Compact Notation Derivation of Reaction Mechanism

The aim of this project is to obtain the compact notation of the reaction mechanism.

Assumption: The given four reactions are assumed to take place in both forward and reverse directions.

`CO+O_2 <=>CO_2+O`

`O+H_2O<=>OH+OH`

`CO+OH<=>CO_2+H`

`H+O_2<=>OH+O`

Let `K_(fN)`and `K_(rN)`be the forward and reverse reaction rate of the respective reaction N (1, 2, 3 & 4).

Then the rate of change of concentration for each of the species is as defined below.

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

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

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

`(d[O])/dt=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[H_2O])/dt=-K_(f2)[O][H_2O]+K_(r2)[OH][OH]`

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

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

Now let us denote the above equations in the compact notation.

Let \'i\' be the reaction index.

Reaction	Reaction Index (i)
`CO+O_2 <=>CO_2+O`	1
`O+H_2O<=>OH+OH`	2
`CO+OH<=>CO_2+H`	3
`H+O_2<=>OH+O`	4

Let \'j\' be the species index. It includes all the species involved in the above reaction.

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

Let us denote the compact notation for the species present in the reactants side.

`ν_(ji)^\'` represents the number of moles of species \'j\' in the reaction \'i\' on the products side

`ν_(ji)^(\'\') represents the number of moles of species \'j\' in the reaction \'i\' on the reactants side`

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

The reaction rates for each of the reaction is obtained by the following equation:

`q_i = k_(fi) \\prod_{j=1}^{N}X_j^(v_j^\')- k_(ri) \\prod_{j=1}^{N}X_j^(v_j^(\'\'))`

`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`

`q_1 = K_(f1)[CO]^1[O_2]^1-K_(r1)[CO_2]^1[O]^1`

`q_2 = K_(f2)[CO]^0[O_2]^0[CO_2]^0[O]^1[H_2O]^1[OH]^0[H]^0-K_(r2)[CO]^0[O_2]^0[CO_2]^0[O]^0[H_2O]^0[OH]^2[H]^0`

`q_2 = K_(f2)[O]^1[H_2O]^1-K_(r2)[OH]^2`

`q_3 = K_(f3)[CO]^1[O_2]^0[CO_2]^0[O]^0[H_2O]^0[OH]^1[H]^0-K_(r3)[CO]^0[O_2]^0[CO_2]^1[O]^0[H_2O]^0[OH]^0[H]^1`

`q_3 = K_(f3)[CO]^1[OH]^1-K_(r3)[CO_2]^1[H]^1`

`q_4 = K_(f4)[CO]^0[O_2]^1[CO_2]^0[O]^0[H_2O]^0[OH]^0[H]^1-K_(r4)[CO]^0[O_2]^0[CO_2]^0[O]^1[H_2O]^0[OH]^1[H]^0`

`q_4 = K_(f4)[O_2]^1[H]^1-K_(r4)[O]^1[OH]^1`

The net production rates for each of the species is obtained as below:

`w_i = \\sum_{i=1}^{L}v_(ji)^Tq_i`

Let us evaluate the production rate for each of the species:

`w_1 = (d[CO])/dt = -q_1-q_3`

`w_1 = (d[CO])/dt = -(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)`

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

`w_2 = (d[O_2])/dt = -q_1-q_4`

`w_2 = (d[O_2])/dt = -( K_(f1)[CO]^1[O_2]^1-K_(r1)[CO_2]^1[O]^1)-(K_(f4)[O_2]^1[H]^1-K_(r4)[O]^1[OH]^1)`

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

`w_3 = (d[CO_2])/dt = q_1+q_3`

`w_3 = (d[CO_2])/dt = (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)`

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

`w_4 = (d[O])/dt = q_1-q_2+q_4`

`w_4 = (d[O])/dt = ( K_(f1)[CO]^1[O_2]^1-K_(r1)[CO_2]^1[O]^1)-(K_(f2)[O]^1[H_2O]^1-K_(r2)[OH]^2)+(K_(f4)[O_2]^1[H]^1-K_(r4)[O]^1[OH]^1)`

`(d[O])/dt=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]`

`w_5 = (d[H_2O])/dt = -q_2`

`w_5 = (d[H_2O])/dt = -(K_(f2)[O]^1[H_2O]^1-K_(r2)[OH]^2)`

`(d[H_2O])/dt=-K_(f2)[O][H_2O]+K_(r2)[OH][OH]`

`w_6 = (d[OH])/dt = 2q_2-q_3+q_4`

`w_6 = (d[OH])/dt = 2( K_(f2)[O]^1[H_2O]^1-K_(r2)[OH]^2)-(K_(f3)[CO]^1[OH]^1-K_(r3)[CO_2]^1[H]^1)+( K_(f4)[O_2]^1[H]^1-K_(r4)[O]^1[OH]^1)`

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

`w_7 = (d[H])/dt = q_3-q_4`

`w_7 = (d[H])/dt = (K_(f3)[CO]^1[OH]^1-K_(r3)[CO_2]^1[H]^1)-(K_(f4)[O_2]^1[H]^1-K_(r4)[O]^1[OH]^1)`

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

Thus the production rates obtained from compact notation are same as those obtained analytically.