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:

Notation for REACTIONS:

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.