Week 5 - Compact Notation Derivation for a simple Mechanism

Derivation of reaction rate of species

Reaction mechanism:

`CO + O_2 => CO_2 + O`

`O + H_2O => 2 OH`

`CO + OH => CO_2 + H`

`H + O_2 => OH + O`

The first step is to identify the species and elementary reactions with index j and i respectively

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

The elementary reactions are identified with i=1,2,3 and 4

Using the indices, we define the number of moles of species in reactant and products

`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],[0,0,0,0,0,1,0],[0,0,1,0,0,0,1],[0,0,1,0,0,0,1])`

Note that indices j and i represent the species and elementary reactions and not the standard row and column indices. For our convenience, the species are represented along columns and elementary reactions are represented along rows.

The next step is to calculate the difference of mole species between reactant and products

`v_(ji)` = `v_(ji)^('') - v_(ji)^'`

`v_(ji)` = `([-1,-1,1,1,0,0,0],[0,0,0,-1,-1,1,0],[-1,0,1,0,0,-1,1],[-1,0,1,0,0,-1,1])`

We define net reaction rate as

`q_i = k_(fi)pi_(j=1)^N[X_j]^(v_(ji)^') - k_(ri)pi_(j=1)^N[X_j]^(v_(ji)^('')`

Therefore, net reaction rate for first reaction is

`q_1 = k_(f1)[CO]^1[O_2]^1[CO_2]^0[O]^0[H_2O]^0[OH]^0[H]^0 - k_(r1)[CO]^0[O_2]^0[CO_2]^1[O]^1[H_2O]^0[OH]^0[H]^0`

This can be written as

`q_1=k_(f1)[CO_2][O_2]-k_(r1)[CO2][O]`

Similarly,

`q_2=k_(f2)[O][H_2O]-k_(r2)[OH]^2`

`q_3=k_(f3)[CO][OH]-k_(r3)[CO_2][H]`

`q_4=k_(f4)[H][O_2]-k_(r4)[OH][O]`

Now, net production rate of species is calculated as

`omega_j=sum_(i=1)^Lq_i.v_(ji)`

Using this, the production rate for each species can be calculated as

`omega_1 = v_(11).q1 + v_(12).q2 + v_(13).q_3+v_(14).q_4`

`(d[CO])/dt =omega_1 = (-1).(k_(f1)[CO_2][O_2]-k_(r1)[CO2][O])+ 0.(k_(f2)[O][H_2O]-k_(r2)[OH]^2)+(-1).(k_(f3)[CO][OH]-k_(r3)[CO_2][H])+0.(k_(f4)[H][O_2]-k_(r4)[OH][O])`

`(d[CO])/dt =-(k_(f1)[CO_2][O_2]+k_(r1)[CO2][O]-k_(f3)[CO][OH]+k_(r3)[CO_2][H]`

Similarly,

`(d[O_2])/dt = omega_2 =-k_(f1)[CO_2][O_2]+k_(r1)[CO2][O]-k_(f4)[H][O_2]+k_(r4)[OH][O]`

`(d[CO_2])/dt = omega_3 = k_(f1)[CO_2][O_2]-k_(r1)[CO2][O] + k_(f4)[H][O_2]-k_(r4)[OH][O]`

`(d[O])/dt = omega_4 = k_(f1)[CO_2][O_2]-k_(r1)[CO2][O]-k_(f2)[O][H_2O]+k_(r2)[OH]^2+ k_(f4)[H][O_2]-k_(r4)[OH][O]`

`(d[H_2O])/dt = omega_5 = -k_(f2)[O][H_2O]+k_(r2)[OH]^2`

`(d[OH])/dt = omega_6 = k_(f2)[O][H_2O]-k_(r2)[OH]^2 - k_(f3)[CO][OH]+k_(r3)[CO_2][H]+k_(f4)[H][O_2]-k_(r4)[OH][O]`

`(d[H])/dt = omega_7 = k_(f3)[CO][OH]-k_(r3)[CO_2][H] -k_(f4)[H][O_2]+k_(r4)[OH][O]`

Thus, using compact notations, the reaction rates for species are derived.