Week 5.1 - Compact Notation Derivation for a simple Mechanism

Aim: The objective of this challenge is to derive the reaction rate of ODE's of a simple reaction mechanism and represent it in the form of compact notations. The compact notations for the reactions listed below will be derived.

Reaction Mechanism

`CO + O_2 hArr CO_2 + O`

`O + H_2O hArr OH +OH`

`CO + OH hArr CO_2 + H`

`H + O_2 hArr OH + O`

Solution:
Before arriving at the compact notations, the number of species in each of the reactions must be identified. Here j represents the number of species and i represents the number of reactions.

Species j

1. `CO`
2. `O_2`
3. `O`
4. `OH`
5. `H`
6. `CO_2``H_2O`

Reactins I :

1. `R_1`
2. `R_2`
3. `R_3`
4. `R_4`

Now that we have identified the number of species and the number of reactions, we now move on to obtaining the stoichiometric coefficient matrix for both the reactant and the products side. These are represented by `V'_(ji)`and `V''_(ji)`.

With this, the net stoichiometric coefficient `V_(ji)` is calculated where `V_(ji) = V''_(ji)- V'_(ji)`

Matrix `V_(ij)` is given by = `[[-1,-1,1,0,0,1,0],[0,0,-1,2,0,0,-1],[-1,0,0,-1,1,1,0],[0,-1,1,1,-1,0,0]]`

From the above data we can calculate the net reaction rate which is denoted by q

`q_i = K_(fi) . pi_(j=1)^7[X]_i - K_(ri).pi_(j=1)^7[X]_i`

where ,

`q_i` is the reaction rate

for Reaction 1 :

`q_1 = K_(f1) . [CO].[O_2] - K_(r1).[CO_2].[O]`

for Reaction 2 :

`q_2 = K_(f2) . [O].[H_2O] - K_(r2).[OH].[OH]`

for Reaction 3 :

`q_3 = K_(f3) . [CO].[OH] - K_(r3).[CO_2].[H]`

for Reaction 4 :

`q_4 = K_(f4) . [H].[O_2] - K_(r4).[OH].[O]`

now,

`omega_j = sum_(i=1)^L v_(ji).q_1`

`omega_1 = v_11.q_1 + v_12.q2 + v_13.q_3 + v_14.q = -q_1-q_3`

`omega_2 = v_21.q_1 + v_22.q2 + v_23.q_3 + v_24.q = -q_1-q_4`

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.

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so on

With the help of the above equation, `W_j` matrix (product rate of each species) can be calculated. Wj is the summation of the product of the stoichiometric coefficient matrix and the rate of each reaction.
The final form is represented below.

`[[omega_1],[omega_2],[omega_3],[omega_4],[omega_5],[omega_6],[omega_7]]` = `[[-1,0,-1,0],[-1,0,0,-1],[1,0,1,0],[1,-1,0,1],[0,-1,0,0],[0,2,-1,1],[0,0,1,-1]]` `X` `[[q_1],[q_2],[q_3],[q_4]]`