Week 5 - Compact Notation Derivation for a simple Mechanism
Objective: To derive the reaction rate ODEs of a simple reaction mechanism using compact notations and write down the desired matrices.
Given: The simple reaction mechanism is as follows:
`CO + O_2 -> CO_2 + O`
`O + H_2O->OH+OH`
`CO+ OH -> CO_2 +H`
`H+ O_2->OH+O`
Theory:
Compact Notations- This helps us to solve the reaction mechanism which includes large nos. of species, intermediary reactions with the aid of the computer.
Notation for SPECIES:
| j | Species |
| 1 | CO |
| 2 | O2 |
| 3 | CO2 |
| 4 | O |
| 5 | H2O |
| 6 | OH |
| 7 | H |
Notation for REACTIONS:
| i | Reaction |
| 1 | R1- CO + O_2 -> CO_2 + O |
| 2 | R2- O + H_2O->OH+OH |
| 3 | R3- CO+ OH -> CO_2 +H |
| 4 | R4- H+ O_2->OH+O |
For this reaction mechanism:
`sumV_(ji)^'=sumV_(ji)^('')` for i= 1,2,3,4,....n
where,
`V_(ji)^'- "Stoichiometric coefficients of reactants"`
`V_(ji)^('')- "Stoichiometric coefficients of products"`
Vji- Keeps a track of nos of species in reaction i.
Matrices for reactants and products are as below:
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]]`
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]]`
`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 generalized form of Net reaction rate for every reaction is written as:
`q_i=K_(fi)prod_(j=1)^N[X_j]^(V_(ji)')-K_(ri)prod_(j=1)^N[X_j]^(V_(ji)^(''))`
where, Kf- forward reaction constant
Kr- reverse reaction constant
We have four reaction mechanisms hence we will have four reaction rates.
q1= Kf1[CO][O2] - Kr1[CO2][O]
q2= Kf2[O][H2O] - Kr2[OH][OH]
q3= Kf3[CO][OH] - Kr3[CO2][H]
q4= Kf4[H][O2] - Kr4[OH][O]
ODE's for all species will be written as:
For CO-
`d/dt(CO)= -q1-q3`
For O2-
`d/dt(O_2)= -q1-q4`
For CO2-
`d/dt(CO_2)= q1+q3`
For O-
`d/dt(O)= q1-q2+q4`
For H2O-
`d/dt(H_2O)= -q2`
For OH-
`d/dt(OH)= q2-q3+q4`
For H-
`d/dt(H)= q3-q4`
All the above equations are ODE's, solving these ODE's will give us a solution for our reaction mechanism, or helps us to solve chemical kinetics.