Week 5.1 - Compact Notation Derivation for a simple Mechanism
A balanced equation for a chemical reaction indicates what is reacting and what is produced, but it reveals nothing about how the reaction actually takes place. The reaction mechanism (or reaction path) is the process, or pathway, by which a reaction occurs.
A chemical reaction usually occurs in steps, although it may not always be obvious to an observer. The decomposition of ozone, for example, appears to follow a mechanism with two steps:
`O_3 rightarrow O_2 + O`
`O + O_3 rightarrow 2O_2`
Species that are produced in one step and consumed in a subsequent step are called intermediates. There are three types of elementary reaction types - Biomolecular Reaction, Unimolecular reaction and Trmolecular reaction. To solve these intermediatory states the equations are changed from ODE to simple matrix form, which can be seen as below -
Reaction Mechanism
`CO + O_2 Leftrightarrow CO_2 +O`
`O + H_2O Leftrightarrow OH +OH`
`CO +OH Leftrightarrow CO_2 +H`
`H + O_2 Leftrightarrow OH + O`
| J | Species |
| 1 | CO |
| 2 | `O_2` |
| 3 | `CO_2` |
| 4 | O |
| 5 | `H_2O` |
| 6 | OH |
| 7 | H |
| i | Species |
| 1 | R1 |
| 2 | R2 |
| 3 | R3 |
| 4 | R4 |
Coefficient matrix for both reactants and products are written as:
Matrix for Reactants -
`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]]`
Matrix for Products -
`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]]`
Net Coefficient Matrix -
`V_(ji) = V_(ji)^('') - V_(ji)^(')`
`V_(ji) = [[-1,-1,0,1,0,1,0,0],[-1,0,0,0,1,-1,1,0],[0,-1,1,0,0,1,-1,0],[-1,0,0,1,0,-1,0,0]]`
Net Reaction Rate is given as
`q_i = k_(fi) Pi_(j=1)^(N) [X_j]^(V_(ji)^(')) - k_(ri) Pi_(j=1)^(N) [X_j]^(V_(ji)^(''))`
it can be expanded as
`q_1 = k_(f1)[O_2][CO] - k_(r1)[O][CO_2]`
`q_2 = k_(f2)[O][H_2O] - k_(r2)[OH]^2`
`q_3 = k_(f3)[OH][CO] - k_(r3)[H][CO_2]`
`q_4 = k_(f4)[O_2][H] - k_(r4)[O][OH]`
The production rate of each of the species is calculated as
`omega_j = Sigma_(i=1)^(L) V_(ji) q_i`
after expansion,
`omega_1 = V_(11) q_1 + V_(12)q_2 + V_(13)q_3+V_(14)q_4`
after expansion
`omega_1 = k_(r1)[CO_2][O]-K_(f1)[O_2][CO]+k_(r4)[O][OH]-K_(f4)[O_2][H]`
`omega_2 = K_(f1)[O_2][CO]-k_(r1)[CO_2][O]-k_(r3)[CO_2][H]+K_(f3)[CO][OH]`
`omega_3 = K_(r2)[OH]^2-k_(f2)[H_2O][O]`
`omega_4 = K_(f1)[O_2][CO]-k_(r1)[CO_2][O]+k_(r2)[OH]^2-K_(f2)[O][H_2O]+K_(f4)[O_2][H]-k_(r4)[O][OH]`
`omega_5 = -K_(f1)[O_2][CO]+k_(r1)[CO_2][O]-K_(f3)[CO][OH]+k_(r3)[CO_2][H]`
`omega_6 = K_(f1)[O_2][CO]-k_(r1)[CO_2][O]-K_(f3)[CO][OH]+k_(r3)[CO_2][H]`
`omega_7 = 2*K_(f2)[O_2][H_2O]-k_(r2)[OH]^2+k_(r3)[CO_2][H]-K_(f3)[CO][OH]+K_(f4)[O_2][H]-k_(r4)[O][OH]`
Also this production rate is equal to the rate of change of each species
`omega_i = d/dt[X_j]`
From the above relation it can also be concluded as :
`[[omega_1],[omega_2],[omega_3],[omega_4],[omega_5],[omega_6],[omega_7]] = [[d/dt[CO]],[d/dt[O_2]],[d/dt[CO_2]],[d/dt[O]],[d/dt[H_2O]],[d/dt[OH]],[d/dt[H]]]`
Here reaction mechanism equations are converted into coupled ODE's which can be interpreted in the machine language and can be solved further.