Week 5 - Compact Notation Derivation for a simple Mechanism

AIM: To derive the reaction rate ODEs of a simple reaction mechanism by using the compact notation and to write down the desired matrices.

Reaction Mechanism:

In chemistry, a reaction mechanism is the step by step sequence of elementary reactions by which overall chemical change occurs.

The following are the given Reaction Mechanism,

CO + O2 -> CO2 + O

O + H2o -> OH + OH

CO + OH -> CO2 + H

H + O2 -> OH + O

Derivation of each Species:

d/dt[CO] = -kf1[CO][O2] + kr1[CO2][O] - kf3[CO][OH] + kr3[CO2][H]

d/dt[O2] = -kf1[CO][O2] + kr1[CO2][O] - kf4[H][O2] + kr4[OH][O]

d/dt[O] = -kf2[O][H2O] + kr2[OH][OH] - kf4[H][O2] + kr4[OH][O] 

d/dt[OH] = - kf3[CO][OH] + kr3[CO2][H] - kr2[OH][OH] + kf2[O][H2O] + kf4[H][O2] - kr4[OH][O]

d/dt[H] = - kf4[H][O2] + kr4[OH][O] - kr3[CO2][H] + kf3[CO][OH]

d/dt[CO2] = kf1[CO][O2] - kr1[CO2][O] - kr3[CO2][H] + kf3[CO][OH]

d/dt[H2O] = -kf2[O][H2O] + kr2[OH][OH]

Assume, 'j' represents Species and 'i' represents no. of reactions.

i  Species

1	CO
2	O2
3	CO2
4	O
5	H20
6	OH
7	H


i  Reactions

1	R1
2	R2
3	R3
4	R4

Stoichiometrix for the reactant and product side are Vji' & Vji''

Therefore, Stoichiometric Co-efficient is Vji = Vji'' - Vji'

Vji' = (1 1 0 0 0 0 0

        0 0 1 0 0 0 1

        1 0 0 1 0 0 0

        0 1 0 0 1 0 0)



Vji'' = (0 0 1 0 0 1 0

         0 0 0 2 0 0 0

         0 0 0 0 1 1 0

         0 0 1 1 0 0 0)



Vji =    Vji'' - Vji'

Vji   = (-1 -1  1  0  0 1 0

          0  0 -1  2  0 0 0

         -1  0  0 -1  1 1 0

          0 -1  1  1 -1 0 0)

Production rate:

q1 = kf1[CO][O2] - kr1[O][CO2] 

q2 = kf2[O][H2O] - kr2[OH][OH]

q3 = kf3[CO][OH] - kr3[H][CO2]

q4 = kf4[O2][H] - kr4[O][OH]



ω1(CO)  = -q1 - q3 = -(kf1[CO][O2] - kr1[O][CO2]) - (kf3[CO][OH] - kr3[H][CO2])

ω2(O2)  = -q1 - q4 = -(kf1[CO][O2] - kr1[O][CO2]) - (kf4[O2][H] - kr4[O][OH])

ω3(CO2) = q1 + q3 = (kf1[CO][O2] - kr1[O][CO2]) + (kf3[CO][OH] - kr3[H][CO2])

ω4(O) = q1 - q2 + q4 = ((kf1[CO][O2] - kr1[O][CO2])) - (kf2[O][H2O] - kr2[OH][OH]) + (kf4[O2][H] - kr4[O][OH]) 

ω5(H2O) = -q2 = -(kf2[O][H2O] - kr2[OH][OH])

ω6(OH)  = 2q2 - q3 + q4 = 2*(kf2[O][H2O] - kr2[OH][OH]) - (kf3[CO][OH] - kr3[H][CO2]) + (kf4[O2][H] - kr4[O][OH])

ω7(H) = q3 - q4 = (kf3[CO][OH] - kr3[H][CO2]) - (kf4[O2][H] - kr4[O][OH])

from the above equations we can calculate production rate of each species.

(ω1   (-1  0 -1  0
 ω2    -1  0  0 -1    (q1
 ω3     1 -1  0  1     q2
 ω4  =  0  2 -1  1   * q3
 ω5     0  0  1 -1     q4)
 ω6     1  0  1  0
 ω7)    0 -1  0  0)