Compact Notation Derivation for a simple Mechanism

AIM

The aim is to derive reaction rate ODE of a simple reaction mechanism

DERIVATION OF REACTION RATE

Consider the following reactions, lets number the following reactions.

CO + O2 -> CO2 + O (1)

O + H2O -> OH + OH (2)

CO + OH -> CO2 + H (3)

H + O2 -> OH + O (4)

And now lets list all the species of hydrocarbons that are taking place in the reaction

1. O2
2. H20
3. CO2
4. OH
5. CO
6. H
7. O

Lets define `v_(ij)'`be the stoichiometric coefficient matrix for the reactants in which the 'i' represents the reaction. 'j' denotes the number of moles of the species coefficient`

`v_(ij)'=[[1,0,0,0,1,0,0],[0,1,0,0,0,0,1],[0,0,0,1,1,0,0],[1,0,0,0,0,1,0]]`

Now we have to determine the stoichiometric coefficient matrix `v_(ij)''`for the products side as well

`v_(ij)''=[[0,0,1,0,0,0,1],[0,0,0,2,0,0,0],[0,0,1,0,0,1,0],[0,0,0,1,0,0,1]]`

Now equation for net stoichiomeric coefficient matrix

`v_(ij)=v_(ij)''-v_(ij)'`

`v_(ij) = [[-1,0,1,0,-1,0,1],[0,-1,0,2,0,0,-1],[0,0,1,-1,-1,1,0],[-1,0,0,1,0,-1,1]]`

So the net reaction rate `q_i`can be written as

`q_i = k_(fi) prod_(j=1)^(N) [X_j]^(v_(ij)^') - k_(ri) prod_(j=1)^(N) [X_j]^(v_(ij)^(''))`

where,

`k_(f1)`is the forward reaction rate

`k_(r1)`is the backward reaction rate

`[X_j]^(v_(ij)^'` is the concentration of species per unit volume of reactants

`[X_j]^(v_(ij)^(''))` is the concentration of species per unit volume of products

Now substituting the values for 'i' in the net reaction rate equation we obtain

`q_1 = k_(f1) [O_2][CO] - k_(r1) [CO_2][O]`

`q_2 = k_(f2) [H_2O] [O] - k_(r2) [OH]^2`

`q_3 = k_(f3) [OH] [CO] - k_(r3) [CO_2] [H]`

`q_4 = k_(f4) [O_2] [H] - k_(r4) [OH] [O]`

Now we define the net production rate `omega_j`of 'j' species is written as

`omega_j = sum_(i=1)^L v_(ij)q_i`

`omega_j=d/dt [X_j]`

Substituting the values for 'j', we can write the net production rate

`omega_j = v_(1j) q_1 + v_(2j) q_2 + v_(3j) q_3 + v_(4j) q_4`

O2 CONCENTRATION

`d/dt [O_2] = -(k_(f1) [O_2][CO] - k_(r1) [CO_2][O]) - (k_(f4) [O_2] [H] - k_(r4) [OH] [O])`

H20 CONCENTRATION

`d/dt [H_2O] = -(k_(f2) [H_2O] [O] - k_(r2) [OH]^2)`

CO2 CONCENTRATION

`d/dt [CO_2] = k_(f1) [O_2][CO] - k_(r1) [CO_2][O] + k_(f3) [OH] [CO] - k_(r3) [CO_2] [H]`

OH CONCENTRATION

`d/dt [OH] = 2(k_(f2) [H_2O] [O] - k_(r2) [OH]^2) - (k_(f3) [OH] [CO] - k_(r3) [CO_2] [H]) + k_(f4) [O_2] [H] - k_(r4) [OH] [O]`

CO CONCENTRATION

`d/dt [CO] = - (k_(f1) [O_2][CO] - k_(r1) [CO_2][O]) - (k_(f3) [OH] [CO] - k_(r3) [CO_2] [H])`

H CONCENTRATION

`d/dt [H] = k_(f3) [OH] [CO] - k_(r3) [CO_2] [H] - (k_(f4) [O_2] [H] - k_(r4) [OH] [O])`

O CONCENTRATION

`d/dt [O] = k_(f1) [O_2][CO] - k_(r1) [CO_2][O] - (k_(f2) [H_2O] [O] - k_(r2) [OH]^2) + k_(f4) [O_2] [H] - k_(r4) [OH] [O]`

In order to evaluate the forward and backward reaction rates, the established relations are to be used

`k_(fi) = A_i T^(beta_i) exp(-E_i/(R_c T))`

`k_(ri) = k_(fi)/k_(ci)`

`k_(ci) = k_("pi")(P_(atm)/(RT))^(sum_(k=1)^k v_(ki))`

`k_("pi") = exp((Delta S_i^o)/R - (Delta H_i^o)/(RT))`

CONCLUSION

1. The concentration of each species was derived
2. Reaction rates was determined using compact notations