Compact Notation Derivation for a

simple Mechanism

Given Reaction Mechanism

CO + O2 -> CO2 + O

O + H2o -> OH + OH

CO + OH -> CO2 + H

H + O2 -> OH + O

Compact Notation:The above mentioned four reaction mechanisms were formed into the Matrix
for system understanding. Finally, we will be getting two matrixes, these are No.of Species and No.of Reactions. Here j is depicted the number of speciesand i is depicted the number of Reactions. These matrixes Mentioned below,

i=`[[CO],[O2],[CO2],[OH2],[O],[OH],[H]]` No.Of species has contained in total Mechanisms.
J=`[[1],[2],[3],[4]]`No. of Reaction Mechanisms.Now we will be forming Reactentent side Stochimetric Matrix (Vij'),
and Product side Stochimetric Matrix(Vij''). Here 'i' will depict the number of species that have been containing in the number of the reaction mechanism and j will depict the number of reaction mechanisms. Here single prime (')indicates the reactant side and double prime ("") indicates the product side.

Reactent side Stochiometric Matrix

V′ij=`[[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]]`

Product side Stochiometric Matrix

V′′ij=`[[-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]]`

Vij=V′′ij-V′ij

Vij=`[[-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 above matrix defines The number of moles of species has contained in a particular reaction.

For Example CO,

In the reaction mechanism, CO has contained both the reactant side and the product side, from the reactant side one mole of CO at the first reaction mechanism and third reaction mechanism, remaining equations are (2 & 4) zero that we have seen from Vij' matrix. At the product side, CO has been zero in all the equation because the product side doesn't have CO.

Similarly, written all species from all reaction mechanisms that we have seen above the V'ij matrix. And similarly, have written product side Vij''.

The Net Reaction Rate:

`underset(i)(q)`=`[underset(fi)(K)prod_(j=1)^N[underset(j)(X)]^(underset(ij)(V')]-underset(ri)(K)prod_(j=1)^N[underset(j)(X)]^(underset(ij)(K')]]`

q_i = Net Reaction rate for each reaction mechanism.

i = No.of Reactions

j= No.of species

X= Concentration of Species

Vij'= Reactent side stochiometric Matrix.

Vij''=Product side stochiometric Matrix.

`underset(fi)(K)`= Forward Reaction

`underset(ri)(K)`= Reverse Reaction

Here, we have written the net Reaction rate of each reaction equation

`underset(1)(q)=underset(f1)(K)[CO]^1[O2]^1-underset(r1)(K)[O2]^1[O]^1`

`underset(2)(q)=underset(f2)(K)[O]^1[H2O]^1-underset(r2)(K)[OH]^2`

`underset(3)(q)=underset(f3)(K)[CO]^1[OH]^1-underset(r3)(K)[CO2]^1[H]^1`

`underset(4)(q)=underset(f4)(K)[O2]^1[H]^1-underset(r3)(K)[O]^1[OH]^1`

The Net Production Rate:

`underset(j)(W)`=`sum_(j=1)^Lunderset(ij)Vunderset(i)q`

j is species index

`underset(j)(W)`=`[[underset(1)(W)],[underset(2)(W)],[underset(4)(W)],[underset(5)(W)],[underset(6)(W)],[underset(7)(W)]]`= `[[(d(co))/dx],[(d(o2))/dx],[(d(co2))/dx],[(d(o))/dx],[(d(H2o))/dx],[(d(OH))/dx],[(d(H))/dx]]`

`underset(1)(W)`=`V11q1+V12q2+V13q3+V14q4`

`underset(CO)(W)` =`-[underset(f1)(K)[CO]^1[O2]^1-underset(r1)(K)[CO2]^1[O]^1]-[underset(f3)(K)[CO]^1[OH]^1-underset(r3)(K)[CO2]^1[H]^1]`

`underset(2)(W)` =`V21q1+V22q2+V23q3+V24q4`

`underset(O2)(W)` =`-[underset(f1)(K)[CO]^1[O2]^1-underset(r1)(K)[CO2]^1[O]^1]-[underset(f4)(K)[O2]^1[H]^1-underset(r4)(K)[O]^1[OH]^1]`

`underset(3)(W)` =`V31q1+V32q2+V33q3+V34q4`

`underset(CO2)(W)` =`[underset(f1)(K)[CO]^1[O2]^1-underset(r1)(K)[CO2]^1[O]^1]-[underset(f3)(K)[CO]^1[OH]^1-underset(r3)(K)[CO2]^1[H]^1]`

`underset(4)(W)`=`V41q1+V42q2+V43q3+V44q4`

`underset(O)(W)`=`[underset(f1)(K)[CO]^1[O2]^1-underset(r1)(K)[CO2]^1[O]^1]-[underset(f2)(K)[O]^1[H2O]^1-underset(r2)(K)[OH]^2]+[underset(f4)(K)[O2]^1[H]^1-underset(r4)(K)[O]^1[OH]^1]`

`underset(5)(W)` =`V51q1+V52q2+V53q3+V54q4`

`underset(H2O)(W)` =`-[underset(f2)(K)[O]^1[H2O]^1-underset(r2)(K)[OH]^2]`

`underset(6)(W)` =`V61q1+V62q2+V63q3+V64q4`

`underset(OH)(W)` =`2[underset(f2)(K)[O]^1[H2O]^1-underset(r2)(K)[OH]^2]-[underset(f3)(K)[CO]^1[OH]^1-underset(r3)(K)[CO2]^1[H]^1]+[underset(f4)(K)[O2]^1[H]^1-underset(r4)(K)[O]^1[OH]^1]`

`underset(7)(W)` =`V71q1+V72q2+V73q3+V74q4`

`underset(H)(W)`=`[underset(f3)(K)[CO]^1[OH]^1-underset(r3)(K)[CO2]^1[H]^1]+[underset(f4)(K)[O2]^1[H]^1-underset(r4)(K)[O]^1[OH]^1]`