Week 5.1 - Compact Notation Derivation for a simple Mechanism
AIM:-
To derive the reaction rate ODEs & production of each species for the simple reaction mechanism given below.
CO2+O2<=>CO2+O
O+H2O<=>OH+OH
CO+OH<=>CO2+H
H+O2<=>OH+O
EXPLANATION :-
Reaction rate or the rate of reaction is the speed at which a rate chemical reaction takes place. We define reaction rate as the speed at which reactants are converted into products Chemical Kinematics is the part of physical chemistry that studies reaction rates.
These rates can be modelled as an ODE as a function of the concentration of the species involved in the reaction.The rate of product formation is actually the rate of depletion of the reacting species. Hence, we arrived at a system of coupled ODE's which can be solved by various integration techniques. This can be feasible only if there are a few reaction rate equations, but in reality reactions are complex in narture because of many intermediate products and hence, multuple ways to complete a given reaction. Thus, the number of ODE's can span upto thousands, hence we use computational methods for this.
Compact Notations:-
Let us efine indices of the reactants and products species and the reactions
j Species:- CO, O2,CO2,O,H2O,OH & H
i Reaction:-
CO2+O2<=>CO2+O
O+H2O<=>OH+OH
CO+OH<=>CO2+H
H+O2<=>OH+O
where, j and i refers to the index of species and index of reactions respectively
For the mechanism, we have 7 species and 4 reactions. We define `Vunderset(ji)'`, which is the coefficient matrix where each element denotes the coefficient of the species on the reactant side in the reactions
`Vunderset(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 `V''underset(ji)` is defined , which is having coefficients of species on the product side in the reactions.
`V''underset(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]]`
Next, we form `Vunderset(ji)`= `V'underset(ji)`-`V''underset(ji)`
`Vunderset(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]]`
The net rate of reaction (q) for each reaction is given as `q_t=k_(fi)*N_prod(j=1)[X_j]overset(v'_(ji))`-`k_(ri)*N_prod(j=1)[X_j]overset(v''_(ji))`
where, N is the number of species in the reaction mechanism
`q_1=k_(f1)[CO]^1[O_2]^1[CO_2]^0[O]^0[H2O]^0[OH]^0[H]^0-k_(r1)[CO]^1[O_2]^1[CO_2]^0[O]^0[H2O]^0[OH]^0[H]^0`
hence, similarly we have,
`q_1=k_(f1)[CO][O_2]-k_(r1)[CO_2][O]`
`q_2=K_(f2)[O][H_2O]-k_(r2)[OH]^2`
`q_3=k_(f3)[CO][OH]-k_(r3)[CO_2][H]`
`q_4=k_(f4)[O_2][H]-k_(r4)[O][OH]`
Now, as we have reaction rates, the production rates can be determined as:
`omega_j=sum_(i=1)^Lv_(ji)*q_i`
where, L is the number of reactions in the mechanism. Hence
`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`
The production rate is also rate is also teh sum of the change of species with time
`omega_j=(d[X_j])/dt`
Hence we can form the rate of change of each species as a matrix
`((d[CO]/dt),(d[O_2]/dt),(d[CO_2]/dt),(d[O]/dt),(d[H_2O]/dt),(d[OH]/dt),(d[H]/dt))=((omega_1),(omega_2(,(omega_3),(omega_4),(omega_5),(omega_6),(omega_7))`
`{d[CO]/dt}=k_{r1}[CO_2][O]-k_{f1}[CO][O_2]+k_{r3}[CO_2][H]-k_{f3}[CO][OH]]`
`{d[O_2]/dt}=k_{r1}[CO_2][O]-k_(f1}[CO][O_2]+k_{r3}[CO_2][H]-k_{f3}[CO][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_{f4}[O_2][H]-k_{r4}[O][OH]`
`{d[H_2O]/dt}=k_{r2}[OH]^2-k_{f2}[O][H_2O]`
`{d[OH]/dt}=2k_{f2}[O][H_2O]-2k_{r2}[OH]^2+k_{r3}[CO_2][H]-k_{f3}[CO][OH]+k_{f4}[O_2][H]-k_{r4}[O][OH]`
`{d[H]/dt}=k_{f3}[CO][OH]-k_{r3}[CO_2][H]+k_{r4}[O][OH]-k_{f4}[O_2][H]`
Hence, a system of coupled ODE's are formed which can be solved by coding any numerical technique.