Chemical reaction rates macroscopic description

Section 11.1 The Macroscopic Description of Chemical Reaction Rates... 

The low-temperature chemistry evolved from the macroscopic description of a variety of chemical conversions in the condensed phase to microscopic models, merging with the general trend of present-day rate theory to include quantum effects and to work out a consistent quantal description of chemical reactions. Even though for unbound reactant and product states, i.e., for a gas-phase situation, the use of scattering theory allows one to introduce a formally exact concept of the rate constant as expressed via the flux-flux or related correlation functions, the applicability of this formulation to bound potential energy surfaces still remains an open question. 

Chemical kinetics deals with quantitative studies of the rates at which chemical processes occur, the factors on which these rates depend, and the molecular acts involved in reaction processes. A description of a reaction in terms of its constituent molecular acts is known as the mechanism of the reaction. Physical and organic chemists are primarily interested in chemical kinetics for the light that it sheds on molecular properties. From interpretations of macroscopic. kinetic data in terms of molecular mechanisms, they can gain insight into the nature of reacting systems, the processes by which chemical bonds are made and broken, and the structure of the resultant product. Although chemical engineers find the concept of a reaction mechanism useful in the correlation, interpolation, and extrapolation of rate data, they are more concerned with applications... 

Understanding chemical reactions has been a major preoccupation since the historical origins of chemistry. A main difficulty is to reconcile the macroscopic description in which reactions are rate processes ruling the time evolution of populations of chemical species with the microscopic Hamiltonian dynamics governing the motion of the translational, vibrational, and rotational degrees of freedom of the reacting molecules. 

The knowledge of the two-minima energy surface is sufficient theoretically to determine the microscopic and static rate of reaction of a charge transfer in relation to a geometric variation of the molecule. In practice, the experimental study of the charge-transfer reactions in solution leads to a macroscopic reaction rate that characterizes the dynamics of the intramolecular motion of the solute molecule within the environment of the solvent molecules. Stochastic chemical reaction models restricted to the one-dimensional case are commonly used to establish the dynamical description. Therefore, it is of importance to recall (1) the fundamental properties of the stochastic processes under the Markov assumption that found the analysis of the unimolecular reaction dynamics and the Langevin-Fokker-Planck method, (2) the conditions of validity of the well-known Kramers results and their extension to the non-Markovian effects, and (3) the situation of a reaction in the absence of a potential barrier. 

Ray Kapral came to Toronto from the United States in 1969. His research interests center on theories of rate processes both in systems close to equilibrium, where the goal is the development of a microscopic theory of condensed phase reaction rates,89 and in systems far from chemical equilibrium, where descriptions of the complex spatial and temporal reactive dynamics that these systems exhibit have been developed.90 He and his collaborators have carried out research on the dynamics of phase transitions and critical phenomena, the dynamics of colloidal suspensions, the kinetic theory of chemical reactions in liquids, nonequilibrium statistical mechanics of liquids and mode coupling theory, mechanisms for the onset of chaos in nonlinear dynamical systems, the stochastic theory of chemical rate processes, studies of pattern formation in chemically reacting systems, and the development of molecular dynamics simulation methods for activated chemical rate processes. His recent research activities center on the theory of quantum and classical rate processes in the condensed phase91 and in clusters, and studies of chemical waves and patterns in reacting systems at both the macroscopic and mesoscopic levels. 

Another problem that has been tackled by multivariate statistical methods is the characterization of the solvation capability of organic solvents based on empirical parameters of solvent polarity (see Chapter 7). Since such empirical parameters of solvent polarity are derived from carefully selected, strongly solvent-dependent reference processes, they are molecular-microscopic parameters. The polarity of solvents thus defined cannot be described by macroscopic, bulk solvent characteristics such as relative permittivities, refractive indices, etc., or functions thereof. For the quantitative correlation of solvent-dependent processes with solvent polarities, a large variety of empirical parameters of solvent polarity have been introduced (see Chapter 7). While some solvent polarity parameters are defined to describe an individual, more specific solute/solvent interaetion, others do not separate specific solute/solvent interactions and are referred to as general solvent polarity scales. Consequently, single- and multi-parameter correlation equations have been developed for the description of all kinds of solvent effects, and the question arises as to how many empirical parameters are really necessary for the correlation analysis of solvent-dependent processes such as chemical equilibria, reaction rates, or absorption spectra. 

The following discussion represents a detailed description of the mass balance for any species in a reactive mixture. In general, there are four mass transfer rate processes that must be considered accumulation, convection, diffusion, and sources or sinks due to chemical reactions. The units of each term in the integral form of the mass transfer equation are moles of component i per time. In differential form, the units of each term are moles of component i per volnme per time. This is achieved when the mass balance is divided by the finite control volume, which shrinks to a point within the region of interest in the limit when aU dimensions of the control volume become infinitesimally small. In this development, the size of the control volume V (t) is time dependent because, at each point on the surface of this volume element, the control volnme moves with velocity surface, which could be different from the local fluid velocity of component i, V,. Since there are several choices for this control volume within the region of interest, it is appropriate to consider an arbitrary volume element with the characteristics described above. For specific problems, it is advantageous to use a control volume that matches the symmetry of the macroscopic boundaries. This is illustrated in subsequent chapters for catalysts with rectangular, cylindrical, and spherical symmetry. 

Several different approaches have been utilized to develop molecular theories of chemical kinetics which can be used to interpret the phenomenological description of a reaction rate. A common element in all approaches is an explicit formulation of the potential energy of interaction between reacting molecules. Since exact quantum-mechanical calculations are not yet available for any system, this inevitably involves the postulation of specific models of molecules which only approximate the real situation. The ultimate test of the usefulness of such models is found in the number of independent macroscopic properties which can be correctly explained or predicted. Even so, it must be remembered that it is possible for incorrect models to predict reasonably correct macroscopic properties because of fortuitous cancellation of errors, insensitivity of the properties to the nature of the model, relatively large uncertainties in the magnitudes of the properties, or combinations of such effects. 

Mechanisms. Mechanism is a technical term, referring to a detailed, microscopic description of a chemical transformation. Although it falls far short of a complete dynamical description of a reaction at the atomic level, a mechanism has been the most information available. In particular, a mechanism for a reaction is sufficient to predict the macroscopic rate law of the reaction. This deductive process is vaUd only in one direction, ie, an unlimited number of mechanisms are consistent with any measured rate law. A successful kinetic study, therefore, postulates a mechanism, derives the rate law, and demonstrates that the rate law is sufficient to explain experimental data over some range of conditions. New data may be discovered later that prove inconsistent with the assumed rate law and require that a new mechanism be postulated. Mechanisms state, in particular, what molecules actually react in an elementary step and what products these produce. An overall chemical equation may involve a variety of intermediates, and the mechanism specifies those intermediates. For the overall equation... 

This chapter treats the descriptions of the molecular events that lead to the kinetic phenomena that one observes in the laboratory. These events are referred to as the mechanism of the reaction. The chapter begins with definitions of the various terms that are basic to the concept of reaction mechanisms, indicates how elementary events may be combined to yield a description that is consistent with observed macroscopic phenomena, and discusses some of the techniques that may be used to elucidate the mechanism of a reaction. Finally, two basic molecular theories of chemical kinetics are discussed—the kinetic theory of gases and the transition state theory. The determination of a reaction mechanism is a much more complex problem than that of obtaining an accurate rate expression, and the well-educated chemical engineer should have a knowledge of and an appreciation for some of the techniques used in such studies. 

The information needed about the chemical kinetics of a reaction system is best determined in terms of the structure of general classes of such systems. By structure we mean quahtative and quantitative features that are common to large well-defined classes of systems. For the classes of complex reaction systems to be discussed in detail in this article, the structural approach leads to two related but independent results. First, descriptive models and analyses are developed that create a sound basis for understanding the macroscopic behavior of complex as well as simple dynamic systems. Second, these descriptive models and the procedures obtained from them lead to a new and powerful method for determining the rate parameters from experimental data. The structural analysis is best approached by a geometrical interpretation of the behavior of the reaction system. Such a description can be readily visualized. 

The goals of microscopic theories of chemical rate processes are to define the range of validity of the macroscopic law, to provide microscopic expressions for the rate coefficients and to extend the kinetic description to smaller distance and time scales. Traditional approaches model the isomerization reaction by motion of a particle of mass m across a potential barrier in a thermal bath of temperature T. Transition state theory (TST) [2] assumes an equilibrium distribution at the barrier top the rate is given by the equilibrium one-way flux across the barrier. The rate thus obtained has the form ... 

Oscillations and chaos are observed frequently in chemical systems. Most of the experimental investigations of these phenomena have been carried out for well-stirred systems where spatial degrees of freedom are assumed to play no role. If this is the case the system may be described in terms of chemical rate equations for a small number of macroscopic chemical concentrations. The periodic or chaotic attractors typically have low dimensions and can be characterized using the tools of dynamical systems theory [20]. Chemical systems may also display spatiotemporal oscillations and chaos. If spatial degrees of freedom are important the appropriate macroscopic model is the reaction-diffusion equation. The attractors may have high dimension and the theoretical description of such spatiotemporal states is less well developed. 

All of these phenomena arise out of the random reactive and elastic collision events in the system and a fundamental understanding of how macroscopic, self-organized chemical structures appear must be based on descriptions that go beyond the macroscopic, mean-field rate laws or reaction-diffusion equations. In this section we use the reactive lattice-gas method to examine how molecular fluctuations influence oscillatory and chaotic dynamics. In particular we shall show how system size, diffusion, reactions and fluctuations determine the structure of the noisy periodic or chaotic attractors. 

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