6DETERMINATION OF COMPLEX REACTION MECHANISMS

space resolution, may require probabilistic rather than deterministic interpretations and analyses. This type of statistical approach is presented in chapters 7 and 8 and in parts of chapter 13, where some elements of Bayesian statistics are introduced. The inference of causality may then be difficult or impossible, as many mechanisms may lead to the same joint probability distribution of all events in the system.

If measurements in time and space can be made at smaller and smaller intervals, then ultimately we come to molecular events, such as reactive collisions, and to a sequence of causal connectivities. To some this seems an obvious fact, to others it is a belief; the pursuit of these issues enters the field of philosophy.

A final point needs to be made regarding taking measurements in time. Suppose we sample a signal y(t) at a sampling rate of . Nyquist showed that if there exists a critical frequency fc = 1/(2 ) for a system such that measurements are limited to frequencies smaller than fc, then the function y(t) is completely determined by these measurements.

Acknowledgments Much of this book is based on our research, with appropriate references, and we copy freely from our publications with additions, deletions, and changes. The use of “we” refers to the respective authors of the referenced section. The contributions of our coauthors are gratefully acknowledged.

References

[1]Berry, R. S.; Rice, S. A.; Ross, J. Physical Chemistry, 2nd ed.; Oxford University Press: Oxford, 2000.

[2]Ljung, L. System Identification: Theory for the User, 2nd ed.; Prentice-Hall: Englewood Cliffs, NJ, 1999.

[3]Nelles, O. Nonlinear System Identification: From Classical Approaches to Neural Networks and Fuzzy Models; Springer: Berlin, 2001.

[4]Broekstra, G. C-analysis of C-structures: representation and evaluation of reconstruction hypotheses by information measures. Int. J. Gen. Sys. 1981, 7, 33–61.

[5] Marriot, F. H. C. The Interpretation of Multiple Observations; Academic Press: San Francisco, 1979.

[6]Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T. Numerical Recipes in C; Cambridge University Press: New York, 1988.

[7]Otnes, R. K.; Enochson, L. Applied Time Series Analysis: Basic Technique; Wiley: New York, 1978.

[8]Michael, G. Biochemical Pathways; Elsevier/Spektrum Akademischer Verlag: Heidelberg, 1998.

2

Introduction to Chemical

Kinetic Processes

It is useful to have a brief discussion of some kinetic processes that we shall treat in later chapters. Some, but not all, of the material in this chapter is presented in [1] in more detail.

2.1Macroscopic, Deterministic Chemical Kinetics

A macroscopic, deterministic chemical reacting system consists of a number of different species, each with a given concentration (molecules or moles per unit volume). The word “macroscopic” implies that the concentrations are of the order of Avogadro’s number (about 6.02 × 1023) per liter. The concentrations are constant at a given instant, that is, thermal fluctuations away from the average concentration are negligibly small (more in section 2.3). The kinetics in many cases, but far from all, obeys mass action rate expressions of the type

dA	= k(T )Aα Bβ . . .	(2.1)
dt

where T is temperature, A is the concentration of species A, the same for B, and possibly other species indicated by dots in the equation, and α and β are empirically determined “orders” of reaction. The rate coefficient k is generally a function of temperature and frequently a function of T only. The dependence of k on T is given empirically by the Arrhenius equation

k(T ) = C exp−Ea /RT

(2.2)

7

8DETERMINATION OF COMPLEX REACTION MECHANISMS

where C, the frequency factor, is either nearly constant or a weakly dependent function of temperature, and Ea is the activation energy.

Rate coefficients are averages of reaction cross-sections, as measured for example by molecular beam experiments. The a priori calculation of cross-sections from quantum mechanical fundamentals is extraordinarily difficult and has been done to good accuracy only for the simplest trimolecular systems (such as D + H2).

A widely used alternative approach is based on activated complex theory. In its simplest form, two reactants collide and form an activated complex, said to be in equilibrium. One degree of freedom of the complex, a vibration, is allowed to lead to the dissociation of the complex to products, and the rate of that dissociation is taken to be the rate of the

reaction. The rate for the forward reaction is
kf =	kBT	(KV ) [A][B]	(2.3)
	h

where kB is Boltzmann’s constant and KV is the equilibrium constant in terms of concentrations for the complex with one vibrational degree of freedom removed. On the right-hand side of eq. (2.3) there is frequently added a transmission coefficient to attempt to make up for the simplifications in that equation, due to (1) not all activated complexes dissociate to products, some dissociate back to reactants; (2) there may be quantum mechanical tunneling through the activation barrier separating reactants from products; and (3) the activated complex may not be at equilibrium and nonequilibrium effects may have to be taken into account.

Reactions among ionic species require explicit consideration of the interactions of the electric charges on the ions. We cite only the primary salt effect formulated by Brønsted

and Bjerrum [1, p. 920]. For a bimolecular reaction between two ions we have
AZA + BZB → [(AB)]ZA+ZB → products	(2.4)

where ZA and ZB are the charges on the ions A and B respectively and (AB) is the activated complex. With the use of activated complex theory and the expression of the equilibrium constant for eq. (2.4) in terms of activities, we can write

K =	aAB	=		[(AB)] γAB			(2.5)
	aAaB		[A] [B] γAγB

where the γ ’s are activity coefficients, and the rate coefficient in the forward direction is

kf =	kB T	K	γAγB			(2.6)
	h		γAB
In dilute solutions the activity coefficient of ion i in water at 25◦C is
log γi = −0.509Zi2I 1/2,			I = 21 i		Zi2mi	(2.7)

with mi denoting molality. I is called the ionic strength. Hence the rate coefficient for the forward direction is

kf =	kB T	K · 10	1.018ZAZB I 1/2
			(2.8)
	h

For reaction between two ions of like charge the rate coefficient increases with increasing ionic strength; for reaction between two ions of unlike charge the rate coefficient decreases with increasing ionic strength.

INTRODUCTION TO CHEMICAL KINETIC PROCESSES

9

2.2Disordered Kinetics

There are systems in which rate coefficients may vary randomly according to some probability distribution. Consider an amorphous solid in which an active intermediate is formed by radiation chemistry. The rate of reaction of that intermediate depends on the specific immediate surroundings of the intermediate, which may vary with location in the amorphous solid. The rate coefficients vary in space in the sample but not in time and such a system is said to have static disorder.

In systems with dynamic disorder, rate coefficients vary randomly in time; the structure of the environment of a reaction site varies as the reaction proceeds. Examples include protein–ligand interactions which vary in time as the protein changes its molecular configuration in time. Other examples are chemical processes in the atmosphere and aquatic environments. (For further analysis of disordered kinetics, see chapter 12.) A system may display both static and dynamic disorder depending on the external conditions. For example, in the case of protein–ligand interactions the reaction rates (rate coefficients) are random because a protein can exist in many different molecular configurations, each characterized by a different reaction rate. At low temperatures the transitions among different configurations are negligibly small (frozen out) and the system shows static disorder. For higher temperatures transitions among different configurations do occur, and the system shows dynamic disorder. For a few references on disordered kinetics, see [2]–[5].

2.3Fluctuations

We make a distinction between internal, or thermal, fluctuations and external fluctuations, those due to variations in the environment of the system. Thermal fluctuations are present in every system at nonzero temperature. For systems at equilibrium the absolute magnitude of the fluctuations, say in concentration of a chemical species, increases with the number N of molecules of that species in a given volume, and is proportional to N1/2. The relative fluctuations therefore decrease proportional to N−1/2. At equilibrium the probability of a fluctuation of small magnitude is given by a Gaussian (normal) distribution. These relations lead to better understanding of the concept of a macroscopic system in which fluctuations are small compared to average quantities [1].

For nonequilibrium systems the probability of a fluctuation is given by the solution of the master equation. The absolute and relative fluctuations, at least for some nonequilibrium systems, are of similar magnitude to equilibrium systems.

As the number of molecules in a system at a given volume increases, the averages of the concentrations, averaged over the probability distribution, begin to obey mass action kinetics. For systems with mass action kinetics, whether at equilibrium or not, relative fluctuations tend to zero in the thermodynamic limit (the limit of large N at constant density N/V ), and are said to be nonintermittent. Rate processes in disordered systems have qualitatively different behavior. There is no universal behavior in the thermodynamic limit, but relative fluctuations tend to a constant nonzero value in that limit. These fluctuations are said to be intermittent and they make a significant contribution to the average values of the concentrations. For further references, see [6,7].