Page 159 - Modelling in Transport Phenomena A Conceptual Approach

5.3. RATE OF GENERATION LN MASS TRANSPORT 139

Each chemical species, Ai, is the sum of the chemical elements, Ej, such that

t
= CP~~ (5.3-5)
E~
j=1
where Pji represents the number of chemical elements Ej in the chemical species
Ai, and t is the total number of chemical elements. Substitution of Eq. (5.3-5)
into Eq. (5.3-4) gives

(5.3-6)

Since all the Ej are linearly independent3, then

(5.3-7)

Equation (5.3-7) is used to balance chemical equations.

Example 5.1 Consider the reaction between N2 and H2 to form NH3

a1 N2 + a2 H2 + a3 NH3 = 0
Show how one can apply Eq. (5.3-7) to balance this equation.

Solution

If Ai =N2, A2 =H2 and A3 =NH3, the above equation can be expressed as

ai Ai + a2A2 + a3 A3 = 0
If we let El = N (j = 1) and E2 = H (j = 2)) then Eq. (5.3-7) becomes

alEl1 +~~2E12+~~3E13 for j = 1
0
=
a1 E21 + a2 E22 + a3 E23 = 0 for j = 2
The expression
n
aizi = a1z1 + a222 + ... + O*Xn
i=l
where {ai,az, ..., an} is a set of scalars, is called a linear combination of the elements of the set
...,
S = (~1~x2, q,} . The elements of the set S is said to be linearly dependent if there exists a set
of scalars {ai, a2, ..., an} with elements ai not all equal to zero, such that the linear combination
n n
aizi = 0 holds. If aixi = 0 holds for all ai = 0, then the set S is linearty independent.
i=l i=l