Week 5.1 - Compact Notation Derivation for a simple Mechanism

AIM: To derive the reaction rate Ordinary Differential Equation(ODE) of the following reaction mechanism using compact notation

Solution:

There are 4 reactions and 7 Species in the reaction mechanism. Let 'i' be the Reaction index and 'j' be the Species index

V_ji^’ be the matrix that express number of moles of each species 'j' in every reaction of 'i' for the reactant side.

V_ji^’' be the same matrix for the product side

Matrix form of Reactant

Matrix form of Product

V_ji be the matrix that represent net number of moles each species 'j' in the reaction 'i'

For the First reaction i = 1

For the Second reaction i = 2

For the Third reaction i = 3

For the Fourth reaction i = 4

The net production rate of w_j species of 'j' which can be written using compact notation as

L --- Total number of reaction in the reaction mechanism

---- Net Production rate of species 'j'

Substituting 'j' with respective species index, the net production rate of the species can be obtained

For j = 1

For j = 2

For j = 3

For j=4

For j=5

For j=6

For j=7

Equation in Matrix Form