1.3 Rate laws of overall and elementary reactions
In general the equation describing the dependency of the reaction
rate r on the concentrations ci of reaction components i is called the
kinetic equation of the reaction:
r = f(c A ,c B ,c D ,c E ,...,c N )
(1.16)
Depending on the reaction the function f can be quite simple but
sometimes it may as well be rather complicated. It usually depends
on the concentrations of the components participating in the main
reaction but it can also depend on the concentrations of catalysts
and inhibitors. For the time being we will neglect this possibility.
For many reactions in ideal systems (gaseous reactions and
reactions in liquid phase) r is found to have the general form of rate
law given in Eq. (1.17).
r = k[ A ] [ B] ....[ L] = kcαAcβB……cλL
α
β
λ
(1.17)
In this rate equation (rate law) k is the reaction rate constant (rate
coefficient) and α, β and λ are often integers or half-integers. The
rate constant k is a function of temperature and pressure, but the
pressure dependency of k is small and is usually ignored. The
reaction is said to have order α in respect of the component A,
order β in respect of the component B, etc. and parameters α, β and
λ are called partial orders of the reaction.
The sum α + β +…+ λ = n is called the overall order (or simply
order) of the reaction. Since r has the units of concentration over
time, k has the units (concentration)1-n time-1. Therefore the rate
constant of a first order (n = 1) reaction has the units s-1 and is
independent of the units used for concentration which is a very
important general property of a first order reaction.
For elementary reactions (unit steps) only we can obtain the rate
law immediately from the chemical reaction equation. For example
if the forward reaction given by Eq. (1.1) can be considered as an
elementary reaction its rate is given by
r = k[ A ]
− νA
[ B] − ν = k[ A ] a [ B] b
B
(1.18)
where νA and νB are the general stoichiometric numbers of the
reactants involved in the elementary reaction given by Eq. (1.1) i.e.
-νA = a and -νB = b. Then the negation of the sum of the general
stoichiometric numbers of the reactants
− ( νA + ν B +... ) = −
∑ νi
(1.19)
i = reactant
is called the molecularity of the elementary reaction.
Some experimentally observed rate laws for homogeneous
reactions are given below:
H2 + Br2 → 2HBr
2N2O5 → 4NO2 + O2
H2 + I2 → 2HI
r=
1
2
k[ H 2 ][ Br2 ]
1 + j[ HBr ][ Br2 ]
(1.20)
r = k[N2O5]
(1.21)
r = k[H2][I2]
(1.22)
In general the rate law must be determined from experimental
measurements of reaction rate and cannot be deduced from the
reaction stoichiometry (except for in the special case of an
elementary reaction).
The reaction equation, Eq. (1.1) gives the overall stoichiometry of
the reaction but does not tell us very much about the mechanism by
which the reaction occurs. Each step in the mechanism of the
overall reaction is an independent reaction and is called an
elementary reaction (unit step). The most simple reaction consists
of one single elementary step. The classical example of such
elementary reaction is the reaction described by Eq. (1.22). It is
explained by the following molecular level reaction mechanism
A + B → A---B → C + B
where A---B denotes the so called activated complex first formed
between the reactant molecules A and B before the final
rearrangement to product molecules C and B.
The formally very similar reaction given by Eq. (1.20) is much
more complicated and is called a complex (composite) reaction. A
complex chemical reaction consists of at least two (and usually
more) elementary steps. A specific characteristic of the complex
reactions is that the exponentials (partial orders) in the rate law and
stoichiometric coefficients in the reaction equation do not fit. In
Chap. 6 we shall give a more detailed analyze for the mechanism
of this reaction.
An important concept is that some reactions can be characterized
by pseudo orders and pseudo constants. For example for the
hydrolysis of sucrose:
C12H22O11 + H2O → C6H12O6 + C6H12O6
Sucrose
Glucose Fructose
(1.23)
one finds that the rate is given by the rate law, r = k[C12H22O11].
However, since the solvent, H2O participates in the reaction, one
could expect that the rate law would preferably have the more
general form, r = k' [C12H22O11] [H2O]v.
Because H2O is always present in the reaction vessel in great
excess, [H2O]v is essentially constant and the rate law has the
apparent form r = k[C12H22O11], where k = k´[H2O]v. The reaction
is said to be of pseudo first order although the kinetic data indicates
for v ≈ 6. The reaction rate constant k is respectively called the
pseudo reaction rate constant.
The number of molecules which react in an elementary step is
called the molecularity of the elementary reaction. Molecularity is
defined only for elementary reactions and should not be used to
describe overall reactions that consist of more than one elementary
step. In fact the only elementary reaction which exists is the
bimolecular one. However, all the types of reactions given below
are considered to be elementary reactions. The unimolecular
reaction and the trimolecular reactions, however, are in fact simple
reactions from the kinetic point of view since they can be explained
by trivial combinations of bimolecular reactions. Examples of each
case are given below:
A → products
unimolecular reaction
A + B → products
2A → products
A + B + C → products
2A + B → products
3A → products
bimolecular reactions
trimolecular(termolecular) reactions.
No elementary reactions involving more than three molecules have
been suggested because of the very low probability of collision of
more than three molecules at the same moment of time, which of
course is the perquisite of a chemical reaction.
The rates of chemical reactions are obtained from measurements of
concentrations of reaction mixture components as a function of
time. Analytical chemistry methods may be applied to samples
taken from reaction mixture when the reaction can be stopped
suddenly. This may be done e.g. by rapid cooling of the samples
for high-temperature reactions.
Physical methods are especially useful for determining the rate of a
chemical reactions because they offer the possibility to measure
continuously the advancement of the reaction. Spectroscopic
methods are most generally used techniques. An important
characteristic of any measurement method is its response time. The
measuring device must obviously respond much more rapidly than
the concentrations are changing. Pulsed lasers have opened up new
opportunities for investigation of very fast reactions. Especially
reactions occurring in the picosecond (10-12 s) range may be studied
in this way.
To study gas reactions at high temperatures it is necessary to heat
the gas to the reaction temperature very quickly because the
reaction then occurs rapidly. This may be accomplished by means
of shock tube in which a shock wave is used to heat the gas with
high speed.
Photochemical reactions may be initiated suddenly by a light pulse
from a flash lamp or laser. In the flash photolysis method, a
reaction vessel is exposed to a very high intensity flash of visible or
ultraviolet light. The flash dissociates the molecules in the sample
and the concentrations of these species are then determined over a
period of time using subsequent flashes at much lower intensities.