Week 5 - Compact Notation Derivation for a simple Mechanism
Aim
To derive the reaction rates for the reaction meachanism given below using compact notatins method.
Rates Of Reactions for Multistep Mechanism
`CO + O2 harr CO2 + O`
`O + H2O harr OH + OH`
`CO + OH harr CO2 + H`
`H + O2 harr OH + O`
`k_f` and `k_r` be the rate of forward reation and rate of backward(reverse)reaction.
`d/dt [CO]`= `-k_(f1)[CO][O_2]+k_(r1)[CO_2][O]-k_(f3)[CO][OH]+k_(r3)[ CO_2][H]`
`d/dt [O_2]`= `-k_(f1)[CO][O_2]+k_(r1)[CO_2][O]-k_(f4)[H][O_2]+k_(r4)[ OH][O]`
`d/dt [CO_2]`= `k_(f1)[CO][O_2]-k_(r1)[CO_2][O]+k_(f3)[CO][OH]-k_(r3)[ CO_2][H]`
`d/dt [O]`= `k_(f1)[CO][O_2]-k_(r1)[CO_2][O]-k_(f2)[O][H_2O]+k_(r2)[OH][OH]+k_(f4)[H][O_2]-k_(r4)[ OH][O]`
`d/dt [H_2O]`= `-k_(f2)[O][H_2O]+k_(r2)[OH][OH]`
`d/dt [OH]`= `k_(f2)[O][H_2O]-k_(r2)[OH][OH]-k_(f3)[CO][OH]+k_(r3)[CO_2][H]+k_(f4)[H][CO_2]-k_(r4)[OH][O]`
`d/dt [H]`= `k_(f3)[CO][OH]-k_(r3)[CO_2][H]-k_(f4)[H][CO_2]+k_(r4)[OH][O]`
By using the Compact Notation we get the following
Here j is the species index and i for is the reaction index
| j | Species |
| 1 | `CO` |
| 2 | `O_2` |
| 3 | `CO_2` |
| 4 | `O` |
| 5 | `H_2O` |
| 6 | `OH` |
| 7 | `H` |
| i | Reactions |
| 1 | R1 |
| 2 | R2 |
| 3 | R3 |
| 4 | R4 |
`v'ji` = Reactants. Moles of j in the reaction i.
`v''ji` = Products. Moles of j in the reaction i.
The below are the number of moles in the reactants and in the products of the reactions
`v'ji` = `[[1,1,0,0,0,0,0],[0,0,0,1,1,0,0],[1,0,0,0,0,1,0],[0,1,0,0,0,0,1]]`
`v''ji` = `[[0,0,1,1,0,0,0],[0,0,0,0,0,2,0],[0,0,1,0,0,0,1],[0,0,0,1,0,1,0]]`
vji=`v''ji - v'ji`
vji = `[[-1,-1,1,1,0,0,0],[0,0,0,-1,-1,2,0],[-1,0,1,0,0,-1,1],[0,-1,0,1,0,1,-1]]`
The reaction rates are formed by formula given below
`q_i = k_(fi)prod_(j=1)^N[X_j]^(v'ji) - k_(ri)prod_(j=1)^N[X_j]^(v''ji)`
`q_1` = `k_(f1)[CO]^1[O_2]^1 - k_(r1)[CO_2]^1[O]^1`
`q_2` = `k_(f2)[O]^1[H_2O]^1- k_(r2)[OH]^1[OH]^1`
`q_3` = `k_(f3)[CO]^1[OH]^1- k_(r3)[CO_2]^1[H]^1``-(k_(f1)[CO]^1[O_2]^1 - k_(r1)[CO_2]^1[O]^1) -(k_(f3)[CO]^1[OH]^1- k_(r3)[CO_2]^1[H]^1)`
`q_4` = `k_(f4)[H]^1[CO_2]^1- k_(r4)[OH]^1[O]^1`
The net production rate is obtained by using the below
`omega_j = sum_(i=1)^L v_(ji)^t q_i`
where `omega` is the net production rate of the species.
j is the index of the species number.
L is the number of reactions
`omega_j = d/dt[X_j]`
`[[omega_1],[omega_2],[omega_3],[omega_4],[omega_5],[omega_6],[omega_7]]` = `[[-1,0,-1,0],[-1,0,0,-1],[1,0,1,0],[1,-1,0,1],[0,-1,0,0],[0,2,-1,1],[0,0,1,-1]]`* `[[q_1],[q_2],[q_3],[q_4]]`
`omega_1` = `-q_1-q_3`
`omega_2`= `-q_1-q_4`
`omega_3` =`q_1+ q_3`
`omega_4` = `q_1-q_2+q_3`
`omega_5` = `-q_2`
`omega_6` = `2q_2- q_3 + q_4`
`omega_7` = `q_3 + q_4`
On substituting the reactions of `q_1,q_2,q_3 & q_4` we get the species production rate.
`omega_1` = `d/dt [CO]`= `-(k_(f1)[CO]^1[O_2]^1 - k_(r1)[CO_2]^1[O]^1)-(k_(f3)[CO]^1[OH]^1- k_(r3)[CO_2]^1[H]^1)`
`omega_2` = `d/dt [O_2]` =`-(k_(f1)[CO]^1[O_2]^1 - k_(r1)[CO_2]^1[O]^1)-(k_(f4)[H]^1[CO_2]^1- k_(r4)[OH]^1[O]^1)`
`omega_3` = `d/dt [CO_2]` =`k_(f1)[CO]^1[O_2]^1 - k_(r1)[CO_2]^1[O]^1 + k_(f3)[CO]^1[OH]^1- k_(r3)[CO_2]^1[H]^1`
`omega_4` = `d/dt [O]` = `k_(f1)[CO]^1[O_2]^1 - k_(r1)[CO_2]^1[O]^1-(k_(f2)[O]^1[H_2O]^1- k_(r2)[OH]^1[OH]^1)+k_(f3)[CO]^1[OH]^1- k_(r3)[CO_2]^1[H]^1`
`omega_5` = `d/dt [H_2O]` = `-(k_(f2)[O]^1[H_2O]^1- k_(r2)[OH]^1[OH]^1)`
`omega_6` = `d/dt [OH]` = `2(k_(f2)[O]^1[H_2O]^1- k_(r2)[OH]^1[OH]^1) -(k_(f3)[CO]^1[OH]^1- k_(r3)[CO_2]^1[H]^1) +k_(f4)[H]^1[CO_2]^1- k_(r4)[OH]^1[O]^1`
`omega_7` = `d/dt [H]` = `k_(f3)[CO]^1[OH]^1- k_(r3)[CO_2]^1[H]^1 + k_(f4)[H]^1[CO_2]^1- k_(r4)[OH]^1[O]^1`
Therefore we have obarined the net production rate of the species using compact notation.