Deriving Reaction rate ODE s of a simple reaction mechanism

Objective:

To derive the reaction rate ODE\'s for a simple reaction mechanism and to use the compact notation to represent the desired matrices.

Given:

Reaction Mechanism:

1. `CO+O_2 harr CO_2+O`
2. `O+H_2O harr OH+OH`
3. `CO+OH harr CO_2+H`
4. `H+O_2 harr OH+O`

Chemical Kinetics:

Chemical kinetics deals with study of the rate of chemical reactions and it provides information about the reaction mechanism , transition state and number of steps involved in obtaining the products.

Reaction Rate:

Reaction rate is the measure of change in concentration of reactants or the change in concentration of products depending on time.It is also the speed at which the reaction occurs and depends on Temperature,pressure.

Reactions:

`CO+O_2 harr CO_2+O`

`O+H_2O harr OH+OH`

`CO+OH harr CO_2+H`

`H+O_2 harr OH+O`

There are 7 species and 4 reactions involved in the mechanism.

1. `(d[CO])/dt=-k_(f1)[CO][O_2]+k_(r1)[CO_2][O]-k_(f3)[CO][OH]+k_(r3)[CO_2][H]`
2. `(d[O_2])/dt=-k_(f1)[CO][O_2]+k_(r1)[CO_2][O]-k_(f4)[H][O_2]+k_(r4)[OH][O]`
3. `(d[O])/dt=k_(f1)[CO][O_2]-k_(r1)[CO_2][O]-k_(f2)[O][H_2O]+k_(r2)[OH][OH]+k_(f4)[H][O_2]-k_(r4)[OH][O]`
4. `(d[H_2O])/dt=-k_(f2)[O][H_2O]+k_(r2)[OH][OH]`
5. `(d[OH])/dt=k_(f2)[O][H_2O]-k_(r2)[OH][OH]-k_(f3)[CO][OH]+k_(r3)[CO_2][H]+k_(f4)[H][O_2]-k_(r4)[OH][O]`
6. `(d[CO_2])/dt=k_(f1)[CO][O_2]-k_(r1)[CO_2][O]-k_(f3)[CO][OH]+k_(r3)[CO_2][H]`
7. `(d[H])/dt=k_(f3)[CO][OH]-k_(r3)[CO_2][H]-k_(f4)[H][O_2]+k_(r4)[OH][O]`

`k_(fi)`-reaction rate constant for forward reaction

`k_(ri)`- reaction rate constant for reverse reaction

+\'ve sign indicates increase in the concentration of species where as -\'ve sign indicates the decrease in the concentration of species.

Compact notation:

Species:

Reactions:

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

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

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

`v_(ij)^\'`is the matrix that represents the stoichiometric coefficients of each species\'j\' in every reaction \'i\' on the reactant side.

`v_(ij)^\'\'`is the matrix that represents the stoichiometric coefficients of each species\'j\' in every reaction \'i\' on the products side.

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

`q_i`- Net reaction rate for every reaction in the reaction mechanism.

`[X_j]^(v_(ji)^\')`-Concentration of species on reactants side.

`[X_j]^(v_(ji)^\'\')`-Concentration of species on products side.

N-total number of species

`q_1=k_(f1)[CO]^1[O_2]^1[O]^0[H_2O]^0[OH]^0[CO_2]^0[H]^0-K_(r1)[CO]^0[O_2]^0[O]^1[H_2O]^0[OH]^0[CO_2]^1[H]^0`

`q_1=k_(f1)[CO][O_2]-k_(r1)[CO_2][O]`

similarly neglecting the stoichiometric coefficients with value 0,

`q_2=k_(f2)[O][H_2O]-k_(r2)[OH][OH]`

`q_3=k_(f3)[CO][OH]-k_(r3)[CO_2][H]`

`q_4=k_(f4)[H][O_2]-k_(r4)[OH][O]`

`w_j=sum_(i=1)^L v_(ji) q_(i)`

`w_j`is the net production rate of Species \'j\' for every reaction i.

L is Total no of reactions in the mechanism.

`w_(1)=-(k_(f1)[CO][O_2]-k_(r1)[CO_2][O])-(k_(f3)[CO][OH]-k_(r3)[CO_2][H])`

`w_2=-(k_(f1)[CO][O_2]-k_(r1)[CO_2][O])-(k_(f4)[H][O_2]-k_(r3)[OH][O])`

`w_3=-(k_(f1)[CO][O_2]-k_(r1)[CO_2][O])-(k_(f2)[O][H_2O]-k_(r2)[OH][OH])-(k_(f4)[H][O_2]-k_(r4)[OH][O])`

`w_4=-(k_(f2)[O][H_2O]-k_(r2)[OH][OH])`

`w_5=-(k_(f2)[O][H_2O]-k_(r2)[OH][OH])-(k_(f3)[CO][OH]-k_(r3)[CO_2][H])-(k_(f4)[H][O_2]-k_(r4)[OH][O])`

`w_6=-(k_(f1)[CO][O_2]-k_(r1)[CO_2][O])-(k_(f3)[CO][OH]-k_(r3)[CO_2][H])`

`w_7=-(k_(f3)[CO][OH]-k_(r3)[CO_2][H])-(k_(f4)[H][O_2]-k_(r4)[OH][O])`

`k_(fi)=A_iT^(beta_i)exp((-E_i)/(R_cT))`

`k_(ri)=(k_(fi))/(k_(ci))`

`k_(ci)=k_(p i)(P_(atm))/(RT)`

`k_(p i)=exp((DeltaS_i^0)/R-(DeltaH_i^0)/(RT))`