Second-order reactions, classes examples

An analytical method provides the solution in closed form , i.e. a formula can be given for the time evolution of the concentrations. This possibility only exists for a rather limited class of differential equations, even in the special subclass of kinetic differential equations. Methods of how to derive solutions can be found in the book by Kamke (1959) or in problems books like those by Filippov (1979), Matveev (1983), or Krasnov et al. (1978). The special case of kinetic differential equations is treated by Rodiguin Rodiguina (1964) who published the explicit solution for many first order reactions, and by Szabo (1969), who collected results for second order reactions too (together with realistic chemical examples). 

Few reactions are truly first-order, in that they involve decomposition of a molecule without intervention of a second molecule. The classic example of a true first-order reaction is radioactive decay, such as 222Rn — 218Po + a-particles. In the atmosphere, by far the most important class of first-order reactions is photodissociation reactions in which absorption of a photon of light (hv) by the molecule induces chemical change. Photodissociation, or photolysis, reactions are written as... 

Because this is an introductory text, we have chosen our examples so that they will fit one of these three simple models. In some cases, however, reaction kinetics can be quite complicated and the appropriate model might be none of the three we tried here. If none of these three plots were linear, we would have to conclude that the reaction was not zero, first, or second order. Similar integrated rate law models can be derived for other cases, but this is beyond the scope of an introductory class. 

A wide variety of problems are amenable to the Redfield methodology in addition to those discussed here. Some of the most important, in our view, are as follows. First, problems involving the interaction of strong laser fields with a condensed-phase system are often difficult to solve because the construction of a small, physically intuitive zeroth-order quantum subsystem Hamiltonian is difficult the numerical methods described above will make it possible to expand the size of the quantum subsystem and allow the problem to be attacked much more easily. A second class of problems involves relaxation of complex systems (e.g., vibronic or vibrational relaxation of a molecule in a liquid) [42,43, 72]. A third class of problems would be concerned with chemical dynamics in which the system could not be described easily by a single reaction coordinate, for example, general proton transfer reactions [98] or the isomerization of retinal in bacteriorhodopsin [120]. A low-dimensional system probably is adequate for these cases, but a nontrivial number of quantum levels will still be required. 

The second class of metallic solutes is represented by the less electropositive metals. Here, the situation is the reverse of that discussed above. Sodium amalgam is widely used in industry and in the laboratory and is a good example of this class. Upon addition of mercury to liquid sodium, the reactivity of the sodium toward aqueous solutions is vastly reduced, and reaction with hydrogen is slower by an order of magnitude than that for pure sodium. This fact is important in the operation of the Solvay cell for the industrial production of sodium hydroxide by electrolysis of brine, in which sodium amalgam forms one of the electrodes. In such amalgams, valency electrons from the conduction band of liquid sodium, which would normally be responsible for its chemical reactivity, are partially localized on the mercury atoms, thus inhibiting the reactivity of sodium. 

Two schemes have been proposed which systematize the available stability constant data. In addition to data contained in the compilations given at the end of this chapter, more qualitative evidence, based for example on the results of displacement reactions, has been included in arriving at the generalizations. Historically, the first scheme is that due to Chatt and Ahrland who pointed out that electron acceptors may be placed in one of three classes. Class-a metals, the most numerous, form more stable complexes with ligands in which the coordinating atom is a first-row element (N, O, F) than with those of an analogous ligand in which the donor is a second-row element (P, S, Cl). Class-b has the relative stabilities reversed. It is not difficult to extend the stability relationships to include heavier donor atoms. Class-a behaviour is, then, typified by a stability order... 

The classes of reactions which produce structure range from complex redox chains [1-4,9], through moderately non-linear reactions of the type A + B C [7,9], to simple first order isomerizations [7,12]. Examples are the photoreduction of Fe(III) [1,6] and the air oxidation of reduced methylene blue [5] (very complex), protonation of methyborange [9] (second or quasi-first order), and photochromic isomerization [7]. See Fig. 4. 

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