Week 5.1 - Compact Notation Derivation for a simple Mechanism

AIM: To obtain the reaction rate ordinary differential equations of a simple reaction mechanism using compact notation.

Introduction:

The reaction rate of a chemical reaction is the pace at which a reaction takes place. The reaction rate depends on several factors such as temperature, pressure, type of reaction, and the concentration of a species in the reaction. The Rate law is utilized in order to obtain the rate of the equation, with respect to the concentration of different reacting species.

Determining the reaction rate of a chemical reaction is important when it comes to optimizing a process that involves certain reactions. The speed of each of the reactions can be calculated, and then observed in detail. Having this information will allow us to improve the process, or help in observing the shortcomings of a system.

Theory:

Reaction mechanisms:

A complete,balanced chemical reaction appears to be direct and simple. However, the reaction seen is the final condensation of multiple small reactions, known as elementary reactions. Hence, a complete chemical reaction is complex in nature,and can be referred to as a complex reaction. Elementary reactions can be slow or fast in nature, and the products obtained from these reactions may or may not be present in the final complex reaction. These species in the products are known as intermediates. Some products obtained from an elementary reaction may be used as a reactant in the next or a reaction taking place later, hence forming a different product. Some species are catalysts for a reaction, and are not present in the final product. But, they are necessary for the reactions to take place.

A reaction mechanism is comprised of multiple elementary reactions.

Compact notations:

Compact notations are used in order express the reaction mechanism in a more systematic manner and can be used to create a mathematical expression for use in a computer program.

Formulae and notations used:

• j = species type
• i = reaction number
• `v_(ji) ' =`Stoichiometric coefficient matrix of forward reaction
• `v_(ji)'' =`Stoichiometric coefficient matrix of reverse reaction
• `q_i = kf_i*(prod_(j=1)^N(X_j)^(v_(ji)')) - kr_i*(prod_(j=1)^N(X_j)^(v_(ji)''))`

MECHANISM:

The following notations have been assigned for the species and reactions respectively:

(A) Reactions(i) :

Reaction number (i)	Reaction
1	`CO + O_2 rightarrow CO_2 + O`
2	`O + H_2O rightarrow OH + OH`
3	`CO+OH rightarrow CO_2 + H`
4	`H + O_2 rightarrow OH + O`

(B) Species (j):

Species number (j)	Species
1	`CO_2`
2	`O_2`
3	O
4	`H_2O`
5	CO
6	OH
7	H

Stoichiometric coefficient matrix of forward reaction:

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

Net reaction rate of the reaction:

`q_i = kf_i*(prod_(j=1)^N(X_j)^(v_(ji)')) - kr_i*(prod_(j=1)^N(X_j)^(v_(ji)''))`

`kf_i = text(Rate constant for forward reaction)`

`kr_i = text(Rate constant for reverse reaction)`

`X_j = text(Species concentration)`

The reverse reaction portion of the equation will be neglected, as there are no reverse reactions taking place, and all of the elementary reactions are forward reactions.

Hence, the reaction rate for all of the elementary reactions are:

`q_i = kf_i*(prod_(j=1)^N(X_j)^(v_(ji)'))`

`q_1 = k_(f1)*[CO_2]^(v_(11)')*[O_2]^(v_(21)')*[O]^(v_(31)')*[H_2O]^(v_(41)')*[CO]^(v_(51)')*[OH]^(v_(61)')*[H]^(v_(71)')`

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

`q_1 = k_(f1)*[O_2]^(1)*[CO]^(1)`

Similarly,

`q_2 = k_(f2)*[CO_2]^(v_(12)')*[O_2]^(v_(22)')*[O]^(v_(32)')*[H_2O]^(v_(42)')*[CO]^(v_(52)')*[OH]^(v_(62)')*[H]^(v_(72)')`

`q_2 = k_(f2)*[CO_2]^(0)*[O_2]^(0)*[O]^(1)*[H_2O]^(1)*[CO]^(0)*[OH]^(0)*[H]^(0)`

`q_2 = k_(f2)*[O]^(1)*[H_2O]^(1)`

`q_3 = k_(f3)*[CO_2]^(v_(13)')*[O_2]^(v_(23)')*[O]^(v_(33)')*[H_2O]^(v_(43)')*[CO]^(v_(53)')*[OH]^(v_(63)')*[H]^(v_(73)')`

`q_3 = k_(f3)*[CO_2]^(0)*[O_2]^(0)*[O]^(0)*[H_2O]^(0)*[CO]^(1)*[OH]^(1)*[H]^(0)`

`q_3 = k_(f3)*[CO]^(1)*[OH]^(1)`

`q_4 = k_(f3)*[CO_2]^(v_(14)')*[O_2]^(v_(24)')*[O]^(v_(34)')*[H_2O]^(v_(44)')*[CO]^(v_(54)')*[OH]^(v_(64)')*[H]^(v_(74)')`

`q_4 = k_(f4)*[CO_2]^(0)*[O_2]^(1)*[O]^(0)*[H_2O]^(0)*[CO]^(0)*[OH]^(0)*[H]^(1)`

`q_4 = k_(f4)*[H]^(1)*[O_2]^(1)`