Week 5.1 - Compact Notation Derivation for a simple Mechanism
Aim :
To derive the reaction rate of ODE's of a simple reaction mechanism using compact notation .
Reaction Mechanism
- CO + O2 ⇌ CO2 + O
- O + H2O ⇌ OH + OH
- CO + OH ⇌ CO2 + H
- H + O2 ⇌ OH + O
Section I . Species list & reaction notation :
| j | Species |
| 1 |
`CO` |
| 2 |
`O_2` |
| 3 |
`CO_2` |
| 4 |
`O` |
| 5 |
`H_2O` |
| 6 |
`OH` |
| 7 |
`H` |
| i | Reaction number |
| 1 |
`R1 : CO + O_2 harr_(K_(r1))^(K_(f1)) CO_2 + O` |
| 2 |
`R2 : O + H_2O harr_(K_(r2))^(K_(f2)) OH + OH` |
| 3 |
`R3 : CO + OH harr_(K_(r3))^(K_(f3)) CO_2 + H` |
| 4 |
`R4 : H + O_2 harr_(K_(r4))^(K_(f4)) OH + O` |
Section II . Rate of change of each species :
- Species mentioned within the square brackets signify the concentration .
- `K_(fi)` = Rate of forward reaction (will be found as a factor with reactant concentration)
`K_(ri)` = Rate of forward reaction (will be found as a factor with product concentration) - The negative sign in front of the rate of reaction signifies the reduction of the respective species concentration.
Species 1 :
`d/dt[CO]=-k_(f1)[CO][O_2]+k_(r1)[CO_2][O]-k_(f3)[CO][OH]+k_(r3)[CO_2][H]`
Species 2 :
`d/dt[O_2]=-k_(f1)[CO][O_2]+k_(r1)[CO_2][O]-k_(f3)[H][O_2]+k_(r3)[OH][O]`
Species 3 :
`d/dt[CO_2]=k_(f1)[CO][O_2]-k_(r1)[CO_2][O]+k_(f3)[CO][OH]-k_(r3)[CO_2][H]`
Species 4 :
`d/dt[O]=k_(f1)[CO][O_2]-k_(r1)[CO_2][O]+k_(f2)[O][H_2O]-k_(r3)[OH][OH]+k_(f4)[H][O_2]-k_(r4)[OH][O]`
Species 5 :
`d/dt[H_2O]=k_(f2)[O][H_2O]-k_(r2)[OH][OH]`
Species 6 :
`d/dt[OH]=k_(f2)[O][H_2O]-k_(r2)[OH][OH]-k_(f3)[CO][OH]+k_(r3)[CO_2][H]+k_(f4)[H][O_2]+k_(r4)[OH][O]`
Species 7 :
`d/dt[O]=k_(f3)[CO][OH]-k_(r3)[CO_2(r4)[OH][OH]+k_(f4)[H][O_2]-k_(r4)[OH][O]`
The equations above are "COUPLED ORDINARY DIFFERENTIAL EQUATIONS"
Section III . Reactants & Products :
Matrix `v_(ji)^'` will contain the stoichiometric data of the reactants alone
Matrix `v_(ji)^('')` will contain the stoichiometric data from products alone
`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))`
`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))`
Section IV . Skew `(q_i)`:
If `q_i` is positive ,reaction is taking place in the forward direction .
If `q_i` is negative ,reaction is taking place in the reverse direction .
`q_i=k_(fi)Pi_(j=1)^N [X_j]^(v_(ji)^')-k_(ri)Pi_(j=1)^N [X_j]^(v_(ji)^(''))`
`k_(fi)`= Rate of forward reaction
`k_(ri)`= Rate of reverse reaction
`[X_j]`= Concentration per unit volume of species .
`q_1=k_(f1)[X_1]^(v_11^') [X_2]^(v_12^') [X_3]^(v_13^') [X_4]^(v_14^') [X_5]^(v_15^') [X_6]^(v_16^') [X_7]^(v_17^')-k_(r1)[X_1]^(v_11^('')) [X_2]^(v_12^('')) [X_3]^(v_13^('')) [X_4]^(v_14^('')) [X_5]^(v_15^('')) [X_6]^(v_16^('')) [X_7]^(v_17^(''))`
`q_1=k_(f1)[CO]^(1) [O_2]^(1) [CO_2]^(0) [O]^(0) [H_2O]^(0) [OH]^(0) [H]^(0)-k_(r1)[CO]^(0) [O_2]^(0) [CO_2]^(1) [O]^(1) [H_2O]^(0) [OH]^(0) [H]^(0)`
`q_1=k_(f1)[CO][O_2]-k_(r1)[CO_2][O]`
Similarly ,
`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]`
Section V . Production rate for species `(omega_f)`:
`omega_f=sum_(i=1)^(L)v_(ji)*q_i`
where,
- `omega_f`= Production rate for each and each species.
- `v_(ji)=v_(ji)^('')-v_(ji)^(')`
- L = Number of reactions = 4
- j = Denotes species number(Column index)
- i = Denotes reaction number(Row index)
- `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))`
For species 1 : `[CO]`
`omega_1=sum_(i=1)^4 v_(1i)q_i`
`omega_1= v_(11)*q_1+v_(12)*q_2+v_(13)*q_3+v_(14)*q_4`
`here, v_(11)=-1; v_(12)=0;v_(13)=-1;v_(14)=0;`
`omega_1=-(k_(f1)[CO][O_2]-k_(r1)[CO_2][O])-(k_(f3)[CO][OH]-k_(r3)[CO_2][H])`
For species 2 : `[O_2]`
`omega_2=sum_(i=1)^4 v_(2i)q_i`
`omega_2= v_(21)*q_1+v_(22)*q_2+v_(23)*q_3+v_(24)*q_4`
`here, v_(21)=-1 ; v_(22)=0 ; v_(23)=0 ; v_(24)=-1 ;`
`omega_2=-(k_(f1)[CO][O_2]-k_(r1)[CO_2][O])-(k_(f4)[O_2][H]-k_(r4)[O][OH])`
For species 3 : `[CO_2]`
`omega_3=sum_(i=1)^4 v_(3i)q_i`
`omega_3= v_(31)*q_1+v_(32)*q_2+v_(33)*q_3+v_(34)*q_4`
`here, v_(31)=1 ; v_(32)=0 ; v_(33)=1 ; v_(34)=0 ;`
`omega_3=(k_(f1)[CO][O_2]-k_(r1)[CO_2][O])+(k_(f3)[CO][OH]-k_(r3)[CO_2][H])`
For species 4 : `[O]`
`omega_4=sum_(i=1)^4 v_(4i)q_i`
`omega_4= v_(41)*q_1+v_(42)*q_2+v_(43)*q_3+v_(44)*q_4`
`here, v_(41)=1 ; v_(42)=-1 ; v_(43)=1 ; v_(44)=1 ;`
`omega_4=(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])`
For species 5 : `[H_2O]`
`omega_5=sum_(i=1)^4 v_(5i)q_i`
`omega_5= v_(51)*q_1+v_(52)*q_2+v_(53)*q_3+v_(54)*q_4`
`here, v_(51)=0 ; v_(52)=-1 ; v_(53)=0 ; v_(54)=0 ;`
`omega_5=-(k_(f2)[O][H_2O]-k_(r2)[OH]^2)`
For species 6 : `[OH]`
`omega_6=sum_(i=1)^4 v_(6i)q_i`
`omega_6= v_(61)*q_1+v_(62)*q_2+v_(63)*q_3+v_(64)*q_4`
`here, v_(61)=0 ; v_(62)=2 ; v_(63)=-1 ; v_(64)=1 ;`
`omega_6=2*(k_(f2)[O][H_2O]-k_(r2)[OH]^2)-(k_(f3)[CO][OH]-k_(r3)[CO_2][H])+(k_(f4)[O_2][H]-k_(r4)[O][OH])`
For species 7 : `[H]`
`omega_7=sum_(i=1)^4 v_(7i)q_i`
`omega_7= v_(71)*q_1+v_(72)*q_2+v_(73)*q_3+v_(74)*q_4`
`here, v_(71)=0 ; v_(72)=0 ; v_(73)=1 ; v_(74)=-1 ;`
`omega_7=(k_(f3)[CO][OH]-k_(r3)[CO_2][H])-(k_(f4)[O_2][H]-k_(r4)[O][OH])`
Since we know that ,
`omega=d/dt[X_j]`= Net production rate of particular species .
Finally we arrive at the following equation ,
`[(omega_1),(omega_2),(omega_3),(omega_4),(omega_5),(omega_6),(omega_7)]=((d/dt[CO]),(d/dt[O_2]),(d/dt[CO_2]),(d/dt[O]),(d/dt[H_2O]),(d/dt[OH]),(d/dt[H]))`
The above equations are same as the one found in Section II,
When we look at the above matrix we can see that we have obtained the rate of change of each species using a matrix notation .
Now this can easily be made into a computer program .