Compact Notation Derivation for a simple Mechanism

Aim: To derive the reaction rate ODEs of a simple reaction mechanism

`CO + O2 implies CO2 + O`

`O + H2O implies OH + +OH`

`CO + OH implies CO2 + H`

`H + O2 implies OH + O`

Compact Notations:

Matrix form for reactants:

`v\'ji = [[1,0,0,0,0,1,0],[0,1,1,0,0,0,0],[0,0,0,0,1,1,0],[1,0,0,1,0,0,0]]`

Matrix form for products:

`v\'\'ji = [[0,0,1,0,0,0,1],[0,0,0,0,2,0,0],[0,0,0,1,0,0,1],[0,0,1,0,1,0,0]]`

`vji = v\'\'ji - v\'ji`

`v ji = [[-1,0,1,0,0,-1,1],[0,-1,-1,0,2,0,0],[0,0,0,1,-1,-1,1],[-1,0,1,-1,1,0,0]]`

The reaction rates for eacc reactions can be derived from the formula:

`qi = kfi*prod_(j=1)^N[Xj]^(v\'ji) - kri*prod_(j=1)^N[Xj]^(v\'\'ji)`

kfi denotes reaction rate coefficients for forward reaction in ith reaction while kri denotes reaction rate coefficients for reverse reaction in ith reaction

`q1 = kf1*[CO]^1*[O2]^1 - kr1*[CO2]^1*[O]^1`

`q2 = kf2*[O]^1*[H2O]^1 - kr2*[OH]^2`

`q3 = kf3*[CO]^1*[OH]^1 - kr3*[CO2]^1*[H]^1`

`q4 = kf4*[H]^1*[O2]^1 - kr4*[OH]^1*[O]^1`

The net production rate of a species J is given by:

`omegaj = sum_(i=1)^L vji*qi`

for example for species 1 that is O2,

`omega1 = omegaO2 = v11*q1 + v12*q2 + v13*q3 + v14*q4 = -q1 - q4`

`omega1 = -(kf1*[CO]^1*[O2]^1 - kr1*[CO2]^1*[O]^1) - (kf4*[H]^1*[O2]^1 - kr4*[OH]^1*[O]^1)`

Similarly generating a matrix for all species production rae, we get:

`[[omega1],[omega2],[omega3],[omega4],[omega5],[omega6],[omega7]] = [[-1,0,0,-1],[0,-1,0,0],[1,-1,0,1],[0,0,1,-1],[0,2,-1,1],[-1,0,-1,0],[1,0,1,0]] * [[q1],[q2],[q3],[q4]]`