Week 5.1 - Compact Notation Derivation for a simple Mechanism

STUDY ON COMPACT NOTATION DERIVATION
FOR SIMPLE MECHANISM

Aim:

⦁ The aim of this study is to derive the reaction rate of ODEs of a simple reaction mechanism & production of each species for the simple reaction mechanism as described below.
⦁ Also, the compact notations for the reactions and the reactant and product matrices are to be derived in this study.

Reactions:

`CO + O_2 <=> CO_2 + O`
`O + H_2O <=> OH +OH`
`CO + OH <=> CO_2 + H`
`H + O_2 <=> OH + O`

PROCEDURE:

• Firstly, we need to determine each and every species involved in all these reactions taking place. So, we can use the notation as 'i' for the reactions taking place and 'j' for the species involved in these reactions.

Reaction, i	Species, j
`R_1`	`CO`
`R_2`	`O_2`
`R_3`	`O`
`R_4`	`OH`
	`H`
	`CO_2`
	`H_2O`

• Secondly, we can obtain the overall coefficient matrix, `T_(ji)` by subtracting the coefficient matrix of products, `(T_(ji))_(Products)` and the coefficient matrix of the reactants, `(T_(ji))_(Reactants)`, i.e.,
  `T_(ji) = (T_(ji))_(Products) - (T_(ji))_(Reactants )`
  `(T_(ji))_(Reactants) = [[1,1,0,0,0,0,0],[0,0,1,0,0,0,1],[1,0,0,1,0,0,0],[0,1,0,0,1,0,0]]`
  `(T_(ji))_(Products) = [[0,0,1,0,0,1,0],[0,0,0,2,0,0,0],[0,0,0,0,1,1,0],[0,0,1,1,0,0,0]]`
  Overall;
  `(T_(ji)) = [[-1,-1,1,0,0,1,0],[0,0,-1,2,0,0,-1],[-1,0,0,-1,1,1,0],[0,-1,1,1,-1,0,0]]`

• With the help of these matrices, we can be able to calculate the net reaction rate for each and every reaction, `q_i`, which is given by the difference in the reaction rate of forward reaction to the reaction rate of reverse reaction.
  `q_i = q_(foward) - q_(reverse)`
  Which implies;
  `q_1 = k_(f1)[CO][O_2] - k_(r1)[CO_2][O]`
  `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]`

• And, with the help of the above equations, we can determine the net production rate of each and every species as follows;
  `omega_j = sum_(i=1)^n(T_(ji)q_i)`
  where `omega_j` is the production rate of each species and
  `n` is the total number of reactions

• Finally, by substituting the values for eacha nd every species in every reactions, we can derive a set of equations for all species which can be interpreted into the below matrix form;
  `((omega_1),(omega_2),(omega_3),(omega_4),(omega_5),(omega_6),(omega_7)) = [[-1,0,-1,0],[-1,0,0,-1],[1,-1,0,1],[0,2,-1,1],[0,0,1,-1],[1,0,1,0],[0,-1,0,0]]((q_1),(q_2),(q_3),(q_4))`

Results:

Thus, the study to derive the reaction rate of ODEs of a simple reaction mechanism & production of each species for the simple reaction mechanism was conducted successfully.