Week 5.1 - Compact Notation Derivation for a simple Mechanism
Compact Notation Derivation for a simple Mechanism
Aim:
Derive the reaction rate ODEs of a simple reaction mechanism using compact notation.
Description:
Chemical kinetics: Chemical kinetics is also known as reaction kinetics. It is the branch of physical chemistry that is concerned with understanding the rates of chemical reactions. it concerns how rates of chemical reactions are measured and predicted, and how reaction-rate data can be used to deduce probable reaction mechanisms. It is to be contrasted with thermodynamics, which deals with the direction in which a process occurs but in itself tells nothing about its rate. Chemical kinetics includes investigations of how experimental conditions influence the speed of a chemical reaction and yield information about the reaction mechanism and transition states, as well as the construction of mathematical models that also can describe the characteristics of a chemical reaction.
Reaction rate:
The reaction rate or rate of reaction is the speed at which a chemical reaction takes place. Reaction rates can vary dramatically. For most reactions, the rate decreases as the reaction proceeds. A reaction's rate can be determined by measuring the changes in concentration over time.
Reaction Mechanism:
A reaction mechanism is the step-by-step sequence of elementary reactions by which overall chemical change occurs.
`CO + O_2 ⇌ CO_2 + O`
`O + H_2O ⇌ OH + OH`
`CO + OH ⇌ CO_2 + H`
`H + O_2 ⇌ OH + O`
Compact notation is a useful tool to describe the properties of each element of a set, rather than writing out all elements of a set.- Reaction 1 :- `CO + O_2 ⇌ CO_2 + O`
- Reaction 2 :- `O + H_2O ⇌ OH + OH`
- Reaction 3 :- `CO + OH ⇌ CO_2 + H`
- Reaction 4 :- `H + O_2 ⇌ OH + O`
- `CO`
- `O_2`
- `CO_2`
- `O`
- `H_2O`
- `OH`
- `H`
There is a total of 7 species participating in the reaction.
Reactants stoichiometric coefficient matrix (`nu_(ij)^'`):-
`nu_(ij)^' = [(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)]`
Products stoichiometric coefficient matrix (`nu_(ij)^('')`):
- `omega_1 = -q_1-q_3`
- `omega_2 = -q_1-q_4`
- `omega_3 = q_1+q_3`
- `omega_4 = q_1-q_2+q_4`
- `omega_5 = -q_2`
- `omega_6 = 2q_2-q_3+q_4`
- `omega_7 = q_3-q_4`
ODE'S
- `{d[CO]}/{dt} = - k_{f1}[CO][O_2] + k_{r1}[CO_2][O] - k_{f3}[CO][OH] + k_{r3}[CO_2][H]`
- `{d[O_2]}/{dt} =- k_{f1}[CO][O_2] + k_{r1}[CO_2][O] - k_{f4}[O_2][H] + k_{r4}[O][OH]`
- `{d[CO_2]}/{dt} = k_{f1}[CO][O_2] - k_{r1}[CO_2][O] + k_{f3}[CO][OH] - k_{r3}[CO_2][H]`
- `{d[O]}/{dt} = k_{f1}[CO][O_2]-k_{r1}[CO_2][O] - k_{f2}[O][H_2O] + k_{r2}[OH]^2+ k_{f4}[O_2][H] - k_{r4}[O][OH]`
- `{d[H_2O]}/{dt} = - k_{f2}[O][H_2O] + k_{r2}[OH]^2`
- `{d[OH]}/{dt} = 2k_{f2}[O][H_2O] - 2k_{r2}[OH]^2 - k_{f3}[CO][OH] + k_{r3}[CO_2][H] + k_{f4}[O_2][H] - k_{r4}[O][OH]`
- `{d[H]}/{dt} = k_{f3}[CO][OH] - k_{r3}[CO_2][H] - k_{f4}[O_2][H] + k_{r4}[O][OH]`