Week 5.1 - Compact Notation Derivation for a simple Mechanism

AIM:

Derive the reaction rate ODEs of a simple reaction mechanism and write down the desired matrices.

INTRODUCTION:

In this assignment we are expected to derive reaction rate ODEs for the given simple reaction mechanism and write down the desired matrices which may include the Stoichiometric coefficient matrix on the reactants side (`nu_(ji)'`), Stoichiometric coefficient matrix on the products side (`nu_(ji)''`) and Net Reaction rate matrix, etc.

Note:-

'i' is reaction index and

'j' is species index.

Also, i=1,2,3,......L

and j=1,2,3,......N

THEORY:

Chemical Kinetics:

It is the branch of physical chemistry that is concerned with understanding the rates of chemical reactions. It is to be contrasted with thermodynamics, which deals with the direction in which process occurs but in itself tells nothing about its rate.

As mentioned in the video lecture, there are multiple reasons as to why to study Chemical Kinetics. These are:-

1. Ignition Delay: It helps us to determine the time at which the ignition starts in the combustion system.
2. Rate of work done: It is also called as the heat release rate.
3. Emissions: It is used to study, analyze and reduce the pollutants produced.

For predicting reaction pathway, there are two important aspects that need to be factored into the calculations apart from the composition of the species involved is:

1. Operating conditons such as Pressure and Temperature.
2. Rate of Reaction.

Here, the rate of reaction is a bit complex to determine.

Necessary concepts and topics that are a must know for this assignment are:

Rates of Reactions: Reaction Rate, in chemistry, is the speed at which a chemical reaction proceeds. It is often expressed in terms of either the concentration (amount per volume) of a product that is formed in a unit of time or a concentration of a reactant that is consumed in a unit of time. Reactions rates can vary dramatically. There are two kinds of rates of reaction, namely - Rate of Forward reaction (`k_(fi)`) and Rate of Reverse Reaction (`k_(ri)`).

Rate of change of concentration of a particular species is indicated as: `d/dt[X_j]`; where [`X_j`] represents the concentration of that species.

Net Reaction Rate: This informs us the direction in which the reaction takes place, i.e. for a negative value of net reaction rate, the direction is reverse and for a positive value, the direction is said to be forward. The formula for net reaction rate is given as:

`q_i` = `k_(fi)``Pi_(j=1)^N``[X_j]^(nu_(ji)')`- `k_(ri)``Pi_(j=1)^(N)``[X_j]^(nu_(ji)'')`

Net Production Rate: The expression for net production rate is given by -

`omega_j` = `Sigma_(i=1)^L``nu_(ji)*q_i`

here, `nu_(ji)` = `nu_(ji)''` - `nu_(ji)'`

Further, from Arrhenius' Law we know that,

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

where, `A_i` is pre-exponential factor

`E_i` is Activation Energy

T is Operating temperature

`R_c` is the characteristic gas constant

`beta_i` is the temperature exponent.

MECHANISM:

Here, let the reaction index 'i' for the given reactions be,

`CO+O_2<=>CO_2+O`---(1)

`O+H_2O<=>2OH`---(2)

`CO+OH<=>CO_2+H`---(3)

`H+O_2<=>OH+O`---(4)

and the species index 'j' for the species present be,

Species Index (j)	Species
1	CO
2	`O_2`
3	`CO_2`
4	O
5	`H_2O`
6	OH
7	H

Here, the Stoichiometric Coefficient Matrix for the reactants side is:

`nu_(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]]`

and the Stoichiometric Coefficient Matrix for the products side is:

`nu_(ji)''=[[0,0,1,1,0,0,0],[0,0,0,0,0,1,0],[0,0,1,0,0,0,1],[0,0,0,1,0,1,0]]`

As we know that,

`nu_(ji)=nu_(ji)''-nu_(ji)'`

`therefore``nu_(ji)=[[-1,-1,1,1,0,0,0],[0,0,0,-1,-1,1,0],[-1,0,1,0,0,-1,1],[0,-1,0,1,0,1,-1]]`

Now it is known that,

`omega_(j)=sum_(i=1)^(L)nu_(ji)*q_i`

Here, L=4;

`therefore``omega_1=nu_(11)*q_1+nu_(12)*q_2+nu_(13)*q_3+nu_(14)*q_4`

`omega_2=nu_(21)*q_1+nu_(22)*q_2+nu_(23)*q_3+nu_(24)*q_4`

`omega_3=nu_(31)*q_1+nu_(32)*q_2+nu_(33)*q_3+nu_(34)*q_4`

`omega_4=nu_(41)*q_1+nu_(42)*q_2+nu_(43)*q_3+nu_(44)*q_4`

By substituting the values of `nu_(ji)` we get,

`omega_1=-q_1-q_3`

`omega_2=-q_1-q_4`

`omega_3=q_1+q_3`

`omega_4=q_1-q_2+q_4`

Now, there is something that we need to know about the net production rate (`omega_j`) of the species,

`omega_j=d/dt[X_j]`

i.e. the net production rate is nothing but rate of concentration change of that species.

Thus,

`[[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]]]`

This is what we call an ODE system.

Hence, in this we have derived the reaction rate ODEs for the given simple reaction mechanism along with the necessary matrices. The whole purpose of this compact notation was to make the data suitable as an input for our computer program so as to perform the complex calculations that are way complicated for us to perform manually. Therefore, we use this compact notation which is in suitable format to be used for calculations by a computer code and compute the results.