Week 5 - Compact Notation Derivation for a simple Mechanism

#Aim:To create compact notation derivation of reaction rate ODEs for a simple mechanism.

Reaction Mechanism

CO + O2 -> CO2 + O

O + H2o -> OH + OH

CO + OH -> CO2 + H

H + O2 -> OH + O

Compact Notation:

Total species=7==>>j =1,2,.....7.

Total no of reactions=4==>>i=1,2...4.

Since ,Reaction is taking in only forward direction ,so only forward reaction rate would be considered ,reverse reaction rate won't be considered.

Kf=Forward reaction rate

It varies from `Kf_1` to `Kf_4` for i=1 to 4

`v_j`=stoichiometric coefficient for the species participating in the reaction

`v_(ji')`=Matrix of species on Reactant side ,formed from moles of species `s_j` in reaction i

`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]]`

Since,`v_(ji'')`=0

Therefore,

`v_(ji)=v_(ji'')-v_(ji')`

`v_(ji)=-v_(ji')`

`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]]`

`q_i`=Rate-of-progress variable

`q_i=Kf_iprod_(j=1)^N[X_j]^(v_(ji'))-Kr_iprod_(j=1)^N[X_j]^(v_(ji''))`

since`Kr_i`=0

So,

`q_i=Kf_iprod_(j=1)^N[X_j]^(v_(ji'))`

`q_1=Kf_1[CO]^(v_(11'))[O_2]^(v_(12'))`

`q_1=Kf_1[CO]^(1)[O_2]^(1)`

`q_1=Kf_1[CO][O_2]`

Like this,

`q_2=Kf_2[O][H_2O]`

`q_3=Kf_3[CO][OH]`

`q_4=Kf_4[O_2][H]`

w=Net production rate of particular species

`w_j=sum_(i=1)^Lv_(ji)q_i`

`w_1=v_(11)q_1+v_(12)q_2+v_(13)q_3+v_(14)q_4`

`w_1=-Kf_1[CO][O_2]-Kf_3[CO][OH]`

Since,

`w_j=d/(dt)[X_j]`(Rate of change of concentration of species with time)

So,

`d/(dt)[CO]=-Kf_1[CO][O_2]-Kf_3[CO][OH]`

Similary,

`d/(dt)[O_2]=-Kf_1[CO][O_2]-Kf_4[O_2][H]`

`d/(dt)[CO_2]=0`

`d/(dt)[O]=-Kf_2[O][H_2O]`

`d/(dt)[H_2O]=-Kf_2[O][H_2O]`

`d/(dt)[OH]=-Kf_3[CO][OH]`

`d/(dt)[H]=-Kf_4[O_2][H]`