Week 5.1 - Compact Notation Derivation for a simple Mechanism

                                               Compact Notation Derivation for a Simple Mechanism

Objective:

To derive the reaction rate ODEs of a simple reaction mechanism learnt in the chemical kinetics lectures.

Reaction Mechanism                 i

CO + O2 ⇌  CO2 + O              R1

O + H2o ⇌  OH + OH              R2

CO + OH ⇌  CO2 + H              R3

H + O2    ⇌  OH + O               R4

j=species

kf=forward reaction coefficient   kr=reverse reaction coefficient

`d[[CO] ]/dt =-kf1[CO][o2]+kr1[CO2][O]-kf3[CO][OH]+kr3[CO2][H]`

`d[[O2]]/dt=-kf1[CO][O2]+kr1[CO2][O]-kf4[h][O2]+kr3[OH][O]`

`d[[CO2]]/dt=kf1[CO][O2]-kr1[CO2][O]+kf3[CO][OH]-kr3[CO2][H]`

`d[[O]]/dt=kf1[CO][O2]-kr1[CO2][O]-kf2[O][H2O]+kr2[OH][OH]+kf4[H][O2]-kr4[OH][O]`

`d[[H2O]]/dt=-kf2[O][H2O]+kr2[OH][OH]`

`d[[OH]]/dt=2*(kf2[O][H2O]-kr2[OH][OH])-kf3[CO][OH]+kr3[CO2][H]+kf4[H][O2]-kr4[OH][O]`

`d[[H]]/dt=kf3[CO][Oh]-kr3[CO2][H]-kf4[H][O2]+kr4[OH][O]`

Defining the reactant matrix

vji'=`[[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]]`

product matrix

vji''=`[[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]]`

net reaction rate

vji=vji''-vji=`[[-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]]`

we have 

`i=(kfi`j=1 prod^N [Xj]^(Vji' ))-(kri j=1 prod^N[Xj]^(Vji'')`

there fore

from i 1 to 4

`q1=kfq[CO]^1[O2]^1-kr1[CO2]^1[O]^1`

`q2=kf2[O]^1[H2O]^1-kr2[OH]^2`

`q3=kf3[CO]^1[OH]^1-kr3[CO2]^1[H]^1`

`q4=kf4[O2]^1[H]^1-kr4[O]^1[OH]^1`

since for each species there is an production rate

then the rate of concentration change of a particular species is given by

`Wj=d[[Xj]]/dt`

by summing up

`Wj=sum^L(i=1)vj^tqi`

where L is the number of reaction

we will get a stiff ODE system

`((w1),(w2),(w3),(w4),(w5),(w6),(w7))=[[-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]]*((q1),(q2),(q3),(q4))`

w1=`d[[CO]]/dt=-q1-q3`

`d[[CO] ]/dt =-kf1[CO]^1[O2]^1+kr1[CO2]^1[O]^1-kf3[CO]^1[OH]^1+kr3[CO2]^1[H]^1`

w2=`d[[O2]]/dt=-q1-q4`

`d[[O2]]/dt=-kf1[CO]^1[O2]^1+kr1[CO2]^1[O]^1-kf4[H]^1[O2]^1+kr3[OH]^1[O]^1`

w3=`d[[Co2]]/dt=q1+q3`

`d[[CO2]]/dt=kf1[CO]^1[O2]^1-kr1[CO2]^1[O]^1+kf3[CO]^1[OH]^1-kr3[CO2]^1[H]^1`

w4=`d[[O]]/dt=q1-q2+q4`

`d[[O]]/dt=kf1[CO]^1[O2]^1-kr1[CO2]^1[O]^1-kf2[O]^1[H2O]^1+kr2[OH]^2+kf4[H]^1[O2]^1-kr4[OH]^1[O]^1`

w5=`d[[H2O]]/dt=-q2`

`d[[H2O]]/dt=-kf2[O]^1[H2O]^1+kr2[OH]^2`

w6=`d[[OH]]/dt=2q2-q3+q4`

`d[[OH]]/dt=2*(kf2[O]^1[H2O]^1-kr2[OH]^2)-kf3[CO]^1[OH]^1+kr3[CO2]^1[H]^1+kf4[H]^1[O2]^1-kr4[OH]^1[O]^1`

w7=`d[[H]]/dt=q3-q4`

`d[[H]]/dt=kf3[CO]^1[OH]^1-kr3[CO2]^1[H]^1-kf4[H]^1[O2]^1+kr4[OH]^1[O]^1`

Conclusion;

Since this ODE is a stiff ODE we need implicit scheme to solve while explicit scheme like euler won't work out

By following this method we can predict  reaction pathways by deriving the ODE Reaction rate