1. Global and elementary reactions
A global reaction, such as CH4 + 2 O2  CO2 + 2 H2O, describes the initial and final states of the reactive process, but not the way it occurs at molecular level.
The reactive process is composed of several elementary reactions involving intermediate species, which may be stable molecules, elements or radicals, e.g., C2H2, H2, H, OH.
A reaction is referred to as elementary when it cannot be broken into smaller steps, occurring at molecular level as described by the chemical reaction equation, e.g., OH + H2  H2O + H. In general, a single chemical bond is broken in an elementary reaction, while a new one is formed.
A reaction mechanism is the set of elementary reactions by which the global chemical reaction occurs. The reaction mechanisms for combustion reactions may involve hundreds of elementary reactions.
Chemical kinetics
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2. Reaction rate of a global reaction
Consider the following global reaction: A + B + C  D + E + F + ...
Experience shows that the reaction rate of species A may be written as
Units of k : [s-1 (kmol/m3)1-(a+b+c+…)]
k – reaction rate constant (mathematically, it is not constant)
a, b, c,… – orders of reaction with respect to A, B, C,…
a+b+c ,…– global order of reaction
The reaction rate is a measure of how fast a chemical reaction proceeds. The greater the reaction constant, the faster will be the conversion of reactants into products. The negative sign is a consequence of the decrease of [A] as the reaction proceeds.
The units of the reaction constant depend on the global order of reaction.
In general, the exponents a, b, c,… and the reaction rate constant for global reactions are obtained by fitting of experimental data, and are valid only for a limited range of pressure and temperatures.
Chemical kinetics
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3. Reaction rate of an elementary reaction
Elementary reactions may be classified as unimolecular, bimolecular or trimolecular depending on the number of molecules in the reactants.
Unimolecular reactions:
Bimolecular reactions:
Trimolecular reactions:
A  products
Units of k – [s-1]
A + B  products
Units of k – [m3 kmol-1 s-1]
A + A  products
A + B + C  products
Units of k – [m6 kmol-2 s-1]
A + A + B  products
Chemical kinetics
Combustion

4. Reaction rate of an elementary reaction
Inert species, identified by M, may participate in chemical reactions, behaving as a catalyzer:
The chemical state of M is slightly different in the reactants and in the products. If A and B are radicals and C is a stable species, the energy released when species C is formed is absorbed by M as kinetic energy.
The previous expressions for the reaction rates assume that the volume of the reactive system remains constant. If that is not so, they must be modified, e.g., in the case of a bimolecular reaction with reactants A and B:
A + B + M  C + M
zi: catalytic efficiency of the ith species
Chemical kinetics
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5. Temporal variation of species concentration in an elementary reaction
The temporal variation of a species concentration in an elementary reaction may be found if the initial concentrations are available.
Unimolecular reaction: A  products
Integrating subjected to ([A])t=0 = [A]o
yields
Bimolecular reaction A + B  products
[A] – [A]o = [B] – [B]o
Chemical kinetics
Combustion

6. Temporal variation of species concentration in an elementary reaction
If [B]o  [A]o then
Integrating the left hand side from [A]o to [A] and the right hand side from 0 to t yields
and, after some algebra,
Chemical kinetics
Combustion

7. Collision frequency Chemical kinetics
The reaction constant may be theoretically estimated for elementary reactions using the collision theory.
Collision frequency (number of collisions per unit time) for two identical molecules (one molecule travels with speed v and the other is at rest) whose mean free path is large compared with the diameter.
NAV=6.0231026 molecules/kmol – Avogrado number
n NAV / V - number of molecules per unit volume
If the two molecules travel with the same mean velocity the collision frequency is given by
kB = 1.3811023 J/K is the Boltzmann constant and m is the mass of one molecule.
Chemical kinetics
Combustion

8. Collision frequency Chemical kinetics
If the two molecules are different, A and B, the collision frequency is given by
In a reactive system, the collision frequency of molecules of species A with molecules of species B is given by
The probability of a collision to originate a chemical reaction depends on the product of two factors: an energetic factor and a geometrical factor
Only the molecules with kinetic energy greater than a certain threshold, referred to as activation energy, react and yield products
The frequency of collisions between molecules of kinetic energy greater than the activation energy is given by the Boltzmann factor: exp(-Ea/RoT)
sAB = 0.5 (sA + sB)
Chemical kinetics
Combustion

9. Activation energy Chemical kinetics
The activation energy is the least amount of energy for a chemical reaction to take place
The intermediate state that is formed during the conversion of reactants into products, at the point of maximum energy, is the activated complex.
The activation energy of the direct reaction is lower than that of the inverse reaction for exothermic reactions, and vice-versa for endothermic reactions.
The more exothermic a reaction is, the lower is its activation energy
Chemical kinetics
Combustion

10. Arrhenius equation Chemical kinetics
The reaction rate of a chemical species A may be written as
[kmol/m3.s]
ZAB/V [molecules / m3 s] – number of collisions between molecules of species A and B per unit time and unit volume
S – geometrical factor
Introducing the expression for ZAB yields
Theoretical reaction rate constant for an elementary bimolecular reaction
Chemical kinetics
Combustion

11. A – pre-exponential factor
Arrhenius equation
The theoretical reaction rate constant is not quantitatively accurate, and does not allow the calculation of either Ea or S
The reaction rate constant for a limited range of temperatures may be approximated by (Arrhenius equation)
A – pre-exponential factor
Dependence of the rate constant on T
Chemical kinetics
Combustion

12. Modified Arrhenius equation
The Arrhenius equation is an approximation, being sometimes inaccurate for a wide temperature range. Although different values of A and Ea may be used for low and high temperature, it is more common to use the following modified form of Arrhenius equation:
The exponent b accounts for the dependence of the term that multiplies the Boltzmann factor on the temperature
Although A, b and Ea may be theoretically estimated, they are generally experimentally obtained, and are tabulated for the most common reactions
The experimental uncertainty is sometimes high
The modified Arrhenius equation may be applied to unimolecular, bimolecular and trimolecular elementary reactions
Chemical kinetics
Combustion

13. Reaction rates in SI units
The reaction rate of an elementary bimolecular reaction between species A and B is expressed as follows in SI units:
[kg/m3.s]
Expressing the rate constant according to the Arrhenius equation yields
Chemical kinetics
Combustion

14. Reaction rates Chemical kinetics
Evolution of reaction rate with temperature at constant pressure
At low temperatures the reaction rate is negligible
At moderate temperatures the reaction rate increases significantly with the increase of temperature, due to the dominance of the exponential term
As the temperature further increases, the decrease of the molar fractions of the reactants compensates the increase of the exponential term, and a maximum value of the reaction rate is achieved
The reaction rate drops significantly with further increase of the temperature, since the exponential term tends to unity while the molar fractions tend to zero
Chemical kinetics
Combustion

15. Reaction rates for a set of elementary reactions
Consider the following set of elementary reactions
H2 + O2  HO2 + H (R1)
H + O2  OH + O (R2)
OH + H2  H2O + H (R3)
H + O2 + M  HO2 + M (R4)
The reaction rates of O2 and H are given by
Chemical kinetics
Combustion

16. Reaction rates for a general reaction mechanism
m = 1, 2, ..., M
M – number of reactions
N – number of species
- stoichiometric coefficients of reactants
- stoichiometric coefficients of products
Mass conservation implies that with for all m
The reaction rate of species i is given by
[kmol/m3s]
Chemical kinetics
Combustion

17. Reaction rates for a general reaction mechanism
The time evolution of the chemical composition of a reactive system may be found by solving the system of N simultaneous ordinary differential equations for the reaction rates of the species:
i = 1, 2, …, N
given at t = 0, and assuming that the species concentrations are only a function of time
The equation of conservation of energy needs also to be solved if the temperature is unknown
This system of equation is generally stiff, because reaction time scales differ significantly depending on the species
CHEMKIN is a well-known commercial software that can be used to solve chemical kinetics problems (
Chemical kinetics
Combustion

18. Relation between reaction constants and the equilibrium constant
Consider the following elementary reaction:
The reaction rate is equal to zero in equilibrium conditions, so that
Hence, the equilibrium constant Kc is equal to the ratio of the forward to the backward reaction constants
Chemical kinetics
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19. Simplification of reaction mechanisms Steady state assumption
Consider the following two elementary reactions:
A  B (R1)
B  C (R2)
Suppose that kB >> kA, i.e., species B is strongly reactive and, once formed via the relatively slow reaction R1, is rapidly consumed via the fast reaction (R2).
Then, the concentration species B is very small and its consumption rate is approximately equal to its formation rate, and the following steady state assumption may be used for species B:
Chemical kinetics
Combustion

20. Simplification of reaction mechanisms Steady state assumption
The steady state assumption is typically applicable to very reactive radicals whose concentration, after increasing in the beginning of the reaction, remains approximately constant, due to fast consumption immediately following their formation.
Example: Zeldovich mechanism, responsible for the formation of NO from the oxygen present in the air
O + N2  NO + N (R1)
N + O2  NO + O (R2)
From the knowledge that k2 >> k1, the steady assumption may be used for N:
Chemical kinetics
Combustion

21. Simplification of reaction mechanisms Partial equilibrium assumption
A few elementary reactions with high reaction rates may achieve a quasi equilibrium state, while other reactions, with lower reaction rates, may be still far from the equilibrium
As an example, the reactions involved in the formation of NO are generally much slower than the reactions of formation of the main products (CO2, H2O). Consider again the Zeldovich mechanism:
The steady state assumption for the N atom yields
Chemical kinetics
Combustion

22. Simplification of reaction mechanisms Partial equilibrium assumption
The concentration of the very reactive O atom may be determined assuming chemical equilibrium of the following three reactions (realistic assumption at atmospheric pressure and temperatures above 1800 K):
H + O2  OH + O
O + H2  OH + H
OH + H2  H2O + H
This yields
Chemical kinetics
Combustion

23. Reaction mechanism of hydrogen
Number Reaction A [mol, cm3, s] b Ea [kJ/mol]
1 O2 + H  OH + O 2.00
2 H2 + O  OH + H 5.06
3 H2 + OH  H2O + H 1.00
4 OH + OH  H2O + O 1.50
5 H + H + M  H2 + M 1.80
6 O + O + M  O2 + M 2.90
7 H + OH + M  H2O + M 2.20
8 H + O2 + M  HO2 + M 2.30
9 HO2 + H  OH + OH 1.50
10 HO2 + H  H2 + O 
11 HO2 + H  H2O + O 3.00
12 HO2 + O  OH + O 
13 HO2 + OH  H2O + O2 6.00
14 HO2 + HO2  H2O2 + O2 2.50
15 OH + OH + M  H2O2 + M 3.25
16 H2O2 + H  H2 + HO2 1.70
17 H2O2 + H  H2O + OH 1.00
18 H2O2 + O  OH + HO2 2.80
19 H2O2 + OH  H2O + HO2 5.40
Chemical kinetics
Combustion

24. Chain reactions Chemical kinetics
Chain reactions usually consist of many repeating elementary steps, each of which has a chain carrier. Once started chain reactions continue until the reactants are exhausted.
Initiation reactions: reactions that form radicals from stable species
H2 + M  H + H + M
H2 + O2  HO2 + H
Chain propagation reactions: reactions with the same number of radicals in the reactants and in the products
H2 + OH  H2O + H
HO2 + H  OH + OH
H2O2 + O  OH + HO2
Chemical kinetics
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25. Chain reactions Chemical kinetics
Chain branching reactions: reactions with more radicals in the products than in the reactants
O2 + H  OH + O
H2 + O  OH + H
Chain termination reactions: reactions that form stable species from radicals
H+ OH + M  H2O + M
Chemical kinetics
Combustion