An Overview of Chemical Reaction Analysis - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 100

These equations become more illustrative if we consider the initial values (subscript 0)
before the reaction takes place. Equation (3.55) leads to

n NO þ 2 n N 2 ¼ n NO ;0 þ 2 n N 2 ;0 (3.57)

which can be rearranged to give
n NO n NO ;0 ¼ 2 n N 2 n N 2 ;0

(3.58)

or

n NO n NO ;0

1

n N 2 n N 2 ;0

1=2

¼ (3.59)

or

D n NO

n NO ¼ D n N 2

(3.60)

n N 2

In general, for any single reaction, we can write
D n j
n j ¼

n ext ; c j (3.61)

and we can write N S

1 independent combinations of these relations among N S chemical
species in a reaction to relate the changes in the number of moles of all species to each other.
There is therefore always a single composition variable that describes the relationship among
all species in a single reaction. In the preceding Eqn (3.61) , we defined n ext as the relation
between the n j 's,

n j ¼ n j; 0 þ v j n ext (3.62)

We call the quantity n ext the number of moles extent.
For simple problems, we most commonly use one of the reactants as the concentration
variable to work with and label that species A to use C A as the variable representing compo-
sition changes during reaction. We also make the stoichiometric coefficient of that species
v A ¼

1.
Another way of representing a single reaction is the fractional conversion f, a dimensionless
quantity ranging from 0, before reaction, to 1, when the reaction is complete. We define f
through the relation

n A 0 1 f A

n A ¼ (3.63)

To make 0 1, we have to choose species A as the limiting reactant so that this reactant

disappears and f A approaches unity when the reaction is complete. We can then define all
species through the relation

f A

D n j
n j ¼

n A 0 f A ; c j (3.64)

For multiple reactions, we need a variable to describe each reaction. Furthermore, we
cannot in general find a single key reactant to call species A in the definition of f A . However,