2
Reaction kinetics in gases
October 8, 2014
In a reaction between two species, for example a fuel and an oxidizer, bonds
are broken up and new are established in the collision between the species. In
the reaction between fuel and oxidizer the enthalpy of formation of the products
are lower than those of the reactants causing heat release.
2.1
Molecular kinetics
In realty very few reactions proceed directly from reactants to final products in
a single reaction step. Instead the reaction proceed along a pathway of intermediate species, formed in so called elementary reactions, where in each step
collisions between two or in rare cases three bodies take place. Some steps in
the chain are extremely fast and some are slow and determine the rate at which
the net reaction proceeds. A setup of elementary reactions necessary to describe
the global reaction is called a reaction mechanism. Reactions that proceed
in just one step are very rare. (One of the overall reactions that for very long
was thought to proceed in just one step was I2 + H2 −−→ 2 HI).
Let us study the collision between two molecules in some detail. We look at
the situation where A (a single atom) and BC ( two atoms) collide, react and
form AB and C. In the beginning of the process BC vibrate with a constant
amplitude and the sum of potential and kinetic energy of the vibrational motion is constant. Figure (5) shows the potential energy stored in the molecule
BC for various internuclear distances. As the A molecule approaches the BC
!'
0.5
0.0
-0.5
-1.0
1.0
1.2
1.4
1.6
1.8
2.0
2.2
rBC/"
Figure 5: Normalized potential energy Ψ� vs the normalized inter-nuclus distance r/σ. The dashed lines show the different vibration levels
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Ψ
EA
EA
∆E
A B-C
A-B-C
A-B C
Figure 6: Potential energy Ψ vs reaction coordinat
molecule the vibrational amplitude increases and an activated complex A-B-C
is formed. In some cases the reaction proceeds and two new molecules AB and
C are formed. Figure (6) visualizes the process in a diagram with the reaction
coordinate (abscissa) and potential energy (ordinate). The figure shows the en−→
ergy necessary to form the activated complex EA (measured per mole, mainly
stored in the translation (kinetic) energy), and energy released as heat, ∆E,
←−
forming the products. To reverse the reaction a larger amount of energy EA is
required.
We have discussed interaction between molecules in an elementary reaction.
Let us revert to a more global view, the reaction between one mole of fuel A
and b mole of oxidizer B forming c mole of products C
A + bB −−→ cC.
Experiments show that the rate at which the reaction proceeds can be expressed
as
d[XA ]
= −kG (T )[XA ]n [XB ]m
dt
where [Xi ] is the molar concentration of species i [kmole/m3 ], kG (T ) the global
reaction rate coefficient and n, m the reaction order in species A and B.
The global order of the reaction is n + m. The exponents n and m as well as
the coefficient kG are determined from experimental data and may change over
the interval of interest. In the global reaction n and m may have any value, it
is only in an elementary reaction they have to be a positive integer.
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2.2
2.2.1
Elementary reactions
Bimolecular reactions
As been mentioned earlier most elementary reactions in combustion are bimolecular, two molecules collide and react to form products. A general reaction would
be
A + B −−→ C + D.
The rate of the process is
d[A]
= −kbi [A][B]
(27)
dt
The reaction rate coefficient kbi is a function of temperature and has, to a certain
degree, a theoretical bases.
The rate at which species A reacts per time and per volume is proportional
the number of collisions between A and B molecules. However all collisions
do not lead to a reaction. Only those that involve molecule pairs having a
kinetic energy higher than the necessary activation energy EA may participate
in the process. In addition the potential barrier that has to be overcome is not
fixed but depends on the orientation and geometry of the colliding molecules
at the moment of impact expressed as a so called steric factor p. The fraction
of particles having high enough energy is proportional to exp(−EA /(RT ) (see
eq.(11)). A suggestion is thus that the rate equation could be expressed by
the product of steric factor, exponential factor and collision number given by
eq.(15)
�
√
dnA
RT
−EA /(RT )
2
= −p · e
2 2πσAB nA nB
.
(28)
dt
MAB
Using nA = NA [A], nB = NA [B], where NA is the Avogadro’s number
(molecules/mole), we get
�
√
d[A]
RT
−EA /(RT )
2
= −pNA · e
2 2πσAB [A][B]
.
(29)
dt
MAB
Comparing the different factors in the this relation with eq.(27) give us an
expression for kbi
�
√
RT −EA /(RT )
2
kbi = pNA · 2 2πσAB
e
(30)
MAB
usually written as a frequency factor, A, times the exponential factor
kbi = A · e−EA /(RT )
(31)
called the Arrhenius law (first arrived at by van’t Hoff (1887) but named after
Arrhenius).
2.2.2
Termolecular reactions
One may expect that two atoms originating from dissociated molecules would
react in the same way as molecules do. However the high energy released in the
dissociation process, which usually take place at high temperature, has to be
liberated from the newly formed molecule before it falls a part. Both dissociation
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energy and kinetic energy have to be transferred to a third body in the collision
to obtain a stable product. This is a rare event and thus termolecular reactions
have a a low probability. A third body is not necessary in molecular collision
because both the kinetic energy which is usually lower and the excess energy
could be split between the different internal energy modes and bonds.
The general form of a termolecular reaction is
A + B + M −−→ AB + M
where M is the third body taking up excess energy. The reaction is found as
the revers of the unimolecular reaction at low pressure in important combustion
reactions like
H + H + M −−→ H2 + M,
H + OH + M −−→ H2 O + M.
The rate equation could be written
d[A]
= kter [A][B][M]
dt
2.2.3
(32)
Unimolecular reactions
Some decomposing reactions are just proportional to the concentration of the
species itself. If this is the case the molecule is normally complex. The necessary
energy to split the molecule is provided through collisions and the energy is
stored in the vibration modes. Randomly energy transfer between different
vibration modes may result in energy concentration above activation level in
one of the modes causing the bond to break sooner or later. If the molecule is
complex or the concentration of molecules is high there will be a balance between
activated and non activated molecules and this balance is not displaced if there
is a drain of activated molecules due to decomposition.
equilibrium
decomposition
−
�
normal molecules −
�
−−
−−
−−
−−
−−
−−
−
− activated molecules −−−−−−−−−→ products
The rate of decomposition of species A will have the following expression
d[A]
= −[A]νe−E/(RT )
dt
where ν is the frequency factor. The equation resembles the Arrhenius law.
2.2.4
Reactions independent of concentration
Several important combustion reactions are independent of species concentration. Those reactions are mainly catalytic and depend to what extent the catalytic surface is covered by molecules. At high species concentration the surface
is saturated and the reaction rate is independent of pressure.
2.3
Net reactions rates
In the preceding discussion we have mainly looked at the forward reactions but
of course each elementary forward reaction has a reverse reaction. Considering
all forward and reverse elementary reactions give us an expression for the net
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production rates. Let us take a common reaction in combustion, the oxidation
of hydrogen
2 H2 + O2 −−→ 2 H2 O
This reaction has a number of elementary reactions four of which are written
below to give the feeling for the technique.
k+1
−
�
H2 + O 2 −
�
−−
−
− HO2 + H
(I)
k−1
k+2
−
�
H + O2 −
�
−−
−
− OH + O
(II)
k−2
k+3
−
�
OH + H2 −
�
−−
−
− H2 O + H
k−3
(III)
k+4
−−
�
H + O2 + M �
−−
−
− HO2 + M
k−4
(IV )
..
.
Now writing the rate equation for molecular oxygen
d[O2 ]
= k −1[HO2 ][H] + k −2[OH][O] + k −4[OH2 ][M]−
dt
−k +1[H2 ][O2 ]−k +2[H][O2 ]−k +4[H][O2 ][M]
Similar equations are obtained for [H] etc., a number of first order differential
equations. A way of writing the elementary reaction is:
N
�
j=1
�
−
�
νji
Xj −
�
−
−
N
�
��
νji
Xj
j=1
�
��
where νji
, νji
are the stoichiometric coefficients of the reactants and products
respectively for the N species Xj in reaction i(i=1,2,3...L ).
A compact way of expressing the net production rates of each species Xj in
a multistep reaction mechanism is given by
ω˙j =
where
L
d[Xj ] �
=
νji qi
dt
i=1
(33)
��
�
νji = νji
− νji
and the qi , called the rate of progress is
qi = k+i
N
�
j=1
�
[Xj ]νji − kri
15
N
�
j=1
��
[Xj ]νji
2.3.1
Equilibrium
The concept of forward and revers reactions
k+
−
�
A+B−
�
−
−C+D
k−
gives us the reaction rate of species [A]
d[A]
= −k+ [A][B] + k− [C][D].
dt
In steady state we have
d[A]
= 0 ⇒ k+ [A][B] = k− [C][D]
dt
which could be rewritten
k+
[C][D]
=K=
k−
[A][B]
(34)
where K is an equilibrium constant.
In general the equilibrium of the global gas reaction
k+
−−
�
aA + bB �
−
− cC + dD
k−
can be written
Kp =
[C]c [D]d
[A]a [B]b
(35)
where [C] is the normalized partial pressure, pp◦i
(The relation is called the law of mass action and was proposed by Guldberg and Waage in 1864)
A derivation of eq.(35) is given in chapter 4.
2.3.2
Reaction rate dependance on pressure
Let us start with a 1st order reaction
d[A]
= −k1 [A]
dt
and look at its total pressure dependance. This is done by changing from [Xi ]
to partial pressure. We have the ideal gas law
pA = [A]RT
and
pA =
With the mole fraction
[A]
[XA ]
p= �
p.
[A] + [B] + · · ·
[Xi ]
[X ]
� i = χi
[Xi ]
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the reaction rate for species A for a first order reaction will be
d[A]
p
= −k1 [A] = −k1
χA
dt
RT
and for a νth order reaction
d[A]
p ν
= −k[A][B] · · · = −k(
) χA χB · · ·
dt
RT
showing that a νth order reaction has a reaction rate dependance on p and ν
d[A]
∼ pν .
dt
(36)
Sometimes it is practical to express the chemical reaction rate in mole fraction
instead of mole concentration. The rate equation of order ν and the relation
pi = χi p = [Xi ]RT
give
p dχA
p ν
)
= −k(
) χA χB · · ·
RT dt
RT
(const. p, T). Thus the pressure dependance on a reaction of order ν expressed
in mole fraction becomes
dχA
∼ (p)ν−1
(37)
dt
(
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