Compact Notation Derivation for a simple Mechanism

Aim: To derive the reaction rate ODEs and production for each species in the0 given reaction mechanism.

Objective

The objective in this work is to determine the net reaction rate for all four reactions using reactant and product matrices. After that, the production of each species is determined with respect to net reaction rate using matrix notation.

                                                           Reaction Mechanism

Introduction:

A reaction mechanism in chemistry is the step-by-step sequence of elementary reactions that results in total chemical change. A chemical mechanism is a theoretical hypothesis that attempts to define in detail what happens at each stage of a chemical process. In most circumstances, the specific processes of a response are not visible. The proposed mechanism was chosen because it is thermodynamically viable and has experimental support in isolated intermediates (see next section) or other quantitative and qualitative reaction features. It also specifies which bonds are broken in each reactive intermediate, activated complex, and transition state (and in what order), and the ties that are made (and in what order). A thorough mechanism must also explain why the reactants and catalyst were chosen, the stereochemistry seen in reactants and products, the products generated, and the quantities of each.

Theory

Reaction Mechanism: Mechanism for any reaction is defined as collection of number of processes that explains the overall reaction.

• This is the actual method of completion of reaction as it gives the number of broken bonds and the number of steps involved.
• In a mechanism the positions of all atoms (stereochemistry), role of solvent molecules and the energy of the system is specified.
• It also helps in describing the reaction intermediate, activated complex and transition state involved in the whole reaction.
• A mechanism accounts for all the reactants used, the function of a catalyst, all products formed and their amount.
• From a mechanism the rate law can be deduced

Chemical Kinetics: Information about the mechanism of a reaction is often provided by the use of chemical kinetics to determine the rate equation and the reaction order in each reactant.

Consider the following reaction for example:

CO + NO2 → CO2 + NO

In this case, experiments have determined that this reaction takes place according to the rate law  `r=k[NO2]^2`  . This form suggests that the rate-determining step is a reaction between two molecules of NO2. A possible mechanism for the overall reaction that explains the rate law is:

2 NO2 → NO3 + NO (slow)

NO3 + CO → NO2 + CO2 (fast)

Each step is called an elementary step, and each has its own rate law and molecularity. The elementary steps should add up to the original reaction. (Meaning, if we were to cancel out all the molecules that appear on both sides of the reaction, we would be left with the original reaction.)

When determining the overall rate law for a reaction, the slowest step is the step that determines the reaction rate. Because the first step (in the above reaction) is the slowest step, it is the rate-determining step. Because it involves the collision of two NO2 molecules, it is a bimolecular reaction with a rate r which obeys the rate law `r= k[NO_2(T)]^2`

Other reactions may have mechanisms of several consecutive steps. In organic chemistry, the reaction mechanism for the benzoin condensation, put forward in 1903 by A. J. Lapworth, was one of the first proposed reaction mechanisms.

Mechanism

The species are considered in the order of their appearance in the reaction mechanism.

The reactions are taken in the same order as displayed in the reaction mechanism.

• Definition of stoichiometric coefficient matrices for reactants and products

We have:

`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^''=[[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]]`

Where `v_ji^'` and `v_ji^''` are stoichiometric coefficient matrices reactants and products respectively. The indices “i” and “j” help us read the matrices as number of moles of Sj in ith reaction.

The net stoichiometric coefficient matrix is obtained as follows:

`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]]`

• Net reaction rate ()

The net reaction rate is defined as follows:

`q_i=k_fi ∏_(j=1)^N[X_j]^(v_ji^' ) -k_ri ∏_(j=1)^N[X_j]^(v_ji^'' )`

In the above equation, the different quantities are:

`k_fi` is the forward reaction rate

`k_ri` is the backward reaction rate

`[X_j]^(v_ji^' )`  is the concentration of species per unit volume of reactants

`[X_j]^(v_ji^'' )` is the concentration of species per unit volume of reactants

  N is the total number of reactions

Considering the given reaction mechanism, the net reaction rate for each reaction is as follows:

1. `q_1=k_f1 [CO][O2]-k_r1 [CO2][O]`
2. `q_2=k_f2 [O][H2O]-k_r2 [OH]^2`
3. `q_3=k_f3 [CO][OH]-k_r3 [CO2][H]`
4. `q_4=k_f4 [H][O2]-k_r4 [OH][O]`

• Production rate (`w_j`)

`w_j=∑_(i=1)^Lv_ji q_i =d/dt [X_j ]`

“j” denotes the species index and L is the total number of reactions

By replacing the values of “i” we have:

`w_j=v_j1 q_1+v_j2 q_2+v_j3 q_3+v_j4 q_4`

1. Concentration of CO

 `d/dt [CO]=-(k_f1 [CO][O2]-k_r1 [CO2][O])-(k_f3 [CO][OH]-k_r3 [CO2][H])`

2. Concentration of O2

 `d/dt [O2]=-(k_f1 [CO][O2]-k_r1 [CO2][O])-(k_f4 [H][O2]-k_r4 [OH][O])`

3. Concentration of CO2

 `d/dt [CO2]= k_f1 [CO][O2]-k_r1 [CO2][O]+k_f3 [CO][OH]-k_r3 [CO2][H]`

4. Concentration O

 `d/dt [O]=k_f1 [CO][O2]-k_r1 [CO2][O]-(q_2=k_f2 [O][H2O]-k_r2 [OH]^2 )+k_f4 [H][O2]-k_r4 [OH][O]`

5. Concentration of H2O

 `d/dt [H2O]=-(k_f2 [O][H2O]-k_r2 [OH]^2 )`

6. Concentration of OH

 `d/dt [OH]=2(k_f2 [O][H2O]-k_r2 [OH]^2 )-(k_f3 [CO][OH]-k_r3 [CO2][H])+k_f4 [H][O2]-k_r4 [OH][O]`

7. Concentration H

 `d/dt [H]=k_f3 [CO][OH]-k_r3 [CO2][H]-(k_f4 [H][O2]-k_r4 [OH][O])`

CONCLUSION

Finally, the concept of reaction mechanism is explained and the reaction rate ODEs and production are derived.

REEFERENCES

• https://en.wikipedia.org/wiki/Reaction_mechanism#:~:text