# 9.2: Untitled Page 218

- Page ID
- 18351

## Chapter 9

If we consider a set of K elementary reactions involving the species indicated by *A * 1 *, * 2 *,...,N *, we encounter a set of *net rates of production* that are designated by I

*R*

II

, *R*

III

, *R *, …, K

*R *. Associated with each elementary reaction is a condition *A*

*A*

*A*

*A*

of *elementary stoichiometry* that we express as

*Elementary Stoichiometry*

*A * *N*

*k*

(9‐31)

*N R*

0 *, *

*J * 1 *, * 2 *,...,T , *

*k * I, II, ..., K

*JA A*

*A * 1

The sum of the K elementary net rates of production for species *A * is the total net rate of production for species *A * indicated by

*k * K

*k*

*R*

*R*

(9‐32)

*A*

*A*

*k * I

Since *N*

is independent of *k * I, II,....,K , we can sum Eq. 9‐31 over all K

*JA*

reactions and interchange the order of summation to recover the local stoichiometric condition given by

*A * *N*

*k * K

*A * *N*

*k*

*N*

*R*

*N R*

0 *, *

*J * 1 *, * 2 *,...,T *

(9‐33)

*JA*

*A*

*JA A*

*A * 1

*k * I

*A * 1

Clearly when there is a single elementary reaction, the *elementary* *stoichiometry* is identical to the *local stoichiometry*.

*9.1.2* *Mass action kinetics and elementary stoichiometry* In this section we want to summarize the concept of mass action kinetics and indicate how it is connected to elementary stoichiometry. As an example we consider a system in which there are four participating molecular species indicated by *A*, *B*, *C*, and *D*. The chemical kinetic schema for one possible reaction associated with these molecular species is indicated by *k* I

Elementary chemical kinetic schema I:

*A * *B * *C * *D * (9‐34) Here we have avoided the use of *k *, *k *, etc., to represent rate coefficients and instead 1

2

we have employed a nomenclature that makes use of *k *, *k *, *k*

, etc., to identify the

I

II

III

chemical reaction rate coefficients. At this point we need to *translate* this *picture* to

405

an *equation* associated with mass action kinetics and then explore what can be extracted from this picture in terms of elementary stoichiometry. According to the rules of mass action kinetics, the chemical kinetic translation of Eq. 9‐34 is given by

Elementary chemical reaction rate equation I:

I

*R*

*k c c *

(9‐35)

*A*

I *A B*

Here we have used the first species in the chemical kinetic schema as the basis for the proposed rate equation, and this represents a reasonable convention but not a necessary one. One should remember that binary collisions dominate chemical reactions and that ternary collisions are rare. This means that we expect the sum of the integers and to be less than or equal to two. Often there is a second reaction involving species *A*, *B*, *C*, and *D*, and we express the second chemical kinetic schema as

Elementary chemical kinetic schema II:

II

*k*

*B * *C *

*D *

(9‐36)

This second chemical kinetic schema leads to a chemical reaction rate equation of the form

Elementary chemical reaction rate equation II:

II

*R*

*k c c *

(9‐37)

*B*

II *B C*

In general we are interested in the *net rate of production* which is given by the sum of the elementary rates of production according to

I

II

I

II

*R*

*R * *R , *

*R*

*R * *R*

*A*

*A*

*A*

*B*

*B*

*B *

(9‐38)

I

II

I

II

*R*

*R * *R , *

*R*

*R * *R*

*C*

*C*

*C*

*D*

*D*

*D*

At this point we need *stoichiometric information* to develop useful chemical reaction rate equations. Since stoichiometry is associated with the conservation of *atomic species*, we need to be very careful when using a representation in which there are no identifiable atomic species. The translation associated with kinetic schemata and elementary stoichiometry must be consistent with Axiom II. In terms of *stoichiometry*, we identify the meaning of Eq. 9‐34 as follows:

moles of species *A * react with moles of

species *B * to form moles of species *C * and

moles of species *D *.

To make things very clear, we consider the *highly unlikely* prospect that 8 moles of species *A * react with 3 moles of species *B *. This would lead to the condition

406