coverage (or stoichiometric coefficients) is somewhat arbitrary. For example, the initial coverage of new soot particles can be obtained from the hydrogen-to-carbon ratio observed in experiments. However, because nucleation reactions have to conserve surface sites on particles and elements, the values of stoichiometric coefficients are subjected to some limitations. In this section, a pseudo nucleation reaction is used as an example to illustrate how these limitations can be derived from conservation laws.

Consider a pseudo nucleation reaction:

C_x1 H_y1 +C_x2 H_y2 => _C(B) C(B)+ _C(S) C(S)+ _H(S) H(S)+ _(S) (S)+ _H H+ _CH CH

The particle core is represented by the only bulk species product C(B), which is a single carbon atom. The inception particle class is equal to the stoichiometric coefficient of C(B), C(B). C(S) and H(S) are respectively elementary carbon and hydrogen on the particle surface. (S) denotes an empty surface site. Both H and CH are gas phase products from the nucleation process.

Conservation of element C sets the relation among stoichiometric coefficients of carbon-containing products

(19–20)

Similarly, the relationship of stoichiometric coefficients of H-containing products can be obtained by conserving the element H

(19–21)

The connection between stoichiometric coefficients of the nucleation reaction and site density of the dispersed material is established via the conservation of surface sites on new particles. Because all particles created by this nucleation reaction are exactly the same, they have the same particle class , thus the same surface area, . Accordingly, if the nucleation rate is [mole/cm³ -sec], the production rate of new surface area can be calculated as

(19–22)

Because the site density on the particle surface is [mole/cm² ], the production rate of new surface sites where new surface species can be accommodated is given by

(19–23)

On the other hand, the production rate of all new surface species on the particles is

(19–24)

Since each surface site has to be occupied by a surface species, production rate of new surface sites must match that of all surface species, that is,

(19–25)

or

(19–26)

Equations Equation 19–30 , Equation 19–31 and Equation 19–37 serve as general constraints on the stoichiometric coefficients in the nucleation reaction. Note that there are 6 stoichiometric coefficients but there are only 3 constraints. This means, given the same set of precursors, the nucleation reaction can have more than one set of products that can satisfy the conservation of elements and surface sites. The final form of the nucleation reaction will be determined by additional information such as size and hydrogen-carbon ratio of the new particles and gas phase products detected in experiments.

For the pseudo nucleation reaction under consideration, assume that all carbon atoms in the precursors become the particle core, that is,

(19–27)

and

(19–28)

Consequently, Equation 19–21 and Equation 19–26 are reduced to

(19–29)

and

(19–30)

Furthermore, if the hydrogen-to-carbon ratio of the new particle is known, the stoichiometric coefficients can be obtained as

(19–31)

(19–32)

and

(19–33)

Since both must be non-negative, a constraint on the particle hydrogen-to-carbon ratio is obtained

(19–34)

When

(19–35)

gas phase H species is generated from the nucleation reaction. If, on the other hand,

(19–36)

some surface sites on the new particles will be open. However, the determination of also depends on the availability of surface site density data. If a reliable value of is known, the stoichiometric coefficient of open surface site (S) is computed from Equation 19–33 . Alternatively, if and values are better known, can be derived from

(19–37)

If and data are known, the final form of the pseudo nucleation becomes

(19–38)