Week 5.1 - Compact Notation Derivation for a simple Mechanism
Aim : To write a derivvation of compact notation for a simple mechanism to form expressions for production rates of different species
Introduction : Each reaction has several intermediary reactions which take place between initial and final steps of reactions. These all make up a reaction mechanism,
we have 8 species and 4 reactions for the given reaction mechanism.
Now we have to represent the species and the reactions in the terms of a compact notation
Reaction Mechanism
CO2 + O2 --> CO2+ O
O + H2O --> OH + OH
CO + OH --> CO2 + H
H + O2 --> OH + O
Compact notation :
Species
| i | Reaction No. | Reactions |
| 1 | R1 | CO2 + O2 --> CO2+ O |
| 2 | R2 | O + H2O --> OH + OH |
| 3 | R3 | CO + OH --> CO2 + H |
| 4 | R4 | H + O2 --> OH + O |
| j | Species |
| 1 | CO |
| 2 | O2 |
| 3 | CO2 |
| 4 | O |
| 5 | H2O |
| 6 | OH |
| 7 | H |
Here we are giving notation of "j" represents the number of species and "i" represents number of reactions.
Here for the given set of reactions,we need to derivate the each reaction with respect to each species.
Then finally, after doing this, we can have the derived equation in which we can extract the matrix from them.
In writings we follow soem notations, kfi which represents the the forward rate reaction and kri represents the reverse rate reaction. The subscript "i" denotes the reaction number.
`omega`represents the production rate for a particular species`q_t`is the reaction rate for
each reaction.
vji ' = Stoichiometric coefficient matrix of reactant`upsilon_{ji} = upsilon_{ji}'' - upsilon_{ji}'`
vji " = Stoichiometric coefficient matrix of product
`upsilon_{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]]`
`upsilon_{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]]`
`upsilon_{ji}` =`upsilon_{ji}''` - `upsilon_{ji}'`
`upsilon_{ji}` = `[[-1,-1,1,1,0,0,0],[0,0,0,-1,-1,2,0],[0,0,1,0,0,-1,1],[0,-1,0,1,0,1,-1]]`
Reaction rates :
`q_t = kf_{i} prod_(j=1)^N [X_{j}]^{upsilon_{ji}'} -kr_{i} prod_(j=1)^N [X_{j}]^{upsilon_{ji}''}`
Accordingly,
For reaction 1:
`q_1 = kf_1[CO]^1[O_2]^1 - kr_1[CO_2]^1[O]^1`
For reaction 2 :
`q_2 = `-kf_2[O]^1[H_2O]^1 + kr_2[OH]^2``
For reaction 3:
`q_3 = kf_3[CO]^1[OH]^1 - kr_3[CO_2]^1[H]^1`
For reaction 4:
`q_4 = `kf_4[H]^1[O_2]^1 - Kr_4[OH]^1[O]^1`
The Production Rates of a species : -
`omega_j = sum_(i = 1)^L V_{ji} . q_t`
here L represents the length of species.
The production rate of each species is shown given below :
`((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]]`
CO Concentration :
`omega_1 = -1(q_1) + 0(q_2) + q_3(-1) + 0(q_4)`
= `omega_1 = - kf_1[CO]^1[O_2]^1 + kr_1[CO_2]^1[O]^1 -kf_3[CO]^1[OH]^1 + kr_3[CO_2]^1[H]^1``
`omega_2 = -1(q_1) + 0(q_2) + 0(q_3) -1(q_4)`
= `kf_1[CO]^1[O_2]^1 + kr_1[CO_2]^1[O]^1 - kf_4[H]^1[O_2]^1 + Kr_4[OH]^1[O]^1`
`omega_3 = 1(q_1) + 0(q_2) + 1(q_3) + 0(q_4)`
= `kf_1[CO]^1[O_2]^1 - kr_1[CO_2]^1[O]^1 + kf_3[CO]^1[OH]^1 - kr_3[CO_2]^1[H]^1`
`omega_4 = 1(q_1) - 1(q_2) + 0(q_3) + 1(q_4)`
= `kf_1[CO]^1[O_2]^1 - kr_1[CO_2]^1[O]^1 - kf_2[O]^1[H_2O]^1 + kr_2[OH]^2 + kf_4[H]^1[O_2]^1 - Kr_4[OH]^1[O]^1`
`omega_5 = 0(q_1) + -1(q_2) + 0(q_3) + 0(q_4)`
= `-kf_2[O]^1[H_2O]^1 + kr_2[OH]^2`
`omega_6` = 0(q_1) + 2(q_2) -1 (q_3) + 1(q_4)
= `-2kf_2[O]^1[H_2O]^1 + 2kr_2[OH]^2 -kf_3[CO]^1[OH]^1 + kr_3[CO_2]^1[H]^1 + kf_4[H]^1[O_2]^1 - Kr_4[OH]^1[O]^1`
`omega_7 = 0(q_1) + 0(q_2) + 1(q_3) + 1(q_4)`
= `kf_3[CO]^1[OH]^1 - kr_3[CO_2]^1[H]^1 + kf_4[H]^1[O_2]^1 - Kr_4[OH]^1[O]^1`
The plus and minus sign in front of each term tells us about whether the forward or backward reaction would increase or decrease the concentration. when due to a reaction the concentration of species increases, a plus sign is used and if concentration decreases a minus sign is used.
Results :
1. The concentration of each species was derived.
2. Reaction rates was determined using compact notations