Compact Notation Derivation for a simple Mechanism

Reaction Mechanism:-

`CO + O_2 rightarrow CO_2 + O`

`O + H_2O rightarrow OH + OH`

`CO + OH rightarrow CO_2 + H`

`H + O_2 rightarrow OH + O`

Deriving the rate of reaction mechanism:-

Compact Notation:-

Reaction Notation:-

`v_(ji)\'` for reactant and `v_(ji)\'\'` for product

Matrix form for Reactant:-

`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]]`

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

`v_(ji) = [[-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 general form of writing the reaction rate equation

`q_i = k_(fi) prod_(i=1)^N [X_i]^(v_(ji)\') - k_(ri)prod_(i=1)^N[X_i]^(v_(ji)\'\')`

`q_i rightarrow` reaction rate for each reaction

`k_(fi) & k_(ri) rightarrow`forward and reverse rate coefficient for `i^(th)` reaction

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

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

The production rate of individual species can be given as

`omega_j =sum_(i=1)^L v_(ji)q_i`

`omega_j` is the production rate of particular species

`omega_1 rightarrow CO`

`omega_1 = v_11q_1 + v_12q_2 + v_13q_3 + v_14q_4`

From `v_(ji)` matrix `v_11 = -1` and `v_13 = -1`rest are zero.

`omega_1 = -q_1 -q_3`

`omega_1 = frac{d(CO)}{dt} = -(k_(f1)[CO][O_2] -k_(r1)[CO_2][O])-(k_(f3)[CO][OH]-k_(r3)[CO_2][H])`

`omega_2 rightarrow O_2`

`omega_2 = v_21q_1 +v_22q_2 + v_23q_3 +v_24q_4`

From `v_(ji)` matrix `v_21 = -1 , v_24 = -1`and rest are zero.

`omega_2 = -q_1-q_4`

`omega_2 =frac{d[O_2]}{dt} = -(k_(f1)[CO][O_2]-k_(r1)[CO_2][O])-(k_(f4)[O_2][H]-k_(r4)[O][OH])`

`omega_3 rightarrow CO_2`

`omega_3 = v_31q_1 + v_32q_2 +v_33q_3 +v_34q_4`

From `v_(ji)` matrix `v_31 = 1 , v_33 = 1` and rest all are zero

`omega_3 = q_1 +q_3`

`omega_3 = frac{d(CO_2)}{dt} = (k_(f1)[CO][O_2] -k_(r1)[CO_2][O])+(k_(f3)[CO][OH]-k_(r3)[CO_2][H])`

`omega_4 rightarrow O`

`omega_4 = v_41q_1 +v_42q_2 +v_43q_3 +v_44q_4`

From `v_(ji)` matrix `v_41 = 1 , v_42 = -1 , v_43 =0 and v_44=1`

`omega_4 = q_1 +q_2 +q_4`

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

`omega_5 rightarrow H_2O`

`omega_5 = v_51q_1 +v_52q_2 +v_53q_3 +v_54q_4`

From `v_(ji)` matrix `v_52= -1`and rest all are zero

`omega_5 =-q_2`

`omega_5 = frac{d[H_2O]}{dt} = -(k_(f2)[O][H_2O]-k_(r2)[OH]^2)`

`omega_6 rightarrow OH`

`omega_6 = v_61q_1+v_62q_2 + v_63q_3 + v_64q_4`

From `v_(ji)` matrix `v_61 = 0, v_62 = 2,v_63 =-1 and v_64 = 1`

`omega_6 = 2q_2-q_3 +q_4`

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

`omega_7 rightarrow H`

`omega_7 = v_71q_1 +v_72q_2 +v_73q_3 +v_74q_4`

From `v_(ji)` matrix `v_73 = 1 , v_74 = -1`and rest all zero

`omega_7 = q_3-q_4`

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

Writing the equation in matrix form

`[[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,-1,0,0],[0,-1,0,0],[0,2,-1,1],[0,0,1,-1]][[q_1],[q_2],[q_3],[q_4]]`

Hence, we have derived the rate of mechanism for above reaction.