Compact Notation Derivation for a simple Mechanism
Objective:- To derive compact notations for chemical reaction mechanism
Rate of reaction:
Reaction rate or a rate of reaction is a term generally used in chemistry which defines the speed at which a chemical reaction proceeds. It is often expressed in terms of either the concentration (amount per unit volume) of a product that is formed in a unit of time or the concentration of a reactant that is consumed in a unit of time. 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 ODEs which can be solved by various integration techniques.
Alternatively, it may be defined in terms of the amounts of the reactants consumed or products formed in a unit of time. This reaction rates varies very drastically and dependent upon the nature of reacting substances. So, it is difficult to compute reaction rates. While using ODE’s, due to number of intermediate reaction mechanism and complex nature it is confusing and hectic to solve number of ODE’s (sometime more than hundreds of ODE’s). So, we can use computational methods for solving such reactions. In order to solve this (ODE's) reaction using computational method, we have derived some compact notations which are given below.
Reaction mechanism is as follows-
`CO +O_2underset(k_(r1))(overset(k_(f1))(<=>))CO_2+O`
`O +H_2O underset(k_(r2))(overset(k_(f2))(<=>)) OH+OH`
`CO +OH underset(k_(r3))(overset(k_(f3))(<=>)) CO_2+H`
`H +O_2 underset(k_(r4))(overset(k_(f4))(<=>)) OH+O`
All of the above reactions from mechanism are reversible reactions so every reaction has two reaction rates. Which are forward reaction rate and reverse reaction rate.
Here,
`k_(fi)=`Forward reaction rate for `i^(th)` reaction
`k_(ri)=`Reverse reaction rate for `i^(th)` reaction
Compact notations for various species are given as follows,
Here subscript 'j' denotes species index.
|
Species index 'j' |
1 | 2 | 3 | 4 | 5 | 6 | 7 |
| Species | `CO` | `O_2` | `CO_2` | `O` | `H_2O` | `OH` | `H` |
Compact notations for various reactions are given as follows,
Here subscript 'i' denotes reaction index.
|
Reaction index 'i' |
Reaction |
| 1 |
R1:- `CO +O_2 <=>CO_2+O` |
| 2 | R2:- `O +H_2O <=>OH+OH` |
| 3 |
R3:- `CO +OH <=>CO_2+H` |
| 4 | R4:- `H +O_2 <=>OH+O` |
`V_(ji)'="Stoichiometric coefficients of reactants species"`
Reactant matrix is given below
`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)''="Stoichiometric coefficients of product species"`
Product matrix is given below
`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]]`
So now calculate net stochiometric coefficient matrix,
`V_(ji)=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]]`
As we know reaction rate 'q' for `i^(th)` can be calculated by following formula,
`q_i=k_(fi)prod_(j=1)^N[X_j]^(V_(ji)') - k_(ri)prod_(j=1)^N[X_j]^(V_(ji)'')`
Where,
`[X_j]^(V_(ji)'` = Concentration of species per unit volume of reactant
`[X_j]^(V_(ji)''` = Concentration of species per unit volume of product
By using above formula for net reaction rate we get,
`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)[H][O_2]-k_(r4)[OH][O]`
Now, the net production rate of species is given by-
`omega_j=sum_(i=1)^L V_(ji)q_i`
Where,
`omega_j=`Production rate for `j^(th)` species
L= Number of reactions (Here L=4)
In matrix form it is calculated as
`[[omega_1],[omega_2],[omega_3],[omega_4],[omega_5],[omega_6],[omega_7]]=[[-1,0,-1,0],[-1,0,0,-1],[1,0,1,0],[1,-1,0,1],[0,-1,0,0],[0,2,-1,1],[0,0,1,-1]][[q_1],[q_2],[q_3],[q_4]]`
Hence, we get production rate for species as follows-
`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`
Also we know,
`omega_j=(d[X_j])/(dt)`
Therefore it can be written in matrix format as-
`[[(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]]`
Reaction rate ordinary differential equations (ODE's) for species of reaction mechanism are as follows-
For `CO:`
`(d[CO])/dt=-k_(f1)[CO][O_2]+k_(r1)[CO_2][O]-k_(f3)[CO][OH]+k_(r3)[CO_2][H]`
For `O_2:`
`(d[O_2])/dt=-k_(f1)[CO][O_2]+k_(r1)[CO_2][O]-k_(f4)[H][O_2]+k_(r4)[OH][O]`
For `CO_2:`
`(d[CO_2])/dt=k_(f1)[CO][O_2]-k_(r1)[CO_2][O]+k_(f3)[CO][OH]-k_(r3)[CO_2][H]`
For `O:`
`(d[O])/dt=-k_(f1)[CO][O_2]+k_(r1)[CO_2][O]-k_(f2)[O][H_2O]+k_(r2)[OH]^2+k_(f4)[H][O_2]-k_(r4)[OH][O]`
For `H_2O:`
`(d[H_2O])/dt=-k_(f2)[O][H_2O]+k_(r2)[OH]^2`
For `OH:`
`(d[OH])/dt=2(k_(f2)[O][H_2O]-k_(r2)[OH]^2)-k_(f3)[CO][OH]+k_(r3)[CO_2][H]+k_(f4)[H][O_2]-k_(r4)[OH][O]`
For `H:`
`(d[H])/dt=k_(f3)[CO][OH]-k_(r3)[CO_2][H]-k_(f4)[H][O_2]+k_(r4)[OH][O]`
By solving above ODE's using coding approach we are able to compute rate of reaction quickly without any hectic efforts.