Compact Notation Derivation for a simple Mechanism

AIM:

To derive the reaction rate ODES of given simple reaction mechanism.

Rates of reaction for Multistep Mechanism:

We have a four-step mechanism, Frist we will be forming the compact rotation for the reaction rates of mechanisms.

Reactions:

CO + O2 -> CO2 + O

O + H2o -> OH + OH

CO + OH -> CO2 + H

H + O2 -> OH + O

we will be using the compact notation where \'j\' will depict the number of species and I will depict the number of reactions

Compact Notations:

Kf_i and kr_i represent the following the forward and backward rate coefficients respectively for the i^th reaction. \'omega\' denotes the production rates for the j^th species and qi defines the rate of progress variable for an i^th elementary reaction.

Gives a set of reactions are only forward reactions. So reverse reaction rates are not considered.

`(d[CO])/(dt)=-kf_1[CO][O_2]-kf_3[CO][OH]`

`(d[O_2])/(dt)=-kf_1[CO][O_2]-kf_4[H][O_2]`

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

`(d[O])/(dt)=-kf_2[O][H_2O]`

`(d[H_2O])/(dt)=-kf_2[O][H_2O]`

`(d[OH])/(dt)=-kf_3[CO][OH]`

`(d[H])/(dt)=-kf_4[H][O_2]`

Derivation:

Matrix Form of the forward reaction

`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]]`

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

here `v_(ji)\'\'=0`

`v_ji\'\'` is reverse reaction rate`

so, `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]]`

Here j = 1 to 7

i = 1,2,3,4.

The net production rate of each species in a multistep mechanisum

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

where `q_i=kf_iprod_(j=1)^N[X_j]^(v_(ji)\')-kr_iprod_(j=1)^N[X_j]^(v_(ji)\'\')`

`q_1=kf_1[CO][O_2]`

`q_2=kf_2[O][H_2O]`

`q_3=kf_3[CO][OH]`

`q_4=kf_4[H][O_2]`

Using the above notataions we can derive all the compact notations as shown below

Consider species CO

`w_1=V_11*q_1+V_13*q_3`

`w_1=-kf_1[CO][O_2]-kf_3[CO][OH]`

`ODE=(dw_1)/(dt)=(dCO)/(dt)`

As shown above we can derive the compact notations of ODE\'s for all the species.