CHEMICAL KINETICS
10.2 SECOND-ORDER REACTIONS
10.3.3 Reversible Reactions
No reaction goes to completion. Reactions we call “complete” are those in which the concentration of, for example, reactant A is negligible or not detectable in the reaction
A→B
This is the case when the standard free energy of A is much larger than that of B, so that the reaction is accompanied by a significant decrease in free energy and leads to B more stable than A.
In many cases, something closer to a free energy balance exists. We write the reaction as an equilibrium characterized by an equilibrium constant Keq(Chapter 7).
Kinetic rate constants kf and kbfor forward and back reactions compete to establish the equilibrium balance:
A
kf
←→
kb
B kf
kb
=Keq= B A
When A and B are mixed in concentrations A and B that are not the equilibrium concentrations, either the forward or the back reaction is faster than the other and the system is displaced so as to approach equilibrium.
The rate of depletion of A,
−d A dt
obs
=kfA−kbB
is smaller than the rate that would be found if the reaction went to completion or if we had some mechanism for removal of B immediately as it is formed. The rate of
approach to equilibrium is always less than either the forward reaction or the back reaction because kf and kbwork in opposition to one another.
For simple reversible systems, A←→B A+B←→C A+B←→C+D etc.
equations can be worked out that relate the observed rate constant kobsfor the approach to equilibrium to kf and kb, the elementary reactions that contribute to it (Metiu, 2006).
The concept of equilibrium as the result of opposite forward and back reactions is called the principle of detailed balance; for example, at equilibrium,
kfA−kbB=0
The equations implied by the principle of detailed balance rest on the assumption that the concentrations of reactant and product vary in the simplest possible way. This assumption is often violated and must never be taken for granted.
Some enzyme-catalyzed reactions, including the famous Michaelis–Menten (Houston, 2001) mechanism, are examples of more complicated reaction mecha- nisms. Despite their complexity, they can often be broken down into elementary steps and equilibriums. The kinetics of complex reactions can sometimes be simplified by regarding one component of the reaction as a constant during part of the chemical process. This is the steady-state approximation (Metiu, 2006).
An especially important class of reaction mechanisms is that of the chain reactions in which one molecular event leads to many, possibly very many, products. The classic example for chemists is production of HBr from the elements:
H2+Br2 →2HBr for which we might guess the rate law to be
1 2
d [HBr]
dt =k [H2] [Br2] wrong but that guess would be wrong. Instead, the rate law is
1 2
d [HBr]
dt =k [H2] [Br2]1/2 right
OTHER REACTION ORDERS 153 An acceptable mechanism for this reaction depends upon three elementary reaction types, initiation, chain, and termination:
Initiation Br2+M→k1 2Br· +M Chain Br· +H2→k2 HBr+H·
Chain H· +Br2
k3
→HBr+Br·
Termination Br· +Br· →k−1Br2
The point of this proposed mechanism is that the cyclic chain steps can continue indefinitely because the Br·free radicals used in the first chain step are produced in the second chain step. Initiation can be by a rather unlikely process such as collision with a high-energy molecule M in the first step or impact of a photon from a flame or spark as in chain explosions. Though initiation may not occur very often, it can have a large effect. A hydrogen–oxygen explosion is an example. A very small spark can cause a very large explosion. The HBr chain, though it may yield many molecules for one initiation step, does not, of course, go on forever. Some step such as recombination of the Br·free radicals terminates the chain or we run out of Br2or H2and the reaction stops.
If we assume that initiation and termination are rare events by comparison to the chain steps, for every Br·used up in the first step of the chain, one is produced in the second chain step, so the amount of free radical present at any time during the reaction is constant:
d [Br·]
dt =0
This is an example of the steady-state hypothesis. Making this assumption, a few lines of algebra (Houston, 2006) lead to the correct rate equation given above:
1 2
d [HBr]
dt =k [H2] [Br2]1/2
If, by a more complicated mechanism, two or more reactive species are produced on each step, the amount of reactive species may increase rapidly. For example, if two reactive species are produced at each step, the geometric series, 1, 2, 4, 8, 16,. . .is followed. If the chain steps are fast, this kind of mechanism takes place with explosive violence. This type of mechanism is characteristic of nuclear fission bomb reactions.
Free radical and (controlled) chain reactions are also characteristic of some bio- chemical reactions, and they can behave in ways that are beneficial or detrimen- tal to the organism. A hydroperoxide RCOO·free radical chain may destroy lipid molecules in a cell wall by lipid peroxidation with disastrous results for the cell.
Free radicals derived from tocopherol (vitamin E) antioxidants act as sweepers in the blood, interfering with chain propagation, thereby slowing or preventing cell degra- dation. Free radicals have been detected in rapidly multiplying natal or neonatal cells
(beneficial) but they are also found in cancers undergoing uncontrolled reproduction.
The biological role of free radicals is complex and not completely understood.
10.4 EXPERIMENTAL DETERMINATION OF THE RATE EQUATION