Compact Notation Derivation for a simple Reaction Mechanism
AIM - Develop a compact notation for the given reaction mechanism so that it can be used in a computer program.
The reactions in the mechanism are -
`CO + O_2 => CO_2 + O ....................R1`
`O + H_2O => OH + OH ....................R2`
`CO + OH => CO_2 + H ....................R3`
`H + O_2 => OH + O ....................R4`
In the above mechanism, the species are CO, O2, CO2, O, H2O, OH, H
Using the compact notation, the species (j) and reactions(i) can be represented as under -
| j | Species |
| 1 | `CO` |
| 2 | `O_2` |
| 3 | `CO_2` |
| 4 | `O` |
| 5 | `H_2O` |
| 6 | `OH` |
| 7 | `H` |
| i | Reaction |
| 1 | R1 |
| 2 | R2 |
| 3 | R3 |
| 4 | R4 |
The coefficient matrices of the the reactant(`nu_(ji)'`) and product(`nu_(ji)''`) sides are -
`nu_(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]]`
`nu_(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]]`
`nu_(ji)'` - Number of moles of species j in reaction i on the reactant side
`nu_(ji)''` - Number of moles of species j in the reaction i on product side
In the above matrices, the total number of rows is equal to the total number of reactions in the mechanism while the total number of the columns is equal to the total number of species in the mechanism.
A new matrix `nu_(ji)` is defined as -
`nu_(ji) = nu_(ji)''-nu_(ji)'`
Thus, for these set of reactions, we get,
`nu_(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]]- [[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]]`
`rArr nu_(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]]`
Net reaction rate (`q_i`) of each reaction i, defines the direction of progress of the reaction (forward or reverse). It is defined mathematically as -
`q_ i = k_(fi)overset(N)underset(j=1)(prod)[X_j]^(nu_(ji)') - k_(ri)overset(N)underset(j=1)(prod)[X_j]^(nu_(ji)'')`
In the above equation,
`k_(fi)` - Rate of forward reaction i(using Arhenius rate law)
`k_(ri)` - Rate of reverse reaction i (using basic thermodynamic relation between `k_(fi),k_(fi) and equilibrium constant K_P`)
N - total number of species (N=7 in this case)
Thus,
`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)`
`rArr q_1 = k_(f1)[CO][O_2] - k_(r1)[CO_2][O]`
`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)`
`rArr q_2 = k_(f2)[O][H_2O] - 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)`
`rArr q_3 = k_(f3)[CO][OH]-k_(r3)[CO_2][H]`
`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)`
`rArr q_4 = k_(f4)[H][O_2]-k_(r4)[OH][O]`
Finally, to obtain the system of coupled ODE for the reaction mechanism, we define,
`dot omega_j` - Net production rate of each species j
`dot omega = fracddt[X_j] = overset(N)underset(j=1)(sum)nu_(ji)q_i`
`nu_(ji) = nu_(ji)'' - nu_(ji)'`
Here N is the total number of species (N = 7 in this case).
For species number 1 (j=1) i.e. `CO`,
`dot omega_1 = nu_11q_1 + nu_12q_2 +nu_13q_3 + nu_14q_4`
From the `nu_(ji)`matrix defined above, we get
`nu_11 = -1; nu_12 = 0;nu_13 = -1;nu_14 = 0`.
Thus, `dot omega_1 = -1*q_1 -1*q_3 = -q_1-q_3`
`rArr dot omega_1 =-(k_(f1)[CO][O_2] - k_(r1)[CO_2][O])-(k_(f3)[CO][OH]-k_(r3)[CO_2][H])`
For species number 2 (j=2) i.e. `O_2`,
`dot omega_2 = nu_21q_1 + nu_22q_2 +nu_23q_3 + nu_24q_4`
`nu_21 = -1; nu_22 = 0;nu_23 = 0;nu_24 = -1`.
Thus, `dot omega_2 = -1*q_1 -1*q_4 = -q_1-q_4`
`rArr dot omega_2 =-(k_(f1)[CO][O_2] - k_(r1)[CO_2][O])-(k_(f4)[H][O_2]-k_(r4)[OH][O])`
Similarly,
`dot omega_3 = q_1+q_3`
`rArr dot omega_3 =(k_(f1)[CO][O_2] - k_(r1)[CO_2][O])+(k_(f3)[CO][OH]-k_(r3)[CO_2][H])`
`dot omega_4 = q_1-q_2+q_4`
`rArr dot omega_4 =(k_(f1)[CO][O_2] - k_(r1)[CO_2][O])-(k_(f2)[O][H_2O] - k_(r2)[OH]^2)+(k_(f4)[H][O_2]-k_(r4)[OH][O])`
`dot omega_5 = -q_2`
`rArr dot omega_5 =-(k_(f2)[O][H_2O] - k_(r2)[OH]^2)`
`dot omega_6 = 2q_2-q_3+q_4`
`rArr dot omega_6 =2(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])`
`dot omega_7 = q_3-q_4`
`rArr dot omega_7 =(k_(f3)[CO][OH]-k_(r3)[CO_2][H])-(k_(f4)[H][O_2]-k_(r4)[OH][O])`
Thus, using this compact notation method, a very simple representation of the reaction rate equations can be developed in matrix form which can then be solved using non linear equation solvers.
The final form of the rate equation thus can be written as a column vector in the following form
`dot omega = [[dot omega_1],[dot omega_2],[dot omega_3],[dot omega_4],[dot omega_5],[dot omega_6],[dot omega_7]] = [[fracddt[CO]],[fracddt[O_2]],[fracddt[CO_2]],[fracddt[O]],[fracddt[H_2O]],[fracddt[OH]],[fracddt[H]]]`