Week 5 - Compact Notation Derivation for a simple Mechanism

Reaction Mechanism :-

CO + O2 -> CO2 + O

O + H2o -> OH + OH

CO + OH -> CO2 + H

H + O2 -> OH + O

Here kf is rate of forward reaction and kr is rate of reverse reaction

Now we use two notation i and j with number for reactions and species .The below table shows the value of i and j for different species and reactions

j	species
1	CO
2	O_2
3	CO_2
4	O
5	H_2O
6	OH
7	H

Table No.1

i	Reaction
1	R1
2	R2
3	R3
4	R4

Table No.2

The Rate changes of each species is given as follows

`(d[CO])/dt=-kf_1[CO][O_2]+kr_1[CO_2][O]-kf_3[CO][OH]+Kr_3[CO_2][H]`

`(d[O_2])/dt=-kf_1[CO][O_2]+kr_1[CO_2][O]-kf_4[H][O_2]+Kr_4[OH][OH]`

`(d[CO_2])/dt=kf_1[CO][O2]-kr_1[CO_2][O]+kf_3[CO][OH]+Kr_3[CO_2][H]`

`(d[O])/dt=kf_1[CO][O_2]-kr_1[CO_2][O]-kf_2[O][H_2O]+Kr_2[OH][OH]+kf_4[H][O_2]-kr_4[OH][O]`

`(d[H_2O])/dt=-kf_2[O][H_2O]+kr_2[OH][OH]`

`(d[OH])/dt=2*kf_2[O][H_2O]-2*kr_2[OH][OH]+kf_4[H][O_2]-Kr_4[OH][O]-kf_3[CO][OH]+kr_3[CO_2][H]`

These are also called coupled ODE

Reactant Matrix

it designated V'ij

Products Matrix V"ij

The net stoichiometrix is taken is differen of the products and reactants matrix

V"ij-V'ij

Now the net reaction rate is  calculated for each reaction

where X_j are species

The net reaction for reaction are 

`q_1=kf_1[CO][O_2]-kr_1[CO2][O]`

`q_2=kf_2[O][H_2O]-kr_2[OH][OH]`

`q_3=kf_3[CO][OH]-kr_3[CO2][H]`

`q_4=kf_4[H][O_2]-kr_4[OH][O]`

Production Rate

`omega_j=sum_(i=1)^Lv_(ji)q_i`

where `omega` is net production rate

L- No of reaction

i-index for reactions

j-index for species

`ω_1 = −q_1 − q_3`
`ω_2 = −q_1 − q_4`
`ω_3 = q_1 + q_3`
`ω_4 = q_1 − q_2 + q_3`
`ω_5 = −q_2`
`ω_6 = 2q_2 − q_3 + q_4`
`ω_7 =q_3 − q_4`

The production rate is also is equal to rate of the change in species

`omega_j=(d[X_J])/dt`

Hence we can get a system of coupled ODE equation that is compatable for computer to solve using numerical integration