Compact Notation Derivation for a simple Mechanism

In this project, the reaction rate ODEs of a simple reaction mechanism is derived and compact notations are used so that the ODEs can be solved using a computer program.

Four reactions are taking place namely `R_1, R_2, R_3 and R_4` involving the species `CO`, `O_2`,`CO_2`,`O`,`H_2O`, `OH`and `H`. They are assigned to i and j and represented below.

Reactions:

i	Reaction
1	R­­1
2	R­­2
3	R3
4	R4

Species:

j	Species
1	CO
2	O2
3	CO­2
4	O
5	H­2O
6	OH
7	H

Below the actual reactions are represented :

`R_1:`

`CO + O_2 underset(K_(r_1))(overset(K_(f_1))(harr)) CO_2 + O`

`R_2:`

`O + H_2O underset(K_(r_2))(overset(K_(f_2))(harr)) OH + OH`

`R_3:`

`CO + OH underset(K_(r_3))(overset(K_(f_3))(harr)) CO_2 + H`

`R_4:`

`H + O_2 underset(K_(r_4))(overset(K_(f_4))(harr)) OH + O`

All the reactions above are reversible and in the above reactions, `K_(f_n)` is the forward reaction rate of `n^(th)`reaction and `K_(r_n)` is the reverse reaction rate of `n^(th)` reaction.

Reactants stoichiometric coefficient matrix:

`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]]`

Products stoichiometric coefficient matrix:

`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]]`

Matrix obtained by subtracting the product stoichiometric coefficient matrix by the reactant stoichiometric coefficient matrix:

`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]]`

In the above matrices i denotes the row and j denotes the column of the matrix.

Net reaction rates corresponding to `R_1,R_2,R_3 and R_4` are given below:

`q_1 = K_(f_1)[CO]^1[O_2]^1 - K_(r_1)[CO_2]^1[O]^1`

`q_2 = K_(f_2)[O]^1[H_2O]^1 - K_(r_2)[OH]^2`

`q_3 = K_(f_3)[CO]^1[OH]^1 - K_(r_3)[CO_2]^1[H]^1`

`q_4 = K_(f_4)[H]^1[O_2]^1 - K_(r_4)[OH]^1[O]^1`

Production rate for individual species represented by compact notations:

`w_1\" for \" CO,`

`w_1 = v_(11)q_1 + v_(12)q_2 + v_(13)q_3 + v_(14)q_4`

From `v_(ji)`matrix, `v_(12)` and `v_(14)`is 0 and `v_(11) and v_(13)` is -1, so

`w_1 = - q_1 - q_3`

`w_1 = - (K_(f_1)[CO]^1[O_2]^1 - K_(r_1)[CO_2]^1[O]^1) - (K_(f_3)[CO]^1[OH]^1 - K_(r_3)[CO_2]^1[H]^1)`

`w_2\" for \" O_2,`

`w_2 = v_(21)q_1 + v_(22)q_2 + v_(23)q_3 + v_(24)q_4`

From `v_(ji)`matrix, `v_(22)` and `v_(23)`is 0 and `v_(21) and v_(24)` is -1, so

`w_2 = - q_1 - q_4`

`w_2 = - (K_(f_1)[CO]^1[O_2]^1 - K_(r_1)[CO_2]^1[O]^1) - (K_(f_4)[H]^1[O_2]^1 - K_(r_4)[OH]^1[O]^1)`

`w_3\" for \" CO_2,`

`w_3 = v_(31)q_1 + v_(32)q_2 + v_(33)q_3 + v_(34)q_4`

From `v_(ji)`matrix, `v_(32)` and `v_(34)`is 0 and `v_(31) and v_(33)` is 1, so

`w_3 = q_1 + q_3`

`w_3 = (K_(f_1)[CO]^1[O_2]^1 - K_(r_1)[CO_2]^1[O]^1) + (K_(f_3)[CO]^1[OH]^1 - K_(r_3)[CO_2]^1[H]^1)`

`w_4\" for \" O,`

`w_4 = v_(41)q_1 + v_(42)q_2 + v_(43)q_3 + v_(44)q_4`

From `v_(ji)`matrix, `v_(43)` is 0,`v_(41) and v_(44)` is 1 and `v_(42)` is -1, so

`w_4 = q_1 - q_2 + q_4`

`w_4 =(K_(f_1)[CO]^1[O_2]^1 - K_(r_1)[CO_2]^1[O]^1) - (K_(f_2)[O]^1[H_2O]^1 - K_(r_2)[OH]^2) + (K_(f_4)[H]^1[O_2]^1 - K_(r_4)[OH]^1[O]^1)`

`w_5\" for \" H_2O,`

`w_5 = v_(51)q_1 + v_(52)q_2 + v_(53)q_3 + v_(54)q_4`

From `v_(ji)`matrix, `v_(51)`,`v_(53)` and `v_(54)` is 0 and `v_(52)` is -1, so

`w_5 = - q_2`

`w_5 = - ( K_(f_2)[O]^1[H_2O]^1 - K_(r_2)[OH]^2)`

`w_6\" for \" OH,`

`w_6 = v_(61)q_1 + v_(62)q_2 + v_(63)q_3 + v_(64)q_4`

From `v_(ji)`matrix, `v_(61)`is 0,`v_(62)` is 2, `v_(63)` is -1 and `v_(64)` is 1, so

`w_6 = (2xxq_2) - q_3 + q_4`

`w_6 = (2xx(K_(f_2)[O]^1[H_2O]^1 - K_(r_2)[OH]^2)) - (K_(f_3)[CO]^1[OH]^1 - K_(r_3)[CO_2]^1[H]^1) + (K_(f_4)[H]^1[O_2]^1 - K_(r_4)[OH]^1[O]^1)`

`w_7\" for \" H,`

`w_7 = v_(71)q_1 + v_(72)q_2 + v_(73)q_3 + v_(74)q_4`

From `v_(ji)`matrix, `v_(71)`,`v_(72)` is 0, `v_(73)` is 1 and `v_(74)` is -1, so

`w_7 = q_3 - q_4`

`w_7 = (K_(f_3)[CO]^1[OH]^1 - K_(r_3)[CO_2]^1[H]^1) - (K_(f_4)[H]^1[O_2]^1 - K_(r_4)[OH]^1[O]^1)`

Upon simplying the `w_n` we get the production rate of respective species, the use of compact notation allows us to solve the production rate of species using a computer program saving much time as compared to solving manually.