Week 5.1 - Compact Notation Derivation for a simple Mechanism

Objective:-

To derive the reaction rate ODE of a simple reaction mechanism.

considering given reactions,

List of all species of hydrocarbons that are taking place in the reaction

1.O2

2.H2O

3.CO2

4. OH

5. CO

6. H

7. O

Defining `upsilon underset(i j')`be the stoichiometric coefficient matrix for the elements of the reactants in which the I represents the reaction and j denotes the number of moles of the species coefficient.

`upsilon underset(i j')` = `[[1,0,0,0,1,0,0],[0,1,0,0,0,0,1],[0,0,0,1,1,0,0],[1,0,0,0,0,1,0]]`

Let's determine the stoichiometric coefficient matric `upsilon underset(i j'')`for the products side as well

`upsilon underset(ij'')` = `[[0,0,1,0,0,0,1],[0,0,0,2,0,0,0],[0,0,1,0,0,1,0],[0,0,0,1,0,0,1]]`

The equation for the net stoichiometric coefficient matrix is

`upsilon underset(i j)` = `upsilon underset(i j'')` - `upsilon underset(i j')`

`upsilon underset(i j)` = `[[-1,0,1,0,-1,0,1],[0,-1,0,2,0,0,-1],[0,0,1,-1,-1,1,0],[-1,0,0,1,0,-1,1]]`

So, the net reaction rate `qunderset(i)` can be written as

`qunderset(i)` = `Kfi prod_(j=0)^N [Xunderset(j)()]^[upsilon ij'] - Kri prod_(j=0)^N [Xunderset(j)()]^ [upsilon ij'']`

Where,

`k underset(fi)` is the forward reaction rate

`k underset(ri)` is the backward reaction rate

`[Xunderset(j)()]^[upsilon ij']` is the concentration of species per unit volume of reactants

`[Xunderset(j)()]^ [upsilon ij'']` is the concentration of species per unit volume of products

Now substituting the values for 'i' in the net reactionn rate equation we get,

`qunderset(1)() = Kf1[O2][CO] - Kr1[Co2][O]`

`qunderset(2)() = Kf2[H2O][O] - Kr2[OH]^2`

`qunderset(3)() = Kf3[OH][CO] - Kr3[CO2][H]`

`qunderset(4)() = Kf4[O2][H] - Kr4[OH][O]`

Now we define the net production rate `omega underset(j)()` of 'j' species is written a`

`omegaunderset(j)() = sum_(i=1)^N upsilon underset(ij)() qunderset(i)`

`omegaunderset(j)() = d / dt [X underset(j)()]`

Substituting the values for 'j', we can write the net production rate,

`omegaunderset(j)() = upsilonunderset(1j)() qunderset(1)() + upsilonunderset(2j)() qunderset(2)() + upsilonunderset(3j)() qunderset(3)() + upsilonunderset(4j)() qunderset(4)()`

O2 Concentration:-

`d/dt[O2]` = -( Kf1[O2][CO] - Kr1[Co2][O]) - (Kf4[O2][H] - Kr4[OH][O])

H2O Concentration:-

`d/dt[H2O]` = -(Kf2[H2O][O] - Kr2[OH]^2)`

CO2 Concentration:-

`d/dt[CO2]` = Kf1[O2][CO2] - Kr1[Co2][O] + Kf3[OH][CO] - Kr3[CO2][H]

OH Concentration:-

`d/dt[OH]` = 2(Kf2[H2O][O] - Kr2[OH]^2)) - (Kf3[OH][CO] - Kr3[CO2][H]) + Kf4[O2][H] - Kr4[OH][O]

CO Concentration:-

`d/dt[CO]` = ( Kf1[O2][CO] - Kr1[Co2][O]) - (Kf3[OH][CO] - Kr3[CO2][H])

H Concentration:-

`d/dt[H]` = Kf3[OH][CO] - Kr3[CO2][H] -( Kf4[O2][H] - Kr4[OH][O])

O Concentration:-

`d/dt[O]` = Kf1[O2][CO] - Kr1[Co2][O] -(Kf2[H2O][O] - Kr2[OH]^2) + Kf4[O2][H] - Kr4[OH][O]

In order to evaluate the forward and backward reaction rates, the established relations are to be used,

Kfi = `Aunderset(i)()Toverset(betai)() exp(-(Eunderset(i)())/(Runderset(c)()T))`

Kfi = `(Kfi)/ (Kci)`

kci = Kpi `((P underset(atm)()) / (RT))^(Sigmaoverset(k)()underset(k=1)()upsilonkt`

Kpi = `exp (frac{nabla Soverset(o)underset(i)()} {R} - frac{nabla Hoverset(o) underset(i)()} {Rt})`

Conclusion:-

1. The concentration of each species was derived.

2.Reaction rates was determined using compact notations.