A n introduction to the 35th Annual Chemical Engineering Symposium of the A C S Division of Industrial and Engineering Chemistry, to be held at the Illinois Institute o f Technology, December 72-73, 7968
Mathema ticaI Aspects
of
Chemical Reaction
RUTHERFORD ARlS
are two approaches to the mathematical aspects There of the kinetics of chemical reactions. Traditionally, mathematics has been regarded as a valuable tool for the elucidation of the course of a chemical reaction and the prediction of its effects and consequences. But more recently the function of mathematics in the refinement of physical concepts and in the articulation of the current paradigm has been appreciated and developed. The program of this symposium is intended to be balanced between these two approaches and to avoid the mundane and banausic in the applications which are to be discussed. Though all of the contributions speak to the problem clarifying the foundations of chemical kinetics, there are two groups of papers that are particularly concerned with this question. Some years ago Dr. Bartholomay pointed out that chemical change is ultimately a stochastic process and could be so represented. The relation between this representation and the statistical physics of intramolecular transitions will also concern him in his talk on “Stochastic Formulations in Chemical Kinetics.” I n his work ‘‘On the Thermochemistry of Reacting Mixtures,” Dr. Bowen will use the principles of rational thermodynamics and draw on his previous work on mixtures to present a theory of reaction in nonlinear elastic materials and of the growth and decay of waves propagated in such media. Dr. Sellers’ work on the “Combinatorial Aspects of Chemical Reaction” has already elucidated the algebraic structure lying behind the notion of the mechanism of a reaction. In his paper he will discuss the application of combinatorial analysis to the problems of enumerating all the modes of reaction consistent with a given set of rules of mechanism and of generating a set of symbols, one for each mode. In working “Toward an Axiomatization of Chemical Equilibria,” Dr. Shapiro will show that the essential propcrties of equilibrium systems can be related to a set of linear constraints on a free energy function which is to be minimized. The axioms that can be laid down for the free energy function will be related to the laws of thermodynamics. A second group of papers is concerned with the pseudosteady-state hypothesis. Dr. Silveston’s “Examination of the Stationary State Hypothesis” will be by a compu-
AUTHOR Rutherford Aris is Professor of Chemical Engineering at the University of Minnesota, Minneapolis, Minn. Professor Aris is Program Chairman of the mmposium he introduces here. 20
INDUSTRIAL A N D ENGINEERING CHEMISTRY
developed a tational study of the steps of a photo-initiated free-radical reaction. Dr. Higgins’ discussion of the “Theory of the Steady State” will also use the computer to show that the hypothesis often provides a good approximation to the exact kinetics. It need hardly be said that here is no attempt to be comprehensive in this program, for the pseudo-steady-state could well support a full-scale meeting of itself. The bankruptcy of the popular distinction between pure and applied is clearly exposed by a third group of papers. Dr. Coleman’s work “On the Stability of Solutions of General Evolution Equations” will relate the stability criteria of thermostatics to the realities of a dynamical situation and indicate why the concepts of thermostatics are so successful. This clearly addresses itself to the foundations, but it involves the notion of stability, a notion that has been “applied” in many other directions of which two will be considered by Drs. Gavalas and Luss. Dr. Gavalas will use the recent theory of nonlinear operators to discuss “Nonlinear Phenomena in Glow Discharges and Other Autocatalytic Reactions.” Dr. Luss, with Dr. Lee, will use the maximum principle of parabolic partial differential equations to show how the regions of “Stability of a Chemical Reaction with Intraparticle Diffusion of Heat and Mass” may be determined. Another group might be made of the remaining papers by noting their emphasis on kinetics and the evolution of chemical change. Dr. Wei’s work on the analysis of monomolecular reaction systems is well known, and here, in a n interesting paper co-authored with Dr. J. C. W. Kuo, he will extend it to the “Analysis of Exactly Lumpable Systems.” Because of the enormous number of species present it is often desirable to lump many species together in a group, and he will discuss the conditions under which this can properly be done. Dr. Zimmerman’s treatment of “Replication Kinetics for Biological Macromolecules” touches on another area that would justify a whole symposium to itself. He will be particularly concerned with the unwinding of the parent helix to expose template sites and the kinetic implications of this. The “Interactions of Reaction and Diffusion in Open Systems,” to be discussed by Dr. Scriven and Mr. Othmer, will also show how impossible it is to drop each paper into a definite compartment. They use stability analysis to classify the behavior that may be seen in multicomponent systems, giving a detailed analysis of those with two and three components. It is hoped this program will provide a n interesting melange of papers and provoke lively discussions that will open up new avenues of advance in the mathematical basis of chemical kinetics. Those who attend may be assured that the kindness and attention we shall receive from our hosts a t the Illinois Institute of Technology will be in complete contrast to the impressions of the Chicago scene given by the events of last August, and we look forward to enjoying their hospitality.
dimensionless grout, lately? If you have, and you’re just about to call it the Humperdinck Number, ho€d on a minute. Someont? else may have used and named the group already. To find out who, you need a couple of I&EC reprints: Dimensionless Groups Part I (I&EC 58(3), 46-60, March 1966) Dimensionless Groups Part I1 (I&EC 60(3), 71-78,March 1968) In these two articles, G. D. Fulford and J, P. Catchpole alphabetically list 285 dimen sionles s groups with their symbols, definitions, fields of significance and use, and original references. Special tables also list the groups under their constituent variables and. the exponents to which they are raised, Order from: Reprints ACS Publications, Dcpt. G 1155 16th St. N.W. Washington, D.C.20036
0Dimensionless Groups4 (1966) 0Dimensionless Groups41 (1968)50 cents each $ total remittance enclosed, payable to American Chemical Society
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