Compact Notation Derivation for a simple Multi-step Mechanism

The objective is to derive the reaction rate of ODEs of a simple mechanism.

The chemical kinetics is the study of reaction mechanism subjected to different operating conditions which is based on equilibrium chemistry. The compact notation for the below rates of reaction for multistep mechanism are shown below as an example so that they can be easily inculcated in the computer programs for various computaional purposes.

Rates of reaction mechanism for multistep mechanism

`CO+O_2->CO_2+O`

`O+H_2O->OH+OH`

`CO+OH->CO_2+H`

`H+O_2->OH+O`

Compact notation

From the above equation we know that there are total 7 species and 4 reactions which are listed below.

The reactions are all forward reactions i.e., irreversible & only rate of forward reaction term is present. we can develop a system of coupled ODE (ordinary differential equation) based on the reaction. The different ODE\'s developed with each species are listed below as follows.

`(d[CO])/dt =-k_(f1)[CO][O_2]-k_(f3)[CO][OH]`

`(d[O_2])/dt=-k_(f1)[CO][O_2]-k_(f4)[H][O_2]`

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

`(d[O])/dt=-k_(f2)[O][H_2O]`

`(d[H_2O])/dt=-k_(f2)[O][H_2O]`

`(d[OH])/dt=-k_(f3)[CO][OH]`

`(d[H])/dt=-k_(f4)[H][O_2]`

where, `k_(fi)`is rate of forward reaction & `k_(ri)` is rate of reverse reaction (which is zero for this case since its irreversile reaction) for each reaction.

These are the rate of concentration of each species & `k_(fi)` taken as -ve for forward reaction as the concentration of that species decreases as the reaction proceeds further. Here, the each equation above is ordinary differential equation which can be further solved using the following notations.

Derivation

let say number of moles of species \'j\' in reaction \'i\' as `v_(ji)`

Matrix form of stoichiometric coefficients of all the species in reactants side

`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]]`

similarly,

Matrix form for products side stoichiometric coefficients of all the species

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

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

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

then, the reaction rate(`q_i`) (or) skew is calculated from these matrices using the below formula

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

where,

i,j are number of reactions of species

[X] is concentration of each species

`k_(fi),k_(ri)`are rate of forward & reverse reaction.

As an example, `q_1=k_(f1)[CO][O_2]`

Now, lets define `w_j` which is net production rate of species in a multistep mechanism.

`w_j=sum_(j=1)^n(v_(ji))(q_i)= (d[X_j])/dt`

where, `q_i` is net reaction rate

j = 1 to 7 species defined in the table above; i = 4 which is number of reactions

this equation is used to denote ODE\'s in order to compute the chemical mechanisms as program code. This is how we compac the actual equations into such notaions for ease of programing & computing.