Week 5.1 - Compact Notation Derivation for a simple Mechanism

AIM:

To derive the reaction rate ODEs & production of each species for the simple reaction mechanism as shown.

Derivation:

First we will start with the compact notation for the given species and the reaction as shown:

Species Notations:

j	Species
1	CO
2	`underset(2)(O)`
3	`Cunderset(2)(O)`
4	O
5	`underset(2)(H)O`
6	OH
7	H

Reaction Notation:

i	Reactions
`underset(1)(R)`	`CO + underset(2)(O) hArr Cunderset(2)(O) + O`
`underset(2)(R)`	`O + underset(2)(H)O hArr OH + OH`
`underset(3)(R)`	`CO + OH hArr Cunderset(2)(O) + H`
`underset(4)(R)`	`H + underset(2)(O) hArr OH + O`

Now we will create the reactant side matrix for the compact notation

`v'underset(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 the product side matrix will be created as shown:

`v''underset(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]]`

Thus,

`vunderset(ji)() = v''underset(ji)() - v'underset(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 can be calculated as

`qunderset(i)() = kunderset(fi)()overset(N)(Pi)underset(j=1)()[Xunderset(j)()]overset(v'underset(ji)())() - kunderset(fi)()overset(N)(Pi)underset(j=1)()[Xunderset(j)()]overset(v''underset(ji)())()`

So the net reaction rate for r1 becomes

`qunderset(1)() = kunderset(f1)()[CO][Ounderset(2)()] - kunderset(r1)()[COunderset(2)()][O]`

Similarly net reaction rate for other reactions is

`qunderset(2)() = kunderset(f2)()[O][Hunderset(2)()O] - kunderset(r2)()[OH][OH]`

`qunderset(3)() = kunderset(f3)()[CO][OH] - kunderset(r3)()[COunderset(2)()][H]`

`qunderset(4)() = kunderset(f4)()[H][Ounderset(2)()] - kunderset(r4)()[OH][O]`

Finally the product rate can be defined as:

`wunderset(j)() = overset(L)(sum)underset(i=1)()vunderset(ji)()qunderset(i)()`

THus the product rates for the reactions can be written as

`wunderset(1)() = vunderset(11)()qunderset(1)() + vunderset(12)()qunderset(2)() + vunderset(13)()qunderset(3)() + vunderset(14)()qunderset(4)()`

`wunderset(2)() = vunderset(21)()qunderset(1)() + vunderset(22)()qunderset(2)() + vunderset(23)()qunderset(3)() + vunderset(24)()qunderset(4)()`

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Thus the equation in matrix form becomes:

`[[wunderset(1)()],[wunderset(2)()],[wunderset(3)()],[wunderset(4)()],[wunderset(5)()],[wunderset(6)()],[wunderset(7)()]] = [[-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]]^TX [[qunderset(1)()],[qunderset(2)()],[qunderset(3)()],[qunderset(4)()]]`