FASE I: Conocimiento Preliminar
2.11 HERRAMIENTAS DE LA AUDITORIA DE GESTIÓN
Only a relatively few kinetic schemes for complex reactions have differential rate expressions which can be integrated straightforwardly. Examples are the following.
A!k1 P1
A!k2 P2
THE KINETIC ANALYSIS FOR COMPLEX REACTIONS 79
The differential equations describing this scheme are
d½A=dt¼k1½A þk2½A ð3:55Þ þd½P1=dt¼k1½A ð3:56Þ þd½P2=dt¼k2½A ð3:57Þ On integration these give
½At¼ ½A0expfðk1þk2Þtg ð3:58Þ
½P1t¼ k1
k1þk2½A0f1exp½ðk1þk2Þtg ð3:59Þ
½P2t¼ k2 k1þk2
½A0f1exp½ðk1þk2Þtg ð3:60Þ
andk1andk2can both be determined directly from the experimental data.
A!k1 I!k2 P
and this gives differential equations
d½A=dt¼k1½A ð3:61Þ þd½P=dt¼k2½I ð3:62Þ þd½I=dt¼k1½A k2½I ð3:63Þ which on integration give
½At ¼ ½A0expðk1tÞ ð3:64Þ
½Pt ¼ 1 k2k1
½A0fk2½1expðk1tÞ k1½1expðk2tÞg ð3:65Þ
½It ¼ k1 k2k1
½A0fexpðk1tÞ expðk2tÞg ð3:66Þ Again the rate constants can be found directly from the experimental data, and it is also possible to draw graphs showing how [I] varies with time for assumed relative values ofk1andk2(see Figures 3.14 and 3.15 below).
Other schemes that can be easily integrated include A!k1 I1!k2 I2!k3 P
A!k1 I
IþX!k2 PþX A!k1
k1
I!k2 P
3.18.1 Relatively simple reactions which are mathematically complex
One problem for kineticists is that only a relatively slight increase in complexity of the kinetic scheme results in differential equations which cannot be integrated in a straightforward manner to give a manageable analytical expression. When this happens the differential equations have to be solved by either numerical integration or computer simulation. This is a mathematical limitation of the use of integrated rate expressions which is not apparent in the kinetic scheme. Typical schemes which are mathematically complex areAþB!k1
k1
I!k2 P AþB!k1
k1
I1!k2
k2
I2!k3 P
These mathematically complex kinetic schemes should be compared with the mathematically straightforward schemes:
A!k1
k1
I!k2 P A!k1
k1
I1!k2
k2
I2!k3 P
These schemes are not much less complex chemically, but can be integrated easily.
The important distinction between the two categories lies in the first reversible step.
In the mathematically simple case the forward step is first order, while in the complex case it is second order. Schemes which are first order in all steps are mathematically simple and can be integrated in a straightforward manner. When there are second order steps, the situation is normally more complicated.
3.18.2 Analysis of the simple scheme A
!k1I
!k2P
This sort of analysis is very important in the formulation of the steady state approximation, developed to deal with kinetic schemes which are too complex mathematically to give simple explicit solutions by integration. Here the differential rate expression can be integrated. The differential and integrated rate equations are given in equations (3.61)–(3.66).
THE KINETIC ANALYSIS FOR COMPLEX REACTIONS 81
3.18.3 Two conceivable situations
Remember:reactions involving more than one step often have one step that is slow compared with the other steps, and the overall rate is limited by the rate of this step, and can be approximated to the rate of this slow step. When this occurs, the slow step is termed the rate-determining step. Not all reactions have a rate-determining step. Sometimes the individual reactions have comparable rates, and the overall rate cannot be approximated to the rate of any individual step.
For the scheme
A!k1 I!k2 P there are two conceivable situations.
1. k2k1. This implies that the production of the intermediate I is the rate-determining step, or the slowest step, in the sequence. Once the intermediate is formed it is very rapidly converted to product P. Graphs of the consequent concentration dependences of A, I and P on time can be drawn (Figure 3.14). The magnitudes of [I] have been very grossly magnified to enable them to be placed on the graph.
time [I]
[A]
[P]
concentration
Figure 3.14 Graph of concentration versus time for reactant, product and intermediate in the reaction A!k1 I!k2 P wherek2k1
Whenk2k1there is always a build-up of intermediate to avery very low, but steady, value of the concentration, which remains almost constant at this value until reaction is virtually complete, when it falls back to zero. The build-up to the steady plateau often occurs so very rapidly that it can only be followed using the specialized techniques of fast reactions. Determination of the very low concentrations of the intermediate is also difficult, and often cannot be achieved, though the modern laser-induced fluorescence methods are helping here.
2. k2k1. This implies that the intermediate is formed from A much faster than it is converted to P. The graph, Figure 3.15, shows that the intermediate concentra-tion never builds up to a steady value, and that the magnitudes of the intermediate concentrations are always relatively large, and comparable to either [A] or [P]
depending on the stage of reaction. Towards the end of reaction [I] tails off to zero.
A similar result is obtained ifk2k1.
These two situations show totally different behaviour, and the graphs demonstrate two points:
in some reactions there is a build-up of the intermediate to an almost constant value; in others there is not;
time concentration
[P]
[A]
[I]
Figure 3.15 Graph of concentration versus time for reactant, product and intermediate in the reaction A!k1 I!k2 P wherek2k1
THE KINETIC ANALYSIS FOR COMPLEX REACTIONS 83
the criterion for build-up to a plateau is that the intermediate must be present in very, very low concentrations. This implies that it must always be removed in a very fast reaction, where in this casek2k1. In general the intermediate can be removed in one reaction, or in a series of reactions, but these must befast.