Compact Notation Derivation for a simple Mechanism

OBJECTIVE:Derive the reaction rate ODEs of a simple reaction mechanism.To use the compact notation and find out the desired matrices.

Reaction Mechanism:

CO + O2 `harr_(k_(r1))^(k_(f1))` CO2 + O

O + H2o `harr_(k_(r2))^(k_(f2))` OH + OH

CO + OH `harr_(k_(r3))^(k_(f3))` CO2 + H

H + O2 `harr_(k_(r4))^(k_(f4))` OH + O

`k_f`=Rate of forward reaction `k_r`=Rate of backward reaction

The rate of change of concentration can be derived as follows:

The rate of change of concentration for O2:

`(d[O2])/dt=-k_(f1)[O2][CO]+k_(r1)[CO2][O]-k_(f4)[O2][H]+k_(r4)[OH][O]`

The rate of change of concentration for CO:

`(d[CO])/dt=-k_(f1)[O2][CO]+k_(r1)[CO2][O]-k_(f3)[CO][OH]+k_(r3)[CO2][H]`

The rate of change of concentration for CO2:

`(d[CO2])/dt=k_(f1)[O2][CO]-k_(r1)[CO2][O]+k_(f3)[CO][OH]-k_(r3)[CO2][H]`

The rate of change of concentration for H2O:

`(d[H2O])/dt=-k_(f2)[H2O][O]+k_(r2)[OH][OH]`

The rate of change of concentration for O:

`(d[O])/dt=k_(f1)[O2][CO]-k_(r1)[CO2][O]-k_(f2)[O][H2O]+k_(r2)[OH][OH]+k_(f4)[O2][H]-k_(r4)[OH][O]`

The rate of change of concentration for OH:

`(d[OH])/dt=k_(f2)[H2O][O]-k_(r2)[OH][OH]+k_(f4)[H][O2]-k_(r4)[OH][O]-k_(f3)[CO][OH]+k_(r3)[CO2][H]`

The rate of change of concentration for H:

`(d[H])/dt=k_(f3)[CO][OH]-k_(r3)[CO2][H]-k_(f4)[H][O2]+k_(r4)[OH][O]`

Species index is denoted as j,

j-1=O2, j-2=CO,j-3=CO2,j-4=H2O,j-5=O,j-6=OH,j-7=H

Reactions index is denoted as i.

i from 1-4 ,Reactions R1, R2, R3,R4

Compact notation matrix:

Stoichometric coefficient matrix of Reactants side

`v_(ji)'=[[1,1,0,0,0,0,0],[0,0,0,1,1,0,0],[0,1,0,0,0,1,0],[1,0,0,0,0,0,1]]`

Stoichometric coefficient matrix of Products side

`v_(ji)''=[[0,0,1,0,1,0,0],[0,0,0,0,0,2,0],[0,0,1,0,0,0,1],[0,0,0,0,1,1,0]]`

Difference of the matrices

`v_(ji)=v_(ji)''-v_(ji)'=[[-1,-1,1,0,1,0,0],[0,0,0,-1,-1,2,0],[0,-1,1,0,0,-1,1],[-1,0,0,0,1,1,-1]]`

Net reaction rate for each reactions

`q_i=k_(fi)Pi_(j=1)^N[X_j]^(v_(ji'))-k_(ri)Pi_(j=1)^N[X_j]^(v_(ji'')`

`q1=k_(f1)[CO][O2]-k_(r1)[CO2][O]`

`q2=k_(f2)[O][H2O]-k_(r2)[OH][OH]`

`q3=k_(f3)[CO][OH]-k_(r3)[CO2][H]`

`q4=k_(f4)[H][O2]-k_(r4)[OH][O]`

Net production rate:

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

Net production rate for each species:

`omega_1=-k_(f1)[O2][CO]+k_(r1)[CO2][O]-k_(f4)[O2][H]+k_(r4)[OH][O]`

`omega_2=-k_(f1)[O2][CO]+k_(r1)[CO2][O]-k_(f3)[CO][OH]+k_(r3)[CO2][H]`

`omega_3=k_(f1)[O2][CO]-k_(r1)[CO2][O]+k_(f3)[CO][OH]-k_(r3)[CO2][H]`

`omega_4=-k_(f2)[H2O][O]+k_(r2)[OH][OH]`

`omega_5=k_(f1)[O2][CO]-k_(r1)[CO2][O]-k_(f2)[O][H2O]+k_(r2)[OH][OH]+k_(f4)[O2][H]-k_(r4)[OH][O]`

`omega_6=k_(f2)[H2O][O]-k_(r2)[OH][OH]+k_(f4)[H][O2]-k_(r4)[OH][O]-k_(f3)[CO][OH]+k_(r3)[CO2][H]`

`omega_7=k_(f3)[CO][OH]-k_(r3)[CO2][H]-k_(f4)[H][O2]+k_(r4)[OH][O]`

The Final ODE system becomes:

`((omega_1),(omega_2),(omega_3),(omega_4),(omega_5),(omega_6),(omega_7))`=`(((d[O2])/dt),((d[CO])/dt),((d[CO2])/dt),((d[H2O])/dt),((d[O])/dt),((d[OH])/dt),((d[H])/dt))`

The above ODE system can be integrated by any integration package for example ODE45 of matlab