Compact Notation Derivation for a simple Mechanism
Aim:
To derive the reaction rate ODEs of given simple reaction mechanism.
Rates of reactions for multistep Mechanism:
We have a four step mechanism. First, we will be forming the compact notation for the reaction rates of mechanisms.
Reactions:
`CO + O2 -> CO2 + O`
`O + H2O-> OH + OH`
`CO + OH -> CO2 + H`
`H + O2 -> OH + O`
We will be using the compact notation where j will depict the number of species and i will depict the number of reactions.
Compact Notation:
j Species i Reactions
1 `CO` 1 R1
2 `O2` 2 R2
3 `CO2` 3 R3
4 `O` 4 R4
5 `H2O`
6 `OH`
7 `H`
`kriandkfi`represents the forward and backward rate coefficients respectively, for ith reaction. `ω` denotes the production rates for jth spices and `qi` defines the rate of progress variable for ith elementary reaction.
Given set of reactions are only forward reactions. So reverse reactions rates are not considered.
`(d[CO])/dt=−kf1[CO][O2]−kf3[CO][OH]`
`(d[O2])/dt=-kf1[CO][O2]−kf4[H][O2]`
`(d [ C O 2 ])/ dt = 0`
`(d [ O] )/dt = − k f2 [ O ] [ H2O]`
`(d [ H 2 O ])/ dt =− k f2 [ O ] [ H 2 O ]`
`(d [ O H ])/ dt = − k f3 [ C O ] [ O H ]`
`(d [ H ] )/dt =− k f4 [ H ] [ O 2 ]`
Derivation:
Matrix Form of forward reaction
`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]]`
`v ji = v ji\'\' − v ji\'`
here `vji\'\'=0`, `vji\'\'`is reverse reaction rate
so `vji = -vji\'`
`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]]`
where j= 1 t0 7
i= 1,2,3,4
The net production rate of each species in a Multistep Mechanism:
`ωj=sum_(i=1)^Lvji*qi`
where `q i = k fi prod_( j=1)^N [ X j ]^( vji\') − k ri prod_( j=1)^N [ X j ]^ (vji\'\')`
`q1=kf1[CO][O2]`
`q2=kf2[O][H2O]`
`q3=kf3[CO][OH]`
`q4=kf4[H][O2]`
using the above notataions we can derive all the compact notations as shown below
Consider species CO
`w1 = V11*q1+V13*q3`
`w1 =−kf1[CO][O2]−kf3[CO][OH]`
ODE = `(dw1)/dt=(dCO)/dt`
As show above we can derive the compact notations of ODE\'s for all the species.