Week 5.1 - Compact Notation Derivation for a simple Mechanism
Aim: - To derive the reaction rate ODEs of a simple reaction mechanism.
Objective: - To derive the reaction rates ODEs of the following mechanism -
1. CO + O2 ⇌ CO2 + O
2. O + H2o ⇌ OH + OH
3. CO + OH ⇌ CO2 + H
4. H + O2 ⇌ OH + O
Theory: -
Reaction rate is the speed of formation/depletion of a certain species in chemical reaction.
Any chemical reaction is muti-step process in which take place with a reaction mechanism consisting of several intermediatery step/products. These increase the dependency of a reaction rates on one or more species concerntration.
To deal with these some Reaction models are used to simulate the reaction and get the desired result.
A system of ODE's of reaction rates are formed using these models, which can be fed to different software packages to find the reaction rates and simulate the reaction.
There are 7 different species and 4 elementry reaction steps in the above reaction mechanism. they are -
| Species (j) | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| Name | CO | O2 | CO2 | O | H20 | OH | H |
Reactions (i) : R1 R2 R3 R4
The coefficient matrix for the reactant (`v_(ji)'`) i.e. Number of moles of species j in the reaction i for reactant is -
`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]]`
Similarly, coefficient matrix for product (`v_(ji)''`) i.e. Number of moles of species j in the reaction i for product is -
`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]]`
Now
`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]]`
Now, the net Reaction rate for 4 reaction can be calculated by
`q i = k f _i ∏_( j = 1 )^7 [ X j ] ^(ν _( j i)') − k r_ i ∏_( j = 1 )^7 [ X j ] ^(ν _( j i)'')`
On putting the right indicies, we get
`q 1 = k f_1[CO]^1[O2]^1[CO2]^0[O]^0[H2O]^0[OH]^0[H]^0 − k r_1[CO]^0[O2]^0[CO2]^1[O]^1[H2O]^0[OH]^0[H]^0`
`=> q1 = kf_1[CO][02] - kr_1[CO2][O]``
Similarly we get
`q 2 = k f _2[O][H2O] − k r _2 [OH]^2`
`q 3 = k f_ 3[CO][OH] − k r _3 [CO2][H]`
`q 4 = k f_4[O2][H] − k r_4 [O][OH]`
Now rate of production can be written as
`ω_j = (d[X_j])/(dt)`
And rate of prouction can also be determined by -
`ω_j = ∑_(i = 1)^4 ν_ (j i) q_ i`
which implies that
`[[w_1],[w_1],[w_1],[w_1],[w_1],[w_1],[w_1]] =[[q_1,q_2,q_3,q_4]]*[[-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]]`
On further solving by putting reaction rate values, we get
`(d[CO] )/(dt) = kr_1[CO2][O]−kf_1[CO][O2]+kr_3[CO2][H]−kf_3[CO][OH]`
`(d[O2])/(dt) = kr_1[CO2][O] − kf_1[CO][O2] + kr_4[O][OH] − kf_4[O2][H]`
`(d[CO2])/(dt) = kf_1[CO][O2] − kr_1[CO2][O] + kf_3[CO][OH] − kr_3[CO2][H]`
`(d[O])/(dt) = kf_1[CO][O2] − kr_1[CO2][O] + kr_2[OH]^2 − kf_2[O][H2O] + kf_4[O2][H] − kr_4[O][OH]`
`(d[H2O])/(dt) = kr_2[OH]^2 − kf_2[O][H2O]`
`(d[OH])/(dt) = 2kf_2[O][H2O] − 2kr_2[OH]^2 + kr_3[CO2][H] − kf_3[CO][OH] + kf_4[O2][H] − kr_4[O][OH]`
`(d[H])/(dt) = kf_3[CO][OH] − kr_3[CO2][H] + kr_4[O][OH] − kf_4[O2][H]`
Above is the system of Reaction rate ODE or given Reaction mechanism