(2) The rate laws of chain reactions:
(2-1) Example of a chain reaction having a simple rate law - The Rice-Herzfeld mechanism for the pyrolysis of ethanal in the absence of air.(3) Special case: Explosions
A chain reaction is one where an intermediate formed in one step generates an intermediate in a subsequent step, then that intermediate generates another intermediate, and so on. The intermediates in a chain reaction are called chain carriers. In radical chain reactions, radicals (i.e. species with unpaired electrons) act as carriers. Other carriers include neutrons (in nuclear fission) and ions.
Chain reaction involved several types of steps (elementary reactions):
(2-1) Example of a chain reaction having a simple rate law - The Rice-Herzfeld mechanism for the pyrolysis of ethanal in the absence of air.
The pyrolysis (i.e. thermal decomposition) of ethanal (acetaldehyde, CH3CHO) is found to exhibit a simple rate law where:
Some ethane is also detected.
This reaction proceeds according the Rice-Herzfeld mechanism where:
According to steady state approximation, the net rate change of the intermediates may be set to zero, i.e.:
i.e. (i)+(ii) =>
and hence the steady-state concentration of the methyl radical is:
i.e. from equation (b), the rate of formation of methane (i.e. the rate of the overall reaction) is given by:
in accordance with three 3/2 order observed experimentally.
NOTE: Although this mechanism captures the principal features of the reaction, it does not accommodate the formation of various by products such as propanone and propanol. Some of these by-products are formed in significant quantities. All this indicates that the true mechanism for pyrolysis of ethanal must be more complicated than the Rice-Herzfeld mechanism.
As we had seen in the introductory section of chemical kinetics, although the stoichiometric equation for the reaction between hydrogen and bromine to give HBr looks very simple, the rate law is very complicated indicting a complex mechanism (see fig 1) i.e.:
Fig. 1: The A schematic representation of the mechanism of the reaction between hydrogen and bromine. Note how the reactants and products are shown as arms to the circle, but the intermediates (H and Br) occur only within the circle. Similar diagrams are used to depict the action of catalysts.
The proposed mechanism for this reaction (see Fig. 1.) is:
Note regarding step (d), termination: The third body M removes the energy of recombination. Other possible termination steps include the combination of H atoms to form H2 and the combination of H and Br atoms to from HBr. However it turns out that only Br atom recombination it important.
Through this mechanism, it may be deduced that the net rate of formation of HBr (product) is given by:
where the concentrations of the intermediates may be determined by invoking the steady state approximation, i.e.:
i.e. from (ii) + (i) we obtain:
which substituted back in (i) gives:
or: dividing 'top' and 'bottom' by k`b :
Thus, the net rate of formation of HBr may be written solely in terms of reactants, i.e.:
which simplifies to:
as predicted by the empirical rate law:
(1) The presence of [HBr] in the denominator is a sign that HBr is acting as an inhibitor, and reducing the rate of formation of product. Likewise, the presence of [Br2] stems from the role of Br2 in the removal of the reactive hydrogen radicals from the chain.
(2) In the examples we have considered, the observed rate laws are reproduced by the mechanisms. In the past, that used to be essentially the end of the calculation (but not of the experimental investigation). However, we can make use of computers to integrate the approximate rate law numerically, and hence predict the time dependence of the HBr concentration. (See Fig. 2). Mathematical software may also be used to integrate the original coupled rate laws without needing to invoke the steady-state approximation.
Fig. 2: The numerical integration of the the HBr rate law, can be used to explore how the concentration of HBr changes with time. These runs began with stoichiometric proportions of hydrogen and bromine; the curves are labelled with the value of 2k'-1.
(3) HBr may also be produced through in a process where the initiation step is photochemical. In this case, the initiation step is of the form:
whilst the other steps are as in the thermal process (b-d).
where represents the rate at which photons of the appropriate frequency are absorbed by the volume in which the reaction occurs. In this case, the rate of formation of HBr in this case is given by:
The verification of this is provided here.
(a) A thermal explosion is due to the rapid increase of reaction rate with increasing temperature. If the energy released by an exothermic reaction cannot escape, the temperature of the system rises and the reaction goes faster. The acceleration of the rate results in a faster rise of temperature, so the reaction goes even faster... catastrophically fast.
(b) A chain-branching explosion may occur when there are chain-branching steps in a reaction, for then the number of chain centres grows exponentially and the rate of reaction may cascade into an explosion.
An example of both types of explosion is provided by the reaction between hydrogen and oxygen:
Although the net reaction is very simple, the mechanism is very complex. A chain reaction is involved, and the chain carriers include . Some steps are:
The occurrence of an explosion depends on the temperature and pressure of the system, and the explosion regions for the reaction are shown in Fig. 3:
Fig. 3: The explosion limits of the hydrogen
+ oxygen reaction. In the explosive regions the reaction proceeds explosively
when heated homogeneously.