Advances in Catalysis and Related Subjects, Volume 13

ADVANCES IN CATALYSIS AND RELATED SUBJECTS

VOLUME 13

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ADVANCES IN CATALYSIS AND RELATED SUBJECTS VOLUME 13

EDITED BY

D. D. ELEY Nottingham, England

P. W. SELWOOD Evanston, Illinois

PAULB. WEISZ Paulsboro, N e w Jersey

ADVISORY BOARD

A. A. BALANDIN Moscow, U . S. S. R .

P. H. EMMETT Baltimore, Maryland

G. NATTA IMilnlro, Ita1.y

J. H.

DE

BOER

Delft, T h e Netherlands

J. HORIUTI Sapporo, Japan

E. K. RIDEAL London, En gland

P. J. DEBYE Ithaca, N e w York

W. JOST Gottingen, Germany

H. S. TAYLOR Princeton, N e w Jersey

1962

ACADEMIC PRESS, NEW YORK AND LONDON

COPYRIGHT 0 1962, BY ACADEMIC PRESSINC. ALL RIGHTS RESERVED

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United Kingdom Edition Published by

ACADEMIC PRESS

INC.

(LONDON) LTD.

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PRINTED IN THE UNITED STATES OF AMERICA

Contributors R. COEKELBERGS, Ecole Royale Militaire, Institut Interuniversitaire des Sciences Nucle'aires, Brussels, Belgium A. CRUCQ,Ecole Royale Militaire, Institut Interuniversitaire des Sciences Nucle'aires, Brussels, Belgium A. FARKAS, Houdry Process and Chemical Company, Marcus Hook, Pennsylvania

A. FRENNET, Ecole Royale Militaire, Institut Interuniversitaire des Xciences Nucle'aires, Brussels, Belgium

L. H. GERMER, Bell Telephone Laboratories, Murray Hill, N e w Jersey G. A. MILLS,Houdry Process and Chemical Company, Marcus Hook, Pennsylvania CHARLESD. PRATER, Socony Mobil Oil Company, Incorporated, Research Department, Paulsboro, New Jersey

F. S. STONE,Department of Physical and Inorganic Chemistry, University of Bristol, Bristol, England JAMES WEI, Socony Mobil Oil Company, Incorporated, Research Department, Paulsboro, hTew Jersey

PAUL B. WEisz, Socony Mobil Oil Company, Incorporated, Research Department, Paulsboro, New Jersey

V

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Quo Vadis Catalysis A Preface Some years back one of our colleagues, leaning back in his chair with a gaze towards the future, was heard dreamily t o utter the words: “Some day we shall know the secret to catalysis.” Somehow this moment has haunted our memories. We remember the brief moment of euphoria which seemed to overcome us a t the thought of this beautiful prospect; and then the moments of reflection which agonizingly continued to suggest that as surely as this thought was beautiful, there was something erroneous buried in its beauty. Years later the words would still echo in our memory. We suppose the reason being t h a t i t reflected a question which many (including ourselves) may have loosely or seriously entertained. Was there going t o be a last volume of the Advances in Catalysis, containing a single and final chapter on the secret? Can we expect a simple answer to why a catalyst X should have the power of making A react with B? I n the whole of “chemistry” we have been dealing with the basic question of why two molecules A and B are reactive to a certain degree ; or, why they are reactive a t all. Why is A more reactive toward B than toward B’? What is the cause of this chemical specificity? I n the very process of exploring these questions we have created an inorganic chemistry, an organic chemistry, a biochemistry, ionic reaction mechanisms, quantum chemistry, molecular orbital models, Ligand field theories, and the like. We seem to be content (or a t least moderately so) to recognize that, although the unifying theory to all chemical reactivities is available to us, in principle, in the form of polyelectronic wave-mechanics, its complexity is too great to be manageable much beyond the simplicity of a hydrogen molecule. Hence; we silently accept the need for staking out certain limited areas within the myriad of various types of atomic arrangements (molecules), wherein certain not-so-general but manageable mechanistic models and theories aid us in our understanding and extrapolation of chemical experiences. Should we seek to study the reason for reactivity of A to B in the presence of X with more intense demands for a simple unifying answer? Obviously, in this task, we take on all of chemistry above, except that each n-body problem (reactivity of A and B ) in chemistry, now becomes a t least an ( n 1)-body problem in catalytic chemistry (A and B and X). This recognition should calm our ambitions a little. But perhaps the “presence of X” involves a special phenomenon quite different from the problems of chemical reactivities among A, B, B’, etc.?

+

Vii

...

PREFACE

Vlll

Indeed, the originally suggested concept of catalysis took on the apparent attire of strict chemical noninvolvement; the catalyst X was not “consumed” but merely [‘present.” It seems we have matured to the realization that any influential effects must imply involvement through some sort of force fields between catalyst and reaction partners; we acknowledge these forces to be electronic and therefore chemical in nature, and thus we imply the existence of a t least temporary chemical complex or bond formation with the catalyst. Clearly then, the n-body problem of chemistry (e.g., of A and B) becomes a t least an ( 2 n 1)-body problem (A, B ; X ; AX, BX) of catalytic chemistry (even before we worry about such strictly additional problems as energy heterogeneity of sites, polyfunctional catalysis, side reactions, etc.). This logical sequence of concepts suggests a simple definition of catalysis, namely: A chemical rate process i s catalyzed when it requires the formation of a steady state concentration of a chemical combination of at least one of the reaction partners with another chemical agent, as the catalyst. For example, with X catalyzing the transformation A B -+ AB, we may have

+

+

A+XeAX B AX e h B

+

+X

A+XeAX B+XeBX AX BX e ABX, ABX, + A B + X + X ,

+

or several other possible variations as regards detail, but not in principle. Having arrived a t this definition by small steps of reasoning we find that we have been well anticipated by ,J, A. Christiansen in an earlier volume of these Advances in the course of a more rigorous analysis in “The Elucidation of Reaction Mechanisms by the Method of Intermediates in Quasi-Stationary Concentrations” (Vol. V, p. 31 1 ff .) . With the events reduced to a set of chemical reactions, what happens to the historical concept that catalyst must not be “used up”? Once the steady state concentration of complex or compound involving catalyst has been produced, (and which itself can be considered as reversibly recoverable after contact), there is no net consumption. Continuing catalyst consumption would take place if the catalyst complex is not physically retained in the reaction space, but this represents an incidental (and well known) “engineering” circumstance. The heterogeneous catalyst facilitates this purely “engineering” circumstance. We might

PREFACE

ix

add the observation that in the case of a solid catalyst it becomes inherently difficult to measure or even detect the initially produced steadystate quantity of “reacted” catalyst, as the surfaces alone (or only parts thereof) are involved in this “consumption.” As we cast the catalytic reaction into a sequence of fairly ordinary steps of chemical interaction kinetics, we return to contemplate our simple “secret”: We conclude that there is no fundamental differencc between attacking the problem of catalytic reactivity and the entire scope of chemical reactivity, but the former must involve relatively greater complexity; if we put side-by-side the boxes which each contain one of the various fields of endeavor concerning molecular processes (inorganic chemistry, organic chemistry, biochemistry, enzymology, quantum chemistry, etc., etc.) , we find ourselves-as catalysis researchers-defining for ourselves not an additional vertical box but a certain horizontal slice through all of them. I n some ways we claim to exercise a “unifying” action across them, for the definition of the catalytic process describes the relatively simple unifying principle which defines the slice we make across the “disciplines” or “sub-disciplines” of molecular interactions. Perhaps this principle is the only simple secret, and beyond this we cannot hope to get simpler than the whole of chemistry. So we do not feel too bad that the final chapter is not a t hand; and that on the contrary, we are-in this six-chapter volume of the Advances in Catalysis-penetrating deeply into the heart of quite a number of major portions of the broad realm of catalysis: An intensive review (F. S. Stonc) examines experimental work and interpretation of interactions where the reactants (A, B, etc.) are some of the simple gases, oxygen, hydrogen, carbon monoxide, and carbon dioxide, and the catalyst (X) is one of a few selected oxide solids. We are carried to vivid realization of the importance of electronic phenomena by the experiences of photon-influenced chemisorption and catalysis on these solids. Another chapter (R. F. R. Coekelbergs, A. Crucq, and A. Frennet) carries us into the relatively new field of radiation catalysis, where A, B, . . . , and X are not a closed or thermal system, but receive discrete forms of energy; specifically where the solid catalyst acts as a transducer for passing the energy of high energy nuclear radiation to a gaseous chemical reaction system. As we have noted, the set of reaction rates which interconnect various chemical species are the most important properties in any catalytic experience, or investigation; i t is a pleasure therefore to devote a large section (J. Wei and C. D. Prater) to notable advances in the analysis and interpretation of measured transformation rates in terms of the actual individual reaction rate parameters between transforming species

X

PREFACE

of a complex system (and any system with more than two interconverting materials is complex!). Although this work may appear to be highly mathematical to those who only glance at the print, i t is, in fact, highly descriptive and physically most meaningful for direct use by the experimentalist. I t s meaning and practical utility span the entire field of all rate process studies and kinetics from chemistry to enzymology. Broad are the implications and application of the principles of polystep reactions on polyfunctional catalyst combinations (P. B. Weisz) . Here we deal with reaction sequences in which two catalyst species X and Y (or more) participate in one set of reaction sequences. Some of the general principles combine thermodynamics and physical parameters to yield important information and criteria for such rate processes, generally whether they occur in hydrocarbon transformations, organic chemistry, in a petroleum plant or in a living cell. We have seen much work done with such ‘(pet” chemical systems as hydrogen-deuterium exchange, hydrocarbon conversion, or carbon monoxide oxidation, and have felt that new insights may well be gained from studies in depth of more varied chemical systems; the transformations group (isocyanates) are an example of such a class of the -N=C=O (A. Farkas and G. A. Mills). They are involved in some large scale chemistry of present-day polymer technology. Theoretically, this molecular system begins to bring along many of the subtle effects of electronic and steric detail on reactivity which attain full magnitude in biochemical systems. Inasmuch as catalytic chemistry involves very special chemical complexes, in small concentrations, and in special places like only the surface of a solid, the development of new techniques suitable for the development of new information is always of great importance to the field. The development of low energy electron diffraction by back-reflection from surfaces to a new and powerful research tool (L. H. Gernier) marks a recent such advance reported in this volume. It presents another potentially important route to direct (‘inspection” of the structural detail of the surface of X or the complexes AX, on solids. The image of the catalytic researcher is assuming a n ever increasing variety of arms and legs as i t becomes a superposition of very many individual images, which coincide strongly only in the common core of the “catalyzed” rate process-as described by perhaps a definition as we tried above. We expect to find some of the most exciting and rewarding developments resulting from an ever increasing amount of “coupling” between disciplines. It will be the continuing goal in this series to call upon relevant progress in many areas of investigative endeavor. July, 1962

P. B. WEISZ

Professor W. E. Garner

It was a sad moment for his friends and colleagues in chemistry departments throughout the world, to read of the death of Professor Garner on March 4th, 1960. He was particularly well known to catalytic chemists, by his papers over some thirty years, and by his effective contributions t o the series of conferences on catalysis which were initiated by the Faraday Society Discussion a t Liverpool in 1950. Many readers of this notice will remember hearing his paper presented to the First International Conference on Catalysis a t Philadelphia in 1956, and will recall with pleasure his characteristically modest, yet persuasive contributions in discussion. Garner’s influence extended far beyond his own research group a t Bristol, his leadership and inspiration being felt over a wide circle of scientists. As befitted a student of Professor F. G. Donnan, Garner possessed wide interests, and was a connoisseur of painting, silver, and ceramics. A pleasant recollection is that of a visit in his company, and th a t of a colleague, to the Washington Art Gallery. Characteristically, Garner’s comments were few in number, but possessed th a t illuminating quality expected from a true collector. A kindly man, and of equable temperament, Garner’s judgment in chemical problems and University affairs was eagerly sought by his lecturers and students, as they successively secured professorships or distinction in industrial science and government. H e was unsurpassed as a chairman of committees, where his natural sympathy for the viewpoints of his fellow members ensured easy cooperation. Professor William Edward Garner was born in 1889 and studied chemistry a t the University of Birmingham. In 1913 he was awarded an 1851 Exhibition Fellowship to work with Professor Tammann a t Gottingen. Returning to England a t the outbreak of the First World War, he worked a t Woolwich Arsenal on problems concerning high explosives. H e retained an interest in this subject for the whole of his life, publishing important papers, both on flames and gaseous explosions and on the decomposition of solid aeides. I n 1919, Garner returned to Birmingham University as an assistant lecturer, but soon left for University College, London, where he worked in Professor F. G. Donnan’s department until 1927. Garner’s work during this period covered a very wide field of activity and included an interest in the physical chemistry of biological systems. This interest saw fruition some twenty years later in the encouragement of similar studies in his department a t Bristol and the commencement of a School of Biological Chemistry in that University. In xi

xii

PROFESSOR W. E. GARNER

1927 he was appointed Leverhuline Professor of Physical and Inorganic Chemistry a t the University of Bristol. I n the period up to 1939 he was cspecially active in the fields of gaseous explosions, heterogeneous catalysis, and heats of adsorption, and the kinetics of solid decomposition reactions. His studies of nucleation in solids, which he related to the general theories of solid state physics, put this subject on a precise basis. His calorimetric studics of adsorption on metallic oxides are classical, and formed a springboard for his subsequent intensive development of this subject. During the 1939-1945 war, Garner’s department was largely devoted to government explosives research, while he played a big role in this field. H e was appointed Chief Superintendent of Armaments Research in 1944, and his war-time efforts were recognized when he was made a C.B.E. in 1946. From 1945, until his retirement in 1954, Garner’s research efforts were largely in the field of chemisorption and catalysis on metal oxides, The observations of Garner, Gray, and Stone on the effects of adsorbed gases on the semiconductivity of cuprous oxide, formed the growing point for a thorough study of certain oxide systems using the methods and concepts of solid state physics. He was particularly happy correlating the findings of these newer approaches with those of the classical calorimetric method. Garner also gave active encouragement to similar studies on metals, alloys, and enzymes. Garner took the lead in organizing many Faraday Society Discussions, the repercussions of which would resound in his department in the following weeks. As an example, shortly after the war, Garner became convinced that free radicals played a role in biological reactions, but the present author had regretfully to report that the para-ortho conversion of hydrogen was too insensitive to test this view. However, with the advent of electron spin resonance techniques this has become a fruitful field of research. Garner’s great success as a laboratory director was due to his ability to stimulate both discussion and experiment on current problems of this type. Garner was elected a Fellow of the Royal Society in 1937, and received a number of other honors. I n recent years he traveled in France, Spain, Belgium, the United States, and other countries, either to attend meetings, or to lecture, and i t is the present author’s impression that he much enjoyed these travels. Although he took a lead in organizing the recent Faraday Society Discussion in Kingston, Ontario, he could not he encouraged to attend in person. Just before the meeting he was taken ill a t his home in Bristol, and a telegram was sent from those present a t Kingston to wish him well, but, unfortunatelv his recovery was of short duration. D. D. ELEY June, 1961

Contents CONTRIBUTORS .

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V

PREFACE .

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vii

PROFESSOR W . E . GARNER.

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1

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xi

. Chemisorption and Catalysis on Metallic Oxides BY F . S. STONE

I . Introduction . . . . . . . . . . . I1. The Adsorption and Oxidation of Carbon Monoxide . . I11. The Uptake of Oxygen by Metals and Metallic Oxides . IV . The Emergence and Significance of the Electronic Factor V . Adsorption and Catalysis on Doped Oxides . . . . VI . Photoadsorption and Photocatalysis . . . . . . VII . Conclusion . . . . . . . . . . . . References . . . . . . . . . . . .

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1 5 21 27 35 40 49 50

BY R . COEKELBERCS. A . CRUCQ. A N D A . FRENNET I . Introduction . . . . . . . . . . . . . . I1. Experimental Studies of Some Irradiated Heterogeneous Systems . . I11. General Degradation Scheme of Radiation Energy in Solids . . . IV . Radiation Catalysis . . . . . . . . . . . . . . . . . . V . Some Comments About the Experimental Results VI . General Conclusion . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . .

55 56 80 110 126 129 130 134

.

2 . Radiation Catalysis

3 . Polyfunctional Heterogeneous Catalysis

BY PAIJLB . WEISZ I . Introduction . . . . . . . . I1. Principles of Polystep Catalysis . . .

. .

. .

. .

I11. The Technique of Physically Mixed Catalyst Components IV . Some Major Polystep Reactions of Hydrocarbons . . V . The Petroleum Naphtha “Reforming” Reaction . . . VI . Other Polystep Reactions . . . . . . . . VII . Conclusions . . . . . . . . . . . . References . . . . . . . . . . . . ...

Xlll

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. . .

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. . .

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137 138 156 157 175 179 188 189

xiv

CONTENTS

4 . A N e w Electron Diffraction Technique. Potentially Applicable to Research in Catalysis

BY L . H . GERMER I . Experimental Apparatus I1. Oxygen on Nickel . References . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

192 193 201

5 . The Structure and Analysis of Complex Reaction Systems BY JAMES WEI AND CHARLES D . PRATER

I . Introduction . . . . . . . . . . . . . . I1. Reversible Monomolecular Systems . . . . . . . . . I11. The Determination of the Values of the Rate Constants for Typical Reversible Monomolecular Systems Using the Characteristic Directions . . . . . . . . . . . . . . . . I V . Irreversible Monomolecular Systems . . . . . . . . . V . Miscellaneous Topics Concerning Monomolecular Systems . . . VI . Pseudo-Mass-Action Systems in Heterogeneous Catalysis . . . VII . Qualitative Features of General Complex Reaction Systems . . . VIII . General Discussion and Literature Survey . . . . . . .

204 208

. . . . . . . . . . . . . . .

364

I . The Orthogonal Characteristic System . . . . . . . . I1. Explicit Solution for the General Three Component System . . . I11. A Convenient Method for Computing the Characteristic Vectors and Roots of the Rate Constant Matrix K . . . . . . . IV . Canonical Forms . . . . . . . . . . . . . V . List of Symbols . . . . . . . . . . . . . References . . . . . . . . . . . . . . .

364 372

Appendices

244 270 295 313 339 355

376 380 381 390

.

6 Catalytic Effects in Isocyanate Reactions

BY A . FARKAS A N D G . A . MILLS I . Introduction . . . . . . . . . . . . . . 393 I1. Polymerization of Isocyanates . . . . . . . . . . 395 I11. Reactions of Isocyanates with Compounds Containing Active Hydrogen 401 IV . Applications . . . . . . . . . . . . . . 441 References . . . . . . . . . . . . . . . 443 AUTHORINDEX SUBJECTINDEX

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447 455

Chemisorption and Catalysis on Metallic (Oxides F. S. STONE Department of Physical and Inorganic Chemistry, Un,iversitu of Bristol, Bristol, England Page

I. Introduction.. . .......................................... 1 11. The Adsorption n of Carbon Monoxide .............. 5 A. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 R. Heats of Adsorption and Complex Formation ............... 6 C. Related Results from Other Experimental Methods.. . . . . . . . . . . . . . . . . . . 11 D. The Catalytic Oxidation of Carbon Monoxide a t Low Temperatures. 111. The Uptake of Oxygen by Metals and Metallic Oxides.. . . . . . . . . . . . . . . B. Different Forms of Chemisorbed Oxygen.. . . . . . . . . . . . . .

A. Semiconductivity Changes During Chemisorption ..................... 27 B. The Boundary-Layer Theory of Chemisorption . . . . . . . . . . . . . . 30 V. Adsorption and Catalysis on Doped Oxides. .

. . . . . . . . . . . . . . . . . . . 35

B. The Oxidation of Carbon Monoxide over Doped Nickel Oxide Catalysts. . 36 C. Other Catalytic Studies with Doped Oxides.. . . . . . . . . . . . . . . . . VI. Photoadsorption and Photocatalysis ........................... 40 A. Introduction.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

E. Photoeffects with Nickel Oxide.. . . . . . . . . .................... VII. Conclusion. .. ................................................... References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........................

49 49

50

I. Introduction The study of the various reactions of carbon monoxide, hydrogen, and oxygen at oxide surfaces holds a particularly important place in the development of research in heterogeneous catalysis. Not only are the well-established technical aspects of these reactions continuously monitored by those engaged in chemical industry, but the chemist interested in fundamental studies of the interaction of gases with oxides naturally turns to the behavior of these gases because of the combination of high reactivity and molecular simplicity which they afford. Finally, for the chemical physicist, 1

2

F. S. STONE

they offer a unique opportunity to establish the links between the phenomena of catalysis and the theory of the solid state. The present article aims to trace the course of research in this field with special reference to the work carried out a t Bristol since the establishment by Garner in 1928 of a research group in chemisorption and solid state chemistry. The main emphasis will be on the developments of the postwar period and their counterparts in current research, so it is appropriate in the Introduction to summarize some of the more important results of the early period. The great technological achievements in catalysis during the first quarter of the century were very largely based on empirical work, and it was not until the period from 1925-1930 that much fundamental research on oxide catalysts began to take shape. Highly significant among the empirical developments of the nineteen-twenties was Patart’s discovery of the marked efficiency of a mixture of zinc oxide and chromium oxide for the synthesis of methanol, and it was this observation which stimulated Garner to begin a study of the interaction of carbon monoxide and of hydrogen with ZnO Crz03and other oxides. This work soon led to the conclusion that these gases could be adsorbed on oxides either “reversibly,” in the sense that they could be recovered chemically unchanged on heating the oxide, or “irreversibly,” meaning that they could only be recovered on heating as carbon dioxide or water respectively. In the case of ZnO CrZO3, for example, Garner and Kingman ( 1 ) showed that some of the carbon monoxide or hydrogen taken up at room temperature and low pressure was evolved as such on heating to 100-200°, but was then slowly readsorbed (Fig. 1). On further heating to 350°, the adsorbed gas was recovered a s carbon dioxide or water, respectively. An important feature of the work was the study of the heats of adsorption, using a calorimetric technique already initiated some years earlier by Garner and successfully used in his classic studies of oxygen adsorption on charcoal (2). By this means it was shown that the heat of adsorption of the “reversibly” adsorbed gas lay in the range between 10 and 30 kcal./mole, establishing that this adsorption was chemical rather than physical in character. It was not, therefore, to be classified with the low temperature adsorptions of hydrogen being discussed a t about the same time by Benton and White ( 3 ) and by Taylor (4), which had much lower heats and were physical in nature. Although a t the time (1931) the results of Garner and Kingman tended somewhat to obscure the pressing issue of the distinction between physical adsorption and Taylor’s ‘lactivated” adsorption, their true significance, more readily appreciated in retrospect, is that they were the first clear results to establish the existence of more than one type of chemisorption for reducing gases on oxides.

-

3

CHEMISORPTION A N D CATALYSIS ON METALLIC OXIDES

Later studies by Garner and his co-workers showed that the fraction of carbon monoxide or hydrogen reversibly chemisorbed a t room temperature varied from oxide to oxide. Zinc oxide was shown to be a case where the adsorption of carbon monoxide a t room temperature was completely reversible. The heat of adsorption, determined both calorimetrically ( 5 ) and isosterically ( 6 ) ,was in the range 12-20 kcal./mole. For several other oxides, however, notably chromia, Mnz03 and Mm0 3 Crz03,the heat of adsorption of carbon monoxide was higher and the chemisorption was 150

100

--a. e a

Y) u)

e

n 50

0

10

20

30

40

so

Time (min.)

FIG. 1. Desorption and readsorption of hydrogen on ZnO . Cr2O3.

irreversible. Moreover, these cases of irreversible chemisorption of carbon monoxide a t room temperature were found to leave the surfaces unsaturated with respect to oxygen. The ability to take up oxygen, small in extent when studied before the adsorption of carbon monoxide, was found subsequently to be appreciable. In addition, the amount of oxygen adsorbed after CO treatment corresponded in several cases to one-half the volume of the preadsorbed CO. These interesting observations assumed a more quantitative significance when set alongside the values of the heats of adsorption obtained concurrently with the volumetric measurements. These are summarized in Table

TABLE I Heats of Adsorption of Carbon Monoxide, Carbon Dioxide, and Oxygen on Oxides of Zinc, Chromium, and Manganese (All Heats i n kcal./male) ~

Heat of adsorption Oxide

Authors

of

co

Qco

ZnO . CrzOa ChOa

MnzOa Mn20,. CrzOs

Garner and Veal. Dowden and Game+ Garner and WardC Wardd

44 29 67 46

Heat Of adsorption of oxygen after CO

+

-45 -110 48 78

~~~~

Heat of adsorption Heat of >@a* of a mixture adsorption of co J 5 0 2 of coz

Qo,*

Garner, W. E., and Veal, F. J., J . Chem. Soc. p. 1487 (1935). Dowden, D. A., and Garner, W. E., J . Chem. SOC.p. 893 (1939). c Gamer, W. E., and Ward, T., J . Chem. SOC. p. 857 (1939). dWard, T., J . Chem. SOC.p. 1244 (1947). a

QCO

~~~

+

QCCO+Y~O~)

66 84 91 85

66 82 85

QCO

+ WQot* -

s P

QCO,

15 18 23 20

QCO.

51 66 68 65

20

3

5

CHEMISORPTION AND CATALYSIS ON METALLIC OXIDES

I ( 7 ) , where reference is also given to the original papers (5, 8-10). QCO is the heat of adsorption of carbon monoxide on an “oxidized” surface, i.e., one which had been pretreated with oxygen a t 450°, evacuated, and cooled t o room temperature. Qo,* is the heat of adsorption liberated when oxygen was subsequently adsorbed, and Q(cO+%o2, the heat liberated per mole of gas taken u p when a mixture of 2CO:1O2 was admitted to a n “oxidized” surface. Note here the agreement with the corresponding QCO $~Qo,*sum. QCO, is the heat of adsorption of COz on a n “oxidized” surface. With the exception of ZnO Cr203,where the CO heat may be low on account of a contribution from reversible adsorption (cf. Fig. l), the $ ~ Q O , * - QCO, is very close to the heat of the reaction quantity QCO CO(g) $
+

-

+

+

+

02-

Me2+

I

02-

f

02-

Me2+

----)

02-

I1

Me2+

02-

I11

Thus, according to this model, irreversible adsorption of carbon monoxide implied that an anion vacancy was created. Analogous experiments were attempted with hydrogen, but results were very much less complete. It was clear, however, that the surfaces after irreversible adsorption of hydrogen were not unsaturated in the same way as with carbon monoxide.

II. The Adsorption and Oxidation of Carbon Monoxide A. INTRODUCTION The studies of Garner and his co-workers in the years 1928-1939, which had established the existence of two types of carbon monoxide and hydrogen chemisorption on oxides and which identified “irreversible” chemisorption with incipient reduction, were followed in the immediate postwar period by an intensive study of the properties of copper oxide (12-15). The work was later extended to nickel oxide (16) and cobalt oxide ( 1 7 , l S ) .With each of these oxides it was established that carbon monoxide was capable of reacting not only with lattice oxygen, but also with adsorbed oxygen. The concept of irreversible chemisorption involving a carbonate ion and ulti-

6

F. S. STONE

mate reduction remains an acceptable explanation of the facts for the interaction of CO with the oxides a t temperatures above lOO-ZGO", but a t room temperature other processes are to be distinguished. It is the purpose of this section t o describe the purely chemical aspects of this work and to illustrate their bearing on the catalytic oxidation of carbon monoxide a t low temperatures. The research as a whole provides a detailed example of the classical approach in heterogeneous catalysis to the fundamental problem of correlating catalytic activity with the stability of adsorbed species.

B. HEATS OF ADSORPTION AND COMPLEX FORMATION Most of the work to be discussed in this section has been carried out on oxide films about 100A. in thickness present on a base of underlyiiig metal. The parent metal has first been formed as granules by processes designed to minimize the oxide content. Copper was prepared by the reduction of an aqueous suspension of copper hydroxide with hydrazine. Nickel and cobalt were prepared in the vacuum system by thermal decomposition of nickel oxalate and cobalt formate, respectively. The metals have then been reduced in situ with hydrogen. Finally the oxide film has been prepared by controlled oxidation, usually at low pressure. The surface areas of the oxides lay in the range between 1 and 10 m.2/g. With the exception of one calorimeter containing oxidized copcer foil ( I S ) , heats of adsorption have been measured using calorimeters of TyFe If described by Garner and Veal (19). A typical heat liberation of 0.5 calorie, corresponding, for examgas with a heat of adsorption of 22.4 ple, to the adsorption of 0.5 ~ mof .a ~ kcal./mole, could be measured to an accuracy of 1%. The method of using an oxide film on a metal base had two important advantages. First, it enabled the oxide surface to be cleaned more readily after exposure to oxygen. Oxygen, sometimes difficult to desorb on baking out, was in the present case simply incorporated into the oxide film a t the expense of some underlying metal. After studies of CO adsorption, outgassing may be accompanied by some reduction of the surface: in such cases oxygen has t o be admitted to the oxide to replenish the surface. It is obviously important to be able t o remove the oxygen rcmaining as a chemisorbed film, and some of the earlier studies (see Section 1) were open to objection on these grounds. The second advantage came in the adsorption calorimetry. The high thermal conductivity of the oxide-coated granules (relative to pure oxide) assisted heat distribution within the catalyst mass. We begin with a discussion of results obtained on copper (cuprous) oxide. Magnetic studies (20) have confirmed that the method of preparation yielded only cuprous oxide. On a surface free from adsorbed oxygen, carbon monoxide could be adsorbed reversibly a t 20". If, however, oxygen had been preadsorbed at 20°, the rapid adsorption of carbon monoxide

CHEMISORPTION AND CATALYSIS ON METALLIC OXIDES

7

was found to be accompanied by a much larger liberation of heat. I n place of 20 kcal./mole, for example, for the heat of adsorption of carbon monoxide on a baked-out surface, Garner, Stone, and Tiley (15) found 49 kcal./mole for the corresponding heat on the surface carrying adsorbed oxygen. Mutatis mutandis, preadsorption of carbon monoxide was found to enhance the heat of adsorption of oxygen from 55 kcal./mole to 100 kcal./mole. It was clear that chemical interaction had occurred. The product was quite stable in the presence of excess oxygen, but in the presence of excess carbon monoxide there was a slow distillation of carbon dioxide into the adjacent cold trap. We may very simply see that the assumption of a conversion of adsorbed oxygen into carbon dioxide by the attack of carbon monoxide from the gas phase would provide a heat even greater than the observed exalted heat of 49 kcal./mole. The heat of the reaction CO(g) $5Oz(g) = COs(g) is 67 kcal. and the heat of dissociative adsorption of oxygen on baked-out copper oxide is 55 kcal./mole, so from O(ads) = the difference 67 - ($5 X 55) we see that the reaction CO(g) COz(g) is 39 kcal. exothermic. The product is in fact mainly in the adsorbed state, so a realistic estimate of the heat of interaction requires the molar heat of adsorption of carbon dioxide to be added. If we set this a t 20 kcal./mole (cf. Table I),we conclude that the heat of adsorption of carbon monoxide on an oxygenated surface should be 59 kcal./mole, if reaction to give COz(ads) were complete. By a similar calculation it follows that the molar heat of adsorption of oxygen on to the surface containing preadsorbed carbon monoxide should be 2(67 - 20 20) = 134 kcal. if all the adsorbed carbon monoxide is converted to adsorbed COZ. The observed heat was 100 kcal./mole. It was noted that the reactivity of adsorbed oxygen towards carbon monoxide decayed with time, so the fact that the observed exalted heat is less than the calculated one for the case of CO on the oxygenated surface is understandable in these terms. The lack of agreement with the oxygen heat is less satisfactory. It was reasonable, therefore, to inquire if some intermediate state other than COZ(,,,,might not be entering into the picture. Some key experiments on the adsorption of carbon dioxide on the copper oxide catalyst provided the clue to a possible alternative. It was found that carbon dioxide could be adsorbed on the oxide only if oxygen had been preadsorbed or if oxygen and carbon dioxide were admitted together. There was negligible adsorption of carbon dioxide on the outgassed oxide. The greatest amounts of COz were adsorbed when oxygen and COZ were admitted together. By following the rate of liberation of heat in an adsorption calorimeter it was confirmed that oxygen with its high heat of 55 kcal./mole was adsorbing first. The only reasonable conclusion to be drawn was that carbon dioxide required to be present in some kind of surface complex with oxygen in

+

+

+

8

F. S. STONE

order to achieve stability on the surface. By increasing the ratio of COZ/OZ in the gas mixture admitted to the oxide, greater capture of oxygen by COz was obtained and the ratio of CO2/02 in the adsorbed state ultimately approached 2 : 1. Since oxygen is almost certainly present on the surface mainly as atoms, the most likely formula for the complex appeared to be cO3.* I n the light of these experiments with COz adsorption, it was of interest to see whethcr better agreement with the experimental heats of interaction could be obtained on the basis of a reaction leading to the COS complex rather than to C02,,,,. The heat of adsorption of CO2 on to a preoxygenated surface was found to be 21 kcal./mole. Using, as before, the two other experimental heats of 55 and 20 kcal./mole for oxygen and for CO respectively on the baked-out surface, the heats t o be expcctcd for formation of the C 0 3 complex on copper oxide are readily calculated as follows (15): (1) Reaction of CO with preadsorbed oxygen = + 28 kcal. + = cos(,,,,+ 21 kcal. CO(,) + %OZ(~) = COZ(~) + 67 kcal. (ii) + (iii) - (i), Co(,)+ = COS(~,~) + 60 kcal. O(nds)

COZ,~)

O(ads)

(9 (ii) (iii)

20Sds)

(2) Reaction of

0 2

with preadsorbed carbon monoxide =

co(ad8)+ 20 kcal.

0 ~ coCd8) ~ ~ = )

96 kcal. co+ade)

Co(,)

(i)

+ (ii) + (iii) - (iv),

+

(iv)

These heats are to be compared with the experimental values of 49 kcal. and 100 kcal. respectively. There is no improvement in the agreement for the CO heat, where the observed time dependence of the reactivity of adsorbed oxygen enters as a complication, but the oxygen heat is in very much better accord with the concept of COa complex formation. The surface reaction of carbon monoxide and oxygen on nickel oxide (16) and on cobalt oxide (I?’) was also investigated by the same method of

* It is appropriate here to emphasize the distinction between this species, which we shall call the “C03 complex,” and Garner’s “carbonate ion” discussed in Section I. The essential distinction visualized is that the co3 complex is formed by interaction with adsorbed oxygen, while Garner’s carbonate ion is formed by interaction with oxide ions of the lattice. Both will be negatively charged. I n the event that a distinction cannot be drawn in a particular circumstance between chemisorbed oxygen ions and oxide ions of the lattice, the difference degenerates towards a semantic one. For the purpose^ of the present discussion, however, we wish to keep in mind the concept of interactions with adsorbed oxygen, and we shall therefore use the term “COOcomplex.”

9

CHEMISORPTION AND CATALYSIS ON METALLIC OXIDES

adsorption calorimetry. With these oxides the additional experiment was made of admitting oxygen after CO or COz adsorption had taken place on a surface containing presorbed oxygen. A clear distinction resulted. Oxygen was found to be adsorbed after the CO adsorption, but not after the COz adsorption. Thus the adsorption of CO on an oxygenated surface releases sites for further oxygen adsorption; the adsorption of COz, on the other hand, does not do so. This is the result to be expected if, as with copper oxide, a complex is formed which has a formula C 0 3 and which occupies one site, according to the simple scheme: (1) Adsorption of CO on an oxygenated surface. 0

0

0

0

I

I

I

I

co +

0

CO3

I

I

0

!

1-

0

0

Oxygen can be adsorbed after the complex has formed. (2) Adsorption of COz on an oxygenated surface. 0

0

I

I

0

I

0

I

co2

0

I

--+

CO,

I

I

I

Oxygen cannot be adsorbed after the complex has formed. The main evidence for COI complex formation, however, comes from a quantitative investigation of the heats liberated when the gases have been adsorbed on the various types of pretreated surface. For the sake of 'clarity these heats have been collected together in Table 11 for each of the three TABLE I1 Experimental Values of the Heats of Adsoiption of Carbon Monoxide, Carban Dioxide, and Oxygen in kral./mole on Surfaces of Copper, Nickel, and Cobalt Oxide in Diferent States of Pretreatment Adsorption of carbon monoxide

Oxide cuzo NiO COO

Adsorption of carbon dioxide

Adsorption of oxygen

Outgassed Presorbed Outgassed Presorbed Outgassed Presorbed Presorbed CO? surface oxygen surface oxygen surface C0 a C e b d f 9 20 26 20

49 88 52

28 22

21 37 22

55 43 59

100 100 95

71 74

oxides. The case of copper oxide has already been discussed; nickel oxide and cobalt oxide provide more complete data since it was found possible to adsorb some carbon dioxide on their outgassed surfaces. First it should be noted that the mere existence of an exaltation of the heat of oxygen adsorption for a surface containing presorbed COS (Table 11) is strong

TABLE I11 Comparison of Calculated and Experimental Heats of Adsorption for the Models of CO2,,,, and C03(ada) Formation' Heat of adsorption of oxygen on a surface carrying presorbed carbon monoxide (kcal./mole)

Heat of adsorpt,ion of CO on a surface carrying presorbed oxygen (kcal./mole)

0bserved

Calculated Oxide

CUZO NiO coo a

b

On the basis of formation of CO*(ada)

On the basis of formation of C03(ada)

(h-;+c)

(h-i+d) 60 82 59

59b 73 59

+

On the basis of formation of cot,&, b

49 88 52

Observed

Calculated

2(h - a

134* 138 138

h, the heat of the reaction CO(g) j40z(g) = COz(g) is taken as 67 kcal./mole. Using an assumed value of 20 kcal./mole for C .

P

On the basis of formation of CO~W,,

+ c)

u,

e 0 Z

M

f 96 100 99

*J

100 100 95

11

CHEMISORPTION AND CATALYSIS ON METALLIC OXIDES

evidence per se that a complex more oxygenated than COs(,,,, is produced. Designating now the heats in the successive columns of Table I1 as a, b , c, d , el j , and 9, and the heat of combustion of CO(n)(67 kcal. t o CO,,,,) as h, we may repeat for NiO and COO the calculation of heats of interaction for the two possible cases of CO,,ada)and COa,,,,, formation, and compare them with the corresponding experimentally observed heats. This is done in Table 111, where the figures for CuzO which we have already discussed or COa,,,,, is overwhelmare also included. The evidence vis-2,-vis CO?(ads) ingly in favor of the cO3 complex. Finally, we may note that the heat of formation of the COa complex from gaseous oxygen and gaseous carbon monoxide may be computed in four ways from the experimental data given in Table I1 (17). The details are shown in Table IV. I n view of the TABLE IV Heats of Formation of the COI Complex (kcal.) CO(d f OW = C03cada)

Oxide Cur0 NiO

coo

a

+f

e+b

e ,+h+d

120 126 115

104 131 111

116 126 119

h+c+i 131 126

'

Mean 113 129 118

difficulties inherent in standardizing the surfaces for the seven independent measurements, the consistency obtained for the heat of formation on a given oxide as computed by the four methods is extremely satisfactory. We shall use these heats of formation later (Section I1,D) in discussing the mechanism of the sustained catalysis of carbon monoxide oxidation on these oxides a t room temperature.

METHODS C. RELATEDRESULTSFROM OTHEREXPERIMENTAL During the last five years a number of researches in other laboratories have added substantially to our knowledge of the interaction of carbon monoxide and carbon dioxide with oxygen a t oxide surfaces, especially nickel oxide. There is support for the idea presented in the foregoing sections that these gases can produce on the surface of the oxide a species with formula C03,but opinions differ as to its precise description and in particular the extent to which lattice oxide ions, as opposed to adsorbed oxygen ions, are involved. The more important of these researches are summarized and critically examined in this Section. MentJion will be made first of the important studies of Winter, using heavy oxygen, which were reviewed by him in a recent volume of this series ( 2 1 ) . Winter's work has extended over a number of oxides, but refer-

12

F. S . STONE

ence will only be made here to his studies on nickel oxide and copper oxide (22), since these are the most relevant to the present discussion. Winter employed the oxides as films on a metal base, following the technique described in the previous section. By contacting the outgassed nickel oxide film with oxygen containing 30% of 0lsa t 540°, complete exchange of all the surface oxygen ions of the oxide was shown to occur. The surface was then outgassed a t 540" and cooled. When carbon monoxide was subsequently admitted at a few centimeters pressure, no measurable exchange of oxygen was observed between CO and the surface below 200".Similarly, carbon dioxide showed no exchange a t these temperatures. There was no reason, however, to suppose that the gases were not adsorbed. When stoichiometric CO/>$OZ mixtures were admitted, catalysis to carbon dioxide was measurable a t 50", but only an extremely small fraction of the heavy oxygen was released into the gas phase. It would appear, therefore, that carbon monoxide and carbon dioxide are adsorbed at the surface of nickel oxide a t low temperatures without appreciable interplay with the oxygen ions of the oxide, as indeed had been assumed in our discussions in the preceding Section (11,B). The behavior of copper oxide was altogether different. Winter showed that both carbon monoxide and carbon dioxide readily exchanged their oxygen with that of the whole oxide surface a t room temperature, and there was even some exchange a t -78". The only difference between the method of experiment here and with nickel oxide was that whereas the NiO had been pretreated with heavy oxygen and then outgassed at 540°, the cuprous oxide had been pretreated and outgassed a t a much lower temperature, viz., 200" or lower. If, however, we assume that no adsorbed oxygen remains after this treatment, the occurrence of oxygen exchange implies that a t least some of the carbon monoxide and carbon dioxide has been adsorbed in a form where it is in close association with the oxide ions of the lattice. Nevertheless, in the case of the CO experiments, the adsorbed species concerned is able to "remember" to give back carbon monoxide after the exchange has taken place. In the absence of adsorbed oxygen, the formation of a COa complex of the type envisaged in the preceding section cannot account for these results. Winter has therefore suggested that the adsorbed species which is formed is Garner's carbonate ion (see Section I), B

A

cu+

CU'

c u + 02- Cu'

02-

cu+ Cuf

02-

Cu+ Cu+ 02- c u +

cu+

Cu+ 02-

cu+

02-

CHEMISORPTION AND CATALYSIS ON METALLIC OXIDES

13

where State A is assumed to be reached after the adsorption of CO and State €3 after the adsorption of COz. There is, however, a distinction to be drawn. Garner considered that once State A was formed on an oxide, carbon monoxide was desorbable only as CO,. Winter, on the other hand, postulates that State A on cuprous oxide undergoes dissociation to give predominantly carbon monoxide, even at - 78". It is doubtful if this latter course is energetically feasible. Consider the following cycle, where we represent by the symbol the analogue of an F-center in CuZO, i.e., an anion vacancy containing two electrons : CO2-

+

4H +

CO@,

+ 202-

Let the dissociation process envisaged by Winter have an endothermicity AH. Let AHox represent the change in heat content when two surface F-centers are filled by oxygen. We can estimate AH,, as 70 kcal. by taking a mean between the heat of dissociative adsorption of oxygen on cuprous oxide (55 kcal./mole) and the heat of formation of cuprous oxide (82 kcal./mole of oxygen). -AHco, is the heat of formation of CO, (67 kcal.). AHc0,Z- is the change in heat content when State B is attained by the reaction of COz with cuprous oxide. It will be bracketed by the heat of adsorption of CO, on cuprous oxide and the heat of formation of cuprous carbonate. We do not know either of these, hut it would be surprising if they differed appreciably from 25 kcal. Substituting these values it follows from the cycle that AH = 57 kcal., compared with only 25 kcal. for the dissociation to COZ envisaged by Garner. The dissociation of carbonates to give COn is barely feasible at room temperature : it seems extremely improbable, therefore, that dissociation to CO could occur a t low temperatures. We believe that an alternative explanation should be sought for Winter's observations, a t least as far as the exchange with carbon monoxide is concerned. And why is cuprous oxide so much more active than NiO in exchanging its oxygen with carbon monoxide? No consideration has been given so far to the likely geometry a t the oxide surface, but this will surely be an important factor influencing the mechanism of oxygen exchange. Cuprous oxide is a cubic crystal with a = 4.28A., but the structure is a rare one. With the exception of the rather unstable silver oxide, CuzO is unique among oxides in having a body-centered anion lattice with the cations lying in an alternating manner on the body diagonals so as to give

14

F. S. STONE

them tetrahedral coordination about the anions. The (001) plane, therefore, presents either planes of oxygen ions or, altcrnatively, planes of coppcr ions, and the same is true of the (111) plane. Layers containing both copper and oxygen do not feature in either of these planes. This is significant for the geometry of the oxide surface: it means, for example, that a t the cleaved surface obtained on the (001) plane there is an equal probability of finding oxide ions in “protruding” and in “buried” positions, respectively (Fig. 2). The readjustments of lattice distances to be expected a t the surfaces of ionic and partially ionic crystals are not likely to change this

3

I

3

1

$.*

a 6

-*

2

3.03A.

FIG.2. Diagrammatic representation of a cross section through the (001) plane at the surface of a cuprous oxide crystal. KEY:0 Oxygen atoms; Copper atoms; - - “Protruding” oxide surface; . . . . “Buried” oxide surface. (The atoms are not all in thc plane of the paper. Atoms marked with the same number are in the same plane, and if atoms designatcd “2” are regarded as being actually in the plane of the paper, those marked “1” are in a plane 1.07 A. above i t , and those marked “3” in a plane 1.07 A. below.)

pattern appreciably for cuprous oxide. For a surface prepared, as in Winter’s experiments, by heating in oxygen and outgassing a t a temperature not greater than 200°, we may expect the proportion of (‘buried’’ oxide surface to become converted to “protruding” surface, so that all the surface oxide ions are likely t o be in protruding positions. These ions are singularly well placed for exchange with adsorbed gases, and the very high efficiency of the reaction with CO and COz is understandable in these terms. We may also notice in passing that there is scarcely a distinction between oxygen ions which have been adsorbed on to (‘buried” oxide surface and lattice oxide ions on a “protruding” oxide surface. Reversible adsorption of CO in a position straddling the cuprous ions brings the molecule into a position where oxygen exchange could reasonably be expected to occur with low activation energy. A similar argument will apply to the (111) plane. On the (011) plane the surface will almost certainly consist of a face which contains

15

CHEMISORPTION AND CATALYSIS ON METALLIC OXIDES

copper and oxygen atoms in equal numbers. This surface, viewed from above, has the distribution shown in Fig. 3. The unique structure of CuzO again provides oxygen atoms in positions, e.g., along the “corridors” AB, which are easily accessible to reversibly chemisorbed CO and COZ. Thus, on this interpretation, there is good reason to expect an easy exchange of all the surface oxygen on each of the three main faces of cuprous oxide under the conditions of Winter’s experiments. With sintered nickel oxide, on the other hand, we have the rock-salt structure, though with a very slight rhombohedra1 distortion below 60”. No comparable favorable positions for oxygen exchange exist on the (001) or (011) faces. On the (111) plane, where faces consisting solely of oxygen ions can occur in principle, the

0

0 0

0

0

0

0

0

0

A

B

0

0 0

0

0 0

0

0

0

0

O

A

B

0

0 0

0

0

0

0

0

0

0

FIG.3. Plan view of an (011) face of cuprous oxide. All the atoms are in the plane of the paper. 0-xygen atoms; .-copper atoms.

nickel sites (necessary perhaps for initial chemisorption) are rather inaccessible. A much lower activity in oxygen exchange with CO and COz may be expected, in agreement with the experimental result. If nickel oxide is prepared in a sufficiently finely divided form, with 20% or more of the oxygen atoms in the surface, arguments based upon the presentation of individual low index faces will obviously fail. It is very instructive to see the influence which this has on adsorption and exchange. Such an oxide has been studied in detail by Teichner and his co-workers (23-25), and oxygen exchange experiments have recently been carried out

16

F. S. STONE

by Klier (26). Teichner has prepared the oxide by vacuum decomposition of precipitated nickel hydroxide a t 200”;his method gives specimens whose surface areas lie in the range from 100-150 m.2/g., compared with 1-5 m.”g. for the specimens of Dell and Stone (16) and Winter (22).I n contrast to sintered nickel oxide, carbon monoxide and carbon dioxide both exchange their oxygen with this active oxide a t room temperature, indicating appreciable labilit,y of the oxide ions in adsorption. Also, as shown first by Teichner and Morrison (23) and subsequently confirmed by the studies a t Lyons (2.4,25),carbon monoxide can be adsorbed to about 25% coverage (at 40 mm. pressure), compared with 1% coverage (at less than 1 mm. pressure) on the sintered oxide used by Dell and Stone. The pressure difference accounts for some of the discrepancy, but in view of the results on the isotopic exchange it is natural to enquire if it is not the labile oxide ions which are mainly responsible for the enhanced adsorption of CO. As with cuprous oxide, the chemisorption of CO in association with single oxide ions is not incompatible with the observed reversibility. When CO is preadsorbed on the active nickel oxide and oxygen subsequently admitted, Teichner et al. (25) have shown that interaction takes place to form a n adsorbed complex which is to be distinguished from adsorbed carbon dioxide. This conclusion is reached by observations of color and conductivity changes. It is, of course, the same conclusion as was reached (see Section I1,B) on the basis of calorimetric studies on the sintered oxide, and by reference to the stoichiometry Teichner and his co-workers also believe the complex to have a formula CO,. Further support for the CO, complex comes from their studies of the adsorption of C 0 2 on the surface containing preadsorbed oxygen, again one of the methods of preparation used by Dell and Stone (16). Differences entered, however, when Teichner and his colleagues attempted the preparation of the complex from presorbed oxygen by reaction with CO or from presorbed C 0 2 (present a t 10-15% coverage) by reaction with oxygen. The preadsorbed oxygen reacted to give adsorbed C02, and the preadsorbed C02 was unreactive towards oxygen. The explanation of this discrepancy probably resides in the fact that CO and C02 were adsorbed a t much higher coverages than in the experiments with the sintered oxide, where it was the oxygen sites which were in the majority (27). There are two possible structures for a negatively charged COs complex. The first, by analogy with bicarbonate and carbonate ions, is a planar one with the carbon atom at the center of a triangle of oxygen atoms. It is unlikely that a triangular arrangement of oxygen atoms would be adsorbed flat on the cubic (001) or (011) faces of the oxides we have discussed, so the most likely mode of adsorption of such an ion will be

CHEMISORPTION AND CATALYSIS ON METALLIC OXIDES

17

-0

\ ,/C

l It

0

-

Me

Eischens and Pliskin (28) have actually provided evidence for such a “bicarbonate ion” structure chemisorbed on nickel oxide from studies of infrared absorption, but they attribute their spectrum to carbon dioxide bonded through a lattice oxide ion. For the co3 complex we would need to specify bonding of COZthrough an adsorbed oxygen ion. The alternative with single point attachstructure is a peroxidic one [-CO-0-0-I-, ment either through the carbon atom or the terminal oxygen atom. Such an ion would be a powerful oxidizing agent, as the C 0 3 complex is, but so little is known about percarbonate ions that one cannot speculate further on this structure.

D. THECATALYTIC OXIDATIONOF CARBONMONOXIDE AT Low TEMPERATURES The activity of copper oxide in this reaction a t 20” has been known since the classic researches on hopcalite in World War I, and Jones and Taylor (29) had communicated a t length on the subject already in 1923. The detailed work on the mutual interaction of adsorbed carbon monoxide and oxygen described in the preceding sections provided a new opportunity to reassess the activity of the oxide and offer suggestions for the mechanism of the catalysis. It was apparent a t quite an early stage that there was difficulty in explaining the catalysis in conventional terms. A residue of GOz could always be removed from the catalyst on heating in vucuo after an oxidation experiment. This implied that COZ had been held strongly on the catalyst, but a simple Langmuir-Hinshelwood mechanism was inconsistent with this in that there was no apparent inhibition in the catalytic reaction. Added to this was the difficulty that, if Copqdl)was a stable state reached in the catalytic reaction, why could it not be reached (cf. Section I1,B) when COz gas was admitted to the outgassed oxide? A mechanism involving the COO complex obviates these difficulties, and the following scheme (15) may be proposed: (a) Adsorption of carbon monoxide and oxygen, followed by reaction to form the COS complex:

co(sds)

+

co(g)

= co(ada)

02(,)

= 20(ads)

20bde) =

COa(,,,,,

(1) (2)

(3)

18

F. S. STONE

(b) Reaction of the complex with excess adsorbed CO:

forming two molecules of carbon dioxide.* After evacuating unreacted gas, the concentration of C03,,d.,prevailing in the steady state remains on the surface. On heating the oxide in uacuo, the complex decomposes to give carbon dioxide and oxygen, so the observations on the residue of COz are explained. The mechanism is schematically illustrated in Fig. 4,which also summarizes some of the data discussed in Section I1,R. The reaction path through the C 0 3complex is shown in heavy lines, and the experimental results from Tables I1 and IV enable the levels to be set quantitatively. On the left-hand side of the diagram for cuprous oxide route d is shown (Table 11) for the preparation of the complex. It emphasizes a novel feature of the mechanism, namely that C02 can autosensitize the oxidation by providing a second route to the reaction intermediate. This principle could be of general importance in sustaining catalysis a t low temperatures. In more familiar terms the COs complex may be regarded as a special type of active site, generated and maintained by the reactants themselves. Winter (22) has also studied the CO-oxidation on cuprous oxide a t room temperature. He has confirmed the presence of C02 in the gas desorbed after oxidation experiments by mass spectrometer analysis, yet also finds no poisoning of the reaction by COZ.He proposes a mechanism in which CO first reacts to form the carbonate ion and an anion vacancy (cf. Section 11,C). The filling of the anion vacancy by oxygen is then considered to promote decomposition of the carbonate ion to carbon dioxide. While the whole of the surface was active in oxygen exchange with GO and COz (q.v.), only 10% of heavy oxygen from the labeled oxide appeared in COZ during CO-oxidation with a 2 : 1 mixture a t 15". Winter interprets this result by postulating that only a small fraction of sites is active for the catalysis. However, it is not easy to understand why some carbonate ions, suitably accommodated with oxygen in their adjacent sites, should not decompose to COZ in the oxidation reaction whcn they apparently do so in the exchange reaction with COZ.We would prefer to attribute the low extent of exchange to the fact that in place of lattice oxygen freshly adsorbed oxygen is playing the major role in the catalytic reaction. The oxidation of carbon monoxide by cobaltous oxide a t low tempcratures shows many similarities to the reaction on cuprous oxide, and thr mechanism involving the CO, complex again accords with the experimental

* The occurrence of Reaction (4) was independently confirmed by preparing the complex by route d of Table I1 and dccomposing it with excess CO.

CHEMISORPTION AND CATALYSIS ON METALLIC OXIDES

,...... .,

19

20

F. S. STONE

facts. An additional test for decomposition of the complex has been devised in this case (18).By making a calorimetric study of the incremental addition of CO to a surface carrying adsorbed oxygen it has been possible to follow the process by which reaction (4) takes over from the reaction

co(,)f 2 o ( a d s ) = Co+ada).

(5)

For the first five increments, the heat of formation of the complex was found to fall slowly with coverage, indicating a fall in the heat of formation of the complex, but 110 carbon dioxide was produced in the adjacent cold trap. A sudden fall of 20 kcal./mole in the heat liberated by an increment was then registered, and at the same time a volume of carbon dioxide was produced in the trap which was greater than the increment of CO admitted. Knowing from the first increments the heat of formation of the complex and knowing the standard heat of formation of COZ,the sudden fall in heat to be expected when reaction (4) takes over from reaction (5) can be calculated. This is 19 kcal./mole, in good agreement with the experimental value. Turning to nickel oxide, the thermochemical data of Tables I1 and I V show that the oxidation via the C03 complex should be more difficult than on cuprous oxide or cobaltous oxide. The various reaction paths are shown schematically in Fig. 4. Because of the increase in the heat of formation of the complex, reaction (4) is now strongly endothermic and the reaction will be subjcct to poisoning. The poisoning effect has been confirmed experimentally, not only in our own studies, but also by Roginskii and COS(,,,, = Tselinskaya (30) and by Winter (21, 22). The reaction CO(,, 2CO2,,, by a Rideal-type mechanism (the dotted line in Fig. 4) can still account for some catalysis, and the activity would also be enhanced if the bond strength of C 0 3 with the surface were weakened. This is quite likely if co3 can be produced at increasingly higher coverages, as evidenced by the above results for COO, and Teichner et al. (25), who have recently studied the oxidation over their high area NiO catalyst a t 35", explain their results in terms of the COS complex route in this way. Regarding these latter results, it is interesting that the authors find a catalytic activity which increases after successive regenerations. This may be due to the progressive destruction of the labile oxide ions, attenuating their interaction with CO and favoring the rcaction of adsorbed carbon monoxide with adsorbed or gaseous oxygen. In our own studies we were unable to decompose the isolated CO, complex by dosing with CO in the absence of the reaction mixture. This may mean, as we have suggested elsewhere ( I @ , that the catalysis observed when the stoichiometric mixture is present is proceeding by some other mechanism, possibly involving only a few sites. Winter (21, 22) takes this view in interpreting his results. Alternatively it

+

CHEMISORPTION A N D CATALYSIS ON METALLIC OXIDES

21

is quite possible that the coverage is sufficiently high under the conditions of catalysis that the decomposition can occur. One of the few methods which in principle offer scope for the direct study of intermediates in chemical reactions is absorption spectroscopy, as applied, for example, in homogeneous reactions using flash photolysis. The experimental problem of placing sufficient absorbing molecules in the incident beam in heterogeneous processes was first solved satisfactorily by Eischens, and, appropriately enough, the first attempt to identify a reaction intermediate in catalysis by this method was made for this case of CO oxidation over nickel oxide. CO and CO, both have very high absorption coefficients in the infrared, and Eischens and Pliskin (28)observed one band a t 4.56~only while the oxidation reaction was in progress. They assigned this tentatively to CO, adsorbed through one of its oxygen atoms or CO adsorbed on a preadsorbed oxygen atom. Although this assignment is not a COO complex, we may note that CO with a preadsorbed oxygen atom is a likely precursor. The observation of one special band during a reaction does not, of course, prove that absorbing species to be the vital reaction intermediate, but the method, perhaps in association with isotopic tracer techniques, is obviously extremely valuable. Courtois and Teichner (31) are currently using the infrared method to provide further information on this system. Our aim in this section has been to prove the existence of a surface CO-oxygen complex, to establish its heat of formation and then to assess the evidence for its participation as the reaction intermediate in CO oxidation. The application of arguments based on isolated chemisorption experiments in discussing the mechanism of a delicately balanced catalytic reaction is always a calculated risk, but we have tried to show here that the method is most powerful if the behavior of all the various possible combinations of preadsorption and dosing can be fitted to a consistent picture.

Ill. The Uptake of Oxygen by Metals and Metallic Oxides A. KINETICSAND

THE

ROLEOF NONSTOICHIOMETRY

The uptake of oxygen by metals a t temperatures above 500" usually obeys a parabolic law, but the trend a t lower temperatures is towards an initial rapid reaction followed by a very slow uptake. At room temperature the reaction usually amounts to a few layers only. The kinetics most commonly observed in the region of room temperature are of the RoginskyZeldovich (32) type

dY = a exp (-by) dt

22

F.

s.

STONE

where q is the upt,ake, arid a and b are constants, the relation" usually being examined in the integrated form

A

q = b [In ( t

+ i)+ In a b ] .

(7)

The mcchanism here is by no mrans certain, but it is generally discusscd either as the migration of cations under the influence of the electric field provided by chemisorbed oxygen ions (34),or as a simple place exchange (35). The uptakr of oxygen is, of course, strongly exothermic and any supposed distinctions between the electronic configurations of different metals are subordinated beneath the effects of the high chemical affinity betwren almost all metals and oxygen. By the same token, the difficulty of dissipating the heat of the rapid oxidation makes for discrepancies ill the experimrntal assessment of limiting uptakes. However, the heat of formation of the limiting oxide film has been measured in the case of powders of copper, nickel, and cobalt (36,37); with the exception of the very first quantities of gas taken up, the heat liberated during the formation of the limiting few layers is close to the heat of formation of bulk oxide. Much of the work on evaporated films also leads to this conclusion (38). The uptake of oxygen by oxides shows a much wider range of phenomena. The greatest quantities are adsorbed on those oxides in which the metal ions can be oxidized to a higher valenry state (e.g., MnO, COO,CuzO, UO,), but it will readily be recognized that the reactions must be considered within the context of nonstoichiometry and the stability of higher oxides. The prewar work of Wagner and his school (39) established that oxides such as CuzO, COO, and N O were rendered nonstoichiometric by heating in oxygen at high temperatures, the oxygen excess arising because of the presence of cation vacancies. The nonstoichiometry is accompanied by p-type semiconductivity. The mechanism of formation of the oxygen cIxCcss must prcsum:hly bc (I) chemisorption of oxygen as ions, with the formation of thc equivalent number of Cu2+or Ni3+ions ( 2 ) incorporation of oxygen, viz., Inovrment of cations into the layer of chemisorbed oxygen, gencrating new adsorption sites and leaving vacancies behind, and (3) diffubion of the vacancies into the bulk. At low temperaturcs the last process with its high activation energy will no longer occur. Diffusion a t the surface, however, is less adversely affectedby a fall in temperature and the possibility exists that between 0" and 100" (1) may be followed by (2). This is borne out by studies of the adsorption of oxygen on cuprous oxide (supported on copper metal) where, already a t room temperaturr, more

* In the oxidation field this equation was first used by Tammann and Koster (SS), but we shall refer to i t in this article as the Roginsky-Zeldovich equation, since we are mainly concerned hcre with chemisorption.

CHEMISORPTION A N D CATALYSIS ON METALLIC OXIDES

23

than a monolayer is adsorbed a t pressures below 1 mm. The kinetics of this chemisorption have been studied using a microbalance (40). The activation energy for the monolayer region is 6.8 kcal./mole, but thereafter the Roginsky-Zeldovich kinetics accord with an activation energy which increases linearly with uptake a t the rate of 1.1 kcal. per monolayer. The rate of the uptake decays rapidly: a space charge is produced because the generated vacancies are not able to diffuse into the interior. If the oxygen gas above the oxide is removed and the temperature is raised, the vacancies diffuse t o the metal-oxide interface and the activity of the surface towards oxygen adsorption a t room temperature is regenerated. Cobaltous oxide films on cobalt behave similarly to cuprous oxide. In this case heats of adsorption have been measured as far as saturation (18). The postmonolayer uptake of oxygen (the incorporation stage) is accompanied by a fall in the heat of adsorption and a tendency towards reversible chemisorption. Nickel oxide, on the other hand, shows a lower activity in oxygen chemisorption, chiefly due, it is thought, t o the greater difficulty of regenerating the surface (16). Engell and Hauffe (41) have shown, however, that a t higher pressures (30 t o 200 mm.) a second stage in the uptake can be detected kinetically at 25” and this is attributed to incorporation obeying Eq. (7). Uranium dioxide is another case where more than a monolayer of oxygen is taken up a t room temperature, as shown by Anderson, Roberts, and Harper (4%’).This oxide, which has a fluorite lattice, is known to exist a t rather higher temperatures as a nonstoichiometric oxygen-excess oxide with the extra oxygen ions in interstitial sites (43). As with cuprous oxide, one may imply that chemisorption a t 20” is to be looked upon as incipient formation of the appropriate stable nonstoichiometric state, so that, for U02, it is presumed that a postmonolayer uptake is made possible b y virtue of the oxygen first adsorbed having subsequently entered interstitial positions (4%’).The same logarithmic law [Eqs. (6) and (7)] was observed.

B. DIFFERENT FORMSOF CHEMISORBED OXYGEN Let us now summarize the various ways in which oxygen can be adsorbed on oxides. We may have (a) physically adsorbed 0 2 molecules, (b) chemisorbed O2 molecules (02-ions), (c) chemisorbed 0- ions, (d) chemisorbed 0 2 - ions, and (e) oxygen which during the act of chemisorption enters anion vacancies, so becoming indistinguishable from lattice oxide ions. Even this list is not complete; we have referred in the preceding section to interstitial oxygen, and also to the process whereby cations move into the layer of adsorbed oxygen, so changing its binding energy. We must also contend with “active” oxides in which the oxide ions themselves are unusually labile (see Section I1,C) and in many respects very little different from chemisorbed oxygen.

24

F. S. STONE

The distinctions most easily drawn are those between physically and chemically adsorbed oxygen. Beebe and Dowden (44) showed that this could be achieved calorimetrically. With chromia a t - 183", a slow liberation of heat without any further uptake of oxygen gave evidence of the transformation from the physically adsorbed to a chemisorbed state. The heat of physical adsorption was 4 kcal./mole, but for the chemisorbed state they derived 25 kcal./mole. A very similar transformation of physically adsorbed oxygen (3-4 kcal./mole) to chemisorbed oxygen (20-55 kcal./mole) has been observed calorimetrically with uranium dioxide a t - 183" by Ferguson and McConnell (45). We are more concerned in this review with distinctions between the various chemisorbed states of oxygen. Some information comes from the absolute values of heats of chemisorption of oxygen when the adsorption is studied under different conditions. Thus, referring again to chromia, we may note that while Beebe and Dowden (44) found a heat of 25 kcal./mole for chemisorption a t - 183")the value at 0" was 50 kcal./mole. One cannot perhaps state unequivocally that these two widely different heats represent intrinsically different modes of oxygen chemisorption (because there may be a strong dependence of the heat of adsorption on coverage), but the result is certainly suggestive. Dowden and Garner ( 8 ) ,moreover, observed that the heat of adsorption of oxygen on chromia was 35 kcal./mole on a partially-reduced chromia surface, but 55 kcal./mole on the same surface after further reduction and outgassing. Some further insight into different states of oxygen chemisorption has come from the work on copper oxide, where a time-dependence of the reactivity of chemisorbed oxygen towards CO and COZ was ascribed to the transformation of the oxygen from a reactive to an unreactive form (14,15).Attempts were made a t this time but although a transient has t o detect a reactive form magnetically (46), been reported (477, we have not succeeded in recent years in repeating this result with diamagnetic cuprous oxide (20). Nevertheless, the most likely reactive species would certainly appear to be the paramagnetic O&), since the observed heat of adsorption (55 kcal./mole) is rather too great for a molecular chemisorption and this species could then become converted to O:&, presumed inactive. Studies of semiconductivity have shed additional light on this problem, and these are discussed in Section IV,A. Oxygen exchange studies are also valuable, and Winter has admirably summarized his views on the various states of chemisorbed oxygen on oxides in the review already mentioned (21). A clear case of different forms of chemisorbed oxygen is provided by recent studies of zinc oxide (48,49).The quantities of oxygen adsorbed by zinc oxide are very small, much less than 1% coverage, but the uptake can be conveniently studied a t low pressures using a Pirani gauge. The adsorp-

CHEMISORPTION AND CATALYSIS ON METALLIC OXIDES

25

+

tion obeys the Roginsky-Zeldovich equation p = C{log (t t o ) - log t o ] , and Fig. 5 shows the variation of C with temperature for different preparations of zinc oxide. Adsorption is rapid and prevalent around room temperature, and again above 300". Desorption studies have confirmed this pattern of behavior. The isotopic equilibration reaction l8OZ leOz = 2180160 has been found to proceed a t 20", so this seems to rule out the ascribing of the low temperature adsorption to O,,, It is suggested th a t the form predominating at room temperature is OGdn), and that above 300" is

+

Temperature

("C)

FIG.5. Rate of adsorption of oxygen on zinc oxide at various temperatures. Values of C in the equation g = C[log (t to) - log to], corrected to unit surface area of 1 m.2/g. and a standard pressure of 2.5 X 10-1 mm. [After T. I. Barry, Proc. 2nd Intern. Congr. on Catalysis,1960, p. 1449. Technip, Paris, 1961.1

+

O:&. The high temperature form is considered to be stabilized by the drawing of interstitial zinc into the surface under the influence of the electric field of adsorbed oxygen ions. Experiments by Thomas (50) on the diffusion of interstitial zinc and by Allsopp and Roberts (51) on the sintering of zinc oxide in oxygen show that this is feasible. The large oxygen adsorptions sometimes observed 011 ZnO specimens particularly rich in excess zinc are to be attributed to this latter form of adsorption, which under those conditions can occur at temperatures lower than 300". In short, one is dealing once again with a chemisorption which is t o be regarded as incipient oxidation.

26

F. S. STONE

It is interesting to remark that the concept of two different forms of oxygen chemisorption on zinc oxide has recently received support froin a quite independent type of experimental study. Thus Kokes (62) has observed that the ability of adsorbed oxygen to quench the electron resonance signal from ZnO a t g = 1.96 is critically dependent on the temperature a t which oxygen has been adsorbed. The signal (studied a t 24’) was quenched very much more effectively by chemisorption a t 25” than by chemisorption a t 400”. Kokes goes on to show that the dependence of the signal on coverage is consistent with a chemisorption a t 25” which withdraws electrons and is of the O&,) type, but a t 400’ a type of chemisorption is present which removes interstitial zinc, thus confirming in a very large measure the conclusions drawn earlier from the studies of kinetics. C. THE REACTIVITYOF ADSORBEDOXYGEN We have rcferrcd a t some length in Section I1 to the reactions of adsorbed oxygen to form complexes with carbon monoxide and carbon dioxide. A number of other interactions with adsorbed oxygen have been examined calorimetrically a t room temperature, notably those with hydrogen and with ethylene. The prewar work a t Bristol revealed a number of cases where higher heats of hydrogen adsorption were observed on “oxidized” surfaces than on “redurod” surfaces, e.g., chromia (8) and ZnO * Crz03 (5), but these are probably not to be classified as interactions with adsorbed oxygen. With cuprous oxide, however, the enhancement of the heat of hydrogen adsorption froin 27 kcal./mole for an evacuated surface to 42 kcal./mole for an oxygenated surface (15) is almost certainly an interaction similar in type to thohc we have discussed for CO and COZ. As far as ethylene is concerned, the most interesting results have been obtained with cohaltous oxide ( I S ) . The heat of adsorption of ethylene on outgassed COO is 13 kcal./mole, but values up to 80 kcal./mole have been obtained on surfaces carrying presorhcd oxygen. Successive doses of ethylene yield progressively lower heats of interaction (Table V). It is interesting to speculate on the TABLE V Heats n.f Adsorption n j Ethylene at 30” on Cobalt Oxide Carrying Presorbed Oxyqen Increment

1 2 3 4 5

Volume of ethylene taken up (cm.a)

Heat liberated per mole of ethylene taken up (kcal.)

0.084 0.116 0.111 0.158 0.081

80 79 60 37 27

CHEMISORPTSON AND CATALYSIS ON METALLIC OXIDES

27

nature of the product. Three significant stages of partial oxidation are given by the formation of (1) ethylene oxide, (2) acetaldehyde, and (3) formaldehyde. By making reasonable estimates of the heat of adsorption of these products (let us say, tentatively, 20 kcal./mole in each case), one may calculate from a knowledge of the relevant heats of formation and the heat of adsorption of oxygen (60 kcal./mole) the heats of interaction per mole of ethylene taken up in each case. For ethylene oxide formation, CL&(,) f O(&) = C&O(,d,), the heat is 15 kcal. [not 2 kcal. as previously stated ( I S ) ]; for acetaldehyde formation, -40 kcal. ; for formaldehyde formation C2H4(,) 2 0 ( a d a ) = 2CHz0(ads),the heat is about 100 kcal. Thus one may judge that one molecule is often reacting with two atoms of adsorbed oxygen in the initial increments, but with later increments t,he main reaction is with single atoms to give adsorbed acetaldehyde or ethylene oxide.

+

IV. The Emergence and Significance of the Electronic Factor A. SEMICONDUCTIVITY CHANGESDUI~ING CHEMISORPTION The equilibrium nonstoichiometry of oxides a s a function of oxygen pressure at high temperatures can be conveniently studied by measurcmcnts of their semiconductivity, a method much exploited by Wagner arid his co-workers (39) in the nineteen-thirties, and inasmuch as chemisorption of oxygen and other gases is held to involve electron transfer and the formation of ions the same method should be applicable to studies of adsorption and catalysis, even a t low temperatures. Dubar (59) showed in 1938 that oxygen and moist air affected the semiconductivity of cuprous oxide a t room temperature, but the first systematic researches adopting this line of approach in adsorption on oxides belong to the immediate postwar period. In this connection, much interest followed the introduction by Gray (54, 55) of an experimental technique in which oxides were prepared for adsorption and conductivity studies by the oxidation of evaporated metal films. Copper oxide was chosen for the first studies, both on account of the substantial knowledge already available concerning its electrical conductivity and because of the related work on the oxidation of copper and reduction of copper oxide in progress in Bristol a t that time. Figures G and 7 (13) summarize several important results obtained with cuprous oxide films a t 200' and 20" respectively. Exposure to oxygen a t a few microns pressure is seen to be accompanied by an abrupt fall in resistance (ix., rise in conductivity) showing that the concentration of current cnrricrs-positive holes in the case of Cu,O-has increased. Oxygen is therefore being adsorbed as negative ions. The fact that recovery of the original Conductivity is possible on evacuation a t 200°, but riot a t 20',

28

F. S. STONE

is evidence that the gas is quite strongly adsorbed. Hydrogen a t 200" produces a sharp rise in resistance, indicating the formation of a positively charged ion in the reversibly chemisorbed state. The same is true of carbon monoxide, but here there is a z3ccndary process, manifested by the reversal of the conductivity change and by the fact that the resistance of the film does not return to the initial value on evacuation. This agrees with the

+

1

constant i n i t i i resistance

1

1

1

1

1

1

1

1

1

;lo 2030405060708090100 application dvacuum

1

1

1

1

1

1

140

-

time (min.)

FIG.6. Resistance changes during the adsorption and desorption of gases on cuprous oxide at 200". A-Oxygen; %-Hydrogen; C-Carbon monoxide.

view that carbonate ion formation can succeed the reversible adsorption of carbon monoxide at 200" and that some reduction to COs occurs on evacuation. The reversal in the presence of CO is absent at 20" : only a small rise in resistance suggestive of CO+ adsorption is observed. When, however, the surface contains presorbed oxygen, the large enhancement in conductivity produced by the oxygen is almost completely destroyed by CO a t 20" (Fig. 7). These are the conditions under which the calorimetric work

CHEMISORPTION AND CATALYSIS ON METALLIC OXIDES

29

shows interaction and complex formation. Finally, we may note that in the presence of CO and oxygen the same steady state is approached during the catalytic reaction whether the initial state was reached by treatment with CO or treatment with oxygen. These observations provided very important background to the developments described in Sections II,B and II,D, where the hypothesis of a common COa complex, capable of being prepared by various routes and active as an intermediate in the catalytic reaction, was put forward. constant

0

10

20

30

40

50

60

70

80

90

100

time (min.)

FIG.7. Resistance changes during the reaction between carbon monoxide and oxygen on cuprous oxide at 20".

The conductivity method offers considerable scope for analysis of the kinetics of adsorption and desorption, particularly of oxygen, and Gray and his co-authors (56-59) have made this the main theme of several of their publications. One of the general conclusions (58) is that the conductivity being measured is essentially that of a surface zone where, under the conditions of well-outgassed surfaces, the number of adsorbed particles subsequently taken u p is related to the square of the conductivity, in agreement with the Fowler-Wilson model. Under conditions of surface saturation with oxygen, on the other hand, degeneracy can be expected and the concentration of adsorbed particles may then vary as the first power of the conductivity. These conclusions are well supported, for example, by the respective kinetics of adsorption and desorption of oxygen at nickel oxide surfaces (58). A satisfactory knowledge of the relationship between the

30

11’. S. S T O N E

observed conductivit,y change and the number of chemisorbed particlrs is vital to quantitative studics of adsorption using this technique, and without this information there is inevitably a risk of making false deductions from observations of conductivity changes alone. Chemisorbed gases do not invariably influence the conductivity, and the same gas chemisorbed in different forms or at different sites may change the conductivity in the one case and not in the other. Below lOO”, for example, hydrogen chemisorbed on zinc oxide (60) and on nickel oxide (68, 61) does not affect the semiconductivity, but a t higher temperatures there is a marked change in conductivity when the gas is chemisorbed. With this having been said, it remains to emphasize that conductivity studies during chemisorption can often have a very important qualitative significance. Thus an observed conductivity change in the prcscnce of a gas amounts to proof of chemisorption. Also we have already drawn attention in Fig. 6 to the obvious distinction which the method affords for adsorption in the donor and acceptor forms respectively. Thus oxygen is always adsorbed as a negatively charged species, giving a rise in conductivity for p-typc oxides such as cuprous, nickel, and manganous oxides, and a fall in conductivity with n-type oxides such as zinc oxide and titania. Convcrscly, hydrogen and carbon monoxide (in so far as they affect the conductivity when chemisorbed) are first taken up on oxides as electron donors, as also are alcohols (62, 63).* The invariant chemisorption of oxygen in the acceptor form enables in its turn coriclusions to be drawn about whether surfaces on which it is chemisorbed with a change in conductivity are p-type or n-type. This is useful in the sulfide field where deviations from stoichiomct,ry can often occur in both directions, and where the surface may consequently have the opposite conductivity type from thc bulk (64). B. THEBOUNDARY-LAYER THEORY OF CHEMISORPTION Any transfer of electrons giving rise to changes of semiconductivity during chcmisorption must be controlled, inter uliu, by the concentration of electroils or holes available in the semi-conductor. The boundary-layer theory of chemisorption (66) is built within the framework of this entirely physical model of the chemisorption act. The gas being adsorbed is represented solely as a donor or acceptor of electrons: the adsorbent is represented as a conventional semiconductor with a given concentration of ionized donor or acceptor centers and whose ability to participate in chemisorption is otherwise uniquely determined by the height of the Fermi level.

* This leaves on one Ride the question of “complete” or “partial” electron transfer. In the case of some chemisorptions, the change in conductivity arises simply from a polarization a t the surface. The terms “acceptor” and “donor” chemisorption are nevertheless convenient ways to indicate the direction of the change.

CIIEMISOIWTION AND CATALYSIS ON METALLIC OXIDES

31

The model has various shortcomings in that it takes no account of such physico-chemical aspects of adsorption as variations in bond type and various physical refinements such as the existence of surface states and the problems associated with degeneracy are also outside its scope, but its cardinal virtue is that it is capable of precise analysis. One of its main predictions in which we have been interested (66) is that for depletive chemisorption, i.e., chemisorption which involves the removal of the majority carriers from the semiconductor, coverage a t equilibrium should be severely restricted. There is now sufficient experimental evidence to show that this pattern is quite well followed, both for donor gases on p-type oxides (e.g., little chemisorption of hydrogen on Cu20, in contrast to ZnO) and for acceptor gases on n-type oxides (e.g. little chemisorption of oxygen on ZnO and TiO,, in contrast to Cu20). In a recent study with oxygen on zinc oxide (48),where the objections to the boundary-layer model are minimal, a quantitative correlation has been attempted. With an excess zinc content of about 1 p.p.m. (i.e., donor density = 5 x l0l6 cm.+), the observed limiting coverage with oxygen at 20" was found to be approximately 0.01% on an oxide whose surface area was 12 m.2/g. The boundary layer theory for this case gives the equations

where e is the percentage coverage at, electrostatic equilibrium, I is the thickness of the boundary layer in cm., no is the donor density per and V , in electron volts is the height of the potential barrier above the Fermi level. If all the donors are ionized and if all the available electrons are trapped by adsorbed oxygen, we may set the thickness of the boundary layer equal to the mean radius f of the zinc oxide particles. Thus for a surface area of 12 me2/g.and spherical particles, f = I = 450A. This determines V,, and it follows from equations (8) and (9) that e should not exceed 0.029;b. This is in good agreement with the experimental value. The rate of depletive chemisorption will decrease as the height of the potential barrier increases and it may be shown that for small values of V, the kinetics should conform to the Roginsky-Zeldovich equation. This is also observed (Section II1,B). Finally we may mention that the boundary-layer theory enables an interpretation to be given of the photoeffects with zinc oxide and oxygen: this is discussed in Section V1,B. Hauffe has extended the boundary-layer theory of chemisorption to catalytic reactions and has shown the way in which the position of the Fermi level may be expected to influence reactions with well-defined ratedetermining steps. Wolkenstein's theory of catalysis on semiconductors,

32

F. S. STONE

which is more fuiidameiital and rather wider in scope than the treatments based on the boundary layer concept, also regards the position of the Permi level as the most dominant factor in chemisorptiori and catalysis on oxides. Both of these theories have been admirably reviewed by the respective authors in earlier volumes of this series (6'7, 68), and both have had a very great stimulus in formulating ideas on the electronic factor in oxide catalysis.

C. ACTIVITYPATTERNS IN CATALYTIC REACTIONS Another approach to the relationship between electronic structure and catalysis has been the search for catalytic activity patterns based on electron configuration and semiconductor type. The first of these patterns to be established was in nitrous oxide decomposition (69-71), as illustrated in Fig. 8. This series, with one or two exceptions, divides remarkably into

FIG.8. The relative activity of oxides for the decomposition of nitrous oxide, showing the temperature a t which reaction first becomes appreciable.

three groups. The p-type oxides (CuzO, COO, MnzOa, and NiO) are clearly the best catalysts, n-type oxides (A1~03,ZnO, CdO, TiOz, Fez03, Gaz03) are among the least effective, and certain insulators (e.g., MgO, CaO) occupy an intermediate position in the series. There is clearly strong evidence from these groupings for a catalytic mechanism which is controlled by electronic and ionic factors. The significant steps in this particular reaction are : N20(,)

+ e (from catalyst) =

N20
NzO&d = N Z ( -k ~ ) OGdJ)

followed by Oeda) =

or Osds)

+ NzO(g)

+

J~OZ,,, e (to catalyst)

= N z ( ~4-) Oi&) OzOe) = O Z ( ~ ) e (to catalyst)

+

(10) (11) (12)

(13)

(14)

Reactions (13) and (14) will be significant when the coverage of chemisorbed oxygen is particularly high (YO,72).Engell and Hauffe (73) assume that the rate of decomposition is controlled exclusively hy the desorption of oxygen

CHEMISORPTION AND CATALYSIS ON METALLIC OXIDES

33

(for which there is substantial evidence) and in alignment with the boundary-layer theory the high activity of the p-type oxides can then be attributed t o their low Fermi level. In other words, the electrons furnished in the desorption reaction are in these cases readily absorbed by the positive holes in the valence band. Another point of view focuses attention on the adsorption reactions, and considers the greater activity of the p-type oxides to arise from their greater proven propensity for oxygen chemisorption (70).As far as the more recent studies of individual oxides are concerned, the emphasis is against rate-determining desorption. Thus Rheaume and Parravano (74) consider that, for Mn203, the formation of adsorbed oxygen by Reaction (11) is rate-determining, and Winter (75) concludes from his work on nickel oxide that the actual formation of the weakly adsorbed NzO- [Reaction (lo)] is rate-determining. The intermediate activity of the insulator oxides MgO and CaO is rather puzzling, though the fact that the alkaline earths form stable ionic superoxides is suggestive. The most recent activity series to appear (76) places MgO and CaO less active than the n-type oxides, so there remains some doubt on this point. The activity series for CO oxidation shows a similar trend of p-type oxides very much more active than n-type oxides, and for the recombination of oxygen atoms a t oxide surfaces, studied recently by Greaves and Linnett (77))a pattern of catalysis is again distinguishable in which the most active oxides are those of copper, magnesium, manganese, nickel, cobalt, and iron, with zinc oxide, chromium oxide, and gallium oxide very much less active. The similarity of this sequence with the series in Fig. 8 is quite remarkable. Except for the anomalous magnesium, however, we may note with Greaves and Linnett that the most active catalysts also correlate with the presence of cations with unpaired d-electrons, from d9 (Cu2+)down to d4 (Mn3+). But chromia ( d 3 ) is relatively inactive. The case for correlations of activity with helectron configuration has been strongly advocated by Dowden (78). There is much experimental support for this view in the field of chemisorption and catalysis on metals, and Dowden has also sought to apply similar principles to transition metal oxides. The case is strongest for reactions involving hydrogen, since this gas, in contrast to oxygen, is not invariably chemisorbed by simple electron transfer. We have already mentioned (Section IV,A) that hydrogen chemisorption on zinc oxide and nickel oxide is without effect on the conductivity below lOO", and the inference is that a weak form of chemisorption is involved, possibly by covalence through the metal ions. The best hydrogen reaction to examine for an activity pattern is the HZ-DZ exchange. This reaction has been studied in an exploratory way by Dowden, Mackenzie, and Trapnell (79),and they have pointed out that the earlier tentative suggestions that [In-typeness)' (as in ZnO and in reduced chromia) was

34

F. S. STONE

a guide to high activity in hydrogen reactions (80) vannot be upheld. They intcrprct their results instead as an example of the "twin peak" patterii of transitioii metal ion properties, the very low activity a t Fez03 representing the stability of the d5 configuration. There is some doubt about the validity of certain of their experimental results: in our experience, for example, nickel oxide does not show activity in IIz-Dz exchange until it, is superficially reduced. We have therefore attempted to place the investigation on a somewhat sounder experimental basis by studying the sequcnc.e d', d2, and d3 on sintered specimens of Tiz03,VzO3, and chromia (81).This series has the merit of a common crystal field for the cation (corundum structurr) and a n appreciable resistance to reduction by hydrogen. Preliminary results have shown unexpected differences in the specific activity of the three oxides in H2-DZexchange: VzO3 was the poorest catalyst below 250°, but on account of a high activation energy it became the best catalyst a t 350'. Thus our results so far do not support the idea of a simple correlation with the d-electron number. We may also note here a report (82) that the halides of the 3d transition metals do not show a twin peak activity pattern in IIZ-Ds exchange, and the same is evidently true for the oxides in the hydrogen-oxygen reaction (83). I n the latter reactlion, Popovskii and Roreskov have found that thc d3 and d8 oxides (Cr& Mn02, and NiO) were less active than the d6, d', and dg oxides (CoaOl and CuO), and Cr20j (8)was even less active than FezOs ( d 5 ) . Although the experimental support for t~ simple correlation of the catalytic activity of transition metal oxides with the number of d electrons is up to now somewhat scanty, the more basic concept of crystal fields as a potential factor in oxide catalysis is undoubtedly worthy of close examination. I t is especially significant when we remember that it also affords a n opportunity for a more penetrating appraisal of the semiconducting property. The behavior of the d-orbitals when the cations are brought into the symmrtrical environment of the crystalline oxide may be illustrated by reference to the studies on nickcl oxide. It was recognized many years ago that an anomalous situation existed here, since if the 3d wave functions of the Ni3+ions overlapped to form a band conductivity would be expected for stoichiometric NiO because the unfilled 3d8 configuration of the cations would give an unfilled band. De Boer and Verwey (84) therefore suggested that conductivity in nonstoichiometric NiO should be explained by a model of localized levels, with electrons jumping from Ni2+to Ni3+ions, the latter owing their existence to the presence of nickel ion vacancies as postulated by Wagner (39). The justification for this idea of localized levels in NiO has been given in reccnt years by Morin (85). He has pointed out that the absorption bands arising from the splitting of the d-level of the free ion into dr and de levels when isolated Ni2+ions are in the octahedral ligand

CHEMISORPTION AND CATALYSIS ON METALLIC OXIDES

35

field of six water molecules are also present a t very similar frequencies in nickel oxide, where the Ni2+ions are in the octahedral crystal field of oxide ions. It is extremely likely, therefore, that the 3d wave functions are also very largely localized on the cations in nickel oxide. I n Tiz03,on the other hand, the same kind of evidence shows that the 3d wave functions almost certainly overlap and a band is formed. Yet Ti2O3at low temperatures is also a p-type semiconductor in the sense that its conductivity increases with temperature and its thermoelectric power is positive (86). The origin of the p-typeness, however, is altogether different from that of nickel oxide. Even within a group of p-type oxides where the localized model is appropriate, differences in the nature of the hole conductivity may be expected according as the holes are in the d y level, as for Mn3+ (d4) in nonstoichiometric MnO, or in the de level, as for Ni3+ (d’) in NiO. Because of the difference in occupancy of the &orbitals, there will also be differences in the ability of the ions to form directional bonds with adsorbed molecules. It is clear, therefore, that p-typeness in oxides is not of itself a sufficiently profound criterion. The foregoing considerations remind us that electron exchange a t the surface is determined to a large extent by “chemical” factors: they also mean that changes in the Fermi level of the oxide due to various agencies (e.g. temperature, dopents) will tend to be specific rather than subject t o broad generalization. We shall have occasion to see this in Section V. This necessarily limits the scope of purely physical theories such as that of Hauffe (67’). Also, although Wolkenstein (68) makes formal allowance for these factors in recognizing “weak” chemisorption, until this concept is developed in more chemical terms it would seem that his approach is similarly limited. But a t the same time we should recognizc that the correlation of activity with d-electron configuration is equally subtle. The most satisfactory approach would appear to be one which gives an appraisal of the influence of the crystal field of oxygen ions on the formation of chemical bonds during the actual chemisorption act. Dowden has begun t o examine this problem, and in a recent, very stimulating paper he and Wells (87)have discussed the activation energies for various catalytic reactions on this basis. From the same point of view we have recently conducted an experimental study of the adsorption of oxygen on nickel oxide under the influence of irradiation in the d-d (crystal field) bands. We defer a n account of this until Section VI,E.

V. Adsorption and Catalysis on Doped Oxides A. INTRODUCTION The type of activity pattern discussed in Section IV,C was that in which activity for a given reaction is examined among a series of metallic oxides

36

F. 6. STONE

which are chemically distinct from one another in the conventional sense. In this section we discuss the behavior which arises when a given oxide is chosen and changes in its catalytic activity are studied a s a function of doping with small quantities of altervalent ions. I n 1948 Verwey and his co-workers (88) established that lithium ions incorporated into nickel oxide produced an equivalent number of Ni3+ ions and so enhanced the electrical conductivity. Later, from measurements of the Seebeck effect,Parravano (89)confirmed that in the presence of lithium the Fermi level of nickel oxide is indeed depressed in accordance with the increased concentration of positive holes. For trivalent additions, Hauffe and Block (90) have shown that the incorporation of small amounts of Cr3+ions decreases the conductivity of nickel oxide: one infers accordingly that the hole concentration is decreased and that the Fermi level is raised. This is therefore an attractive situation with which to examine the influence of the height of the Fermi level on catalytic activity. The most appropriate n-type oxide for analogous studies is zinc oxide. OF CO B. THEOXIDATION

OVER

DOPEDNICKELOXIDE CATALYSTS

This has been the most widely studied of all reactions on doped oxides. Parravano (91) showed that the addition of lithium to nickel oxide led t o an increase in the activation energy of the reaction, while the substitution of ions such as Cr3+reduced the activation energy. Later, Schwab and Block (93) reported the opposite effect. The temperature ranges studied by Parravano (20"-250") and by Schwab and Block (250"-400") did not overlap, and several authors (67, 68, 80, 93, 94) have since speculated on the discrepancy with this distinction in mind. The common feature of these speculations is the invoking of the existence of different forms of adsorption for CO and oxygen and different rate-determining steps in the two regions of temperature : we have already discussed several of the possibilities in earlier sections. At the same time, however, there was a need for more precise data, since the work of both Parravano and of Schwab and Block left room for improvement in a number of respects, not least of which was the employment of a rather low firing temperature (640' and 850" respectively) during catalyst preparation. It is obvious that the experiment, is only meaningful if the dopents are in true solid solution. The low temperature region (20"-350") has now been reinvestigated by Keier, Roginskii, and Saaonova (95) and the high temperature region (300"-450") by Dry and Stone (96). These studies have strikingly confirmed the behavior revealed respectively by the earlier investigations in that, a t low temperatures, doping with lithium raises the activation energy. For chromium additions, Dry and Stone show that the activation energy rises less abruptly than found by Schwab and Block (Fig. 9) and this is attributed to a higher firing tempera-

37

CHEMISORPTION AND CATALYSIS ON METALLIC OXIDES

ture (lOOOo) having produced a more homogeneous distribution of chromium ions in the bulk. Keier, Roginskii, and Sazonova also found a rise in activation energy on adding chromium, in disagreement with Parravano. Of greater significance is the fact that the new work has shown that the changes in activation energy are matched by substantial changes in the frequency factor. Thus although the activation energy a t high temperatures was observed by Dry and Stone (Fig. 9) to vary from 11.5 kcal./mole (for 2.8% Li) to 17.6 kcal./mole (for 5% Cr), the frequency factor rose monotonically in good accord with the theta rule. The balance of the two fac-

'I

* I

0

I4

IZ

10

0 0

--

0

0 I

I

I

I

I

S

4

3

1

1

atom % chromium

NiO

I

I

I

I

I

I

2

3

4

S

atom % lithium

FIG.9. Apparent activation energies for the oxidation of carbon monoxide on doped nickel oxide. 0 G. M. Schwab and J. Block [Z.physik. Chem. (Frankjurt) [N.S.] 1, 42 (1954)l;0 M.E.Dry and F. S. Stone [Discussions Faraday Soc. 28, 192 (1959)j.

tors is such as to make the 2.8% Li catalyst only about five times as active as the 5% Cr catalyst at 350". This is reminiscent of the result observed by Wagner for the influence of gallium on the catalytic activity of zinc oxide in NzO decomposition (69, 7 1 ) .It raises the following paradox. If we assume that the favored rate when the Fermi level is low is due to a rate-determining donor reaction in CO oxidation, e.g., CO = GO+ e, or GO+ 202- = COZe, then why does a change which increases the concentration of holes by several orders of magnitude decrease the frequency factor? Winter (97') has suggested on the basis of oxygen exchange studies that at high hole concentrations the reaction becomes channeled through a few very reactive oxygen sites. The invoking of active centers is certainly the most convenient way round the difficulty. It may be that one should interpret

+

+

+

38

F. S. STONE

the influence of one partner (oxygen) as predominating on IC, and look for the influence of the other (CO) primarily on E. Elaborating on this, one may recall that the kinetics of the reaction are first order in carbon monoxide and zero order in oxygen. If a reaction between adsorbed CO and adsorbed or lattice oxygen is rate-determining, the apparent activation energies of Fig. 9 will contain a contribution from the heat of adsorption of CO, and it is possible that the variation in the activation energy reflects the variation in this heat of adsorption. For this to be so, the heat of adsorption of CO would be required to increase by about 6 kcal./mole on going from 5% Cr to 2.8% Li. For a donor reaction, the heat of adsorption should increase as the Fermi level of the oxide is decreased (85), and we may note that a t 400" Parravano (89) did indeed observe a spread of about 7 kcal./mole as the Fermi level was lowered during lithium doping. The converse should apply in oxygen chemisorption, and Cimino, Molinnri, and Pomeo (98) have explained the results of their detailed study of magnetocatalytic effects a t the NBel point of nickel oxide in terms of such a decrease in the heat of adsorption of oxygen as the Fermi level is lowered. In our case, however, the observed zero order kinetics for oxygen imply that this effect, if present, is not large enough to make itself felt. The high heat of adsorption of oxygen, relative to that of CO, is evidently sufficient to maintain a high coverage of this gas. Zlegarding the reversal of dopent effect with temperature, one must draw attention again t o the fact that the reaction is self-poisoning a t low temperatures (Section 11,D).In addition, the variations in specific activity observed by Keier e2 al. (95) were very large. It is likely that the lowered activity of the lithiated nickel oxide at these temperatures is to be explained on the grounds that it adsorbs one or other of the species participating in the reaction much more strongly than the pure oxide. It is not possible to say which specks this is. Keier and Kutseva (99) have reported that lithium doping (provided the quantities are not too large) enhances oxygen and COZ adsorption a t room temperature. Winter (75) has reported similar results for oxygen. Our own experience (100) of chemisorption on doped nickel oxide also indicates that oxygen is strongly chemisorbed a t room temperature on litl.iium-roritainiug specimens (which had been prepared a t 1000" and outgassed at 500") but weakly adsorbed on chromium-doped specimens. These results do not agree with the arguments developed a t the end of the preceding paragraph. There is no basis for considering oxygen chemisorption to be other than an acceptor reaction, whatever the temperature, so one is led to the coiiclusioii that boundary-layer theory is not the right foundation on which to base the discussion, at least at low temperature. For CO adsorption, Keier and Kutseva (99) find a weaker adsorption on the oxide a t room temperature when it is lithiated, but Winter (97) has given a

CHEMISORPTION AND CATALYSIS ON METALLIC OXIDES

39

few results for lithium-containing NiO a t 150” which indicate that a t this temperature there is an enhancement of CO adsorption with respect to the pure oxide. Some new evidence for poisoning phenomena in the low-temperature range has come from a recent study of CO oxidation by Bielanski et al. (iOl),who observed activation energies below 250” which were dependent on the partial pressure of CO in the reaction mixture. The absolute activity decreased with increasing CO pressure. I n the course of experiments rather analogous to those shown in Fig. 7, they showed that there was a change in conductivity to the “CO level” during catalysis in CO-rich mixtures, but the “oxygen level” was maintained in oxygen-rich mixtures.

C. OTHER CATALYTIC STUDIES WITH DOPED OXIDES Hauffe, Glang, and Engell ( 7 i ) and also Winter (75) have studied the influence of lit,hium additions on NzO decomposition over nickel oxide. An increase in the p-typeness of the oxide by this means increases the activity, in agreement with the activity pattern discussed in Section IV,C. The same trend of increasing activity for this reaction in the direction n -+p (decrease of Fermi level) is also shown by the results of Schwab and Block (92) for small additions of lithium and gallium to zinc oxide. Doped zinc oxide has also been studied by Molinari and Parravano ( i O 2 ) in Hz-DZ exchange, and by Cimino et al. (103) in hydrogen and deuterium chemisorption. The most active catalyst for exchange, the well-conducting galliurn-doped oxide, gave the lowest equilibrium coverages in chemisorption. This confirms the role of weak chemisorption for H2-D2exchange, as commonly discussed in metal catalysis. The influence of dopents has also been studied with respect t o the oxidation of hydrogen and of ethylene (96). Using nickel oxide as the host oxide, a trend very similar to that for CO oxidation (Fig. 9) was observed for hydrogen, though the activation energy range (22 kcal./mole for 5% chromium to 17 kcal./mole for 2.8% lithium) was rather higher. The reaction proceeds more slowly than CO oxidation, and it is interesting that the greater resistance to catalysis is reflected entirely in the increased activation energies. The values of the frequency factor are in fact about an order of magnitude greater than in CO oxidation. Regarding the mechanism, the same difficulties of interpretation arise as in the high-temperature CO oxidation, and we shall not speculate on them further. For the oxidation of ethylene to COZ and water, which was studied at, temperatures above 400”, no systematic influence of dopent was found. Lithium and chromium doped oxides were both more active than the pure nickel oxide and both showed decreased activation energies. This could be construed as evidence against the idea that oxidation to ethylene oxide is rate-determining, since

40

F. 5. STONE

such a reaction would have much in common with the oxidation of CO and might therefore have been expected to show a similar influence of dopent.

VI. Photoadsorption and Photocatalysis A. INTRODUCTION A further way in which the relative concentrations of electrons and holes in a n oxide may be changed is by irradiation in the fundamental absorption band of the solid. One may visualize, using the band picture, that an absorbed quantum excites an electron from the valence band across the energy gap which separates it from the conduction band, and the independent existence of an electron liberated into the conduction band and a hole free to move in the valence band manifests itself in a n applied electric field as photoconductivity. If the electron and hole which constitute the excited state remain bound together as an exciton, photoconductivity is not observed unless or until some additional agency leads to its dissociation. I n the event, however, that there is the means to create electrons and holes in this way by light or by y-ray absorption, one may expect, if ideas concerning the role of electrons and holes as reactants in adsorption and catalysis are soundly based, that irradiation will stimulate “photoadsorption,” “photodesorption,” and “photocatalysis,” just as it produces photoconductivity. Furthermore, one may anticipate that, inasmuch as photoconductivity decays very rapidly once the exciting radiation is cut off , due to the recombination of electrons and holes, so photoadsorption and related effects will only be detectable during sustained irradiation. Donor or acceptor centers present in an oxide, due for instance to iioiistoichiometry, will play an important part in determining the nature and magnitude of surface photoeffects both because of their ability to trap electrons or holes selectively and because of the possibility that the distortion to the periodicity of the lattice which they entail will enable absorption to occur outside the range of fundamental absorption. I n this section this field of study is reviewed with reference to work on zinc oxide and titanium oxide. Two comprehensive articles on the electronic properties of zinc oxide (104) and of rutile (105) serve very well to provide the physical background. B. PHOTOADSORPTION AND PHOTODESORPTION AT ZINC OXIDESURFACES It is fifty years ago that Eibner (106) first reported the photoactivity of zinc oxide, but we shall refer here only to the more recent studies which bear on the general theme of this article. In 1954 Myasnikov and Pshezhetskii (107’) carried out a simple but elegant experiment in which two zinc oxide specimens were mounted side by side in the same evacuated vessel.

CHEMISORPTION AND CATALYSIS ON METALLIC OXIDES

41

One specimen was irradiated with ultraviolet light, and its conductivity increased; simultaneously, however, a decrease in conductivity was ohserved for the other specimen which had been shielded from the radiation. The immediate conclusion is that oxygen has been transferred from the irradiated specimen t o the other one. This phenomenon of oxygen release from ZnO under the stimulus of irradiation has since been described by a number of investigators (4s)1OS-113), several of whom have studied the process manometrically. The conditions under which the build-up of pressure occurs are such that photolysis of zinc oxide can be ruled out, and a t the same time several types of test have established that the effect is not simply thermal. It is clear that we are dealing here with genuine photodesorption. The existence of photodesorption may be understood in the following way. Photoconductivity studies (104)show that irradiation within the fundament,al absorption band of zinc oxide leads to the production of electrons and holes. Other things being equal, these are free to participate in adsorption and desorption, and will react with oxygen as follows:

Now the reaction of mobile holes with presorbed oxygen, which causes desorption, will not be entirely compensated for by the increase in adsorption rate due to the photoelectrons, since the latter have to overconie the potential barrier in the boundary layer (Sec. IV,B). The outcome is an adjustment of the adsorption equilibrium in favor of more oxygen in the gas phase, i.e., photodesorption. The possibility of adsorbed oxygen participating as a molecular ion in these reactions is not, of course, ruled out : our choice of O;ds) as the active species a t 20" is based on the evidence described in Section II1,B. It is now known from the work of Fujita and Kwan (llW),of Terenin and Solonitzin (113),and from our own studies (48)that, although photodesorption of oxygen is to be regarded as the most typical process a t zinc oxide surfaces, it is possible under certain circumstances to obtain an over-all photoadsorption of oxygen. An example is shown in Fig. 10 of a reversal in the photoeffect caused by increasing the excess zinc content of the oxide (48). I n Fig. 10a the normal photodesorption property is made apparent by an increase in desorption rate during a thermal desorption of oxygen a t 20".After heating in 1mm. hydrogen a t 520' for a short time and cooling t o 20", oxygen was introduced and chemisorption took place. After rapidly removing the oxygen remaining in the gas phase and isolating the system from the pumps, thermal desorption of oxygen could again be fol-

42

F. S. STONE

lowed. The photoeffect in this case, however, was photoadsorption (Fig. lob). Finally, by heating in 1 atmosphere of air at 520" arid prolonged outgassing a t 400°, it proved possible to restore the original Characteristics of photodesorption (Fig. 10c). Fujita and Kwan (11%')found photoadsorp-

time ( m i 3 FIG.10. Photoeffects during oxygen dcsorption with a n oxidized and reduced specimen of zinc oxide. (a) before treatment in a reducing atmosphere; (b) after heating in hydrogen at 520"; (c) after subsequent treatment in an oxidizing atmosphere a t 520". Arrow indicates when irradiation was begun.

tion of oxygen at 18" when the specimen of ZnO had been pretreated for 1 hour in 10 mm. oxygen a t 300" and then cooled before evacuation. Tercniii and Solonitzin (113) also observed the phenomenori on "oxidized" zinc oxide, but they do not give details. The possibility of photoadsorption resulting from the excitation of gaseous oxygen can be ruled out bccause the gas does not absorb in the near ultravioIet. Wolkenstein and Kogan (114)have given a generalized treatment of the influence of irradiation on chemisorption 011 semiconductors, arid their theory does in fact provide for the dual existence of photodesorption arid photoadsorption in one and the same chemical system. This treatment relates the direction of the photoeffect to the concentrations of adsorbed particles in the "weak" aiid "strong" forms of chemisorptiori which characterize Wolkenstein's theory of catalysis. P'or the case of oxygen (an electroil

CHEMISORPTION AND CATALYSIS ON METALLIC OXIDES

43

acceptor) interacting with zinc oxide, this treatment requires photodesorption to be observed when the irradiation produces a relative change in the concentration of electrons An/no which is small compared t o th a t of holes Ap/po, while photoadsorption should occur when the reverse obtains. Now in zinc oxide, a typical n-type conductor, no is greater than PO, so for An and A p of comparable order of magnitude, An/no is less than A p / p o and photodesorption of oxygen should occur as the normal behavior, in line with experiment. The brief interpretation of photodesorption which we gave above might in fact be called a very much simplified picture of Wolkenstein and Kogan’s treatment. On this basis, however, photoadsorption should not occur on zinc oxide unless the Fermi level can be severely depressed. This state of affairs cannot exist under the reduced conditions where photoadsorption has been found in our experiments, and it is also questionable whether Fujita and Kwan’s preoxygenation (119) can have influenced the Fermi level sufficiently to uphold Wolkenstein and Kogan’s mechanism. We must therefore look for alternative explanations of photoadsorption of oxygen. I n appreciating the distinction between photodesorption and photoadsorption it should be remembered that oxygen chemisorption is peculiarly extensive and strong on specimens of zinc oxide rich in excess zinc (Section II1,B). With this in mind, there appear to be three other mechanisms by which, a t least in a qualitative way, photoadsorption of oxygen can be imagined t o occur. First, one may suppose that for an oxide with a high density of donor centers, the potential barrier which develops during oxygen chemisorption (Section IV,R) may be sufficiently high to prevent electrostatic adsorption equilibrium being attained. Morrison (115) was the first to point out this possibility. Quanta absorbed in the surface layers of the zinc oxide crystals will generate electrons there whose potentiality for reaction with physically adsorbed oxygen is much greater than that of the thermal electrons which have to penetrate the full thickness of the boundary layer. Irradiation may therefore be a means of stimulating adsorption which had been prematurely arrested by Morrison’s “pinch-off” effect. A second interpretation follows from the assumption, based on photoconductivity studies (226),that a t high zinc contents the trapping of holes on interstitial Zn+ ions slows down electron-hole recombination sufficiently t o permit the adsorption reaction (15) to predominate. Thirdly, one may consider the role of the tightly bound oxygen. If we are correct in assigning a configuration close t o to this oxygen, we may postulate that it too can act as a trap for holes, being converted to O&, and that photoadsorption may again manifest itself through t,he increased concentration of electrons which prevails. The second and third interpretations become linked together through Fujita and Kwan’s work if we assume that at 300” and above

44

F. S . STONE

chemisorbed oxygen draws interstitial zinc towards the surface. Their oxide was presumably carrying OEi8 on its surface. It is noteworthy th a t the negative photoconductivity effects reported by Myasnikov and Pshezhetskii (107) and by Borissov and Kanev (117) are explicable as photoadsorFtion of oxygen, though neither of these authors mention this possibility in their interpretations. The conclusioii that photoadsorption of oxygen is associated with the presence of O&h) and subsurface interstitial zinc has prompted us to examine the photoeffect above 300" (218)where this second type of oxygen is considered to ocrur (Section III,B and Fig. 5 ) . It was found that at 400"

65

63

-

I

I

I Dark

I

I

I

I

I

I

Irradiated

I

Dark

Time (min.) FIG.11. Photoeffects during the adsorption of oxygen on zinc oxide at 400" in two different ranges of pressure.

and low pressures (-10 microns and below), photodesorption of oxygen occurred, but at, higher pressures (-100 microns) the photoeffect became converted to adsorption. Figure 11 shows an example of this behavior. It is considered that at the lowest pressures the concentration of adsorbed oxygen a t t,he surface is not sufficiently great to establish the field effect. The adsorbed gas is less strongly bound under these conditions and shows the photoeffect (desorption) characteristic of the state at 20". At higher

CHEMISORPTION AND CATALYSIS ON METALLIC OXIDES

45

pressures the strong adsorption becomes established and photoadsorption appears. The experiments illustrated in Fig. 11have also been studied using zinc oxide doped with lithia or chromia, the oxides having been fired a t 800". There was no change detectable in the dark adsorption, but the photoeffects were found to increase in the direction ZnO-Cr -+ ZnO -+ ZnO-Li. This is the direction of decreasing conductivity but increasing interstitial zinc content (119).We tentatively attribute this trend in the photoproperty to an extension of the absorption edge to longer wavelengths a s the excess zinc content is increased, the oxide thereby absorbing more of the radiation incident from the arc. Scharowsky (120) has shown that increasing the zinc content of ZnO extends the edge considerably. Radiation-induced desorption and adsorption of oxygen has also been observed using very high energy quanta. Klingman (121) has irradiated zinc oxide with 200 kv. X-rays and has observed a rise of conductivity indicative of oxygen desorption. Barry (49) has recently reported the effect of irradiating zinc oxide with y-radiation during the adsorption and desorption of oxygen. At room temperature and below, the effect of y-radiation in situ from a strong 6oCosource was to promote desorption of oxygen from a surface carrying oxygen preadsorbed a t room temperature, but adsorption of oxygen a t a surface carrying oxygen preadsorbed at about 300". These results are in very close agreement with those obtained using ultraviolet irradiation. The effect of ultraviolet and visible radiation on carbon monoxide adsorption on zinc oxide has also been studied (118). Both photoadsorption and photodesorption have been found, depending on the temperature, and there is a quite different pattern of behavior if oxygen has been preadsorbed. Reaction to give oxidized complexes, presumably of the COp,,,,, COS,,,,, or carbonate ion type, can readily be stimulated under irradiation. For oxide outgassed a t 400", the extensive adsorption of CO in the temperature range - 100" to 25", well known from the work of Garner and Maggs ( 6 ) , was found to be unaffected by irradiation, but photodesorption was observable between 100" and 200". It is interesting that this is the region of temperature where CO adsorption on zinc oxide begins to show a conductivity change (122). No effects attributable to irradiation were found with carbon dioxide and zinc oxide.

REACTIONS PHOTOSENSITIZED BY ZINC OXIDE C. CATALYTIC Although the photosensitizing action of zinc oxide suspensions has aroused a good deal of interest over the years, notably for the production of hydrogen peroxide, very little attention has been given in academic studies t o ZnO photosensitization in the dry state. The simplest photo-

46

F.

s.

STONE

catalysis t o study is the equilibration reaction for the oxygen isotopes, 02'8 O2I6 2018016,since this reaction merely involves dissociative adsorption of oxygen and associative desorption. This has been briefly studied a t Rristol (48),and it has been fouild that for zinc oxide which shows the characteristics of oxygen photodesorption, irradiation decreased the rate of isotopic equilibration. On the other hand, for zinc oxide which showed photoadsorption of oxygcii, irradiation increased the rate of equilibration. This isolated result indicates that adsorption of oxygen is rate-determining for this reaction on zinc oxide, a t least a t low temperatures. The result does not show unequivocally that dissociated species are the photoactive ones, sirice an adjustmerit of the O&,,, concentration through irradiation would also affect the concentration of dissociated species [e.g. O& and O 4 , 4 if these are all in equilibrium, but it is nevertheless suggestive. The oxidation of carbon monoxide may also be photocatalyzed by zinc oxide, though in view of the complicated nature of both the oxygen and the CO photoeffects it is not easy to establish the mechanism. Studies have been made over a wide raiige of pressure (10 microns to 100 mm.) and a t temperatures up to 400" (12.9). The dark reaction is appreciable a t 400°, but, as illustrated in Fig. 12, irradiation from a mercury arc gives a severalfold increase in the rate. The kinetics of both the dark and photoreactions are first order with respect to CO. Oxygen has much less effect on the rate, but it is interesting to see that while an increase in oxygen pressure favors slightly the dark reaction (13' --+ C' in Fig. 12), it decreases the rate of the photocatalysis (B + C in Fig. 12). At room temperature the reaction is best studied a t low pressures using a Pirani gauge to measure the change in pressure. Here also the photocatalysis was most strongly developed by working with mixtures which were rich in carbon monoxide. The mechanism of the catalysis, however, is almost certainly different under the two sets of conditions. Experiments with filters have shown that light of wavelengths between 400 arid 450 mp, that is, wavelengths rather longer than the usually accepted position of the absorption edge a t 385 rnp, are effective in the photocatalysis. Tereiiiii and Soloiiitziri (115)have shown this t o be the case also for oxygen photodePorptjon, where they report having followed a decreabirig photoactivity as far as 500 mp. Excess zinc is known to extend the position of the edge, as also does a rise in temperature (log),but the reason for the extension of the photoactivity into the visible with these powdered specimens of zinc oxide is not yet understood. However, in that the photocatalysis as well as the adsorption photoeffects were found to be more proiiounced with lithium-doped zinc oxide than with uridoped or chromium-doped ziric oxide, we may tentatively attribute the effect to excess zinc. It is worth noting in passing that, according to Ititchey and

+

~

47

CHEMISORPTION AND CATALYSIS ON METALLIC OXIDES

Calvert (l24), sintered cuprous oxide present as a thin layer on bulk copper also shows photocatulysis of CO oxidation when irradiated with ultraviolet light at 25". These authors observed that the addition of sulfur or antimony in solid solution, which increased the conductivity of the oxide, decreased the photoeffect, but the reason for this particular choice of dopents is obscure.

A

E E

Y

n

a

I

Prc:. 12. The dark and photocatalyxed oxidation of carbon monoxide by zinc oxide at 400". A, B, and C-Photocatalysis. A', B', arid C'-Dark reaction. pco Initial partial pressures (mm.) po2

A

B

57.01 32.76

32.23 32.30

C 20.92 56.00

A' 56.25 30.62

B'

C'

28.70 33.80

30.77 60.57

Myasnikov (125) and Komuro, Fujita, and Kwan (126) have observed photosensitization during the oxidation of isopropyl alcohol in the presence of irradiated zinc oxide, but the reaction has not yet been studied in much detail. Klingman (121) has reported a sensitized oxidation of propane in the presence of zinc oxide being irradiated with X-rays. Finally, Barry and

48

F. S. STONE

Roberts (127) have noted some cases of a significantly greater amount of methanol formation during the synthesis a t 1 atm. over zinc oxide which was being irradiated in situ by y-rays from a 350-curie cobalt source.

D. PHOTOEFFECTS WITH TITANIUM DIOXIDE Kennedy, Ritchie, and Mackenzie (128) have studied the photo-effects between oxygen and TiOz in some detail. The characteristic property is photoadsorption. The uptake of oxygen a t 25" was irreversible and extensive, amounting to about 1 monolayer. A parabolic law was obeyed and the rate was found to be proportional to light intensity. 365 mp was the most active radiation of the mercury arc, but 406 mp and 436 mp were also active. It is interesting that the visible radiations should be active, since the absorption edge for Ti02 is very close to 400 mp. This would appear to be another case where some agency, again possibly point defects, allows absorption a t wavelengths longer than the edge to take place in the surface layers of a finely divided solid. The photoadsorption of oxygen on TiOz has recently been reinvestigated by Dr. F. Romero-Rossi in the author's laboratory (118) and the results of Ritchie et al. (128) have been confirmed and extended. Radiation of wavelength longer than 400 mp was not active in this case, possibly because rutile was used instead of anatase, but in almost all other respects the two studies agree very well. There is, however, one striking difference. An increase in temperature in our case decreased the rate of photoadsorption. Other things being equal, one would attribute this to an increased rate of recombination of holes and electrons as the temperature is raised. Zinc oxide, for instance, shows such an effect. Ritchie and his co-workers, however, observed that an increase in temperature from 6" to 41' increased the rate of photoadsorption, and they calculated an activation energy of 5.4 kcal./mole. Assuming that two steps are involved

+

OZ(~) e

: OqSds) : 20?ida) I I1

and that the rate-determining step involves 11, the apparent activation energy of the reaction contains the heat of adsorption of the process OZ,, = O,,,. If this heat decreases with coverage the possibility exists that a negative temperature coefficient for the observed process can arise a t low coverages, where the heat of adsorption is larger, but a positive one a t higher coverages. This is the direction of change required to explain the discrepancy, since Ritchie et al. studied the photoadsorption a t about 50 mm. pressure, while our studies were made at pressures below 0.05 mm. The suggest that the rate-deterparabolic law kinetics ( A p proportional to

d)

CHEMISORPTION A N D CATALYSIS ON METALLIC OXIDES

49

mining step is an activated diffusion of chemisorbed oxygen over the surface of the oxide to sites where the gas can be irreversibly incorporated. Photoeffects with Ti02 in carbon monoxide, hydrogen, and water vapor have been sought (118, 128), but none have been found. Marked photoreactions do occur, however, with nitric oxide and with the dye chlorazol sky blue (128, 129).

E. PHOTOEFFECTS WITH NICKEL OXIDE As may be inferred from our discussion in Section LV,C, there is a special interest in nickel oxide in view of the existence of well-characterized d-d bands lying on the long wavelength side of the fundamental absorption (130).We have recently examined this region for activity in photoadsorption and photodesorption of oxygen. The work has revealed the extremely interesting fact that while irradiation in the fundamental absorption band is quite ineffective in photoadsorption or photodesorption, irradiation in the d-d region beyond 650 mfi stimulates photodesorption of oxygen (131). A mechanism of photodesorption similar to that advanced for zinc oxide is therefore ruled out, and attention necessarily becomes focussed on the ideas which we were discussing a t the end of Section IV,C. When oxygen ions are chemisorbed on the rocksalt lattice of NiO, surface Ni2+(d8)ions complete their octahedra of oxygen neighbors and a gain in crystal field stabilization energy results. The energy change per molecule of oxygen chemisorbed is likely to be greatest on the (011) plane where the Ni2+ions on the bare surface normally have a fourfold coordination. We therefore propose the following mechanism for the photodesorption. Irradiation in the region indicated excites the six-coordinated surface nickel ions carrying adsorbed oxygen from their ground state (3A2,) to the excited 3T1, state. In this activated state the sixfold coordination is much less stable than the corresponding ground state of tetrahedral coordination (3T1).Thus the two adsorbed oxygen atoms completing the octahedron of a Ni2+ion on a (011) plane have a high probability of being rejected into the gas phase, i.e., of being photodesorbed. The result which we have just discussed again illustrates the value of photostudies in providing, a t least conceptually, a convenient bridge between theory and experiment. It suggests that this type of study applied to oxides containing transition metal ions could be quite useful in formulating ideas on the possible role of crystal fields in heterogeneous processes.

VI1. Conclusion I n the various sections of this article an attempt has been made to approach the questions of the adsorption and the catalytic reaction of simple gases, with oxygen and carbon monoxide as principal examples,

50

F. S. STONE

from a wide variety of directions. On the basis of experimental studies, the theme to which one continually returns is the versatile role of individual gases in being able to exist in the chemisorbed state even on a given oxide in different forms. These correspond to different states of catalytic activation, and it is evident also from the work on the interaction of carbon monoxide and oxygen that this diversity extends to the intermediate complexes involved in catalysis. One may trace the origins of this idea a long way back in the study of adsorption and catalysis, but it is essential for an understanding of catalytic mechanisms that our knowledge of this property should be as complete as possible. We hope in this review to have indicated some of the ways in which thorough experimental studies, stimulated whenever possible by leads from theoretical work, are helping to achieve this end.

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50. Thomas, D. G., Phys. and Chem. Solids 3, 229 (1957). 51. Allsopp, H. J., and Roberts, J. P., Nature 180, 603 (1957); Trans. Faraduy Sac. 66, 1386 (1959). 52. Kokes, R. J., J. Phys. Chem. 66,99 (1962). 53. Dubar, L., Ann. Phys. 1111 9, 5 (1938). 54. Gray, T. J., Nature 162, 260 (1948); Proc. Roy. Sac. (London) A197, 314 (1949). 65. Gray, T. J., in “Semiconducting Materials” (H. K. Henisch, ed.), p. 180. Butterworth, London, 1951. 56. Gray, T. J., and Savage, S, D., Discussions Faraday Sac. 8, 250 (1950). 57. Derry, R., Garner, W. E., and Gray, T. J., J . chim. phys. 61, 670 (1954). 58. Gray, T. J., and Darby, P. W., J . Phys. Chem. 60, 201 and 209 (1956). 59. Gray, T. J., and Savage, S. D., Discussions Faraday Sor. 28, 159 (1959). 60. Kubokawa, Y., and Toyama, O., J . Phys. Chem. 60, 833 (1956). 61. Bielanski, A,, Deren, J., Haber, J., Sloczynski, J., and Wilkowa, T., Bull. acad. polon. sci. (Ser. sci. cham. geol. geogr.) 7, 5 (1959).

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espafiol fis. y qulm (Madrid) B60, 35 (1954). 63. Bielanski, A., Deren, J., Haber, J., and Sloczynski, J., Proc. 2nd. Intern. Congr. on Catalysis, 1960, p. 1653. Technip, Paris, 1961. 64. Stone, I?. S., Proc. Srd Intern. Congr. on Reactivity of Solids, Madrid, 1966, 1, 641 (1957). 66. Aigrain, P., and Dugas, C., Z. Elektrochem. 66, 363 (1952); Hauffe, K., and Engell, H. J., ibid. 66, 366 (1952); Weisz, P. B., J . Chem. Phys. 20, 1483 (1952); 21, 1531 (1953); Takaishi, T., Z. Naturforsch. lla, 286 (1956); Krusemeyer, H. J., and Thomas, D. G., Phys. and Chem. Solids 4, 78 (1958). 66‘. Stone, F. S., in “Chemisorption,” Proc. Chem. SOC.Symposium, 1956 (W. E. Garner, ed.), p. 181. Academic Press, New York, 1957. 67. Hauffe, K., Advances in Catalysis 7, 213 (1955); 9, 187 (1957). 69. WolkenRtein, Th. (Volkenstein, F. F.), Advances in Catalysis 9, 807 and 818 (1957); 12, 189 (1960). 69. Wagner, C., J. Chem. Phys. 18, 69 (1950). 70. Dell, R. M., Stone, F. S., and Tiley, P. F., Trans. Faraday SOC.49, 201 (1953). 71. Hauffe, K., Glang, R., and Engell, H. J., Z . physik. Chem. (Leiptig) 201,223 (1952). 72. Amphlett, C, B., Trans. Faraday SOC. 60, 273 (1954). 73. Engell, H. J., and Hauffe, K., Z.Elektrochem. 67, 776 (1953). 74. Rheaume, L., and Parravano, G., J. Phys. Chem. 63, 264 (1959). 76. Winter, E. R. S., Discussions Faraday Soc. 28, 183 (1959). 76. Saito, Y., Yoneda, Y., and Makishima, S., Proc. 2nd Intern. Congr. on Catalysis, 1960, p. 1937. Technip, Paris, 1961. 77. Greaves, J. C., and Linnett, J. W., Trans. Faraday SOC.66, 1346 (1959). 78. Dowden, D. A, J . Chem. SOC.p. 242 (1950); in “Chemisorption,” Proc. Chem. SOC. Symposium, 1956 (W. E. Garner, ed.), p. 3. Academic Press, New York, 1957. 79. Dowden, D. A., Xlackenzic, N., and Trapnell, B. M. W., Proc. Roy. SOC.(London) A237, 245 (1956). 80. Stone, F. S., in “Chemistry of the Solid State” (W. E. Garner, ed.), p. 367. Academic Press, New York, 1955. 81. De, K. S., and Stone, F. S., unpublished data (1961). 82. Brennan, D., Discussions Faraday SOC.28, 219 (1959). 83. Popovskii, V. V., and Boreskov, G. K., Problemy Kinet. i Katal. Akad. N a u k S.S.S.R. 10, 67 (1960); Chem. Abstr. 66, 11040 (1961). 84. de Boer, J. H., and Verwey, E. J. W., Proc. Phys. SOC.(London) 49, (extra part), 66 (1937). 85. Morin, F. J., Bell System Tech. J . 37, 1047 (1958). 86. Pearson, A. D., Phys. and Chem. Solids 6, 316 (1958). 87. Dowden, D. A., and Wells, D., Proc. 2nd Intern. Congr. on Catalysis, 1960, p. 1499. Technip, Paris, 1961. 88. Verwey, E. J. W., Haayman, P. W., and Romeijn, F. C., Chem. Weekblad 44, 705 (1948). 89, Parravano, G., J. Chem. Phys. 23, 5 (1955). 90. Ilauffe, K., and Block, J., 2.physik. Chem. (Leipzig) 198, 232 (1951). 91. Parravano, G., J . Am. Chem. Sor. 76, 1452 (1952). 92. Schwab, G. M., and Block, J., 2. physik. Chem. (FrankftLrt) IN. S.] 1, 42 (1954). 93. Parravano, G., and Boudart, M., Advances in Catalysis 7, 50 (1955). 04. Schwab, G. M., and Block, J.,Z.Elektrochem. 68, 756 (1954).

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96. Keier, N. P., Roginskii, S. Z., and Sazonova, I. S., Izvest. Akad. N a u k S.S.S.R., Otdel. Fiz. Nauk 21, 183 (1957). 96. Dry, M. E., and Stone, F. S., Discussions Faraday SOC. 28, 192 (1959). 97. Winter, E. R. S., Discussions Faraday Soe. 28, 213 (1959). 58. Cimino, A., Molinari, E., and Romeo, G., 2. physik. Chem. (Frankjurt) [N. S.] 16, 101 (1958). 99. Keier, N. P., and Kutseva, L. N., Izvest. Akad. N a u k S.S.S.R., Otdel. K h i m . N a u k 23, 797 (1959). 100. Gale, R. L., and Stone, F. S., unpublished data (1960). 101. Bielanski, A., Deren, J., Haber, J., and Sloczynski, J., 2. physik. Chem. (Frankfurt) IN. S.]24, 345 (1960). 102. Molinari, E., and Parravano, G., J . A m . Chem. Soc. 76, 5233 (1953). 103. Cimino, A., Molinari, E., and Cippolini, E., Proc. 2nd Intern. Congr. on Catalysis, 1960, p. 263. Technip, Paris, 1961. 104. Heiland, G., Mollwo, E., and Stockmann, F., Solid State Phys. 8, 193 (1959). 106. Grant, F. A., Reve. Modern Phys. 31, 646 (1959). 106. Eibner, A., Chem. Ztg. 36, 753 (1911). 107. Myasniltov, 1. A., and Pshezhetskii, S. Y., Doklady Akad. N a u k S.S.S.R. 99, 125 (1954). 108. Putseiko, E. K., and Terenin, A. N., Problemy Kinet. i Katal. Akad. N a u k S.S.S.R. 8, 53 (1955); Chem. Absti. 60, 1453 (1956). 109. Heiland, G., 2. Physik. 142, 415 (1955). 110. Melnick, D. A., J . Chem. Phys. 26, 1136 (1957). 111. Medved, D. B., J . Chem. Phys. 28, 870 (1958). 112. Fujita, Y., and Kwan, T., Bull. Chem. Soc. J a p a n 31, 380 (1958); J . Research Inst. Catalysis, Hokkaido Univ. 7, 24 (19600). 113. Solonitzin, Y. P., Zhur. Fiz. X h i m . 32,2142 (1958); Terenin, A. N., and Solonitzin, Y . P., Discussions Faraday SOC.28, 28 (1959). 114. Wolkenstein, Th. (Volkenstein, F. F.), and Kogan, S. M., J . Chim. Phys. 66,483 (1958). 116. Morrison, S. R., Advances in Catalysis 7, 259 (1955). 116. Mollwo, E., Ann. Physik. [6] 3, 230 (1948). 117. Borissov, M., and Kanev, S., 2. physik. Chem. (Leipzig) 206, 56 (1955). 118. Romero-Rossi, F., and Stone, F. S., unpublished data (1961). 119. Hauffe, K., and Vierk, A. L., 2. physik. Chem. (Leipzig) 8196, 160 (1950). 120. Scharowsky, E., 2. Physik. 136, 318 (1953). 121. Klingman, W. H., Ind. Eng. Chem. 62,915 (1960). 122. Kubokawa, Y., Bull. Chem. Soc. J a p a n 33, 740 (1960). 123. Romero-Rossi, F., and Stone, I?. S., Proc. 2nd Intern. Congr. on Catalysis, 1960, p. 1481. Technip, Paris, 1961. 124. Ritchey, W. M., and Calvert, J. G., J . Phys. Chem. 60, 1465 (1956). 126. Myasnikov, I. A., Zhur. Fiz. Khim. 31, 2005 (1957). 126. Komuro, I., Fujita, Y., and Kwan, T., J. Research Inst. Catalysis, Hokkaido Unit!. 7, 73 (1959). 127. Barry, T. I., and Roberts, R., Nature 184, 1061 (1959). 128. Kennedy, D. R., Ritchie, M., and Mackenzie, J., Trans. Faraday Soc. 64, 119 (1958). 129. Goodeve, C. F., and Kitchener, J. A., Trans. Faraday SOC.34, 570 and 902 (1938). 130. Newman, R., and Chrenko, R. M., Phys. Rev. 114, 1507 (1959). 1S1. Haber, J., and Stone, F. S., Proc. Chem. Soc., p. 424 (1961).

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Radiation Cata I ysis R. COEKELBERGS, A. CRUCQ,

AND

A. FRENNET

Ecole Royale Militaire Institut Interuniversitaire des Sciences Nucldaires Brussels, Belgium Page I. Introduction.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Experimental Studies of Some Irradiated Heterogeneous Systems. . . . . . . . . . . 56 A. Some Properties of Microporous Solids. . . . . . . . . . . . . . . . . . B. Evaluation of G in Heterogeneous Systems.. . . . . C. Nitrous Oxide Radiolysis.. .................... D. Nitrogen Fixation.. .......................... E. Methane Radiolysis. .......................... . . . . . . . . . . . 68 F. Ammonia Synthesis by Beta Rays.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 G. Ethylene Polymerization Induced by Gamma Ra H. Results by Other Workers.. ..................... . . . . . . . . . . . . . . . . 75 111. General Degradation Scheme of Radiation Energy in Solids.. . . . . . . . . . . . . . . 80 A. Radiation of Energy Larger than 1 kev., ............................. 81 13. Radiation of Energy Smaller than 1 k ..................... 97 C. Permanent and Transient Effects. . . . IV. Radiation Catalysis. . . . . . . . . . . . . . . . . . ...................... 110 A. General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Catalyst Activation. . . . . . . . . . . . . . . . . . ..................... 115 C. Energy Transfer.. . . . . . . . . . . . . . . . . . . . V. Some Comments About the Experimental Results.. . . . . . . . VI. General Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendices A and B . . . .......................... 130 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I . Introduction Since a few years ago, the attention of several scientists (1) has been drawn to the influence exerted by corpuscular and electromagnetic radiations upon certain heterogeneous systems. Consequently, a growing interest in a new research field appeared. This field is known as radiation catalysis and it is concerned with a set of typical phenomena quite different in nature from those usually encountered in homogeneous radiation chemistry. These phenomena are brought about by irradiating a solid phase which plays the role of a catalyst. In this respect, one can distinguish two major groups of experimental studies. A first group (2-7) is concerned with the modification of catalytic 55

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R. COEKELBERGS, A. CRUCQ, AND A. FREN N ET

activity of solids irradiated prior to the introduction of the reactants. Through comparison with the activity of the same nonirradiated solid an increase or decrease of catalytic activity is observed, and this change is related to modifications of certain surface properties. To the second group belong those works in the course of which entire heterogeneous systems, solids and reactants, are irradiated. To this type of research belong different studies which have been carried out in our laboratory. A few years ago, experiments were undertaken to study the utilization of the energy of fission fragments in processes of chemical synthesis. To this end several gas phase reactions have been carried out in the presence of various solids characterized by a large surface (8-12). These solids, which did not present any catalytic activity in the absence of radiations, were shown to influence, sometimes to a considerable extent, radiation-induced chemical processes. This finding is in good agreement with observations made by other authors (13-20) under different circumstances. As will be seen later, the influence of the radiations on the heterogeneous reactions may vary widely and depends on a set of factors among which we mention the area and nature of the solid, the composition of the gaseous reaction phase, and the nature of the radiation. It is our purpose in this paper to place emphasis on those experiments in the course of which both solid and reactants are irradiated a t the same time. The experimental part is only devoted to these works. To this end we give, mainly, descriptions of experiments carried out in our laboratories. Results by other authors are considered as well, but no systematic survey has been made of all the studies concerned with this problem. In the theoretical part of this essay, energy degradation processes in solid are first considered. The conclusions drawn in this way allow us to propose a scheme of interpretation for the results obtained in the field of radiation catalysis and to establish a set of previsions, which are to be considered as work hypotheses and may eventually be used for establishing a more general theory.

I I . Experimental Studies of Some irradiated Heterogeneous Systems

A. SOMEPROPERTIES OF MICROPOROUS SOLIDS This subsection describes a few physical properties of the catalysts mentioned in the part of this paper related to the experimental works of our laboratory. The data given here are not essential for complete understanding of this paper; however, they are necessary to calculate radiochemical yields (see Section II,B and Appendices A and B), and are also useful for supporting certain points of the discussion. The solids used by other authors

RADIATION CATALYSIS

57

cited in this paper often possess different characteristics; discussion of these solids and their characteristics is found in the original papers. Our experiments are carried out in the presence of various types of microporous solids. They consist principally of alumina (CA), silica (CS), and activated charcoal (EC). For irradiation purposes with fission fragments, uranium is incorporated as an oxide by impregnation or coprecipitation methods. Microporous solids with variable uranium content are made available in this way with, as supporting phase, alumina (CAU), silica (CSU), and activated charcoal (ECU) (8, 9 ) , respectively. Quite apart from the chemical composition, the knowledge of the physical structure of these solids is particularly important. For this reason a series of systematic measurements have been carried out on these solids of granulometry between 15 and 30 mesh. These measurements deal with the following. The B.E.T. area. The apparent density expresses the ratio of weight to volume for a large number of grains loosely piled up without compression. The mercury density is determined by measuring the volume accessible to mercury under atmospheric pressure; this volume only consists of pores with a radius exceeding 6p. The helium density is calculated by measuring the volume accessible to helium. This is the true density and is also spoken of as the crystallographic density. The spectrum of pore dimensions. Pores of radius 10-300 A. are studied by analyzing, according to the C. Pierce method, the nitrogen desorption isotherms at the temperature of liquid nitrogen (21). For pores of radius between 300 A. and 611, the mercury porosimetry under high pressure (22, 23) is used. This set of measurements gives a rather precise over-all picture of pore distribution of the various solids. Table I shows a set of numerical data concerning the surface, the density, and the total pore volume of the different catalysts used. These values are completed by the spectrometry of the pores. For example, Fig. 1 (plain lines) shows the variation of total pore volume (V,) as a function of pore radius ( T ) , measured in angstroms in the case of microporous solids of the CSU, CAU, and ECU type. The dotted line, corresponding to a CAU type, shows pore distribution as a function of pore radius, the maxima permitting the evaluation of those pores which occur most frequently in the microporous structure. In the example chosen, the pores of approximately 35 A. are the most frequent ones. Another very important maximum is located at 300 A. Similar studies were made for the other microporous solids. In the case of silica (CS and CSU), a maximum number of micro-

58

R. COEKELBERGS, A. CRUCQ, AND A. F R E N N E T

pores possess an average radius of 25 A,, whereas for active charcoal (EC and ECU) the majority of micropores have radii smaller than 10 A. It can be shown that, in the examples mentioned, pores with radii smaller than 30 A. exceed 50% of the total porous volume. Table I also indicates that the volume of the solid nearly equals the volume of the pores (pores of radii smaller than 6 p ) . In this way an approximate picture of the comTABLE I Physical Characteristics of the Porous Solids

CAa CS* ECc CAUd

csue

ECUf ESA@

0 0 0 12 13 28 0

0.7-0.8 0.7-0.8 0.5 0.7-0.8 0.7-0.8 0.7 0.7-0.8

0.65-0.7 0 . 8 -0.85 1.25-1.1 0.75 0.9 0.75 0.8

0.254.3 0 . 4 -0.35 0 . 3 -0.35 0.25 0.4 0.35 0.4

0.4 0.45 0.55 0.5 0.5 0.4 0.4

150- 250 500- 750 1000-1250 150- 250 400- 600 650- 900 300- 400

_ _ _ ~

Porous alumina. * Porous silica. Active charcoal. Porous alumina impregnated with uranium. Porous silica impregnated with uranium. f Active charcoal impregnated with uranium. 0 Molecular sieve 5A. Ir Apparent density: ratio of weight to volume of a large number of grains loosely piled up without compression. ’ Mercury density: calculated by measuring the volume accessible to mercury; this volume consists only of pores with a radius exceeding 6p. 2 Helium density: calculated by measuring the volume accessible to helium; this volume includes all the micropores. Volume of pores the radius of which is less than 6p.

plete microporous struct)ure can be obtained. It consists of “macrograins” measuring approximately 1 mm. (15-30 mesh). These in turn consist of “micrograins” measuring some ten angstroms separated by micropores of comparable size. Irradiation, as shown below, does not alter these charact’eristicsin any appreciable way; a few cases are cited that need to be singled out. This fact is especially important in the case of irradiation with fission fragments. Indeed, as shown below, the passage of these particles through solid matter produces structural modifications commonly known as radiations damage (24). These have recently been investigated in the cage of alumina, silica,

3

0 W 0,

RADIATION CATALYSIS

3 3

a a

mv) w w u -

I

L

I

'

59

--a

-0

n L

60

R. COEKELBERGS, A. CRUCQ, AND A. FRENNET

and activated charcoal; important structural modifications appear only for energy amounts exceeding loz3e.v. g.+.

B. EVALUATION OF G IN HETEROGENEOUS SYSTEMS When studying radiation-induced reactions it has become customary to express results in terms of the G of the reaction. This is a n arbitrary unit expressing a yield. It measures the number of molecules that react for every 100 e.v. supplied to the system as ionizing radiation. It can be evaluated either with respect to the number of reagent molecules th a t disappeared or with respect to the number of molecules produced in the course of the reaction. One may distinguish two classes of radiation-induced reactions depending upon their G value (25): (1) Reactions with small G values, between 0.1 and 10. Endothermic reactions belong to this class as well as some exothermic reactions not obeying chain mechanisms. This will often be the case for high activation energy reactions, e.g., the ammonia synthesis (G = 1) (26, 27). (2) Reactions with high G values. In this case, the G varies widely and may attain several hundred thousand units. These reactions, always exothermic, follow chain mechanisms; to this class belong polymerization reactions and some halogenation reactions. I n this paper, when homogeneous systems are considered, the corresponding G values will be represented by the symbol Ghom. Their computation requires only the estimation of the energy dissipated in the reaction medium and does not present any particular difficulty. For heterogeneous systems, such as solid-gas systems, where the reacting phase is gaseous, the problem is more complicated. One has to take into account the distribution of the absorbed energy between the different phases. To start with, we assume that the energy absorbed by the solid is lost for the reaction. For this reason, we must distinguish two G values: (I) the apparent G value (Gapp) computed with respect to the total energy supplied to the system as a whole, and (2) the actual G value (G,,J computed with respect to the energy effectively supplied to the reacting phase. Comparison between Ghom and Gectfor the same reaction permits the measurement of the influence exerted by the presence of a solid upon the course of a radiation-induced reaction. Unfortunately, Gaotcomputations have been impossible u p to the present time. For evaluation of the energy distribution between various phases to become possible, it is necessary to make certain assumptions. These hypotheses, as will be seen later (Sections I V and V), are but partly justified. Therefore, Gact is no longer mentioned in this paper. The quantity calculated with the following hypotheses is called G,,,.

RADIATION CATALYSIS

61

The heterogeneous system gas-microporous solid is considered as a homogeneous gas-solid mixture. I n this case, the following assumptions are made. (1) The radiation energy distribution between the phases is given by the ratio of the corresponding stopping powers. (2) For a given constituent, the stopping power is proportional to the product of molecular stopping power and molecular concentration; (3) The molecular stopping power is equal to the sum of atomic stopping powers. These in turn are commonly considered to be proportional to the atomic numbers. It should be stressed that for these assumptions to be valid, the range of incident radiation into matter must be large compared to the dimensions of solid phase particles. The micrograins of our supports measure several tens of angstroms whereas the range of the fission fragments reaches some ten microns. Thus these particles traverse several thousands of micrograins and micropores. The above mentioned assumptions are thus valid in the case of fission fragments. Even more so, they are valid in the case of other radiation types (beta, gamma, etc.), the range of the latter exceeding largely that of fission fragments. However, the quantitative approach to the problem of the repartition of radiation energy between the solid phase and the gaseous phase requires the exact knowledge of the gas concentration in the micropores. We have been able to show that for certain gases (NzOamong others) the concentration of the gas in the pores cannot be calculated simply by application of the perfect gas law. It is necessary to take into account gas adsorption by the solid, even for experiments carried out a t temperatures exceeding the critical temperature of the considered gas. Conditions prevalent in our experiments and, more particularly, high pressure render the determination of adsorption isotherms difficult; consequently, the latter are not known in most cases. Two extreme G,,, values can thus be calculated corresponding on one hand to complete adsorption of the reagents, and on the other hand to complete absence of adsorption. In the case of total adsorption, calculations are simple, whatever the type of radiation. I n the case of no adsorption, the calculations are different, depending upon experimental conditions, and more specially upon the type of radiation. Indeed, in the case of gamma irradiation, a n important fraction of the gas is enclosed in the pores; in this portion of the gas a certain amount of radiation energy is dissipated and induces a homogeneous radiochemical reaction ; the over-all result needs to be corrected for this particular contribution. Considering fission fragments irradiation, a fraction of the energy is dissipated outside the macrograins. In the ideal case of a spherical grain, it can be shown that the corresponding fraction is 1.6y0(28). However, if account is taken of both the

62

R. COEKELBERGS, A . CRUCQ, A N D A. F R E N N E T

density of the dissipated energy as a function of the distance to the surface of the grains, and the extent of the cont,act surface between grains, the irradiation of the thin gas film enveloping the “macrograins” may be considered as heterogeneous. I n order to render the text easier to read, detailed calculations and their justifications have been carried out in Appendices A and B. For the results, the two extreme values of G,,, are often mentioned. Which of these two values is to be considered as most probable is indicated whenever such a choice may be substantiated by known values of reagent adsorption.

C. NITROUS OXIDE RADIOLYSIS 1. Homogeneous Radiolysis

The homogeneous NzO radiolysis has been studied by various authors using different radiation types (fission fragments, alpha particles, beta rays, gamma rays) (29-31). Recently Moseley and Truswell (32) have published a comprehensive survey of this subject. The various homogeneous radiolysis results are usually in very good agreement. The G of decomposition, except for some X-ray irradiations (SS), show a remarkable independence towards the radiation type and the pressure, provided that the latter value is greater than 300 mm. Hg and its value is approximately 12. Decomposition products are Nz, NOz, and 0 2 , in the ratio 1:0.48:0.12.The value of G in Nz formation is between 8 and 9.7, the latter value being the most probable one, and the first one corresponding to a reaction characterized by a high extent of reaction. 2. Heterogeneous Radiolysis by Fission Fragments

Sealed quartz vessels of 3 ml. capacity, containing 1 g. of microporous substance impregnated with uranium of the CSU or CAU type and 50 ml. S.T.P. NzO, are exposed to the flux of thermal neutrons of a nuclear reactor. In such conditions and if calculated according to the above assumptions (Section II,B), the percentage of the energy of the fission fragments which is directly dissipated in the gaseous phase is between 3.5 and 9%, depending on the unknown amount of NzO adsorbed by the solid. Without making a complete survey of results published elsewhere (8, 9), we retain here only some very characteristic results. I n the absence of radiation, the microporous system used gives no NzO decomposition even a t temperatures 50°C. higher than the temperature (SOOC.) a t which radiolysis was performed. The percentages of nitrogen, oxygen, and NzO produced in the case of heterogeneous radiolysis are the same as those for homogeneous radiolysis. Nevertheless, a slight dispersion with respect to Oz values has been observed. The results show the reaction

63

RADIATION CATALYSIS

G value, as calculated with respect to the produced nitrogen, to be constant for extent of NzO decomposition attaining 10%. As shown in Table 11, G,, is indeed much higher than Ghom. TABLE I1 Heterogeneous NzO Radiolysis by Fission Fragments Nature of the support"

csu csu CAU

Of

the support (m.2 g.-l)

Amount of Produced irradiated NzO dose Nzb (mole x 10-3) (e.v* X 'O") (moles x 10")

456 317 300

2.1 2.1 2.1

1.37 1.37 1.33

35.5 29 37.5

Gap,'

1.56 1.13 1.69

Ggaac

1847 1449 18-50

1 gram of support in all cases. for the contribution of the reactor background (gamma radiation, fast neutrons). Calculated with respect to the produced nitrogen. The two cited figures are extreme values. The first one is calculated by assuming that the total amount of N10 is adsorbed in the pores; the second by assuming that no adsorption occurs. Low pressure measurements indicate important adsorption. The first value seems therefore more correct. Detailed calculations are carried out in appendices A and B.

* Corrected

E f e c t due to the nature of the microporous solid. Several experiments carried out with microporous supports of comparable surface but of a differentnature have shown the reaction G value to be distinctly influenced by the nature of the surface. The two last results of Table I1 make this situation apparent in the case of alumina and silica microporous supports; the reaction G value for alumina is 50% higher than for silica. An analogous effect has been systematically observed in the course of all our studies carried out on various systems. 3. Heterogeneous Radiolysis by Gamma Rays

Irradiations of NzO alone, as well as in the presence of porous silica (CS), alumina (CA), arid silica and alumina impregnated with uranium, have been carried out with a CoB0source of gamma radiation. Results from a few experiments are cited in Table 111. In this instance, once more, the rate of NzO decomposition is seen to be increased, sometimes to a considerable extent, by the presence of a microporous solid substance.

4. Firsl Conclusions The most striking first conclusions are the high Gga8 values, which are 3-10 times as high as the corresponding homogeneous values. If it is taken

64 Total fixed nitrogen on

0 CSU-irrodiatian

of air

(0,).700 x IO'M

@ @ @

Total N fixed moles x 16"

CSU- irrodiation of N;1.55 0, (0,)=2000 IO'M CAU- irradiotion of air (0,)=450x 10-eM CAU- irradiation of N,-1.55 OI (0,)*2000x I d %

408

,' 0 '

0

,'

30

20

10

0.5

I

Dose evx lo"

Fra. 2. Nitrogen fixation by fission fragments on porous catalysts.

into account that irradiations with gamma rays and with fission fragments are carried out, respectively, a t 30°C. and 80"C., and that NzO adsorption is consequently more important in the first case than in the second, then the G,, values may be considered to be of the same order for both types of radiation. It should again be pointed out that no NzO decomposition takes place if, in the absence of radiation, NzO is introduced a t 80"C., in the presence

65

RADIATION CATALYSIS

TABLE I11 Heterogeneous NzO Radiolysis by Gamma R a y s

Nature Of of the the support“ support (m.2 g.-l) none

-

CA

456 223 560 186

csu CAU cs

Amount of irradiated NzO (mole X 10-3) (e.v. 2.25 2.25 2.25 2.25 2.25

x

dose

Produced Nzb (moles X 10-6)

GW,

7.2 26 35.8 44.5 37.4

7.7 2.45 3.38 4.20 3.53

0.57 6.35 6.35 6.35 6.35

GBB2

27-107 37-177 47-242 38-200

I gram of support in all cases. Calculated with respect to the produced nitrogen. c See footnote “d,” Table 11. Q

of the microporous solid submitted to preliminary irradiation. Increased efficiency results only from the irradiation of the heterogeneous system; the presence of a microporous solid finally results in the better utilization of the available radiation energy. Bearing in mind the above mentioned assumptions for the calculation of the energy repartition between phases, things seem to happen as if, in

0.4

0.3 Irradiation of air (pressure 25 ato)

0.2

0.I

0

5

10

FIG.3. Nitrogen fixation on CSU by fission fragments-G course of the reaction.

15

20 Per cent of consumed 0,

values; evolution in the

66

R. COEKELRERGS, A. CRUCQ, AND A. FREN N ET

the extreme cases cited, an important fraction, eventually reaching 50% of the energy absorbed by the solid, was transferred to the gaseous phase.* Nevertheless, much care is needed when considering this transfer concept, which results from comparison between Ghom and G,,,. This comparison is only valid when homogeneous and heterogeneous mechanisms are identical. In the course of the subsequent discussion (Section V), this question will he raised again. D. NITROGEN FIXATION Nitrogen fixation from air constituents has drawn the special attention of research workers interested in the possibility of chemical synthesis by means of the kinetic energy from fission fragments (34-38).

1. Homogeneous Irradiation Harteck and Dondes (34), followed by other workers, have systematically studied the homogeneous reaction using either fission fragments irradiation or the radiation (gamma, fast neutrons) from a nuclear reactor. The systems used as sources for the fission fragments were either uranium oxide films, or very fine grains of these same oxides which can eventually give suspensions. Quartz fiber systems (38) in which uranium oxide is incorporated have been used as well. The radiochemical nitrogen fixation reaction which is realized by means of these systems is usually considered as homogeneous. The results of these workers put a t our disposal a few values of Gho!,,. Unfortunately these values remain rather fragmentary. Harteck and Dondes show the Ghom value for NOz formation in the gamma and neutron flux of a nuclear reactor to increase from 0.28 a t 1/30th atm. to 5 a t 25 atm. Moseley and Edwards (37) working a t atmospheric pressure find in the case of fission fragments irradiation a G h o m of 0.9 for nitrogen fixation. They mention this value to be variable on the path of the fission fragment and to show a maximum of 9 a t the end of the range. Highest G’s for nitrogen fixation from atmospheric air cited in the literature are 6.6 (38) and 7.3 (36). Moreover, a theoretical study by Harteck and Dondes (38) shows G = 7 to be a maximum value for the homogeneous reaction.

2. Heterogeneous Irradiation by Fission Fragments Numerous nitrogen fixation experiments b y fission fragments irradiations in the presence of microporous solids have been carried out in our laboratory

* In a previous study (8) our attention was particularly drawn to the possible high efficiency utilization of the kinetic energy of the fission fragments. For this reason we did not hesitate to make a quantitative estimate of this transfer. This variable is a purely formal one as its purpose is solely t o facilitate over-all energy calculations.

67

RADIATION CATALYSIS

(9).We merely state here those results bearing a relationship to our present viewpoint. Irradiation was carried out with atmospheric air or with mixtures having the composition 02/Nz = 1.55. The pressure was approximately 25 atm. The microporous solid was uranium impregnated alumina or silica (CAU and CSU). After the reaction nitrous nitrogen, nitric nitrogen and total fixed nitrogen (NO, N203 Nz06)were determined. Figs. 2 and 3 and Table I V show some results a t an irradiation temperature in the reactor of 80°C. These figures show the total fixed nitrogen as a function of the total energy absorbed by the heterogeneous solid-gas system. In these experiments, the percentage of transformed 0 2 is between 10 and 50%; this corresponds to energy doses smaller than loz3e.v. It was mentioned previously that the microporous solid remains practically unaltered under such conditions. Table I V gives a few Gap,, and G,, values

+

+

TABLE IV Heterogeneous Nitrogen Fixation by Fission Fragments Irradiation Nature

Irradiation of a mixture Oz/Nz = 1.55

Irradiation of air

nf thp

CSU Cf\U

0.4 0.25

3.7-10.3 2.7-6.9

0.27 0.15

2.5-6.9 1.6-4.2

0.13 0.17

1.5-3.8 1.9-4.9

0.13 -

1.5-3.8

-

Values corresponding to a very low percentage of transformed 0 2 . 0 2 transformation. The two cited figures are extreme values. For calculating the first one, we assume that the reactants are completely adsorbed on the solid, and for the second one, that there is no adsorption a t all. This latter case being more probable, the second value therefore seems more correct. Detailed calculations are carried out in Appendices A and B. a

* Values corresponding to a 20%

for nitrogen fixation corresponding to the beginning of the reaction and to 0 2 transformation extent of about 20%. The G's are seen to be greater for air irradiations than for the OdNz = 1.55 mixture. This difference is generally more pronounced in the presence of silica than with alumina. The G,,, values between 1.6 and 10.3 are to be compared to the above mentioned 6.6 and 7.3 values of Ghom. A marked difference exists between the behavior pattern of silica and alumina, in the case of prolonged irradiation; this particular feature is worth mentioning, even if it is not immediately relevant to the problem under examination. While the nitrogen fixation is progressing, the G,, of the reaction decreases in bothrcases. This quite normal decrease is more apparent as equilibrium is approached , when the opposite nitrogen oxide decomposition is more important. Nevertheless, decrease of G is seen to be much more rapid for silica than for alumina, especially for energy amounts

68

R . COEKELBERGS, A. CRUCQ, AND A . FRENNET

greater than e.v. This phenomenon is probably closely related to the evolution of the microporous structure; in the case of alumina the result is a marked decrease in surface and porosity for amount of energy greater than e.v. (24). By way of conclusion, although it is very difficult to have an exact evaluation of the interaction between solid and gaseous phase, it seems clear that the course of the reaction is influenced by the presence of a microporous solid. This phenomenon depends markedly upon the nature of the solid.

E. METHANERADIOLYSIS We have carried out methane radiolysis experiments under widely varying conditions (9). The most important results are obtained either by homogeneous gamma ray irradiations with a Co60source or by heterogeneous irradiations with this same gamma ray source or with fission fragments. Alumina (CAU), silica (CSU), and uranium impregnated charcoal (ECU) have been used as microporous solids. Table V and Fig. 4 summarize the principal data that have been obtained. In these experiments the percentage of transformed CHI does not exceed 10%. This latter value corresponds to energy doses of e.v., dissipated by the fission fragments. In the case of homogeneous irradiations the energy dissipated into the reaction system is equal to 3 X 1020e.v. Among the products, the most important are hydrogen and ethane, to a lesser extent ethylene and propane, and finally a complex mixture of unidentified hydrocarbons. The composition of these hydrocarbons undergoes strong changes in the course of irradiation; however, their proportion with respect to the amount of light hydrocarbons (ethane, propane, ethylene) that are produced, does not exceed 10% for the considered extent of radiolysis. The G values with respect to CH, disappearance are also indicated, These values are obtained through the over-all carbon and hydrogen balance of the radiolysis products. A 20% uncertainty results from the presence of the unidentified hydrocarbons. We immediately point out that our results for the homogeneous phase arc in good agreement with those found in the literature. With beta irradiation (2 M.e.v.), Lampe (39) has obtained the following yields: G(-CH4)

7.6

5.7

G ( H 2 ):

G(C2Ea)

2.1

which are to be compared with our respective values of 6.7, 5.2, and 2.7. The data of Table V point out clearly both the effect produced by the nature of the microporous solid for a given type of irradiation and the influence due to the nature of the radiation for a given type of solid. In the case of irradiation with fission fragments, the GBa8corresponding

TABLE V Heterogeneous Methane Radiolysis Fission fragments irradiation in the presence of:

csu

CAU

Gamma irradiation in the presence of:

ECU

csu

CAU

ECU

0.43 6 3-14.8 0.28 4.4-9.6 0.14 2.24.8 0.01 0.2-0.4 0.01 0.2-0.4 0.01 0.24.5

1.57 26-122 0.54 9-28 0.45 7.5-23 0.07 1.2-5.5 0.08 1.3-7.5 0.05 0.8-4.5

0.3 5-0 0.14 2.34 0.085 1.4-0 0.01 0.2-0 0.02 0.3-1.5 0.01 0.2-0.9

0.16 2.8-0 0.09 1.64 0.03 0.5-0 0.01 0.2-0 0.01 0.2-0.9 0.01 0.2-0.9

0.5

0.83

Homogeneous gamma irradiation ~~

0.43 7-15 0.51 8.3-17.9 0.11 1.8-3.9 0.01 0.2-0.4 0.01 0.2-0.4 0.03 0.5-1.1 0.22

0.82 13-29 0.84 13.4-29 0.13 24.5 0.01 0.2-0.4 0.02 0.3-0.7 0.015 0.24.5 0.15

0.61

0.34

7 5.2 2.7 0.3 0.1 0.1 0.52

G1-CE,) values, corresponding to CH4 disappearance, are given with a 20% uncertainty resulting from the imprecision on the over-all carbon and hydrogen balance. b Two values are cited, corresponding to the extreme cases either of total adsorption of the reactants, or of total absence of adsorption. This latter case and, thus, the value cited in second place seem more probable.

70

R. COEKELBERGS, A. CRUCQ, AND A. FRENNET

to Hz apparition and CH, disappearance are seen to be the greatest in the presence of alumina (CAU) and the smallest in the presence of activated charcoal (ECU). Now the proportion of produced ethane and ethylene seems to be rather constant and thus independent from the nature of the microporous solid. The influence of the latter on the G for production of H2 and disappearance of CH, must originate in the proportion of heavy (C,) hydrocarbons formed. This proportion seems thus to be strongly influenced by the nature of the support. (0)

H,

formotion on

(ijCAU @ csu

(bl C,H,

formotion on

@ ECU

@ CAU

0 csu @

ECU

FIG.4. Methane radiolysis by fission fragments.

For gamma irradiation, the influence of a given solid is not the same as in the case of fission fragments irradiation. The G,,, for H2 production and for CH, disappearance, are seen to be the greatest in the presence of silica (CSU) whereas they are the smallest in the presence of alumina (CAU) and active charcoal (ECU). These are very important facts because they clearly show the effects of radiation catalysis to be essentially variable. The course of the reaction is not always influenced in a favorable way and in some instances a real inhibition may even occur. F. AMMONIA SYNTHESIS BY BETAR A Y S At the time of this writing, several experiments related to NH, synthesis are on hand in this laboratory. This study is still far from finished ; nevertheless, we have deemed it interesting to mention some preliminary results (10).

71

RADIATION CATALYSIS

This synthesis is carried out under the influence of beta irradiation from a 1 curie SrgnYgOsource. Gaseous reactants are taken in a n approximate Hz/PU’*= 2 ratio. They are introduced at atmospheric pressure in the reaction chamber in the presence of microporous alumina. The walls of the pyrex vessel have a thickness of 50 mg./cm.2. The energy dissipated into the reaction medium is measured by chemical ferrous sulfate actinometry and by direct measurement with a proportional window counter. Irradiation is carried out long enough so as to dissipate an energy dose of approximately loz1e.v. into the system. Irradiation temperature is 25’C. If we admit, according to the above mentioned assumptions, that the beta ray energy distribution between phaPes is proportional to the stopping power, 0.0470 of the energy is found to be directly dissipated into the gaseous reaction phase. Table VI summarizes a few results. Several reference measurements have shown that the applied technique does not allow recuperation of more than 25% of the ammonia on the catalyst. The values of G,,, and G,, of Table VI are thus seen to be minimum values that are, in fact, to be multiplied by a factor of a t least 4. TABLE VI Ammonia Synthesis by Beta Rays in the Presence of Alumina

SBET (m.2 g.-l) 107” 20h -

20h

Dose (e.v.)

Measured NH3 (moles X 1018)

GPP

48.

12.6 X lozo 14.6 X lozo 14.9 X lozo

3 1.5 0.6

2.4 1 0.4

486 260 100

Alumina outgassed during 100 hrs. a t 450°C. Same alumina as in Q,outgassed during 24 hrs. a t 1000°C. With respect to the alumina not only the extent but also the nature of the surface varied. a

b a,

The particularly high values of G,,, are very striking. We remember that the G for NH3 synthesis in homogeneous phase under the influence of alpha particles have been measured by different authors (26, 27). Under experimental conditions similar to ours, values between 0.9 and 1.5 have been obtained. If, in our experiments, GPa8and Ghom were of the same order of magnitude, the produced NH3 would remain practically undetectable. This example illustrates quite remarkably the role played by the solid in “radiation catalysis.” The efficiency of the heterogeneous reaction induced by radiation is indeed several hundred times greater than foreseen. This study is still in a preliminary stage; nevertheless, we believe the heterogeneous mechanism to be very different from the homogeneous one. In the course of the heterogeneous reaction a quantity of water is formed which

72

R.

COEKELBERGS, A. CRUCQ, AND A. FRENNET

is equal to the amount of ammonia produced. On the contrary, in the case of irradiation of the catalyst in the presence of Nz or Hz alone no water formation is observed. G. ETHYLENE POLYMERIZATION INDUCED BY GAMMA RAYS

A recent study by Mechelynck-David (11, 12) on ethylene polymerization induced by gamma radiation, has pointed out the influence of the presence of various solids upon the course of the reaction. The solids used are either large surface solids, like silica, alumina, active charcoal, 4-5 A. molecular sieves, or small surface solids like sintered silica, quartz wool and powdered zinc oxide. Experiments are carried out a t 30°C. and 25 atm. in a 5 X lo5roentgen hr.-1 radiation field. For the same total dissirated energy of 3 X lo5rads., the percentage of transformed ethylene varied between a few per cent and ninety per cent, depending upon the nature of the solid. The Ghom of ethylene polymerization (G = 68) indicates short chain mechanism. The presence of small surface solids does not exert any noticeable effect. Using ZnO, Ggas*has the same value as Ghom. In presence of quartz fiber a slightly superior G,,,* value of 100 results whereas in the case of sintered silica this value is 220. Large surface active charcoal (1000 m.2 g.-') inhibits the reaction; Ggns*becomes inferior to 20, even for very small percentages of transformed ethylene. If, on the contrary, 10% uranium oxide is incorporated into this same active charcoal, the inhibition effect vanishes and Ggna*takes a normal value of 120. I n the case of other large surface solids (silica, alumina, molecular sieves) the Ggae*(-CzH4) is 15-30 times as great as the value obtained without any solid addition. In the case of silica for instance, for a 50% ethylene transformation extent, the Ggas* reaches a value greater than 3000. The corresponding Gap,, is approximately 350. In the case of alumina comparable G values are obtained. These results suggest the following remarks. (1) The experiments under examination are carried out at temperatures higher than the critical ethylene temperature. The observed effects cannot, possibly result from any ethylene liquefaction (capillary condensation) within the solid pores. (2) The observed increase of the yields cannot be explained only by an increase of the initiation rate. If all the energy dissipated into the solid is quantitatively used for the initiation process through a solid-gas transfer phenomenon, and if the heterogeneous polymerization mechanism is the same as the homogeneous one, the G,, cannot exceed largely the Ghom. * Ggasis calculated by this author with the same hypotheses as ours (cf. Section 11,B). Ethylene is completely adsorbed on the solid; the evaluation can thus be made exactly.

73

RADIATION CATALYSIS

As a matter of fact, Gap,, in extreme cases, is 4-5 times as great as Ghom. As a consequence the heterogeneous mechanism is necessarily thought of as different from the homogeneous one, and probably related to the extent of the surface. This hypothesis is confirmed by the following results. TABLE VII Heterogeneous Ethylene Polymerization Induced by Radiation InJluence of the Catalyst A m o u n t Amount of LMS 5A. (g.) 0.1 0.4 0.7 1 1.5 2

Percentage Of

polperized ethylene 0.66 13.7 41.5 41.7 30.8 14.2

GW - (C2H4) 76 1582 4776 4785 3545 1631

41 355 679 498

254 89

Surface Efect. As seen before, 5 A. molecular sieves (LMS) increase considerably the yield of radiation induced ethylene polymerization. In the next experiments, constant amounts of CzH4 (0.415 moles) are irradiated in the presence of increasing amounts of 5 A. molecular sieves. Table VII and Fig. 5 show the corresponding results. For a constant energy dose, the fraction of polymerized ethylene, expressed as a function of solid addition, is seen to possess a maximum corresponding approximately to

f FIG.5. Effect of the weight of an additive on the course of the C2H4 polymerization.

74

R. COEKELBERGS, A. CRUCQ, A N D A . F R E N N E T

1 g. of added solid. Assuming that the area of an adsorbed ethylene molecule is 15 A.2, the ethylene amount necessary for covering with a monolayer the whole surface (350 m.2) of one gram of 5 A. molecular sieve is 0.4 mole, which is approximately the quantity used in these experiments. As the working temperature exceeds the critical ethylene temperature, capillary condensation is not to be considered. It seems thus quite probable that the first part of the curve on Fig. 5 should correspond to a monomolecular ethylene layer covering the entire molecular sieve surface. Admitting the gaseous phase polymerization rate to be negligible as compared to surface polymerization, which, in view of the preceding results, seems plausible, then the reaction rate clearly grows with increasing surface. Whereas, on the contrary, the second part of the curve should correspond to an unsaturated ethylene surface. I n this case the "holes" in the adsorbed layer may be thought of as decreasing the rate of chain propagation. Temperature Eflect. The experiments concerning temperature effects are carried out between -80°C. and +60"C. in the presence of a 5 A. LMS. Comparison of various experiments with constant ethylene and solid quantity submitted to equal amounts of gamma energy, a t temperatures beyond 9°C. which is the critical ethylene temperature, shows the polymerization rate to decrease with increasing temperature. Below 9°C. down to -8O"C., the polymerization rate varies slowly and seems to be slightly affected by temperature. These results, and particularly the existence of a negative temperature coefficient beyond 9O"C., may be explained if again ethylene polymerization in the presence of microporous solids is thought of as a surface phenomenon a t the expense of adsorbed molecules. Indeed, a t temperatures beyond 9"C., no capillary condensation can take place, and a temperature increase simply results in a concentration decrease of the adsorbed ethylene. Below 9°C. practically all the ethylene can be shown to be condensed within the solid micropores. This, to a certain extent a t least, allows for the fact of the small influence of the temperature upon ethylene polymerization rate. Let us also mention that, a t -8O"C., various ethylene irradiations were carried out by gamma rays in the absence of any microporous solid. No polymerization occurred. The microporxs solid is thus seen to take a prominent part in the polymerization phenomenon. These combined facts seem to prove sufficiently that the gamma-induced polymerization in the presence of microporous solids obeys a mechanism of its own, which is different from the homogeneous one. Ethylene polymerization, as is well known, can be carried out thermally in well chosen microporous solids containing a metal oxide ( 4 0 , d I ) . The question arises whether

RADIATION CATALYSIS

75

a parallel exists between thermal polymerization and gamma ray induced polymerization in the presence of these same large surface solids. In both cases, the polymerization might be considered to occur in the adsorbed phase, the nature of the initiation process being the only difference. In the case of thermal polymerization, the metallic oxide produces the initiation. The initiation occurring in the case of gamma ray induced polymerization might proceed through the creation of active centers generating ions or radicals, thus rendering superfluous the presence of the oxide. I n any case the existence of a large surface is a requisite for a rate increase. Nevertheless, this condition is not sufficient in itself because inhibition occurs in the presence of active charcoal. This inhibition by active charcoal has been mentioned previously in the case of methane radiolysie. This example illustrates very well the influence of the nature of the surface on the course of the reaction. Similar observations have been made in the case of thermal and catalytic polymerization of styrene, acrylonitrile, and ethylene (42-44).

H. RESULTSBY OTHERAUTHORS This subsection summarizes some results of radiation catalysis obtained in other laboratories. It is not our purpose to make a complete survey of the corresponding literature, but we want to mention these characteristic result,^, which may prove valuable for the subsequent discussion. 1. Pentane Radiolysis by Gamma Rays

Several papers by Allen and co-workers are related to pentane radiolysis under the influence of Co60 gamma rays in the presence of large surface solids (13,44).These experiments are carried out a t room temperature and results are compared to those of liquid pentane irradiation (45).The solids used are silicagels, certain synthetic zeolithes like the “13X molecular sieve” and “5 A. molecular sieve.” In certain cases and expecially with 13 X molecular sieve the nature of the ions is modified through exchange between Na+ ions from the molecular sieve and Ca++, CO”, and Mn++ ions. Substances containing an oxide are used a s well. As expected, radiolysis products are complex. Apart from Hz, which is the principal gaseous product, these authors have identified saturated and unsaturated hydrocarbons up to Clo. Among them, isodecane is most abundant; approximately 50y0of the reacting pentane is to be found under this form. Pentane is also shown to undergo no appreciable decomposition when put in the presence of these previously irradiated solids. This proves the effect t o result from simultaneous irradiation of pentane and solid, thus

76

R. COEKELBERGS, A. CRUCQ, AND A. F R E N N E T

excluding an intermediate catalytic state through solid activation, and leading to a decomposition in the absence of any radiation. Comparison between homogeneous and heterogeneous irradiations shows the considerable effect brought about by the presence of the solid. Among the various solids, silica and samples containing cobalt give the highest decomposition rates. The corresponding Gg,, are considerably higher than the Ghom. Caffrey and Allen (IS), in the case of irradiation in the presence of silicagel, for instance, cite G,,, values for Hz formation 12 times as great as the corresponding Ghomvalues. These authors consider that things happen as if a great part of the energy absorbed by the solid transferred to the adsorbed pentane; they draw attention, moreover, to the fact that the nature and the proportion of the reaction products are influenced by the nature of the solid. The authors consider that their experimental results are insufficient to give a satisfactory explanation of the observed phenomena. They nevertheless believe that the energy transfer phenomenon is probably related to the migration, towards adsorbed pentane, of “electron hole pairs” created in the solid phase, the pentane thereby becoming ionized. Moreover, they consider this transfer phenomenon to be closely related to the nature of the solid which subsequently influences also the evolution of the radicals and ions formed at the surface. 2. Methanol Synthesis by Gamma Rays In the above mentioned studies, the considered solids exerted no influence in the absence of radiation. Methanol synthesis and cyclohexanol dehydration (Section II,H,3) are on the contrary studied in the presence of other solids which already, in the absence of radiation, have certain catalytic properties. The influence of Co60 gamma radiation upon methanol synthesis from carbon monoxide and hydrogen under atmospheric pressure, with ZnO as catalyst, has been studied by Barry and Roberts (20). Various types of zinc oxide are used which are characterized by different stoichiometries. As a consequence considerable differences appear with respect to both catalytic activity and sensitivity to radiation. Positive radiation influence is only obtained in the case of ZnO samples containing a little zinc excess. Under these conditions, the influence of the radiations results in an increase of yield in the limited 160-230°C. temperature range, where thermal reaction, if slow, is possible. Below 160°C. no thermal reaction takes place; beyond 230”C., it is rapid; in both cases no radiation effect is observed. Increase of yield at about 200°C. can be characterized by a G equal to one, if calculated with respect to the energy dissipated into the zinc oxide.

RADIATION CATALYSIS

77

The thermal catalytic activity of zinc oxides containing a large excess of zinc is smaller; for these catalysts no radiation influence is observed. These authors consider the increase of the methanol formation rate to result from the direct contribution of electrons and positive holes produced by gamma irradiation in the solid. The “free carriers” are able to modify the adsorption equilibria of Hz and CO, because these reactants, according to the authors, are adsorbed as ions on the surface. They consider that the observed unit G may be explained by admitting 20% of the electrons produced by radiation in the solid to be effective for catalytic reaction, 20 e.v. being necessary for the production of one electron. 3. Cyclohexanol Dehydration by Bela Rays Balandine et al. (15, I S ) have studied cyclohexanol dehydration in the presence of catalyst composed of MgS04 and Na2S04 mixed in variable proportions. These experiments are carried out at a temperature between 325 and 420°C. These authors compare the catalyst activity in the absence and then in the presence of radiations. In a first series of experiments, irradiation is carried out with NazSOd labeled with known quantities of S F which emits 0.15 M.e.v. beta rays. With a denoting cyclohexene yield for a given catalyst, in the absence of at a temperature T and during a time t and a+ denoting cyclohexene yield for a catalyst containing S36,the other experimental conditions remaining the same, the authors express the increase of the reaction yield as A = 100[(a+ - a)/a]%. In Fig. 6, A is seen to vary proportionally with the logarithm of beta energy dissipated into the reacting medium. Considerable effects result and the increase of the reaction yield may be greater than 200%. These authors have also reported the activation energy under irradiation to be decreased by 1 or 2 kcal. depending upon the amount of beta energy dissipated per unit time. Analogous results are obtained when introducing in the system Ca46as CaCl2.A recent study carried out at Oak Ridge (46) under the same experimental conditions has confirmed these results. The purpose of another series of experiments by these authors is to control whether irradiation of the system by an exterior beta source yields analogous results. The reacting system containing the nonradioactive catalyst, is exposed to the 800 k.e.v. electrons of an accelerator. Under such conditions irradiation is seen to exert no more influence. No explanation for this phenomenon is given by the authors. The latter result seems to us rather surprising. When comparing the two series of experiments, the sole difference is seen to result from either the presence or the absence of S F in the catalyst, or from a different repartition of the beta radiations in the heterogeneous system.

78

R. COEKELBERQS, A. CRUCQ, AND A. FRENNET

It seems unlikely that the presence of SaKalone might create any particular effect. Moreover, the authors have experimentally shown that the disintegration products of the S36and the total amount of beta radiation received by the catalyst did not exert any influence. The latter point has also been confirmed in the above mentioned American study (46) where it is shown that preirradiation of the catalyst with a 10l2 e.v. g.-I gamma dose does not modify the catalytic activity. 30C

20c

IOC

C Log o f the specific activity

FIG.6. Dehydration of cyclohexahol on magnesium and sodium sulfates.

Concerning the hypothesis of a different repartition of beta radiations within the system, it is difficult to express an opinion since knowledge of the exact experimental conditions of the irradiation by 800 k.e.v. electrons is still lacking. However, it seems possible to explain these differences of behavior in the following way: on one hand, the range of 800 k.e.v. electrons is indeed between 250 and 300 mg./cm.2; on the other hand, the description by the authors of the experimental device shows beta rays going through four successive windows, one of which is 100 mg./cm.2. Under such conditions, beta radiation may influence only a thin catalyst layer, thus considerably limiting the observed effects.

RADIATION CATALYSIS

79

4. Heterogeneous Radiolyses in the Liquid Phase

Before finishing this survey of experimental facts, some works carried out in the liquid phase will be considered. The research work of Vesselovsky (17) is particularly important in this field. This author compares the influence of ultraviolet and gamma rays on the formation and the decomposition reactions of hydrogen peroxide in the presence of certain semiconductors. When oxygen-saturated alkaline aqueous solutions are irradiated by gamma rays, the introduction of zinc oxide into the reaction medium results in a four- to fivefold increase of the yield of hydrogen peroxide, as compared to the homogeneous reaction yield. However, the amount of added zinc oxide is always small enough to induce only a negligible (2%) variation of the absorbed dose. The presence of the semiconductor seems to account for a better utilization of the energy from the incident radiation. Hence, the semiconductor acts as an heterogeneous sensitizer. For this behavior of the semiconductor under gamma irradiation, Vesselovsky proposes an explanation analogous to the mechanism accounting for the zinc oxide influence in the photosensitized formation of hydrogen peroxide; he considers the gamma energy to undergo an efficient transformation into electronic energy of the semiconductor. He characterizes this phenomenon by a “multiplication coefficient,” which measures the increase in the number of excited electrons in the semiconductor per absorbed gamma. Vesselovsky calculates that, in the case of ZnO, energy absorption from CoGogamma radiation (1.23 M.e.v.) must correspond to a multiplication factor of 4 X lo6, since the electron excitation from valency band to conduction band requires 3 e.v. This 3 e.v. value corresponds to the zinc oxide absorption band approximately located at 3850 A. Experimental results show the ‘(multiplication coefficient” to be twice as high as this theoretical value. The author believes that excited electrons of half this energy also contribute to the reaction initiation, the latter resulting from the fixation of one electron by each adsorbed 0 2 molecule. The study of the decomposition reaction of hydrogen peroxide in the presence of ZnO leads to analogous conclusions. Haissinsky and Duflo (18) and Preve and Montarnal (19) report similar phenomena. They have studied, among other reactions, the benzene oxidation in alkaline solution, yielding phenol, the potassium iodide oxidation in aqueous solution, and the uranous sulfate oxidation in aqueous solution. These reactions have been carried out in the presence of various semiconductors, i.e., ZnO, ThOz, Ti02, CoS, Co203,&03,under the influence of X-and gamma rays. I n the case of benzene oxidation these solids do not catalyze the reaction

80

R. COEKELBERGS, A. CRUCQ, AND A. F R E N N E T

in the absence of radiation. The amount of catalyst present as a suspension in the solution is always small enough for the variations of the total dose adsorbed by the reacting system to remain negligible. The increase of the yield, as compared to the homogeneous reaction, is generally proportional to the amount of catalyst. Oxidation of benzene into phenol under the influence of X- and gamma rays is characterized by a Ghom value of 3.1. The G of the heterogeneous reaction takes for constant concentration of the various catalysts the following values: ThOz: 6.5; ZnO: 4.6; ZnS: 4.5; CuO: 5.9; CaO: 3.1; Coz03:6.6; AIZO3:3.1; MgO: 3.1; TiOP: 5.5. I n the case of potassium iodide and uranous sulfate oxidation induced by radiation, the effect of the presence of various solids such as CoS, ThOz, and Nbz06,which are catalysts for these reactions in the absence of radiation, has been investigated. The increase of the oxidation of uranous sulfate always remains small, and never exceeds a factor of two, whereas in the case of potassium iodide oxidation the presence of Tho2 increases the G from 3.2 to 10. For purposes of general interpretation these authors resort to the above mentioned Vesselovsky theory.

111. General Degradation Scheme of Radiation Energy in Solids We begin this section with a general survey of the principal modes of interaction between matter and radiation. This will allow an evaluation of the repartition of the absorbed energy for each particular type of effect. Only those radiations most frequently used in radiation chemistry are considered here and therefore we limit ourselves to radiation of energy smaller than a few million electron volts. Thus we shall be interested with the following types of radiation: gamma rays, beta rays, protons, deutons, alphas, fission fragments, and fast neutrons. The following scheme illustrates the various interaction possibilities between radiation and matter: nucleus of target 7

incident radiation

7 nuclear reaction atom displacement L bremsstrahlung

-+

’

electrons of target 7 ionization I excitation

It is not our purpose to make a detailed study of nuclear reactions. Such reactions indeed are only induced to an appreciable extent by very high energy radiation (>>M.e.v.) and by thermal neutrons; these two cases are not considered here. However, gamma radiation consecutive to radiative capture as well as beta, alpha, and other radiations resulting from other nuclear reactions will be studied here. Moreover, allowance has to be made

81

RADIATION CATALYSIS

for the production by these nuclear reactions of impurity atoms, which may exert an important influence on the properties of a solid. The latter case will be discussed later (Section III,C,l). This study is made up of three major parts. The first one (Section II1,A) is essentially devoted to the primary processes resulting from the passage of the incident radiation through matter. This concerns the energy domain beyond 1 k.e.v. In a second part (Section II1,B) we shall consider the complex and still obscure process of absorption of secondary small energy radiations in nonmetallic solids as well as subsequent degradation of the potential energy stored in these solids as lattice or ionic energy. In a third part (Section II1,C) we make an over-all balance of the effects produced by radiation in the solids, and we emphasize the character, permanent or transient, of these effects.

A. RADIATION OF ENERGY LARGERTHAN 1

K.E.V.

1. Gamma Rays

The energy dissipated by direct interaction between the gamma rays and the target nuclei is negligible as compared to the energy dissipated through the interaction with the electron cloud (47).As yet, little is known about atom displacement consecutive to gamma radiation (48).A certain number of displacements, however, may result in an indirect manner through the secondary betas emitted when the gamma is absorbed. The phenomena related to the gamma bombardment of matter belong nearly exclusively to either the Compton or the photoelectric effect. The incident gamma energy is shared between these two effects. As each element constitutes ? particular case, it would be necessary to consider a lot of examples in order to give a thorough description of the phenomenon. At the condition of making certain-eventually important-approximations, it is possible to derive an order of magnitude for the energy repartition in the case of direct interaction between gamma and matter. The subsequent absorption of the secondary radiations as well as the evolution of ions and excited states that have been created, will be considered later. The variations for the respective cross section for Compton (UC) and photoelectric (up) effects as a function of the energy of incident gamma radiation are known for a great number of elements (47, 49, 50). Figure 7 illustrates these variations for air, aluminium, copper, and lead. The Compton effect is the sole contributor in the case of high energy gamma rays whereas the photoelectric effect alone accounts for the dissipation of energy of soft gamma. The curves which show the variations of the cross sections for the Compton and photoelectric effects intersect at an energy Ez which is a characteristic of the target element. The variations of this particular point

82

R. COEKELBERGS, A. CRUCQ, AND A. F R E N N E T

(M.e.v)

16’

EZ I

E(M.ew)

FIG.7. Variation of the Compton (UJ,photoelectric (up), and total mass absorption coefficients with energy E of the incident gamma ray, for different substances. (a) air; (b) aluminum; (c) copper; (d) lead. x-x-x-x up;--- -u.3; (ut = ,Tc UP) (After R. D. Evans “Handbuch der Physik,” Vol. 34. Springer, Berlin, 1958.)

+

plotted as a function of atomic number 2 are seen to follow the regular curve of Fig. 8. A relatively simple description of the energy decay process of the incident photon is possible if the following approximation is made: for a given element the Compton effect alone is assumed to account for the phenomenon

83

RADIATION CATALYSIS

I

0.90

0.80 -

0.70

-

0.60

-

0.50 -

0.40 -

Q30 -

0.200.10 -

01

I

I

I

I

I

I

I

I

I

20 30 40 50 60 70 80 so FIG.8. Variation of Ez with the atomic number of the absorber. (After R. D. Evans, in “Handbuch der Physik,” Vol. 34. Springer, Berlin, 1958.) 10

when the incident gamma energy is greater than Ez, whereas only the photoelectric effect is retained for values smaller than Ez. Let us consider for instance the absorption of a 1 M.e.v. photon in alumina. I n this case, Ez is equal to approximately 80 k.e.v. (49).According to the above approximation, only the Compton effect has first to be considered. It is well known that this effect is important when the incident gamma energy is great compared to the electron binding energy, even to that of the deepest (e.g., K level) levels of the irradiated element. Consequently the electron binding energy cannot influence the gamma interaction probability. In other words, all the electrons may be considered equivalent. Therefore, electrons are freed from the various levels proportionally,

-

I

0.8

0.6

0.4

0.2 E,( M.e.v )

-

FIG.9. Mean energy E of the Compton scattered gamma ray as a function of the energy EOof the incident gamma ray. (After E. Segrb, “Experimental Nuclear Physics,” Vol. I, part 2, p. 321. Wiley, New York, 1952.)

85

RADIATION CATALYSIS

according to the degree of occupation of these levels. This results in the creation of an ion, whereas the incident photon is scattered with a certain loss of energy. The wavelength of the scattered photon is shown to be given by X = Xo AX, where Xo is the wavelength of the incident photon, and A x is defined by the approximate formula AX = 0.024 (1 - cos S), 0 being the angle between the direction of the incident and scattered photons (48, 49, 51). Figure 9 (4Y)shows the mean scattered photon energy plotted against the incident photon energy. With the aid of this graph, we see th a t the over-all effect of the first interaction between a 1 M.e.v. gamma and alumina results in the production of a 550 k.e.v. gamma, a 450 k.e.v. beta, and an ion. As a first approximation the scattered gamma and the beta energy thus contain all the incident energy. The potential energy, corresponding to the removal of one electron from an atom of one of the alumina constituants, is indeed very small: the energy corresponding to K , transition is 1500 e.v. for aluminium and 525 e.v. for oxygen. The secondary beta radiation and the ion will be neglected now; they will be studied later (Sections III,A,2 and III,B,l). Let us consider absorption of the 550 k.e.v. scattered gamma ray. This gamma ray decays in turn as a consequence of the Compton effect; this same process goes on until the energy of the last scattered gamma approaches Ez.At this moment the over-all degradation scheme may be represented as follows:

+

Gamma 1 M.e.v.

1

Lon

Gamma 550 k.e.v. Beta 450 k.e.v.

Gamma 200 k.e.v. Beta 110 k.e.v. Ion

-+

+

1

Gamma 310 k.e.v. Beta 240 k.e.v. -+ Ion

150 k.e.v. k.e.v. -+

1

Gamma 120 k.e.v. Beta 30 k.e.v. -+ Ion

Gamma 100 k.e.v. Beta 20 k.e.v.

4

At this moment the gamma energy is such that the interaction probability by photoelectric effect becomes greater than by Compton effect. Attention has to be drawn to the fact that, electron removal from the K level will take place in a preferential way (47, 49) when resonance conditions are approached. If we make a balance at this particular moment of the degradation process we see that the incident gamma energy Eo (1 M.e.v.) is statistically shared between (1) 7 betas, their energy ranging between (E0/2) (500 k.e.v.) and EZ (80 k.e.v.), and (2) 7 ions, the sum of their potential energy being less than 0.5% of Eo. For a target made from heavy elements, this photoelectric effect is seen

86

R. COEKELBERGS, A. CRUCQ, AND A . FRENNET

to occur already for higher threshold EZ values of the energy of the successively produced gammas. The energy for electron removal from the K level for a given element grows simultaneously but remains neverthelcss always considerably smaller than the characteristic EZ value. Consequently, it can be shown that the potential energy carried away by the ions does not greatly exceed 10% of the incident gamma energy. To summarize, the major part (> 90%) of the absorbed energy of gamma radiation consecutive to primary interaction with matter is, a t a definite moment, to be found as secondary electrons; this fraction increases as the atomic number of the elements from the target decreases. 2. Beta Rays

The interaction between beta rays and matter is a more complex problem than the corresponding one for gamma rays. Three effects need to be considered ; they are in order of importance : atom displacements, bremsstrahlung, and ionization. Atom Displacements. Atom displacements resulting from direct collision with incident beta radiation give rise, as will be seen later, to dissipation of but a small fraction of the total radiation energy. I n the subsequent discussion this displacement problem will nevertheless be considered frequently; for this reason we will study it here in somewhat more detail. As a general rule, the incident particle requires a minimum energy En, in order to produce a displacement. This energy is variable, depending upon both the type of radiation and the mass of the atom to be displaced. In the case of beta radiation the following formula applies (52) :

I n this relation me is the mass of the electron, M the mass of the displaced atom, and E d the minimum energy that the atom must poseess for displacement to take place. Most authors accept an average value of 25 e.v. for this displacement energy E d (47, 48, 53). For instance, displacement of a hydrogen atom is possible with an 11 k.e.v. beta, whereas for aluminium atom 300 k.e.v. are required. For displacement of atoms whose atomic number is beyond 40, a beta must possess an energy greater than 1 M.e.v. Let us now consider the displaced atom. It carriers an energy that can be transferred to neighboring atoms, producing either ionizations or lattice vibrations. The first mechanism is effective only for energies exceeding a definite and always high value; this will be discussed further (Section 111,A,5). I n the second case the energy is stored in a portion of space sufficiently small for a local temperature rise to take place; this, in turn, induces permanent atom rearrangements within the solid. This effect is

RADIATION CATALYSIS

87

spoken of as “temperature spikes.” Distinction is generally made between two cases. (1) The first case, that of the so-called “thermal spike,” is one in which energy received by the atom is smaller than 2 E d .The energy of the primary knocked atom is progressively transmitted to its neighbors; this process takes approximately sec. (one vibration). This problem is generally treated assuming the crystal to be a continuous medium; the energy of the primary knocked atom is transmitted in an isotropic manner (54) by a diffusion mechanism. It is also possible to define a particular spherical volume in which the atoms have received sufficient energy for the average local temperature to be greater than the melting temperature of the irradiated material. Let us consider, for instance, a primary knocked atom receiving 20 e.v. If the fusion temperature of the irradiated material is approximately 1000”K.,an average of 0.1 e.v. per atom is necessary for increasing the surrounding temperature to about 1000°K. I n these conditions, 200 atoms are brought a t this temperature. If one takes into account the short lifetime of the “thermal spike” (1O-l1 sec.), all the atoms undergo a rapid cooling so that a great number of lattice defects produced by the heating, are quenched. ( 2 ) The second case, that of the so-called “displacement spike” is one in which the knocked atom receives an energy greater than 2Ed. The primary knocked atom can travel a certain length until its energy becomes smaller than 2Ed”. Along its path, secondary knocked atoms may be produced. This process may be repeated, till the energy of these atoms becomes lower than 2Ed. Each of these is then the center of a “thermal spike.” Little experimental information is available regarding the energy fraction of beta radiation dissipated in displacements. Seitz (53) estimates this fraction to be smaller than 0.1%. Tible VIII gives a few experimental values for germanium irradiation (54, 55). These values show, for an energy of loz2dissipated under the form of beta radiation, that the number of displaced atoms is between l O I 4 and 1016;if the same dose is dissipated into one gram of germanium, the resulting concentration of displaced atoms is between and Bremsstrahlung. Every electrically charged particle which slows down,

* The average energy imparted by a moving atom with energy E and mass M I to a “knocked atom” with mass Mz is given by the formula (54) Z = E

2M1Mz (MI M2Y

+

If the moving atom and the target atoms are the same, ?,? = E / 2 . For atoms carrying an energy E smaller than 2Ed, the energy given, upon collision, to other atoms is smaller than Ed; as a consequence they cannot any longer produce displacement.

88

R. COEKELBERGS, A. CRUCQ, AND A. F R E N N E T

emits electromagnetic radiation. This phenomenon is called “bremsstrahlung” in the particular case of a high energy particle decelerating in the field of the nucleus of an atom. This effect accounts for the dissipation of TABLE VIII (54, 56) Number of Displaced Atoms upon Beta Irradia.tion of Germanium Beta energy (M.e.v.)

Temperature (OK.)

Number of displaced atoms for each incident electron

1.5 4.5 3

90 90 298

0.092 0.196 0.106

an appreciable proportion of the total energy, and becomes greater as the elements that constitute the target become heavier. If E b is the energy carried away as bremsstrahlung and Eo the energy of the incident beta, the relation (49) Eb _ -

Eo

EOZ 1600m,c2

in which me is the electron mass and c the light velocity, shows the fraction of total energy dissipated as bremsstrahlung to increase linearly with incident beta energy Eo and atomic number Z of the irradiated element. For a 1 M.e.v. beta ray absorbed in different targets, the following numerical values result : &/Ea

Nature of target

Atomic number

2% 8% 15%

A1 Sn Pb

13 50 82

The photon spectrum of bremsstrahlung is continuous. The theoretical calculations show the number of emitted photons to decrease continuously as their energy increases and goes to zero when the photon energy is equal to the incident beta. The result of this evaluation, in the case of an aluminium target irradiated by beta rays from T1204,is expressed by the curve (full lines) of Fig. 10. If this curve is corrected for self-absorption in the target, which is a n important phenomenon for low-energy photons (A > 1 A.), even in the case of a thin target, we get the dotted curve of Fig. 10. This curve, starting at the origin, presents a maximum for an energy value which is a characteristic of both the energy of the incident beta and the nature

89

RADIATION CATALYSIS

of the target, then decreases monotonically (56).The agreement with experiment remains always within 20%. The subsequent absorption of the bremsstrahlung photons of energy greater than 1 k.e.v. has been studied in Section III,A,l. The photons of lower energy will be considered later. Interaction with the Electron Cloud. The interaction with target electrons causes the dissipation of the major part of the incident beta energy.

6 z-.

c a C

+

.-C

c c

2n 4 62

.-+

-a a

2

Energy of Brernsstrohlung photon -L

Energy of incident Beta ray

0.2

04

0.6

0.8

FIG. 10. Theoretical bremsstrahlung spectra from a thin target (50 mg. cm.-I) of aluminium, irradiated by beta rays from a TIzo4 source. Theoretical. -- -Theoretical corrected for self-absorption of the target. (After J. F. Cameron and J. R. Rhodes (M).) ~

The target electrons behave like free electrons a s long as the energy of the incident beta radiation remains large compared to the energy (Ek) required for the removal of the electron from the K level of the target atoms. Consequently within this energy range, which is more or less large depending upon the nature of the target, the incident beta radiation undergoes progressive degradation into a series of secondary beta rays until their energy reaches the limiting Ek value. Moreover, whenever a secondary beta

I

90

R. COEKELBERGS, A. CRUCQ, A N D A. F R E N N E T

appears, an ion is created. The mean energy transferred to a target electron may be evaluated as a function of the energy of the incident beta (57). The theoretical analysis of the interaction between target electrons and beta rays of energy less than II:k is very intricate. However, the formation of secondary betas and ions remains the process that experimentally accounts for dissipation of nearly the whole of the incident energy, until the beta energy reaches a value of about 100 e.v. As in the case of gamma irradiation the potential energy stored by the ions does not exceed 10% of the incident energy. I n brief, the energy of incident beta rays undergoes progressive degradation. Secondary beta rays carry away the major part, exceeding sometimes 80% of the incident energy. A fraction of the incident energy is converted into bremsstrahlung photons. These will be much less absorbed than the incident and secondary betas, and, depending upon the shape of the target, a n important fraction of the energy of these photons might not be absorbed. At least, a certain number of displacements result from direct interaction of the beta particle with target atoms; however, as yet, it remains impossible to carry out a systematic control of the theoretical evaluations of the number of displaced atoms. I n the case of the microporous solids used in our experiments the problem is still more complex. The grain dimensions indeed are sometimes smaller than 100 A., and consequently the proportion of surface atoms is important and may exceed 10%. The theoretical considerations that have been developed in the case of atoms in the bulk (48,54) as well as in the case of surface atoms of crystals (sputtering) (5&),do not seem to apply directly to the microporous solids. 3. Photons, Deutons, Alphas

The energy repartition between the various effects produced by these particles, is not as well known as in the preceding cases. When these particles possess an energy lower than a certain threshold value, which is 0.1 M.e.v. for photons, and 1 M.e.v. for alpha particles, they can capture electrons from the target, thus being neutralized (47). They behave then like hot atoms with a very high energy content, and not any longer like electrically charged particles. This electron capture phenomenon is still obscure, thus rendering more intricate every theoretical approach. Let us first mention that only a very small fraction of the proton, deuton, and alpha energy is dissipated as bremsstrahlung (47). The processes that will account for energy dissipation are thus essentially atom displacements and ionization and excitation phenomena.

RADIATION CATALYSIS

91

The minimum energy required for the production of displacements may be calculated by means of the same approximate methods used in the case of beta rays; this minimum value, however, is much smaller in the present case, because the particles under consideration are lo3to lo4 times heavier (52, 54). Although quantitative determination remains impossible, an estimation made by Seitz (54) considers the energy fraction dissipated as displacement to be lower than 0.1%. Let us also point out that certain displaced atoms receive an energy sufficient for the creation of displacement spikes. Some experimental information about atom displacements in metals and semiconductors is available at the present time. Table I X shows values obtained in the case of germanium irradiation. The number of displaced atoms, for dissipated dose of loz2e.v., is seen to be between 1OI6 and 1017 (54, 55) With regard to the interactions with the electron cloud, we draw attention to two important features: (1) in contrast to beta irradiation, a clear distinction is possible between the incident radiation and the secondary one. The latter indeed consists of electrons and photons; (2) statistically, the energy transmitted to an electron of the target by an incident particle depends upon the ratio of the masses of both interacting particles. Consequently, protons, deutons, and alphas of about 1 M.e.v. can impart to an electron a maximum energy of only about 1 k.e.v. It results that, in the case of targets constituted of elements with atomic number greater than 10, irradiated with particles whose energy is smaller TABLE IX Number of Displaced Atoms Produced by Alpha Particles and Deutons in Germanium Type of particle

Particle energy (M.e.v.)

Number of atoms displaced per incident particle

a

5.3 5 9.5

39 47 8

01

D

than a few million electron volts, the outer electrons only may interact with the incident particles. Moreover, for protons, deutons, and alphas whose energy is notably greater than 1 k.e.v., excitation is an important way of energy dissipation as compared to ionization, whereas in the case of beta rays, this process becomes effective only for values of the energy smaller than 100 e.v. More than 99% of the energy of protons, deutons, and alpha particles is used in the creation of secondary beta radiation, ions and excited states. Secondary beta rays whose energy is between 10 k.e.v. and 100 e.v. have

92

R. COEKELBERGS, A. CRUCQ, AND A. FREN N ET

already been considered (Section III,A,2). Low energy electrons as well as ions and excited states will be studied later (Section 111,B).

4. Fission Fragments The study of radiation damage caused by fission fragments is of considerable practical interest. For this reason the problem has received closer attention than in the case of protons and alpha particles (48, 52-54). Energy dissipation from fission fragments results both from direct interaction with the lattice and with target electrons. The latter in all cases receive more than 95% of the energy of the fission fragment.* In view of the mass ratio between electron and fission fragment (= 2.105), the maximum energy that an electron can receive is 400 e.v., the average value being 100 e.v. With regards to the collision, Ozeroff evaluates that in uranium, the fission fragments dissipate 5% of their energy through displacement collisions. In his evaluation he considers different energy ranges of the fission fragment, specified as follows. (1) For energy higher than 10 M.e.v. the electrical charge of the fission fragment is about 10 and the fraction of the dissipated energy through displacements decreases with the atomic mass of the target. (2) Below a threshold of 2 M.e.v. the fission fragment is almost neutralized and the major part of his energy is dissipated through displacements over a distance of approximately one micron? in a solid. (3) For energy ranging from 2 M.e.v. to 10 M.e.v., the average value of the electrical charge is not apparent; nevertheless, the author, for the value of simplification applies the same hypothesis than for energy higher than 10 M.e.v. energy. We will thus distinguish two ranges according to whether the energy is higher or lower than 2 M.e.v. Let us first consider the atoms enclosed within a cylinder whose length is the range of an energetic (2 M.e.v.) neutral atom of the same atomic number as that of the fission fragment and whose radius is approximately that of a thermal spike. If one takes into account both the density of the dissipated energy and the average value of the energy received by a knocked atom it is seen that the distance that separates two consecutive displaced atoms, is smaller than the diameter of a thermal spike of spherical symmetry as defined above. The volume with cylindrical symmetry may t1,us be considered as one large thermal spike which involves about lo7 atoms. Each of these atoms, at a definite moment, possesses an average energy of the

* Immediately after the fission, the light fragment possesses an energy of 95 M.e.v., whereas this value is 65 M.e.v. for the heavy one. t A fission fragment has a range of a few tens of microns in a solid.

RADIATION CATALYSIS

93

order of tenths of an electron volt, which corresponds to a temperature of several thousand degrees centigrade. Important radiation damage results from this temperature increase (cf. Section 111,AJ2).For one gram of solid of atomic mass 20 to be affected by this phenomenon, [(6 X lOZ3)/2O]X lo-’ = 3 X 10l6 fission fragments are necessary. This represents a total energy dose of 3 X 10l6 X 80 M.e.v. = 2.4. loz3e.v./g. If we now consider that part of the range of the fission fragment where the latter still possesses an energy greater than 2 M.e.v., different cases must be distinguished, depending upon which kind of solid is used as target. a. Conductors. In this case the energy received by the electrons is rapidly dissipated through interactions with conduction electrons. Before being transmitted to the lattice this energy is spread into a large volume; the increase of temperature that corresponds to lattice energy in this volume is consequently very small. However, a certain number of atoms that are displaced along this part of the trajectory of the fission fragments induce displacement spikes. For light element targets, the energy fraction dissipated in this way is negligible. I n the other extreme case of uranium, a Qheoretical approach shows this energy fraction to be 3% of the total energy (59). b. Insulators. As in the case of conductors, the primary step is electronic excitation. However, as no thermal conduction electrons are present, a large part of the energy received by these excited electrons is directly transmitted to the lattice within a cylindrical volume centered along the path of the fission fragment. Let us consider insulating substances like the silica or alumina we used in the experimental part. As an extreme case, we assume that all the energy transmitted to the electrons appears as lattice energy, within a cylindrical volume whose length is the range of the fission fragment (2.5 X cm.) and whose radius is the range of the secondary electrons (-loy6 cm.).* The number of atoms involved in this new type of thermal spike is between 108 and lo9. An elementary calculation shows that each atom contained in this volume receives a few tenths of an electron volt; this energy corresponds to a temperature of several thousand degrees centigrade (60). The amount of dissipated energy that is required for 1 g. of solid of atomic mass 20 to undergo important radiation damage is thus between loz1and loz2e.v. Thus, it can be foreseen that a dose of approximately loz2e.v. g.-l will affect in an appreciable manner the properties of an irradiated solid. This evidently is an extreme that has to be compared to this other in which only the thermal spike of the end of the trajectory is considered. We have seen that doses of 1 0 2 4 e.v. g.-l are then necessary for solid structure

* The range of the secondary electrons (maximum energy of about 400 e.v.) is between 100 and 200 A. in a solid of middle atomic mass.

94

R. COEKELBERGS, A. CRUCQ, AND A. FRENNET

(B.E.T. area, porosity, . . .) to be affected. In this latter case, if we consider the first part of the trajectory, the energy of the secondary electrons directly transferred to the lattice remains insufficient to cause radiation damage. A more or less important fraction of this energy will be imparted to the target electrons, thus producing electronic excited states as it will be seen later. The evaluation of this energy fraction is not possible a priori. Nevertheless as shown by the exyerimental results, it may be important. Indeed, as already mentioned, marked structure modifications appear for doses of approximately loz3e.v. g.-'. Electron microscope examination by Burlein and Mastel (61) has shown the diameter of the track of a fission fragment in uranium oxide to measure 150 A. This dimension corresponds nearly to that of the micrograins of the microporous solids which were used. Consequently, the temperature of the surface of a certain number of micrograins that are in direct contact with the gaseous reactants may be raised to a very high value (more than 1000°C.); a t this temperature kinetic and thermodynamic considerations applicable a t the over-all macroscopic temperature cease to be valid. 5. Neutrons We shall limit ourselves to fast neutrons; the only effects these particles can produce are atom displacements. The theoretical model of the hard sphere collision is particularly suitable to the case of this neutral particle. The mean energy l? imparted to an atom knocked by a neutron is given by the relation

2M

E

=

(I

+ M)zX E

where E is the energy of the incident neutrons and M the mass of the knocked atom (54). Figure 11 (50) shows that fast neutrons are able to impart by collision an important fraction of their energy to light atoms. These energetic knocked atoms may in turn either produce secondary displacements, or impart energy to electrons. The relative magnitude of these two effects is related to a parameter which expresses the importance of the screening effect of the electrons upon the charge of the nucleus. The value of { is given (54) by

where Z is the atomic number of the target elements, l? is the mean energy received by one atom of the target, R h is the Rydberg constant (13.5 e.v.), a h is the Bohr radius of the hydrogen atom, and a = USZ-"~. For { smaller than unity, the screening effect is small; things happen as

95

RADIATION CATALYSIS

if the energetic atom was positively charged. This atom will loose a fraction of its energy upon interaction with the target electrons. This fraction is all the more important as becomes smaller. On the contrary, for [ equal or greater than unity, the screening effect grows important and the theoretical model of the hard sphere collision

-E E

l

0.5

0.2

-I

10

0.5~16'

0.2.Id'

10' 0.SxldP

I

I

I

I

5

10

20

50

I

I

100

200 M-

FIG.11. Mean energy energy E.

d received by an atom of mass M knock by a fast neutron of

becomes applicable to the knocked atom as well. Interactions with the lattice, therefore, account for the entire energy dissipation. Table X gives, for a few elements, the threshold energy value beyond which energy deperdition upon interaction with target electrons ceases to be negligible (55). Combined information from Table X and Fig. 11 shows that neutrons which possess an energy of approximately 1 M.e.v. impart a relatively high energy to very light atoms (1 < 2 < 10); these atoms then dissipate most of their energy interaction with electrons. For example,

96

R. COEKELBERGS, A. CRUCQ, AND A. F R E N N E T

in the living tissues (49) whose hydrogen content is great (>65%), the fast neutrons dissipate their energy principally through ionization and excitation; in this case the mean energy imparted to the electrons is of the same order of magnitude as the mean energy imparted to the protons. If, on the other hand, the target is made of heavy elements (2 > 30), the energy received by the primary knocked atom is completely transmitted to the lattice as displacement spikes or as thermal spikes. TABLE X Minimum Value of the Eneryy of a Moving Atom for Interaction with the Electrons of the Target Atoms to be Possible *We Of target

Moving atom

Diamond Diamond A1 A1 cu cu

C Deuton Al Deuton cu Deuton

Energy (e.v.) 1.5 x 2.0 x 1.7 X 1.3 x 5.0 x 1.6 x

104 103

10' 103 104 103

Let us finally mention that the total number of atoms which undergo displacements on collision show a maximum value for elements of mass 50 (48). From these considerations, it results that in the case of alumina and silica (2 'v 12) 90% of the energy of the fast neutrons produce ionization and excitations. 6. S u m m a r y

At this stage of the degradation of the radiation energy in solids, it is interesting to draw up an energy balance sheet. We have indeed seen that, quite independently of the type of radiation, the dissipated energy is found under the four following forms. Lattice energy. This energy is essentially the energy which is dissipated through displacements and related phenomena (temperature spikes, lattice defects, etc.); it always corresponds to a very small fraction of the dissipated energy, except for fast neutron and fission fragment irradiations. Ion energy. Ions contain in potential form an energy that may reach about 100 k.e.v. in the case of gamma and beta irradiations of heavy element targets. The corresponding value does not exceed a few k.e.v. in the case of irradiations by protons, deutons, and alpha particles, and a few hundred e.v. for fission fragment irradiations. Electron energy. Electrons carry away a fraction of the incident energy

97

RADIATION CATALYSIS

as kinetic energy, the value of which may be as high as a few hundred k.e.v. with gamma and beta irradiation; a few k.e.v. with protons, deutons, and alpha particles, and a few hundred electron volts with fission fragments. Photon energy. Photons are produced exclusively by bremsstrahlung. The latter phenomenon takes place only in the case of beta irradiation. However, it also occurs in the case of gamma irradiation and, to a lesser extent in the case of irradiations by protons, deutons, and alpha particles, as a consequence of secondary beta rays emitted by these particles. TABLE X I Repartition of the Dissipated Energy According to the Produced Effects Nature of the radiation

2 5 6

Lattice energy Ions Soft @-rays

Gamma

Beta

Lattice energy Ions Soft /3-rays Bremsstrahlung photons

Proton Deuton (I particle

Lattice energy

Fission fragment

Lattice energy Ions Soft @-rays

Neutron

Percentage of the energy dissipated in target of

Produced effects

Ions Soft prays

Lattice energy Ions Soft prays

-0 -0

-1OOOJO -0

-0 -100% -0

6<2<30 -0 -1% -99%

-

10-4 % -1% -94% 4-5%

-

-lo-*%

-100%

1 100% 1

2%

2-3 %

-98%


Z>30

98%

1-99%

The repartition of the incident energy between the four forms just described is given in Table XI. This repartition is influenced by the atomic number of the elements constituting the target. For this reason, we distinguish the cases of light (2 < 6) intermediate (6 < 2 < 30) and heavy (2 > 30) elements.

B. RADIATION OF ENERGY SMALLER THAN 1 K.E.V. In the first part we limited ourselves to the effects resulting from interaction between matter and radiations of energy greater than 1 k.e.v. A t a

98

R. COEKELBERGS, A. CRUCQ, AND A. F R E N N E T

definite stage of degradation, the incident energy is recovered, as seen before, under the form of ions, and of low energy ( E < k.e.v.) electrons and photons; lattice defects also contain a small fraction of the incident energy. We will carry on here the study of the degradation of the energy of these ions, electrons and photons. The properties of the irradiated solid, that exert negligible influence on the absorption of high energy radiation, must be taken into account in this part devoted to the absorption of low energy radiation. Let us recall that we are exclusively interested in this paper with nonmetallic solids (semiconductors and insulators) ; frequently, however, we will use the band theory of solids. The lattice defects and their influence on the properties of solids will be studied later (Section III,C,l).

1. Ions I n general, ions result from the removal of an electron which received an energy amount exceeding the extraction energy of this particular electron. We shall distinguish ordinary ions and excited ions. Ordinary ions. The removed electron is a valency electron. These ordinary ions may originate from atoms situated normally arranged on the lattice. The formation of an ordinary ion gives rise to a charge defect in the valency band, more commonly termed “positive hole.” If the electron is removed from an atom corresponding to an impurity level (impurity or interstitial atoms), an acceptor level is produced. Excited ions. The removed electron is a core electron. The excited ions decay to their ground state after a time of approximately sec. (optical transition) ;ordinary ions are produced in this way. This problem of the return of excited states to the ground state is familiar to spectroscopists. The selection rules indic:zte that the transition probability is the greatest between neighboring levels. The emitted photon, however, may be immediately absorbed through the Auger effect, (62, 6‘3). An electron is then emitted with an energy equal to the difference between the energy of the photon and the binding energy of this electron. Two electrons are now missing in the atom; this process, which may be repeated several times, is known as the cumulative Auger effect. By this phenomenon ions which carry several positive charges are produced. It must be mentioned that the yield of Auger electrons becomes more important as the corresponding transition energy becomes smaller. In Fig. 12 the fraction of K , transitions giving rise to Auger electrons is plotted against the atomic number of the elements. In the case of light element targets, it is seen that only a few per cent of the energy of the excited ions that have lost a K electron, gives rise to photons. The problem is quite different for the heavy elements; here the energy of the emitted photons is always greater than 1 k.e.v. and may sometimes

99

RADIATION CATALYSIS

reach 100 k.e.v. However the L, M , and N transitions of these heavy elements are analogous to the K transitions for light elements. The ion formation is also influenced by the nature of the incident radiation. It has indeed-been shown"previous1y (Section II1,A) that beta and

20

40

60

80

100 - z

FIQ. 12. Yield of Auger electrons from the K level for the elements. (After H. L. Hagedoorn and A. H. Wapstra (63)).

gamma rays of about one M.e.v. are able to remove electrons from the target regardless of the levels involved. This does not hold any more in the case of photons, small energy electrons, protons, deutons, and alpha particles. Indeed these particles can only impart to the electron an energy

100

R. COEKELBERGS, A. CRUCQ, AND A. F R E N N E T

smaller than a few thousand electron volts and, therefore, they can never remove an electron whose extraction energy exceeds this value. Similarly, the maximum energy that fission fragments can impart to an electron is a few hundred electron volts; as a consequence, they can only modify the outer shell of the atom. One may thus conclude that most of the excited ions originate from shells of small extraction energy except for heavy elements when irradiated by high energy beta and gamma rays. Under such conditions, only a few per cent of the total energy stored up by the excited ions will be emitted as photons. The remainder is carried away by the electrons. Let us point out that a t this stage of the energy degradation, the excited ions may contain a fraction greater than 10% of the dissipated energy. The various positive charges of the ion that underwent the cumulative Auger effect can be considered as positive holes in the valency band, if the ion originates from a normal lattice atom. If, on the contrary, the ion with multiple positive charges originates from an impurity atom, then these charges correspond to an equal number of acceptor levels. 2. Electrons

We consider here the effects produced b y electrons with energy smaller than a few hundred electron volts. We have seen that these electrons carry away more than 80% of the total energy dissipated by the various types of radiation, with the exception of fast neutrons in the presence of heavy element targets. Let us remember that these electrons are obtained (I) by gamma and beta irradiation, after a great number of successive processes, (2) by proton, deuton, and alpha irradiations, after a small number of successive processes, and (3) directly by fission fragment irradiations. I n a solid, these electrons may impart their energy either to the target electrons, or to the lattice. The problem of these interactions cannot, as yet, be solved theoretically. Nevertheless, we will try to estimate qualitatively the repartition of the energy for the two extreme cases of a semiconductor and of an insulator. The essential difference between these two types of solids arises from both the number of conduction electrons and the value of the energy gap. Every interaction with a conduction electron is effective and results in a n exchange of energy. In the conduction band which consists of a continuous set of levels, the electrons may indeed receive arbitrarily small amounts of energy, whereas the interaction of the incident electron with those belonging to levels lower than the conduction band do not always result in electronic excitation. If AE represents the energy difference either between the valency and conduction band, or between one of these bands and an impurity level, or also between two impurity levels, electronic excitation may be produced only if the incident electron

RADIATION CATALYSIS

101

has an energy equal to or greater than 2AE; the eventual excess of energy corresponding to transition between these levels may be transferred to the lattice. I n all the other cases the energy exchange takes place only with the lattice. Since, in an insulator, the energy gap, and consequently AE, have high values, the energy fraction imparted to the lattice is, in general, more important in this type of solid than in a semiconductor. Regarding the excited electrons, two possibilities have to be considered. (1) The received energy is greater than the extraction energy E . An ion and an electron with positive total energy are thus produced. If Eo is the energy of the incident electron then a continuous spectrum of secondary electrons is obtained, ranging from Eoup to (E0/2) - E. Unless they escape from the solid these electrons will finally return to the conduction band after dissipation of their excess of energy through ionizations, excitations, or interactions with the lattice. (2) The received energy is smaller than the extraction energy. Excited electronic states are then produced. (a) I n many cases, the energy imparted to the electron is sufficient for the latter to be promoted to the conduction band. When the electron originatesfrom the valency band or from an impurity level, a positive hole or an acceptor level is simultaneously produced. The irradiation is then able to modify in a transient way the concentration of the conduction electrons or the pairs of free carriers (conduction electrons and positive holes). The return of these excited states to the ground state is accompanied by a release of energy in the form of photons and phonons (64) (lattice energy). This energy also may be used either to induce chemical reactions in the solid itself or as will be seen later, to influence certain catalytic processes occurring a t the surface of the solid. The free carriers also may be trapped by lattice defects during a certain while, producing, for instance, F centers and V centers. On the surface a weakly adsorbed molecule may act as a lattice defect and trap a free carrier; this may result in the creation of a strong chemisorptive bond which generally will be a t the origin of catalytic phenomena (cf. Section V). (b) There exist allowed levels situated a little below the conduction band which do not give rise to conduction phenomena; their existence has been evidenced by spectroscopy. An electron brought in that state remains bound to the corresponding positive hole. The system of the electrons with its associated positive hole is called a n exciton (65). It is generally admitted that the migration rate of an exciton is about the same as that of thermal electrons (lo7 cm. set.-'). If it does not get trapped by an imperfection in a metastable state, its lifetime will approximately be that of an optical transition (66). It then returns to its ground

102

R. COEKELBERGS, A. CRUCQ, AND A. FRENNET

state with photon and phonon emission; occasionally ionization of defects ( F , V , . . . centers) may also take place. We have seen that most of the radiation energy is finally recovered as electrons with energy lower than a few hundred electron volts. These in turn produce excited electronic states (free carriers, excitons) . A fraction of the incident energy is directly transferred to the lattice as heat; this fraction, whose estimation i s difficult, is more important in insulators than in semiconductors.

3. Photons

It has been shown that photons of energy smaller than 1 k.e.v. are produced in a solid by incident radiations. These photons may have different origins. (1) Let us recall that with incident beta radiation of sufficient energy, an important percentage, up to 15%, of the total dissipated energy is converted into bremsstrahlung photons; as shown by Fig. 10 a large part of these bremsstrahlung photons possess a small energy. By an indirect process, incident gamma radiation may also produce bremsstrahlung. Indeed, 90% of the gamma energy is transformed into secondary beta rays of high energy; finally their bremsstrahlung yield is of the same order of magnitude as that of beta radiation. The other particles give no bremsstrahlung photons by direct interaction with matter; but secondary beta radiation resulting from protons, deutons, and alpha irradiations, as well as from interaction between target atoms and hot atoms produced b y collision with fast neutrons, may produce such a bremsstrahlung; the yield, however, is small, since it is proportional to the incident beta energy. The fraction of the total energy dissipated as a continuous photons spectrum thus amounts only to a few per cent. (2) Excited ions returning to their ground state constitute another source of photons. We have seen that the practical yield of the transitions giving rise to low energy (< k.e.17.) photons is smaller than lo%, the remainder being absorbed by Auger effect. However, the potential energy contained by the excited ions may represent more than 10% of the total dissipated energy; approximately 1% of the total energy is thus recovered, a t some definite stage of the energy degradation, in the form of a characteristic spectrum of the elements constituting the irradiated target. The wavelengths of this spectrum range from approximately one angstrom (X-ray spectrum) to several thousand angstroms (visible region). (3) Photons, at last, may be created whenever the excited electronic states (discussed in Section III,B,2) return to the ground state (recombination of charge carriers, excitons). An important fraction, generally more

RADIATION CATALYSIS

103

than SO%, of the total dissipated energy is converted into electronic excited states. A fraction of this energy, whose estimation is impossible a t the present time, is converted into photons. These are emitted as a band spectrum characteristic of the irradiated solid (67) with energies ranging from a fraction of an electron volt up to some 20 e.v. The spectrum of photons emitted by the solid is then constituted of two parts: the first one is characteristic of the elements constituting the solid, the second one characterizes the solid structure. These photons may be absorbed by the solid. Attention must be drawn to the fact that the stopping power of matter with regard to these photons is not necessarily great, although their energy is low; their absorption spectra show indeed tl characteristic series of maxima and minima and between two consecutive extrema, the variations of the absorption coefficient may attain several powers of ten. In the energy range beyond 20 e.v. these absorption maxima are the so-called “absorption edges” which are characteristic of the nature of the elements constituting the target. Below 20 e.v., there exist additional absorption bands which are characteristic of the structure of the irradiated solid. The absorption of these photons give rise to a certain number of phenomena that we have already studied: (1) formation of an ion and of an electron with positive energy; (2) formation of excited electronic states (free carriers, excitons, etc.) ; (3) formation of phonons. When the excited states return to the ground state, a photon may again be emitted with an energy equal to or, most often, lower than the energy of the primary absorbed photon. I n brief, only a few per cent of the total dissipated energy is converted into photons whose energy is between 20 e.v. and a few thousand electron volts; on the other hand, a very important fraction of the incident energy may give rise to photon of energy less than 20 e.v.

4. Summary At this last stage of the degradation process, the whole energy absorbed by the nonmetals is recovered in the following five forms: (1) structural imperfections, (2) thermal lattice energy, (3) photon energy, (4) excited electronic states, and (5) electrons of positive energy. Let us also point out that electrons and photons may eventually escape from the solid. It has also been shown above that, at a given moment, a certain fraction of the absorbed energy, generally more impo‘rtant in a semiconductor than in an insulator, is stored in excited electronic states. This fraction, which cannot be evaluated in the case of fission fragment irradiation, represents for all types of radiations, except in the case of neutron irradiation, the major part of the incident energy. I n the case of fast neutron irradiation, this fraction is very small for heavy element targets; it becomes even more

104

R. COEKELBERGS, A. CRUCQ, A N D A. FRENNET

important as the atomic number of the target becomes smaller, and, in the particular case of organic solids, the fraction constitutes the largest part of the dissipated energy. Secondary photons, principally those of energy between a fraction of an electron volt and about 20 e.v., may also constitute a t a given moment a very important fraction of the dissipated energy. It has been shown th a t the absorption of this electromagnetic radiation by the solid essentially produces excited electronic states, the excess being lost in the form of thermal energy. However, if one takes into account both the values of the . absorption coefficients and the geometry of the solid it is seen that a great number of these photons may not be absorbed by the solid; the latter case will be considered later (cf. Section IV,C,S). Excited electronic states thus give rise to photon emission with a yield smaller than unity; on the other hand, absorption of these photons produces, in turn, excited electronic states, also with a yield smaller than unity. Consequently, if one neglects the possibility for the photons to escape from the solid, a quasi-equilibrium is established between these two forms of energy between which the near totality of the incident energy is recovered, However, every conversion from one form to the other is accompanied by a release of thermal energy. If the irradiated system does not use the energy in either of these forms for certain definite purposes, such as chemical reaction for instance, the totality of this energy will be finally converted into heat.

C. PERMANENT AND TRANSIENT EFFECTS This subsection is devoted to the study of the influence of the various radiation-induced imperfections, on the properties of nonmetallic solids, taking into account the lifetime of these imperfections. From this particular point of view, and in a general way, one may distinguish between structural and electronic imperfections. To the first group, belong all the imperfections which modify the lattice periodicity; they comprise lattice defects (Section III,C,l) and dislocations, which are only mentioned as a reminder in this paper. Electronic imperfections are described in Section III,C,2. 1. Lattice Imperfections

The lattice defects belong to the following four types: (a) vacancies, (b) interstitial atoms, (c) replacements, and (d) impurity atoms. The first three types of defects have a common origin, namely, collisions. These collisions have been shown to produce both displacements and “temperature spikes.” An atom which has received sufficient energy on collision, may leave its normal position in the lattice; in this way a vacancy is created.

RADIATION CATALYSIS

105

The displaced atom may either be trapped by another vacancy, or stopped in an interstitial position. It may also give rise to a replacement; this situation occurs when a moving atom, after having knocked and expelled another atom from its normal position, does not possess sufficient energy to escape from this vacancy. We have seen that the “temperature spike” involves a relatively large number of atoms, which, after having been superheated, undergoes a subsequent rapid cooling. A certain number of atoms are thus quenched in metastable positions, increasing the local density of defects. Impurity atoms, if present, originate in eventual nuclear reactions. Vacancies. The character, either anionic or cationic, of the vacancies, depends upon the type of displaced atom. Anionic vacancies can capture one or more electrons, thus creating F , F’, . . . centers. These centers account for the characteristic coloration of many irradiated crystal. A cationic unoccupied position may trap a positive hole, thus producing a V center. Interstitial atoms or ions. These atoms or ions, which occupy metastable positions in the lattice, constitute either donor or acceptor levels depending upon their nature. Replacements. If the solid is constituted by two or several elements, it may occur that an atom or an ion with cationic character replaces an atom or ion of anionic character, and vice versa. The former atom then constitutes an impurity. Impurity atoms resulting from nuclear reactions. Their influence on the properties of a solid is generally very similar to that of the same impurities introduced in a chemical way. Only thermal neutrons are able to produce them in any appreciable amount; even in this case, and as far as the irradiated material does not contain elements with a large cross section, like boron-10 or lithium-6, the concentration of the produced impurities quite generally remains negligible. The first three types of defects correspond to metastable equilibrium state and have generally a long lifetime. These, in the following, will be termed quasi-permanent defects. Their disappearance, related to their mobility, which, in general, is the greatest in the case of vacancies, requires a certain activation energy; therefore, their lifetime, which depends upon the temperature, may attain several months at room temperature (68).The lifetime of impurity atoms is obviously infinite unless they are radioactive. In a general way, the lattice defects may considerably influence the properties of the solids; catalyst activation, among other phenomena, results principally from theee defects (cf. Section IV,B). The obtained effects, however, depend upon the type of irradiated solid. In the case of semiconductors, the lattice defects may be considered as

106

R. COEKELBERGS, A. CRUCQ, AND A. FRENNET

new impurity levels; according to the position of these levels in the forbidden band, the defects will act either like “donor” or “acceptor” centers, or will be inefficient from the standpoint of conduction. The fraction of the imperfections which depends upon the type of solids and which influences the concentration of one or both types of free carriers will be called “efficient” imperfection. So, for instance, James and Lark-Horowitz (69) have shown vacancies obtained b y irradiation of germanium and silicon to possess an acceptor character, whereas the corresponding interstitial atoms possess a donor character. Moreover, lattice defects may act like trapping centers with respect to the free carriers. They may also behave as recombination centers for the excess free carriers; these latter problems will be discussed further in the next paragraph (1111C,2). The order of magnitude of the produced effects for a given amount of dissipated energy depends also upon the nature of the radiation. Let us assume that, as in most of the experiments described in the experimental part, a dose of e.v. g.-l has been dissipated. It has been shown that doses of this order consisting of protons, deutons, or alpha to lo1’ defects per gram, whereas for an idenparticles generate from loL6 tical dose of beta rays the corresponding values are between 1014and 10l6. I n intrinsic semiconductors, considerable effects are possible, provided that the concentration of charge carriers remains small in comparison with the values of 1014and 1OlT just cited. Depending on whether the “donor” or “acceptor” type of defect is predominant, transformation may result into n- or p-type extrinsic semiconductor. For extrinsic semiconductors two cases are possible. If the efficient defects produced by the irradiation possess the same character (donor or acceptor) as the pre-existing defects or impurities, i.e., if they modify the concentration of the majority carriers, the effect of the irradiation is generally quite small in this type of solid, and the concentration of pre-existing imperfections is frequently of the order of 10’’ to lo1* g.-I; this is larger than the concentration of the defects created by irradiation. If the produced defects influence the concentration of the minority carriers, very important modifications of the properties of the solid occur. Indeed, if one takes into account that the product of the number of conduction electrons and of positive holes is a constant at a given temperature (np = N 2 ) , a variation of the number of minority carriers necessarily entails a simultaneous variation of the number of majority carriers. Since the concentration of the minority carriers may be very low, even smaller than 1O1O g.-l, it is possible, in contrast with the preceding case, to obtain considerable effects; modification of the type of conductivity of the sample may result for relatively small doses of radiative energy. For instance, in the case of germanium, where the product np is equal to 6 X loz6,the “aweptor” levels

RADIATION CATALYSIS

1Oi

(vacancies) produced by irradiation play a dominant role (YO). Upon irradiation of n-type germanium, a great number of positive holes are created; in this way, the concentration of n carriers and consequently the conductivity, both decrease rapidly; if the dissipation energy dose is sufficient for the production of about 1013 efficient “acceptor” defects, the semiconductor becomes intrinsic, whereas its conductivity passes through a minimum; upon further irradiation the conductivity increases, but becomes of the p-type. The case of insulators and, more part,icularly, of porous solids (silica, alumina) that we used, is, insofar as the principles involved are concerned, very similar to the case of semiconductors, with regard to the creation and the influence of lattice defects. A very small number of free carriers are present in insulators, and, therefore, it seems that relatively small energy doses are able to appreciably modify their properties. However, the energy gap between valency and conduction bands is very large and the various phenomena are liable to be more intricate. It is probable that the lattice defects artificially created by irradiation exert a strong influence in both the “trapping” and carrier recombination phenomena; later on, this topic will be discussed further. In brief, the production of defects corresponds to new impurity levels in the irradiated solid. Their lifetime, quite generally, is rather long and may even possess a quasi-permanent character a t room temperature. 2. Electronic Imperfections

I t has been seen in Section III,B,2 that in general more than 80% of the whole dissipated energy appears at a given moment as electronic excitation. The excited states have generally short lifetime; they are therefore considered here as transient, compared to the lattice defects, which are indeed quasi-permanent. The influence of these various electronic imperfections upon the properties of solids, and more specially upon their catalytic properties is still little known. However, a simplified view of the problem results when considering only the pairs of excess free carriers produced in the course of irradiation. These tend to recombine and then stationary concentration depends simultaneously on both the recombination time and the intensity of the incident radiation. As soon as the irradiation ceases, this stationary concentration rapidly recovers the thermal equilibrium value. The problem hence reduces to the determination of the conditions under which the stationary concentrattion of the free carriers under irradiation will differ notably from the thermal equilibrium value. The simple hypothesis of radiative recombinations occurring directly between the electrons of the conduction band and the positive holes of the valency band, with simultaneous photon emission seems only valid in

108

R. COEKELBERGS, A. CRUCQ, AND A. F R E N N E T

a few cases (64). Indeed, the recombination rates, if calculated according to this hypothesis, agree with the experimental values only in those cases where the number of excess free carriers, produced by irradiation, is great in comparison with the number of pre-existing free carriers; in all other cases these recombination rates are much lower. As a rule, the recombination proceed through impurity levels acting as recombination centers. These levels may be situat.ed relatively far from the valency and conductivity bands, in contrast with the impurity levels which account for extrinsic conductivity. For instance, the impurity levels of nickel and copper in germanium, which are situated 0.22 e.v. above the valency band, act like recombination centers (64). Whichever the mechanism may be, any recombination process is usually characterized by its recombination time. Under a no-equilibrium condition, for instance, when excess free carriers are created by irradiat,ion, this time T is defined by:

where n is the number of excess carriers per unit volume, ( d n / d t ) R is the recombination rate, i.e., the effective number of recombinations per unit volume and unit time. If, for a given irradiation, A represents the number of pairs of free electrons and positive holes formed per unit time and per unit volume, there results at a given time : dn -=A - (n/~) dt During irradiation, the steady state condition is written ( d n l d t ) = 0, from which it results that n = AT. This relation permits evaluation of the stationary concentration of the excess free carriers, provided that the value of T is known. In the semiconductors, the recombination times as a rule are small; in most cases, their values are between see. However, and a considerable increase of the recombination times may result from the presence of certain imperfections (impurities, lattice defects) pre-existing in the solid or generated by the radiations which act like trapping centers of the free carriers. The following calculations are therefore only valid in the absence of this trapping phenomenon. Let us assume, as first approximation, that a pair of free carriers is produced for every 10 e.v. dissipated in the solid. This latter value is only an order of magnitude (71); it takes account of two parameters. (1) The first is the fraction of the dissipated energy used up for the production of carriers; a part of this fraction is lost in the form of photons and of electrons with positive energy, which escape from the solid, whereas another part is transformed into heat.

109

RADIATION CATALYSIS

(2) The second is the average value of the energy necessary for a valency electron to be promoted to the conduction band; this value takes account of the width of the valency band as well as of the repartition of the electronic population in the various levels of the valency band. In this way, the steady state concentration of excess free carriers may be easily evaluated for various values of 7 . The corresponding results are set out in Table XII. Let us note that even for irradiation intensities as high as 10'' e.v. set.-' ~ m . -which ~ is the highest value cited in the experimental part, the stationary concentrations of excess carriers does not exceed 1014g.-l and most often remain smaller than this value. TABLE XI1 Stationary Concentration of Excess Free Carriers Under Irradiation IrradiiLtion intensity (e.v. sec.? c m . 3

7

8ec. 10'2 10'3 1014

sec.

10+ Rec. 108 109 10'0

In the absence of any trapping phenomenon, the data of Table XI1 must be compared to the carrier concentrations in the various types of solids in the absence of radiation. In intrinsic semiconductors, this concentration hardly exceeds 1014g.-l; in very pure germanium, for instance, this value is equal to 2.5 X 1013. In extrinsic semiconductors the majority carrier concentration is generally between 1015and 1Ol8 g.-l; the corresponding value for minority carriers is generally much lower and may be as small as lo8. Unless very high radiation intensities are used, it is thus seen that in the absence of trapping phenomenon, the influence of radiation upon the number of carriers can become appreciable only in the case of extrinsic semiconductors; in other cases only the concentration of minority carriers is affected. For insulators, the phenomena of free carrier recombinations are not different with regard to the principles involved. However, the trapping may take here an important part; therefore the recombination times are often very long. The influence of radiation upon the number of carriers is thus the most important in this type of solids, this number being very small before irradiation.

3. Summary In brief, we have seen that the radiations give rise to both quasi-permanent effects and transient effects.

110

R. COEKELBERQS, A. CRUCQ, A N D A. F R E N N E T

The quasi-permanent effects principally result from the produced lattice defects, which may induce considerable modifications of the electronic structure of the solid, and even are able to convert a n n-type semiconductor into a p-type, and vice versa. Insofar as the electronic structure influences the catalytic properties, these properties will also be modified. The transient effects consist essentially in the creation of excited electronic states and more particularly of free carriers. It can be shown that in the case of insulators, the properties related to the concentration of free carriers may undergo considerable modifications. In the case of extrinsic semiconductors only the properties depending on the concentration of carriers minority are altered, except in the case of irradiation of very high intensity. Finally, the properties depending on the concentration of carriers are the least modified in the case of intrinsic semiconductors. As will be seen further, modifications of the free carriers’ concentration will probably involve new catalytic properties. But one has to stress th a t other states of electronic excitation should be taken into account, for instance, excitons. Furthermore, under irradiation, the distribution of the electrons among all the characteristic energy levels of the solid do not correspond to the thermal distribution given by the Fermi-Dirw statistics. Let us also indicate that the electronic imperfections, transient by nature, may sometimes possess a quasi-permanent character with regard to the trapping phenomenon. As already mentioned, the different considerations which have been made above involve some rough approximations. Therefore, it is still difficult to foresee their quantitative influence on the solid properties and more particularly on their catalytic properties. Finally, we have seen that the relative importance of the two kinds of imperfections (lattice defects and electronic imperfections) depends strongly upon both the nature of the radiation and the kind of irradiated solid. The proportion of lattice defects increases with both the mass of the ionizing particle and the atomic number of the elements constituting the target. With neutron irradiation, up to 100% of the energy may be dissipated for producing lattice defects in targets constituted of elements whose atomic number is greater than 30.

IV. Radiation Catalysis A. GENERAL Two cases should be distinguished when studying the influence of radiation upon catalytic processes; the first one considers preliminary irradiation of the catalyst in the absence of reactants, whereas in the second one the catalyst and the reactants are simultaneously irradiated.

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The variations of catalytic activity which are observed in the first case have a quasi-permanent character; they result from structural and electronic imperfections of long lifetime. Preliminary irradiation thus produces an activation of the catalyst. By this process new catalytic properties are imparted to the irradiated solid; this addition, however, does not entail any modification of the thermodynamic laws applicable to the reacting system. The new catalysts created in this way, may modify the reaction rate and the reaction mcchanism, or even orient the reaction system towards the formation of new products. I n the case of simultaneous irradiation of the catalyst and the reactants, we are most interested in the phenomenon related to the imperfections with transient character, i t . , essentially to the electronic imperfections. However, this does not mean that we do not take account of the quasi-permanent imperfections. Three groups of phenomena may be distinguished here, depending upon the heterogeneous system under consideration. (1) The modifications brought about in the solid only serve to create a new catalyst whose properties are related essentially to the nature and the stationary concentration of the imperfections with transient character. In this case the thermodynamic laws which determine the evolution of the reaction remain valid. Only the kinetics of the reaction may undergo modifications. The concept of G, as defined in radiation chemistry has only little meaning in this case. The phenomenon involved here can thus be considered as catalyst activation. (2) A fairly large fraction of the energy absorbed by the solid may be transferred to the gaseous reactants by three possible transfer mechanisms: (a) by electronic excited states, (b) by temperature spikes, and (c) by selective photon absorption. Here the reaction is really radiation induced: a relatively important fraction of the radiation energy is transformed into chemical potential energy. Just as in the case of homogeneous radiation chemistry, it is possible to carry out reactions which are unfavorable from a thermodynamical point of view. In that case, assuming that the total energy absorbed by the solid is transferred to the gaseous reactants, and on condition that the homogeneous mechanism and the heterogeneous one are identical, the Gappmay not exceed the Ghom. However, it should be noted that in the majority of the endothermic reactions induced by radiation, the energy is utilized with a very poor yield. On the basis of thermodynamic considerations one may calculate a maximum G value that we have termed G,, (25). G,,, is equal to (100/H) where H is the reaction enthalpy, expressed in e.v., at the temperature of the experiment. For most of the homogeneous endothermic reactions induced by radiation, the ratio (Gh,,/G,,) amounts to a few per cent (65). Assuming that the presence of the solid results in a more efficient utilization of the radiation energy,

112

R. COEKELBERGS, A. CRUCQ, AND A. F R E N N E T

then the Gappmay in certain cases be notably greater than the Gh,,, but can never exceed the G,,. If certain exothermic reactions with high activation energy are considered, the transfer concept may also acquire a microscopic meaning. It can indeed be applied to certain elementary endothermic steps of the reaction. (3) Here we consider radiation induced reactions, the initiation of which occur exclusively in the homogeneous phase. I n this case the solid may only influence the evolution of the excited species, ions or radicals, which were first produced in the homogeneous gaseous phase by the incident radiation. The fixation on the surface of these entities may lead to the formation of new activated complexes. In certain cases the phenomenon results in a modification of the mechanism; a n orientation of the reaction takes place that may give rise to products different from the ones obtained in the homogeneous reaction. I n other cases a better degree of utilization of the excited species m a y be obtained; as a result, the G of a given reaction is increased without exceeding the G,, if the reaction is endothermic. This latter aspect in many cases may be very important, and its thorough investigation may contribute to a better understanding of the elementary mechanisms of certain heterogeneous reactions. We shall not go further into this problem here. In this study, we are indeed mainly interested in the particular states in which a solid is brought under the influence of radiation allowing it to play upon the course of the reaction, an active role, different from that of a simple surface brought in the presence of reactants or of excited species. In this chapter we will thus distinguish two major parts depending upon the nature of the phenomena to be studied; the first is related t o catalyst activation, either through preliminary irradiation or during irradiation, whereas the second is related to energy transfer. It should be noted again that the present paper deals essentially with heterogeneous systems under irradiation, which is the reason why in the experimental part no examples of catalyst activation through preliminary irradiation are mentioned. But the latter mode of activation is narrowly related to activation under irradiation, so that we have preferred to discuss simultaneously these two points in the present chapter. Before we undertake a systematic study of these various points, there is some interest to reconsider here the conclusions of the preceding subsection (Section II1,C) from the more particular standpoint of the catalysis. The present state of the catalysis theory as well as the great variety of phenomena observed in radiocatalysis do not allow, a s yet, to bring out precise conclusions. Nevertheless, it is possible to infer some ideas which may become the starting point for a general theory. First of all, it is important to notice that the balance sheet of the effects

RADIATION CATALYSIS

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produced by radiations in solids, as established above, does not make any difference between the surface and the bulk of the solid. As a matter of fact this distinction may prove important in several cases and modifications of surface properties, which are decisive as regards to catalysis, may sometimes be very different from the modifications created in the bulk of the solid. However, in a general study, it is difficult to take account of such a difference; in order to simplify the problem, we shall assume that the conclusions of Section II1,C are entirely applicable to the surface. This hypothesis must be taken into account when considering the subsequent reasoning. The electronic theory of catalysis (7'2) distinguishes two main classes of heterogeneous reactions : N-type or acceptor reactions, which are catalyzed by electrons, and P-type or donor reactions, which are catalyzed by positive holes. Let us point out, however, that certain complex reactions may decompose into various successive simple steps, some being of the donor type, others of the acceptor t'ype. Whether a given reaction belongs to one or the other class depends on both the donor or acceptor character of the reactants that are preferentially adsorbed, and on the repartition of the electronic population in the bands and in the various impurity levels of the catalyst surface. This repartition, which is characterized by the position of the Fermi level of the surface, influences the catalytic activity of the solid with respect to a given reaction, and more particularly its capacity of chemisorption, the proportion of the various forms of chemisorption (including weak adsorption), the reactivity of the chemisorbed species and the selectivity of the catalyst. Radiation has been seen to produce in solids, and consequently a t their surface, both structural imperfections and excited electronic states. The structure defects, which constitute new impurity levels, induce in a quasipermanent manner a new equilibrium repartition of the electronic population in the various levels. They modify the position of the Fermi level, and therefore the catalytic activity. On the contrary, the creation of excited electronic states and particularly of pairs of free carriers results transiently in a repartition of the electronic population, differentfrom the thermal one. The potential energy, stored this way in the surface of the solid, may give rise to new catalytic processes. Radiations may produce various effects, and we shall try to bring out their common features. In a first case, the nature of the adsorbed species and the character of their bonding with the surface remain unaffected by the irradiation which exerts its influences only upon two factors-the number of adsorbed particles and their reactivity; a variation of the latter may modify the energetic balance of certain catalytic processes. These two factors may, depending upoii circumstances, vary either separately or simultaneously. Thus the

114

R. COEKELBERGS, A. CRUCQ, AND A. FRENNET

existence, under irradiation, of numerous excited electronic states disturbs the electron distribution a t the surface and influences, therefore, the number of chemisorbed particles. But at the same time these excited states constitute a source of potential energy that can be recovered a t the moment of adsorption; this may, for instance, give rise to an energy transfer. The creation of impurity levels (vacancies, interstitial atoms) either identical to the pre-existing ones, or of a different nature but with the same donor or acceptor character, may produce similar effects. In these various cases, the effect of irradiation is purely puantzlative. The mechanism of the reaction remains unchanged since the adsorbed species are identical; only tlhe kinetic constants of the reaction (frequency factor and activation energy) are modified. Several other effects are possible. The irradiation may cause t,he nature of the bonding between the adsorbed reagents and the surface to vary. This generally results either from a profound perturbation of the character of the irradiated solid, due to the in.croduction of new impurity levels with a character different from that of the pre-existing ones, or from appreciable modifications of the proportion of both kinds of free carriers. In this case, the effects of irradiation is pualitatioe, which means that the consequences upon the over-all catalytic phenomenon may be the following: (1) appearance or complete disappearance of the catalytic properties of a solid with respect to a given chemical system; (2) mechanism modifications for the same over-all reaction, and (3) different over-all reactions starting with the same reactants. I n the following discussion, in order to give a correct general description of the phenomena, we limit ourselves, in most cases, to a mere distinction between qualitative and quantitative effects, in the precise meaning we just have defined. Before going further, however, two remarks should be made. First the importance of certain structural modifications of the surface of the irradiated catalysts should be stressed. Among these modifications we cite the sometimes considerable transformations of the physical texture of the solid (modifications of the pore spectrum, etc.) (24,73) due to sintering, as well as the modifications of the chemical nature of the surface (decomposition of oxides, of silanol groups, . . .) (74).These structural modifications may have a particularly important influence in the case of large surface solids such as the silica and alumina used in our experiments. Second, we point out that the described effects occur only in the cases where the surface itself takes an active part in the process under examination. These cases are the most frequent ones. It will nevertheless be shown later, more particularly when studying the energy transfer that the surface takes only a little part in certain phenomena. This will especially be true

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in the case of transfer through selective photon adsorption and, to a lesser extent, in the case of transfer through “temperature spikes.” B. CATALYST ACTIVATION 1. Catalyst Activation by Preliminary Irradiation

Experimental studies, most of them recent, have shown the catalytic activity of certain solids towards some reactions to be considerably influenced by an irradiation previous to the chemical reaction (2-7). These modifications of catalytic activity are diminishing with time, and finally disappear after a lapse of time dependent upon the temperature; at room temperature, this activation may last several months after the irradiation. Among the imperfections responsible for activation, the structural imperfections probably take a prominent part only when the number of defects produced by the irradiation is not negligible compared to the number of pre-existing imperfections, as in the case of irradiations with particles such as neutrons and fission fragments, for example. Furthermore, these imperfections should be “efficient”; this means that, owing to their donor or acceptor character as well as their position in the forbidden band, they modify effectively the character of the surface, by influencing, for instance, the number of free carriers. Electronic imperfections with long lifetime result from the trapping of the free carriers. The produced effects depend both upon the nature and the number of the pre-existing impurity levels and of those created by radiations. These effects consist in an appreciable modification of the equilibrium concentration of both kinds of free carriers. Only the electronic imperfections can explain a catalyst activation by gamma and beta radiations which produce only a very small number of structural defects. Before studying the possible effects for different types of solids, it is necessary to emphasize the importance of several factors, such as the type of the radiation and the nature and the quantity of pre-existing impurities and defects, which determine the relative importance of the structural and electronic imperfections. Concerning extrinsic semiconductors, it has been seen in Section III,B,l that in general none of the two types of defects exerts any observable influence upon the number of majority carriers. Consequently, appreciable catalytic effects are only liable to occur in those cases where the produced structural and electronic defects influence the number of minority carriers. It was noticed in that case that the character of the semiconductor may even be changed; one may therefore expect important qualitative effects to occur in this way. In intrinsic semiconductors, it has been shown that the impurity con-

116

R. COEKELBERGS, A. CRUCQ, AND A. FRENNET

centration is always extremely small, and the influence of electronic imperfections generally negligible. Activation can only result from structure defects. Insofar as those are efficient, and provided that their number is sufficient, the semiconductor becomes extrinsic, of the n- or p-type, depending upon the donor or acceptor character of the defects. The catalytic effect in this case may be either qualitative or quantitative, depending upon both the character of the created defect and the type of chemisorption existing before irradiation. It has been seen above that through irradiation the properties of an insulator may be considerably modified, because of both the created lattice defects and the possible trapping of the excess carriers. In this case of considerable modification, important catalytic effects may be expected, as well qualitative as quantitative ones. They will be widely varied in character, depending upon the nature of the radiation and of the pre-existing impurities. These results are obtained even with very small doses of dissipated energy. Let us recall that in certain cases modifications of the chemical nature of the surface and of the physical structure of the catalyst need to be taken into account. Some examples in the recent literature support these theoretical views. Kohn and Taylor ( 4 ) have carried out several experiments with silica and alumina irradiated with gamma rays and with fast neutrons. They generally test the catalytic activity of those solids with respect to the exchange H2-D2. In the presence of‘gamma irradiated silica, the half exchange time, which exceeds 100 days before irradiation, decreases to a few hours immediately after irradiation. One month after irradiation, this value has increased again to approximately a few days. This decrease in catalytic activity is more rapid when raising the temperature. The catalytic activity induced by fast neutrons is more important and more lasting than that induced by gamma rays. These authors attribute the effects produced by neutrons to displacements, whereas those produced by gamma rays are attributed to trapping. They also emphasize the prominent part taken by the pre-existing impurities. These, for example, influence the hydrogen adsorption capacity of irradiated silica. By way of conclusion, we should like to insist upon the interest of experimental works in the field of catalyst activation through preliminary irradiation, We believe that new experimental evidence in this particular field will give clearer insight into many problems of catalysis; for instance, important information could be gained concerning the real nature of the “catalytic sites.” In this respect, we must cite an important result obtained by Kohn and Taylor in one of the studies cited above (4). These authors have attempted to identify the “sites” responsible for the HZ-DZ exchange. They

RADIATION CATALYSIS

117

have shown that the colored F centers produced by the irradiation of silica, although they are not involved in the exchange process, are closely related to the hydrogen absorption phenomenon. 2. Catalyst Activation under Irradiation

It has been shown in Section I1 that catalyst irradiation in the presence of reactants can modify to a considerable extent the kinetics of the catalytic reaction. Some experimental results as well as theoretical ideas concerning a connected subject, namely, the influence of visible and ultraviolet light upon the adsorptive and catalytic properties of the semiconductors, have been published recently (72). To a hrge extent the conclusions of these studies remain valid in the case of irradiation with particles and photons of high energy. Indeed, it has been seen (Section II1,C) that in most cases, the near totality of the energy dissipated by the light as well as by the high-energy radiation, transforms into electronic excited states (pairs of free carriers, excitons, etc.). The analogy between ultraviolet and gamma irradiation has been demonstrated by the works of Vesselovsky (Section I1,H ,2). Although the energy fraction used for the production of structural defects is generally small compared to the fraction creating excited electronic states, the effects resulting from these structural imperfections cannot be neglected when studying the activation under irradiation. The most important factor indeed is the relative importance of the stationary concentration of the electronic excited states as compared to the concentration of the structural defects. The ratio between these concentrations depends both upon the nature of the radiation and the irradiation intensity. Indeed, for a given dissipated dose, the stationary Concentration of electronic excited states depends on the intensity of the radiation, whereas the concentration of the lattice defects depends on the dose of dissipated energy. The latter concentration exert an influence relatively the more important the longer the exposure to a given type of radiation. I n this chapter are successively studied the modifications under irradiation of the catalytic properties of intrinsic semiconductors, extrinsic semiconductors, and insulators. In order to simplify the reasoning we assimilate all the electronic excited states to pairs of free carriers. Intrinsic semiconductors. Let us remember that the concentration of carriers in this type of semiconductor is relatively small (between 1O1Oand l O I 3 ) . We have seen that in order to appreciably modify a concentration of 1013 carriers (case of germanium), intensities of 1016-1020 e.v. cm.3 sec.-’, are required depending upon the lifetime of these carriers. Let us point out that the creation of pairs of carriers does not modify the intrinsic character of the semiconductor. The created lattice defects on the contrary, if

118

R. COEKELBERGS, A. CRUCQ, AND A. FRENNET

efficient, can modify the character of the semiconductor, which becomes, therefore, extrinsic. Depending upon the nature and the intensity of the radiation, one or the other of the two types of imperfection takes the prominent part; this results in a corresponding difference in the catalytic behavior of the solid. If the electronic defects prevail, there is observed in the most favorable cases only a slight increase in the number of both positive and negative carriers; as a consequence, the difference, if any, between catalytic properties under irradiation and without irradiation, can only be of a quantitative nature. If, on the contrary, modifications due to lattice defects predominate, not only quantitative differences may occur, but qualitative ones as well. I n that case effects may be observed for activation under irradiation which are practically identical to those observed for activation through preliminary irradiation, except if the lattice defects created by irradiation are either themselves excited or are possibly influential on the recombination time of the charge carriers. Extrinsic semiconductors. In extrinsic semiconductors the number of majority carriers is very large, always exceeding largely 10l6cm.+. On the contrary, the very much smaller number of minority carriers can be influenced by irradiation to an appreciable extent; their stationary concentration may in certain cases be multiplied by a factor of several powers of ten. The minority carriers which are thus produced may be at the origin of qualitative modifications of the catalytic activity. For instance, Romero-Rossi and Stone (75) who studied the CO oxidation in the presence of ZnO, have shown CO not to be chemisorbed in the dark, whereas under the influence of ultraviolet light, CO forms a strong bond with the surface by capturing the created positive hole. They propose a reaction mechanism where one of the steps is the recombination COads+ Ond8-+ COz; in this hypothesis the chemisorbed reactants behave likc recombination centers for the excess free carriers. These authors have also observed that the irradiation does not exert any influence upon the catalytic activity of certain zinc oxides which contain a great excess of interstitial zinc atoms. They explain this fact, which has been experimentally confirmed by Barry and Roberts (L'O), by admitting a competition between CO molccules and interstitial zinc atoms, for the capture of the positive holes. We have also seen that, although the lattice defects created by irradiation have a direct influence only upon the concentration of the minority carriers, they nevertheless are able to modify the or p character of the semiconductor as well. This effect can only be observed in the case where the structure defects give rise to a greater number of minority carriers than those resulting from the pairs of excess carriers created by irradiation. It is thus seen that whether the type of predominating defects are structural or electronic, the resulting catalytic effect may be only a qualitative

+

RADIATION CATALYSIS

119

one. The two modes of activation, by preliminary irradiation and in the presence of radiation, give rise to analogous effects, which may become identical if, under irradiation, the lattice defects play the principal role. Insulators. In insulators the number of pre-existing carriers is always very small. As a consequence, considerable effects are to be expected. The irradiated solid may behave transiently as an extrinsic or an intrinsic semiconductor, depending upon whether the trapping of one or the other type of carrier modifies or not, to any appreciable extent, the relative concentration of both types of carriers. Moreover, structural imperfections created by irradiation may play here an important role, either because of their own character or in consequence of their intervention in the phenomenon of trapping of the excess carriers. Numerous factors are thus liable to play a role in the case of insulators; as a rule, important modifications of activity occur which are qualitative as well as quantitative, but it is impossible to go into more details. 3. Summary

Table XI11 summarizes the conclusions of Section IV,B and contains a comparison between the effects of the activation either through preliminary irradiation or under irradiation. Two border line cases are considered: the case of irradiation by low energy photons, which induce no structural imperfections and the case of irradiation by fast neutrons, for which the energy fraction used for the creation of structure defects most often exceeds the one used for the creation of electronic defects. The following conclusions may be brought out: (1) The insulators present the most interesting activation possibilities, whichever the intensity and the type of radiation. (2) For solids other than insulators activation through preliminary irradiation is dependent upon either the presence of pre-existing impurities or the creation of lattice defects; for these solids the difference between the catalytic effects produced by both modes of activation is small. Attention is once more drawn to the fact that the present conclusions are only valid within the limits of the approximations that have been made. We emphasize the fact that certain modifications of the catalyst surface conditions cannot be theoretically foreseen and have been therefore neglected.

C. ENERGY TRANSFER The energy transfer from an irradiated solid towards the gaseous phase, which can result in a modification of the thermodynamical equilibrium of the considered system, clearly distinguishes this aspect of the radiation catalysis from the activation process. This transfer phenomenon is related

TABLE XI11 Comparison Between the Activation of Catalysts by PrarioUS Irradiation and the Activation by Simultaneous Irradiation; Summary of the Possible Efeds Nature of the incident radiation Photons

Neutrons

Type of activation

Type of solids Insulators

Previous irradiation

Very important qualitative and quantitative effects, even for small doses; these effects are connected with the pre-existing impurities. Simultaneous Very important qualitative and irradiation quantitative effects even for weak intensities; effects are independent from the preexisting impurities. May be very important but deDifference pends often on the impurities. Previous irradiation

Very important qualitative and quantitative effects even for small doses.

Simultaneous Very important qualitative and irradiation quantitative effects even for low irradiation intensities. Difference

May be very large because of some electronic excitations during irradiation.

Intrinsic semiconductors

Extrinsic semiconductors

No effects.

No effects, except if the concentration of minority carriers is modified by trapping, thereby giving important qualitative effects.

Null effects or small quantitative effects for high irradiation intensities.

Modifications of the number of minority carriers may cause important qualitative effects.

None or very small.

Essentially bound with preexisting impurities. Only quantitative.

Possible transformations into extrinsic semiconductors; important qualitative and quantitative effects. Possible transformation into extrinsic semiconductors, important qualitative or quantitative effects. None or small.

Possibility to modify the character of the semiconductor,possible important qualitative effects. Possibility to modify the character of the semiconductor,possible important qualitative effects. None.

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to the possibility of recuperation of a more or less important fraction of the radiation energy dissipated in the solid in the form of potential chemical energy. It is only observed in the case of simultaneous irradiation of solid and reactants. We shall here successively consider the three possible modes of transfer. 1. Energy Transfer by Excited Electronic States

The first step of a catalytic reaction always consists in a strong adsorption of one of the reactants upon the surface of the solid. As an example, let us consider the case of CO oxidation on a nonmetallic solid. Various mechanisms have been proposed for this reaction (75-79). A first possible scheme is the following: 0 2

Osds-

$-

+2 8 cows

+ 20nda+

COZ

+8

The strong adsorption of an Ozmolecule occurs by electron capture. Another equivalent mechanism, where the CO molecule undergoes strong adsorption by capture of a positive hole can be written:

co + @ coad,'

+

+

COad.' c o Z g n 8

$02

+@

In both schemes, the free carriers (electron or positive hole) needed for establishing the strong adsorption bond is recovered in the final step of the mechanism and thus may induce a great number of reactions, if under the prevailing experimental conditions the reaction is thermodynamically possible. An irradiation of the solid which increases the number of free carriers is thus seen to result in an activation of the catalytic properties of this solid. Let us now consider the inverse COz decomposition reaction. The first step may consist in the capture of an electron:

coz + 0 cozdd.--+

Since it is strongly endothermic, it does not seem possible to carry on the reaction further without a supplementary energy supply. But, if the solid is simultaneously irradiated, large amounts of excess carriers of opposite sign may be produced, and the adsorbed COZmolecule may act as recombination center, giving rise to the following reaction. COZ,d,

+ €9

-

ICOzl* -+ co

+0

This process gives out a quantity of energy generally in the same order of magnitude as the one used for the creation of the pair of carriers. As a good

122

R. COEKELBERGS, A. CRUCQ, AND A. FRENNET

estimate, the average values of this energy is equal to the energy gap between the valency and conduction bands. The COzmay receive a relatively important fraction of this energy, which can suffice to allow decomposition to occur. This constitutes an energy transfer. One may compare this phenomenon to the Auger effect in which the photon energy that is normally emitted by an excited ion is absorbed by this ion. In this respect, we note that such a transfer phenomenon has been observed in our laboratory for COzdecomposition a t room temperature in the presence of alumina irradiated with beta rays from SrgO.This work was not mentioned in the experimental part of this paper because at the time that part was written, only preliminary results were obtained which were insufficient to go further into the mechanism of transfer. The concept of energy transfer exhibits its entire significance in the case of endothermic reactions. Nevertheless, a transfer can be envisaged at a microscope scale, when one of the steps of an over-all exothermic process is endothermic. In a quite general manner, the scheme of an energy transfer through electronic excited states may be written in one of the following forms:

+ €3

AB AB.ds-

+@

4 4

AB.da[AB]* -+ A

+B

-+AB+hv +M-+X+Y

or otherwise, CD CD,d.+

+ CB + €3

4 4

CDadn'

+

[CD]*+ C D -+CD+hv +M-+X+Y

Recombination must not necessarily take place between an adsorbed molecule and a carrier of opposite sign. For instance, we may have:

+e B +@ A

Aads-

+ Bada'

-+

A ~ ~ ~ -

+ Bada'

+ [AB]* -+X

+Y

In any case, the energy released through this mechanism of transfer must have an average value in the order of the energy gap between the valency and conduction bands, but may not exceed the maximum extraction energywof~an'electronpfromthe valency band. If the AH of an endothermic reaction or of the endothermic step of a reaction is greater than this

RADIATION CATALYSIS

123

maximum extraction energy, and if abnormally high G values are observed for this reaction as well, these facts cannot be explained by a transfer through excited electronic states and must originate in another phenomenon. Insulators for which the forbidden region often has values exceeding 200’kcal. mole-’, will prove the most efficient ones, for the transfer by electronic excited states. Intrinsic semiconductors with a much smaller forbidden zone, of the order of 15-30 kcal., can give rise to this kind of phenomena only for a limited number of reactions. Extrinsic semiconductors require an examination in each particular case since the value of the forbidden region varies within large limits. Therefore, the nature of the solid plays the leading part because it fixes the value of the energy gap. The role of impurities is less important, and this constitutes an important difference with respect to activation phenomena. 2. Energy Transfer by Temperature Spikes The ‘%emperaturespike” transfer is evidently concerned with irradiations only by fission fragments (Section III,A,4) and fast neutrons (Section III,A,5) ; indeed, only these particles release an important fraction of this energy to the lattice in the form of temperature spikes, and this released energy corresponds to short-lived (10-*-10-11 sec.) temperature increases of several thousand degrees, involving a great number of atoms. In the case of fission fragment irradiation this number varies between lo6 to lo9;the corresponding value for neutrons is between lo2and lo4. In case the temperature spike includes surface atoms, the adjoining gas is brought at a very high temperature as well. In the case of transfer through electronic excited states, energy is imparted to a given atom or molecule which forms a part of a system in thermal equilibrium, the latter being defined by the mean temperature; whereas in the case of energy transfer through temperature spikes, groups of gaseous molecules are brought to a temperature different from the mean temperature. In the first case, one has to deal with a perturbation of the Maxwell-Boltzmann distribution, and therefore the laws of thermodynamic do not apply; whereas in the second case these laws remain valid, but the local temperature in the limited zone under the influence of the spike is taken into account. A calculation of this type has been carried out by Walton for the decomposition of potassium iodate and nitrate by fission fragments (80). As a consequence, it is seen that, whereas the energy transfer through excited states favors indistinctly both endothermic and exothermic reactions with a high activation energy, the transfer through temperature spikes, on the contrary, essentially applies to reactions which are thermodynamically possible at high temperature, and thus most often to endothermic reactions. If, however, some radicals produced by the temperature spikes have a longer lifetime in the considered

124

R. COEKELBERGS, A. CRUCQ, AND A. FRENNET

reacting system than the temperature spike itself, the initiation of certain exothermic reactions with a high activation energy then becomes possible. 3. Energy Transfer by Photons

Photon sources of energy smaller than, or of the order of, 1 k.e.v., as seen above, (1) are bremsstrahlung, (2) return to the ground state of the excited ions, and (3) are electronic excited states returning to their fundamental state; this last process is the recombination of carrier pairs. A part of the photons produced by the two first phenomena have energies between a few electron volts and a few thousand electron volts: the whole of these photons, represent at most a few per cent of the total dissipated energy (Section III,B,3). Photons produced by the third phenomenon have an energy lower than 20 e.v. ; at a certain stage of the degradation process of the radiation energy these photons may represent a very considerable fraction of the total dissipated energy, but no definite order of magnitude can be fixed. The absorption spectra of these photons have already been shown to display a series of characteristic extrema. The position of the absorption maxima depends upon the nature of the elements constituting the solid and the reactants and also upon the nature of the chemical bonds between these elements. Between tt maximum and a minimum, the absorption coefficient may vary by a factor exceeding lo3.From this, it results that a selective absorption of these photons either by the solid, or by the reactants, is possible. The preferential absorption by the solid leads to the formation of excited electronic states (pairs of carriers, excitons, etc.) whose influence on catalytic properties have already been studied. When these excited states do not induce any particular catalytic property, this energy is lost with respect to the studied reaction. The preferential absorption by the reactants and consequently the initiation of the reaction, if any, may take place either in the gas phase or in the adsorbed phase. It seems then to be very probable that the surface will influence the course of the reaction, a t least one step being heterogeneous. Nevertheless, when the reaction is initiated in the gas phase, it is possible that all the steps of the reaction take place in homogeneous phase; the reaction is, then, quite similar to the homogeneous chemical reaction, particularly with respect to the mechanism and the nature of the products. For an evaluation of the order of magnitude of the energy transferred by this process from the solid to the gas phase, it is necessary to know both the respective absorption spectra of the solid and of the gas in the considered energy domain, as well as the emission spectra of the solid in this same domain. The absorption spectra of the majority of gases are generally well

RADIATION CATALYSIS

125

known in the visible and in the near ultraviolet ranges. In this same domain, the number of known spectra is more limited for solids, the alkali halides being the best investigated ones. For wavelengths shorter than 2000 A., information concerning the absorption spectra of the gases is relatively scarce, whereas very little is known about the solids. Nevertheless, the order of magnitude of the absorption coefficients is such that a solid thickness ranging from a few tenths of to a few microns decreases by a factor of two the intensity of the transmitted light. Finally, the emission spectra of the irradiated solids are practically unknown. This is the reason why as yet, no experimental evidence is available in support of the hypothesis of energy transfer through selective photon absorption. Let us finally stress the fact that the photons whose wavelength corresponds to this energy range are the sole particles which can be selectively absorbed; the above mentioned phenomena will never occur for the other types of radiation studied in this paper.

4. Conclusions At the present time, the existence of energy transfer phenomena appears to be clearly demonstrated in an experimental way. However, the transfer mechanisms we propose, even if they are plausible, nevertheless should be looked upon as different hypotheses that cannot be, as yet, either confirmed or invalidated. It is probable that, in certain cases, only one particular transfer mechanism is possible, but it is also quite plausible that, in other cases, the three transfer mechanisms exist simultaneously. In this respect we also note that, whichever is the transfer mechanism, its existence is related to the granulometry and the porosity of the irradiated solid. It is to be expected that grain dimensions will affect the transfer by temperature spike the most, and the transfer by excited electronic states the least. In this latter case, if account is taken of the mobility (-lo7 cm. sec.-l> and of the lifetime of the excited states, the influence of grain dimensions upon transfer can only depend on the probability for a charge carrier or an exciton to be trapped by a surface molecule. If this probability is great, no influence of the granulometry can be observed below a certain value of the grain dimension. In the case of transfer by temperature spikes the grain dimensions do not exert any more influence when they become smaller than the spike dimensions (a few hundred angstroms at a maximum). The case of photon absorption is intermediate and depends on the value of the absorption coefficient. Finally let us point out the particular interest of the transfer mechanism which should allow a much more efficient utilization of the absorbed energy of the incident radiation than in homogeneous radiation chemistry. This could be especially the case of the transfer by electronic excited states in

126

R. COEKELBERGS, A. CRUCQ, AND A. FRENNET

which one may thus hope to observe value.

Gap,

values approaching the

Gmax

V. Some Comments About the Experimental Results Our intention is to examine the experimental results described in Section

I1 in the light of the general ideas just developed. For two reasons, a limited discussion of the experimental facts should be made. First, the considerations put forward in Section I V do not yet constitute a real theory; second, the available experimental results are still incomplete and allow only of a partial justification of the theoretical ideas. The N20 decomposition is particularly interesting. This reaction has been thoroughly investigated, thermally in the homogeneous phase (81,82) and in presence of catalysts (83-85), as well as under the influence of radiations (30-34). Within the limits of experimental error, homogeneous and heterogeneous radiolysis yield the same products in the same proportions (N2,02,N02in the ratio 1:0.38 :0.14) ; whereas thermal decomposition, homogeneous as well as cataIytic, furnishes only N2 and 0 2 , to the exclusion of nitrogen oxides. With regard to thermal decomposition on semiconductors and insulators, the following mechanisms are generally proposed :

The adsorption stcps (1) and (la) are generally more rapid than reactions (2) and (2a) which are favored by the presence of positive holes. For this reason, semiconductors of the p-type are generally better catalysts than insulators, whereas n-type semiconductors are the least efficient ones. Concerning the heterogeneous NzO radiolysis two essential features are to be stressed: the abnormally high values of G , and, in contrast to thermolysis, the formation of nitrogen oxides. Because this latter process is endothermic, we must admit the existence of an energy transfer from the solid phase towards the reactants, following one or the other mechanism which have been proposed in Section IV,C. One may, for instance, consider a transfer through electronic excited states. In this hypothesis, the heterogeneous radiolysis proceeds, like the heterogeneous thermolysis, through the adsorption step (1); we do not retain step (1 bis) which, in principle, cannot lead to the formation of nitrogen oxides. Recombination with a positive hole can be expressed either by reaction (2) or by the following reaction (3) : NzOaa.- 4- 0 .+ NO 4- N

(3)

RADIATION CATALYSIS

127

Comparison of the two elementary processes (2) and (3) shows the latter to be the most endothermic one. Indeed, decomposition of N20yielding NO and N entails the rupture of the N=N bond, which requires 85 kcal. mole-' (3.6 e.v. molecule-'); whereas for N20 decomposition into Nz and 0, the breaking of the NO bond requires only 38 kcal. mole-1 (1.6 e.v. molecule-I). This explains that thermally, in the absence of radiation, reaction (2) is always considerably favored, compared to reaction (3). Under irradiation , however, a great number of excess free carriers are produced in the conduction and valency bands; these carriers tend to recombine. In this respect the adsorbed N2O molecule may behave like a recombination center. This phenomenon can be accounted for by considering the adsorbed NzO molecule to be an acceptor level. Under this hypothesis, the N20 chemisorption results from the capture, by the weakly adsorbed molecule, of an electron from the conduction band. At the moment of recombination with a positive hole from the valency band, a variable amount of energy can be recovered, depending on the position of the level constituted by the adsorbed N20 molecule. For the silica and alumina we have utilized, the width of the forbidden region is about 10 e.v.; process (3) which only requires 3.6 e.v. may thus become possible. These considerations do not necessarily imply that the mode of transfer through excited electronic states is the only possible one. It is certain that, in the case of irradiation by fission fragments, a transfer through temperature spikes is possible. Anyhow, a transfer through selective photon absorption is also possible. In the case of the insulators that we utilized, the energy gap between valency and conduction bands has a value of about 10 e.v. as mentioned above; now NzO displays an important absorption band in the vicinity of 10 e.v. (1236 A.). Homogeneous photolysis of N20 by radiation of this wavelength, moreover, has been shown (86) to produce the same proportion of nitrogen oxides as those obtained by radiolysis. I n the absence of precise data concerning the emission spectra of irradiated silica and alumina, energy transfer through photons thus remains quite plausible. In conclusion, the existence of an energy transfer from solid to gas seems to us clearly established in the case of NzO radiolysis; but the mechanism of this transfer cannot, as yet, be determined. It is difficult to explain by an energy transfer mechanism the result concerning ammonia synthesis, and more particularly the high value of the ratio (Ggas/Ghom). Indeed, if account is taken of the fact that 75% of the produced ammonia remains adsorbed on the catalyst, GaPpis practically equal to Ghom. The transfer hypothesis requires that the near totality of the energy adsorbed by the solid is transferred to the gaseous reactants; this is hardly conceivable. Moreover, water is formed in quantities always equal to the quantities of produced ammonia; this occurs exclusively when

128

R. COEKELBERGS, A. CRUCQ, AND A. FRENNET

nitrogen, hydrogen, radiation, and a catalyst are simultaneously present. This leads us into the belief that the heterogeneous synthesis induced by radiation proceeds through steps which are entirely different from those commonly adopted for homogeneous radiochemical synthesis. Moreover alumina does not catalyze the thermal synthesis of ammonia. These various considerations show that, as a consequence of irradiation, the solid has acquired new properties, and has thus become a catalyst. It is therefore an example of catalyst activation through radiation. Hydrocarbon radiolysis a;y well as ethylene polymerization induced by radiation are generally admitted to proceed by radical mechanisms (87, 88, 39). This is confirmed by the observed inhibition of these reactions by active charcoal. As relat8ed by Mechelynck-David (11, 12) and other workers (89),the active charcoal possesses groups with a quinonic structure, which are typical inhibitors for radical reactions. The various results concerning ethylene polymerization and methane and pentane radiolysis lead us to the belief that the solid takes only a negligible part in the initiation process of the reaction. On the contrary, the subsequent evolution of the excited species, i.e., the ions and radicals generated by the radiation is strongly influenced by the presence of the solid. An orientation of the reaction seems thus to happen, which in the case of the radiolysis of methane and pentane leads to a different stoichiometry for the obtained products. In the case of ethylene polymerization the preliminary adsorption of the gas on the solid increases the propagation rate of the reaction chain, or decreases the number of radicals recombinations. It should be pointed out that, in the presence of silica, results obtained by Allen as well as those obtained in this laboratory have shown the yields of the methane and pentane radiolyses, expressed as G,,,, to be considerably higher than in the homogeneous phase. These facts can easily be accounted for by some of the transfer mechanisms we have considered ; however, processes which result in a better utilization of the excited species are not to be excluded. The phenomena of radiation catalysis observed in the course of cyclohexanol dehydratation, methanol synthesis, and water radiolysis, constitute without any doubt cases of activation under irradiation. With regard to the methanol synthesis from CO and H2 in the presence of ZnO, we believe that the small activation effects observed under irradiation to be attributable to the contribution of a new reaction mechanism, more precisely to the possibility of chemisorbing CO, in the form CO+, through capture of a positive hole generated by irradiation. In this respect, let us note that, as already pointed out by Romero-Rossi and Stone (75), the interstitial excess zinc atoms compete with CO molecules for the capture of these positive holes; therefore, when a nonstoichiometric catalyst

RADIATION CATALYSIS

129

containing a large zinc excess is irradiated, no activation phenomenon is occurring. With regard to the work of Vesselovsky on the hydrogen peroxide decomposition in the presence of ZnO, one has to stress the analogy, put in evidence by this author, between the ultraviolet irradiation and the gamma irradiation. He has verified that the used zinc oxide possesses an absorption band for protons of about 3 e.v.; this value corresponds to the energy of the transition between the valency and the conduction bands. He has shown, moreover, that from the standpoint of activation during irradiation, a gamma photon is equivalent to a number of ultraviolet photons which are equal, within a factor 2, to the energy ratio (E/Euv) of these two photons. This constitutes clear evidence in support of the mechanism we have proposed for the degradation of the radiation energy. The results of cyclohexanol dehydration experiments in the presence of various sulfates show insulators to possess interesting activation possibilities, which differ depending on whether the irradiation is carried out before or during the chemical reaction. Concerning nitrogen fixation, it is difficult to draw clear conclusions. It is certain that the presence of a solid as well as its nature both influences the course of the reaction; no explanation of the mechanism involved has been forthcoming.

VI. General Conclusion Quite independently of any theoretical idea, the catalyst activation by preliminary or simultaneous irradiation, and the energy transfer between the different phases of the heterogeneous irradiated systems, are now experimentally tangible. However, the number of experimental results available a t this time does not permit a general theory of the influence of radiation upon the catalytic properties of solids to be established. The various ideas which have been put forward should be considered as plausible work hypotheses, which are to be proven valid or invalid by further experimental evidences. Before ending this article, however, we should like to state again the most important ideas. In a general way, it appears that radiation catalysis will contribute in an important manner to a better insight of the catalytic processes. Experimental work in the field of catalyst activation appears to be a rapid and reliable inroad to information about catalytic sites; not withstanding the amount of results accumulated in the field of classical catalysis, only fragile hypotheses have been put forward as yet on this point. For this reason, we believe that activation studies, for instance, will be able to furnish valuable

130

R. COEKELB:ERGS, A. CRUCQ, AND A. FREN N ET

information concerning the role played in the catalytic processes by certain defects and impurities, and by certain excited electronic states. In the same way, the study of transfer phenomena, more particularly by electronic excited states, might, constitute a powerful tool for the investigation of the active role played by catalysis in the energetic phenomena which characterize certain elementary catalytic processes insofar as the above enunciated assumption proves to be correct. We should also stress the fact that irradiation, under certain circumstances, may communicate new specific properties to catalysts, thereby largely extending the already available range of catalysts. In this respect, insulators seem to be of particular interest. Finally, it is noteworthy that in the particular field of radiation chemistry, radiation catalysis may permit a better utilization of the radiation energy, and considered from this standpoint, might constitute a source of practical applications of the radiations.

ACKNOWLEDGMENTS We wish to thank Mr. J. Sohier, IngBnieur Civil de 1’Ecole Royale Militaire, for his helpful assistance in the prepa,ration of the manuscript. In the course of our studies, the samples have been irradiated at the “Centre d’Etude de 1’Energie NuclBaire” (MOI,, Belgium). The results of irradiation with fission fragments using microporous solid supports were conducted by the nuclear research team of the “SociBtB Belge de 1’Aeote e t des Produits Chimiques du Marly” working in collaboration with the “Centre d’Etude de 1’Energie NuclBaire.”

Appendix A. CALCULATION OF G,,, FOR NITROGEN APPEARANCEIN THE CASE OF GAMMA IRRADIATION OF NzO IN THE PRESENCE OF CSU 1. Experimental Data. Amount of catalyst: 1 g. CSU (13.291, U, 86.8% SiO,) Volume of reaction vessel: 3 ml. Volume occupied by the solid: dHcl = 0.4 ml. Volume accessible to the gas: (3 - 0.4) ml. = 2.6 ml. Pore volume: - dH0-l = [0.9 0.41 ml. = 0.5 ml. Amount of NzO introduced: 2.25 X 10-3 moles Total dissipated dose: 6.35 X lozoe.v. Amount of produced nitrogen: 26 X moles = 1.56 X 1019molecules Ghom for nitrogen formation: = 8

-

2. First Case Considered: T h e NzO Adsorption on the Solid i s Complete. The gamma energy dissipated into the heterogeneous system is distributed

131

RADIATION CATALYSIS

between the two phases in accord with the hypothesis put forward in Section II,B which assumes this repartition to be finally dependent upon the number of electrons present in each phase. 2.25 X loF3moles NzO electron-gram. One g. represent 2.25 X [(2 X 7) 81 = 4.95 X CSU represents

+

132 x 10-3 868 238 92 -k 28 (2 X 16) [14

+

+ (2 X 8)] = 48.6 X

loM2electron-gram

The energy dissipated in the gaseous phase is hence equal to 4.95 x 10-2 6.35 X lozoX (48.6 4,95)

e.v. = 5.9 X 10’’ e.v.

+

1.56 x 1019 102 = 2, Ggas= 5.9 x 1019 3. Second Case Considered: No NzO Adsorption Occurs o n ihe Solid. The volume accessible to the gas is equal to the volume of the vessel (3 ml.) diminished by the volume of the solid ( V , = dHB-l= 0.4), which is equal to 2.6 ml. This volume includes (1) the pore volume V , = 0.5 ml., and (2) the volume Vi, either situated above the grains of the solid (these are disposed a t the bottom of the vessel) or interspersed with the grains of the solid. The irradiation of the gas inside the pores must be considered as heterogeneous, owing to the pores’ dimensions, which are of the order of the mean free path of the molecules a t the considered pressure. Within Vi on the contrary, the irradiation must be considered as homogeneous, indeed, the distance which separates the gas from the surface of the solid, is lo6to lo6 times greater than the mean free path under the prevailing conditions. As in the preceding case, 5.9 X l O I 9 e.v. are dissipated into the gaseous phase; but one part of this energy serves to initiate a homogeneous reaction, whereas another part initiates a heterogeneous reaction. These two parts are in the ratio of the two quantities of gas which obey either mechanism; this ratio is (Vi/V,) since it has been assumed that NzO is not adsorbed. Consequently, the energy which serves for the initiation of the homogeneous reaction is equal to: 5.9

x

1019 x Vi

vi v,e.v. = 5.9 x 1019: 2 1 = 4.8 X 1019e.v.

+

2.6

If we take Ghom equal to 8, the amount of nitrogen produced in the homogeneous radiolysis will be equal to : 4.8 X 1019 X 8 X

= 3.84

X lo1*molecules Nz

132

R. COEKELBERGS, A. CRUCQ, A N D A. FRENNET

Subtraction of this quantity from the experimentally produced amount of nitrogen gives the amount of nitrogen produced by the heterogeneous reaction :

1.56 X 1019 - 3.84 X 1018 = 1.18 X 1019 molecules Nz The energy used for the initiation of the heterogeneous reaction is equal to :

5.9 X 1019 - 4.8 X 1019 = 1.1 X 1019 e.v. There results for G,,

Owing to the lack of more precise data concerning N20 adsorption, G,, is between 27 and 107.

B. CALCULATION OF GgB, FOR NITROGEN APPEARANCE IN THE CASE FISSION FRAGMENT IRRADIATIONS OF N2O IN THE PRESENCE OF CSU

OF

1. Experimental Data. Amount of catalyst: 1 g. CSU (13.2% U, 86.8% Si02) Volume of reaction vessel: 3 ml. Volume occupied by the solid: dHe-l = 0.4 ml. Volume accessible to the gas: Vi = (3 - 0.4) ml. = 2.6 ml. Volume of pores: V , = 0.5 ml. moles Amount of N20 introduced: 2.10 X Total dissipated dose: 1.37 X 1021e.v. moles = 2.13 X l O l 9 moles Amount of nitrogen produced: 35.5 X

2. First Case Considered: The N 2 0 Adsorption in the Pores of the Solid i s Complete. The energy of the fission fragments is shared between both the solid and the gas contained in the pores according to the hypothesis of page 61, which assumes this repartition to be finally proportional to the number of electrons present in each phase. 1 g. CSU represents 48.6 X lo2 electron-grams (cf. Appendix A). 2.1 X moles NzO represent 4.62 X electron-gram. The energy dissipated in the gaseous phase is equal to

3. Second Case Considered: N o N 2 0 Adsorption Takes Place on the Solid. Since the fission fragments in this case have their origin in the support

133

RADIATION CATALYSIS

itself and since their range is very small, (about 25p in a solid, about 20 mm. in a gas under standard conditions) it follows that only a fraction of the gas is irradiated. The energy dissipated into the gas is the sum of two terms. (1) The first of these is the energy dissipated in the gas contained in the pores. This term depends on the gas concentration and, owing to the absence of any adsorption phenomenon, depends finally on both the total amount of the gas which has been introduced and on the ratio volume of the pores total volume accessible to the gas (2) The second is the energy of those fission fragments which are generated in the outermost annulus of the macrograin and thus retain sufficient energy to escape from it. A calculation shows that in the ideal case of spherical grains of granulometry between 15 and 30 mesh, 1.6% of the fission energy is dissipated outside the grains. The greatest part in this energy is dissipated in the immediate neighborhood of the grain (loop); a close contact is thus seen to exist between the irradiated gas and the solid and consequently the irradiation may be considered as heterogeneous. The amount of NzO contained in the pores is equal to 2.1

x

10-3

ml. x 0.5 -- 0.4 X 2.6 ml.

moles NzO

+

This represents 0.4 X (2 X 7) 8 = 0.88 X lop2 electron-gram. One gram of CSU represents 48.6 electron-grams. The fraction of the energy dissipated in the gas contained in the pores is equal to o’88 48.6 X 0.88

=

1.74%

= 2.38 X 1019 e.v. This corresponds to 1.37 X loz1X 1.74 X The energy carried away by the fission fragment which escape from the grain is

1.37 X 1021 X 1.6 X

= 2.19 X 1019e.v.

The total energy dissipated into the gaseous phase is (2.38

+ 2.19)1019e.v. = 4.57 X 1019e.v.

It results that G,, is equal to 2*13 ‘O” 102 4.57 x 1019

=

46.6

Owing to the lack of more precise data concerning NzO adsorption, G,, is between 19 and 47.

134

R . COEKELBERGS, A. CRUCQ, AND A . FRENNET

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135

3%.Moseley, F., and Truswell, A. E., A.E.R.E., R. 3078 (1960). 33. Burtt, B. P., and Kircher, J. F., Radiation Research 9, 1 (1958).

34. Harteck, P., and Dondes, S., Nucleonics 16, No. 8, 94 (1957). 36. Moseley, F., Dawson, J. K., Long, G., and Sowden, R. G., Proc. %ndIntern. Con]. Peaceful Uses Atomic Energy, Geneva, 1968 8, 252 (1959). 36. Aerojet General Nucleonics, TID-5693 (1959). SY. Moseley, F., and Edwards, A.E.R.E., C/R 2710 (1958). 38. Harteck, P., and Dondes, S., in “Large Radiation Sources in Industry,” Warsaw, 1959, Vol. I, p. 231. Intern. Atomic Energy Agency, Vienna, 1960. 39. Lampe, F. W., J.Am. Chem. SOC.79, 1055 (1957). 40. Caylord, N. C., and Mark, H. F., “Linear and Stereoregular Addition Polymers.” Interscience, New York, 1959. 41. Friedlander, H. N., J . Polymer Sci. 38, 91 (1959). 4%. Kraus, G., Gruver, J. T., and Rallman, K. W., J . Polymer Sci. 36, 564 (1959). 43. Bonnet, J. B., and Heinrich, G., Compt. rend a d . sci. 246, 3341 (1958). 44. Feeler, M., and Field, E., Ind. Ens. Chem. 61, 155 (1959). 46. Denies, A. E., and Allen, A. O., J . Phys. Chem. 63, 879 (1959). 46. O.R.N.L., 2993, p. 183 (1960). 4Y. Bethe, H. A., and Ashkin, J., in “Experimental Nuclear Physics” (E. Segr6 ed.), Vol. I, part 2, pp. 166-357. Wiley, New York, 1952. 48. Kinchin, G. H., and Pease, R. S., in “Reports on Progress on Physics” (A. C. Stickland, ed.) Vol. XVIII, p. 1. Physical Society, London, 1955. 49. Eine, G. J., and Brownell, G., “Radiation Dosimetry.” Academic Press, New York, 1956. 60. White, G. R., N.B.S. Report 1003 (1954); 583 (1957). 61. Evans, R. S., in “Handbuch der Physik”. Vol. 34, p. 218. Springer, Berlin, 1958. 6%. Brooks, H., Ann. Rev. Nuclear Sci., 215 (1956). 65. Seitz, F., Discussions Faraday Soe. 6, 271 (1949). 64. Seitz, F., and Koehler, J. S., in “Solid State Physics” (F. Seitz, D. Turnbull, eds.), Vol. 11, p. 307. Academic Press, New York, 1956. 66. Dienes, G. J., and Vineyard, G. H., “Radiation Effects in Solids.” Interscience, New York, 1957. 66. Cameron, J. F., and Rhodes, J. R., Conference on the Use of Radioisotopes in the Physical Sciences and Industry, Copenhagen, 1960, RICC/14. 67. Massey, H. S., and Burhop, E. H., “Electronic and Ionic Impact Phenomena.” Oxford Univ. Press, London and New York, 1952. 68. Thommen, K. Z., Z.Physilc 161, 114 (1958). 69. Ozeroff, J., AECD 2973 (1949). 60. Harwood, J. I., Haussner, H., Morse, J. G. and Rauch, W. G., “Effects of Radiations on Materials.” Rheinhold, New York, 1958. 61. Burlein, T. K., and Mastel, B., RW 61656. 6.8. Compton, A. H., and Allison, S. K., “X-Ray in Theory and Experiment.” Van Nostrand, New York, 1935. 63. Hagedoorn, A. L., and Wapstra, A. H., Nuclear Phys. 16, 46 (1960). 64. Shockley, W., “Electrons and Holes in Semiconductors.” Van Nostrand, New York, 1950. 66. Frenckel, J., Phys. Rev. 37, 17, 1276 (1931). 66. Apker, L., and Taft, E., Phys. Rev. 79, 964 (1950); 82, 786, 814 (1951). 6Y. Leverens, H. W., “Introduction to Luminescence of Solids.’’ Wiley, New York, 1950.

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R. COEKELBERGS, A. CRUCQ, AND A. FRENNET

68. Stone, F. S., in “Chemistry of Solid State” (W. Garner, ed.), p. 20. Butterworths,

London, 1955. 69. James, H. M., and Lark-Horowite, K., 2.Physik Chemie 198, 107 (1951). 70. Lark-Horowite, K., and Fan, H. Y., “Report of the Bristol Conf. on Defects in Crystalline Solids,” p. 232 Physical Society, London, 1955. ‘71. Kennedy, P. J., Proc. Roy. SOC. 8263,37 (1959). ‘72. Wolkenstein, T., Advances in Catalysis 12, 189 (1960). 73. Dalmai, G., Imelik, B., and Seguin, M., J. chim. phys. 68, 292 (1961). ‘74. Terenin, A., and Solonitzin, Y., Discussions Faraday SOC.28, 28 (1959). ‘76. Romero-Rossi, F., and Stone, F., in “Actes du deuxieme congres international de catalyse, Paris, 1960,” Vol. 2, p. 1481. Technip, Paris, 1961. ‘76. Schwab, G. M., and Drikos, A., 2. Physik Chem. (Leipzig) B62, 234 (1942). ‘77. Garner, W. E., J. Chem. SOC.p. 1239 (1947). 78. Garner, W. E., Gray, T., and Stone, F. S., Proc. Roy. SOC. A197, 294 (1949), ‘79. Garner, W. E., Stone, F. S., and Filey, E. F., Proc. Rou. Sci. A211, 445 (1952). 80. Bertocci, U., Jacobi, R. B., and Walton, G. N., Intern. Conf. Chem. Effects of Nuclear Transformations, Prague Comm. CENT/17 (1960). 81. Friedman, L., and Bigeleisen, J., J. Am. Chem. SOC.76, 2215 (1953). 82. Johnston, H. S., J. Phys. Chem. 19, 663 (1951). 83. Rheaume, L. and Parravano, G., J. Phys. Chem. 63, 264 (1959). 84. Winter, E. R. S., Discussions Faraday SOC.28, 183 (1959), Advances in Catalysis 10, 196 (1958). 86. Hauffe, K., Advances in CataEysis 7, 213 (1955). 86. Zelikoff, M., and Aschenbrand, L. M., J. Chem. Phys. 22, 1680 (1954); Threshold of Space, Proc. Conf. Chem. Aeronautics, p. 99 (1956). 87. Gevantman, L. H., and William, P. R., J. Phys. Chem. 66, 569 (1952). 88. Swallow, A. J., “Radiation Chemistry of Organic Compounds.” Pergamon, New York, 1960. 89. Drushel, H. V., and Hallon, J. V., J. Phys. Chem. 62, 1502 (1958).

Poly f unctio na I Hete roge neous Cat a lysis PAUL B. WEISZ Socony Mobil Oil Company, Incorporated Research Department Paulsboro, New Jersey Page I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 11. Principles of Polystep Catalysis. . .............. 138 A. Single or Multifunctional Catalysts. ................................. 138 B. Intermediates and Reaction Sequences.. .

D. Mass Transport in Polystep Reaction

. . . . . . . . . . 144

. . . . . . . . . . 153 IV. Some Major Polystep Reactions of Hydrocarbons.. . . . A. Reactivity for Isomerieation of Paraffins.. ............................ 158 B. Reactivity for Hydrocracking of Paraffins. . . ............. 162

D. Reactivity for Cyclohexane.. . . . . . . . . . . . . . . ......................... E. Aromatisation of Alkylcyclopentanes .................................

V. The Petroleum Naphtha “Reforming” Reaction. ......................... VI. Other Polystep Reactions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. The Xylenes-Ethylbeneene Interconversion. ..........................

169 170 175 179 179

............. 189 1. Introduction Reaction mechanisms which involve successive reaction steps, and therefore chemical intermediates, have been discussed in many areas of chemical experience. They are notably familiar in biochemistry, where metabolic and synthesis reactions occur as a result of chains of successive reaction steps. Often, individual steps are catalyzed by various and different enzymes. Similarly, the concept of catalysis of successive reaction steps by different catalytic centers or materials has been suggested early in the history of man-made catalysis-for example, in the case of the Fischer137

138

PAUL B. WEISZ

Tropsch catalyst (see, e.g., ref. 1). On various occasions such cooperative action has been hypothesized (see, e.g., the review by Natta and Rigamonti, 2 ) to explain the performance of solid catalytic materials prepared from more than one chemical ingredient, and presumed not to constitute a single, homogeneous chemical composition (“Mehrstoffkatalysator”). The existence of such action, cooperative thTough the mediation of reaction intermediates, has been largely speculative. Its direct and specific demonstration has been difficult, since other modes of coaction by different chemical components of a catalytic mass can occur, such as, for example, electronic modification of a solid composition by an “impurity”; or, catalysis by the boundary structure existing between two distinct chemical phases. During recent years, studies of a number of hydrocarbon transformations catalyzed by porous solid oxides containing a transition metal, notably platinum, have evolved some concrete examples and demonstrations of truly polystep catalytic reactions. Specifically, these reactions have been shown to be performed by catalysts which contain geometrically separate and different catalyst components, each of which catalyzes separate steps. The chemical intermediates exist as true compounds, although often at undetected concentrations. The term “true” is used in this context to characterize the intermediate as a normal chemical species, existing independently of, and desorbed from, the catalyst phase, and subject to ordinary physical laws of diffusion. From such hydrocarbon reaction studies emerges an understanding of some of the characteristics of such polystep catalytic reactions, and of some of the basic physical requirements which must be fulfilled in order for the purely formal kinetic scheme of successive reactions to be operative in physical reality.

II. Principles of Polystep Catalysis A. SINGLEOR MULTIFUNCTIONAL CATALYSTS If a catalyst mass contains only one type of catalytic site we shall call it a monofunctional catalyst. By one ‘(type” is meant that every catalytic site or surface exhibits the same qualitative catalytic property as to what reaction or reaction steps it can catalyze. We shall concern ourselves only, of course, with reaction steps which are thought to be relevant to the reaction examined. For example, we normally assume that platinum/charcoal is a monofunctional catalyst in the hydrogenation of olefins. (For the present purpose we need not be concerned about the quantitative equivalence of every Pt-surface site, i.e., whether or not there is uniformity or a spectrum of catalytic effectiveness for the same reaction among different platinum sites.)

P OLYFUNCTIONAL HETEROGENEOUS CATALYSIS

139

The term “monofunctional” does not exclude the possibility that the intrinsic activity of a catalytic material may be influenced by chemical contact with a second material. Thus the intrinsic hydrogenation activity of a metal may differ depending on the nature of the support. Such effects may be due to varying degrees of metal dispersion, or due to more profound effects of electronic interaction which modify the electronic properties of the metal. In such cases, although the activity of the metal may depend on the nature of the support, the locus of activity is still at the metal constituent, and we still have a case of monofunctionality. In contrast, we shall see that in a paraffin isomerization system a platinum on silica-alumina catalyst is a multifunctional, specifically, a bifuno tional catalyst; the platinum sites catalyze distinctly different reactions and reaction steps than do the silica-alumina sites; neither catalyze the reactions of the other component; furthermore, both types of reactions are relevant to accomplish the over-all reactions of the desired conversion system . In a polyfunctional catalytic solid, we shall refer to the materials or sites responsible for distinctly different reactions or reaction steps as catalyst components.” ((

B. INTERMEDIATES AND REACTION SEQUENCES 1. De$nition

It is important to recognize the specific meaning of the term “intermediate” in this context. The use of the term will not relate to the concept of “surface complex,” or “activated complex”; for, in this case, at least in heterogeneous catalysis, the catalyst, or a part of it, is structurally combined or, by specific force-fields, is interacting with a reaction-participating molecule. In contrast to this meaning, the term “intermediate” here will refer to a chemical species that is produced by the catalyst as a desorbed, normal chemical species, i.e., one that has its own name, structure, and thermodynamic properties normally associated with independent chemical compounds. 2. Physical Meaning of Reaction Sequences

The concept of reaction intermediates is linked intimately with any picture of reaction kinetics that includes successive reactions, such as A P B e C (scheme I)

where A , 3,C are gas phase (or liquid phase), i.e., desorbed species. A much discussed example is the sequence cyclohexane-cyclohexene-benzene encountered in the dehydrogenation catalysis by chromia-alumina (3).

140

PAUL B. WEISZ

Such a formal presentation as scheme I, which actually symbolically describes transformations between gas phase species, and which are indeed the observables for the investigator, must be recognized to be a simplified model of a more complex situation, A

B

i-Tl--I

-II ---

C

D- 1

A S F ; ? B S ~ C S! (scheme 11)

where AS , BS, and GS are the surface-bonded species, and the dashenclosed area defines a “black box” which contains the actual processes involving the catalytic surface. For a description of the course of reaction of the change A to products B and C , the behavior of this scheme I1 can sometimes be treated satisfactorily by an analogue such as scheme I. This particular analogue will be valid, for example, when the rate-constants of the gas-surface steps are large compared to the rate-constants of interconversion of the adsorbed species. Yet the same “black box” described by scheme I1 can behave in a manner described by the scheme

r--1

A

+I

L--J

I

<

B

, that is c

(scheme 111)

A ’? ks\

C

namely, when equilibration of the reactant A to the various surface species is very fast compared to desorption of products B and C . De Boer and Van der Borg ( 4 ) have shown how a number of cases of scheme I1 are approximated by various gas phase analogues. They point out that a more generally applicable formal analogue is the scheme B ‘4

<
C (scheme IV)

of which schemes such as I or I11 above are special cases. Another article of this volume (5)will discuss more generally and thoroughly the problem of applying kinetic models of free-molecular species and its application to heterogeneous catalysis. For the present discussion it is important to realize that scheme I as well as scheme I11 arise out of a qualitatively identical set of mechanistic steps, as represented by 11. The species B appears in the form of a gas phase “intermediate’, in scheme I primarily because of the applicability of this

POLYFUNCTIONAL HETEROGENEOUS CATALYSIS

141

formal analogue. An examination of the detailed physical situation, scheme 11, however, places some important limitations on the nature of B as a truly necessary “intermediate” : a molecule once adsorbed and reacted to the form BS may desorb but may also react further to CS. The magnitudes of the various rate constants express the relative probability for these events. Molecules of C can arise which never existed in the gas phase as B. The species B is not by necessity a gas phase intermediate. In fact, there is no concrete significance to B being an (‘intermediate” as long as the entities BS which transform to C are chemically indistinguishable as to whether they were created from AS by surface reaction, or from readsorption on S of a gas phase molecule B. This situation exists in the case of a monofunctional catalyst system, where only one type of catalytic site S exists which can form only one type of BS complex. On the other hand, where the conversion steps A to B and B to C are catalyzed by two qualitatively different and distinct catalytic sites SI and SZ,scheme I1 becomes A

B

.

--.

/xi- , \y. ,

i-TL----I A S i + BSi

BSz

L,,--J

C Ma-1

CS,

I

(scheme V)

aFd thus B attains unqualified significance as a true intermediate in a bifunctional catalyst system. 3. The Potential Role of Quasi-Intermediates of Monofunctional Reaction Sequences

There remains an important potential role, however, for the gas phase product B in a monofunctional catalyst reaction sequence: we can profoundly affect the rate of conversion to the monofunctional reaction product C by “operating on” the gas phase B molecules. If we remove B from the gas phase at a fast enough rate we can decrease the steady-state concentration of B , and thus of the species BS which generates C. Thus, although the concentration of B in the usual reaction A + C may be so small as to remain unnoticed or undetectable, the provision of a large enough rate for its removal can effectively divert the previous path of the monofunctional reaction. Such rate of removal of B may be provided by the presence of another catalyst material which removes B by conversion to a new species. Thus, we shall call the product B of the monofunctional reaction a “quasiintermediate” because it has the potential of becoming a true intermediate in a polyfunctional catalysis system. This phenomenon is an important one connected with catalyst selectivity, as will be discussed later.

142

PAUL B. WEISZ

C. TRIVIAL AND NONTRIVIAL POLYSTEP REACTION We shall now examine specifically the nature of the polyfunction reaction sequence involving a true intermediate, where the analogue of scheme I is applicable. If a chemical transformation A + B is known to be accomplished in the presence of catalytic material X , and the conversion B 4 C is known to proceed on catalyst Y , then the reaction X

Y

A+B-+C (scheme VI)

can obviously be accomplished in successive reaction zones : A

+

-m+-pJ+ B

c

(scheme VII)

If high conversion can be achieved for each step under similar catalytic conditions, the two reaction zones may be combined and a basically similar over-all conversion would be expected. Such a case of a polystep chemical reaction can be considered trivial. On the other hand, if the first reaction step is greatly limited in attainable conversion for reasons of thermodynamic equilibrium, i.e., X A

Y

*k i B ka C; ki

+

K

=

k&’

<< 1,

(scheme VIII)

then a consecutive operation of scheme VII above cannot lead to anything more than negligible over-all conversion. Nevertheless, the formal mathematical treatment of the kinetic scheme (VIII) permits any desired net reaction rate and therefore any desired conversion. For example, treating all steps as first-order steps, the maximum concentration of B is 1

[B1=

(l/K)

+ (kz/k,)

and can be arbitrarily small. On the other hand, the rate of the over-all reaction is

and the value of k” can be made arbitrarily large by appropriate choice of the rate constants kl and kz. The magnitude of the attainable conversion

143

POLYFUNCTIONAL HETEROGENEOUS CATALYSIS

can then be greater than the product of the conversions individually attainable in separate successive steps,

EAC

CAC

>> CAB x EBC

(3)

This will serve as a dejnition of the nontrivial polystep reaction. As an example, a hydrocarbon reaction might be carried out in such a manner that the first reaction step be a dehydrogenation to the respective olefin. Such a reaction is characterized by a thermodynamic equilibrium constant for the dehydrogenation step, K

R-CH2-CHz

+ H2

R-CH=CHz

which is shown by the data in Fig. 1for the hexane-hexene equilibrium. EQUILIBRIUM CONVERSION 8 8:1 H,:HC

200 PSIG.1500 6S.l.G. Io i

- 10-a 10%

10-L

- lo-* 105

-lo-* 10.5

-lo-' 10% -lo-' 10-A

-10'' 10% 0.F 1

200

1

1

"

'

300

1

'

1

"

'

400 TEMPERATURE

1

'

500 *C.

FIG.1. Equilibrium constants and some typical equilibrium conversions for the dehydrogenation of n-hexane to hexenes.

144

PAUL B. WEISZ

The maximum attainable conversion of paraffin to a corresponding olefin is obtained from [olefin] [paraffin]

K [H21catm.)

At 435OC the olefin concentration is found to be only about 0.02% at 30 atm. partial pressure of hydrogen. Thus, if we were to carry out paraffin isomerization by successive and separate steps of dehydrogenation of n-paraffin to n-olefin, followed separately by skeletal isomerization of the n-olefin produced to iso-olefin (and subsequent rehydrogenation), the over-all conversion of such a scheme couId be, at best, 0.02%. Thus, the paraffin isomerization, if accomplished in a bifunctional reaction system with a high conversion as might be described by formula (2), is an example of a nontrivial case as defined by (3) above.

D. MASS-TRANSPORT IN POLYSTEP REACTION Why is it that the operation (VII) does not accomplish the high rate of conversion which the formal derivation of (1) and (2) allows for the scheme VI? The over-all reaction rate is limited by the rate of transport of intermediate product B from the generation zone to the sites for re-reaction, and this process is not taken into account in deriving (2). I n the case of successive reaction zones, with a reactant flow rate F , the rate of transport of intermediate will be F E A B , and over-all conversion, therefore, will be limited to this rate. The necessity of molecular transport must obviously arise whenever the reaction sites X and Y are not in geometrically identical locations, and it is precisely this condition that characterizes an important requirement for the multifunctional catalyst. In the case of a catalyst solid that consists of a composite of X-sites and Y-sites in a single reactor zone the physical transport of intermediates between X - and Y-sites must proceed by a diffusion prccess, which then becomes an important and integral link in the chain of reaction events. 1. The Simplest Model of Separate Catalytic Surfaces

Let catalytic X-sites exist on a plane, and catalytic Y-sites on another plane located parallel to the first, at a distance z = L in space. For the consecutively catalyzed reaction scheme VIII, intermediate R molecules must now diffuse from z = 0 to z = L through a medium having diffusivity D. We shall then have (in the steady state; all steps taken as first-order; dN/dt = over-all rate per unit surface area) : For the first reaction step

POLYFUNCTIONAL HETEROGENEOUS CATALYSIS

dN _ - k&l] - k1"B],o; dt

kl K =kl'

145 (4)

For the second reaction step

dN = kz[B],~ dt From the law of diffusion

From these equations we can eliminate [B]*o and [B],-L, and obtain the actual rate as

to be compared with formula (2) for the case of purely chemical kinetics. Thus the effectiveness factor q is

It follows from k" t 00 that there is an absolute ceiling to attainable rate, regardless of how high the effective catalytic rate constant might be, of

(T)

max

=

[A]KD L

=

[BeplD I;

(7)

2. Model of Multifunctional Porous Solid Catalyst Systems In the case of the multifunctional porous catalysts, such as are familiar in hydrocarbon reactions, the situation is somewhat different from that in the model above. In the model above, the diffusion problem is confined to a volume of space where catalytic activities (the sources and sinks) occur only a t its boundaries. I n the present case a volume element of (porous solid) space is permeated by both diffusive resistance as well as distributed catalytic sources or sinks. For this case a convenient model for analysis is pictured in Fig. 2. The catalyst solid consists of a mixture of two distinct types of component particles, one component containing catalytic sites of type X only in its pore structure, the other containing sites of type Y only. This model is one that has been experimentally realized (see Sections 111-V). A mechanical mixture is made of the two catalyst components in powder form, from which the larger granules are composited by pressing or extruding. In such a

146

PAUL B. WEISZ

model we can consider the gas spaces between the component particles us “short circuits,” i.e., offering no diffusional resistance compared to the intraparticle diffusional problem within each component particle : Effective diffusivities in free gas space are one to two orders of magnitude larger than those within high-surface-area solids, and the average interparticle MICROSCOPIC COMPONENT

* PART I CL E s ‘

I

I

CATALYST SOLID(‘PELLET*) A=B--C

~~-

FIG.2. Model of a two-component catalytic solid, consisting of separate particles of catalyst compounds X and Y .

distances are smaller than the particle diameters by nearly an order of magnitude (e.g., compare the dimensions of the voids between packed spheres with the diameters of the spherical particles). The problem of mass transport of the intermediate molecules between X and Y catalyst sites can thus be formulated as an intraparticle diffusion problem, involving separately the particles X and those of Y by conventional analysis. For each of the X- and Y-particle systems the concentration of the intermediate species B has to satisfy a diffusion equation DVZ[Bj

dB * dt =0

(8)

with B-generation (+), and B-consumption (-) within the X - and Ysystems, respectively; in addition, we have two conditions connecting the systems. (1) The steady state rates of generation of B in the X-system and consumption in the Y-system are equal, and are identical with the over-all reaction rate

2= dt

(3, (g) =

-

Y

(9)

(2) There is a common boundary condition for the X - and Y-systems:

at rx = Rx,and r y = RY:[Blx = [B]Y= Bo,

(10)

i.e., the concentration at the particle boundaries is equal and is the concentration in inter-particle space. Let us now examine the polystep reaction of scheme VIII.* We shall assume the kinetics of all steps to be first-order in the respective reactants.

* Component particle sizes and diffusivities will be taken as equal for both component systems.

POLYFUNCTIONAL HETEROGENEOUS CATALYSIS

147

For the Y-system we are dealing with the basic case of the rate of consumption of the reactant B that is the intermediate in the over-all reaction,

which, placed into (8), results in the well known form (references 6-8) for the utilization factor q (which indicates what fraction of the diffusionuninhibited reaction rate is attained). 3

1

where

For the X-system, we must take into account the forward and backward reaction rates,

_ dB - klA - kl'B dt

With

and B,, being the equilibrium concentration of B in A

dt

=2!!

K

$B,

we can write

( [ B e q] [B])

This rate, when substituted into (8), results in a solution for qx of the same functional form as (12Y) with

-

(PX =

R

d&

The functional form of q in each case is such that negligible diffusion effects will result when (P < 1. This condition can also be expressed in terms of measured parameters: For the Y-system, from (13Y), (llY), and (9), d--N 1 R2 < dt [B] D and for the X-system, from (13X) (llX), and (9),

dN

1 R"<1 dt [Beq]- [B]D

148

PAUL B. WEISZ

3. A General Criterion for Polystep Reaction Systems

Regarding the over-all reaction effect, the kinetics (1) and (2) lead to

[B]approaching [Beq]if the first reaction step is already adequately fast, so that (14Y) of the Y-system will be the limiting condition, with B + Beq. On the other hand, the kinetics (1) and (2) leads to [B]<< [Beq]when the first reaction step is limiting, and thus when (14X) is controlling, we have Be, - B + Beq. It follows, in view of (lo), that the expression

applies generally to the system as a whole. The relative insensitivity of this type of diffusion criterion to particle shape and to assumption of exact kinetics, has been discussed in connection with the macroscopic reactant diffusion problem on catalyst granules ( 7 ) . The condition (15) i s a general order-of-magnitude criterion dejining the physical conditions of intimacy between the component systems for no masstransport inhibition. I t defines a requirement for realizing the formal kinetics of polystep reactions. We can rewrite (15) in terms of a partial pressure requirement for the intermediate species, since

where

[Beq]= [mole~/cm.~],T

=

[OK.],

Ps = [atm.]

We obtain

Figure 3 shows typical requirements for the maximum size of catalyst component particles for a typical magnitude (8) of diffusivity (D = 2 X 10-3 cm.2/sec.), and reaction rate ( d N / d t = moles/sec. crn.9, as a function of the equilibrium concentration (expressed as partial pressure) which the chemical intermediate can attain. It is interesting to note that a degree of intimacy of a magnitude which is still realizable by mechanical mixing, such as R = cm. = l p can still support a reaction if a gas phase intermediate can be produced with a partial pressure as low as lo-’ atm. rSuch considerations serve to demonstrate thct polystep reactions m a y easily

POLYFUNCTIONAL HETEROGENEOUS CATALYSIS

149

proceed with intermediates at concentralions jar below the limit of experimental detection. Aside from defining the size of distinct component particles in a mixed catalyst composite, R also defines a maximum allowable magnitude for the

0.I

I

10

too

200

400

1000 jJ

48 24 I2 6 100

MESH SIZE

FIG.3. Intimacy requirement, in terms of component particle size, for typical conditions of reaction rate (10-6 moles/sec cm3),as a function of equilibrium vapor pressure of intermediate.

degree of heterogeneity in a catalyst mass as regards the distribution of regions or patches of catalytic activity of the two types.

E. SELECTIVITY IN THE POLYFUNCTIONAL CATALYST We have noted in Section II,B,3 that a single monofunctional catalyst while generating a desired product may give rise to species which we have termed “quasi-intermediates.” The introduction of a different catalyst component Y may then lead to interception of the usual reaction path, diverting old product C into a new product D:

x x ki

A e B + C ki’

ki

150

PAUL B. WEISZ

becomes kl

x c kar

A e B kt‘

y\ka

D (scheme IX)

1. Changing Catalytic Strength of the Intercepting Catalyst Component

Let us examine a typical and relevant example of such a reaction scheme quantitatively, where (kl/k1’) << 1, i.e., the amount of B produced is very small. It is obvious from an inspection of scheme IX that the rates of production of B and C at any one moment will have the ratio (dC/dt)/(dB/dt) = k&,, and thus the amounts converted to these respective products will stand in that ratio, c ( t ) / D ( t )=

(16)

kz/k3

which determines the selectivity in a simple manner. The total conversion r is easily seen to be e = 0.1 1 24

a too

10

1 - exp ( - ~ E T ) ;

20 50 k,

% TOTAL CONVERSION

FIQ.4. Selectivity dependence (conversions to products C and D)of reaction scheme IX on the magnitude of the rate constant ki; for ki = kz = 4,k{ = 100, 7 = 10.

k

-

(k2

- kl’

+

k3)kl

+ kz +

k3

+

From this, with (16), and with A d t ) = C(t) D(t) (neglecting the very small amount of B produced), we can trace the effect of introducing into a given X-catalyst system the “intercepting” catalyst Y having a rate constant k3. As an example, Fig. 4 shows a plot of the conversion to each product vs. the total conversion, as a function of the magnitude of the k3-activity of Y-catalyst when introduced into the monofunctional catalyst system X , having the constants kl = kz = 4, k,‘ = 100, for a residence time r = 10. Thus the new product can be made in amounts increasing with the magnitude of k3, and constituting an increasing amount of drain on the original reaction.

2. Changing Catalytic Strength of the Intercepted Catalyst Component

Suppose it is desired to make the species D of scheme IX, and a given catalytic activity k3 is available for the intercepting component. If now the

151

POLYFUNCTIONAL HETEROGENEOUS CATALYSIS

catalytic strength of the intercepted catalyst system X is varied, we obtain a noteworthy result for the course of the reaction because the X catalyst component is responsible for both the necessary generating step kl as well as for the competing step k2. Figures 5a and 5b show two examples calculated from the same scheme IX and the formulas above, for k, = 1 and for k3 = 2, respectively; since here the catalytic strength of the entire X-system is varied, we have used

0

i -

100

a01 0.1 I

I

10. I

5.0 I

10.0 I

50.0

k,

I

(b)

0

50 100 % TOTAL CONVERSION

FIQ.5. Selectivity dependence (conversions to products C and D ) of reaction scheme IX on the magnitude of the rate constants of catalyst component X at constant Y-component strengths k3 = 1, and ks = 2; for kl/kl' = 1/100, kz/kl = 1, T

=

10.

the constant ratio kl/k,' = 1/100 and k2/kl = 1, with r = 10, and kl varying as indicated. The conversion to D increases and proceeds through a maximum as the strength of the X-system is increased, and the catalytic selectivity thereafter changes in favor of product C . The amount of D producible at the maximum, as well as the total catalyst conversion a t which it occurs, depend on the catalyst strength Ic3 of Y that is available (compare Figs. 5a and 5b).

152

PAUL B. WEISZ

3. “Coupling” through a Side Product It is important to recall (see Section II,B,P and II,B,3) the limitation we must ascribe to the meaning of “quasi-intermediates” as to their being true intermediates in the monofunctional X-system. We must recognize that similar “coupling” of a second catalytic system ( Y ) to a previously monofunctional system can be attained where the latter system can be pictured as one generating a low level side-product ki

k1

A+C&B kr’

x x (scheme X)

The situation may arise reasonably often that, for reasons of an unfavorable thermodynamic equilibrium (kz/kz’ << 1), B is produced at a very small concentration level which may remain undetected, or whose potentially profound significance is not easily suspected. Yet the magnitude of the individual rate constants kz and kZI may well be so great that the reaction can be made to proceed rapidly through B to a new product if a “sink” in the form of an adequately large rate constant for the consumption of B is provided by a new catalyst component; kr

ki

ka

A-tCeB-D kr’

X X Y (scheme XI)

In fact, if the rate constants kz and kz‘ are adequately fast the product B will at any time be a constant fraction of the material C and the behavior of this system will be easily recognized as equivalent to that of ki

K2ka

A-C+ D X Y (scheme XII)

where Kz is the equilibrium constant Kz = kz/kz‘ in scheme XI. If, therefore, for a given monofunctional catalyst subject to scheme X with certain kl, a Y-component is added, draining of C into the new product D can occur in the manner of a consecutive reaction (scheme XII) for which kinetic behavior has been variously analyzed (e.g., in ref. 10). For example, the product compositions of C and D for kl = 0.2, r = 10, and variable (Kzk3)are plotted in Fig. 6.

POLYFUNCTIONAL HETEROGENEOUS CATALYSIS

153

W

w[L a

I

V

z I-

50

V 3

8 rr a z

0

I.o

0.I

10

K,k,/k,

-+

FIG.6. Relative product variation (C and D)in consecutive reaction of scheme XI1 when the effective second step rate constant (Kzks) changes; kl = 0.2, z = 10.

T h u s a low-level “side product” m a y play a role in allowing polystep and polyjunctional catalysis to arise, in that this side-product possesses the potential properties of a quasi-intermediate.

4. Applicability of the D i f u s i o n Criteria The main difference between the cases discussed in this section on selectivity and the previously discussed case of a simple polystep reaction resides in the fact that here the single component which generates the species that becomes the intermediate in the polyfunctional composite can itself generate a distinct product species with appreciable yield. Since the coupling between the Y- and the X-system occurs in any event through mass-transport of intermediates between X-sites and Y-sites, the diffusion criteria already discussed must apply or the kinetic schemes which accomplish “interception” or selectivity control will not be physically and effectively accomplished. The criterion, of formula (15) should be satisfied.*

F. THERMODYNAMICS OF THE POLYSTEP RATEPROCESS 1. Obtaining a n Activity Sequence The intimacy criterion above (15, 15A) involves physical parameters, a reaction rate, and a quantity based on thermodynamic equilibrium. It thus presents a link between rate process variables and thermodynamics.

* Note that circumstances may arise where the intimacy requirement is more stringent than expressed by (15); namely, when kz > kl‘ in scheme IX. Then, (15) should contain the maximum possible concentration B,,, in place of the equilibrium concentration Be,, that is B,, = A[kl/(kl’ k 2 ) ] ,since B,, < Be, = A(kl/k1’). In such a case (15) is a minimal requirement.

+

154

PAUL B. WEISZ

It will be instructive therefore to discuss the polystep kinetics in the light of thermodynamics. For the successive species involved in a polystep conversion, we can examine the successive changes in free energy as we pass from reactants to products. Figure 7 shows this by a plot of positions of the standard free

I

C

t A Ff

Fro. 7. Thermodynamics of the (a) nontrivial and of the (b) trivial polystep reaction.

energies of formation of the species of each step in the two-step reaction, scheme VIII. Of course, the over-all change AFA+c must be favorable, i.e., in most cases negative. The requirement for nontriviality, as discussed in Section IV, translates itself into the requirement that for the nontrivial case, A F f q Bmust lie above AF,,A, as shown by Fig. 7a, while Fig. 7b represents the trivial consecutive reaction. Thus, the step AFA+B, to be nontrivial, must be positive; and the intimacy criterion (15, 15A) indicates how positive it may be for certain physically attainable conditions. For a simple monomolecular transformation, as the intermediate step, we have

P B ,= ~ P A exp ( - A F A + B / R T ) and therefore (15A) becomes

POLYFUNCTIONAL HETEROGENEOUS CATALYSIS

155

It is interesting to plot the magnitude of AF for the “initiating” reaction which is still permissible for certain reasonable size parameters R of a t301

’

,

I

I

I

I

FIG.8. Maximum permissible AF for the initiating reaction ( A + B ) which produces the intermediate.

heterogeneous polyfunctional catalyst system. Figure 8 shows such a plot, for the typical physical magnitudes as employed previously above.

2. Obtaining Selectivity In the case of “interception” (scheme IX), a monofunctional reaction A to B to C is diverted to a new product D by provision of a new reaction path through action of an additional catalyst component. The thermodynamics of the situation is represented by Fig. 9. The figure is self-explana-

FIG.9. Thermodynamics of selectivity in polystep reactions; compare with scheme IX.

tory, but it is particularly noteworthy that in a finite contact operation the product D m a y be made at the expense of C even though the thermodynamic properties are such that C i s more stable than D , i.e., even though the direct equilibrium C D would not enable a n y appreciable conversion to occur f r o m C to D. This is a very useful consideration as it enables one to show that, ~

156

PAUL B. WEISZ

upon introduction of the second catalyst component, the occurrence of the new product D could not have arisen through further conversion of the previous product, in the order A --+ C ---f D.

111. The Technique of Physically Mixed Catalyst Components The simplest direct method for testing the cooperative action of catalyst components consists of a comparison of conversion results between experiments where only catalyst particles of type X , only of type Y , and where a loose mixture of the same amounts of X and Y particles have been placed into the reaction zone, under otherwise similar contact conditions. Polystep action is then indicated if the extent of reaction in the latter case is seen to exceed the sum of the conversions in the two single component contact runs. In a given experiment the observed effect may, of course, be immeasurably small due to diffusion effects discussed in Section I1,D. In that case, a sufficient lowering of the component particle size may uncover the effect. If the effect can be thus demonstrated to exist, a siudy of the dependence of catalytic eflectiveness on the component particle size can be used to yield information concerning the magnitude of vapor pressure of the intermediates, in accordance with Section II,D,3. In an ordinary static bed reactor vessel particles with diameters down to about 5 X low2cm. = 500p can often be used without excessive plugging or pressure drop. For smaller size component particles static bed operation can be achieved by forming larger pellets from the mechanical component mixtures. Conventional pressure-pelletizing or extrusion techniques can be used. In the author’s researches, pressure pelletizing the mechanical mixture in the dry state without added binding agents has been employed to avoid aqueous ionic migration of materials, or interaction with third component materials. While such a procedure may result for some materials, in pellets which are relatively weak mechanically, it is usually adequate enough for research experimentation where no large demands exist for mechanical strength. For porous oxide particles a size-range down to about 60p diameter can be obtained by conventional crushing and grinding techniques. Smaller particle sizes are obtained by conventional ball-milling, thus reaching a size range of the order of 1-5p. Thorough mixing of components before pelletizing is, of course, an important prerequisite, since otherwise the size of heterogeneous domains becomes subject to the diffusion criterion (15) (see Section 11,D13).Such mixing can be achieved by co-ball-milling, or by use of a Waring blender. In research preparations the latter method was preferred in order to preserve strictly mechanical intermingling and minimize spot heating at interfaces. Homogeneity was checked under an optical microscope,

POLYFUNCTIONAL HETEROGENEOUS CATALYSIS

157

In studies of dual-functional catalysis the mixed catalyst technique has many advantages, two of which are mentioned. (1) It allows separate and independent preparation of each component; for example, a platinum preparation can be made in any manner desired in order to obtain a certain platinum activity without regard to what such procedures might do to the acidic properties of the oxide base, this interdependence always being a matter of concern in conventional direct impregnation techniques. (2) A component’s relative activity contribution can be flexibly varied in a perfectly known and controllable manner by simply varying its bulk amount in admixture with the other.

IV. Some Major Polystep Reactions of Hydrocarbons A number of hydrocarbon transformations have been shown by Haensel and Donaldson ( I O ) , Heinemann et al. ( I I ) , and by Ciapetta and Hunter (12) to be catalyzed by solid catalysts in which a transition metal, notably platinum or nickel, has been combined with an “acidic” oxide carrier substance such as, for example, silica-alumina, or halogen containing alumina. These include the isomerization of paraffins, the hydroisomerization of naphthene rings, and the hydrogenative cracking of paraffis.

.I c-hexone c C

0 .+ 0

0)

m

e

R

I I

c-hexene

me-c-Dentone

hexone

iso-hexanes

I

I

- hexene - iso-hexenes

I -me-c-pentene

c-hexodiene

I

benzene

r

isornerizotion function

FIG.10. Formal reaction path representation for dual functional Ca-hydrocarbon conversion, after Mills et al. (13).

Mills e2 al. ( I S ) proposed formal reaction schemes in which conversion proceeds through olefinic reaction intermediates, such as the one illustrated by Fig. 10 for the Cs-hydrocarbons.* They introduced the concept of separate catalytic functions, in terms of an isomerization activity-operative on the olefinic intermediate-associated with the acidic oxide base and a hydrogenation activity associated with platinum. Ciapetta and Hunter

-

* We do not consider the particular link between the cyclic and the aliphatic structures (rnethylcyclopentane hexene) as properly presented by this diagram. We reproduce i t in accordance with the authors’ picture since this reaction step is not-at any rate-involved in the reactions here discussed.

158

PAUL B. WEISZ

(12) in their work on paraffin isomerization proposed that “the isomerization activity of these catalysts is due to compound formation between the nickel and hydrous aluminum silicate,” thus forming a special active catalyst “complex.” Mills et al. (13) proposed that (see Fig. 10) “in order to travel to a compound in a diagonal position, it is necessary to transfer on the catalyst surface from one site to another.” Thus the concept of the “dual functional” hydrocarbon conversion catalyst with distinctly different sites became clearly introduced. In the author’s laboratory, extensive studies were undertaken not only to examine the reaction schemes proposed by Mills et al., but specifically to test the feasibility of catalytic cooperation by chemically unconnected, i.e. , physically separate, catalytic components, wherein the inlermediates are true gas phase species coupling the catalyst components through mass transport jollowing the classical laws of gaseous diffusion, in line with the principles and characteristics discussed in the preceding sections. Experimental work that makes use of physically distinct catalytic materials or components constitutes the most direct route to the testing and study of true polystep reaction mechanisms. The feasibility of coupling through the diffusion process was briefly reported by Weisz (14). Experimental evidence of the cooperative action of catalyst mixtures was mentioned by Mills (15) and Weisz (16) at the First International Congress on Catalysis, and brief reports have appeared by Weisz and Swegler (I?‘), Hindin et al. ( I S ) , and Weisz (19).

A. REACTIVITY FOR ISOMERIZATION OF PARAFFINS The reaction* X

n-paraffin $ n-olefin

Y

X

e iso-olefin F!iso-paraffin

has been studied using the mixed catalyst technique. The “acidic” solid catalysts, such as silica-alumina are very active for the conversion of olejins to skeletal isomers. Evidence for such high reactivity, at relatively low temperatures, can be found, for example, in the early work of Egloff et al. (go), and of Greensfelder and Voge (21). Over conventional silica-alumina cracking catalyst (422 m.2/g. surface area, 11% A1203) such as has been used in the author’s laboratory in some of the mixed particle researches to be described below, contacting 1-hexene with a residence time of 3.5 sec. at atmospheric pressure leads to 43% conversion to skeletal isomers at 300°C (17).

* In this and following reaction schemes participation of hydrogen will be silently assumed and not explicitly shown. The schemes trace the course of the hydrocarbon structures.

POLY FUNCTIONAL H E T E R O G E N E O U S CATALYSIS

159

With such a catalyst available for thc olcfin isomerization step one may hope to find that mechanical combination with a (de-) hydrogenative catalyst, an over-all paraffin isomerization may be accomplished. 1. Simple Demonstration Tests with n-Hexane

The isomerization of mhexane is indeed demonstrable by simply showing the additivity effect of fairly large particle mixtures “poured” into the reactor, even in atmospheric pressure operation with a particle size of 0.8-1.5 mm. diameters of a platinum-bearing material (X) and silica-alumina ( Y ) . In a typical example, the X-catalyst is obtained by aqueous chloroplatinic acid impregnation of a pure silica (obtained from hydrolysis of tetraethylsilicate), drying and calcination in air a t 450°C for 1 hr., and bearing 0.55 wt. % of platinum; the Y-catalyst is the commercial cracking catalyst described above. Table I below illustrates the conversion to isohexanes observed when the reactor is charged with either of the two catalyst components alone, or with a loose mixture of both. The operation takes place at 373”C, at a feed-rate of 17.2 g. of n-hexane per hour, with hydrogen at a 5 : 1 molar ratio of hydrogen to n-hexane. TABLE I Polystep Hexane Isomerization on Coarse Catalyst Mixtures Wt. % Conversion to iso-hexanes

Catalyst charge to reactor 10 0111.3 of silica/Pt ( X ) 10 0111.3 of silica-alumina ( Y ) Mixture of 10 ~ mof. X ~and 10 of Y

0.9 0.3

6.8

The successful interaction of the two catalyst components is apparent from these results of a simple ‘Lpoured”and loose particle mixture. n-olefin step (see Fig. 1) indiThe thermodynamics of the n-paraffin cates a maximum attainable first step conversion of between 0.04 and 0.6%, depending on whether dehydrogenation occurs to predominantly one double-bond isomer like 1-hexene, as one extreme case, or possibly all double bond isomers a t full equilibrium concentration, as the other. Thus the observed action of polyfunctional coaction is not trivial, by the definition of Section II,C, and represents a demonstration which in view of its simplicity, would even lend itself-like others to be discussed below-to classroom demonstration, since product spectra can now be displayed essentially immediately with simple vapor chromatographic techniques. The criterion (15) can be applied to show that the rate of paraffin isomerization demonstrated by the simple experiment above represents a ~

160

PAUL B. WEISZ

heavily mass-transport inhibited rate, far below that which could be potentially attained at smaller component particle size. From the observed conversion (Table I) and the n-hexane feed-rate, the rate per-unit-volume of catalyst component calculate to be diV/dt = 0.5 X moles/se~./cm.~. The maximum possible concentration of olefin intermediates was stated above to be 0.6% of the paraffin vapor pressure and is thus found to be The effective diffusivity of the about [Bes]= 2.4 X 1 0 P mole~/cm.~. silica-alumina material used in the author’s laboratory has been determined independently by methods previously described (8, SZ), and is Deff= ~ m . ~ / s e cThus, . with an average particle size of 2R = 6 X 2X cm., one obtains

i.e., a number far above unity and which we call *. As pointed out by Weisz and Prater (ref. 22, p. 167), it is possible to obtain the magnitude of the catalyst effectiveness factor q from the magnitude of @. In fact, when CP >> 1, one can show that q = l/*, so that we get q = 0.03 for the experiment, i.e., only 3% of possible reactivity was realized. (This is an upper limit since the maximum olefin vapor pressure including all theoretically possible double-bond isomers was assumed .)

2. Demonstralion of the Intimacy Requirement with n-Heptane The applicability of the intimacy criterion has been demonstrated (19) in a series of tests on n-heptane isomerization under conventional operating conditions, i.e., at elevated hydrogen partial pressure where catalyst deactivation is minimized. The reaction was examined over mechanically distinct but mixed particles of X (Pt-bearing particles) and Y (silica-alumina) of varying particle size R (equal for both types of particles) in 50-50 volume proportion as well as over single type catalyst in the reaction zone. The reaction conditions provided a partial pressure of n-heptane of 2.5 atm., of hydrogen of 20 atm., and a residence time of 17 sec. Figure 11 shows conversion to iso-heptanes to be negligible for (0.5 wt. yo)platinum supported on activated carbon (Pt/C) as the only catalyst, and also for (0.4 wt. %) platinum on silica-gel (Pt/SiOz). No detectable conversion was obtained with silica-alumina. A mechanical mixture of either of the Pt-bearing particles with silica-alumina of about 150 rn.”/g. surface area, both in millimeter diameter particle size (1OOOp), immediately resulted in appreciable isomerization (@ SiAl with Pt/C; @ SiAl with Ft/SiOn). I somerization increases rapidly for smaller component particle sizes, of 70p and 5 p diameters. It approaches the performance of a silicaalumina that has been directly impregnated with platinum, and which has

161

POLYFUNCTIONAL HETEROGENEOUS CATALYSIS

the same total silica-alumina surface area in the reactor (a).A plot of isoheptane production vs. component particle size, at 468"C, Fig. l l b shows the close approach to diffusion-uninhibited performance for component particle sizes below 1 0 0 ~ . The conversions in this operation approach equilibrium conversion( at least to the mono-methyl hexanes, which are the major products in dual (b)

(a) 60

468 OC

LIMIT FROM IMPREGNATED, CATALYST

-

v)

w

f

40

I-

n

w

=d\t

2

5 G

I

20

3

IOOOp

--------

0

<

5

475 TEMPERATURE ("C)

450

5-J

I I I I * I 10 100 IOOOp COMPONENT PARTICLE S I Z E (2R)

FIG.11. Isomerization of n-heptane over mixed component catalyst, for varying size of the component particles: (a) conversion vs. temperature; (b) conversion at 468°C vs. component particle diameter.

functional catalytic isomerization), near 470°C. Calculated estimates of particle size from the intimacy requirement can be made, but must take into consideration the approach to equilibrium in the range of operating conversion. For an integral reactor, conventional derivations-for first-order reactions-lead to an expression for (dN/dt)/(l/[A]) anywhere along the reactor, with [A] being the reactant concentration dN _ 1 _

dt [A1

= 1 / In ~

-1-

1 E/EW

where E and ceq are the observed and the equilibrium conversions of the over-all reaction. Since we can write, for the intimacy criterion (15).

I

162

PAUL B. WEISZ

we can also write the criterion [with (IS)] as

With this we can calculate the requirements at, say, 90% approach to equilibrium, corresponding to 40% conversion to iso-heptanes in Fig. 11. The ratio [A]/[BCq]calculated from the thermodynamics of the paraffinolefin equilibrium calculates to between 1 and 4 X lo3. We have 7 = 17 sec., D = 2 X cm.2/sec., e/ceq = 0.9, from which we obtain

= (7

. . . 30) x 104

-

~2

and thus for CP = 1, we obtain the size magnitude R 20 . . . 40p, in good agreement with the observations. The nature of the intimacy requirement is thus well demonstrated experimentally.

B. REACTIVITY FOR HYDROCRACKING OF PARAFFINS A side reaction of paraffin isomerization is that of hydrogenative cracking to lower molecular weight paraffins. Besides hydrogenolysis of hydrocarbons, which we visualize as occurring on metal catalyst sites alone, polystep hydrocracking according to the scheme exemplified by X

hexadecane

Y

X

hexadecenes + 2 octenes (scheme XIII)

2 octanes

can be shown to proceed readily when both Pt-sites ( X ) and acidic sites ( Y )such as of silica-alumina can coact via vapor phase diffusion, in a polystep reaction sequence. Heinemann et al. (il) showed that the hydrogenative cracking of heptane on a dual-functional catalyst leads to butane as a predominant product indicating prevalence of “center-cracking,” which recalls an acid-catalyzed cracking activity. Myers and Munns (5%) presented evidence for the existence of the dual-functional reaction path for hydrocracking leading to center-cracked products, and for a second and distinct mechanism of hydrogenolysis involving the metal sites alone, the latter leading to bond rupture probabilities equally large towards the ends of n-paraffins. From comparisons on n-pentane, n-hexane, and n-heptane, they indicated that the dual-functional hydrocracking mechanism becomes increasingly important with greater molecular weight of the paraffin. 1. Demonstration of the Hydrocracking of Parafins

Work by Weisz and Swegler has shown the dual-functional (scheme XIII) hydrocracking activity to increase so rapidly with molecular weight,, that

163

POLYFUNCTIONAL HETEROGENEOUS CATALYSIS

with n-ClzHza,or n-C16H34, a striking and simple demonstration of the dual-functional hydrocracking mechanism can be made in a simple atmospheric pressure reactor. The method is analogous to that used in the demonstration of the nature of n-hexane isomerization, involving tests with single catalyst components, and with a loosely poured mixture of both components. The components are identical to those used in the n-hexane experiments. The measurements are made at 370°C reactor temperature, at partial pressures of 0.25 atm. of hydrocarbon and 0.75 atm. of hydrogen and 3 sec. residence time. Table I1 below shows conversions observed with only silica-alumina, with only platinum on activated carbon, and with a 50/50 volume mixture of both particles simply poured together into the reactor space. Reactivity of high molecular weight olefin (intermediate) cracking is so high that a high degree of conversion is attainable through vapor phase component interaction, even with the high degree of diffusion inhibition in relatively large component particles. TABLE I1 Polystep Hydrocracking of High Molecular Weight Para-ffinson Coarse Mechanical Catalust Mixtures Conversion Catalyst

~

n-Cd& (to 175°C. E.P:) ~

Silica-alumina Pt-carbon Pt-carbon Silica-alumina

+

a

(%I

Particle size (mm.) 0.8-1.4 0.8-1.4 0.8-1.4

~

n-Cl& (to 275°C. E.P:)

~

2.7 4.0 13.2

2.1 2.5 36.5 -

E.P. = distillation end-point to define products.

Higher reactivities are attained by decreasing the component particle size. Such mixed catalysts were made by mechanically mixing 50 to 1OOp particles of two components: Pure silica bearing 0.7 wt. % of platinum, and 420 m.2/g. silica-alumina cracking catalyst, and compressing to 3/16" X 3/16" cylindrical pellets. Under operating conditions identical to the tests above, but at still lower temperature, 358"C., n-dodecane gave the conversions (to products boiling below 175°C.) of Table 111, in seven successive runs each of 75 min. duration (each interrupted by one-half hour hydrogen flow). The high reactivity of an acidic solid such as the silica-alumina component for the cracking of the high molecular weight ole& has been indicated by previous work (20,21). It was demonstrated for the particular conditions of our work (19), when dodecene-1 was passed over the silica-alumina

164

PAUL B. WEISZ

TABLE I11 Hydrocracking n-CtzHze over Mixed (SilicalPt and Silica-alumina) Catalyst at 60 to l o o p Component Particle Size (r = 3 sec., 358°C.) Run

Conversion (wt. %)

1 2 3 4 5 6 7"

47 47 43 46 42 41 46

2 hrs. hydrogen flow a t 480°C. prior to run 7.

alone, a t atmospheric pressure, 3 sec. residence time, and 330°C.) with the results in Table IV below. Furthermore, the molecular weight distributions of the products from the cracking of dodecene-1 and from the hydrocracking dodecane are characteristically similar. Figure 12 shows approximate carbon number distributions obtained from a run of dodecene-1 cracking over silica-alumina (left TABLE IV n-Dodecene Conversion over Silica-alumina Catalyst (mz../g.)

Conversion (yo) (to 175°C. E.P..)

90 196 420

18 36 47

Catalyst surface area

a

E.P.

=

(T

=

3 sec.) 330°C.) Gas production (wt. %)

3.5

distillation end-point to define products.

side of Fig. 12) and from the average of three hydrocracking runs of dodesilica-alumina) mixed catalyst (right side of cane over the (Pt/carbon Fig. 12). The product distributions are seen to have the same general character. Thus these relatively simple experiments demonstrate some basic features of dual-functional hydrocracking. Acidic catalyst, like silica-alumina exhibits a very high activity for the cracking of high molecular weight olejins. The corresponding paraffins can be cracked under similar conditions by providing the olefin as intermediate product through the action of a separate but diffusionally coupled dehydrogenation catalyst component, such as, for example, platinum.

+

POLYFUNCTIONAL HETEROGENEOUS CATALYSIS

CRACKING OF D O D E C X - I

165

V-HYDROCRACKING OF DOMCAA

I-

0

3

n

20 0 v)

W

-1

0

s w

2

10

I-

4W

K

.o

1 7 ,

11

L

I 2 3 4 5 6 7 8 9 10

I 2 3 4 5 6 7 8 9 10

CARBON NUMBER OF PRODUCT

FIG.12. Relative distribution of carbon number in cracked products from cracking of dodecene-1, and from hydrocracking dodecane.

2. Hydrocracking and Hydrogenolysis The existence of a mode of hydrogenative cracking other than via the dual functional route, for the light hydrocarbons, as was indicated by Myers and Munns (23),can be demonstrated and substantiated by the use of the mixed catalyst technique. By uskg mechanical mixtures of varying proportion of the two catalyst component materials, the relative catalytic strengths can be varied without any danger of changing other qualitative properties of the catalytic sites. Such a study was undertaken with n-heptane under operating conditions identical to those described for the n-heptane isomerization study in Section IV,B,2 above. The (de-)hydrogenation component consisted of Alcoa F-10 alumina impregnated with 0.4 wt. % of platinum and subsequently steamtreated to remove halogen contents. The acidic component was supplied by 150 m.z/g. silica-alumina cracking catalyst. The components were of 5p size, blended, and pelleted in three different volume proportions. Figure 13 shows the results for the ratios of platinum-component to acidcomponent of 25/75, 50/50, and 75/25. It will be noted that the activity for isomerization decreases in that order, i.e., with decreasing acidic component (in spite of increasing Pt-component). Similarly the majority, at least, of cracking to C3 and C4 paraffins (the “center-cracking” products) decreases similarly with decreasing acidic component. This is consistent with the picture that both reactions are rate controlled by the silica-alumina component. However, the converse is true for the appearance of C1 and CZ products, indicating that a major portion of these products is obtained

166

PAUL B. WEISZ

through a different reaction route in which the platinumlalumina component plays the major and rate controlling role. v)

w

s

40

8

20

0 W

/75/25

8 I

-

I

0.8

I-G z +

2

0.4

4, a LL

0

0

W J

RATIO IS Pt-CATALYST VOLUME/Si/AI VOLUME

0

i

0.4

:-*

75/25

J ” O C

I -

0

0.2

01 4; i

450 4i5 TEMPERATURE (OC)

500

FIG. 13. n-Heptane isomerization and hydrogenative cracking over mixed catalysts of varying proportion of platinum and acidic component.

These results and the indications described by Myers and Munns (23) are consistent, therefore, with the following diagram of reaction paths for paraffin hydrocarbons on metal-acidic site dual functional catalysts : Yl

XI

n-paraffin s n-olefin + iso-olefin dXa

YzL

dYa

+ iso-paraffin XI

XaL

cracked paraffins cracked paraffins cracked paraffins (“hydrogenolysis”) (“hydrocracking”) (“hydrogenolysis”) (scheme XIV)

In this scheme, X represents metal sites, and Y represents “acidic” sites. It is suggested that the term “hydrogenolysis” be adopted uniformly to apply to hydrogenative cracking involving the metallic sites alone, while the term “hydrocracking” be used for the dual-functional catalytic conversion. Such terminology would be helpful since the catalytic processes differ both in mechanism as well as nature of products. Hydrogenolysis

POLYFUNCTIONAL HETEROGENEOUS CATALYSIS

167

leads to relatively indiscriminate bond breakage, as compared to preferred breakage of centralized bonds in hydrocracking. When catalysts are provided with acidic activities comparable to those of active conventional cracking catalysts, hydrocracking can be obtained a t a level far above hydrogenolysis, for paraffins of carbon number n much greater than seven. IN POLYSTEP PARAFFIN REACTIONS C. SELECTIVITY

1. Isomerization Selectivity

If we view broadly the past and present work cited and reported above, we can summarize certain observations that lead to an understanding of some of the factors involved in catalytic selectivity of para& conversion. Scheme XIV shows the reaction path of dual functional isomerization with its side-reactions which detract from perfect selectivity. For a given high activity of the acidic component Y, the temperature requirement for (intermediate) olefin isomerization activity drops rapidly as we go from low carbon number to high carbon number paraffins. This is pictured purely qualitatively in Fig. 14. A typical metal component such as platinum-

-

I

OLEFIN CRACKING

W

OLEFIN a

2

w

O IL

I

l

i

l

4

l

8

l

l

12

CARBON NUMBER

l

l

l

16

FIG.14. Qualitative sketch of reactivities for olefin reactions (on acidic catalyst) and paraffin hydrogenolysis (on platinum metal) vs. carbon number of hydrocarbon.

alumina exhibits hydrogenolysis activity at a temperature level which depends on the catalyst preparation but does not vary greatly with n-paraffin size. Consequently, for the isomerization of the lighter paraffin, the temperature requirements (forced upon us by the Y-component) force us to operate under conditions where hydrogenolysis becomes prevalent. Thus for light parafin isomerization, the selectivity depends importantly on the ratio of rate constants involved in steps Y l and X,; i.e., >> 1 is desired. This implies that relative activities of the two independent catalyst components are involved, and may be controllable in the manner discussed in Section II,E,P. One may reduce kx, by decreasing X-component strength to increase said ratio. This reduction of strength should

168

PAUL B. WEISZ

be useful until the generating rate constant kx, (due to the same catalyst sites) becomes inadequately small. This is aptly demonstrated by a laboratory experiment on n-hexane isomerization, by the use of a mixed composite of platinum and acidic catalyst of very high initial platinum activity, which is subsequently progressively lowered by contacting with the catalyst increasing amounts of hydrogen sulfide which lowers platinum activity (XI and X,) alone. Figure 15 shows the changes in effluent product

0

2

3

4

DEACTIVATION OF X- COMPONENT

FIG.15. Observed variation of n-hexane isomerisation selectivity with degree of activity of the X-component in a platinum (X) acidic oxide ( Y ) catalyst.

distribution, as 0, 2, 3, and 4 total units of hydrogen sulfide have been contacted with the catalyst, operating at 373"C., at approximately 2.3 sec. residence time, 0.9 atm. of hydrogen and 0.1 atm. of n-hexane partial pressure. Essentially all hydrogenative cracking is seen to have resulted from hydrogenolysis on Pt-sites, and is progessively suppressed by lowering the platinum rate constants with hydrogen sulfide, until further lowering results in quenching of the olefin producing generating step of the dual functional isomerization. Experimendal details. The reaction is carried out in a microcatalytic reactor substantially as described by Kokes et al. (24). The catalyst consists of 2.8 g. of a pelleted 1 : l volume mixture of particles of Alcoa F-10 alumina impregnated with 0.6 wt. % of platinum and of 420 m."g. silicaalumina cracking catalyst, maintained in hydrogen at a flow rate of 60 cm.3 (S.T.P.)/min. A pulse of 2 ~ m (S.T.P.) . ~ of a 1O:l mixture of hydrogenln-hexane is inserted into the flowing gas stream, passed over the catalyst and through a vapor chromatographic column to record the product spectrum. Metered amounts of gaseous hydrogen sulfide, in units of 1/64 ~111.~ (S.T.P.), are introduced into the hydrogen stream passing to the catalyst, in between successive activity tests, for progressive deactiv-

POLYFUNCTIONAL HETEROGENEOUS CATALYSIS

169

ation of platinum activity. Figure 15 indicates the cumulative number of such units of HzS placed on the catalyst. The pulse technique provides very brief contacts of the catalyst with hydrocarbon charge and allows measurements to be made a t the very high “fresh” platinum activities. As we consider isomerization of paraffins with larger carbon number, the olefin cracking reactions, on Yz, become of increasing importance relative to the hydrogenolysis reaction Xz (see Fig. 14), and the selectivity problem becomes more complex: In addition to the control of k y , / k x , to be sufficiently large, the intrinsic ratio ky,/ky, should be large. The latter is an intrinsic characteristic of the acidic catalyst material and this therefore represents a problem different and independent from the problem of balancing catalyst component strengths. 2. Hydrocracking Selectivity

When the molecular weight of high carbon number paraffins is to be lowered by hydrocracking, with a minimum of light gas production, there also are encountered two types of selectivity problems. Catalyst acidity and operating conditions are now to be chosen (see scheme XIV) to maximize the Yz reactions as leading to the desired products, which generally implies that the Y 1reactions are still faster (see Fig. 14) and tend to push the isomer distribution toward equilibrium. One problem is again a matter of relative component strengths in that now ky,/kx, is to be sufficiently large. After this, the remaining product selectivity problem is a more subtle one relating to the statistics of position of the bond ruptures in the Y2-processwhich determines the heavy to light product ratios of this intrinsic process (e.g., as illustrated by Fig. 12).

D. REACTIVITY FOR CYCLOHEXANE The conversion of cyclohexanes to aromatics is a classical dehydrogenation reaction which will readily take place on many transition metals and metal oxides. On chromia-alumina Herington and Rideal (3) have demonstrated the occurrence of cyclo-olefin intermediate products. Weisz and Swegler (95) have demonstrated the effect on benzene yield of allowing early diffusional escape of cyclo-olefin from the porous catalyst particle. Prater et al. (26) have developed evidence that cyclohexene occurs as a quasi-intermediate in aromatization catalysis over platinum catalyst also, although a t a smaller concentration, because of a larger ratio of effective rate constants kz/kl in the scheme cyclohexane

kl ka + cyclohexene + benzene ki’

for platinum catalysis compared to chromia catalysis.

170

PAUL B. WEISZ

Cyclo-olefins are subject to structural isomerization in contact with acidic catalysts, as Bloch and Thomas (27) and Greensfelder and Voge (21) have shown. Therefore, such catalytic activity when intimately coupled to the aromatization reaction may direct the reaction path to products having five-carbon ring structures which cannot aromatize : Xl xa cyclohexane e cyclohexene -+ benzene

xa

D y

methylcyclopentane S methylcyclopentenes (scheme XV)

The quasi-intermediate cyclo-olefin of the monofunctional dehydrogenation reaction thus becomes a true intermediate in the dual functional naphthene ring isomerization, by a process of interception, as discussed in Sections II,B,3 and II,E,l. Conditions for this interception, leading to methylcyclopentane a t the expense of benzene, must include a sufficiently large ratio of the rate constants Icylkx,, and favorable thermodynamic conditions. These can be realized, for example, at 450°C. under hydrogen pressure conditions such as 20 atm. of hydrogen and 5 atm. hydrocarbon partial pressure. In Table V below are shown the results from converting cyclohexane under these conditions with platinum catalyst containing the acidic component in increasing intimacy. Catalysts B, C, and D are the mechanical mixture catalysts with 1000~and loop component particle size of the n-heptane isomerization study above (see Section IV,A,2). Catalysts A and E are the platinum component alone (Pt/SiOz) and the directly impregnated dual functional catalyst from that study, respectively. TABLE V Diversion of Cyclohexane -+ Benzene Reaction to Methylcyclopentane Formation ~~~

~

Products per 100 parts cyclohexane charged

A B C

D E

Catalyst

Benzene

Pt component alone IOOOp mixture lOOp mixture 5 p mixture impregnated catalyst

85 84 57 59 40

Methylcyclopentane

C-hexane (unconverted)

1.5 6 20

9 8.5

23

8.5

40

7

7.5

E. AROMATIZATION OF ALKYLCYCLOPENTANES This reaction is easily shown to follow the polystep reaction mechanism via diffusingintermediates. Actually the reaction path is already contained

171

POLYFUNCTIONAL HETEROGENEOUS CATALYSIS

in scheme XV, wherein the reactant now is the methylcyclopentane and it is desired to selectively follow the path to benzene. A study of this reaction is interesting for several reasons. By a choice of experimental conditions (temperature and pressures) one can create a transition from a case of a “trivial” to a case of a “nontrivial” polystep reaction. While operating under the former condition, one can obtain the olefinic reaction species in sufficiently large concentrations to demonstrate their identity. Also, phenomena related to catalyst selectivity, i.e., to the choice between alternate reaction paths, can be studied in this case. 1. Demonstration Using Atmospheric Pressure, Large Particle Mixiures

Hindin et al. (18) published data showing benzene formation to proceed readily from methylcyclopentane over mechanical mixtures of platinum bearing particles and silica-alumina, at atmospheric pressure and near 500°C.temperature. Under these conditions the equilibrium constant for conversion of a cyclopentane to a cyclopentene is of the order of unity. Consequently, the first step, if it is catalyzed by X , can itself proceed with TABLE VI Conversion of Methylcyclopentane over Single and over Coarse Particle Mixed Catalysts, at Dehydrogenative Conditions Liquid product analysis (mole %)

y) I I

II

Fc()fc()

-

I

I I

I

I I

10 cc. SiAl 10 cc. Pt/Si02

SiAl

+ Pt/SiOn

’

I I

!

YJt

-

I

;

iy

I

I

I

98 62 65

I

I - - x )

I

I

I I

0 20 14

1

1

0 18 10

I

I

i

I

I

I

I I

‘3 I /I I 0.1 I 0.8 I 10

I

I I

large conversion to “intermediates” a t atmospheric pressure. Thus, a mixture of relatively large component particle size should indeed produce the reaction and represents a case of the “trivial” polystep reaction (Section 11,C). However operating under such conditions allows us, to study the identity and behavior of analyzable quantities of intermediates. Table VI shows results (19) of liquid product analyses from methylcyclopentane (MCP) converted under conditions of p H 2 = 0.8 atm. p ~ c = p 0.2 atm., with X = 0.3 wt. yo Pt on SiOz and Y = 420 m.2/g. silica-alumina, as catalyst components in 0.8-1.4 mm. particle size, and with

172

PAUL B. WEISZ

2.5 sec. residence-time. They demonstrate the production of cyclo-olefin from methylcyclopentane over platinum component alone, and the cooperative conversion to benzene when both components are present. Moreover, under these operating conditions, Pt-component is seen not only to create methylcyclopentene, but also appreciable quantities of methylcyclopentadiene. (The second column contains the sum of both mono-olefin species which were not resolved by the mass-spectrometric procedure.) Furthermore, the principle of interception (Section II,E,1) is well illustrated if we consider the methylcyclopentane -+ methylcyclopentadiene reaction as the platinum catalyzed reaction as being diverted by the Y-component, through the intermediate cyclo-olefin (here grossly detectable).

2. Nontrivial Conditions

A transition to nontrivial polystep conditions can easily be made by increasing pressure and/or lowering temperature. At 380°C., and 12 atm. pressure, equilibrium partial pressure of cyclo-olefin is of the order of atm. and the equilibrium conversion for the first step is then no greater than 0.1%. The required particle intimacy (component particle size) now approaches that in the case of n-heptane isomerization above. Table VII below shows a n experimental demonstration of the mixed component catalyst operation under these conditions. TABLE VII Conversion of Methylcyclopentane under Nomtrivial Polystep Conditions (Hydrogenative Condition) (T = 380°C., p ~ =* 10.3 atm., P M C F = 1.1 atm., 7 = 7.5 sec.)

Mixed catalyst

Pt/Alr03

+ Silica-alumina

Products (wt. %)

Component particle size

{

+

b)

Benzene cyclohexane

c1 - Ce Paraffins

500 5

3.3 20.0

12.5 5.6

When the required intimacy has been provided, the polystep conversion to six-carbon ring products is seen to be successfully accomplished even a t this low temperature by the coaction of the separate catalyst components. I n addition, we observe here again the phenomenon of reducing the rate of production of certain products by providing more intimate contact with a second catalyst component: the hydrogenolysis reaction of C--C: bonds to C1 to Csparaffins is inhibited; i.e., diverted to a new reaction path.

POLYFUNCTIONAL HETEROGENEOUS CATALYSIS

173

3. Selectivity of Alkylcyclopentane Aromatiz‘ation We have thus observed above the following type of selectivity phenomena : Under dehydrogenative conditions (510°C. and atmospheric pressure, as seen in Table IV), the double step reaction over platinum leading to methylcyclopentadiene is intercepted by the introduction of “acidic” catalyst centers ( Y ) ,

(scheme XVI)

to produce six-membered rings at the expense of the products formerly made. Under hydrogenative operating conditions, hydrogenolysis is the major Pt-catalyzed reaction. Its inhibition when the Y-path is introduced can-be similarly explained on the basis of the reaction scheme

cn 5 cfi 2 x

C1 t,o Co paraffins 7 (hydrogenolysis)

cyclohexane

+ benzene

(scheme XVII)

The amount of conversion to six-carbon rings should then depend on provision of an effective and high rate constant for the Y-step (acidic component)-see Section II,E,l-and the selectivity (i.e., minimizing hydrogenolysis products) should be controllable by providing an X-component which is just sufficiently large, but not excessively so (see Section II,E,2).This case is analogous to the case of minimization of hydrogenolysis described for n-hexane isomerization in Section IV,C,l. Using mixed catalysts of platinum/alumina and silica-alumina, and the microcatalytic technique (both as described in Section IV,C,l) each of the above mentioned phenomena can be amply demonstrated. A wide range of effective catalyst acidity (Y) is obtained by varying component particle size of the mixture. A large spectrum for the platinum component strength ( X ) is then obtained for each case by starting with highly active “fresh”

174

PAUL B. WEISZ

I

I

I

I

I

I

-

6 8 10 0 2 4 N.UM0ER OF 1/32 CC PULSES OF H2S

12

+

FIG.16. Reaction of methylcyclopentane to Crring hydrocarbons (benzene cyclohexane), and to hydrogenolysis products, on dual-component mixed catalyst, when platinum component is progressively deactivated by hydrogen sulfide.

Pt/AI + S i / A I

2o 0

0

20 40 60 80 I I0 MCP CONVERSION (WT. %)

+

FIG. 17. Methylcyclopentane conversion to 6-membered ring products (benzene cyclohexane) and hydrogenolysis products, for Pt-component alone, and for mixtures with silica-alumina of two degrees of intimacy, when Pt-component activity is varied.

POLYFUNCTIONAL HETEROGENEOUS CATALYSIS

175

catalyst, and progressively deactivating the platinum function by successive additions of gaseous hydrogen sulfide between methylcyclopentane conversion tests, at 330°C. (other conditions as in Section IV,C,l). For the intimate (5p particle) composite, such control of the X-function results in the results shown in Fig. 16. The “fresh” platinum, under these conditions leads to complete hydrogenolysis of methylcyclopentane. Platinum deactivation controls the reaction, optimizing the isomerization products (benzene cyclohexane) , until the generating step itself becomes insufficient. The results of these studies are collected in Fig. 17 for three levels of effective acidity ( Y ) ,i.e., for the Pt-component alone, and for the mixtures having 500 and 5 p component particle size. Total conversion has been used as a convenient abscissa for comparing the three cases. Note the similarity of the behavior of the second and third cases to those calculated for the analogous model of selectivity control in Section II,D,2; this model is shown in Figs. 6a and b for two levels of activity of the Y-component. The maximum isomerization activity depends on the effectiveness of the acidic Y-component, and is essentially nil without the added Y-component. It appears that the acid catalyzed reaction step is generally the rate controlling step in the dual functional hydrocarbon reactions over platinum containing catalysts, as indicated for the naphtha “reforming” conditions by Weisz and Prater (as), and indirectly concluded from comparisons of aromatization of various naphthenes by Keulemans and Voge (29), and for paraffin isomerization as shown by Sinfelt et al. (SO).

+

V. The Petroleum Naphtha “Reforming” Reaction The “reforming” of petroleum fractions boiling between about 90 and 200°C. to high octane gasolines constitutes one of the largest scale industrial catalytic operations of our times. The quantity processed over platinum catalysts exceeds 2 X lo8 liters/day. A majority of the reactions involved are polystep hydrocarbon conversions (see refs. 10, 11,and the extensive review of the art by Ciapetta et d . ,31). The major objective of reforming is the isomerization of paraffins close to equilibrium, and the production of a maximal amount of aromatics. The octane number is often used as a measure of reforming activity or, alternatively, the temperature needed to accomplish a given octane number for the reformate. Although an octane number may not seem to be a good choice for a variable denoting catalytic activity performance in research or discussions of mechanisms, this turns out to be quite acceptable as a fairly good measure of conversion to aromatics: Fig. 18 shows a collection of data concerning a large variety of reformer gasolines, which, when standardized to a fixed boiling range show a very good correlation between octane num-

176

PAUL B. WEISZ

ber (research octane number-R.O.N.-with 3 ml. tetraethyl lead-TEL) and total aromatics content. Since a t these conventional reforming severity levels the paraffin isomerization proceeds rather closely to equilibrium throughout the range, the major variable of conversion is indeed the degree of aromatics production. Weisz and Prater (28) have pointed out that, while on platinum reforming catalysts alkylcyclohexanes will proceed rapidly to aromatics, the conversion of alkylcyclopentanes to aromatics becomes the key dualfunctional reforming reaction in the neighborhood of 98 (R.O.N. with 3 ml. I

.-. 120 m +

C 6 z

llo-

I

I

I

I

I

-

0

NEAR EAST CRUDE T E X A S CRUDE

NeY

100-

4

90-

I

0

80

I

CALIFORNIA CRUDE t

z

t,

I

I

Pt REFORMATES

d

-

I

I

I

I

I

I

I

I

I

TEL) octane number reforming severity. They have presented evidence that the catalytic strength of the acidic catalyst component is ratecontrolling once the platinum component is sufficiently strong to provide the initiating step, in line with the discussions in Section IV,E. They showed the reforming temperature required to attain 98 R.O.N. to be well correlatable to the acidity measured by a cumene cracking test across a variety of acidic catalyst compositions including platinum on silica alumina, silica-magnesia, and aluminas activated by chlorine, fluorine, and boron. Figure 19 is a reproduction of the correlation originally presented; however, there are now identified compositions (marked with crosses) in which platinum component and ‘(acidic)’components were mechanically separate and were obtained by pelletizing mixtures of -5p particles platinum/silica or platinum/alumina with the acidic components. In this correlation, these reforming catalysts are seen to be indistinguishable from the conventional “impregnated” variety, the indistinguishability being consistent with the conclusion that in the reforming reactions on naphthas too, the two catalyst components can have individual and separate identity

POLYFUNCTIONAL HETEROGENEOUS CATALYSIS

177

FIG. 19. Relationship between temperature requirement for reforming naphtha to 98 O.N. level and the activity of the “acidic” function. Crossed points mark mechanically mixed composites.

and can operate satisfactorily through the mediation of the gas phase intermediates if sufficient diffusional intimacy is provided (32, SS). Figure 20 shows the operational characteristics of two reforming catalysts, one of which (b) is a conventional reforming catalyst containing 0.5 wt. % ’ platinum and 0.5 wt. % fluorine on alumina, and the other (a)

FIG.20. Reforming activity for a twelve day period of operation, for various impregnated and mixed-type catalysts.

178

PAUL B. WEISZ

a 50/50 particle mixture of halogcn-free alumina/l.l wt. % ’ platinurn and alumina/0.7 wt. yoF, over a period of about 12 days of operation (experimental procedures as described in ref. 28). In (c) and (d) of Fig. 20 are pictured the initial activity repsonses of two catalysts in which the “acidic” sites are supplied by silica-alumina of similar “over-all acidic” activity, with (c) being a silica-alumina base of 90 m.2/g. surface area impregnated directly with platinum, and (d) being a 65/35 mechanical mixture of 5 p particles of an aged cracking catalyst of 120 m.2/g. surface area and alumina/platinum. The test conditions (apparatus) are again those described in reference 28. The showing of adequate and equivalent operation in reforming reactions of catalysts composited from physically separate components must not be interpreted, of course, to imply that in all of the possible catalyst compositions, the agents or sites supplying the (de-)hydrogenation function and the ‘Lacid’’function must be or always are independently located. For example, in the case where “acidity’ and (de-)hydrogenation activity are derived from chlorine and platinum, respectively, due to the introduction of PtCle-ions into some aluminas by impregnation from chloroplatinic acid, it is quite likely that the two elements do not appreciably separate in the course of subsequent drying and calcination. However, it appears that, for all purposes studied, such direct proximity is not mechanistically significant. Rather, the mechanistic models that picture the catalytic components as having separate identities have provided a useful and constructive framework for further investigation and understanding of the behavior of this type of catalyst, as evidenced by the development and use of functional tests (28; 31, p. 522; 23) studies of mechanisms of catalytic deactivation, etc. Another example for the use of the mixed catalyst concepts in mechanistic studies can be found in investigations concerning aging mechanisms. For example, experiments, such as those pictured in Fig. 20, using various mixed composition, supply pertinent information. In reforming operations under conditions described above, the initial fast-to-slow deactivation transition [(c) in Fig. 201 is characteristic of platinum on silica base impregnated catalysts, while “flat” or gradually rising temperature requirements are enoountered with good alumina-type reforming catalysts. The observation that the fast-to-slow response occurs also with the silica-alumina alumina/Pt mixed catalyst [(d) in Fig. 201 suggests that this particular initial deactivation response is characteristic of the type of acidic site and not of the platinum support as such. In further confirmation of this, Weisz, Prater, and Swegler (unpublished) have tested mixed compositions using alumina/F as acidic component with platinum on silica and were able to

+

POLYFUNCTIONAL HETEROGENEOUS CATALYSIS

179

obtain “flat” activity responses. Evidence for the role of acidic sites in at least the initial aging processes of reforming catalysts was presented by Weisz and Prater (28),and probable mechanisms discussed by Myers et ab. (34).

One reaction occurring in the reforming process and which has not been discussed in this review is the cyclization-aromatization of paraffins. I ts contribution to the reforming reactions is a secondary one a t the severity conditions so far discussed. When processing to increasingly higher aromatics content this reaction gains increasing importance as a source for additional aromatics. However, the published literature does not as yet offer a clear picture of reaction paths and the specific contributions of the catalyst functions. McHenry and co-workers (35) present data showing a correlation between cyclization-aromatization activity of platinum catalysts and the amount of Pt extractable with H F or acetylacetone, and suggest that platinum in a special complex with the alumina surface has superior activity. They suggest a combination of platinum with a n acidic site a t a “critical spacing”; beyond the point of speculation, however, the observations would not appear capable of more specific interpretation than to show that the extractable form of platinum in their samples is more active; perhaps due to their greater degree of dispersion. A conclusive showing as to whether or not these catalysts have superior activity for their basic (de-)hydrogenative function would be desirable before attempts are made to interpret dehydrocyclization results in terms of special properties beyond the elementary (de-)hydrogenation and “acidic” functions. Tests with cyclohexanes under more or less standard operating conditions are usually inadequate for this because of the difficulties introduced by the very high specific rate constants involved and the resulting major thermal and mass diffusion effects. Very careful and special techniques are required for this (see ref. 68; and 31, p. 522).

VI. Other Polystep Reactions A. THEXYLENES-ETHY LB ENZENE INTERC ONVERSI ON The isomerization reactions among the three xylenes are catalyzed by the acidic catalysts, such as silica-alumina. However, the skeletal transition to ethyl-benzene will not occur (36). Pitts et al. (37) have shown that such reaction wi!l proceed if a (de-)hydrogenation component is also present; furthermore, such conversion is favored by lowering of operating temperature, i.e., by conditions favoring hydrogenation. One is led to the conclusion that the xylenes to ethylbenzene “skeletal” rearrangement takes place in a polystep fashion according to

180

PAUL B. WEISZ

(scheme XIX)

that is, by ring contraction and expansion steps of cyclo-olefin intermediates catalyzed by the acidic component (38) and analogous to those occurring in the hydroisomerization reactions of cyclohexanes and alkylcyclopentanes described earlier above. P. S. Nix and S. Lucki, a t our laboratories, have demonstrated (unpublished) the ability of separate platinum and acidic catalysts, as a mixed composite, to perform the skeletal transition from ethyl-benzene to xylenes under hydrogenative conditions ( p H 2 = 11.8 atm., ~ E = B 1.2 atm.; 427"C, T = 3.3 sec.) with 40% conversion to xylenes. Yet a t the same temperature, but at atmospheric pressure where production of cyclo-olefin intermediates is not favored, they obtained no measurable conversion even with platinum directly impregnated on the silica-alumina.

B. HYDROGEN EXCHANGE BETWEENPARAFFIN HYDROCARBONS Myers et al. (39) have studied the deuterium exchange between two paraffins, i.e., between deuterobutane and butane, on a variety of dualfunctional catalysts (various impregnated types of platinum-acidic oxides). They find that the hydrogen exchange correlated primarily with the (de-)hydrogenation activity of the catalyst, but in addition, an appreciable positive correlation with ((acidic" activity was demonstrated, which led the authors to conclude that, this ('may mean t,hat acid sites and platinum sites coaci to promote the latter effect." It is reasonable to suspect] that a dual functional reaction path exists following the scheme, X

butane: d-butane

+E l d-butane + Ih butene

x Ytl

1 I

I

1

(scheme XX)

such that direr.L exchange interaction among olefin intermediates reacting on acidic sites ( Y ) add to the rict rate of exchange. The existence of such

POLYFUNCTIONAL HETEROGENEOUS CATALYSIS

181

vapor phase transfer mechanism between sites may be subject to test by the mixed catalyst technique. C. ORGANICREDUCTION REACTIONS Rylander and Cohn (40) have reported synergistic effects for the use as catalysts simultaneously of two different platinum or palladium group metals in the reduction of various organic compounds. For example, in Fig. 21 are plotted observations oncerning the rate of hydrogenation of

c ._

E

\ - 50

-E

0

0

I

0.5

I

0 Ru

0.5

I

Pd

J

I

I

FRACTION EACH M E T A L

FIG.21. Rate of butyne-1,4-diol reduction vs. relative composition of u composite Pd and Ru catalyst (after Rylandcr and Cohn, 40).

butyne-1,4-diol using 300 mg. of catalyst, 200 mg. of reactant, and 100 ml. of methanol solvent at ambient conditions, wherein the catalyst consisted of palladium or ruthenium or both, precipitated on carbon in varying proportions. The total weight of the precious metal was 5 wt. % ’ of the catalyst in all cases. Similar effects were demonstrated by them for palladium and ruthenium in the reduction of pyridine, and for ruthenium and platinum in the reduction of nitriles and of nitrobenzenes. Since they show similar although not as large synergistic effects when the two metals are introduced on separate carbon particles, one may be sure that the enhancement is not a n intrinsic property of some chemical combination or contact between the different metals. As the authors point out, the various metals are known to have differing catalytic effectiveness for the reduction of different functional groups. Thus, when reduction takes place via more than one chemical step, such as in the case

IZQZ(-J

221

00 H

H

H

182

PAUL B. WEISZ

one may expect specific effects from combinations of metals each of which contribute different, rate constants to various of the steps. In some cases the intermediates may be well identifiable in analyzable quantities; in other cases they may occur at subdetectable concentrations. It would appear that the mixed catalyst techniques together with quantitative appraisals based on principles outlined above may give more information on the concentration and nature of the intermediates in various cases. By way of illustration, the reaction rates reported in this work were of the order of d N / d t = moles/sec./cc. catalyst. With a liquid phase diffusivity of intermediates of molecular weight similar to that of reactant estimated at D = 5 X cm.2/sec. we have from (15) (see also Section IV,A,l)

-

20R2/[R]

9

From the relative rates observed for separate particles and of coprecipitated metal (assumed to offer complete intimacy for the intermediate) we might estimate @ (see ref. 6 ) as of the order of magnitude 3 in Rylander and Cohn's observation, which leads for various catalyst particle sizes (carbon granules) to an estimate of concentration of intermediate as follows:

-

R

=

PI = H'% =

T

10 10-8 0.2

x

100

x

10-4

20

1000

7 x 10-2 -

/1

rnolcu/~rn.~ in liquid

(x,

These figures suggest that in this case intermediates occurred at potentially analyzable quantities. It seems clear from the interesting observations of Rylander and Cohn (40) that the catalytic possibilities using polyfunctional catalyst should increase rapidly in number when one deals with increasingly complex organic structures, where many more individual reaction steps become possible.

D. UNSUSPECTED QUASI-INTERMEDIATES AND POLYSTEP REACTIONS Certain catalyst materials that may not have been considered to operate as multifunctional catalyst composites may in fact be operative via distinct and separate catalytic sites. Keulemans and Schuit (41) have discussed such possibilities briefly. In this article, we shall not look further into these possibilities; however, inasmuch as we have examined various reactions and reaction sequences, it is very appropriate to point out that there undoubtedly exist reactions which have been considered to be catalyzed essentially by monofunctional systems, but which may in fact undergo polystep conversion. Also, quasi-intermediates may exist, appearing as traces of side products or remaining undetectable, that carry a ready

POLYFUNCTIONAL HETEROGENEOUS CATALYSIS

183

potential to be “operated upon” by a second catalyst component, in the manner described in Section II,E,3.

Example: Cumene Cracking An example for this phenomenon appears to exist in observations reported by Weisz and Prater (28) on cumene cracking. This reaction is usually pictured simply as

When this reaction rate was measured by the rate of molar gas formation in a Schwab type differential reactor (28, 28a) over silica-alumina catalyst, this rate was found not greatly affected by the introduction of a platinum component into the catalyst mass; the observed gas formation rate remained controlled by the acidic activity alone. Yet, an analysis of the gas produced, and of the liquid product, showed a shift in product composition

benzene

+ propylene

methylstyrene

+ hydrogen

Examination of this phenopenon was continued by J. Wei, who was able to demonstrate the same effect under gross conversion conditions, that is, in an integral reactor, at 478°C. Figure 22 shows the results for the extent and constitution of the gaseous product for four cases: (a) with pellets of 420 m.”g. silica-alumina, (b) with pellets of 0.6 wt. yoplatinum on alumina, (c) with a pelleted 5p size particle mixture of the two components (a) and (b), and (d) with a mixture of pellets of each component. Even under these severe conditions of high conversion (60-90~0) where some by product gases are made, the shift of gas composition is amply demonstrated when the two-component mixture is in good mutual intimacy (c). The mere presence of whole pellets of both tyrcs of catalysts does not show this effect (d), hut approximates simple additivity of products from each component. Wei carried out a study with varying component particle size, and found a maximum transition to hydrogen production to occur somewhere near 10-3Op. This corresponded to a vapor pressure of the quasi-intermediate of about lov5atm. by application of the principles outlined in Sections II,D,3 and 111. This would correspond to a free energy of formation of the quasi-intermediate from cumene of about 16 kcal/mol. The quasiintermediate may well be represented as a very low level side product of

+

184

PAUL B. WETSZ I00

FIG.22. Gas production rates and composition observed for cumene cracking with (a) silica-alumina; (b) Pt/AlzOa; (c) intimate 5p mixture of (a) and (b); (d) mixture of pellets of (a) and (b).

the cumene cracking reaction which, however, is produced at a fast rate, SO as to be essentially in equilibrium with cumene on the catalyst when the system is not disturbed by the added reaction path:

1 -7 (SIAI)

cwmene

cumcne/catalyst

(SiAI) -+

benzene

I, L

(SlAl) JT.

(pt)

“B” -+ 12

styrene

+ CsHe

+ HZ

It is interesting to note that the existence of such an intermediate “B” need not be of any concern in the course of study of cumene kinetics, or, restating this, may not be noticeable in any studies of cumene kinetics as such. This demonstrates the power of the mixed catalyst technique as a probing tool which can go far beyond the capabilities of analytical methods in the detection of participating product species (see Section 111). E. ENZYMATIC PROCESSES In a discussion of polystep reaction phenomena, especially aimed a t developing and examining some of thc general principles of such processes, it would seem rather disappointing not to turn briefly to the chemistry of life processes, where stepwise catalytic processes arc the rule rather than a novelty. The biochemical reaction systems contain innumerable polystep reaction sequences with various steps catalyzed by different enzymes. Thus, if

POLYFUNCTIONAL HETEROGENEOUS CATALYSIS

185

various participating enzymes of a reaction sequence are not uniformly and homogeneously dispersed, we have a situation rather analogous to that of the mixed or polyfunctional heterogeneous catalyst systems described in this chapter. We should be able to apply intimacy criteria as in Section II,D to estimate the maximum distance which may exist between enzyme systems involved in the same polystep reaction sequence. If we make the assumption that such enzyme systems must be contained within the biological cell, then such an estimate will define the maximum allowable cell dimension, in the absence of hydrodynamic means for mass transport within it. 1. Cell Dimensions

In Section II,D,l we have developed an expression, formula (7), for the maximum dimension between two parallel planes representing the two different catalytic surfaces of a reaction sequence, containing the reaction rate per unit surface area. Now let us imagine a reaction space (of unit volume) filled with such surfaces of catalyst X and Y . Then, the number of such planar reaction units will be n = 1/L, and the maximum attainable reaction rate per unit volume will be

or, for a given magnitude of reaction rate to be achieved, the maximum allowable dimension for separation of catalyst types is given by

The form of this criterion is identical to that of formula (15) for porous catalyst components. However, there it defines conditions for the onset of reaction inhibition, while here it sets an absolute ceiling. For a biochemical process by enzyme systems located at some distance from each other, we might now estimate the maximum dimension L from the magnitudes of the reaction-rate to be accomplished d N / d t , and the concentration of the intermediate species B,, involved.* We may expcct the diffusivity to be of a magnitude typical for large molecular species in an aqueous medium, i.e., D =i: 10+ cm.2/sec. The metabolic rate of oxygen uptake by various organs, cells, and bac10-8moles/sec. cm.3. teria is of magnitude cc. Oz/min./g., or d X / d t Let us take this as a typical rate magnitude. (Presumably there are many

-

*We will assume that the actual concentration does approach the attainable equilibrium conrentration, i e . , that the enzyme catalyst has developed to fairly optimum activity.

186

PAUL B. WEISZ

specific reaction processes going on at a smaller rate than the over-all respiratory process, which, however, would also be confined to more specific and thus smaller volumina of the body.) We thus find the following dimensions L for various concentrations of reaction intermediates :

[*Acrn.3. L , cm.:

10-12

10-10

10-8

10-0

10-4

10-2

10-6

10-4

10-3

10-2

10-1

1

Now what are typical magnitudes for concentration of intermediate reacting species in oxygen metabolism? Cell respiration is seriously inhibited by 0.5 X moles/cc. of a poisonous ion, such as cyanide, indicating that Some important species must exist at a concentration as low as this (see e.g., ref. 4%').The concentration of cytochrome c in yeast cells is found to be of magnitude moles/cc. With such concentration magnitudes indicated, and assuming that the concentration of an enzyme specific for a given reaction will not greatly exceed in magnitude the concentration of the species to be reacted, we obtain for the maximum allowable dimension

L

-

cm.

which is of the order of actual cell dimensions, i.e., about 5 X 10-4 to 5x cm. It is interesting to note that the criterion, formula (20), contains the quotient ( d N / d t )(l/[Bes]) and to observe that there appear many situations where for a given type of process this ratio of chemical rate to intermediate

p G CYTOCHROME-C/GRAM

FIG.23. Respiration rate and cytochrome c concentration in various types of rat tissue (after Puppenheimer and Hendee, 48).

POLYFUKCTIOSBL HETEROGENEOCS CATALYSIS

187

concentration indeed tends to be cwiistant for vnrious tissue even though each of these quantities may vary over nearly three orders of magnitudr, as, for example, in the measuremeiit,s of Pappcnhcimer and Hendee (43) on respiration rate and cytochrome c contcrrt of various rat tissues from which data Fig. 23 is drawn. These observations might thus suggest that, griicrally, a good fractioir of the dimension of the biological (.ell is involvtd in diffusive transport of an intermediate between two different enzymc’ rryions. This fact is corrsistent with the trend of findings i l l cytology indicating that the activity of specific enzymes is localized within small area of the cell (mitochondria, sarcosomes, etc.; see, e.g., ref. 44).We should note that it is sufficient for only one intermediate reactioii step of a long sequence to bc sub,ject to suc.h transport, and our criteria will apply to thr ovcr-dl reaction.

2. Turnover Numbers

It is difficult to find many specific instances wit,h quantitative information concerning the concentration of iritermcdiatrs. However, the quotient (dN/dt)(l/[Beq])can be transformcd to the “turnover number” T familiar to the biochemist (molecules of reactant rcacted pcr minute per molecule of enzyme), by

n=

molar enzyme concentration molar reactant concentIration

From this and (20) follows

-

for the maximal distance magnitudes. If we now use D 10-6 cm.2/sec., and n of order of magnitude unity, we have

L 5

sx

10-3/1/T

For enzyme reactions characterized by turnover numbers of lo2 to loG (the latter magnitude in more exceptional rases), we obtain

L 6

. ..

mi.

The maximal valuc is again typical of actual c d u l a r dimensions, with the suggestion that some processes, namely, thoscl having the very high turnover numbers require greater proximity, hut, uithin cellular spare dimensions.

188

PAUL B. WEISZ

F. INORGAXIC I~EACTIOM There is little rrason to believe that the phenomrna of stepwise reaction processes and the criteria developed for them above would not be found to be operative and applicable in some circumstances of inorganic chemistry, notably in solid-solid reactions and solid-gas reactions accelerated by a second solid. In a reaction betwrcii solids A and B in a mixture of these solids, a vaporproduct A,, of A may he the agency which contacts B , at R so that the reactivity is based on A

‘4 B ;tA , + A ,

X 13

with the letters above the rate arrows again indicating the location of each procms in complete analogy to schcme VIII. A , may be due to a small vapor pressure of A itself or due to a decomposition pressure, which then would definr thr concentration of the intermediate [A,],in applying criteria such as (15) or (20). Iriterrsting examples and variants in detail (but not variants in principle) are such reactions ah the carbonate transformation “catalyzrd” by the admixture of (+arbon.

or the metal oxide reduction with carbon, involving the cycling of gas phase intermediate (carbon moiioxidc) Me0

++ CO JIPO (Me +)COZ Cnrbon SCO ~2

--f

J

Many situations arc encountered in solid-solid reactions whrrc the concept of the limited point contacts being the sole surfaces of chemical interaction is a difficult one to accept. On the other hand, the numerical magnitudes which thc above criteria indicate for the reaction rates a t even minute intermediate vapor pressures lend credencr to the possibility that these phenomena may provide the “coupling mechanism,” in such cases.

VII. Conclusions Reactions on solid c:at,alysts can proceed by way of several distinct steps of chemical transformation, each catalyzed by distinctly different types of cat,alytic sites. The chemical intermediates can exist as true desorbed species in the phase above the catalyst. The kinetic steps of the sequence of reactions are coupled to each other through the processes of diffusion of these intcrmediates. Thc classical laws of diffusion create a qualitative and quan-

POLYFUNCTlONAL HETEROGENEOUS CATALYSIS

189

titative link between the rate process and the thermodynamics of reaction steps. The criteria and principles involved have general applicability not only in catalytic hydrocarbon transformations, but also in other catalyzed organic reactions, in some of the largest scale industrial processes, in chemical laboratory experience, and in the biochemical world of life processes.

REFERENCES 1. Fischer, F., and Tropsch, N., Ber. 66, 2428, 1923. 2. Natta, G., and Rigamonti, R., “Handbuch der Katalyse,” Vol. 5, p. 412. Springer, Berlin, 1957; (on “Mehrkatalysatoren”). 3. Herington, E. F. G., and Rideal, E. K., Proc. Roy. Sac. A190, 289, 309 (1947). 4. de Boer, J. H., and Van der Borg, R. J. A. M., “Actes du QBme congr. intern. de catalyse, Paris 1960,” Vol. 1, p. 919, Technip. Paris, 1961. 5. Prater, C . D., and Wci, J., Advances t n Catalysis, this volume. 6. Damkohler, G., D. Chem. Ing. 3, 430 (1937); Thiele, E. W., Ind. Eng. Chem. 31, 916 (1939); Zrldowitsch, J. B., Acta Physicorhivn. U.R.S.S. 10,583 (1939); Wheeler, a., Advances i n Catalysts 3, 249 (1951); Weisz, P. B., and Prater, C. D., ibid., 6, 143 (1954). 7 . Weisz, P. B., 2. physik. Chem. (Frankfurt) 11, 1 (1957). 8. Weisz, P. B., Chem. Eng. Progr. Symposium Ser. 66, 29 (1959). 9. Rakowski, A., 2. physik. Chem. 67, 321 (1907) 10. Haensel, V., and Donaldson, G. R., Ind. Eng. Chem. 43, 2102 (1951). 11. Heinemann, H., Mills, G. A., Hattman, J. B., and Kirsch, F. W., Ind. Eng. Chem. 46, 130 (1953). 12. Ciapetta, F. G., and Hunter, J. B., Ind. Eng. Chem. 46, 147, 155, 159, 162 (1953). 13. Mills, G. A., Heinemann, H., Milliken, T. H., and Oblad, A. G., Paper presented a t 121st Am. Chem. Soc. Meeting, Milwaukee, Wisconsin (March, 1952); Ind. Eng. Chem. 46, 134 (1953). 14. Weisz, P. B., Sczence 123, 887 (1956). 15. Mills, G. A., Advances i n Catalysis 9, 639 (1957). 16. Weisz, P. B., Advances zn Catalysis 9, 640 (1957). 17. Weisz, P. B., and Swegler, E. W., Science 126, 31 (1957). 18. Hindin, S. G., Weller, S. W., and Mills, G. A., J. Phys. Chem. 62,244 (1958). 19. Wrisz, P. B., i n “Actes du 2Bme congr. intern. de Catalyse” Paris, 1960, Vol. 1, p. 937, Technip., Paris, 1961. 20. Egloff, G., Morrell, J. C., Thomas, C. L., and Bloeh, H. S., J. Am. Chem. Sac. 61, 3571 (1939). 21. Greensfelder, B. S., and Voge, H. H., Ind. Eng. Chem. 37, 983 (1945). 22. Weisz, P. B., and Prater, C. D., Advances i n Catalysis 6, 143 (1954). 23. Myers, C. G., and Munns, G. W., Jr., Ind. Eng. Chem. 60, 1727 (1958). 24. Kokes, R. J., Tobin, H., Emmett, P. H., J. Am. Chem. Sac. 77, 5860 (1955). 26. Weisz, P. B., and Swegler, E. W., J . Phys. Chem. 69, 823 (1955). 26. Prater, C. D., Smith, R. L., and Wei, J., Symposium of the Philadelphia Catalysis Club, Philadelphia, Pennsylvania (1961). 27. Bloch, H. S., and Thomas, C. L., J . Am. Chem. Sac. 66, 1589 (1944). 28. Weisz, P. B., and Prater, C. D., Advances i n Catalysis 9, 583 (1957). 28a. Prater, C. D., and Lago, R. M., Advances i n Catalysis 8, 294 (1956). 29. Keulemans, A. I. M., and Voge, H. H., J . Phys. Chem. 63, 476 (1959). YO. Sinfelt, J. H., Hurwitz, H., and Rohrer, J. C., J . Phys. Chem. 64,892 (1960).

190

PAUL 13. REISZ

3 1 . Ciapetta, F. G., Dobres, R. M., : ~ n dB:tker, R. \V., rn “Catalysis” (1’. II. Emmett ed.), Chapter 6. Reinhold, New York, 1958. S1. Weisz, P. B., U.S.P. 2,854,400 (1954); 2,854,404 (1954). 33. Prater, C. D., and Weisz, P. H., 1J.S.P. 2,854,404 (1954). 34. Myers, C. G., Lang, W. H., and Wrisz, P. H., I n d . Eng. Chem. 63, 29!) (1961). $6. McHenry, K. W., Bwtolacini, It. J., Brennan, H. M., Wilson, J. L., and Seelig, H. S., paper f44 of Colloid Symposium, 1.38th Meeting Anr. Chem. Sac., iYrw York, 1960. 36. Lien, A. P., Pttper presented at 126th Alwtzng L l n i . Chrnr. Sac., Kurrsus Czty, Alassoun (1954). 37. Pitts, P. M., Jr., Connor, J. E., Jr., Leum, L. X., 17kd. Eng. (‘hrnr. 47, 770 (1955). 38. Pines, I I . , Shaw, A. W., J . A m . (‘hein. Sac. 79, 1474 (1957). 49. Myers, C. G., Sit)t)ett, D. J., and Ciapetta, F. G., J . I’hys. Chern. 63, 1032 (1959). 40. Rylarider, P. N., and Cohn, C., “Actes du congr. intern. de c.atulysr, Paris, 1960,” Vol. I , p. 977. Teehnip., Paris, 1961. 41. Keulemanfi, A. I. M., and Schuit, G. C. A., zn “The Mechanisms of Heterogeneous Catalysis” (J. H. de Boer, ed.). Elsevier, AmstcJrdam, 1960. 62. Fruton, J. S., arid Simmonds, S., “General Biochemistry,” p. 347. Riley, New Yorh, 1959. 43. Pappenheimcr, A . M., Jr., and Hendee, E. D., d . Biol. Chem. 171, 701 (1!147). 44. Hogeboom, G. H., Srhneider, W. C., and Pallade, G. E., J . Biol. (‘hem. 172, 619 (1948); Schneider, W. C., and Hogeboom, H. C., J . Nut/. Cancer Insl. 20, !)6Y (1950); IIogrhoom, G. H., a n d Svhneider, H. C., ibid., 983. Q i m c k

A N e w Electron Diffraction Technique, Potentially Applicable to Research in Catalysis

Cornell l ~ T n i n e r s i tIthaca, ~~, N . Y. Page

I. Experimcsrital Apparatus.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Oxygen on Nickel.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rrfrrrnc.es. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

192 193 201

'l'hcrc are maiiy different reasons for interest in the surface of a crystal. From :I narrow, practical viewpoint, one might mention the economic importaiic~of the reaction of oxygen with the surface of an iron crystal (caorrosion or ruhting), the c4kc.t of surfaw layers upon the performance of various hemiconductor devices, and of roursc caatalytic reactions on crystal surfaces which presumably include all of catalysis. The diffractioii of slow cle(8trons at a crystal surface furiiishes a direct method for study of a crystal surface which is completely or almost clean. Very possibly, also, it can lead to a more detailed and accurate knowledge of the surfwe than can any other method of investigation. This fact has bccii w l l knoivn for more than 30 years. Yet, in spite of the great importance of drtailed understaiidiiig of nearly clean crystal surfaces, and of the effert iveiicss of low-energy electron diffraction in obtaining such knowledge, the workers in this field havc been few and the amount of work relatively small. Only Professor Farnsworth arid his co-workers have been studying clean arid nearly clean crystal surfaces by means of electron diffraction ( 1 ) . Considering the number of workers involved , their accomplishments have been c*onsiderablc,but nevertheless quite trivial when compared with the of the available field of knowledge. It i:, true that the diffraction of rather high energy electrons has been widely used for the study of surface layers, and of thin films. This technique, involving c>lectroiisof energies of the order of 50 kev., has been extensively developed and caommercial clcrtron diff ractiori equipment is now available. Furthermore, high-energy clectroii diffraction is widely used in a great many rwearrh laboratories, and muvh of the work in this field is described in sonic thousands of scientific papers. I n some cases what I am calling high energy electron diffraction has even been used successfully to study a monolayer of atoms or molecules upon a smooth surface. Worthwhile as this 191

192

11. H. CiERMER

type of work is, however, it, is rather iiiappropriate to the study of siiiglc layers of adsorbed atoms, and extensive attempts have riot been made to apply it to their study. More basic investigations of the arrangement of the very first foreign atoms upon crystal surfaces require the use of low energy electrons (-100 ev.) in experimental arrangements which are quite different and rertainly more difficult. I t is just this experimental difficulty which for many years has limited work in this potentially important field to the small group of workers at Brown University. In 1934 Ehreiiberg demonstrated that electrons of a low energy diffraction pattern can be accelerated hy a potential difference of the order of a kilovolt so that the entire pattern can be seen on a fluorescent screen (2). Although this represented a potentially great simplification of experimcrital procedure, Ehrenberg did not develop the technique sufficicritly to make it practically useful. Hc did not discover patterns due to adsorbed atoms, nor did he obtain any new knowledge about a crystal surface. We have recently developed the Ehrenberg idea t o obtain a useful research tool, and the first tests with this equipment have given results even more interesting than we anticipated. Up to the present time it has been uwd in the study of clean surfaces of crystals of tungsten, nickel, and silicon and of adsorbed layers of gases upon these surfaces, only one specieb of gas atoms at a time. An obvious development for the near future is the extension of this work to the study of two gases simultaneously present in the apparatus, and doubtless to the coiiditioiis under which they react on the crystal surface.

I. Experimental Apparatus A sketch of one of the experimental tubes currently used in thc Hell Telephone Laboratories (3) is shown in Fig. 1. A beam of electrons of adjustable speed strikes the surface of the crystal under investigation at normal incidence. Electrons of the resulting diffraction pattern travel in a fieldfree space. Those within about 35" of the back-reflection direction pash through two fine mesh grids and are then accelerated by a potential difference of three or four kilovolts before they impinge on a fluorescent screen. The diffraction pattern can he observed arid photographed by looking past the crystal and directly a t the surface of the screen, as indicated in th r sketch. In Fig. 2 a photograph of the tube is shown with some of thc associated glass connections to a Vacion vacuum pump and to bottles of various gases which can be admitted through Granville-Phillips valves. This is essentially a "bac.k-reflection" diffraction apparatus, with observation limited to a cone of half-angle 35" about the back-reflection dircction. This limiting angle could be increased if the need arose, and it would also be possible by placing the grids and fluorescent screen differently to

L O W EXERGY ELECTROS DIFFRACTION

193

examine diffracted electrons coming from the crystal near grazing. An incident angle different from normal could also be used. Up to the present time the limitation t o normal incidence and to diffraction angles within 35" of the back-reflection direction has not hampered the work we have carried out. CRYSTAL POLISHED

\

ELECTRON LENSES

/

THORIATED FILAMENT

FIG.1. Sketch of low-voltage electron diffraction tube.

The crystal iii the photograph of Fig. 2 can be cleaned by sputtering in argon, and by heating t o high temperatures with the heating current flowing directly through the crystal. The sputtering gcar is not very novel, being modeled after that of J. T. Law (4). Modifimtions to adapt it to the present requirements will be described elsewhere.

I I . Oxygen on Nickel This apparatus has been used in a number of investigations, and others are in progress. All of these are basically quite simple. In fact, they have been purposely designed t o be as simple as possible. It has seemed to us that the first set of investigations must necessarily be concerned with a clean crystal surface, and the second set with atoms or molecules of a single gas upon a clean surface. Only after these have proved successful can one hope to study two gases on a surface. It is possible that the most important applications of this new technique will be in the field of catalysis and will involve gases of two sorts. We have the intention of carrying out such more complex studies, but have not yet got around t o them. The purpose of this paper is to bring the possibilities of the experi-

1!)4

L. H. GEItMER

11’1(,. 2 . Photograph of diffraction tuhr :md some of the associated c>cluipmcnt

mental method to thr attention of a iiew audience. To do this I shall siimmarizr here quite briefly thc observations which have been made in one single set of experiments. Other experiments have been carried out and published (Or), still othcrh arc in press, and some are at prrsrnt partially cwmplrt>ed. The work which I havc chosen to describe briefly is an iiivrstigation of the intrrwtion of oxygcri with a clean cube face of a nickel crystal(6). Diffraction patterns are observed from the crystal face before the admission of oxygen, then again during and after exposure to oxygen at various low pressures and for various t imes, and finally a t different stages while oxygen is being removed by heat.

LOU’ ESERGY ELECTItOY DIFFHACTIOY

195

A typical diffractioii pattern from the clean crystal is reproduced in Fig. 3. The fourfold symmetry of this pattcrn, corresponding t o the symmetry of the surfacc nickel atomh as shown in Fig. 4, is evident in the photograph. Thc azimuth of thc diffraction spots of Fig. 3 is designated A. As the voltage of the priniary clectron bcam is varied cont,iriuously from 0 to 500, diffrac$ioii beams appear alteriintely in the two principal azimuths, A and €3 in Fig. 4. Each beam reaches a maximum iritcrisity a t a particular voltagc arid these volt agcs represent electron wavelengths appropriate to Lauc diffraction beams from t h r rrystal. All of the diffraction beams correspond, withiti thc cxpcrirncwtal errors of the measurements, t o a crystal

Ii’IC;. 3 . Diffraction patterns from a clean (100) face of a nickel crystal. 144 volts, or I .02 A. The bright spots of this pattern are (711) Laue diffraction beams a t their maximum intensity.

iiiner potential of 16 volts. All possible diffraction beams have been found for potentials in the range up to 400 volts, and there are no unexplained hams. To study the adsorption of oxygeii, the gas is admitted through a Granvill(~-€’hillips valve to give a low controlled pressurc. The total exposurr to oxygen is appropriately measured by the product of the gas pressurc atid thc cxposure time. A coiivenierit unit of cxposure is lop6mni. Hg sec. which would be about, the exposure required to cover the crystal surface with a moriolayer if every molecule which collided with t h e surface were t o stic*kt o it. The prcseiicc of oxygeii adsorbed upoii the crystal surface is discovercd by the appearancc of iirw diffraction hcams which were not present before the admission of oxygen and which caiiiiot be accouritcd for by the struc-

196

1,. H. GERMER

ture of the nirkrl crystal. If oxygen atoms were adsorbed on the surface at the positions which would be occupied by an additional layer of nickel atoms, no new diffraction beams would appear and the foreign atoms would make their presence known only by changes in intensities. Adsorption studies would then be much more difficult than they are actually found t o he. The new diffraction beams, which appear when oxygen is admitked, represent lateral spacings between oxygen atoms larger than the smallest spacing between surface nickel atoms, which is interpreted to mean that an oxygen atom is larger than a nickel atom. 0

A

B

0

A-•

0

0

J

A

\

B

FIG.4. Sketch of the surface atoms of the nickel crystal, the orientation being the same a8 that of all of the diffraction patterns reproduced here.

The first new diffraction features appear a t an oxygen exposure of about mm. Hg sets. These first features correspond not t o a monolayer or more than a monolayer, as they well might from the magnitude of the rxposure, but t o only about one-tenth of a single monolayer. The probability that an oxygen molecule which strikes the surface sticks to it is thus much less than unity; in fact, it is about 0.01, which is now the probability that a molecule striking the surface dissociates and earh atom sticks to the surface separately. It has been found that these first-adsorbed oxygen atoms lie in bands parallel t o the two different (110) directions lying on the surface. The structure of these bands of adsorbed atoms is readily determined from the

4X

LOW ENERGY ELECTRON I)IF'FHACTION

197

FIG.5 . Diffraction pattern, a t 200 volts or 0.8G5 A., from oxygen atoms adsorbed upon the nickel surface in thc arrangement of Fig. 7-called the 4-Structure. The two very bright spots are (820) L a w diffraction beams from the nickel lattice a t their maximum intensity.

patterns, but consideration of them will he omit,ted here for the sakc of simplicity (6). With continued exposure to more oxygen a simpler structure develops as a monolayer becomes more nearly filled. The development is simplest when the crystal is maintained a t an elevated temperature of the order of 350°C,

FIG.6. Diffraction pattern, a t 72 volts or 1.44 A., from the nickel surface covered by atoms of oxygen arranged in the 4-Structure (Fig. 7), the same surface which gave the pattern of Fig. 5 . The single spot a t the top is a (511) Laue diffraction beam from the nickel lattice.

198

L. €1. GERMER

or is aiiiicaled occasionally at such a temperature. Under these coiiditioiib a vrry simplr arrangcmciit of adsorbed oxygen atoms is reached at a total cxpoxurr of :tbout 30 X mm. Hg sec. Diffraction patterns ohtaiiird a f t r r slwh cxposiirc :m rcprodurcd in Figs. 5 and 6. In thcsc pattcnis, at

.. .. . . * . o

.

.

.

0

0

0

0

0

0

0

0

0

.. .. .. .. .. .. .. .. .. .. .. ..

FIG.i . Skrtvti o f the arritngement of oxygen atoms in the nickel surface giving riso t o ttw patterns of Fig8. 5 and 6. ‘l’tio Mtrk dots itre surfare nickel atoms, and thr opc’u circles adxortwtl 0.uygc.n atoms.

rcbpectively 200 arid 72 volts, arc all of the possiblc diffractioii branis from a single laycr of atoms arranged as are the atoms in the topmost lnyrr of iiicakcl atoms but with the basic sparing of this layer just doublc that h-

PI(,.8. Diffraction pattern, a t the same electron wavelength as Fig. 5, of a f i l l 1 monolayer of oxygen atoms adsorbed upon the nickel surface-the 2-Strncture as shou II sc1itm:itirally in Fig. 10. twccii thc nivkel atoms. Since the arraiigemerit is similar with thc atom spacings doubled, the number of oxygen atoms in this layer is one-fourth of the number in a layer of nickel atoms. The arrangement is shown in Fig. 7 which can be callcd thc 4-Structure because of this ratio of 4.

LOW ENERGY ELECl‘IWX DIFFKACTIOS

l!N

Further oxygen exposure (total exposure about 100 X lop6mm Hg sec) results in disappearance of some of the iiew diffraction beams yielding thc patterns of Figs. 8 mid 9 which now correspond to the structure shown

PIG.9. Diffraction pattern, a t the same electron wavelength as Fig. 6, of a full moiiolayer of oxygen atoms (Fig. 10). As in Fig. 6, the spot a t the top is a (511) reflection from the nickel lattice.

schematically in Fig. 10, “thc 2-Structure.” No further atoms caii he adsorbed into this first layer, aud the structure of Fig. 10 is therefore called a monolayer.

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 0

0

0

0

0

0

0

0

0 0

0

0

0

FIG.10. Schematic arrangement of oxygen atoms on the nickel surface in the 2-Structure.

The structurc giving rise to the patterns of Figs. 8 and 9 is quite resistaiit to heat, not being changed by heating the crystal for a short time a t 830°C. Heating somewhat above this temperature (86OOC.) causes those rcflectioiis of Fig. 5 which are missing in Fig. 8 to reappear, and correspondingly the patterii a t 72 volts is now like that of Fig. 6 rather than Fig. 9. I t is clear that half of the adsorbed oxygcii atonih mere rcmovcd by this

200

L. H. OERMEU

heating. All of the adsorbed oxygen is removed by heating to a temperature only slightly higher (900OC.). It is because of this observed stability at low temperatures that the adsorbed oxygen is described as atomic rather than molecular. Although no more atoms of oxygen can be adsorbed into the first monolayer on a nickel surface, the arrangement of atoms sketched in Fig. 10 does not, by any means, represent the total amount of oxygen which can be adsorbed. With still further exposure to oxygen a pattern such as those of Figs. 8 and 9 becomes gradually weaker and finally disappears completely. At the same time new and very diffuse diffraction spots appear, becoming gradually stronger as the 2-Structure pattern becomes weaker. Such a new diffuse pattern is shown in Fig. 11. This patterii is due to several

FIG. 11. Diffraction pattern, at. 130 volts or 1.07 A., from several layers of oxygen molecules adsorbed upon thc riirkcl cqxtal. The pattern shows a set of diffuse spots in the B-azimuth at, their maximiini intensity.

layers of oxygen m o l e d e s adsorbed on top of the first monolayer of atoms. They can be completely removed by heat without in any way altering the arrangement of adsorbed atoms in the first monolayer. That these layers are molecules is inferred from this ease of removal. Some information regarding the structure of these molecular layers, and of their stability to heat, has been obtained ( 6 ) )but will be omitted here. An obvious continuation of these experiments is the admission of hydrogen to a surfare covered by oxygen. One can hope to find under what conditions the reaction between these gases is catalyzed by the crystal. Various int,eresting expcrimcnts have been thought of but none has yet been tried.

LOW ENERGY ELECTRON DIFFRACTION

20 1

REFERENCES 1. See, for example, Schlier, R. E., and Farnsworth, H. E., Advances in. Calalysiu 9, 434

(1957). 2. Ehrmberg, W., Phil. Mag. [7] 18, 878 (1934). 3. Germer, L. €I., and Hartman, C. D., Rev. Sci. Instr. 31, 784 (1960). 4. Law, J. T., Phys. and Chem. Solids 14, 9 (1960). 6. Germer, L. H., Scheibner, E. J., and Hartman, C. D., Phil. Mag. [8]6, 222 (1'360). 6. Germer, L. H., and Hartman, C. D., J . A p p l . Phys. 31, 2085 (1960).

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The Structure and Analysis of Complex Reaction Systems JAMES WE1

AND

CHARLES D. PRATER

Socony Mobil Oil Go., Inc., Research Department, Paulsboro, New Jersey Page I. Introduction ....................................... 204 11. Reversible ems. ...... A. The Rate Equations for Reversible Mo B. The Geometry of the System.. . . . . . . . . . . . . . . . . . C. The Structure of Reversible Monomole 111. The Determination of the Values of the Rate Constants for Typical Reversible Monomolecular Systems Using the Characteristic Directions.. . . . . . . 244 A. The Treatment of Experimental Data.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 B. Example of a Three Component System: Butene Isomerieation over Pure ....................................... 247 Alumina Catalyst ponent System. . . . . . . . . C. An Example of a IV. Irreversible Monomolecular Systems. . . . . . . . . . . . . . . . A. Geometric Properties of Irreversible Systems. ..... B. Experimental Procedures for the Determination of Characteristic Directions for Irreversible Systems and Applications to . . . . . . . . . . . . . . . . . . 285 Typical Examples ms . . . . . . . . . . . . . . . . 295 V. Miscellaneous Topics A. Location of Maxima and Minima in the Amounts of Various Species.. . . 295 B. Perturbations on the Rate Constant Matrix.. ........................ 302 C. Insensitivity of Single Curved Reaction Paths to the Values of the Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 Constants VI. Pseudo-Mass-Action Systems in Heterogeneous Catalysis. . . . . . . . . . . . . . . . . 313 A. Some Classes of Heterogeneous Catalytic Reaction Systems with Rate Equations of the Pseudomonomolecular and Pseudo-Mass-Action Form. 313 B. Systems with more than a Single Type of Independent Catalytic Site.. . 332 C. The Hydrogenation-dehydrogenation of CB-Cyclics over Supported Platinum Catalyst as a Pseudo-Mass-Action System.. ..................... 334 VII. Qualitative Features of General Complex Reaction Systems A. General Comments ............................................ 339 B. Constraints.. ......................... . . . . . . . . . . . 340 C. The Equilibrium Point in G x Reaction Systems.. . . . . . . . . 343 D. Liapounov Functions. . . . . . . . . . . . . . . . . . . . . . . . 344 E. Irreversible Thermodynamic 349 the Direction of the Reaction Paths.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..... VIII. General Discussion and Literature Survey. .

APPENDICES I. The Orthogonal Characteristic System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 A. Transformation of the Rate Constant Matrix into a Symmetric Matrix. 364 B. Transformation to the Orthogonal Characteristic Coordinate System.. . . 368 203

204

JAMES WE1 A N D CHARLES D . P R A T E R

C. Proof That the Characteristic Roots of the Rate Constant Matrix K are Nonpositive Real Numbers.. . . . . D. The Calculation of the Inverse Matrix X-l. . . . . . . . . . . . . . . . . . . . . . . . . . . 371 11. Explicit Solution for the General Three Component System, . . . . 111. A Convenient Method for Computing the Characteristic Vectors of the Rate Constant Matrix K.. IV. Canonical Forms.. . . . . . . . . . . . . . . . . V. List of Symbols. . . ......... References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390

I. Introduction I n catalytic and enzyme chemistry we often encounter highly coupled systems of chemical reactions involving several chemical species. It is an important purpose of chemical kinetics to explore and to describe the relations between the amounts of the various species during the course of the reaction, and to relate the concentration changes to a minimal number of concentration independent parameters that characterize the reaction system. Reaction kinetics provide an important part of the understanding of highly coupled systems and, in addition, provide the method for predicting their behavior. As is well known from previous attempts, the behavior of even linear systems containing as few as three reacting species is sufficiently complicated to make their basic dynamic behavior difficult to visualize. Chemical kinetics also plays a basic role in the study of the nature of catalytic activity. Studies of the catalyst and reactants in the absence of appreciable over-all reaction, such as studies of the electronic properties of catalytic solids or optical studies of adsorbed molecular species can provide valuable information about these materials. In most cases, however, kinetic data are ultimately needed to establish the relation and relevance of any information derived from such studies to the catalytic reaction itself. For example, a particular adsorbed species may be observed and studied by a spectral technique; yet it need not play any essential role in the catalytic reaction since adsorption is a more general phenomenon than catalytic activity. On the other hand, kinetics studies can provide information about the variation, as a function of experimental conditions, of the relative number of adsorbed species that play a basic role in the reaction. Consequently, such information may make it possible to identify which, if any, of the adsorbed species studied by the use of a direct analytical technique are relevant to the reaction. As another example, when studies are made of the solid state properties of a given catalytic solid, the question as to which, if any, of these properties are related to catalytic activity must ultimately be answered in terms of consistency with the observed behavior of the reaction system.

ANALYSIS OF COMPLEX REACTION SYSTEMS

205

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 qualitative 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 structural approach will also contribute to the analysis of the thermodynamics of nonequilibrium systems. It is the aim and purpose of thermodynamics to describe structural features of systems in terms of macroscopic variables. Unfortunately, classical thermodynamics is concerned almost entirely with the equilibrium state; it makes only weak statements about nonequilibrium systems. The nonequilibrium thermodynamics of Onsager (Z), Prigogine (2), and others introduces additional axioms into classical thermodynamics in an attempt to obtain stronger and more useful statements about nonequilibrium systems. These axioms lead, however, to an expression for the driving force of chemical reactions that does not agree with experience and that is only applicable, as an approximation, to small departures from equilibrium. A way in which this situation may be improved is outlined in Section VII. The major part of this article will be devoted to a particular class of reaction systems-namely, monomolecular systems. A reaction system of (n) molecular species is called monomolecular if the coupling between each pair of species is by first order reactions only. These linear systems are satisfactory representations for many rate processes over the entire range of reaction and are linear approximations for most systems in a sufficiently small range. They play a role in the chemical kinetics of complex systems somewhat analogous to the role played by the equation of state of a perfect gas in classical thermodynamics. Consequently, an understanding of their behavior is a prerequisite for the study of more general systems. Two subclasses of monomolecular systems will be discussed : reversible and irreversible monomolecular systems. A reaction system will be called reversible monomolecular if the coupling between species is by reversible first order reactions only. A typical example of a reversible monomolecular system is

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JAMES WE1 A N D CHARLES D. PRATER

where the ith molecular species is designated Ai. A reaction system will be called irreversibly monomolecular if some of the species are connected to other species by first order reactions that are irreversible. The presence of completely irreversible steps implies an infinite change in free energy and is consequently an idealization. Nevertheless, many reactions contain steps with a sufficiently large change in free energy so that irreversibility is an excellent approximation for them except in the neighborhood of the equilibrium point. The type of approach to be used and its advantage over the conventional approach is illustrated in Section II,A by a brief discussion of the problem of determining the value of the rate constants from experimental data for reversible monomolecular systems. Our discussion of monomolecular systems will also provide structural information about an important class of nonlinear reaction systems, which we shall call pseudomonomolecular systems. Pseudomonomolecularsystems are reaction systems in which the rates of change of the various species are given by first order mass action terms, each multiplied by the same function of composition and time. For example, the rate equations for a typical three component reversible pseudomonomolecular system are

In Eq. (a), ai is the amount of the species A ; , eij is the pseudo-rate-constant for the reaction from the jth to the ith species and is independent of the amounts of the various species, and 4 is some unspecified function of the amounts of the various species and time. This concept may be further generalized to give pseudo-mass-action systems. These are defined as systems in which all rates of change of the various species are given by mass action terms of various integral order each multiplied by the same function of composition and time. Pseudomonomolecular systems and pseudo-mass-action systems may arise when the reaction system contains quantities of intermediate species that are not directly measured and that consequently, do not appear

ANALYSIS O F COMPLEX REACTION SYSTEMS

207

explicitly in the rate expressions. These unmeasured species may include adsorbed species on the active sites of a solid catalyst; hence, heterogeneous catalytic systems will often follow rate laws of the pseudo-mass-action form. This characteristic of many heterogeneous catalytic systems makes it possible to simplify their treatment by separating the problem into two parts, each of which can be independently studied. The mass action part can be studied as if the system were a homogeneous reaction between the measured species as will be shown in Section II,B,2,i and Section VI. Hence, contrary to first impressions, the understanding and formulation of mass action kinetics for highly coupled systems play an important role in the understanding of heterogeneous catalytic reactions. Some conditions that lead to pseudo-mass-action kinetics in heterogeneous catalysis will be discussed in Section VI. This article is designed to serve a multiplicity of purposes and unfortunately does not escape the weaknesses inherent in such multiplicity. Some comments on the handling and application of this material may prove useful to the reader. Many of those who might find useful application for the results and methods presented herein may have only a limited acquaintance with the linear algebra used in the detailed applications. Consequently, most of this linear algebra is presented in terms of the geometrical concepts arising from the kinetic problem.* The reader, therefore, need not have specialized preparation in linear algebra and the need to consult works on abstract algebra is minimized. Detailed examples are given to provide practice in the use of the procedures. Matrix notation is used for the manipulation of the geometrical interpretation; the computation procedures for the matrix operations are presented in footnotes where they first occur in the text. The development of the main ideas are presented in Sections 11, IV,A, VI,A, and VII. The detailed examples are contained in Sections 111, IV,B, and VI,B and are not necessary for the main development. These examples are built around the determination of rate constants from experimental data. This should not be considered to mean that this is the only, or even the most important, use that can be made of this approach to reaction rate problems. The reader unfamiliar with linear algebra should, on the first reading of the main development, ignore the algebraic formalism as much as possible and think in terms of the geometric interpretations. In this respect Section VI is the most tedious since it involves considerable algebraic

* The geometrical approach, in t e r m of the kinetic problem, to linear algebra should make this useful branch of mathematics more appealing to the experimentalist. In fact, the ease with which the results and methods may be visualized in geometrical t e r m makes it a natural mathematics for the experimentalist.

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JAMES WE1 AND CHARLES D. PRATER

manipulation. This section is, however, of importance to the investigator in heterogeneous catalysis. The reader familiar with linear algebra may obtain the main points of the development from Section II,A, Section II,B,2,c,d,e,g,j, Section lI,C, Appendix I, Section IV,A, Section VI,A, and Section VII. The relation of the results of previous investigations to the results presented in this article is best understood against a background of the complete picture. It is for this reason that few references to previous work will be given before Section VIII, which contains a historical survey and a discussion of the relations of previous results to the results presented in this article.

II. Reversible Monomolecular Systems A. THE RATE EQUATIONS FOR REVERSIBLE MONOMOLECULAR SYSTEMS 1. The General Solution

Let the ith species of a monomolecular reaction system be designated by Ad and the amount by ai. Let the rate constant for the reaction of the ith species to the jth species be kji, i.e., Ai 3 Ai; there will be no rate constants of the form hi. Using this system of notation, the most general three-component monomolecular reaction system is kn

I

k:#

Ar-

A2

(3)

A3 The rate of change of the amount of each species in scheme (3) is given by da1 -= dt

- (kzl

+

k31h

+ klzaz + h a 3

The right side of the set of Eqs. (4)is written so that the various species are in numerical order-all az, and then a3. The negative term on the right of the ith equation of Eqs. (4)is the sum of the reaction rates away from the ith species and the remaining terms are the reaction rates of each. jth species back to the ith species.

ANALYSIS OF COMPLEX REACTION SYSTEMS

209

The structure of Eqs. (4)leads to the generalization for n-component systems,

... ... ... ...

... ...

... ... ... ...

... ... j=l

where the absence of rate constants of the form kii from each summation term is signified by the notation ', i.e., Z'j,lflkj; is the sum of the rate constants k j i for all j from 1 to n except j = i. The general solution (3-6) to a set of linear first order differential equations such as Eqs. ( 5 ) is well known; it is

+ + a , = cmo + a, = + al a2

= c10 = czO

cf10

clle-xlt c21e-xlt

c,le-xlt.

cnle-xlt

+

+ + ... ... . . .+ ...

. . ... . . + c2(m-l)e-X--1t . . . cl(m-l)e-xm-lt

. .

.+ .... . ..

cm(m-l)e-hm-lt

. . . + c,(m-l)e-Xm-lt.

... ..

cl(n-l)e-xn-lt

~ 2 ( ~ - ~ ) e - ~ n - 1 ~

cm(n-l)e-Xn-li

cn(n-l)e-xn-It

where cji and X i are constant parameters related to the rate constants. Procedures for calculating the values of the constants (c, A) from known values of the rate constants can be found in many standard works on chemical kinetics or ordinary differential equations (3-6). Using the values of the constants (c, X ) determined by these procedures the time course of the reaction-that is, the amount a; as a function of time-is easily computed. But the inverse process of determining the rate constants k j i from the experimentally observed time course of the reaction has presented difficulties. 2. Dificulties in Determining the Values of the Rate Constants from Experimental Data

The rate constants k j i may be determined directly from the rate Eqs. ( 5 ) by measuring the initial rates of formation of the various species Aj from pure A i . The difficulties encountered in obtaining the accuracy needed in the chemical analyses for points sufficiently close to zero time

2 10

JAMES WE1 AND CHARLES D. PRATER

limits the use of this method. A complete set of consistent and accurate rate constants will not, in general, be obtained for complex systems. Furthermore, the evaluation of the rate constant from initial slopes is very sensitive to errors in contact time. To derive the rate constant from the general solution [Eq. (S)] using experimental observations of the time course of the reaction requires (1) the determination of the set of constants c and X and (2) the derivation of the rate constants from this set. In the conventional solution, the constants (c, A) are not quantities that are directly measured in an experiment but are usually obtained from curve fitting techniques applied to the experimental data. The hazards of using curve fitting techniques when the data involve more than a single exponential term are often not recognized. Although the constants obtained may give a solution that fits the experimental data of composition vs time, used for their evaluation, as satisfactorily as the true solution, their values may have little resemblance to the true values and they are useless for predicting the course of the reaction for initial compositions differing appreciably from that used in the evaluation of the constants. Unless advantage is taken of special features of the solution, either the data must be excessively accurate or the number of data excessively large for meaningful values to be obtained for the constants in the general case. A detailed discussion of the problem is given by Lanczos (7). Additional discussion will be found in Sections V,C and VIII. In the conventional treatment of the kinetics of monomolecular systems, the explicit relations of the rate constants kji to the set of constants ( c , X) are obtained only in special cases; consequently, even assuming that the constants (c, X) are satisfactorily obtained, the calculation of the values of the rate constants from them is not possible, in general, for the conventional treatment. Although the values of the constants (c, X) are sufficient for determining the composition as a function of time, the rate constants kji are more useful quantities since they are the ones more directly related to basic mechanisms. That these difficulties are well recognized is illustrated in “The Foundations of Chemical Kinetics” by Benson (6) when he writes, The chief difficulties with such complex reaction systems arise not so much from the mathematical solutions but from the application of the solutions to data when the experimental rate constants are unknown. No general methods have yet been devised for such applications, and the case8 treated have been attacked more or less by trial and error and a judicious choice of experimental conditions.

3. Nature

05 the New Method

We shall show that the analysis of the structure of kinetic systems can provide such a general method. Since the new method arises from an under-

ANALYSIS OF COMPLEX REACTION SYSTEMS

211

standing of the structural features of the systems, the search for the method provides an excellent framework for the structural discussion. It must be remembered, however, that the insight obtained from the general analysis is much more broadly useful than merely providing a method for the extraction of the rate constants from experimental data. In the new method, quantities that correspond to the constants cji and X i in Eq. (6) are determined; but in addition, their relation to the rate constants k j i also appears. The method is one which is best suited for the experimentalist since it suggests experimental procedures that yield the necessary information for the determination of the values of the rate constants from a minimal number of data. Furthermore, only a relatively small amount of computation is required to obtain the values of the rate constants from these data. Let us now examine briefly the approach provided by the structural analysis. An examination of Eq. (5) shows that the rate of change of the amount ai of each species depends not only on ai but on the amounts aj of other species as well. Thus, changes in the amount of Aj during the reaction affect the amounts of species A i ; there i s strong coupling between the variables in the set of Eqs. (5). It is this coupling between the variables a; and a j that is the source of the difficulties outlined above. We shall show that a monomolecular reaction system with n species A i can be transformed, by means of appropriate mathematical operations (which involve only addition and multiplication) , into a more convenient equivalent monomolecular reaction system, with n hypothetical new species Bi,which has the property that changes in the amount bi of any species Bi does not affect the amount of any other species Bj. This means that there i s a set of species Bi equivalent to the set of species Ai such that the variables bi in the rate equations for the B species are completely uncoupled. For example, there is a three component reaction system with species Bo, B1, and Bz equivalent to the reaction system Eq. (3) such that

Bo does not react

The rate equations for Eq. (7) are

212

JAMES W E 1 AND CHARLES D. PRATER

They are a set of simple completely uncoupled differential equations. The scheme (7) can be readily generalized to n-component systems. A hydrodynamic analogue for the three component system is shown in

A,

A2

A3

FIG.1. Hydrodynamic analogue of a three component reversible monomolecular reaction system.

Fig. 1 and serves to further illustrate the transformation. The bottoms of three cylinders are connected together by three tubes such that the rate of flow of fluid between each pair of cylinders is proportional to the difference

FIG.2. Hydrodynamic analogue of the equivalent three component system given by scheme (7).

in heights of the fluid; the proportionality constant is analogous to the rate constants for the chemical system. Let the cross section area of the ith cylinder be proportional to the equilibrium amount of the chemical species Ai. I n this model the volume of fluid is analogous to the amount of

ANALYSIS OF COMPLEX REACTION SYSTEMS

213

the species A i in the chemical reaction. This model leads to a set of rate equations similar to the set of Eqs. (4).The transformation of reaction scheme (3) to reaction scheme (7) is analogous to the transformation of the hydrodynamic system of Fig. 1 to the simpler hydrodynamic system given in Fig. 2. This simpler hydrodynamic system consists of a single static cylinder and two unconnected cylinders leaking a t the bottom. It is, obviously, much easier to study than the original system. We shall show (1) that the transformation required to change a given composition from the A to the B system of species (and vice versa) can be easily determined from appropriate experimental data, (2) that the rate constants hi for the B system of species can then be measured, and (3) that the measured rate constants X i for the B system can then be changed to the rat'e constants k j i for the A system by the same experimentally measured transforms obtained in step (1). Thus, the rate constants kji can be derived from experimentally measured rate constants X i and transforms.

B. THEGEOMETRYOF

THE

SYSTEM

1. Some Elementary Geometric Properties of the System

A geometrical interpretation is facilitated by expressing the sets of Eqs. (4) and ( 5 ) in matrix form. This change represents a distinctly new point of view and is not used merely as a shorthand notation for these equations. The three-component system will be used to illustrate some basic properties, followed by the generalization to n components. Equation (4)in matrix form becomes

* The product of a column matrix y having n elements by a square matrix G containing n X n elements yields another column matrix n having n elements. We shall designate (1) the i t h element of the matrix n by qi, (2) the elements of the matrix G in the ith row and j t h column by Gij and (3) the j t h element of y by "/i. The product of a column matrix 'y by the ith row of the matrix G gives the ith element of the column matrix n and is defined as the sum from j = 1 to n of the products of the j t h element of y by the j t h element of the ith row of the matrix G, i.e.,

214

JAMES WE1 AND CHARLES D. PRATER

The column matrices

may be interpreted as vectors in three dimensional space. Let the column matrix

be designated by a. Figure 3 shows a three-dimensional coordinate system with the species A i as axes and a as a vector directed from the origin to the composition point with coordinates (al, a2, as) on their respective axes.

FIG.3. The composition space for the general three-component system. A composition vector a with components al, a2, and a8 is shown.

This set of coordinate axes defines a composition space for the whole reacting system and the vector a, terminating at the composition point, is the composition vector. The column matrix

ANALYSIS O F COMPLEX REACTION SYSTEMS

215

may be written da/dt and interpreted as the time rate of change of the composition vector a in composition space. Thus, instead of considering the amount of each component ai separately as is done in the set of Eqs. (4),the composition of the reacting systems at a n y particular time t i s now treated as a n entity, i.e., the vector a(t). Let the square matrix in Eq. (9) be designated by K.

- (kZl

K=(

+ k2l

ki2

k31)

-

+

(kC2

k3l

k32

k32)

-

(ki3

24- )

(10)

kz3)

The matrix K may be thought of as an operator or transform that changes vectors into other vectors; it may, therefore, be treated as an entity. The set of Eqs. (4)then reduces to the single equation da _ -- K a

dt

There are two constraints on these reaction systems: (1) the total mass of the reaction system is conserved (law of conservation of mass); and (2) no negative amounts can arise. It will be convenient to manipulate the amount of the various species as mole fractions so that the law of conservation of mass is given by

&=1 i=l

Condition (2) gives

ai

20

(13)

for all values of i. Let us examine the geometrical effects of these constraints. The constraint, given by Eq. (12), confines the end of the vector a to the plane passing through the points (1, 0, 0), (0, 1, 0), and (0, 0, 1);the constraint given by Eq. (13) further confines the end of (Y to the equilateral triangle defining that part of this plane lying in the positive octant of the coordinate system A1, Az, and A 3 as shown in Fig. 4. This equilateral triangle will be called the reaction triangle and the plane on which it lies the reaction or phase plane. As the reaction proceeds, the composition point, at the end of the composition vector a ( t ) , moves along the reaction plane towards the equilibrium point a t the end of the equilibrium composition vector a* with component al*, u2*, and a3*. The curve that the composition point traces out as it goes to equilibrium lies on the reaction triangle and is sufficient to describe the composition change during the course of the effective reaction. This curve will be called the reaction path for the particular starting composition a(0). Thus, the reaction plane with one

216

JAMES WE1 AND CHARLES D. PRATER

A2

FIG.4. The composition space for a three-component system showing the reaction ai = 1 triangle to which the end of the vector a(t) is confined by the conditions 2i-1~ and ai 2 0. The equilibrium vector a* is indicated. The curve - - - , lying on the reaction triangle, represents a typical reaction path along which the composition point at the end of the vector a@) moves to the equilibrium point at the end of the vector a*.

FIG.5. Some reaction paths on the reaction triangle for a typical three component system. The rate constants for this system are A,

-

10 u A2

ANALYSIS OF COMPLEX REACTION SYSTEMS

217

dimension less than the composition space is sufficient to describe many properties of the system and will be used often in the treatment to follow. Typical reaction paths on the reaction triangle are shown in Fig. 5 for a typical three-component system. The n-component monomolecular system may be treated in exactly the same manner except that an n-dimensional composition space is used. Although n-dimensional spaces with n > 3 cannot be simply put into pictures, a geometrical language still aids our ability to solve problems using the concepts, language, and techniques of two and three dimensional systems. The set of Eqs. ( 5 ) reduces to a single equation identical to Eq. (11) except that (Y is now the column matrix or vector in n-dimensional space given by

(“)

(Y=

a,

and K is the square matrix given by

-

(ifkjl)

k1z

. ..

kl,

.

9

.

kl,

j =I

1)

...

K=

... ... ...

... .n

kml

k,z

...-

... ... ... ... knl

kflz

,

( X I

j=1

”,). . . ... ...

k,,

.. .-

Ern,

(2‘k j f l ) j=1

Analogous to the three-component system, constraint (12) confines the end of the vector (Y to the (n - 1)-dimensional “plane” passing through the ends of the n unit vectors along the n coordinate axes, Ai.Constraint (13) further limits the composition point at the end of the vector (Y to that part of the “plane” lying in the positive orthant of the n-dimensional coordinate system. This part of the “plane,” which forms the (n - 1)-dimensional equivalent of an equilateral triangle for three components and a tetrahedron for four components, is called a simplex. The reaction paths in this system will be curves lying within the reaction simplex.

218

JAMES WE1 AND CHARLES D. PRATER

2. The Relation ,of the Rate Constants to Geometric Properties of the System

a. Characteristic directions in composition space. As pointed out in Section II,A the source of the difficulty with the solution of Eq. ( 5 ) is in the strong coupling between the variables ai. It was also stated that the difficulty can be overcome by transforming compositions in the system of A species to compositions in an equivalent system of hypothetical species with rate equations containing completely uncoupled variables. We shall now show that this equivalent system of hypothetical species exists and demonstrate its properties. To do this we need a geometrical interpretation of the coupling between the variables ad. According to Eq. (11), multiplying the vector a by the square matrix K is equivalent to computing a new vector that is the time rate of change of a. If the elements of K are converted to dimensionless quantities by dropping the units sec-I, the matrix K becomes an operator that transforms the vector a, by rotating it and changing its length, into a new vector a'

I I

A2

A,

FIG.6. The interpretation of the matrix IS as an operator or transform which changes the vector ri into the vector a'.

(= da/dt with dimensions ignored) in composition space as shown in Fig. 6. This dimensionless K will be used in much of the development to follow without explicit statements to that effect since those instances where the physical dimensions of K are needed are readily apparent. After an increment of time dt, a vector (Y will change into the vector (Y da. Multiplying both sides of Eq. (11) by the scalar dt (now considered dimensionless) gives

+

da = Kadt = a'dt

(16)

ANALYSIS O F COMPLEX REACTION SYSTEMS

219

Since dt is an infinitesimal scalar multiplier of a’,Eq. (16) shows that the vector da is an infinitesimal length of the vector a’. Let us examine what happens when a composition vector containing only one species, the ith, reacts;

a =

()

The set of Eqs. (5) shows that when a pure component reacts it produces changes in the amounts of other components in addition to changes in a i ; hence, the vector d a derived from a pure component vector must contain other components. The vector a’,of which d a is an infinitesimal length, is derived from the vector a by two geometric changes: (1) a change in length which cannot introduce a new component into the vector and (2) a rotation which can. Consequently, pure component composition vectors for any species Ai must always be rotated by K. Thus, the geometrical manifestation of the coupling between the variables in Eq. ( 5 ) is the rotation which pure component composition vectors undergo when transformed by the matrix K. In addition to the pure component vectors, most of the other composition vectors are also rotated by the matrix K.For reversible n-component monomolecular systems, however, there always exists n independent directions in the composition space such that vectors in these directions will undergo only a change in length under the action of K (see Appendix I for proof). These will be called characteristic directions. Let a; be any vector in the jth characteristic direction, then

Kajt

=

-XXjajt

(I@*

where X i is a scalar constant. The vectors a: are called characteristic vectors or eigenvectors and the scalar constants - X j are called characteristic roots or eigenvalues of the matrix K. I n Section II,B,P,k, the characteristic roots of the rate constant matrix K are shown to be the negative of the decay constants X i in the set of Eqs. (6). In Appendix I, C these characteristic roots are shown to be nonpositive numbers. Hence, we shall always write the characteristic roots of the rate constant matrix K as -hj where hj is a positive real number or zero. The negative sign in Eq. (18), which means that the vector a’ undergoes a reflection as well

* In calculating the product of a matrix (or vector) by a scalar quantity, each element of the matrix (or vector) is multiplied by the scalar.

220

JAMES WE1 AND CHARLES D. PRATER

as a change in length under the action of the matrix our arguments. Combining Eqs. (11) and (la), we obtain

K,does not

change

The characteristic directions, therefore, have the very desirable property that the rate of change of a!: depends only on a!:; it is completely uncoupled from vectors along other characteristic directions. The characteristic directions can be interpreted as representing pure components in the following manner. Any set of n independent coordinate axes may be used to provide the components for the representation of a vector as a column matrix. Therefore, the n independent characteristic directions can equally well serve as coordinate axes for composition space instead of the first choice. This first choice was made by interpreting the set of pure components A i to be the coordinate axes; this choice will be designated the natural or A system of coordinates. We shall choose the n characteristic directions as a new set of coordinates and, reversing the procedure in the first choice, interpret them as a set of hypothetical new species, Bj. We shall designate this the characteristic or B system of coordinates. We may also consider B, as a special package of A i molecules because in the reaction they transfer as a unit. Let some particular vector in each of the jth characteristic directions be chosen as a unit vector for this direction. The amount of each of the new characteristic species Bj, expressed as multiples of the unit vector in the j t h direction, will be designated by b j , and the composition vectors expressed as a column matrix in the B coordinate system by 0, i.e., for an n-component system @ is the column matrix

The round brackets of Eq. (14) and the square brackets of Eq. (20) are used to distinguish between column matrices written in the A and B systems respectively. It must always be remembered that we are interpreting a! and @ as different representations of the same vector obtained by changing the coordinate axes while the vector remains fixed in space (alias transformations). This is in contrast to an interpretation in which the coordinate axes remain fixed in space but the vector moves (alibi transformations). An example of

ANALYSIS O F COMPLEX REACTION SYSTEMS

221

the latter is the interpretation of the action of K on Q: as a transformation of a into a new vector a’ in the same coordinate system as shown in Fig. 6 . Figure 7 shows the resolution of a composition vector in both the A and B systems for two components. Negative amounts of the characteristic species

FIG.7. A two dimensional composition space showing a composition vector resolved into components in the coordinate system of the species A1 and A ) and in the coordinate system of the hypothetical species Bo and B1.Note that in the B coordinate system negative concentrations @I) can arise and that the coordinate axes of the B system are not a t right angles to each other.

exemplified by bl in Fig. 7, will cause no difficulty since they do not represent actual chemical species. Note that the B coordinates of the example in Fig. 7 are not orthogonal to each other; the B coordinates are not required to be and, in general, will not be orthogonal. Let the unit vector in the jth characteristic direction expressed as a column matrix in the A system of coordinates be designated xi. Remembering that alpha is always used to designate a composition vector expressed as a column matrix in the A system of coordinates, then, for any vector a] in the jth characteristic direction, we have Bj,

Q:jt=

bI.x.1

(21)

Substituting the value of a:, given by Eq. (21), into Eq. (19), we obtain

222

JAMES WE1 AND CHARLES D. PRATER

since the unit vector x j is constant. Hence,

Thus, the rate of change of the amount of the pure species B , is completely independent of other B species. Since each characteristic direction will give a differential equation in the form of Eq. (23), we have

... ... ...

..

(24)

... ... ...

dbn-l - -Xn-lbn.-l dt

Therefore, the rates of change of the various pure species in the B system are given by the set of simple completely uncoupled differential equations, Eqs. (24), in contrast to the rates of change of the various pure species in the A system, which are given by the set of highly coupled differential equations, Eqs. (5). b. The Solution for Monomolecular Reaction Systems in Terms of the Characteristic Species. The set of Eqs. (24) may be written in matrix form, analogous to Eq. (1l), as

where A is the rate constant matrix for the B system of species equivalent to the rate constant matrix K in the A system. In this case, however, the rate constant matrix is the special n x n diagonal matrix (all diagonal elements are lambda's, all other elements zero)

It is shown in Appendix I that all the characteristic roots are real numbers 6 0 . Therefore, the solution to the set of Eqs. (24) is

ANALYSIS O F COMPLEX REACTION SYSTEMS

223

where bjo is the value of bj at time t = 0. According to Eq. (27), when X j > 0 the amount bj of this species reacts away to zero concentration as t 3 0 0 . The law of conservation of mass must hold for the B system and the amounts bj of all the B species cannot be zero simultaneously. It follows, therefore, that at least one of the characteristic roots, say -Ao, must be zero so that bo = b00 at all times. At equilibrium, (dai*/dt) = 0 for all ai*.Therefore, Ka* = O$ = Oa*; consequently, the equilibrium vector a* is a characteristic vector of the system and has a characteristic root of zero. We shall limit our attention to reversible systems in which it is possible to go from any species A i to any other species Aj either directly or through a sequence of other species. Such systems do not contain subsystems that are isolated from each other and each system has, therefore, a unique equilibrium point. For such systems, there can be no other characteristic vectors with X = 0 since the equilibrium vector, which does not decay, already accounts for all the mass in the system. Let this equilibrium species correspond to the species Bo; then the first equation of Eqs. (24) is replaced by

dbo-0 dt and its solution in Eq. (27) by

bo

=

boo

(29)

Hence, for three component reactions, scheme (3) is replaced by the simple equivalent scheme XI

B1-+ 0

t

(B, does not react)

B2 All the mass in the system is accounted for by the equilibrium species

Bo; the other characteristic species do not account for any mass and must,

224

JAMES W E 1 AND CHARLES D. PRATER

therefore, be excess species that measure the departure of the reacting system from equilibrium. This places certain restrictions on the elements of the unit characteristic vectors, which will now be discussed. The unit characteristic vectors xi are shown in Fig. 8 for a typical three component

FIG. 8. The unit vectors of the B coordinate system of a typical three component reaction showing their resolution in the A coordinate system.

system, where the component of xj along the coordinate A i is designated xij. Since the species B j , j # 0, does not contain any of the mass of the system, the elements of each unit characteristic vector other than xo must satisfy the condition

2

2ij

= 0;

j #0

i=l

Since Bo contains all the mass in the system, the elements of the vector must satisfy Eq. (12); hence,

2

xi0

xo

=1

i=1

It follows that all vectors xj other than xo must contain elements that are negative amounts as can be seen in Fig. 8. They, therefore, cannot lie in the positive orthant of the A coordinate system and by themselves do not represent realizable compositions. The important point is that, in spite of this, the vectors xi are directly determinable in terms of realizable initial composition vectors of certain special reaction paths as will be shown in Section II,B,2,d.

225

ANALYSIS O F COMPLEX REACTION SYSTEMS

c. Transformation of Compositions Between the Natural and Characteristic Coordinate Systems. To take advantage of the simplicity offered by the rate laws of the B species, a method is needed to convert compositions from the A to the B system of coordinates and vice versa. Any vector (Y is equal to the sum of a set of vectors a: along the characteristic directions; that is n -1

a = zai’

(33)

j =O

Substituting the value of a] [given by Eq. (21)] into Eq. (33), we obtain n-1

Writing Eq. (34) in terms of the components of the vectors, we have n -1

al =

2 bjxl:lj

j =O n-1

bizzi

a2 = j=O

(35)

.. ..

n-1

which are the equations for matrix-vector multiplication given in the footnote on page 213. Thus, a =

X@

(36)

where X is the matrix

Since Xj=f)

(38) Xnj

the matrix X is formed by writing the unit characteristic vectors by side;

x = ((XO)

(XI)

(x2)

...

(xn-l>)

xj

side

(39)

226

JAMES WE1 AND CHARLES D. PRATER

where the round bracket on the sides of each vector is used to emphasize that they are written as a column matrix in the A coordinate system and not as a row matrix. Thus, the matrix X,formed from the unit characteristic vectors xj, transforms a composition vector written as @ in the B system into the same composition written as (Y in the A system of coordinates. The matrix to transform the composition vectors from the A to the B system is also needed. It is related to the matrix X in the following manner. Let the matrix that transforms IY into @ be designated X-l; then X-la = @ Substituting the value of

(Y

_

(40)

given by Eq. (36) into Eq. (40), we obtain X-l(X@) = @

(41)

Equation (41) shows that the matrix X-' counteracts the effects of the matrix X on @.It also shows that x-'X

=

I

(42*)

where I is a matrix whose action on a vector is to leave it unchanged and is, consequently, an identity matrix. For a n n-component system, I is the n X n diagonal matrix (diagonal elements unity, others zero)

I=

(i

0 0 0

1

;)

... 0

(43)

. . . . .. . . 0 0 0 0

...

Equation (42) gives the relationship of the matrix X-l to the matrix X. For these particular transformation matrices, there is a simple method for calculating X-' from X that involves a further transformation of the B coordinate system and is given in detail in Appendix I. The calculation is made in the following manner: The diagonal matrix L is computed from

XTD-lX =

("; . ;;j

. . ... " . )-. 0 0 . . . 1,-1

(44)

* Since a matrix with n columns may be considered as composed of n column vectors written side by side as in Eq. (39), the matrix-matrix multiplication needed in Eq. (42) and later may be treated as repeated matrix-vector multiplication. The product of two n x n matrices is another n X n matrix since each matrix-vector multiplication produces another vector.

ANALYSIS O F COMPLEX REACTION SYSTEMS

where XT is the transpose1 of the matrix

0

227

X and ...

0 (45)

... ... The inverse matrix X-l is given by

X-1

=

L-1XTD-1

where L-’ is the inverse of the diagonal matrix L and is given by 1 - 0

...

10

L-1

=

(

1 0 I1

... .. .. ... ...

0 0

-i

(47)

1,-1 1

d. The Relation of the Unit Characteristic Vectors to Straight Line Reaction Paths. The unit characteristic vectors xj are directly related in a simple manner to straight line reaction paths in the reaction simplex. We shall use the general three component system to demonstrate this relation. The general n-component system follows logically from the three component system and will not be discussed. The end of the vector xo lies on the reaction triangle because it represents a real composition. Since the other unit characteristic vectors do not contribute to the mass of the system, they can have no components along the normal to the reaction plane and, therefore, must lie on a plane parallel to it. Hence, if x1 and xz are moved to the end of xo at the equilibrium point E, as indicated by x ’ ~and x’Z in Fig. 9a, they will lie entirely on the reaction plane. $ The transpose of an m X n matrix is the n X m matrix formed from it by interchanging rows and columns. For a vector written as a column matrix, the transpose is the vector written as a row matrix. Let the element in the ith row and j t h column of the matrix G be designated (G)+ The elements of the transpose matrix GT are related to the elements of the matrix G by the equation

(GT)<j= (G)ii

228

JAMES WE1 AND CHARLES D. PRATER

Except for the vector X O , we have not as yet specified the lengths of the vectors xj, which are to serve as the unit vectors of the B system. These

FIG.9. A typical three component system with equilibrium composition point E. The characteristic vectors are &, xl,and XZ. The translations of XI and xz to the end of xo form the vector sums ( ~ ~ ~=( xo 0 )+ X I and a,,(O) = xa XZ. The translated vectors X’I and X‘Z represent straight, line reaction paths along which the initial compositions cu,,(O) and az,(0)go to equilibrium. The extension of these vectors shown by X”I and x”Z in Fig. 9b also represent straight line reaction paths for two other initial compositions corresponding to choices of XI and XZ in the direction opposite to the first choice.

+

will be chosen, for convenience, such that the ends of the vector x ’ ~and X‘Z lie on the boundary of the reaction triangle as shown in Figs. 9a and b. Then the vector sums$ az1(0) = xo x1 (48) a,(O) =

+ xo +

x2

(49)

represent real compositions. At least one element of each vector azi(0)will be zero because of the above choice in length of the characteristic vector xi,i # 0, which causes a,,(O) to terminate on the boundary of the reaction triangle. Substituting the value of bj from Eqs. (27) and (29) into Eq. (34), we obtain

+ bloe-Xlfxl . . . + bm-loe--Xm-lLx,-l. . . + bn-lOe--Xn-ltx,-l

a(t) = booxo

(50)

$ Matrix addition is defined only between two matrices of the same “size,” i.e., between two m x n matrices. The ijth element of the sum is obtained by adding the corresponding ijth elements of the two matrices. Thus, for the special case of a vector, the j t h element of the sum is obtained by adding the corresponding j t h elements of the two vectors.

ANALYSIS O F COMPLEX REACTION SYSTEMS

The initial value of a(t) is a(0) = boOXo

+ blOXl. . . +

b,-PX,-I.

. . + bn.-?Xn-l

229

(51)

Equations (48) and (49) are Eq. (51), for n = 3, with boo = blo = 1, b2O = 0, and with boo = bzo = 1, b? = 0, respectively. Using these values, Eq. (50) gives the equation for the movement of these two composition vectors into a*,

aZ,(t)= xo aZ,(t)= xo

+ + e-x2t~2 e-X1%

(52) (53)

Hence, composition points at the ends of the vectors az,(t) and aZ,(t) shift with time to the equilibrium point along the straight lines x ’ ~and X’Z respectively; the displaced characteristic vectors x ’ ~and x ’ ~ are, therefore., straight line reaction paths for these compositions. All straight line reaction paths must be derived from characteristic vectors displaced along xo since all such paths are expressible in the form of Eqs. (52) and (53). From the generalized form of Eqs. (48) and (49), we have

xi

=

(Uz;(O)

- xo

(54)

We see that the problem of determining the unit characteristic vectors becomes that of determining the composition vectors azi(0), which will be called the ith characteristic composition vector, and the equilibrium composition vector xo. The characteristic composition vectors are completely specified in the composition space alone. Consequently, the reaction time is not needed in the determination of the unit characteristic vectors; they can be determined from a knowledge of the various compositions through which a given initial composition passes on its way to equilibrium and do not depend on a knowledge of the value of the reaction time at which a particular composition occurred. The vectors xland xz could have been chosen to have a direction opposite to the choice made in Fig. 9a. This choice corresponds to the displaced vectors x ” ~and x ” in ~ Fig. 9b. They form with xfland X‘Z two straight lines that extend across the reaction triangle and that intersect at the equilibrium point as shown. Either of the unit characteristic vectors corresponding to x ’ ~and xfflmay be combined with either of the unit characteristic ~ the vector b to give a matrix X vectors corresponding to xfz and x ” and for making the required transformations. e. The Equations for the Reaction Paths in Terms of the Characteristic Species and the Determination of the Characteristic Roots. The displaced unit characteristic vectors that form the straight line reaction paths become the coordinate system for the characteristic species Bj in the

230

JAMES WE1 AND CHARLES D. PRATER

reaction simplex and has the equilibrium point as its origin. Since the reaction simplex has one dimension less than the composition space, one B coordinate is deleted in this description of the system; it is the coordinate corresponding to the species Bo. Since the species Bo does not decay with time and contains all the mass in the system, this deletion does not matter when we describe changes in the system in terms of the characteristic species, and this description is in terms of massless quantities that measure the departure of the system from equilibrium. A vector directed from the origin of this coordinate system to another point in the reaction simplex will describe the system at any moment. The components of this vector in the displaced B system of coordinates are the amounts bj, j # 0 (as shown in Fig. lo). The reaction paths are the paths that the ends of such

B

FIG. 10. The straight line reaction paths aa coordinate axes for the characteristic species Bj, j f 0, in the reaction simplex.

a vector take as it decays to zero length. This decay is described in terms of the components of the vector by the set of Eqs. (27) with the equation for bo omitted. This set of equations is a parametric representation, with time as the parameter, for the reaction path in the coordinate system provided by the straight line reaction path in the reaction simplex. The time, however, may be eliminated by using the amount of one of the B species, say b j , as the parameter. Consider the decay in the amounts of the ith and the jth characteristic species given by Eq. (27) : bi

=

bte-ki"

(55)

ANALYSIS OF COMPLEX REACTION SYSTEMS

231

and

bj

=

bjoe-Ajt

(56)

Taking the logarithm of both sides of Eqs. (55) and (56), eliminating t, and rearranging, we obtain

thus,

bi = gijbp/hj

(58)

where gij is the constant term bio (bj'J)Ai/Aj

Using Eq. (56) to eliminate t from the set of equations of the form of Eq. (55) for i = 1 to n - 1 (i # j ) , we obtain n - 2 equations in the form of Eq. (58), which are the parametric representation of the reaction path in terms of bj and are simple power functions of bj. The characteristic roots are determined by transforming experimental compositions along appropriate reaction paths into the B system of coordinates. Equations (44) and (46) are used to compute the matrix X-' from the matrix X determined from the straight line reaction paths and the equilibrium composition. Each observed composition a(t) is transformed The decay of each bj with time is by the matrix X-l into @(t)[Eq. (a)]. given by the set of Eqs. (27) and the value of --Xi can be determined from the slope of the straight line obtained from a graph of In bj vs time. The above determination of the values of the characteristic roots requires a knowledge of the reaction time. As we have seen from the parametric representation of the reaction path in terms of bj, we need only a knowledge of the compositions along reaction paths to determine the ratios hi/Xj. According to Eq. (57), a graph of In bi vs In bj is a straight line with a slope of Xi/Xj; consequently, any curved reaction path that contains sufficient bi and bj for accurate plotting can be used to determine Xi/Xj. f. Degeneracy i n the Values of the Characteristic Roots. For reversible monomolecular systems there are always n independent characteristic directions (see Appendix I for proof). Nevertheless, different unit characteristic vectors may have the same characteristic root. For any two characteristic species with the same value of the characteristic roots, (Xi/Xj) = 1 and Eq. (58) becomes bi = gilbj

232

JAMES WE1 AND CHARLES D. PRATER

Hence, all reaction paths in this plane become straight lines as shown for a t,hree component system in Fig. 11. For this system the degeneracy in the lambda’s occurs when kI2 = k13, kzl = kza,and k31 = k32 simultaneously. For the general n-component system, any degree of degeneracy m 6 n - 1 may occur. In this case, the region of the reaction simplex in which all reaction paths are straight lines will be a subspace of the reaction simplex and will have the same number of dimensions as the degree of degeneracy.

FIG. 11. Three component system with kl, = k13, k n = k23, and system XI = Xz and all reaction paths are straight lines.

ksl

=

ksl. For this

For example, in a six component system, three equal characteristic roots means that all reaction paths will be straight lines in a particular three dimensional subspace of the five dimensional simplex. The lack of uniqueness in the choice of the n independent characteristic directions, brought about by the existence of an infinite number of straight line reaction paths for the degenerate cases, will cause no difficulty. In an n-component system, let m (m ,< n - 1) characteristic roots be equal. There will be, then, (n - m) characteristic vectors determined uniquely except for sign. The remaining m vectors are chosen from the infinite number of straight line paths in the m-dimensional subspace of the reaction simplex; the best choice to make is an orthogonal set of m straight line reaction paths. g. The Transformation of the Rate Constant Matrix for the Characteristic Species into the Rate Constant Matrix for the Natural Species. The matrix A, whose diagonal elements are the easily measured characteristic roots -Xi, is the rate constant matrix in the B system of coordinates and is

233

ANALYSIS O F COMPLEX REACTION SYSTEMS

analogous to K in the A system of coordinates. Thus, we need to discover the transforms for changing the matrix A into the matrix K.The characteristic directions corresponding to the species Bj have been defined as the direction in composition space in which vectors of arbitrary length undergo only a change in length under the action of K.The n unit characteristic vectors xj are, therefore, related to K by n equations in the form (18) which, when written in terms of the vectors xi,are

Kx.=

- - X . X3.

(59)

3

The scalar constant X j in Eq. (59) is the rate constant Xi for the j t h species in Eq. (30) as shown by the relation between Eqs. (18), (19), (23), (27), and (30). The set of n equations given by Eq. (59) can be written as a single equation in terms of the matrix A [Eq. (ZS)] and X [Eq. (37)]. In view of the interpretation of matrices given by Eq. (39), the n matrix-vector multiplications, Kxj, on the left side of Eq. (59) can be written

K ((xo),(xi), ( X Z ) . . . ( x d ) Multiplying each vector in

=

KX

(60)

X by K gives

K ((xo), (xi), (XZ) . . ( x n - 1 ) ) = (O(Xo),

-X,(X,).

--Xi(Xi),

. . -L-i(X%-i))

=

XA

(61)

by the rule of matrix-matrix multiplication (Footnote, page 226). The matrix A must be written on the right side of the matrix X so that the ith column vector in X will be multiplied by the diagonal element - - X i from A. Hence, the set of n equations, Eq. (59), is equivalent to the single equation

KX

=

XA

(62)

Multiplying each side of Eq. (62) from the right by the matrix X-l, we obtain

KXX-' = XAX-l or

K

=

XAX-'

(63) *

since XX-I = I and KI = K. Equation (63) gives the required transformation for changing the rate constant matrix A for the B system into the rate constant matrix K for

* The order of the arrangement of the matrices in products, such as those occurring in Eqs. (62) and (63), must be maintained since the commutative law of multiplicat,ion does not hold for matrices in general, i.e., PG # GP.

234

JAMES WE1 AND CHARLES D. PRATER

the A system and involves the same transformation matrices X and X-l which effect the changes between (Y and 0. Thus, the matrix K,whose offdiagonal elements are the individual rate constants of Eq. (5), can be calcuated from measured characteristic vectors xi and characteristic roots -Xj. h. Simplification and Advantages of Introducing Relative Values of the Rate Constants. We have seen that the unit characteristic vectors Xj and the lambda ratios, Ai/Ai, can be determined without an explicit consideration of the reaction time-that is, they can be obtained from a knowledge 01 the various compositions through which particular initial compositions pass on their way to equilibrium without regard to the time at which the various compositions occur. We shall show now that the rate constants k j i can be determined to within a constant factor (relative rate constants) from the ratios AJXj and the vectors xi and, consequently, without an explicit consideration of reaction time. This is fortunate since the value of the reaction time required to produce a given composition is usually the least reproducible information obtained about a system. Dividing each element of A [Eq. (26)] by A, and multiplying the entire matrix by A,, we have 0

0

...

0

0

...

0

0

...

0

0

,..

0

- A2

...

0

0

...

0

0

0 --X1 Am

0 A

=

A,

..

0

.. 0 0

-

.. 0

Am

... ...

0

0

0

...

..

..

.

0

0

0

.. .. .. ...

0

...

0

-Xm+l

...

0

Am

...

0

...

0

-i

...

~

0

...

-Afi-1 Am

or A = X,A'

where A' is the matrix on the right of Eq. (64). Substituting Eq. (65) into Eq. (63), we obtain K = A,XA'X-l (66) since A, is a scalar quantity. The matrix X L ~ ' Xis - ~a relative rate constant matrix, which we shall designate

K'

=

XA'X-1

hence,

K

=

A,K'

(67)

ANALYSIS O F COMPLEX REACTION SYSTEMS

235

Any one of the nonzero relative elements ktji of K',say k'lm,may be made equal to unity by dividing each element of K' by ktlm giving a matrix, which will be designated K, and whose elements will be designated by kji. Then,

K' = k'lmK where the element k'lm is the element of the matrix The elements of K are

(69)

K' that is unity in K.

Thus, the jith element of K is the ratio of the true rate constants k j i / k l m for the reaction system. i. Application to Pseudomonomolecular Reaction Systems. It is because relative rate constant matrices can be determined from composition data alone that much of the developments presented for the monomolecular system can be applied to the pseudomonomolecular system. We defined pseudomonomolecular systems in Section I as systems with rate equations of the form

where @I may be a function of time and the amounts of the various species and is the same for each rate equation for a given system. The quantities that are included in I#J have a degree of arbitrariness that allows us to select the pseudo-rate-constants eji for the system so that at least one of them has the value of unity. The quantity I#J may be treated as a function of time, @I(t),since each variable ai of the system is itself a function of time, ai(t). Therefore,

where r is a new time scale with the differential element dr = @I(t)dt. Hence, with the new time scale r , the pseudomonomolecular reaction system behaves like a monomolecular reaction system. We cannot determine this time scale without integrating the set of nonlinear differential equations (71) to obtain the functions aj(t). Nevertheless, since one of the pseudorate-constants 0,; is known to be unity, we do not need any time information to determine the value of these constants; we need only to determine the relative matrix K with the proper element unity, This can be done from composition data alone without regard to reaction time as we have seen,

236

JAMES WE1 AND CHARLES D. PRATER

Conversely, the composition sequence for any initial composition may be determined from the relative matrix as for the monomolecular system. j. Time Contours in the Reaction Simplex. When the time appears explicitly in the equation for the reaction paths, it is as a parameter (see Section II,B,2,e); hence the explicit inclusion of time in the reaction simplex is also parametric and it may be shown by means of contours of constant time as discussed below. The equations for these contours provide a convenient method for computing the reaction paths and for understanding some of the characteristics of these systems. For a given initial composition a(O), there is a corresponding initial composition @ ( O ) given by @(O) = X-la(0)

(73)

and for each composition @(t),there is a corresponding composition a(t) given by = X@(t) (74) The compositions @(t)are given in terms of the initial composition @ ( O ) , by [Eq. (27) in matrix form]

W ) = exp At eco>

(75)

where exp A t is the diagonal matrix

0

0

...

0

... Combining Eqs. (73), (74), and (75), we obtain

a(t) = X(exp At)X-la(O)

(77)

Let Ttl designate the matrix

T" = X(expAtl)X-l

(7@*

* A monomolecular system may be defined in terms of the matrix TI instead of the matrix K.For infinitesimal St, a(6t) = T h ( 0 ) = X(exp nGt)X-la(O) =

x [I

+*at

+*2$

+. .]X-la(0) f

Neglecting higher order terms, a(6t) = [I =

+ KGt]a(O) = a(0) + Ka(0)SL + d a(O)6!

a(0)

This formulation of these systems is useful in many cases. The matrix Tfis a stochastic matrix and the group (TI) is a one parameter linear continuous transformation group,

ANALYSIS O F COMPLEX REACTION SYSTEMS

237

for some particular time t l ; then a(t1) = T'la(O)

(79) Equation (79) shows that T'1 transforms a particular initial composition a(0) into its value a(tl) at time tl. If we have a set of initial composition points a(0) that forms a curve in the reaction simplex at t = 0, this transform will change the original curve into a new curve representing the time contour at time tl containing the composition points a(tl). Hence Eq. (79) gives the constant time contour as a function of the initial composition. When the matrix Ttl is applied to the composition a ( t l ) ,we have, from Eq. (79), T''a(t1) = (T'I)~~(O) = (~(2t1)

(80)

since (Ti1)2= X(exp htl)X-lX(exp Atl)X-'

=

X(exp A(2Ll))x-l

Hence, a((m

+ 1 ) t l ) = T'Ia(mt1)

(81)

where m is a positive integer or zero. Equation (81) may be used to calculate the composition points along a reaction path at successive time intervals At = tl. Near equilibrium the matrix T1lmay give points with closer spacing or (T'I)~may be comthan desired; in this case, either the matrix (T'I)~ puted; these correspond to At = 2tl and to At = 4tl, respectively. In the computation of reaction paths, the relative matrix A' may be used instead of the matrix A. In this case the time t is not actual reaction time but is merely a "bookkeeping" parameter to enable us to calculate successive compositions along the reaction paths. The constant time contours for monomolecular systems have the interesting and useful property of preserving straight lines and relative distances. When the time behavior of two different initial compositions are known, the time behavior of any initial composition between the two may be obtained by linear interpolation. We shall discuss this for three component systems; it generalizes readily to n components. Let a set of compositions a(0,r) lie along the straight line in the reaction triangle connecting the ) a 2 ( 0 ) ;a(0,r)is given by the equation ends of the vectors ~ ( 0and

a(0,r) = (1 - r)a1(0)

where 0 have

6r

<

+ raz(0)

(82) 1. Multiplying Eq. (82) from the left by the matrix Til,we

Ttla(O,r) =

r)

+

= (1 - r)Ttl~l(O) rT"~uz(0)

(83) Only the scalar r is a function of the initial composition; hence, Eq. (83) is the equation for compositions lying on a straight line connecting the a(t1,

238

JAMES WE1 AND CHARLES D. PRATER

ends of the vectors Ttlal(0)and Ttwz(0).Therefore, a set of compositions that lie in a straight line a t t = 0 remain on a straight line time contour for all time, and also a set of compositions lying along the sides of the reaction triangle a t t = 0 continue to lie along the sides of triangular time contours that shrink in size and change orientation as the reaction proceeds to equilibrium (see Fig. 12). The triangular time contours are, however, not

PIG. 12. Constant time contours for a typical three component system. The initial compositions a(0) lie along the boundary of the reaction simplex.

similar triangles. For n-component systems, the shrinking triangular contours become,shrinking simplex contours since Eq. (82) may be generalized to contain n - 1 vectors ai(0) and n - 2 scalar quantities r that depend only on initial composition and not on time. The structure of Eq. (83) shows that the relative distances between composition points are preserved as straight line contours go to equilibrium. The scalar r gives this relative distance, and, since it is not a function of time, a point that is rth of the distance from q(0) to ~ ( 0 a)t time t = 0 will also be rth of the distance between Ttlcul(0)and Ttlcuz(0)at time tl. Ic. Comparison Between the New and the Conventional Solutions. The solution obtained using the B system of coordinates must be equivalent to the general solution, Eq. (6), and the transformation from the B to the A system provides the means for showing the equivalence. Substituting Eqs. (27) and (29) into Eq. (35), we obtain

239

ANALYSIS OF COMPLEX REACTION SYSTEMS

... ...

... ...

...

A comparison of Eqs. (6) and (84) shows that X j and cii in Eq. (6) are equivalent to X j and bjOzij respectively in Eq. (84). Hence, the general solution Eq. (6) is nothing more than the transformation of e(t) to the A system of coordinates. Nevertheless, Eq. (84) represents a gain over Eq. (6) because its constants have interpretations that give their relation to the rate constants kji. Furthermore, the interpretation provides much easier and more accurate methods for the determination of the constants than the usual curve fitting techniques. 3. The Orthogonality Relations Between The Characteristic Vectors

a. The Origin of the Orthogonality Relations. In Eq. (84), the constants bjo are parameters determined from the initial composition and the constants xij and X j are the parameters that are determined from experimental data. There are (n2 n - 1) constants x i j and X i in Eq. (84). There are, however, only [ ( n 2 ) ( n - 1 ) ] / 2 independent constants because additional relationships between the constants x;j are provided by (1) the law of conservation of mass and ( 2 ) the principle of detailed balancing, which requires that kijaj* = kjiai*. In Eq. (84), n characteristic vectors, each containing n elements, are to be determined. The law of conservation of mass imposes the restrictions of Eqs. (31) or (32) on each vector and, consequently, reduces the number of independent elements of each vector to n - 1. This reduces the number of independent constantsinEq. (84) to n(n - 1) (n - 1 ) = ( n l ) ( n - 1). Consequently, there are ( n / 2 ) ( n - 1) relations, as yet undetermined, between the constants xij; these are provided by the principle of detailed balancing and further reduce the number of independent constants xi+ The principle of detailed balancing provides the means for making a further transformation to a third coordinate system in which the characteristic directions are orthogonal to each other. The transformation is discussed in detail in Appendix I, but we have already made use of this orthogonal B system in obtaining the inverse matrix X-l (Section II,B,2,c). The ( n / 2 ) ( n - 1) relations provided by the principle of detailed balancing are the requirements that the unit characteristic vectors xi must be orthogonal to each other after this transformation.

+ +

+

+

240

JAMES WE1 AND CHARLES D. PRATER

The transformation required to change the unit characteristic vectors

xj into the unit vectors jij for the orthogonal B system of coordinates is given by [Eq. (A17), Appendix I] f j = D-%xj (85) where D-’i is the diagonal matrix [Eq. (AlO), Appendix I] 1

...

0

0

... To transform the unit vector j i j back to the unit vector for the nonorthogonal B system, we have [Eq. (AlB), Appendix I] X. =

D%t,

where [Eq. (A9), Appendix I]

da,* 0 DJ.5

=\

o \

...

VGF...

( 0

... ...

0

... 42

0

The dot or inner product of two vectors is a scalar quantity and is, in matrix notation,

(”:)

r f1jj

xzj

* *

. xnj

=

XjTXi

(89)

Xni

where T indicates the transpose of the vector xj. The inner product between two orthogonal vectors is

fjT&

i # j (90) There are Zipln(n - i) = (n/2)(n - 1) independent orthogonality relations in the form of Eq. (90) for an nicomponent system; they are the (n/2)(n - 1) additional relations sought. The orthogonality relation, Eq. (90) may be written in terms of the vectors in the nonorthogonal system; =

0

x.TD-’xi = 0

(91)

ANALYSIS OF COMPLEX REACTION SYSTEMS

241

where D-I is the matrix given by Eq. (45). This equation is obtained as follows: Substitution of the value of fj and t i given by Eq. (85) into Eq. (90) gives

(D-Xxj)TD-Xxi

=

0

Since the transpose of the product of the matrices is equal to the product of the transpose of the individual matrices taken in reverse order, we have xjTD-?4D-Xxi = 0 because (D-s)T = D-% for diagonal matrices. Using D-fsDD-X= D-I , we obtain Eq. (91). We shall now show how these orthogonality relations may be used (1) to correct experimentally measured vectors xi for lack of orthogonality and (2) to determine the region of the reaction simplex in which to search for characteristic composition vectors. b. The Correction of Unit Characteristic Vectors for Lack of Orthogonality. In order to correct a pair of vectors for lack of orthogonality, one of them must be converted to unit length. The square of the length of fj in the A coordinate system is given by =

f.Tf. J

J

1.3

(92)

The required adjustment is (93)

where S j is the orthogonal characteristic vector of unit length in the A system of coordinates. Let us assume that the vector Xi has been determined accurately but errors exist in the vector fj such that XiTfjr

=

eij

(94)

where the prime on the subscript indicates an inaccurate vector. The vector f given by f

=

- Eijji

fj,

(95)

is orthogonal to f i as shown by multiplying both sides of (84) from the left by iiiT;

- eijfiTfi - eij = 0

IiTf = XcTZj,

-

‘{j

(96)

since IiTli = 1 for vectors of unit length. Only the vector considered to be accurate in Eq. (95) need be of unit length in Eqs. (94) and (95). The vector f has been purged of the vector li that 2j1 contained but must have its length adjusted before it is either fj or Xi. This procedure may be used to

242

JAMES WE1 A N D CHARLES D. PRATER

obtain a self consistent set of characteristic vectors by correcting the least accurately determined vectors by those determined with greater accuracy. c. The Determination of the Region of Composition Space in Which to Search for Characteristic Vectors. After a t least one unit characteristic vector, in addition to xo, has been measured, the region of the reaction simplex in which to search for additional characteristic composition vectors may be determined by the use of the transformation to the orthogonal B system of coordinates. Furthermore, the orthogonality relations may be used to reduce by one the number of characteristic vectors that must be measured. Let us consider the calculation of the value of the last characteristic vector when the other n - 1 vectors, lo,1, . . . In+ have been determined experimentally and made self consistent by the above procedure. The last vector, L-lmust , satisfy the n - 1 relations

and the requirement that xn-l is of proper length. These are sufficient requirements for computing the value of xnP1. A vector f 1 orthogonal to one of the known vectors, say 10, is written down. It can be obtained, for example, by making all elements of I0 zero except two , interchanging their position and placing a negative sign before either one of the two-for instance

The vector f l is purged of Til by applying Eqs. (94) and (95) to give fz; repeating the procedure, the vector f2 is purged of Iz to form f3 and so on until P ~ has - ~ been calculated. To calculate Xn--l, the vector fnP1 is transformed to the nonorthogonal system by applying Eq. (87). Since the vector obtained is not of proper length for its end to lie on the boundary of the reaction simplex when translated by an amount XO, its length is adjusted by scaling the elements of the computed vector. The scaling factor is determined from the requirement that a t least one element must be zero and all others equal to or greater than zero when xo is added to the vector. The vector obtained by

ANALYSIS O F COMPLEX REACTION SYSTEMS

243

this scaling process is xnW1. The vector x,-~is converted to ~ X ~ " - ~ (by O ) adding Xo [Eq. (54)l. The same procedure is used for locating the region of search when fewer than n - 1 vectors are known; the process is merely terminated earlier and the estimate will be less precise.

C. THESTRUCTURE OF REVERSIBLE MONOMOLECULAR SYSTEMS The above presentation has centered about the development of a general method for determining rate constants from experimental data. During the course of this development, much information has been obtained on the structure of these systems. Some of this information will be briefly summarized and extended in this section. In the above development the equilibrium point is a structural feature that plays a central role, and it might appear either that its existence has been assumed or that it was implicitly introduced from thermodynamics. This is not the case; it is a consequence of the following: (1) the law of conservation of mass [Eq. (12)], (2) no negative amounts can arise [Eq. (13)], (3) The rate of change of each species is a linear function of the amounts of the various species.

In statement (3), we are not assuming rate Eqs. (5), but only that the rate equations are some linear function of the concentration. These three statements are also sufficient to guarantee that the system will converge to a single equilibrium point over a sufficiently long time provided it is possible to go from any species Ai to any other species Aj either directly or through a sequence of other species. We have seen that the reaction paths do not spiral about the equilibrium point during this convergence to equilibrium. Neither the above three statements nor thermodynamics is sufficient to guarantee that the reaction does not spiral; it is a consequence of (4) the principle of detailed balancing. These four statements are also sufficient to guarantee that this kinetic system is consistent with the second law of thermodynamics, i.e., that the Gibbs free energy of the system decreases as the reaction proceeds to equilibrium for isothermal, isobaric systems. Statements (1) through (4) may be taken as the axiomatic formulation of monomolecular systems and the properties that we have discussed in the above sections are consequences of them. Further discussions of the equilibrium point and the convergence to it will be found in Section VII. We have seen that all n-component reversible monomolecular systems have n - 1 straight line reaction paths and n - 1 decay constants XI.

244

JAMES WE1 AND CHARLES D. PRATER

The location of the straight line reaction paths and the values of the lambda’s depend upon the experimental conditions such as pressure, temperature, nature of catalyst, etc. They are, however, independent of the initial composition used. For a given experimental condition, the entire behavior of the reaction system for all initial compositions is specified when the straight line reaction paths and the decay constants X are known. T h u s , all quantitative and qualitative information about the system i s contained in the location of the straight line reaction paths and the values of the lambda’s. Furthermore, all reaction paths, all time courses, and all rate constants are quickly and simply determined from them. In addition, the general behavior of such systems can be most easily visualized in terms of them; they provide a panoramic view of the entire reaction system and provide the most useful and convenient formulation of the system available. The excess species Bj, j # 0, provide interesting and useful quantities that measure the departure of the system from equilibrium. In addition, the hypothetical B systems of species demonstrate the existence of “transference units,” composed of the natural species Ai that change as a unit during the reaction. Additional discussions of the structure of these systems are given in the last two sections.

I l l . The Determination of the Values of the Rate Constants for Typical Reversible Monomolecular Systems Using the Characteristic Directions

A. THETREATMENT OF EXPERIMENTAL DATA In the discussion to follow, we shall consider the initial composition for a reaction path as lying on the boundary of the reaction simplex for the following reasons: (1) the improved accuracy obtained by using as long a reaction path as possible in the process of locating straight line reaction paths, and ( 2 ) the convenience of having at least one A species present in zero concentration in the initial composition. In addition, we shall have occasion to compare the rates of decay of the B species associated with each straight line reaction path and in such a comparison the species B,, naturally, will not appear (see Section II,B,2,e). A graphical method may be used to locate the straight line reaction paths in the reaction simplex for a three component system. Any convenient composition such as pure A 1 is chosen as an initial composition and its reaction path determined. Sufficiently close to equilibrium the reaction path will be dominated by the B species with the smallest decay constant since the other B species will have decayed to a much greater extent by this time. Consequently, a linear extrapolation of the part of the reaction path near equilibrium back to the side of the reaction triangle gives a new

ANALYSIS O F COMPLEX REACTION SYSTEMS

245

composition containing more of the B species with the smaller decay constant than the original composition and less of the B species with the larger decay constant. This new composition is used as an initial composition and the process repeated gives a third composition still richer in the slowest decaying B species. This convergence process is continued until the reaction path obtained becomes a straight line. This straight line reaction path corresponds to the B species with the smallest decay constant. Methods can be given for converging on the straight line reaction path corresponding to the B species with the largest decay constant; it can, however, be calculated from the straight line reaction path corresponding to the slowly decaying B species and the equilibrium composition using the orthogonality relations given in Section II,B,3. In principle, the graphical method used for the three component system can be used for a four component system since its reaction simplex is a tetrahedron; it is not very convenient, however, to plot reaction paths in three dimensions and for systems with more components this is not available. Consequently, a method for representing a reaction path is needed that does not involve the reaction simplex directly. In the reaction simplex a reaction path is a single curve in an (n - 1)-dimensionalspace. A reaction path also can be specified parametrically by n - 1 curves in two dimensional coordinate systems if the amounts of each of the various components ai(i # j ) is plotted in terms of another one of them, aj, that is monotonic with time. A straight line reaction path in the (n - 1)-dimensionalreaction simplex becomes n - 1 straight lines in this two dimensional graph. The relation of these straight lines to the elements lcii of the unit characteristic vectors are obtained in the following manner. For the n-component system there are n - 1 equations of the form crZ,(t) =

xo

+

erAstx3

(98)

representing the n - 1 straight line reaction paths in the reaction simplex. Let us consider the path corresponding to the lth characteristic vector and write Eq. (98) in terms of the components of this vector; this gives

Using the fact that the equilibrium concentration of the mth component is equal to xm0 and using the j t h equation to eliminate ePArt from the other equations, we obtain n - 1 equations of the form

a",

246

JAMES W E 1 A N D CHARLES D. PRATER

a m = ( a m * - zx mal j *

) ;;

+-aj,

m # j

Thus, a single straight line reaction path in n - 1 dimensions becomes n - 1 straight lines in two dimensions with slopes and intercepts equal to ( ~ m 1 / ~ j and i ) [am* - (~,z/zjz)aj*] respectively. Among the B species present in significant amounts in an arbitrary initial composition of an n-component system there will be, in general, one with a smaller decay constant X than the others. It will be designated the slow B species for this particular initial composition. As for the three component system, the composition along the reaction path near equilibrium contains relatively more of the slow B species and less of the other B species than contained in the initial Composition. For an n-component system, the converging procedure applied to the three component system can be used with the two dimensional representation of the reaction path; it gives us the characteristic composition vector corresponding to the slow B species of a particular initial composition. In order to increase the accuracy, an algebraic least squares fitting of a straight line to the points may be used for calculating the new composition vector rather than the graphical method. The least squares expressions are particularly simple for this case because the straight lines must pass through the equilibrium points, which are considered to be more accurately determined than the composition points along the reaction path. Let (a,) and (aj) designate the average value of the observed amounts a, and aj for those compositions near equilibrium that are to be used to obtain an estimate of a characteristic composition vector. The least squares fitting of a straight line to the points for the two dimensional representation gives n - 1 equations of the form a, aj

- am* - (a,) - am* - aj* (aj) - aj*

Before Eq. (101) can be used to determine initial composition vectors, we need to know how to recognize the boundary of the reaction simplex in the two dimensional representation. Use is made of the requirements that a,,, 0 for all values of m and that on the boundary a t least one of the amounts a, must have the value zero. An examination of the set of Eqs. (101) for each particular case will show which a, goes to zero first as aj increases. There is one additional precaution that must be taken for n-component systems: after a straight line reaction path has been located, the B species that correspond to it must be removed from other initial compositions that are used to locate new straight line reaction paths. Otherwise, the same straight line path will be obtained all over again if this species happened

>

ANALYSIS O F COMPLEX REACTION SYSTEMS

247

to be the slow species for this initial composition vector. Any initial composition can be purged of any B species already determined by use of the orthogonal relations (see Section II,B,3).

B. EXAMPLE OF A THREE COMPONENT SYSTEM:BUTENE ISOMERIZATION OVER PURE ALUMINA CATALYST 1. Experimental Determination of the First and Second Characteristic Vectors

The interconversion of 1-butene, cis-2-butene, and trans-2-butene 1-butene k n 7 Ikai

k21 I( 7 k 1 2

k32

cis-Zbutene

+

trans-2-butene

kza

has been studied by Haag and Pines (8) using pure alumina catalyst; they used conventional methods to estimate the values of the rate constants. More recently, Lago and Haag (9) have applied the method presented in this paper to the determination of the rate constants for the same system. We shall use their data obtained at 230" to illustrate the method as applied to three component reversible systems. In this example complete data will be given so that the computations may be reproduced in detail, as a practice example, by those who desire to do so. Any convenient initial composition such as pure cis-2-butene is used to determine a reaction path to the neighborhood of equilibrium. The approximately straight portion of the reaction path near equilibrium is extrapolated by a straight line back to the side of the reaction triangle, as shown in Fig. 13, to give a new starting composition vector

In the column matrix a, the order of the components are cis-2-butene trans-2-butene

This composition is used as a new starting composition and its reaction path determined near equilibrium. Since we are very near the straight line reaction path, the twelve composition points given in Table I, forming the approximately straight line portion near equilibrium, is fitted to a straight

248

JAMES WE1 AND CHARLES D. PRATER

I - Bu tene

tene FIG.13. Method of converging on ( ~ ~ ~for( 0the) three component system. Pure cis-2butene a =

(8>

is used as the first initial composition.

TABLE I Composition Sequence for the Second Convergence

Total
>

1-butene

cis-2-butene

trans-2-butene

0.1622 0.1776 0.1664 0.1654 0.1690 0.1603 0.1537 0.1571 0.1542 0.1521 0.1525 0.1532

0.3604 0.3769 0.3595 0.3622 0.3671 0.3441 0.3471 0.3464 0.3431 0.3451 0.3408 0.3416

0.4775 0.4455 0.4741 0.4724 0.4639 0.4955 0.4992 0.4965 0.5027 0.5028 0.5067 0.5052

1.9237 0.16031

4.2343 0.35286

5.8420 0.48683

ANALYSIS OF COMPLEX REACTION SYSTEMS

249

line by the least squares Eqs. (101). Using the equilibrium values determined experimentally by Lago and Haag,

and the average values of the composition points given in Table I, we obtain

p::;)

0.0000

The above process is repeated until a sufficiently accurate agreement is obtained between successive straight line extrapolations. The sequence of initial compositions used to converge on this value is: Initial composition New initial composition 0.0000

0.240

/’ // /

The experimental points from the third and fourth initial compositionsused to obtain the new initial composition are given in Tables I1 and 111. All experimental points for the last initial composition are given in Fig. 14.

250

JAMES WE1 AND CHARLES D. PRATER

TABLE I1 Composition Sequence for the Third Convergence I-butene

cis-2-butene

trans-2-butene

0.2289 0.2362 0.1989 0.1895 0.1751 0.1801 0.1557 0.1577 0.1583 0.1509 0.1551 0.1534

0.4606 0.4738 0.4118 0.3915 0.3678 0.3815 0.3478 0.3589 0.3423 0.3395 0.3324 0.3290 0.3314

0.3105 0.2900 0.3894 0.4190 0.4571 0.4384 0.4965 0.4767 0.5000 0.5021 0.5167 0.5159 0.5152

2.3042 0.17725

4.8683 0.37448

5.8275 0.44827

0.1644

Total

The following comments apply to the sequence (105). In preparing the initial compositions one does not, of course, have to match the predicted new initial composition exactly. In those cases where the initial composiTABLE I11 Composition Sequence for the Fourth Convergence

Total
>

1-bu tene

cis-2-butene

trans-2-butene

0.2974 0.2917 0.2800 0.2659 0.2577 0.2444 0.2311 0.2075 0.1938 0.1714

0.5689 0.5642 0.5386 0.5202 0.5043 0.4758 0.4579 0.4281 0.4031 0.3618

0.1337 0.1447 0.1814 0.2139 0.2380 0.2798 0.3110 0.3644 0.4031 0.4668

2.4409 0.2441

4.8229 0.4823

2.7366 0.2737

tions are almost the characteristic composition, care must be exercised not to include points from too early a part of the path in the least squares fitting of the points. To define the value of the characteristic composition

ANALYSIS OF COMPLEX REACTION SYSTEMS I

I

I

I

I

I

I

25 1

I

.5

w W

.4

I-

.I

I

I

.30

I

.40

.50 cis-2-BUTENE

I .60

FIG.14. The composition pointe for the reaction path corresponding to the last initial composition in scheme (105) is plott,edon an expanded scale. The least squares line used to obtain cr,,(O) is shown. Only points with cis-2-butene content <0.60 are included in the least squares fit.

vector, it is sufficient to obtain agreement between two new predicted initial composition vectors within the accuracy required. We shall take the last value obtained in the new initial composition sequence (105) to be the characteristic composition vector since it differs by a t most two units in the third place from the preceding value; that is

)::I:(

a.,(O) = 0.6508

Using the value of a,,(O)and xo given by Eqs. (106) and (104) in Eq. (54), we obtain 0.3492 Xi =

0.1436

(0.0000) 0.6508 - (0.535) 0.3213 =

0.2056

(-:::::!)

(107)

2. Calculation of the Third Characteristic Vector from the First and Second Characteristic Vectors

The vector xz is calculated from the vector xo and x1 using the orthogonality relations (Section II,B,3,c). The value of the matrices D" and D-%,defined by Eqs. (88) and (86) respectively, are computed from the equilibrium amounts ai* given by the elements of the vector xo in Eq. (104). They are

252

JAMES WE1 AND CHARLES D. PRATER

0

0.3789

(108) 0

0.7315

and

D-Ja

=

2.6389 0 0 1.7642 0 1.3670 ( 0

(109)

The vectors xo and x1are converted [Eq. (85)] to unit characteristic vectors for the orthogonal B system by using the values of D-3*, xo, and x1 given by Eqs. (log), (104), and (107), respectively; this gives

and fl =

D+xI

=

( :::z)

(111)

-0.7315

Note that the elements of the vector jio are the diagonal elements of D”” and may be written down at once when D5$has been calculated. Furthermore, since foTto =

1

(112)

this vector is already of unit length in the A system of coordinates; hence,

zo = 3 0

(113)

The length of sZ1 must be adjusted, however, to unit length in the A system of coordinates by applying Eqs. (92) and (93). We have

The vector f l is formed by interchanging the first two elements of reversing their signs and making the third element zero. This gives fl

=

(

-0.5668 0.3789) 0.0000

t,

ANALYSIS OF COMPLEX REACTION SYSTEMS

253

which is, of course, orthogonal to lo.The vector yz, orthogonal to Til, is computed from f l using Eqs. (94) and (95);

and y2

=

fl

+ 0.080799l1 =

(

-0.5262 0.4224) -0.0547

(116)

The vector f z is transformed back to the nonorthogonal system by Eq. (87) ; YZ =

DXyz

=

(

-0.1994 0.2394) -0.0400

(117)

The vector yz is adjusted in length to give xz by multiplying each element in the vector of equation (117) by (0.1436/0.1994) to give

xz =

(

-0.1436 0.1721) -0.0288

The characteristic composition vector a,(O) is an(0) = xz

+ xo =

c:3 0,4937

(119)

Combining the vectors xo, xl, and xz, given by Eqs. (104), (107), and (118), respectively, to form the matrix X,we obtain

X

=

1

0.1436 0.2056 -0.1436 0.3213 0.3295 0.1724 0.5351 -0.5351 -0.0288

(

(120)

3. The Inversion of the Matrix X

The inverse of the matrix X given by Eq. (120) is obtained from Eqs. (44) and (46). The matrix D-' is computed from the equilibrium amounts given by Eq. (104) and is

0

1.8688

254

JAMES WE1 AND CHARLES D. PRATER

Using this in Eq. (44), we obtain

L

=

XTD-lX 0.1436 0.3213 0.5351 0.2056 0.3295 -0.5351)( -0.1436 0.1724 -0.0288

0.1436 0.2056 0.3213 0.3295 0.5351 -0.5351

-0.1436 0.1724 -0.0288

8

)-

6 9638 0 ' 0 3.1123 0 0 1.8688 1.0000 0.0000 0.0000 1.1674 0.0000 0.0000 0.0000 0.2377

Using the elements of the matrix given in Eq. (122) in Eq. (47), we have 1.0000 0.0000 0.0000 0.8566 0.0000 0.0000 0.0000 4.2077

(123)

Hence, from Eq. (46),

X-1

=

L-1XTD-1 0.5351 0.1436 0.3213 0.2056 0.3295 -0.5351 -0.1436 0.1724 -0.0288

(

)

6.9638 0 0 3.1123 0 0 1.8688

0"

and 1.0000 1.0000 1.0000 1.2265 0.8784 -0.8566 -4.2077 2.2579 -0.2264

(124)

We can check whether this is a good inverse by applying Eq. (42); 1.0000 1.0000 1.0000 1.2265 0.8784 -0.8566 -4.2077 2.2579 -0.2264

0.1436 0.3213 0.5351

)(

0.2056 -0.1436 0.1724 0.3295 -0.5351 -0.0288

1.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 1.0000

)

255

ANALYSIS O F COMPLEX REACTION SYSTEMS

4. The Experimental Determination of the Characteristic Root Ratios and the Calculation of the Relative Rate Constant Matrix The inverse matrix X-l is used to transform the compositions (Y in the A system to compositions 0 in the B system for a highly curved path such

bl‘! as the path from pure cis-2-butene or pure 1-butene. Applying Eq. (40) to the compositions a(i) along the reaction path for cis-Bbutene, we obtain the composition @(t) given in Table IIa. Equation (57) shows that In bl is a linear function of In b2 with slope X1/X2. This graph is shown in Fig. 15 for the data in Table IIA; the slope obtained is XJX2 = 0.4769, which gives, for the matrix A’ [Eq. (64)],

.3

.2

.I

.3 .4

,6 .8 I b2

FIG.15. Ln bl vs In bz obtained from pure cis-2-butene initial com-

0.0000 0.0000 0~0000 -0.4769 0.0000 0.0000

(

=

position. The slope of the straight line is XI/X2 = 0.4769

0.0000 0 .oooo - 1.0000

)

(126)

Equation (67) is used with the value of X, X-I, and A’ given by Eqs. (120), (124), and (126), respectively, to compute the relative rate constant matrix K’;

K’=

0.1436 0.3213 0.5351

(

0.2056 -0.1436 0.3295 0.1724 -0.5351 -0.0288

0.0000 0.0000 0.0000 0.0000 -0.4769 0.0000 0.0000 0.0000 -1.0000

)

)(

*

1

1.0000 1.0000 1.0000 1.2265 0.8784 -0.8566 -4.2077 2.2579 -0.2264

(

Performing the indicated multiplication, we obtain -0.7245

0.2381

K’=( 0.5327 -0.5273 0.1918

0.2892

0.0515 0.1736 -0.2251

)

(127)

The relative matrix K is formed by dividing each element of the matrix

K’by 0.0515; this gives

- 14.068 K=(

4.623 1.000 10.344 -10.239 3.371 3.724 5.616 -4.371

(128)

2/56

JAMES W E 1 AND CHARLES D. PRATER

TABLE IIA Butene Isomerization The composition @ ( t ) computed from the experimentally observed compositions. a(t) obtained from an initial composition of pure cis-2-butene.

to

0.0000 tl

0.0422 t2

0.0560

I-Butene

cis-2-But ene

trans- 2-Bu time

FIG.16. Comparison of calculated reaction paths with experimentally observed compoaitions for butene isomeriaation. The points are observed composition and the solid lines are calculated reaction paths.

ANALYSIS O F COMPLEX REACTION SYSTEMS

257

Hence, for the isomerization of butenes over pure alumina catalyst at 230" in an all glass flow reactor, the relative rate constants are 1-butene 10.344/ 74.623 3.7241 1.000 5.616 cis-2-butene f trans-2-butene 3.371

(129)

This reaction takes place on a solid catalyst and is pseudomonomolecular; consequently, the absolute value of the rate constants in the matrix K will in general be a function of the amounts ai and are not computed. The reaction paths, however, may be computed from the matrix Tfl (Section II1B,2,j) calculated from the value of X,X-l, and A' given above. A comparison between observed and computed reaction paths is shown in Fig. 16; the points are the experimentally observed compositions and the solid curves are the calculated paths.

C. AN EXAMPLE OF A FOUR COMPONENT SYSTEM 1. The Use of the Four Component System in Testing the Efects of Experimental Accuracy

We shall demonstrate the determination of the characteristic vectors, characteristic roots and rate constant matrices for systems with a greater number of components than three using as an example a four component system with known reaction rate constants, but we shall pretend that only the experimental data obtained from the system are known. In this manner, the results obtained during the procedures can be not only compared with the actual, accurate description of the system, but the effects of and sensitivity to experimental accuracy may also be studied. The hypothetical four component system used is given by 3

2

with the rate constants shown. The correct values of the characteristic composition vectors and characteristic roots are calculated from the correct

258

JAMES WE1 AND CHARLES D. PRATER

TABLE IV Characteristic Composition Vectors ( ~ ~ ~and ( 0Characteristic ) Roots -xi f o r Hypothetical Four Component System Index i

-xi

a,@)

0.1000

0

0

-5.209

1

2

-36.34

3

-78.45

rate constant matrix by the method given in Appendix IV and are shown in Table IV. Fig. 17 shows the correct straight line reaction paths for this system in the reaction simplex; it is a tetrahedron for four component systems. Experimental composition points along the reaction path, corresponding A4

FIG.17. Reaction simplex for the hypothetical four component system given in text. The straight line reaction paths are shown.

ANALYSIS OF COMPLEX REACTION SYSTEMS

259

to a given initial charge, are the theoretical, that is, ideally correct values calculated from the correct characteristic vectors and roots but perturbed by superimposing on them a Gaussian distribution of random errors with a u of 1% or 0.001 mole fraction, whichever is larger. In practice, this seems to be a reasonable estimate for the accuracy of careful work with clean systems. For example, in the studies of Lago and Haag (9) used above and for t,he hexane isomerization studies of Wiggill ( l o ) , the value of u for their error distribution was approximately one-half per cent. The effects of larger errors can be estimated by the method used in this chapter or by the perturbation method discussed in Section V,B. 2. “Experimental” Determination of the Characteristic Vectors Pure A1 is used as a convenient initial composition. Nine “experimental” compositions along the reaction path obtained for pure Al are given in Table V and are plotted according to the scheme u, vs aj in Fig. 18. The TABLE V Cornposition Points Along the Reaction Path for an Initial Composition of Pure A1

a(0) =

6)

equilibrium values lie along the line indicated by - - - - and the straight line approximations of each set of points in the neighborhood of this equilibrium line are shown. The species AZ and ABincrease in amount as the

260

JAMES WE1 AND CHARLES D. PRATER

amount of A1 decreases; consequently, the straight line portion of their paths near equilibrium must have a negative slope and extrapolate to a positive intercept along the A1 axis. A vertical line, - - - , is erected at the

.6 .5

.4

ai .3

.2

al

FIG.18. The method of obtaining the first new initial composition.

positive intercept nearest the origin of A , in this case at al = 0.330 corresponding to the intercept of the straight line for species Az. The values of the intercepts of the straight lines with this vertical line, along with the value of al, are written as a column matrix

(a);:

0.638

The sum of the elements in this matrix is 1.008 because of the random errors in the composition. Dividing each element in the above matrix by 1.008 gives

The composition (131) is used as an initial composition, “experimental” composition points are determined along the reaction path, and equation (101) used with the composition points near equilibrium to obtain the next

ANALYSIS OF COMPLEX REACTION SYSTEMS

26 1

initial composition. This is repeated until a sufficiently accurate agreement is obtained between successive values of the extrapolation. The sequence is Initial composition

New initial composition

0.3274 0.0397

/ I(

((.oooo) 0.3225

(o.oooo) 0.3274' 0.0397 0.6329

I/

__j

0.0360

/ 0.6415

/

(o.oooo) 0.3225

;(.oooo)0.3214

0.0360 0.6415

0.0382 0.6404

The last three values are very close to the correct value of ( ~ ~ ~given (0) in Table I V and probably represents random wandering about the true value caused by the errors introduced. Although the last value is the closest to the correct value, we might well have stopped at the preceding value. Since we are searching for the effect of the errors, let us take the next to the ( 0 )is, last value for the experimental value of ( ~ ~ ;~that

0.3214 0.0382 0.6404

(133)

262

JAMES WE1 AND CHARLES D. PRATER

The unit characteristic vector x1 is calculated by subtracting xofrom (133). The value of xo = a* given in Table I V is used since we are considering, in all examples discussed, that the equilibrium value is measured more accurately than the individual composition points. We obtain 0.2214 (134)

-0.3618 0.4404

The procedures given in Section II,B13,care now used to determine the new initial composition free from xoand x1that will be used to search for the next characteristic composition vector. The matrices D f 6 , D-.5i, x o are determined from the equilibrium value a* = xo given in Table I V and are

Dt6

=

D-56 =

"i

0 0 0.547723 0 0 0 0.632456 0.447214 0 0

(135)

0 0 1.825740 0 0 0 1.581137 2.236066 0 0

(136)

r37

and jo =

(0.547723) 0.316228 (137)

0.632456 0.447214

Although only four figures are obtained in the experimental characteristic composition, we shall make the characteristic vectors self-consistent to six figures since the accuracy of the method for obtaining the inverse matrix X-1 given in Section II,B,2,c depends on the self-consistency of the characteristic vectors. In addition, the use of six figures will reduce the accumulation of errors caused by the computation procedure. Using Eq. (85) to calculate f l from Eq. (134), we have

)

0.700128

% = ( -0.547722 -0.572055 0.984763

ANALYSIS O F COMPLEX REACTION SYSTEMS

263

Equations (92) and (93) are used to adjust the length of fl to unity; we obtain

0.484615 -0.379123 -0.395966 0.681634 Forming

fl

(139)

from the second and third elements of 1 0 , we have

0.000000

which is orthogonal t o I,. The vector yl is purged of 8 , by applying Eqs. (94) and (95) to give llTyl

=

+O .022899

and

72 =

fl

- O .O2289911 -0.011097

-0.015609 Transforming 72 to the nonorthogonal system, we have

-0.003509 -0.341655

(142)

-0.006981 Adjusting the length of y~so that the second elements become -0.300000, we obtain -0.003081 -0.300000 (143) 0.309211 -0.006130 and a(0) =

XZ’

+ xo =

0.7092 0.1939

(144)

264

JAMES WE1 A N D CHARLES D. PRATER

The composition given by Eq. (144) is used as an initial composition, the reaction path determined, and a new value of a(0) obtained, using Eq. (101); this gives 0.0888 0.7076 0.2036

(145)

This composition vector is purged of any x1 reintroduced by the random errors in the system (Section II,B,3,b);this gives 0.0882 0.7092 0.2026 Since the compositions given by Eqs. (144) and (146) do not differ greatly, we shall use Eq. (146) as the value of ~ ~ ~ (Comparison 0). with the correct value given in Table IV shows that we are indeed close to the true value. The value of x2 obtained from the above purge procedure is

/ -0.011748\

The last unit characteristic vector x3 is calculated from the values of xo,xl,and xz (Section I11B,3,c).We can begin with the value of p given by Eq. (141); this vector, however, contains little fBsince it is already very near the correct value of f 2 . A vector f3may be computed from this value of f z l but it is composed of small elements formed by the difference between large numbers and is, consequently, accurate to only one or two figures; it may be lengthened and again purged of each vector xo, x1, and xz to obtain a correct value of y3. It is more convenient, however, to begin with a new value of 71. This new vector yl is formed from the first and last elements of 80;

/

0.447214) 0.000000 -0.316228

Purging

of i1, we have giTTl

= +0.0011748

265

ANALYSIS OF COMPLEX REACTION SYSTEMS

and

When the vector xz [Eq. (147)] is converted to the orthogonal system and adjusted to unit length, we obtain 1 2 = (

Purging pzof i2, we have

73 =

)

-0.050537 -0.745087 0.665004 0.007820

(

0.445378 -00.017139) .OH237

-0.316833 Transforming

back to the nonorthogonal system, we obtain

1

0.140841\

\ -0.141692/ The elements of y3 are adjusted by the ratio (0.20000/0.14169) to give 0.198799 0.015300 -0.200000

(153)

The characteristic composition vector is

and is close to the correct value. Some comments are needed on the choice of y1in the above development. If we had chosen given by Eq. (148) instead of 71 given by Eq. (140)

266

JAMES WE1 AND CHARLES D. PRATER

in the search for o).tl(O), the initial composition obtained for the first reaction path is composed largely of by and the convergence to cu,(O) is a more lengthy process. In such cases, it may often save work to make another choice of 71 in the hope that the new initial composition will contain much more of 6, and less of b3, as is the case for the choice of yl given by Eq. (140). The matrix X is formed from xl,xz, and x3 given by Eqs. (134), (147), and (153) respectively and the value of xo in Table I V ; we have

x=(

0.198799 0.221400 -0.011748 0.100000 0.300000 -0.300000 -0.300000 -0.014099 0.309178 0.015300 0.400000 -0.361800 0.002571 -0.200000 0,440400 0.200000

)

(155)

The matrix X-' is computed from Eq. (155) using Eqs. (44) and (46). The matrix D-' needed for this is formed from the equilibrium value given in Table I V and is

D-1

0 0 10.000000 0 0 3.333333 0 2.500000 0 0 5.000000 0 0 0

=

(156)

TJsing Eq. (156) in Eq. (44), we have

L

=

XTD-lX

=

1.000000 0.000000 0.000000 0.000000

0.000000 2.087188 0.000000 0.000000

0.000000 0 .oooooo 0.540390 0.000000

0.000000 0 .oooooo 0.000000 0.596457

The elements of the matrix in Eq. (157) are used to form the inverse matrix L-' IEq. (47)]; it is

L-'

The inverse matrix

X-1

=

0 0 0.479113 0 0 0 1.850512 1.676563 0 0

")

=

X-'is

L-1XTD-1

1.000000 1.000000 1.000000 1 000000 1.055007 1.060757 -0,479114 -0.433357 0.023788 1.430344 -0.217397 - 1.850513 0.064133 - 1,676563 3.332992 -0.078793 *

(158)

ANALYSIS O F COMPLEX REACTION SYSTEMS

267

3. The (‘Experimental’’ Determination of the Characteristic Roots and the Calculation of the Rate Constant Matrix IC First, the X ratios will be determined using the equation for the reaction path with bj as the parameter [Eq. (58)l. For this purpose a n initial composition containing sufficient quantities of all B components is needed; the composition used is

0.0550

for which

~ ( o =)

0.0510 0.4980

[

0.371906] 1.000000 -0.659966 -0.679545

Ten “experimental” compositions along the reaction path for this composition are given in Table VI. The values of @(t),obtained by multiplying each a(t) by X-l, are also shown. The graphs of In 61 and In ba as a function I

.04 .06 .08 .I

I

I

.2 .3 .4

I

l l l U

.6 .8 1.0

b2

FIG.19. The determination of k ratios for the four component system. Graph of In b, vs In b2 for the data in Table VI. The slopes are (k,/X,) = 0.151 and (A&,) = 2.24.

of In bz are shown in Fig. 19; the slopes of the lines in this graph are used to form the relative matrix At =

(; 0

;

-0.151

0 -1

0

0 0 -2.24

(162)

268

JAMES WE1 AND CHARLES D. PRATER

TABLE VI The Compositions ( t ) Computed from the Experimentally Observed Composition a ( t ) from the Reaction Path Obtained from the Initial Composition

e

00

1.0000 0.3719 -0.6601 -0.6795

1800

1.0000 0.3763 -0.6431 -0.6567

900

1.0000 0.3678 -0.6359 -0.6147

450

1 1 1

-0.6056 -0.5764

225

0.9999 0.3612 -0.5599 -0.4837

120

1.0000 0.3543 -0.4891 -0.3531

1 1

90

0.9999 0.3382 -0.4574 -0.2743

60

1.0001 0.3484 -0.3597 -0.1814

45

1.0001 0.3203 -0.2950 -0.0955

30

0.9999 0.3054 -0.2022 -0.0390

15

0.9999 0.2638 -0.0621 -0.C049

1 1

1 1

Using Eq. (67), the relative matrix K' is computed from the values of X, X-l,and A' given by Eqs. (155), (159), and (162), respectively;

K' = XA'X-1

=(

- 1.52223

0.02936 0.00274 0.71160 0.08809 -0.57935 0.41150 0.00198 0.01093 0.54866 -0.46811 0.10774 0.00132 0.05387 -0.82132 1.42320

)

(163)

The value of A2 is needed to determine the value of the true K matrix from K'. It may be determined from a graph of In b2 vs time, as shown in Fig.

ANALYSIS O F COMPLEX REACTION SYSTEMS

269

.I

0

.01

.02

.03

.04

.05

.06

TIME

FIG.20. The determination of Xz for the four component system. The graph of In be vs time has a slope of X2 = 36.02.

(20), and is Xz

K

=

=

36.02. Using this value of hz in Eq. (68), we obtain

X2K' =

-54.82 1.06 0.10 25.63 3.17 -20.87 14.82 0.07) 3.88 19.77 -16.86 0.39 1.94 -29.58 51.26 0.04

(164)

The true value of K, obtained from scheme (130), is

-53.00

K=(

3.00 0.00 50.00

1.00

0.00

-21.00 15.00 20.00 -17.00 0.00 2.00

25.00

0.00) 4.00 -29.00

(165)

A comparison of Eqs. (164) and (165) shows that we did surprisingly well when one considers the high sensitivity to errors of such highly connected systems. In some respects, the acid test is to reproduce the zeros for the steps in the center of scheme (130). For these steps, we obtain kal = 0.39, k13 = 0.10, k42 = 0.04, and k24= 0.07. The small values obtained for the rate constants between steps A z and A4 are acceptable but the values obtained between steps A , and A 3 need to be improved. In Fig. 21 we have plotted the values of the four compositions on the right side of the scheme (132) on a highly magnified triangular region of the face of the reaction tetrahedron (Fig. 15) on which these points lie. The correct value of aZ,(O)is given by x and the observed values by 0 . The relative position of the points suggests that the movement of points 2, 3, and 4 is probably a random movement caused by the errors in the composition. Consequently, to obtain more probable values of the rate constants, we should add more cycles to the scheme (132) and use the average

270

JAMES WE1 AND CHARLES D. PRATER

FIG.21. The random movement of the observed estimates of the characteristic composition cu,,(O) about the true value indicated by X. The plot is on a highly magnified triangular region of the face of the reaction tetrahedron on which these compositions are located.

value obtained from the several cycles. Also, more than a single determination should be made of the value cr,(O) and the averages used. In this case, we must always remember to purge each new initial composition of any x1 it contains. When these additional steps are introduced the value obtained for the rate constant matrix K is improved.

IV. Irreversible Monomolecular Systems A. GEOMETRICPROPERTIES OF IRREVERSIBLE SYSTEMS I. N e w Features Introduced by Irreversible Steps

Although systems containing completely irreversible steps are an idealization, reaction systems are very numerous that contain steps with a sufficiently large change in free energy so that they may be approximated quite accurately by irreversible steps. When a species A ; is connected to other species by irreversible steps, its equilibrium amount a;* is zero. When the equilibrium amount ai* of some species is equal to zero, the matrices D-1 and D-'6 do not exist and are not available for transforming the rate constant matrix K into a symmetrical matrix (see Appendix I). In this situation, we have no assurance that n independent characteristic

ANALYSIS OF COMPLEX REACTION SYSTEMS

27 1

directions exist for an n-component monomolecular system. Nevertheless, almost all irreversible systems will have n independent characteristic directions and only in very unusual cases, such as the example given below, will the situation be otherwise. Consequently, most irreversible monomolecular reactions will have completely uncoupled systems of B species equivalent to them. We shall use special examples to show that the following new features may be exhibited by irreversible monomolecular systems: (1) Straight line reaction paths may occur that do not lie within the reaction simplex and cannot be observed in the laboratory. (2) Under very special conditions, degeneracy in the characteristic directions may occur so that a full set of independent coordinate axes cannot be formed from them. (3) When (2) applies, coupling cannot be completely eliminated and the equivalent reaction systems will contain some species coupled by sequences of irreversible steps. (4) Systems with an infinite number of equilibrium points may occur and the particular one to which the system converges will depend on the initial composition used.

This last feature is, of course, an idealization arising from the same idealization used to introduce irreversibility in the first place and its validity depends on the time duration of an experiment. After a sufficiently long time the reaction steps neglected in the irreversible approximation will exert their full influence on the system, causing the reaction to go to a single equilibrium point. 2. Characteristics of Irreversible Systems with a Single Equilibrium Point

Some of the new characteristic features of irreversible monomolecular systems not shown by reversible systems may be demonstrated by the three component system

for which the explicit solution in terms of the rate constants may be easily obtained by conventional methods. The matrix K for this system is

272

JAMES WE1 AND CHARLES D. PRATER

Equation (59) may be used to verify that xo

=

and xz =

t)

(-;)

are characteristic vectors and A. = 0, Al = -kl, and AZ = -kz are characteristic roots of the system. Equation (169) shows that, as long as (k,/kz) # 1, there are three independent characteristic vectors so that the reaction can be transformed into an equivalent completely uncoupled B system. The characteristic compositions obtained by adding xo to x1 and xz are

(171)

and (172)

Equation (172) shows that the straight line reaction path corresponding to xz lies along the side of the reaction triangle connecting u2 = 1 and u3 = 1 as shown in Fig. 22. When (kl/kz) > 1 the first term of e,,(O) is a negative amount; consequently, this characteristic composition vector lies outside the reaction triangle as shown in Fig. 22. The choice in length of the vectors XI and xz are such that the vectors terminate either on a boundary of the reaction triangle or an extension of it. When (k,/kz) < 1 the straight line reaction path corresponding to x1 lies within the reaction triangle as shown in Fig. 23. Typical reaction paths calculated by means

ANALYSIS O F COMPLEX REACTION SYSTEMS

273

FIG.22. The reaction triangle for the system ki

kl

A1 -+ A z + As with (kJk2) = 4. The displaced characteristic vectors x ’ ~and x’z are shown. The vector lies outside the reaction triangle since (kl/kz) > 1.

X’I

of the matrix Ttl (Section II,B,2,j) for these cases are shown in the two figures. When (lcl/kz) 41, we have kl 3 kz, XI 4Xz, and xl+ xz;hence, there are only two characteristic directions and the system is not equivalent to a completely uncoupled system. Monomolecular systems that have too f e w independent characteristic directions can always be expressed as a n equivalent system in which the only coupling i s by sequences of irreversible steps. For example, a seven component system with only four independent characteristic directions might be equivalent to A1

A2

A3

B, --+ Bz -+B3 40 with (XI = XZ = X 3 ) A4

B, 4 0 A6

B, --+ 0 AS

Bs --+ 0 Bo does not react

(173)

274

JAMES WE1 AND CHARLES D. PRATER

Let us examine how this kind of B system arises for the reaction scheme (166) with ( k l / k z ) = 1. We shall select an independent direction in space to replace the missing characteristic direction and complete the set of coor-

FIG.23. The reaction triangle for the system k:

ki

A1 * A2

+ As

with (k1/4) = i. The displaced characteristic vectors lies inside the reaction triangle since (kl/kz) < 1.

x'1

and X'Z are shown. The vector

X'I

dinates needed. This direction, of course, will not have the properties of a characteristic direction. The unit vector in this direction, designated 91, wilLbe selected such that it satisfies the requirement This gives

91 =

(-;)

The unit vector along the two independent characteristic directions of this system are xz and xo given by Eqs. (170) and (168), respectively. The vectors in Eqs. (168), (173), and (170) are used to form the matrix Y, which is (176) 1 -1

275

ANALYSIS O F COMPLEX REACTION SYSTEMS

Since the transformation matrix D-1 does not exist for irreversible systems, we must compute the inverse of Y by conventional methods (11);it is

Y-l=

(; 3)

(177)

For reversible monomolecular systems, the rate constant matrix K is transformed into the diagonal rate constant matrix A by the transformation

X-lKX

(178)

=A

which is obtained by multiplying Eq. (63) from the right by X and from the left by X-l. Transformations in the form of Eq. (178), that is, in the form P-lGP, are called similarity transformations. Let us use the matrices Y and Y-l given by Eqs. (176) and (177)) for a similarity transformation of the matrix K,given by Eq. (167), with kl = kz; this gives

Y-lKY

'X-i: -iz:)f : :) =( -H)=(W -;-8,) =( 1l 0

0

0 1 0

1 -1

kz 0

0

-1

0

(179)

4;

since X2 = kz. The form of the matrix on the right of Eq. (179) shows that, when kl = kz, the scheme (166) is equivalent to

&

kz

ka

Bz 3 0

Bo does not react The rate equations for these reactions are

_ db2 - Xz(b1 - bz) dt

they have the solution

Figure 24 shows the displaced vectors yfl and x'z and typical reaction paths calculated by a matrix Ttl (Section II,B,2)j) in which A is replaced by the matrix on the right of Eq. (179).

276

JAMES WE1 AND CHARLES D. PRATER

FIG.24. The reaction triangle for the system ki

ki

A1 + A2 -+ At with (h/h) = 1. The two characteristic vectors coincide as shown and a new coordinate Y'I is chosen so that the appropriate canonical form is obtained.

The matrix on the right of Eq. (179) is in a canonical form; this particular canonical form will be designated by N. For example, 0

'-A1

0 - A 2 0 0 0 0 0 0

N=

0 0 0 0 0

0 0 0 0 0

0

0 0

, o

0

0

0

0 0 0 A s 0 0 0 - A 4 0

0 0 0 0 0 0 0 0

X4-X4

0 0 0 0 0 0 0

X q

0 0 0

0 0 0

0 0 0 0

0 0 0 0 0

0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - X 4 0 0 0 0 0 0 - A g o 0 0 0 0 O - X e O 0 0 0 0 0 o - x 7 0 0 0 0 0 0 X7--X?o 0 0 0 0 0 X7--x70 0 0 0 0 0 X7-X7

is the canonical form N for a 12 X 12 matrix with threefold degeneracy and fourfold degeneracy in the characteristic vectors with roots X4 and X7 respectively. Proofs will be found in many books on modern algebra ( l a ) that any matrix whose elements are real (or complex) numbers can always

ANALYSIS O F COMPLEX REACTION SYSTEMS

277

be transformed by a similarity transformation into a canonical form called the Jordan canonical form. It is not difficult to prove that the Jordan canonical form can always be transformed by a similarity transform into the canonical form N (see Appendix IV). Hence, all monomolemlar systems with a n insuficient number of characteristic vectors are equivalent to reaction systems containing, at most, sequences of irreversible reactions with the final species in any sequence decaying to zero as illustrated by schemes (173) and (180). When a reaction system has an equivalent rate constant matrix in the canonical form N,the general form of the rate equations and their solutions may be inferred from those of an wcomponent system with m-fold degeneracy in one characteristic direction and p-fold degeneracy in another. For this system there are n - (m p ) = q characteristic directions with rate equations corresponding to complete uncoupling

+

dbi _ -- -Aibi, at

0

6i6

( q - 1)

(184)

there are m equations

and p equations (s = q

+ m)

corresponding to the coupled sequences of the B species. The solutions of tJheseequations are bi = biOe-Ait,

O
(187)

for the set of Eqs. (184), b(,+jl = (b,O

+ b(,+l)Ot...b(q+j)otf/j!)e-aqf,

0

6j 6

(m - 1)

(188)

0

6j 6

(p - 1)

(189)

for the set of Eqs. (185), and

b(,+j, = (b,O

+ Zqatl)Ot.

for the set of Eqs. (186).

. .b(,+j)otj/j!)e-Aa',

278

JAMES WE1 AND CHARLES D. PRATER

3. Characteristics of Irreversible Systems with a n Injinite Number of Equilibrium Points

First we shall use a very simple system to illustrate the characteristics of systems with an infinite number of equilibrium points. Although the analysis of this system is trivial, the essential features found in more complex systems are demonstrated by a simple geometry, which aids greatly in visualizing the behavior of the more complex systems. The reaction scheme that we shall use is

which has the rate constant matrix

The two vectors

and

are independent characteristic vectors with X = 0, but any linear combination of these two vectors is also a characteristic vector with X = 0; hence,

where 0 6 r 6 1. Any pair of vectors given by Eq. (192) may be used as x,, and xl.The other characteristic vector is

and has a characteristic root

A2

+

= (kz~

k31).

ANALYSIS OF COMPLEX REACTION SYSTEMS

Let us turn immediately to a specific example with kzl = 2 and k~ The vector xz is

and we shall choose x1

=

(;)

279 =

1.

(195)

We may obtain a displaced characteristic vector x’z by adding any vector given by Eq. (192) to the vector xz given by Eq. (194). Let us use

This gives

x‘z

=

and

x’1=

(i)

(197)

(i)

The positions of the displaced vectors xtl and x’z in composition space are shown in Fig. 25. The displaced vector x ’ ~does not lie in the plane of the reaction triangle and cannot be a straight line reaction path. This will be true for all choices of xo(r) and xl.For a displaced vector x’i to be in the plane of the reaction triangle, the sum of the elements of azi(0) must be the same as the sum of the elements of the vector b ( r ) used to move x i . This can only occur if the sum of the elements of the vector x i is zero as is the case for characteristic vectors with X # 0 but not for characteristic vectors with X = 0 since these vectors contribute to the mass of the system.

280

JAMES W E 1 AND CHARLES D. PRATER

Thus, only the vector x’2 is available to serve as a coordinate axis for B species in the reaction simplex and only one B species decays. Hence, all reaction paths corresponding to the equilibrium point given by a particular vector xo(r) must be confined to a one dimensional subspace of the reaction triangle, that is, to a single straight line. But each new choice of xo(t) will

A2

FIG.25. The composition space and the reaction triangle for the system A, I!/

Aa

L2

A2

The reaction paths are a set of parallel lines each passing to a different equilibrium point. The displaced vector x’1 does not lie in the plane of the reaction triangle.

lead to a different position of the displaced characteristic vector and to a different set of reaction paths confined to the new displaced vector X’Z as shown in Fig. 25. These displaced vectors x’z will be parallel to each other as shown. In this manner, we obtain sets of reaction paths confined to parallel straight lines in the reaction triangle. Each straight line corresponds to a particular equilibrium point given by the value of &(r) used to move xz to the reaction triangle. From the behavior of this trivial system, we can infer that more complex systems have the following characteristics: (1) An infinite number of equilibrium points occur when two or more independent characteristic vectors have characteristic roots equal to zero. (2) Displaced characteristic vectors with X = 0 do not lie in the same subspace of the composition space in which the reaction simplex

ANALYSIS O F COMPLEX REACTION SYSTEMS

28 1

is located. This, consequently, reduces the number of coordinates available for discussing the behavior of the system in the reaction simplex. (3) The reaction paths corresponding to a given equilibrium point will lie in a subspace of the reaction simplex; each subspace corresponding to a different equilibrium point. (4) These subspaces will be “parallel” to each other in the sense that the same independent directions are orthogonal to each subspace in the set just as the same directions are orthogonal to each line of a set of parallel lines. Let there be m species in an n-component system corresponding to A* and A , of scheme (190) that do not react to any other species but have irreversible reaction steps from other species. In this case, there will be m characteristic vectors with characteristic roots zero and any linear combination of them will also be a characteristic vector with a zero root. There will be an infinite number of equilibrium points lying on the (m - 1)-dimensional “plane” connecting the ends of these m pure component vectors of unit length. There will be n - m characteristic vectors corresponding to B species that decay. When these are displaced along one of the characteristic vectors with X = 0 they will define an (n - m)-dimensional subspace in which all reaction paths go to the equilibrium point corresponding to the particular vector along which the displacement is made. The infinite set of (n - m)-dimensional subspaces, corresponding to the infinite set of equilibrium points, will be “parallel” to each other in the sense that the same m independent directions in space are orthogonal to each of these subspaces. 4. Constraints a. Theory. We saw in Section I1 how the law of conservation of mass, as expressed by Eq. (12), provides a constraint that confines the ends of the composition vector (Y for an n-component system to an (n - 1)-dimensional subspace of the n-dimensional composition space. The reaction paths all lie in this subspace and have no components in the direction orthogonal to it. We saw in the above section how irreversible systems with m species that do not react to other species but have irreversible steps from other species further restrict the reaction paths for each equilibrium point to an (n - m)-dimensional subspace of the composition space. Thus, in such irreversible systems, there must be m - 1 constraints in addition to the law of conservation of mass. Let us see how these constraints are related to the rate constant matrix K and to the characteristic vectors of this matrix. Up to this point we have used the set of characteristic vectors obtained by multiplying column matrices from the left by the matrix K.On the other

282

JAMES WE1 AND CHARLES D. PRATER

hand, we can obtain a set of characteristic vectors by multiplying row matrices from the right by the matrix K. I n this case, we have (199)*

z ~ K= -Xixi

where x i is a row matrix (row vector). The characteristic vectors xi and x i are called the right and left characteristic vectors, respectively, of the matrix K. The characteristic roots -hi obtained with the left characteristic vectors have the same value as the characteristic roots obtained with the right characteristic vectors*. The vectors xi and xi, corresponding to the same characteristic roots -Xi, will be equal only if the matrix K is symmetric; this is seldom the case. Hence, for each right characteristic vector xi with X i = 0 there will be a left characteristic vector x i with X i = 0. All vectors with X = 0 are, of course, invariant with time; those corresponding to the right characteristic vectors xi give the invariance of the equilibrium compositions with time and those corresponding to the left characteristic vectors x i give the constraints on the systems. For reversible monomolecular systems, the left characteristic vector that corresponds to the right characteristic vector xo is xo

='1 1 1 . . . 11'

since the sum of each column of

(200)

K is zero. The inner product

is the constraint imposed by the law of conservation of mass given by Eq. (12). We shall now show that the inner product of any left Characteristicvector x i with X = 0 and cu is invariant with time and is, consequently, the equation for a constraint. We have d

- (Xicu) dt

=

dcu

xi dt

* Multiplying Eq. (62) from the left and from the right by X-I, we obtain X-lK

= AX-1

The length of the column vectors that compose the matrix X are arbitrary insofar as they are defined by Eqs. (59) and (62) and the particular choice is governed by other considerations (Section II,B,2,d). Thus, except for an arbitrary choice in lengths, the left characteristic vectors are the rows of the inverse matrix X-1, and the characteristic roots corresponding to the left and right characteristic vectors are the same.

ANALYSIS OF COMPLEX REACTION SYSTEMS

283

since ziis invariant with time. But

zi-dcu = ziKa

at

= ( ~ i K ) a= 0

(203)

since the associative law holds for matrix multiplication. Thus,

d dt

- (&a) =

0

(204)

and

zicu = constant

(205)

as required. b. A n Example of the Equations of Constraint. We shall determine the equations for the constraints in the reaction system.

which has the rate constant matrix klZ

k41)

- (kl2

=

+

0

k32

k4l

0

k32)

0 0 0 0 0 0

For this matrix, the left characteristic vectors with X

z0 =

=

0 are

i, 1, 1, i 1

and

The equations of the constraints are and

where M is a constant. These are the equations for three dimensional ‘(planes” (three dimensional linear subspaces) in a four dimensional space. The reaction paths

284

JAMES W E 1 AND CHARLES D. PRATER

will lie on the plane (two dimensional linear subspace) of intersection between these t'wo three dimensional subspaces. The equation for this plane in the three dimensional coordinate system of the reaction simplex formed by taking a4 = 1 from Eqs. (210) and (211); it is

The value of M in Eq. (212) is determined by the particular equilibrium point through which the plane passes; from Eq. (211) and as* a4*= 1 we have

+

Hence, in the reaction simplex coordinate system, the equation for the plane of the reaction paths corresponding to the equilibrium point (a3*,a4*) is

5a1 + (1 + k4l

2) + ( a2

k32

k41

(a3 - as*) = 0

(214)

Figure 26 shows a subspace of the composition space for a typical four component system in the form of Eq. (206); it is the subspace that contains the reaction simplex. We show in this figure the plane of intersec-

FIG.26. The subspace of the reaction simplex for a typical four component system with an infinite number of equilibrium points. The plane of intersection of this subspace with the subspace defined by the constraint given by Eq. (211) is shown.

ANALYSIS OF COMPLEX REACTION SYSTEMS

285

tion of this subspace with the subspace given by t h e constraint (211) for the equilibrium point E , = (0, 0, 0.6078, 0.3922) and the rate constant n

B. EXPERIMENTAL PROCEDURES FOR THE DETERMINATION OF RATE CONSTANTS FROM CHARACTERISTIC DIRECTIONS FOR IRREVERSIBLE SYSTEMS AND APPLICATIONS TO TYPICAL EXAMPLES 1. The Determination of Straight Line Reaction Paths Lying Outside the Reaction Simplex I n irreversible monomolecular systems, the location of straight line reaction paths that lie outside the reaction simplex must be determined. These reaction paths are not subject, however, to direct measurement. At the same time, the principle relations between the characteristic vectors for reversible systems-the orthogonality relations-are not available for irreversible systems because the matrix D-1 does not exist. The constraints imposed by the law of conservation of mass and the additional constraints discussed in Section IV,A,4,a for systems with an infinite number of equilibrium points are available, however, as aids. For those systems with all steps irreversible, such as the reaction scheme (166), conventional procedures give the explicit solution in terms of the rate constants kji and in a form that allows the evaluation of exponential terms one a t a time as is required for accurate analysis of the system. Such systems will not be discussed further. For those systems with some steps reversible, such as reaction scheme (206), the principle of detailed balancing again comes to our aid in providing orthogonality relations that apply in a subspace of the composition space. These relations, along with other constraints available in the system provide the additional information needed to compute the location of the straight line reaction paths that lie outside the reaction simplex from those that lie inside the simplex. The use of subspaces in which the orthogonality relations hold implies that the reversible steps of such systems may be treated, in part, separately from the irreversible steps. This introduces into the discussion of such systems the value of the equilibrium composition of the reversible steps in the absence of the irreversible steps. The discussion in Section I1 shows the important roles played by the equilibrium composition in determining the behavior of reversible systems; a knowledge of its value is implicit in any method used to evaluate the constants for these systems regardless of

286

JAMES WE1 AND CHARLES D. PRATER

whether or not any data are used in their evaluation from the neighborhood of the equilibrium point. Hence, much more accurate and convenient determination of the constants will result if the equilibrium composition can be determined directly. This holds, also, for the reversible steps of irreversible systems. In this case, however, the values cannot be determined by the simple process of allowing sufficient time for the various species Bj, j # 0, to decay to a negligible amount; they may, however, be determined from free energy data or better, directly from other reactions in which irreversible steps are absent. In the discussion to follow, we shall consider these equilibrium values as known. 2. Example of a Typical Irreversible Reaction System with a Single Equilibrium Point: the Three Component Butene-butane System of Hamilton and Burwell The three component system cis-2-butene ka

1T

\ hi

klz

7' ksz

butane

trans-2-butene

studied by Hamilton and Burwell (IS) in an investigation of the hydrogenation of dimethylacetylene a t 20" over palladium-on-alumina catalyst provides data to illustrate a typical three component irreversible system. The hydrogenation reactions of dimethylacetylene at 20" over this catalyst is uncoupled in time into two reactions; at first dimethylacetylene is hydrogenated to cis-2-butene and the reaction (216) is suppressed, but as soon as essentially all of the dimethylacetylene disappears, the reaction given by scheme (216) then proceeds. We shall be concerned with only the analysis of this last part of the reaction. Hamilton and Burwell did not obtain good reaction time-composition correlation and, therefore, worked with composition data alone. They determined the relative value of the rate constant by giving a special solution to the problem. We shall use the method developed above to extract the numerical values of the relative rate constant and the explicit expression for the amount of butene as a function of the ratio of trans-2-butene to cis-2-butene for this particular case. The composition data obtained by these investigators are given by the points shown,in \Fig. 27. The equilibrium mole fractions used by them for the components of the reaction cis-2-butene trans-2-butene were obtained from A.P.I. Project 44 data (14); they are 0.22 and 0.78 for cis-2butene and trans-2-butene respectively. These investigators noted that, as the reaction proceeds to pure butane, the ratio of cis-2-butene to trans-2-

ANALYSIS OF COMPLEX REACTION SYSTEMS

287

cis- 2- But en e

trans- 2-Bu tene

FIG.27. The data of W. M. Hamilton and R. L. Burwell, Jr. [Proc. 2nd. Intern. Congr. on Catalysis, Paris, 1960 Paper 441 for the reaction cis-2-butene

1

[

>

butane

trana-2-butene are given by points 0 . The reaction paths calculated by the method of the text are the solid curves.

butene converged to the equilibrium ratio. This means that one characteristic composition vector is

corresponding to the straight line reaction path within the reaction triangle shown in Fig. 27. The order of the components in the vectors for this system will be chosen to be cis-2-bu tene trans-2-butene (butane The characteristic vector xo is &=@

)

288

JAMES WE1 AND CHARLES D. PRATER

and the characteristic vector x1

=

is

XI

Cy,(O)

- Xa

=

( z) -1.00

The matrices D-% and D%[Eqs. (86) and (SS)] for the subreaction cis-2butene ~2trans-2-butene are

and

The first two elements of xz and x1 must be orthogonal to each other when transformed by D-W ;

Interchanging the elements of Eq. (223) and reversing the sign of the second, we have

(-Z) Transforming this back to the nonorthogonal system, we have

Hence, the first two elements of xz are equal in value and opposite in sign. Since the sum of the elements in xzmust be zero, the last element must have the value of zero. The values of the elements in Eq. (224) may be normalized to unity to give x2

Therefore,

X

=

(

=

(-;)

0.22 0.78 -1.00

-a)

289

ANALYSIS O F COMPLEX REACTION SYSTEMS

The inverse X-l, obtained by conventional methods, is 1

I .o

.4

---

.3

-

.2

-

.8

.6

b,

I

I

I

I

Five composition points along the reaction path given in Table VII are converted to the B system by use of Eq. (40). The values of @ obtained are given in Table VII. The graph of lnbl vs In b:! (Fig. 28) gives (Xl/Xz) = 0.215; hence,

I Ill-

K' = XA'X-1

-.06 -

.I0 .08

.04

=

-

.03 .02

.o I

=(

.o

FIG.28. Ln bl vs In bz for evaluating the h ratio for the butene-butane system of W. M. Hamilton and R. L. Burwell. Jr. IProc. 2nd Intern. Congr. on Catalysis, Paris, 1960 Paper 441. The value of ~ X I / X ~ )is 0.215.

(1

0 X 0 -0.215 0

-H)X-l

-0.827 0.173 0 0.612 -0.388 0 0.215 0.215 0

)

(228)

and, making the relative rate constants to butane equal to unity, we have

"=(

-3.85 0.80 0 2.85 -1.80 0 1.00 1.00 0

which gives

cis-2-butene 2.85

1T

\ 1.0

0.80

7 1.0 trans-2-butene

butane

(230)

Hamilton and Bunve obtained 2.6 for the value of the relative rate constant (kZ1/ks1)for which we obtain 2.85. This difference in value is probably caused by the particular selection of data used in each case. The set of reaction paths shown by the solid lines in Fig. 27 were calculated using the matrix T t (Section 111B,2,j). Let us now determine the explicit equation for 1-butene as a function of

290

JAMES WE1 AND CHARLES D. PRATER

TABLE VII Composition Data of Hamilton and Burwell (13) Used to Evaluate X Ratio for the Butene-butane Sustem (216)

e

(Y

0.012

0.033

0.555

the ratio of trans-2-butene to cis-Pbutene. Since only relative rate constants are to be used and the rate constants kJ1 and k32 are equal, we may write scheme (216) as cis-2-butene kzl

1T

\LO

kle

71.0

butane

trans-2-butene

The reaction scheme (231) has the relative rate constant matrix

(

-&21

K=

+ 1) kzl 1

kiz -&z

+ 1)

0

1

its characteristic vectors are

xo

,)I(

=

x1 =

(-), kia

+ kzi

and xz =

(-i)

(233)

29 1

ANALYSIS OF COMPLEX REACTION SYSTEMS

with relative characteristic roots Xo = 0, X1 respectively. The matrix X is

= - 1and Xz =

- (1

+ + kzl) klz

According to Eq. (35), we have (1

al

=

a2

= (1

a3

=

+ k)-'bl + bz

+g) -1

Since (1

+

(k12/kzJ)-l

bo -

+ (1 + bl

=

a1

bz

b1-

(235)

bl

(kZl/k12))--l

+

a4 =

=

1-

1 and bo = 1, we have

a3

and bz = (1

+

g)-l

a1

- (1

+

z)-l

a2

(237)

Hence, according to Eq. (58), we obtain for pure cis-2-butene initial composition

Substituting the values of (238), we have

bl

and bz from Eqs. (236) and (237) into Eq.

or

Equation (239) is the equation obtained by Hamilton and Burwell (13). A comparison of Eq. (239) with Eqs. (236), (237), and (238) shows that they used the relation of In bl vs In (b&) to obtain the rate constant

292

JAMES WE1 AND CHARLES D. PRATER

ratio instead of In bl vs In bz. The explicit equation for the general case with kZ1$ k32may be obtained from the general three component solution given in Appendix 11. 3. An Example of a Typical Irreversible Reaction System With an Infinite Number of Equilibrium Points: A Hypothetical Four Component System

Let us use the reaction scheme (206) to illustrate the use of orthogonality relations in a subspace of the composition space and constraints to determine the missing displaced characteristic vector that lies outside the reaction simplex for systems with an infinity of equilibrium points. The value of the equilibrium composition for Al A z is al* = 0.6000 and a2* = 0.4000. The logical initial compositions to use are mixtures of A1 and A z ; these compositions will converge to the particular straight line reaction path within the reaction simplex shown in Fig. 26. The value of aa(0) that we obtain is 0.7208 cU,,(O)

=

( 00.0000 .2792) 0.0000

The value of the equilibrium point for the reaction plane on which this initial composition lies is determined from the extrapolation of the measured part of the straight line reaction path x’z to its intersection with the edge of the reaction simplex connecting a3 = 1 and a4 = 1 as shown in Fig. 26; the value is

0.6078 0.3922

Subtracting xo from a,(O), we obtain the value 0.7208 -0.6078 -0.3922

The problem is to locate the displaced characteristic vector ~ ‘ that 3 lies outside the reaction simplex, as shown in Fig. 26. The first two elements of each unit characteristic vector must always be consistent with the subreaction A1& A,; hence, these two elements of each unit characteristic vector

ANALYSIS OF COMPLEX REACTION SYSTEMS

293

must be orthogonal under the two dimensional transformation D-S (Section II,B,3). Using the equilibrium value given above in Eq. (86), the transformation matrix D-ta is

D-S =

1.581140

(243)

For the first two elements of xz, we obtain 0.7208

D-S (0.2792)

=

0.930548 (0.441454)

(244)

The vector orthogonal to the vector given by Eq. (244) is formed by interchanging its elements and placing a negative sign before one of them; we obtain

This vector is transformed back to the nonorthogonal system;

Dw

(

-0.441454 0.930548)

=

-0.341949 0.588530)

(

Equation (246) gives the relative sign and the relative values of these first two elements of x3. The constraints are used to determine their absolute value and the values of the other elements of x3.The ends of the vector cr,(O) must lie in the subspace given by [see Eqs. (212) and (214)] a1

+ fzaz + f3a3

=

c1

(247)

where fz, f3, and C1 are constants. This subspace must pass through the points (0, 0, 0.6078, 0.3922) and (0.7208, 0.2792, 0, 0). Using these values in Eq. (247), we have 0.7208

+ 0.2792fz = O.6O78f3

(248)

One more equation is needed to determine the values of fz and f 3 in Eq. (247); it may be obtained from any other initial composition and the equilibrium point to which it converges. An initial charge of pure A1 reacts to the equilibrium point (0, 0, 0.5333, 0.4667). The subspace a1

+ fzaz + fsa3

=

cz

(249)

294

JAMES WE1 AND CHARLES D. PRATER

must pass through the points corresponding to this initial composition and equilibrium point; the constants fz and f3 are the same as given in Eq. (248). Hence, 0.533313 = 1

(250)

Equations (248) and (250) give fz = 1.5000 and f3 = 1.8750; hence, 1.8750~3*=

which gives

C1

= 1.1396

+ 1.50~2+ 1 . 8 7 5 0 ~= 1.1396

(25 1)

From Eq. (246),

We shall make a3 = 0 in a , ( O ) ;hence solving Eqs. (251) and (252) for uzand UI,we have al = -0.7205 and u2 = 1.2401. The sum of the elements of a,,(O) must be unity; thus, a4 = 0.4804. Consequently, -0.7205

0.0000 0.4804

(253)

and

-0.7205 -0.6078 0.0882

(254)

These techniques may be used for general irreversible systems by separating the systems into completely irreversible sequences and subreactions containing reversible steps. For example, the system

is separated into the system

and

ANALYSIS OF COMPLEX REACTION SYSTEMS

295

V. Miscellaneous Topics Concerning Monomolecular Systems A. LOCATION OF MAXIMA AND MINIMA IN THE AMOUNTS OF THE VARIOUS SPECIES 1. Theory The location of maxima and minima in the amounts of various species in an n-component monomolecular system and the condition under which they occur is of considerable interest since to maximize the yield of some species and to minimize the yield of others is the aim of most chemical processes. The number and location of the maxima and minima for a given initial composition in an n-component system may, of course, be determined by calculating compositions along the reaction path. This information, however, may be obtained more easily for a given component ai from the number and location of the intersections of the particular reaction path with an (n - 2)-dimensional linear subspace (generalized “plane”) of the reaction simplex. This generalized “plane” is the locus of points for which (dai/dt) = 0. The “plane” for the ith species will be called the ith isocline. Although the term isocline applies, in general, to any “plane” for which (daildt) is some constant value, in this article we shall always mean the “plane” for which the constant is zero. It will be our purpose to determine the equation for the isocline and to derive the necessary equations for the determination of the number and location of the maxima and minima in the component a
=0 dt

=

hlal

+ k i z a z .. . (-

n

1‘kji> a i . . . + kinan

(258)

j=1

The right side of Eq. (258) is the inner product of two vectors;

kil ki2 *

. . (-

X’kji> . . . kin

i=l

Hence, all vectors orthogonal to the row vector formed by the ith row of the matrix K give extreme values for ai. These vectors define an (n - 1)dimensional linear subspace of the n-dimensional composition space. The intersection of this subspace with the (n - 1)-dimensional linear subspace

296

JAMES W E 1 AND CHARLES D. PRATER

of the reaction simplex is the (n - 2)-dimensional linear subspace that is the locus of points for the admissible maxima and minima in the amounts of ai. The equation for this (n - 2)-dimensional linear subspace in the coordinate system of the reaction simplex, formed by taking ai = 1 as the origin, is obtained by eliminating ai between n,

kilUl

+ kizaz. . . (-) 2’ j=1

kjiUi

. . . + kinan= 0

(260)

and a1+az

. . . +a,=

1

This gives

as the equation for the ith isocline in this coordinate system of the reaction simplex. For a three component system, the (n - 2)-dimensional linear subspace given by Eq. (261) is, of course, a straight line. A typical example is shown

,,&v

If

I N I T I A L COMPOSITION WITH REACTION PATHS ,NJpINING MAXIMA

Fig. 29. The regions of the reaction triangle for a typical three component system in which initial composition lead to either a maxima or minima in species A1 for finite time. The isocline (&I/&) = 0 is shown.

ANALYSIS O F COMPLEX REACTION SYSTEMS

297

in Fig. 29; the three component system given in Fig. 5 is used. Any reaction path that intersects the isocline shown in Fig. 29 will have a n extreme value in the species A1 at the point of intersection; that part of the line for which a1 > al*, the extreme is a maxima and for al < al*, the extreme is a minima. For the particular reaction system shown in Fig. 29, the straight line reaction path x‘Z corresponds to the fast characteristic species; hence, b2

=

g2161””1

where b / X 1 > 1. Therefore, any reaction path will approach the straight line reaction path x ’ ~faster than the isocline, which is a linear function of composition. Consequently, any reaction path originating from an initial composition in the shaded portion of the reaction triangle will cross the isocline and have an extreme value; those originating in the unshaded portions will be monotonic functions of al. Since the equations for the reaction paths are particularly simple in the B system of coordinates, we shall need the general equation for the isocline in this coordinate system. The rate of change of @ is given by [Eq. (25)]

Transforming the composition @ in the derivative into the composition by multiplying from the left by the constant matrix X,we have

a!

We shall define the vector ei to be the row vector with all zero elements except the ith, which is unity; that is

ei = 0 . . . o 1 0 . .. O The element ai of the vector

a!

is given by the inner product

Hence,

Therefore, from Eqs. (262) and (265), we have

(263)

298

JAMES WE1 AND CHARLES D. PRATER

Equation (266) is the equation for the locus of points (duildt) = 0 in terms of the B species. The product eiX is the vector formed by t.he ith row of the matrix X;

and

Hence,

Equation (267) is the equation for the ith isocline in the B system of coordinates. It is the desired “plane” in the reaction simplex since bo drops out automatically. Let us now derive the equation from which the number and location of the maxima and minima may be determined. We shall use the B system of coordinates in this derivation. For convenience, let us define f j and rj to be bj ip. = -

’

b?

and

Using this notation, the parametric equations for the reaction path given by Eq. (58), with bl as the parameter, are

ANALYSIS OF COMPLEX REACTION SYSTEMS

299

Since bj decays exponentially from bjo, 0 6 qj \< 1 for 1 6 j 6 n - 1. In addition, we shall use the convention that bl is the species with the slowest decay constant. Consequently, 1 6 rj for 2 6 j 6 n - 1 and we shall further order the characteristic species, vectors, and roots such that r2 6 r3 6 . . . 6 rnF1,i.e., we shall always designate the roots so that A1

6

A2

6

A).

..6

A,-1.

When Eq. (267) for the isocline is changed to the notation given by Eqs. (268) and (269) it becomes \kl = hz\kz

+

h3q3

+ .. + a

hn-l\kn-l

(27 1)

where

Substituting Eqs. (270) into Eq. (271), we have 9 1 = h291'1

+ h,*k,ls.

. . + hn-l*l'n-l

(273)

The number of maxima and minima and their locations are determined by first locating the solutions of Eq. (273) for ql in the range 0 6 \k1 6 1 and then determining the value of qj,2 6 j 6 n - 1 by the application of Eq. (270). The solutions of Eq. (273) are given by the intersections of the functions fl(\kl) = h2\kp h391r3. . . hn-l*lv-l and fz(\E1) = \kl. First, there is always the solution \kl = 0 that corresponds to equilibrium composition and infinite time. This solution will be neglected in the discussions to follow. When all the constants hj are of like sign, fl(\kl) is a monotonic power function of \kl. For this condition, we distinguish three cases:

+

+

(1) When all constants hi are positive and Zj=zn-l hj 3 1, the function fl(\EI) will cross fz(Sl)once in the interval 0 < \kl 6 1 since rj 2 1 for all 2 6 j 6 n - 1. One maxima or minima exists in the component ai for the particular initial composition used. (2) When all constants hj are positive and Zj-zn-l hj < 1, the function will crossfi(\kl) outside the interval 0 < \kl 6 1 and no observable maxima or minima exists in the species ai for the particular initial composition used. (3) When all constants hi are negative, the function fl(q1) does not intersect f2(\E1) for q1> 0 and no maxima or minima exists in the composition ai for the particular initial composition used.

When the constants hi are not all of the same sign, the number of maxima and minima may range from zero t o n - 2. Since 1 6 r2 6 r3 6 . . . 6 n - 2 extreme values in the component ai are obtained when the

300

JAMES WE1 A N D CHARLES D. PRATER

signs alternate, beginning with a positive hz, and the constants hi are of proper value. Before turning to an example to illustrate this calculation, we shall discuss the special case for an initial composition of pure Al and show that there can be no maxima or minima in az. It is sufficient to show that all constants hj are negative in Eq. (271) for the Ith isocline when the initial composition is pure A z . In Eq. (272) (&/Al) is always positive; hence, if ( x l j / x l 1 ) and (bjo/bf')are of the same sign, all the constants hi will be negative. For a pure initial component A z ,the components bp in the B system are given by X-la(0) = (lth column of the matrix X-l}. But the sign of each element of the lth column of the matrix X-l are the same as the corresponding element in the lth row of the matrix X since X-1 = L-IXTD-1 where LL1and D-l are diagonal matrices with all principal diagonal elements positive. Hence, bjo = (X--l)jl and b1° = (X-l)lz and the ratios (bjo/bf') have the same sign as the ratios (xzj/xll). Therefore, all hi must be negative and the species Al can have no maxima or minima for an initial composition of pure A z . 2. An Example of the Determination of an Extreme Value in a Four Component System

We shall use the hypothetical four component system discussed in Section II1,C to illustrate the method. In particular, the location of the maxima in the component A4 for an initial composition of pure A1 will be computed and compared with the location determined from the calculated reaction paths (Fig. 18). For an initial composition of pure A1, we have

and, using the value of X-I given by Eq. (159), we obtain

1

1.000000\

\

3.332992/

which gives (bzO/bf')= -0.204946 and (b30/b1°) = 3.14209. The values of the ratios of the elements of the various rows of the matrix X [Eq. (155)] are given in Table VIII. From the matrix A' [Eq. (162)], we have (Xz/Xl)

ANALYSIS OF COMPLEX REACTION SYSTEMS

30 1

= 6.6225 and (X3/X1) = 14.8344. Using Eq. (272) and the values of the ratios (x42/241), we obtain

and h 3 = - - - - - b30 - +21.1676 X i $41 bP

Since both constants hj are positive and 2j=2n-i hi > 1, there 's a n extreme TABLE VIII The Ratio of the Elements of the Characteristic Vectors

2

=

!?!

-0.053062

211

?2

=

2

$1.000000

+0.897918

= $0.046997

521

521

=

-0.85455

=

231

%

=

211

-0.042288

531

=

9 = -0.454133

$0.005838

541

241

in the component A4 for an initial composition of pure Al. The location of this extreme is obtained from the solution of

+

!PI = 0.0079237\k16.6225 21.1676?111'4.8344

The graphs of fl(!PJ and fz(?I12) are shown in Fig. 30; the value of Pi obtained is 0.805. Using this value of !Pi in Eqs. (270), we have !Pz = 0.2378 I

I

I

\

c - I

I

I

-x

Y FIG.30. The graph of fi(\kl) example given in the text.

and

fi(%)

used to evaluate Ylfor the four component

302

JAMES WE1 AND CHARLES D. PRATER

and q3= 0.04005. The values of the elements of @ a t which the extreme value occurs may be determined from the definition of qj .Eq. (268)l; we obtain 1 .000000

The value of a corresponding to this

8 is given by

0.0771 0.5492

This locates the maxima in A4 more accurately than the data in Fig. 18. Let us examine the other components A Zand A3. Using the value of the ratios (zij/zil) given in Table VIII, we have for the species A2 and A3 9 1

=

1.3573916.6226 - 2. 1905891'4.8344

and 9 1

+ 1.971191'4.8344

= - 1.15984916.6226

respectively. The reader can verify that these equations have no solution in the interval 0 < P1 6 1 and therefore, Az and A3 have no extreme values as Fig. 18 shows. When Az, A 3 ,and A4are used as pure initial compositions, the extreme values may be obtained from the value of the ratios (zij/zil) given in Table VIII and the value of ratios (bjo/bla) obtained from the second, third, and fourth column of the matrix X-'[Eq. (159)], respectively.

B.

PERTURBATIONS ON THE

RATE CONSTANT MATRIX

1. Theory

Although the relation between the parameters in the general solution, Eq. (6), and the rate constants are obtained for monomolecular systems by the methods given in Sections I1 through IV, the mathematical form of these relations are not the explicit expressions that conventional algebra has accustomed us to using. Consequently, the reader may feel that the treatment of the problem given in this article leaves much to be desired and that the usual explicit expressions for the composition as a function of the rate constants should still be sought so that the dependency of composition on each rate constant can be visualized by merely looking at these

ANALYSIS O F COMPLEX REACTION SYSTEMS

303

explicit expressions. Except for the simplest systems, this viewpoint is not justified. This may be seen by examining the explicit solution given in Appendix I1 for the general three component system; it is al = al*

+ -

u2 = a2*

+

+

1 ~

u2

-

u1

{[(a? - al*)

+ u,(uz0 -

+

[(a? - al*) u2(uz0- az*)]ule-x2f) 1 { - [(aln- al*) ul(azn-

- u1 [(a? - al*)

~

u2

a3 = 1 - (a1

+

+

+ uz(uzO- az*)]e-x2')

a2)

It is not easy to visualize directly how the composition depends on any particular rate constant for such a complex expression as Eq. (274). Consequently, numerical calculations must be made to determine the effect of changes in the values of particular rate constants just as one does in the method given in this article. Using Eq. (274), the composition at various reaction times must be calculated for a given set of rate constants, the values of particular rate constants changed, the calculations repeated, and the results compared. This is not a small task. The same results may be obtained more easily by calculating the characteristic vectors and roots (Appendix 111) for the original and perturbed rate constant matrix K and then comparing the compositions calculated by means of the matrix Ttl [Eq. (78)] corresponding to each rate constant matrix. But the same results may be obtained still more easily by means of a first order perturbation calculation when the changes in the values of the rate constants are relatively small. The equations needed for this perturbation calculation will now be derived. Since almost all monomolecular

304

JAMES WE1 AND CHARLES D. PRATER

systems have completely uncoupled equivalent characteristic systems with diagonal rate constant matrices A, we shall restrict our attention to such systems. Let designate the original rate constant matrix similar to the diagonal matrix A;

*

=

XAX-1

Let this matrix be changed to a new matrix A;

K

(275)

K similar to the diagonal matrix

= %AH-1

(276)

where H is a matrix composed of the characteristic vectors of K but not necessarily of the length given in Section II,B,2,d. This new matrix may be written as

K=K+&

(277)

where R’ is a matrix which will be called the perturbation matrix and E is a scalar constant with a small value. Our objective is to express the characteristic vectors and roots of K in terms of the matrix 9 and the characterwhen the changes in the rate constants are istic vectors and roots of relatively small. The matrices H and A may be written

*

it = x [ r

+ + Ew.. .I

(278)

+ €1’+ e2L” . . .

(279)

E

~

r

and A =A

where H’, H” . . . , L’, 1” . . . are matrices and e is the same small-valued constant as in Eq. (277); the matrices L’, 1” . . . are all diagonal matrices. From Eqs. (276) and (277), we have

(K+ eR)E = HA

(280)

Eliminating H and A from Eq. (280) by use of Eqs. (278) and (279) and collecting terms in powers of el we have

(KA - XA) + E [ ( S Y X

+

+ ax’) - (XL’ + XH’A)]+ + X H ’ L ’ + XL”R)] + . . . = 0

E ~ [ ( S Y H ’ &Err) - (XL”

(281)

Since Eq. (281) must hold for any arbitrary small value of E , the term for each power of E must be zero. Consequently, neglecting all powers of e greater than the first, we have

KX

=

(282)

ANALYSIS O F COMPLEX REACTION SYSTEMS

and QX

+ KE’= Xl’ + XE‘A

305

(283)

Equation (283) is sufficient to determine the elements of the matrices 3’ and 1’. Rearranging Eq. (283) and using Eq. (282), we have

- XSfA = 21’ -

ax

(284)

Multiplying Eq. (284) from the left by X-l, we obtain

A%‘- EfA = 1’ - X-lQX

(285)

The elements of the matrix on the left of Eq. (285) are

(AX’ -

E’b),j

=

- ( E ’ ) i j ( X i - Xi)

(286)

[As in Section 11, the ijth element of a matrix G is designated (G)+] Equation (286) shows that the elements along the principal diagonal (i = j) of the matrix A%’- %’Aare zero. Hence, the elements along the principal diagonal of the matrix on the right side of Eq. (285) are also zero and

(1’)ii = (X-Y=tX)ii

(287)

Since 1’ is a diagonal matrix, the off diagonal elements of X-lQX are given by (X-1aX)ij

=

(E’)ij(Xi

- Xj)

This gives

The diagonal elements of b’ are not determined by Eq. (283); for the perturbation calculation, they may be arbitrarily assigned the value zero for the following reason: The matrix E is formed from the characteristic vectors of the matrix K.The lengths of these characteristic vectors are arbitrary and the particular lengths chosen do not affect the results of similarity transforms (Section IV,A,2) made by E and E-l, since in these transformations both E and E-l occur. The lengths of the vectors of E and E-l will become important only for those transformations in which E or E - I occur singly, as for the transformation of compositions from the A to the B system of coordinates. Only similarity transformations are involved in Eq. (276) and in the calculation of the matrix T t [Eq. (78)]; hence, vectors of any length may be used to form the matrix E for the perturbation calculations. In this case, one element of each vector may be

306

JAMES WE1 AND CHARLES D. PRATER

identical in value to the corresponding element in X. Thus, the diagonal elements of the matrix E’ may be chosen to be zero. Nevertheless, it must always be remembered that for some purposes, the length of the vectors in the matrix E must be adjusted according to the criterion given in Section II,B,2,d to obtain the matrix X. TOsummarize, the characteristic vectors and roots of the matrix K = k €9are given, to first order terms in e, by

+

E

=

XII + EZ’]

(289)

and A

b

=

+ €1‘

where the elements of the matrix E’ are given by and (E’)ij

= (X-15tX)ij, xi

- xj

and the diagonal elements of the diagonal matrix 1’ are given by

(1’)ii

=

(X-lRX)ii

(293)

2. Example o j the Perturbation Calculation: The Butene Isomerization System We shall demonstrate the use of these equations by calculating the effect of a 20% decrease in the rate constants k32 and kz3 of the butene isomerization system discussed in Section I1,B. The rate constants for this system are [Eq. (129)] 1-butene 10.344//4.62~4,4g31 //” 10.344//4.623

3 . \Y 724\y 3.724\ \ 1.OOO

// // cis-2-butene t

5.616 3.371 (2.697)

’ trans-2-butene

where the new values of the rate constants are given by the numbers in the brackets. The rate constant matrix for the system has one element normalized to unity and is, therefore, the relative rate constant matrix K discussed in Section II,B,2,h. The relative rate constant matrix in the B system of coordinates equivalent to K will be designated 1; the relative rate constant matrix A’ that has been used previously is similar to the relative matrix K’(Section II,B,2,h).

307

ANALYSIS OF COMPLEX REACTION SYSTEMS

The values of the matrices X, X-l, and k for this system (given by Eqs. (120), (124), and (128), respectively) are used to calculate the relative matrix i; it is

L=

c

0 -9.2602

0

o

0 0 - 19.4180

(295)

The perturbation matrix R for decreasing the values of the rate constants and k32 is [Eq. (277)]

k23

R

=

0 (0 0

0 5.616 -3.371 -5.616 3.371

(296)

0 0 0 6.3406 1.8484 0 9.0784 2.6465

(297)

The matrix X-IRX is

For

e =

0.2, Eqs. (290) and (293) give

)

0

)

0 0 0 0 + 0 . 2 ( 0 6.3406 - 19.4180 0 0 2.6465

8

0 -18.889 Equation (298) gives (Xl/X2)

=

7'9921 - 0.42310 and, consequently, 18.8887

~

0 (299) -1

Using Eq. (292), we have for the off diagonal elements of the matrix I' ( W 3 2

and

=

19'0784 0.1578 - 0.89314

308

JAMES WE1 AND CHARLES D. PRATER

The matrix 2’ is, therefore, 0

0

0 0.89374

-0.18197 0

(300)

and we have

H

=

X[I

+ 0.2Z’J 0

-0.03639 0 0.17875

1

Hence, 0.1436 0.3213 0.5351

0.1799 -0.1511 0.3603 0.1604 -0.5402 -0.0093

(301)

The inverse of the matrix 2 may be calculated by the method given in Section II,B,2,c. The matrix T t(Section II,B,2,j) is calculated from b , Z-I, and 1, and used to calculate the reaction paths for cis-2-butene, I-Butene

FIQ.31. The effect of a 20% decrease in the value of the rate constants h3 and the butene isomerization system.

k32

for

trans-2-butene, and 1-butene. These reaction paths and the straight line paths for the perturbed system are shown in Fig. 31 along with the corresponding paths for the unperturbed system.

ANALYSIS O F COMPLEX REACTION SYSTEMS

309

C. INSENSITIVITY OF A SINGLECURVED REACTION PATHTO THE VALUESOF THE RATECONSTANTS A short discussion was given in Section II,A,2 on the difficulties of using curve fitting techniques to obtain the values of the parameters in the general solution of the rate equation. In this section, we shall continue the discussion with an illustration of how insensitive a single curved reaction path can be to the values of the rate constants and make some comments on the related question of the use of curve fitting techniques in obtaining these values. The butene isomerization system discussed in Section III,B will be used for the illustration. The true straight line reaction paths are given by the heavy solid lines in Fig. 32. Let us imagine that the straight line reaction paths are rotated until the one corresponding to the slowest decaying characteristic species passes through the pure trans-2-butene corner of the triangle as shown by the straight line reaction paths - - * - in Fig. 32. Also let the straight line reaction paths be rotated so that the paths 9

I -Bu t e n e

FIG.32. The insensitivity of the reaction path for a pure cis-2-butene initial composition to the values of the rate constants.

corresponding to the slow B species passes through the point (0.38,0.62, 0) as shown by the lines - - - - in Fig. 32. Consider an initial composition of pure cis-2-butene; the true reaction path for this composition is also shown by the curve in Fig. 32. As we have seen in Sections I1 and 111, the reaction path for a particular initial composition is completely

310

JAMES WE1 AND CHARLES D. PRATER

determined by the location of the straight line reaction paths and the ratio of the rate constants in the B system (lambda ratios). Hence, we may inquire as to whether the lambda ratios can be adjusted so that reaction systems corresponding to the rotated straight line reaction paths of Fig. 32 give composition points indistinguishable from the true reaction paths within the accuracy of experimentally measured compositions. For the straight line reaction path - - - - for which the path corresponding to the slowest decaying species is rotated towards the reaction path for pure cis-2-butene1 the composition points may be returned to the neighborhood of the true reaction path by a decrease in the rate of decay of the fastest decaying species, i.e., by an increase in the ratio X1/X2 from its true value of 0.4769 (Fig. 15). Let us try the value 0.485. This gives, for the matrix A’,

-

(:

0

A’=

0 - 0 7

-

-!)

(302)

- -

The matrix X obtained from the rotated straight line paths - - - is 0.1653 -0.1436 0.3698 0.1430 0.5351 -0.5351 0.0000

0.1436

X

)

(303)

7

(304)

(

= 0.3213

and has the inverse,

x-1 =

F

-4.8125

1 1 2.1510

-0.8688 0

Using the matrices (302), (303), and (304) in Eq. (78) to compute a matrix = 0.5, we obtain

Ttl for the time interval tl T,,,,

1‘’

=

0.6927 0.0860 0.0309 0.1921 0.7988 0.0692 0.1152 0.1152 0.8999

(

)

(305)

This matrix Ttl is used to calculate compositions for a pure cis-2-butene initial composition [Eq. (Sl)]. The points obtained are shown by o in Fig. 32. As can be seen they lie on the true reaction path within the accuracy of the graph. For the straight line reaction path --- for which the path corresponding to the slowest decaying species is rotated away from the reaction path for pure cis-2-butene, the composition point may be returned to the neighborhood of the reaction path by an increase in rate of decay of the

311

ANALYSIS OF COMPLEX REACTION SYSTEMS

fastest decaying species. This means a decrease in the value of (A&,) 0.4769; let us try 0.46. The matrices A', X, and X-' for this case are

0.1436 0.5351

0.2364 0.2987 -0.5351

-0.1436 0.1969 -0.0533

=

(307)

and

-3.7096

1 0.7734 -0.8320 2.2732 -0.3695

which gives 0.7239 0.0909 0.0195 0.7763 0.0797 0.0728 0.1328 0.9008

(309)

The composition points, computed by use of this matrix Ttl, for pure cis-2-butene initial compositions are given by h in Fig. 32. Again the points lie on the true reaction path within the accuracy of the graph. The composition points calculated for tl = 0.5 from the matrix Ttl for the true reaction system are given by the points in Fig. 32. The matrix Ttl for the true reaction system is 0.7087 0.0892 0.0246 0.1997 0.7854 0.0753 (310) 0.0916 0.1253 0,9001

+

The values of the relative rate constant matrix K for these three indistinguishable cases are

Ktrue

=

Kcasel

=

and

kase2 =

( ( (

-14.07 4.62 10.34 -10.24 3.72 5.62 -4.37 -11.07 3.28 7.34 -7.01 3.73 3.73

-18.23 6.48 14.50 -14.81 3.73 8.33

i::)

i:::)

-3.24

i:)

-6.00

312

JAMES WE1 AND CHARLES D. PRATER

Hence, the three sets of rate constants

8.33 5.62 3.73

have indistinguishable reaction paths within the accuracy of the graph of Fig. 32. Yet these constants in some cases differ by a factor of two. Hence, any appropriate set of constants in this range may be used to compute quite accurate reaction paths for initial conipositions in the neighborhood of the reaction path for pure cis-2-butene. But when we use compositions away from this neighborhood the agreement may be extremely bad as shown by the calculation of compositions along the reaction path for an initial composition of pure 1-butene. These composition points are shown in Fig. 32 and give a spread in compositions of as much as six percentage points in the neighborhood of 20y0 trans-2-butene. This illustrates the hazards of using only a single curved reaction path to determine the rate constants by curve fitting techniques. The situation gets progressively worse as the number of components increases beyond three. In the above calculation no effort has been made to see how wide a range of rate constants can be obtained for a given error in composition and time. We have merely taken the results of the first calculation made for this system. The spread in values of the rate constants may be decreased, in principle, by including the reaction paths or time courses for other initial compositions differing greatly from the first one used. It is, however, much easier to locate the straight line reaction paths directly. In the absence of explicit expressions for the relations between the constants cii and X j in Eq. (6) and the rate constants, an appropriate analog computer may be used to obtain the rate constants by curve fitting techniques. When a single initial composition from a region with curved reaction paths is used, the above discussion shows that rate constants may be obtained that have little resemblance to the true value; for such cases, the time courses and reaction paths may be predicted correctly only in the immediate neighborhood of the path used to obtain the value of the rate constants. Again, reaction paths from widely different regions of the reaction simplex must be fitted in order to converge on the true values with sufficient accuracy for the prediction of all time courses and reaction paths and for use in investigation of basic mechanisms. In this case, it is

ANALYSIS OF COMPLEX REACTION SYSTEMS

313

easier to locate the straight line reaction paths. When this is done, analog computers offer no special advantages for the solution of the problem.

VI. Pseudo-Mass-Action Systems in Heterogeneous Catalysis A. SOMECLASSESOF HETEROGENEOUS CATALYTIC REACTION SYSTEMS WITH RATE EQUATIONS OF THE PSEUDOMONOMOLECULAR AND PSEUDO-MASS-ACTION FORM 1. General Comments

By definition a heterogeneous catalytic reaction involves, at least, the interaction of a molecular species with another entity of the system provided by the catalyst. In reactions using solid catalysts, this interaction is usually considered to lead to adsorbed molecular species on the active surface of the catalyst. For such systems, the amounts of both the adsorbed molecular species and the free molecular species will appear in the general rate equation. In many types of catalytic studies, however, the amounts of the adsorbed species are not measured and rate equations are sought in terms of the amounts of the free molecular species only. Many models proposed for reactions over solid catalysts have algebraic rate equations from which the amounts of adsorbed species may be eliminated for steady state conditions. Such eliminations will, in general, lead to expressions for the rate of change of the ith species that are given by the ratio of two polynomials in the amounts of the various free molecular species. In contrast, mass action kinetics requires that the rate of change of each molecular species be given by a single polynomial in the amounts of the various species. Consequently, except for special circumstances, one might expect that mass action kinetics would play a relatively minor role in studies of heterogeneous catalysis that use free molecular kinetics expressions. Nevertheless, the actual role of mass action kinetics in such studies is much greater than the above argument suggests because, for a large class of models, the polynomial in the denominator of the polynomial fraction is the same for each rate equation correspondingto free molecular species in the system, that is, the rate equations are pseudo-mass-action equations. A pseudomass-action system will be defined as a reaction system in which the rate equations are given by mass action terms of various integral orders multiplied by a function of composition and time, 4, that is the same for all rate equations in the system. In particular, the rate equation for the ith species in a pseudomonomolecular system is given by

314

JAMES WE1 AND CHARLES D. PRATER

where 0,; is the pseudo-first-order rate constant for the reaction from the ith to the j t h species. For the entire system, we have

or

do! dt

-=

@o!

(313)

where 0 is the pseudomonomolecular rate constant matrix. In this section, we shall establish some sufficiency condition for the origin of some pseudo-mass-action systems with particular emphasis on pseudomonomolecular systems. This discussion will not, however, lead to the establishment of necessary conditions for the origin of such systems. 2. A Class of Heterogeneous Catalytic Systems with Rate Equations of the Pseudomonomolecular Form

a. Conditions Specifying the Type of Reaction. Let us examine a class of heterogeneous catalytic systems that satisfies the following conditions : The reaction systems can, in principle, be decomposed into steps with the reaction flux through each step given by mass action expressions in which the rate parameters are independent of the amounts of the reacting species. Only one type of active site contributes to the reaction systems. The number of active sites is constant and equal to N. Free molecular species cannot interact with sites that have adsorbed molecules on them. Adsorbed molecules cannot interact with other adsorbed molecules in the manner of a two dimensional gas. The amounts of the various adsorbed species on the catalyst are invariant with time (steady state conditions). For many uses of pseudo-mass-action and pseudomonomolecular kinetics we do not need to give explicitly the detailed split of the reaction into the mass action steps postulated in condition (1) ;their existence in principle is sufficient. Experimental demonstration that the kinetics is of a pseudomass-action type will give strong support to the hypothesis that such a decomposition ,into mass action steps is possible.

ANALYSIS O F COMPLEX REACTION SYSTEMS

315

I n the discussion to follow, we shall always assume that the amounts of adsorbed species are not measured. We shall show that, when the above conditions apply, the amounts of adsorbed species can be eliminated from the rate equation and that a pseudomonomolecular reaction system is always obtained. Hence, conditions (1) through (6) are sufficient to insure that the rate equations for the reaction systems are pseudomonomolecular. We shall first discuss systems satisfying the above conditions and then examine some systems in which some of the conditions are not satisfied. b. The Pseudomonomolecular Rate Equations Obtained from Detailed Reactions with Simple Coupling. A typical example of a system satisfying conditions (1) through (6) will be discussed to establish the method of approach and the nomenclature to be used. Let there be three free molecular species Ai in the reaction system to be used as a n example; the generalization to systems containing n free molecular species will be made as we proceed. Let us designate the adsorption complex between the species A , and the active catalytic site by 21c and active catalytic sites with no adsorbed species on them by '%. The reaction scheme we shall consider as an example is

Ai

+ 30

k,i

kdi

lr

81

A2 kzl

*

+ 80 ll k d z

ka2

3 2

klz

As+ 80 k3z

*

ka3

LT

kd3

8 3

k23

(314)

where k,i and k d i are the rate parameters for the adsorption and desorption of the species A i respectively and ki, is the rate parameter for the reaction from the adsorption complex % i to the adsorption complex gj. Condition (1) requires that the rate parameters are true rate constants; they are not functions of the amounts of the various free and adsorbed species. Making k,j and k d i independent of the various amounts ai of the various adsorbed species Bi is equivalent to the assumption that the adsorption is of the Langmuir type. Condition (3) requires that

,f a i = N

(315)

i=O

When condition (6) applies, the rates of reaction of the various free species in scheme (314) are given by

3 16

JAMES WE1 AND CHARLES D. PRATER

The steady state condition (6) also requires that (hi

+ kzi)ai - (Laiao +

(kd2

f

k32

f

(kd3

+

k23) a3

kl2)a2

-

k i 2 ~ 2 )=

- (kn2a2aO

(kn3a3ao

+

+ kaaz)

d a1 =0 dt

k23a3

+

k2lal)

da 3 dt

= -=

=

d a2 dt =0

0

(317)

The seven equations given by the sets of equations (316) and (317) and Eq. (315) are sufficient for the elimination of the amounts a i from the rate expression (316). It will be most convenient for our generalization to n components to express these equations in matrix form: 1

1

'N' 0 0

= o

0

I:

I

Let A designate the square matrix of the left of Eq. (318) and let Mji designate the minor formed by deleting the j t h row and the ith column of the matrix A. Then, by Cramer's Rule [Birkhoff and MacLane (15)], we have from Eq. (318)

where ]A]and lMlil are the determinants* of the matrices A and Mli

* The following rules for determinant manipulations will be needed: (1) Interchanging two rows or columns of a determinant changes its sign. (2) The addition of a constant times the i t h row (or column) to the j t h row (or

column) does not change the value of the determinant.

ANALYSIS O F COMPLEX REACTION SYSTEMS

317

respectively. These determinants are, of course, polynomial in the amounts ai and the system is a t least pseudo-mass-action since N / l ~isl the same for each rate equation. Conditions (2) and (4) requires that the amounts ai are located in the same column of the matrix A. Consequently, the deterIM111, 1M12[,and 1M13( are linear functions of the amounts of minants /A], free molecular species, and these systems give pseudomonomolecular rate equations for steady state conditions. Let us generalize the matrix A given in Eq. (318) to systems with n free molecular species and obtain the general expression for the determinants /A1 and lMlil (i = 1 . . . n).To make this generalization, it is advantageous to decompose the matrix A into submatrices. Examination of the matrix A given by Eq. (318) for the scheme given by Eq. (314) shows that for an n component system, we have

(0 . . .0 1 . . . l

1

where I is an n X n unit matrix [Eq. (43)], 0 is a n n X n matrix with all elements zero, U is the n X n rate constant matrix for the adsorbed species, and G is a n n X n matrix arising from the steady state condition. For the three component system given by Eq. (314), these matrices are

and

/

;

\

They are indicated by \ in Eq. (318). The matrices U and G are independent of the amounts' of the various species in the system and are matrices with nonzero elements only on their principle diagonal and the diagonals immediately below and above the principal diagonal;

318

JAMES WE1 AND CHARLES D. PRATER

... .. . ...

0 0

u11

UlZ

0

UZl

UZZ

uZ3

0

.. .

U3Z

.. .

u33

u34

0 0

0 0

0 0

0 0

... ...

0 0 0

0 0 0

... .. .. ..

U(n-1)(n-Z)

(323)

%-1)(n-1)

0

U(n-1)n

Un(n-1)

unn

and

... ... ...

... where, for i = 1 .

. . n, the elements of the matrix =

uii

whenj

=

(i

k(i-l)i

+

U are given by (325)

k(i+l)i

+ l ) , or (i - 1) u . . = - k . .3’ 32

and w h e n j # (i - I), i, or (i

+ l), uji

In Eqs. (325) and (326), Correspondingly,

k01

=0

and F;ii

(327) have no meaning and are zero.

= kdi

(328)

Uii

and

6.. = u3% .. 3%

(329)

The matrices G and U differ only by the terms k d i , which occur in the elements of the principal diagonal of G. We shall use the structure of the matrices U and G to write the generalization of Eq. (319) in the pseudomonomolecular form (311). Let us designate the matrix formed by replacing the j t h row of the matrix G by a row of one’s as G ( ’ ) j and the matrix formed by replacing the jth row of the matrix G by the ath row of the matrix U as G i j (for i = 1 we have G l i with no parenthesis). The position of the unit submatrix I in the matrix A [Eqs. (318) and (320)l show that the determinant [A] is given by

IAI

=

IGI

+

IG(l)ll

kalal

+

/G(1)21

k,Zaz.

..+

lG(l)nl

k,,a,

(330)

ANALYSIS OF COMPLEX REACTION SYSTEMS

. . n), is given by ka2a2 . . . - JGinI kanan

319

and that the determinant IMli/, (i = 1 . (- l)l+'jMljl = - lGil1 k , i ~-

lG'2l

(331)

Since the matrices U and G do not contain the amounts of any species in the system, Eqs. (331) and (319) show that systems with the properties (1) through (6) give pseudomonomolecular rate equations for steady state conditions. The set of equations, (319), (330), and (331) may be written in matrix form for comparison with Eq. (312); we have

Comparison of Eqs. (332) and (312) shows that the pseudomonomolecular rate constant Oii for the reaction from thejth to the ith species is to be identified with 8(3. . = - k

ai. ( G i i (

i# j

(333)

In order to make this identification, however, Eq. (312) shows that we must have jGiil

=

-

2' [GGl i

(334)

The reader can verify that Eq. (334) does indeed apply since each column in the matrix U sums to zero. Hence,

It can be shown that the determinant [ G i j / is always negative and the determinant jGiiJ is always positive so that B i j and - Z'j eji always have the correct sign. The general function 6, is given by +J=

IGI

N

+ Z:

(336) kaiai

IG(~)~I

and may be identified with the adsorption of the various species on the active sites. The polynomial in the denominator of 6, gives information about the adsorption competition between the various reacting molecules

320

JAMES WE1 AND CHARLES D. PRATER

in the system and is of considerable interest. I t will be designated the isotherm polynomial in contrast to @a, which will be called the mass action polynomial for the pseudomonomolecular system. It can be shown that the determinants IGJ and IG(')il are always positive so that all terms in the isotherm polynomial are positive quantities. It is desirable to make one of the pseudomonomolecular rate constants unity by dividing each element of the n X n matrix of Eq. (332) by one of the determinants ka~IGmzl, I # m and multiplying the general function $J by kallGmzl.Let the ijth pseudo-rate-constant for this normalized system be designated elij and the rate constant matrix by 0'. Then

and

Equations (333) and (335) or Eqs. (337) and (338) may be used to determine the explicit relations between the pseudo-rate-constants and the true rate constants since the matrices G and U are in terms of the true rate constants for the system. Let us apply Eqs. (333) and (335) to determine this relation for the reaction in scheme (314). From Eqs. (331), (321), and (322), we have for species A1

Adding various rows in the determinants of Eq. (339) to other rows so as to produce as many zeros as possible and evaluating the resulting determinant, we have

SW3LSAS NOIL383X X31dW03 $0 SISA'IBNB

322

JAMES WE1 AND CHARLES D. PRATER

and

e13= ~~~e~~ Consequently, the principle of detailed balancing holds for the pseudomolecular formulation of scheme (314). Let us normalize the rate constants given by Eq. (343), so that 1 9 ’ ~=~ 1, by incorporating kalk32k21kd3 into the function 4. Hence,

(347)

and

where (349)

c. The Pseudomonomolecular Rate Equations Obtained from Detailed Reactions with More Complex Coupling. I n scheme (314), each free species corresponds to a unique adsorbed species and vice versa. Let us seek a more general scheme that obeys the set of conditions (1) through (6). We may have either adsorbed species that do not correspond to a free species or a single adsorbed species may arise from more than one free species or both. For example, either a scheme may contain sequences A i + 80 Kdi

*

Koi

KUi-1)

%i-1

K( i-1) i

Ai+2

lJ

Ko(i+P)

i2 Ki(i+l)

*

Ki+Ni+l)

K(i+l)i

Bi

%’i+l

K(i+l)(i+Z)

+

%la

Kd(i+Z) K(i+a)(i+z)

Bi+Z

@

%r+3

(350)

K(i+z)(i+3)

where the prime on %’;+I signifies an adsorbed species with no corresponding free species, or a scheme may contain

323

ANALYSIS OF COMPLEX REACTION SYSTEMS

Of course, the number of free species corresponding to a single adsorbed species and the number of adsorbed species in the sequence is not limited to two and one respectively. In addition, a scheme may contain both types of structures. First, we shall show that scheme (350) with intermediate steps that do not correspond to free species is indistinguishable from the scheme A i + go km JT

*

k,(*-l)

B2-l

k(E-l)2

A2+2

kdi

+ 30

TJ h ( t + Z )

kd(%+2)

e

kW2)

*

k(%+V(l+Z)

k(C+2)%

Bi

B2+2

B2+3

(352)

k(ttZ)(~+3)

when no direct information is obtained about the amount of the species B’i+l. Let us look at the steady state equation, in matrix form, for the steps given by scheme (350) ;

=

0

The problem may be restated now in geometrical language. The vector on the left side of Eq. (353) has six elements; it will represent a vector in six dimensional space if none of the elements can be expressed as linear combinations of the other elements. On the other hand, if scheme (352) is to be equivalent to scheme (350), it must be a vector in five dimensional space. Hence, to prove the equivalence of schemes (350) and (352), we need only to show that the vector in Eq. (353) is really in five dimensions rather than six. This may be accomplished by showing that the 6 X 3 matrix in Ey. (353) can be transformed, by the elementary row operations (16) given below, into a matrix in which the third column is of the form

(353)

324

JAMES WE1 AND CHARLES D. PRATER

The elementary row operations needed are (1) the multiplication of a row by any nonzero constant and (2) the addition of any multiple of one row to any other row. Applying these elementary row operations to the matrix on the left of Eq. (353), we obtain Ka(s-1)

K(1-1)~

--1

+

Kd,

0

K(s+3)(+2)

1

K(*+3)1~+2)

-K(s+%)

(Kd&

+

K.Z($+Z)%+Z)

--

--

Ka(.+zI%+z XI(.+l)

K.(*+I)

K.(.+l)

K,(,+l)

-

(1+3)

K(.+Z)(.+Z)

\fJ-

()-Q

(355

The above matrix shows that the element of the vector on t,.e !ft is a linear function of ail ai+2, ai+3, and ao. The functional relationship is obtained by multiplying the column vector on the left by the second row of the matrix (355); it is K(i+i)i

a"+1 = -ai Ki(i+l)

K(i+3)(i+2) K(i+2) (3+3) -ai+2 + Ki(ifl) Ki(i+i)

Ka(i+2)ai+2

+

a0

Ki(i+l)

Hence, only five elements of the vector are independent and the vector is five dimensional instead of six. Thus, scheme (352) is equivalent to scheme (350) when no direct observations are made of the amounts of adsorbed species a'i+l. This treatment may be generalized to include sequences of any number of adsorbed species of the type ~ ' ( i + ~ ) ;any column of the generalized form of the matrix in Eq. (353) may be reduced to the required form (354) if the sum of its elements is zero. This will always be the case for all adsorbed species that do not correspond to free species. Now let us turn to the scheme (351). It also leads to a pseudomonomolecular system. We shall not, however, treat the general case but give the equation for a special case from which the reader may infer the general form of the relation between the pseudo-rate-constants and the rate constants for the detailed kinetic scheme. The scheme to be considered is

kd8

+i T Zo ka.9

AB

The matrix equation obtained for this system, analogous to Eq. (318), is

ANALYSIS O F COMPLEX REACTION SYSTEMS

325

(357)

this gives

/

kc1

- klc

O \

326

JAMES WE1 A N D CHARLES D. PRATER

where

'

+

N

+

+

+

(361)

IGJ I G ( l ) l j k a l a l JG(1)2](ka2a2 kQ3a3) IG(1)31ka4a4 Equation (360) is of the pseudomonomolecular form with =

A

4

eZ1 = -

-

IG21jka1,

C'

Biz = -

=

- (/Gy21 -

j=1

eZ3= - (/G22J - I G ( ) k a 3 , 824 = - IG23jka4

/GZ21ka2,

4

e31 - IG311ka~,

032

e41 = - 1G411k,1,

e42 = -

/G421ka~j

-

[Gl)hg,

e48= -

2' ej3

j =1

= -

(G 1ka3,

,e,

- 1G331ka4

=

-41G421ka3,

2'

j =1

Oj4

=

-

jG43(kQ4

(362)

The reader can verify that the diagonal elements of the matrix @ formed from Eq. (362) are equal to the negative of the sum of the other element in each column as required. Hence, general schemes composed of steps of the type exemplified by the steps in schemes (314), (350), and (351) and satisfying conditions (1) through (6) always give rate equations of the pseudomonomolecular form. d. The Principle of Detailed Balancing for the Class of Reaction Systems. Before we can use the results for monomolecular systems given in Section I1 for the steady state pseudomonomolecular systems, we must prove that the principle of detailed balancing holds for these n-component pseudomonomolecular systems. We shall show that

An equivalent statement for Eq. (363) is that the matrix @D is a symmetric matrix for any n-component system satisfying conditions (1) through (6). For a system with n free molecular species and m surface species, we introduce the n m dimensional grand transition rate constant matrix K so that

+

ANALYSIS OF COMPLEX REACTION SYSTEMS

327

For steady state condition, all (daildt) = 0, and the set of all vectors (a, a) really belong to a subspace of dimension n. The situation here is similar to that discussed in Section VI,A,2,c. It is then possible to find a matrix P so that

where the rectangular boxes are used to show that the matrix P is, in general, not a square but a rectangular matrix and that the column vectors a and a contain different numbers of elements. This box notation will be used throughout this section to indicate the general form and partitioning of the matrices arising in the development. Equation (365) states that, for steady state conditions, the vector a is not independent of, but is a function of the vector a. The method of evaluating the matrix P was given in Sections VI,A,2,b and c. Using Eq. (365), Eq. (364) can be rewritten as

(366)

K

The matrix K may be partitioned into submatrices, and we have

(367)

+

In Eq. (367), the matrix Kb KdP is identically equal to zero since its product with any arbitrary vector (Y is zero. Equation (366) reduces to

328

JAMES WE1 AND CHARLES D. PRATER

that is, to

Equation (369) corresponds to Eq. (332). It will be shown that, when detailed balancing is valid in the grand matrix K of Eq. (364), the collapsed matrix Ka K,P, which is also $I@, obeys detailed balancing. The fact that K obeys detailed balancing is equivalent to the equation

+

where 0 is the diagonal matrix

Introducing the partition of K used above, we have

Because of the symmetry of the matrix KD, we have

and

ANALYSIS O F COMPLEX REACTION SYSTEMS

329

+

We shall use these equations to show that (K, KcP)D is a symmetric matrix. Since K,D is already symmetric, it is sufficient to show that KcPD is symmetric. Since Kb KdP is identically equal to zero, we have

+

p = - K -d 'Kb

(372)

Thus,

KCPD

=

-K&-'&D

(373)

and by Eq. (371), we have

KcPD = -KC&-'%KcT

(374)

The matrix on the right of Eq. (374) is shown to be symmetric as follows: We see that

(KcKd-l%KcT)T= Kc%(Kd-')TKcT

(375)

From Eq. (371), we have

(&%)-I

=

((&%)T)-'

this gives

%-lKd-l = (Kd-l)TD-l Multiplying the above equation from both the right and left by 3,we have

Kd-lSD

=

%(Kd-')T

Hence,

(K&-'%KcT)*

=

KcKd-'DKcT

and the matrix is symmetric. Thus, the matrix K,PD is symmetric. Consequently, the matrix (K, K,P)D is symmetrical, and the principle of detailed balancing holds for pseudomonomolecular rate equations obtained from reaction systems satisfying conditions (1)t,hrough (6) ;we always have 0.. = K..e.. ,I *I 3 % (376)

+

3. A Class of Heterogeneous Catalytic Reaction Systems with Rate Equations of the Pseudo-Mass-,4ction Form Reaction systems with mass action polynomials containing terms of order greater than unity are obtained when interaction is allowed between free and adsorbed molecular species-that is, when condition (4) is relaxed. Such interactions always cause the amount of the free species involved to appear in columns of the matrix A other than the one corresponding to the free sites Uo. Consequently, this introduces terms containing products of the amounts of various species into the mass action polynomial.

330

JAMES WE1 AND CHARLES D. PRATER

A general proof will not be given but instead we shall examine a special example of such a system. This system is

Writing the set of equations in matrix form, we obtain

"I

0 0

= o

0 0

,o

(379)

ANALYSIS OF COMPLEX REACTION SYSTEMS

331

Equation (379) resembles the matrix Eq. (318). The most important change is that the rate constant kzl has been replaced by kn2a2and, therefore, all amounts ad do not occur in the same column of the matrix A, Hence, we shall obtain terms cont,ainingthe products of amounts and the system will not be pseudomonomolecular but will, in this case, contain second order mass action terms. Equation (379) gives

and

Thus, the system is pseudo-mass-action. Evaluating the determinants in the mass action polynomial of Eq. (380), we have IG"1 1G2'1

1G3'1

= knzkd3k32a2,

1G211 =

= -k23kdlkd2, = -kazazJGd3k32,

lGl31

=

knzk32kd3az -kdik32kd2

(381)

1G331 = kdikdzkz3

Substituting the values of the determinant given by the set of Eqs. (381) into the set of Eqs. (380), we obtain

a_ " - 4 { - (kazkaikd3k32)aiaz+ kdik,izh3ka3a3I dt

and the mass action part of the set of Eqs. (382) corresponds to the reaction scheme

+A z - e h @/

A1

(383)

8b

where 0, and o h are the forward and backwards pseudo-rate constants respectively and are given by

of = ka~kalkd3k32

332

JAMES WE1 AND CHARLES D. PRATER

Therefore, if free molecular species interact with adsorbed molecular species and the other conditions (1) through (6) hold, we may infer that the reaction will be pseudo-mass-action but not pseudomonomolecular and that conditions (l), (2), (3), ( 5 ) , and (6) are sufficient conditions for the system to have pseudo-mass-action rate equations in terms of the free molecular species. B. SYSTEMS WITH MORETHAN A SINGLE TYPE OF INDEPENDENT CATALYTIC SITE Let there be more than a single type of independent active site contributing to a reaction [relaxation of condition (2)]. By independence, we mean that the different types of sites and the molecular species adsorbed on them do not interact with each other. For this discussion we shall exclude not only direct interaction but interaction by means of desorbable intermediate passing through the gas phase as discussed by Weisz in this volume of Advances in Catalysis. Nevertheless, there is a kind of coupling between the two sites provided by the free molecular species. This type of system may be exemplified by a model that might apply for the butene isomerization system discussed in Section I1,B. A possible mechanism might involve two types of sites on the alumina catalyst; one type of site that can cause both the double bond shift and the cis-trans isomerization and another type of site that can cause only cis-trans isomerization. We shall consider all the other conditions given in Section VI,A,2,a to hold. The rate of reaction of the species A; will be given by the sum of two independent pseudo-mass-action terms corresponding to the two reactions,

Let us examine this system in greater detail; we shall consider the mechanisms

and

K12

ANALYSIS OF COMPLEX REACTION SYSTEMS

333

where % and 8 represent two independent types of catalyst sites. Since these systems only interact by means of the free molecular species A 1 and A z , we have

and

where (dai/dt)a and (dai/dt)a are the reaction rates for each type of independent sites. Moreover, the reactions (dai/dt)a and (dui/dt)a each satisfy the conditions (1) through (6) and each lead to its own set of pseudomonomolecular rate equations. Hence,

where the sub- and superscripts in Eq. (389) signify corresponding quantities for each independent reaction system. Obviously, Eq. (389) is not the rate equation for a simple pseudomonomolecular system since two functions $8 and $8 are involved. Hence, except for the special cases given below, such proposed mechanisms can be tested by determining whether the reaction paths in all regions of the reaction simplex are given by a single set of relative rate constants. If this is the case, under the conditions of the experiment, either only a single type of site is involved in the reaction to any significant extent, or the adsorptions of all free molecular species in the system are such that only the linear part of their isotherms are involved, or that the strengths of all adsorptions on each type of site are equal. For the linear case, $8 and $8 are constant and the system appears to be truly monomolecular. Whether the system is truly monomolecular or not can be tested by observing the time behavior of the system; if the graph of In b, vs time is curved, then all adsorption steps cannot be in the linear part of the isotherm and we can say that only a single site is involved in the reaction. Lago and Haag have applied the method presented here to studies of the mechanisms of butene isomerization over pure alumina catalyst. The results of their study and the tech-

334

JAMES WE1 AND CHARLES

1).

PRATER

nique used will be given in their forthcoming article and will not be discussed here. The methods and results of this article may be combined with the treatment of the diffusion step given by Weisz (17‘) to study complex systems containing more than one kind of site but with coupling provided by intermediate species passing through the gas phase. OF C s - OVER ~ ~ C. THEHYDROGENATION-DEHYDROGENATION PLATINUM CATALYSTS AS A PSEUDO-MASS-ACTION SYSTEM SUPPORTED

+

Smith et al. (18)have studied the reaction cyclohexane benzene 3Hz over platinum on alumina catalyst. The methods of this article were used to test whether the reaction scheme

is consistent with experimental data. The species Al, Az, and Al are to be identified with cyclohexane, cyclohexene, and benzene, respectively. In scheme (390) the adsorption steps are assumed to be Langmuir-that is, that the rate parameters k,i and IGdi are not functions of the amounts of either adsorbed or free molecular species. This is possible since the platinum on the catalyst used is present as very small particles of approximately 15A or less in diameter. Certainly, for such small particle sizes, the metal will not have the properties of bulk metal and a single adsorption event, leading to catalysis, may well modify the electronic properties of the metal particles so greatly that a simultaneous second such adsorption may not occur on the same particle. The rate equations for scheme (390) may be derived by the method of Section VI,A; they are identical with the set of Eqs. (348) except that wherever the rate constants klz and kZ3 appear they are multiplied by the amounts of hydrogen, [Hz], and the amount of hydrogen squared, [H2lZ, respectively. Thus the rate equations for the scheme are pseudo-mass-action and we have

_ dai - 4’{ ( K31a3[H2IY- al) + e’zl(Kzlaz[Hzl - ad} at

~

~

ANALYSIS OF COMPLEX REACTION SYSTEMS

335

and

If the reaction mechanism is consistent with scheme (390), the reaction paths for all initial compositions should be identical with those obtained from the reaction scheme

8 H2

H2

H

Ho H

/ H H

+ 2H2

In addition, at constant hydrogen partial pressure the system should appear to be pseudomonomolecular and the reaction paths for all initial compositions should be identical with those obtained from the scheme

where A1 is cyclohexane, A2 is cyclohexene, A , is benzene, and pij is the pseudomonomolecular rate constant for this system. Comparison of scheme (393) and Eqs. (391) and (347) shows that p12,p23, p21,and p13 are functions of hydrogen partial pressure. Thus, consistency of the behavior of the reaction with the implications of the scheme can be tested by determining whether the two sets of rate constants derived from schemes (392) and (393) can predict, in the presence and absence of changing the hydrogen partial pressure respectively, the reaction paths for all regions of the reaction simplex. The methods of this article are needed because (l),as we shall see below, the cyclohexene passes from differential to integral conditions at such a small mole fraction that truly differential experiments could not be made, ( 2 ) sufficiently reproducible composition-time correlations are difficult to obtain even with the most careful work, and (3) the rate equation is sufficiently complex so that the ability to study the mass action polynomial separate from the isotherm polynomial becomes critical to the successful

336

JAMES WE1 AND CHARLES D. PRATER

analysis of the consistency of the prediction of scheme (390) with experimental data. Some experimental data obtained by Smith et al. (18) will be used to illustrate the method of approach. These data were obtained with a pressurized all glass flow reactor at a temperature of 399". The catalyst used was 0.05% platinum on q-alumina prepared by impregnating the alumina with chloroplatinic acid. The chlorine introduced by this method of impregnation was removed to reduce the acidity of the catalyst so that the isomerization of cyclohexene to methyl cyclopentane was suppressed. The amount of conversion was varied by changing the amount of catalyst in the reactor. The catalyst bed in the reactor consisted of 15 ml of crushed vycor (60-100 mesh) in which the desired amount of finely crushed (100-200 I

W

5

I

1

I

.0016

X

.0014

3V

.0012

0

.0010

>

z

p

.oooe

V

2

.0006

lL

w

.0004

5

MOLE FRACTION CYCLOHEXANE

FIQ.33. The reaction path obtained with pure benzene initial composition for the reaction over supported platinum catalyst a t 399"C., 20 atm of hydrogen and 5 a t m of hydrocarbon.

mesh) catalyst was uniformly dispersed. The amount of catalyst was less than 150 mg in all cases and the flow rate of hydrocarbon was 0.833 ml/min. The partial pressure of hydrocarbon was 5 atm in all cases. The chemical analysis of the liquid product was made by chromatographic techniques. Let us examine first some data obtained at a constant hydrogen partial pressure of 20 atm. For the conditions of these experiments, the equilibrium mole fraction of benzene, cyclohexene and cyclohexane were measured to be 0.3245, 0.0009, and 0.6746, respectively. The results obtained with an initial composition of pure benzene are shown in Fig. 33. The results indicate that the mole fraction of cyclohexene rises to approximately the equilibrium mole fraction of 0.0009 and then levels out.

337

ANALYSIS OF COMPLEX REACTION SYSTEMS

This fact tells us that one of the characteristic composition vectors is

(:::z)

cuZ,(O) = 0.0009

(394)

where the order of the elements are cyclohexane cyclohexene (benzene

)

The straight line reaction path corresponding to this characteristic composition must be parallel to the side of the reaction triangle connecting pure cyclohexane and pure benzene. It can be shown from the orthogonality relations that the other straight line reaction path must pass through the pure cyclohexene apex of the reaction triangle; hence,

Thus, the matrix

X is 0.6746 -0.6746 -0.6746 0.0000 0.9991 0.3245 0.6746 -0.3245

(396)

and its inverse, obtained from Eq. (46), is

X-l

=

1.0000 1 .0000 1.0000 -0.4815 0.0000 1.0009 -0.0009 1.0000 -0.0009

(

(397)

Transforming the points on the rising part of the curve in Fig. 33 to the B system by multiplying each composition vector by the matrix X-' and plotting In b , vs In bz, we obtain A1 -A2

-0.0079

(398)

Hence, using Eq. (67) to calculate K' and normalizing so that obtain K, we have

K=

(

-1.396 297 0.396 -440 1.000 143

"t:)

-2.48

plal = 1 to

(399)

338

JAMES WE1 AND CHARLES D. PRATER

The theoretical reaction path for pure benzene initial composition is shown by the solid curve in Fig. 33. A comparison of the theoretical curve and experimental points in Fig. 34 shows that the characteristic vectors and

.oos W

z

,004 W

I

s

.003

z 0

d

.002

I+.

w

J

p

.001 .0008 .0006 .0004 .0002

'

MOLE FRACTION BENZENE

FIG.34. The reaction paths obtained from an initial composition of pure cyclohexame and from an initial composition consisting of a mixture containing 0.9949 mole fraction of cyclohexane and 0.0051 mole fraction of cyclohexene.

lambda ratios given by Eqs. (396) and (398) predict quite accurately the reaction path given by initial compositions of pure cyclohexane and a mixture containing 0.9949 mole fraction of cyclohexane and 0.0051 mole fraction of cyclohexene. Let us see if we can predict also the reaction path when the partial pressure of hydrogen changes. From Eq. (347) and the fact that we must replace k23 by [H2I2kz3 shows that

We need to know the value of ( k 2 3 / k d 3 ) H2I2in Eq. (400) before we can predict the reaction paths at a new partial pressure of hydrogen. For the moment let us take it to be negligible compared to unity and calculate

339

ANALYSIS O F COMPLEX REACTION SYSTEMS

reaction paths for pure benzene and pure cyclohexane initial composition at 10, 30, 40, and 50 atm hydrogen partial pressure. These reaction paths are shown in Fig. 35 with some experimental points. We see that the small amount of experimental data is consistent with these calculated reaction paths except for a small departure at high pressure probably caused by a ) / (the JC~ rate ~ ) constants. This slight contribution of the term ( I C ~ ~ [ H ~ ] ~to departure can be used to calculate the value of (k23/kd3). ,0030 W

I W x W

I

s

YU

.0020

z

Gc u

2

.0010

LL

W -I

0

I

0 0

.20

.4 0 .60 MOLE FRACTION BEN ZEN E

.a0

1.0

FIG.35. The reaction paths for pure benzene and pure cyclohexane initial compositions for different hydrogen partial pressures. Experimental points obtained with pure benzene initial composition are shown. The shifting equilibrium points are shown by 0.The experimental points for the different pressures are indicated as follows: A 10 atm; 63 20 atm; IxI 30 atm; A 40 atm; El 50 atm.

Although sufficient data are not given so that firm conclusions may be reached, the reader can see how one may be able to establish whether the scheme (390) is consistent with experimental data and to evaluate the constants B i j in the three mass action polynomials. Furthermore, this can be accomplished without knowledge of the reaction time; this makes the experimentation much easier. With the effects of the mass action polynomials established, the more interesting isotherm polynomial is easier to study. A more complete treatment of this reaction system will be found in the forthcoming article of Smith et al.

VII. Qualitative Features of General Complex Reaction Systems

A. GENERALCOMMENTS In the preceding sections we confined our discussion almost exclusively to systems with monomolecular and pseudomonomolecular rate equations.

340

JAMES WE1 A N D CHARLES D. PRATER

As was noted in Section I, the understanding of these systems is a prerequisite to the treatment of more general complex systems. In the discussion of monomolecular and pseudomonomolecular reactions the qualitative features, such as the straight line reaction paths, the functional form of the equation for the reaction paths [Eq. (SS)], the existence of an equilibrium point, and the existence and nature of the constraints on the system give us a clear insight into the problem. A panoramic view is obtained that enables us to plan a course of action for optimizing the usefulness of the results of experimentation and computation. In this case we were also able to give the quantitative solution to the problem. Unfortunately, the rate equations for most complex systems are nonlinear equations for which such quantitative explicit solutions are not known. The situation parallels our ability to give such solutions for nonlinear differential equations with only one dependent variable or linear equations with many dependent variables. The usefulness of the qualitative features in the treatment of linear systems tells us that we should also seek to discover the qualitative features of the more complex systems. An understanding of the qualitative features of a set of nonlinear differential equations enables one to plan intelligently, efficiently, and accurately the experimentation and computation program needed to supply the quantitative details. I n Section VII, we shall examine some of the known general characteristics of complex nonlinear systems and supply some comments as to how this knowledge may be extended.

B. CONSTRAINTS One feature of the system, which has been Iong recognized [see for example, DeGroot (29)],is that interactions between the various species in the system can lead to additional constraints on the system. For example, consider the reactions 2A1+ A2 2A3 -41 A B ~ S A ~ (401) Although the reaction space for this system is four dimensional, the reaction paths are confined to two dimensional subspaces since only two rates are independent. We may define the quantities E l 1 and Ef2 such that

+

ANALYSIS OF COMPLEX REACTION SYSTEMS

341

The reader will recognize the two quantities 5 ’ 1 and lt2as the degree of advancement introduced by DeDonder (60); they express the two degrees of freedom or the two dimensionality of the subspace to which the composition point is confined during the course of the reaction. Thus, in addition to the constraints introduced by the conservation of mass and by irreversible steps discussed in Section IV,A,4, we have constraints introduced by the interaction between the various molecular species. Constraints may be thought of as invariants of the system; they separate the entire reaction space into subspaces characterized by constant values of the invariants. No point in composition space is accessible to another point unless they have the same value for all invariants. Let us write the pth chemical reaction of a set of p reactions in a complex coupled reaction system, involving n species, as VipfA1

+

vzpfA2

+ . . . vnpfAn

*vi;A1

+

+ . . . v , ; A ~ (403)

~2p6A2

where vi,f and vi,” are the non-negative forward and backward stoichiometric coefficients respectively, many of which may be zero. The stoichiometric coefficientsused here are not the ordinary stoichiometric coefficients of a chemical reaction but are the products of the ordinary coefficients and the molecular weight of the corresponding species. Let us define Yip

=

ViPb

- ViPf

(404)

The law of conservation of mass then gives n

2

Yip

=

0

(405)

i=l

In the reaction scheme (401) each reaction characterizes a degree of freedom. Let us examine the criterion as to whether each of the chemical reactions given by Eq. (403) characterizes a degree of freedom. Let Ep be the (‘degreeof freedom” characterized by the pth reaction. Then, we have P

dai

=

2 vi,dtP P-1

where ai is the weight of species A;. If each EP characterizes a true degree of freedom, the rank or dimensions of the space defined by the matrix v must be q. For a reaction system with n species the matrix can be at most of rank n, even if q > n, since the entire composition space is only of dimension n. Thus, if q > n, some of the reactions are redundant and their number may be reduced at once to n.

342

WL

JAMES WE1 AND CHARLES D. PRATER

The constraints on the system can be found by determining the vectors perpendicular to the matrix v; WZTV

=

0

(407)

or II

The invariants, or equations of constraint, are then given by

Il

= wzTa =

constant

(409)

or n

11

=

C

=

constant

i=l

where a is the composition vector for this system. We see a t once that, since the law of conservation of mass requires that

one of the vectors, WIT, is 1 . \

..1...

1 and that the rank of v is a t /

most n - 1. Let there be n - m invariants in the system; then the set of rate equations of order n are reduced to a set of rate equations of order m,

where l < i < m

Each true degree of freedom can be characterized by a reaction of the form of Eq. (403). Thus, the set of q 6 n chemical reactions can be reduced to a set of m 6 q chemical reactions each characterizing a t,rue degree of freedom tP,(1 p 6 m),and n - m equations of constraint. We are free to assign the value zero to all 5's a t time zero. Hence,

and we can write

ANALYSIS O F COMPLEX REACTION SYSTEMS

343

When all (d,$,/dt) = 0, then all (dad/&) = 0. The reverse need not be true, however, if there are any nonzero vectors which become zero when multiplied by the matrix v. Nevertheless, if the principle of detailed balancing applies, we always have when all (ciai/&) = 0, then all (dti/dt) = 0. In this manner, the dimensions of the problem for the original complex reaction is reduced from rL to m.

C. THEEQUILIBRIUM POINTIN GENERALCOMPLEX REACTION SYSTEMS Just a s for monomolecular systems, the equilibrium points are structural features that play a central role in the discussion of general complex reaction systems. It is not, however, necessary to introduce them into the system as explicit basic assumptions or to introduce them by means of thermodynamics or statistical mechanics; they arise as a consequence of some much more primitive concepts, which are always included in the basic models for closed reaction systems and for many open systems as well. The reader may ask why raise the question as long as the existence of the equilibrium points are assured by some known principles such as those provided by thermodynamics: the reason is that a new point of view and an appreciation of the consequences of implicitly and explicitly known basic characteristics often reveal to us the path to a better understanding of nature and to the solution of a particular problem. We shall now show that the existence of a t least one equilibrium point in a reaction system is assured by the following basic characteristics [We;

(sol: (1) The total mass of the system is conserved; Z iai = constant. (2) No negative amounts can arise: ai >, 0. (3) The rate of reaction of each species is a continuous function of the amounts of the various species in the reaction system: (dai/dt) = Sib1 . . . an). The continuity requirement given by condition (3) is merely a restatement of the every day experience in the laboratory that, as a chemical reaction proceeds, the compositions change “smoothly” from one composition to another and do not jump discontinuously from one composition to another composition differing greatly from the first. Mathematically it is the requirement that a continuous line remains unbroken, and neighboring points remain neighboring points as the reaction proceeds. Just as for monomolecular systems, the set of continuous rate equations for general reaction systems can be thought of as a continuous transformation that moves a composition along a reaction path or a time course (trajectory). Characteristics (1) and (2) tell us that this transformation can never take a Composition point outside the special region of the reaction space

344

JAMES WE1 AND CHARLES D. PRATER

that we have called the reaction simplex. There is a theorem in topology [Lefschetz (22), Courant and Robbins (WS)],called Brouwer’s fixed point theorem which states that in a given region of space there is always at least one point unchanged by a given transformation if this transformation can never move a point in the region to outside the region. This is exactly what we need because the equilibrium point is a point unchanged by the transformation of the chemical reaction and characteristics (1) and (2) prevent the transformation from changing any composition into compositions lying outside the reaction simplex. Hence, the existence of equilibrium points is a consequence of much more primitive characteristic of the reaction system than those provided by thermodynamics or statistical mechanics-namely, characteristics ( l ) , (2), and (3) given above. Brouwer’s fixed point theorem also applies for open reaction systems in which characteristic (1)is replaced by Z ai 6 constant [see Wei (Wl)]. These primitive characteristics are not sufficient to guarantee that the reaction will converge to the equilibrium points for general complex reaction systems. For example, there may exist limit cycles (or periodic paths) where a point returns to its original position periodically and never converges to an equilibrium point. There may also be nonconverging, nonperiodic paths such as can be observed with a pendulum describing Lissajou figures. Consequently, other principles or basic characteristics of the system must be sought as the source of the convergence to equilibrium that is the common experience in chemical reactions. Again we could invoke the principle of thermodynamics, but first let us examine this question from a more general viewpoint in order to obtain a maximum understanding of the problem. This leads to a consideration of Liapounov functions.

D. LIAPOUNOV FUNCTIONS The general convergence properties of differential equations of more than one dimension are best studied in terms of Liapounov functions [Lefschetz (ad)]. In this approach one constructs functions resembling potential functions that have minima at the equilibrium points since one feels that, in a chemical system, all spontaneous changes must be powered by a drop in some sort of potential. The existence of a Liapounov function always signifies a dissipative system where occurrences are irreversible as opposed to conservative systems where occurrences may recur indefinitely. The decreasing Liapounov function is the antithesis of an invariant, which is changeless. A Liapounov function V(a) may also be thought of as a generalized distance function between a composition point and the equilibrium point if V(a*) is defined to be zero. Familiar examples of Liapounov functions are: the Hamiltonian function in a system of mechanical motions where there are nonconservative forces opposing the motions, Boltzmann’s

ANALYSIS OF COMPLEX REACTION SYSTEMS

345

H-function in the statistical mechanics of molecular collision, and the excess entropy (S - S,,) for adiabatic systems in classical thermodynamics. It is important to emphasize that the convergence of a reacting system is guaranteed by the existence of some Liapounov function, not necessarily those of classical thermodynamics. When a Liapounov function is found, it is not unique; any monotonic function of the Liapounov function is another Liapounov function. Formally, a Liapounov function is a function V(al, az . . . a,) of fixed sign that is everywhere positive (or negative) and its time derivative dV aV dai -=ca,-rtl dt

(414)

is everywhere negative (or positive). Therefore, V(dV/dt) is always negative except when all ai = 0. The convergence to an equilibrium point in a complex reaction system is guaranteed if there exists a function B(a1, a2, . . . a,) such that

v -dV
(415)

for this system. Let us compare a general formulation in terms of Liapounov functions with classical thermodynamics, since a Liapounov function, excess entropy ( S - S,,), arises in the second law. Let us look at a most penetrating and logically satisfying formulation of the second law-namely, the axiomatic formulation given by Caratheodory (25). The axiom that Caratheodory invokes is: “In the neighborhood of each point P in the space defined by the equation of state there is a point Q that is inaccessible from P by any adiabatic process.” It is equivalent to boldly asserting that P and Q belong to different equivalent classes (26)and, of course, a function S exists that defines these equivalent classes. The functional forms of the Liapounov functions given by classical thermodynamics are dictated by the first law, that is, by conservation of energy with heat included. For chemical reaction systems, an assertion that leads to Liapounov functions is: “If a reaction path passes through a composition point P, then there is a neighborhood about P to which the reaction path will never again return.” This characteristic is sufficient for convergence; but what determines the form of these Liapounov functions? Herein lies the importance of considering general Liapounov functions because the question is raised as to what basic characteristics or principles are available for determining the form of the functions. Certainly, we are not limited to the conservation of energy alone, but we may seek other characteristics and principles com-

346

JAMES WE1 AND CHARLES D. PRATER

mon,to all members of a particular class of reaction system of interest. A new potential function determined from such characteristics and principles may yield exactly the structural features that give the needed insight into the behavior of the systems. In general, a complex reaction system will have many different Liapounov functions that are not monotonic functions of one another. Let us examine two such functions for reversible monomolecular systems as an illustration. The function

is a Liapounov function

According to the last equation in Appendix IC, we have

60

&TRB

(418)

for arbitrary vectors 6. Hence,

-dVi
(419)

and, since tiT& is a positive number (it is just the square of the length of the vector ti), we have Vl(dVl/dt)6 0 as required. Writing out the product >6tiTii we have

Typical contours of constant V1 are shown by the curves - - - - in Fig. 36 for a typical three component system. Let us now show that, for an isothermal and isobaric reversible monomolecular system, the excess Gibbs free energy (G - &) is a Liapounov function for the system. Let the reactants be considered to be perfect gases; then [Slater (27)]

G-& RT ~

=

C a i l n -ai ai* i

where G is the Gibbs free energy per mole for the monomolecular system and & is the Gibbs free energy per mole at equilibrium. The time rate of change of [(G - G)/RT]is given by

347

ANALYSIS OF COMPLEX REACTION SYSTEMS

FIG. 36. Contours of const5nt values of the Liapounov functions VI (- - -) and the for a typical three component reversible monomoexcess Gibbs free energy G - G (-) lecular system.

since fi is a constant. But

Zi(ciai/&)

d(G/RT) dt

0, and we have

=

i

Equation (422) can be written in matrix form;

d(G'RT) = [In (D-la)lTKa, dt

where [In (D-'a)lT is the row vector ln (ul/ul*) .

,

. In (u,/u,*): I

[In (D-'a)lTD% = [D%In (D-'a)]T, we have

-dG/RT - [Dta In (D-Ia)]T(D-xKD%)D-sa dt

=

[ D g In ( D % ) ] T & t

(424)

The elements in the matrices in Eq. (424) are given by

(D% In D - I ~ =) ~42 In 2 Ui*

(&)j

= Uj/l/?

(K)ii =

-C'

i ( Q j

=

kij

(425) kji

4=, i # j

348

JAMES WE1 AND CHARLES D. PRATER

Substituting these into Eq. (424), we have

since each term in the summation is negative or zero. This shows that the free energy always decreases except in the case where kiiaj - lciiai = 0; that is, when equilibrium is established. Hence, the excess Gibbs free energy (G - &) is a Liapounov function for reactions between perfect gases in isothermal, isobaric systems;

The contours of constant Vz are given by the solid contours in Fig. 36 for a typical three-component system. It is well known that any monotonic function of entropy (or free energy) will serve as an entropy-like (or free-energy-like) function since the contours of constant entropy are unchanged; they are merely rescaled. It is important to note, however, that every reaction path must be consistent with all Liapounov functions, not merely those given by classical thermodynamics. The difference between the functions T'l and Vz above is not trivial as in the case of the rescaled entropy but the functions have very different contours (Fig. 36). The existence of Liapounov functions with contours different from the Liapounov functions of classical thermodynamics is general for complex reaction systems. The interesting possibility arises that one of these nonclassical thermodynamic Liapounov functions may be more useful in discussing general complex reaction systems. In fact, for reaction systems that are neither adiabatic nor isothermal, classical thermodynamics does not supply a Liapounov function. A more general treatment of the Liapounov functions in chemical kinetics is given by Wei (21).

ANALYSIS O F COMPLEX REACTION SYSTEMS

349

E. IRREVERSIBLE THERMODYNAMICS AND THE RELATION OF LIAPOUNOV FUNCTIONS TO THE DIRECTION OF THE REACTION PATHS The individual potential-like Liapounov functions are not sufficient to give us the reaction paths since no rules have been given that relate these functions to the rate of reaction. For example, in isobaric, isothermal systems, thermodynamics requires only that the reaction paths always give a decrease in the Gibbs free energy, and it does not prohibit impossible reaction paths such as those for monomolecular systems shown by the heavy curves -t-- in Fig. 36. In fact, contours of constant Gibbs free energy, given by the solid curves, do not rule out paths inconsistent with the contours of constant Liapounov functions V1 given by the dashed curves in Fig. 36. This can be seen by examining the region between the uppermost curves for Vl and the Gibbs free energy in the Al corner of the reaction triangle of Fig. 36. Let us consider the directions that the reaction paths may take from the point of intersection of the two contours - - and on the left side of the triangle. Let these directions be expressed as vectors drawn from the given point. We see at once that, although the direction vectors lying in the region between the two contours lead to a decrease in Gibbs free energy, they lead to an increase in the Liapounov function V1. This is clearly a violation of the requirement that the reaction paths must be consistent with a decrease in the value of all Liapounov functions. On the other hand, for those regions where the free energy contours are less curved than the V1 contours, some possible reaction paths consistent with the function Vl, are ruled out by the Gibbs free energy. Hence, the directions of the reaction paths for a given complex reaction system are defined within increasingly narrow limits as the number of known classes of Liapounov functions increases. I n fact, if all classes of Liapounov functions are known for a given system, the reaction paths are uniquely determined by the Liapounov functions alone. In the absence of this complete information about the system, some rules relating the direction of the reaction paths to the Liapounov functions are needed. It is because classical thermodynamics does not supply such rules that it gives such little information about rate processes. In classical mechanics, we are accustomed to relating the gradient of a potential to the direction and magnitude of the force F producing the motion of a point; that is, I _

F

= -grad

V(z, y, z )

Can we consider the motion of the composition point in composition space as produced by a “force” that is the gradient of one of the Liapounov func-

350

JAMES WE1 AND CHARLES D. PRATER

tions of the reaction system? This is certainly possible for the monomolecular system. For the moment, let the motion of the composition point always follow exactly the lines of “force” for these systems; then 1 oTAQ = -grad _ do -- F = grad -

2

dt

VB

(429)

Thus, in the B system of coordinates, the scalar

is such a Liapounov function for a monomolecular system. Writing out the product in Eq. (430), we obtain

Vs = A1 - bI2 2

+ -xz2 bzZ + . . .

An-1

__

2

bti

(431)

The lines of constant potential V 3are (n - 1)-dimensional ellipsoids in the orthogonal B system of coordinates and are distorted ellipsoids in the nonorthogonal B system. The reaction paths can be obtained immediately as the lines of “force” normal to the equipotential lines in the orthogonal system. This formulation, however, has the disadvantage that both the rate constants and the amounts of the various species are included in the potential function. Let us examine another Liapounov function for the monomolecular system;

The equipotential lines are now circles in the orthogonal B system of coordinates. The motion of the composition point is given by

9 = A grad Vq dt

(433)

The motion of the composition point is no longer always in a direction normal to the equipotential line, but at an angle determined by the matrix A. This angle changes with position but, since the matrix A is diagonal, the components of the vector Q in each direction are completely independent. Hence, the determination of the reaction path may be made easily from the equipotential lines if A is known. Let us examine the Liapounov functions Vl and V2 given by Eqs. (416) and (427). For Vlr we have di?

1

- = &grad 5 iiT6 dt

(434)

ANALYSIS O F COMPLEX REACTION SYSTEMS

351

The response (deldt) is again not in general normal to the equipotential lines. In addition, since the matrix R is not a diagonal matrix, the response in each direction depends on all other directions and the reaction paths are not easily determined from the equipotential lines. The gradient of the Liapounov function Vz is a nonlinear function of the amounts; it is a vector with elements given by 1 In (aJai*). Consequently, no constant linear transformation exists for relating the response to grad V z . Thus, the use of the Liapounov function V 2actually makes the formulation of the system in terms of the gradient of a Liapounov function more complicated than the original rate equation corresponding to the formulation

+

@ dt

=

K grad -1 aTa 2

(435)

For the monomolecular system the determination of the proper Liapounov function for a proper choice of coordinates reduces the complexity of the problem and leads to the immediate determination of the reaction paths. This formulation is, of course, merely the solution given in Section I1 rewritten in a different form. Its importance lies in the suggestion that the complexity of some classes of nonlinear systems may be reduced by such an approach even if the complete solution is not obtained. The above formulation has some similarity to the formulation used for the irreversible thermodynamics of Onsager (1) et al. Irreversible thermodynamics discusses systems in which more than one irreversible process is taking place such as heat transfer, diffusion, electrical conduction, and chemical reaction. It introduces into classical thermodynamics additional plausible axioms to relate the rates of these processes to the Liapounov functions of thermodynamics. Consider a system that is characterized by a number of parameters (Q1 . . . Qn) where each Q i is a quantity such as temperature, pressure, concentration, etc. The entropy of the system is a given function of all the parameters Q. Suppose the system is closed; then it will move along a reaction path in the Q-space until it reaches the equilibrium point of maximum entropy. Since entropy of the system must increase with each spontaneous change, it is used as the potential function in the Q-space whose gradient is the driving force. Suppose, for a region near the equilibrium point, that the rates are linear functions of the gradient of entropy;

Then, with the help of the principle of microscopic reversibility, it can be proven that the matrix L is symmetric.

352

JAMES WE1 AND CHARLES D. PRATER

It is well known that irreversible thermodynamics is of very limited application in chemical kinetics [see, for example, Prigogine ( 2 ) , DeGroot (19),Denbigh (as)]. Let us examine the reasons. The rates are, in general, not linearly related to the gradient of the entropy. The relation (436) is, however, applied in the neighborhood of the equilibrium point since many functions can be approximated by linear ones in a sufficiently small interval. For monomolecular systems in which the reactants are perfect gases, the gradient of the entropy is identical with -R grad Vz. The elements of the In (ai/ai*)]. Although the gradient of the vector -R grad V2 are -R[1 entropy is a driving “force” in the sense we have used it in t,his section, it is unfortunately not proportional to the rates for any known chemical kinetics and the application of the approximation given by Eq. (436) turns out to be very small. Even if the linear law given by Eq. (436) applies, the irreversible thermodynamics of Onsager et al. still has nothing to contribute to chemical kinetics. Its central theme and source of usefulness is the statement (Onsager relations) that the matrix L is symmetric. This symmetry is a consequence of the principle of detailed balancing-a principle well known and used in chemical kinetics independently of irreversible thermodynamics. A typical proposal of how the Onsager relations provides useful information about chemical reactions proceeds as follows [DeGroot (29)l: The true kinetics of a chemical reaction are very often not known and the principle of detailed balancing cannot be readily applied. In such cases the symmetry of the matrix L can be used to obtain some useful information about the process. One proceeds by noting that the “conjugant forces ( x ) and fluxes ( j ) ” in Eq. (436) are related by

+

dS dt

aS dQi i

(437) i

It is asserted that if one determines pairs of “conjugant forces and fluxes” in the laboratory so that

and

then it follows that the matrix z is symmetric [DeGroot (29)].Hence, reaction systems near equilibrium may be studied by determining “conjugant forces and fluxes” obeying Eqs. (438) and (439) and using the

ANALYSIS O F COMPLEX REACTION SYSTEMS

353

symmetry of the matrix E in reducing the number of parameters needed to specify the system in terms of these “forces” and “fluxes.” Coleman and Truesdell (SO), however, have shown that the matrix need not be symmetric. We shall give here an alternate proof that follows more closely the usual method of approach [see for example DeGroot (29)l. We have given that

ds

-=

jTx

(440)

at

j = Lx

(441)

where L is a symmetric matrix. Let j , x, and L be transformed to new coordinate systems so that

-

j = Zf

(443)

where -7 and f are the fluxes and forces in the new coordinate systems. We want to know whether the matrix E is always symmetric. From Eq. (441), we have jT

=

xTLT

= xTL

(444)

Substituting the value of jTgiven by Eq. (444) into Eq. (440), we obtain

ds

_ _ --

XTLX

dt

(445)

By the same procedure we have

Let P be the linear transformation so that

then Consequently, we have

ds =X dt

~ =~

XX ~ p ~ ~ ~ p X

(449)

It is sometimes thought that the solution to Eq. (449) is uniquely PTETP

=

L

(450)

354

which says that tion is really

JAMES WE1 AND CHARLES D. PRATER

z is symmetric since L is symmetric. But the general solu+w = L (451) PTZTP

where W is any antisymmetric matrix with all principle diagonal elements zero and off diagonal elements such that WT=

-W

(452)

This can be seen by noting that for any vector x XTWX

=0

(453)

Hence, dt

= xT(PT€TP

+ W)x

= XT(PTETP)X

(454)

= XTLX

Z given by

Consequently, the asymmetric matrix =

-

(PT)-lLP-l

(PT)-lWP-l

(455)

may be the matrix corresponding to the particular choice of forces and fluxes which satisfy the requirements of Eqs. (442) and (443). We must conclude that we cannot be certain that the matrix E is symmetric except by measuring its elements; consequently, Onsager relations cannot be used to obtain information about the system. Furthermore, let us imagine that we have measured in the laboratory all elements of the matrix E and have found that the Onsager relations hold for it;

Lij = Ljd

(456)

One might think that we could conclude that

- =as xi

aQi

(457)

and consequently, get some useful information about the system from irreversible thermodynamics. This is unfortunately still not true in general. Suppose that the system of fluxes has constraints, such as the law of conservation of mass in monomolecular reactions, so that all fluxes Ji can not be varied independently. In general let yTj =

0.

(458)

then the value of Xi given by

as xi = cyi + aQi

(459)

ANALYSIS OF COMPLEX REACTION SYSTEMS

355

where C is any arbitrary constant, will satisfy the equation. Hence, unless the dimension of the matrix 1 is reduced to eliminate the constraints on the system, we cannot even conclude that the forces measured are the gradient of the entropy in the neighborhood of equilibrium. It would appear from the above discussion of Liapounov functions in monomolecular systems that there are potential functions better suited for chemical kinetics than entropy since their gradients are proportional to the rates. The Liapounov function V1 is exactly correct for the monomolecular system, and is a better approximation for second or higher order systems than the logarithmic function given by V z , which is used in irreversible thermodynamics.

VIII. General Discussion and Literature Survey In this section, we shall discuss the results presented in the preceding sections as a whole and supply the relation of the development presented in this article to previous investigations. No attempt will be made, however, to give an exhaustive survey of the literature on complex reaction systems. Such a survey may be obtained from standard works on chemical kinetics [Benson (6),Frost and Pearson (6), Laidler @I)] and from review literature such as the Annual Report on the Progress in Chemistry and the Annual Review of Physical Chemistry. The monomolecular reaction systems of chemical kinetics are examples of linear coupled systems. Since linear coupled systems are the simplest systems with many degrees of freedom, their importance extends far beyond chemical kinetics. The linear coupled systems in which we are interested may be characterized, in general terms, as arising from stochastic or Markov processes that are continuous in time and discrete in an appropriate space. In addition, the principle of detailed balancing is observed and the total amount of “material” in the system is conserved. The system is characterized by discrete compartments or states and “material” passes between these compartments by first order processes. Such linear systems are good models for a large number of processes. The application of this model in physics and chemistry has had a long history. We shall give some examples of the early works. The work of Einstein (32) on the theory of Brownian motion is based on a random walk process. Dirac (33) used the model to discuss the time behavior of a quantum mechanical ensemble under the influence of perturbations; this development enables one to discuss the probability of transition of a system from one unperturbed stationary state to another. Pauli (34) [also see Tolman (.!IS)], in his treatment of the quantum mechanical H-theorem, is concerned with the approach to equilibrium of an assembly of quantum states. His equations are identical with those of a general monomolecular

356

JAMES WE1 AND CHARLES D. PRATER

chemical reaction system; the occupancy of a quantum state is analogous to the concentration of a molecular species, the degeneracy of the quantum states forms a diagonal matrix analogous to the matrix D,and the H-function is analogous to the entropy. In the quantum mechanical H-theorem, the coupling constants arise from a perturbation matrix that is Hermitean (a matrix with complex elements which is equal to the transpose of its complex conjugate) ; consequently, the principle of detailed balancing is derived as a theorem. A typical recent example of the application of the model is the irreversible thermodynamics of Cox (36). A recent paper of interest to the readers of this article is that of Thomsen (37) in which he attempts to establish the convention that “microscopic reversibility” is to mean that the matrix K is symmetric, and that “detailed balancing” is to mean that the matrix KD is symmetric. This model is also used for many problems outside of physics and chemistry, for example, card shuffling, power supply, and parking lot congestion problems [Feller (St?)].A more exhaustive survey of the varied application of this model can be found in Feller (39),Bharucha-Reid (40), Bellman (41), Chandrasekhar (.42), and many other books on stochastic processes. Most of the literature on coupled reaction systems in chemical kinetics has been concerned with the solution of special cases of the monomolecular reaction systems that almost always contain no more than three components and with only a few of the rate constants nonzero. The solutions are obtained in a closed form with the values of the constants cji and X i of Eq. (6) expressed as functions of the rate constants kji. As examples we may mention the work of Rakowski (@), Alberty and Miller (44),de Boer and van der Eorg (45), and Jungers et al. (46).This type of approach, however, can be used only for ‘certain special cases. A more general approach is provided by the matrix formulation used by Zwolinski and Eyring (47) in their discussion of the transition between t’he various states of molecules. In this approach the rate constant matrix K is given and t,he values of the constants cji and Xi are determined as discussed in Section II,A and as discussed in greater detail by Benson (6) and by Frost and Pearson ( 5 ) . Although Zwolinski and Eyring (47) write the constant cji in two parts analogous to bPx,i of Eq. (84), no attempt is made to interpret these quantities in physical terms. For general systems with four or more components, this approach is of little use in obtaining explicit expressions for the constants cji and X i in terms of the rate constants kji because of the need to solve algebraic equations of the third or higher degree. Hence, as we have seen, it is not easy to apply this formulation to experimental data when the values of the rate constants are unknown. As typical examples of the application of this formulation to the

ANALYSIS O F COMPLEX REACTION SYSTEMS

357

transitions in multilevel systems, we may mention the papers of Carrington (48) and of Montroll and Shuler (49). Acrivos and Amundson (50, 51) applied matrix algebra to the unsteadystate behavior of stagewise operations in chemical processes. Instead of using a characteristic vector expansion, they emphasize the use of the Sylvester-Lagrange-Buchheim formula (52). Even though this formula is equivalent to the characteristic vector expansion, it is more difficult to manipulate and is not easily related to physical concepts such as straight line reaction paths. There are then really two problems that may best be stated in terms of the matrix notation. We have

-at_ - Ka!

da

The problems are:

(I) Given the rate constant matrix K and the initial composition distribution a(O), find the rates and compositions as a function of time. This is the approach used by Zwolinski and Eyring and discussed by Benson and by Frost and Pearson. It presents no difficulties.

(11) Given the compositions a(t) for various times and various initial compositions, find the rate constant matrix K and the rates (or symmetrically, given the rates da!(t)/dt for various times and various initial rates, find the rate constant matrix K and the compositions). This is the problem first faced by the experimentalist. It is, in general, a difficult problem for complex systems because of the lack of sensitivity of the results to variations in the rate constant matrix K as long as the experimentation is confined to a single or even to a narrow range of initial values. The example given in Section V,C shows that large variations of the rateconstant matrix may fit the data quite well. Carrington (48) has also made an analysis of the error transformation properties of such systems. When only a single initial composition (population of states) is used, he concludes that it is impossible to study systems with more than five or six components and even then with no more than nonzero nearest neighbor transition probabilities. In the development presented in this article, we have always assumed that the best available statistical methods are to be coupled with the structural analysis. For complex systems, which involve spaces of many dimensions, the error magnification obtained in the transformation from the

358

JAMES WE1 AND CHARLES D. PRATER

experimental data points to the values of the rate constants may be prohibitively large. Consequently, any information that will serve to increase the physical and mathematical understanding of the system will provide methods for improving the accuracy with which the rate constants may be determined for a given amount of experimental data. For instance, the amouqt of the various species are very complicated functions of the rate constants k,i for which nonlinear regression methods must be used [Booth and Peterson (53), Peterson (54)] to determine the best values of the parameters from experimental data. If the characteristic directions are used, however, linear methods may be applied and, since the compositions are relatively simple functions of the lambdas and the characteristic compositions as discussed in Section II,B, they provide methods that are less prone to magnification of errors in passing from experimental data to the derived quantities. We may thus distinguish three different statistical approaches to the problem of relating experimental data to parameters in monomolecular models. The most precise method consists of using a large amount of experimental data obtained from a large number of different initial compositions drawn from all regions of the reaction simplex in the estimation of the parameters. This method requires a large amount of both experimental and computational labor. We have seen in Section 11, however, how the use of composition space and the elimination of time can reduce the magnitude of the problem. As we have seen, a method, which is often applied without fully realizing the hazards involved, is to use one series of data points, all from the same initial composition, and then apply nonlinear regression techniques to the data. The results are likely to be poor. Increasing the amount of experimental data available along this time course does not improve the result nearly as much as can be obtained from the same amount of additional data from initial compositions differing greatly from the original. In addition, nonlinear regression methods must still be used. The combination of statistical methods with the physical interpretations of the mathematical structures of the system gives a method that requires much less experimental work than the complete exploration of all regions of the reaction simplex and, furthermore, requires very little computational effort. In addition, linear regression techniques may be used and the mapping of the effects of the errors in the experimental data to their effects on the rate constants (mapping of the experimental data space into the rate constant space) may be easily determined by the methods given in Section V. Also the results are much better than those obtained using a single initial composition. In the work of Carrington (48))if the population of the various rotational states can be changed a t will, the characteristic vectors

ANALYSIS O F COMPLEX REACTION SYSTEMS

359

may be located experimentally and the transition probabilities may be determined more accurately and for much larger systems than are now possible. Let us now turn to a discussion of the structural features of monomolecular systems. We shall begin with the consideration of a most important question, namely, whether or not the constant parameters --Xi of Eqs. (6) and (84)are real negative numbers. If they are complex numbers with negative real parts, the reaction paths will spiral into the equilibrium point as shown in Fig. 36. If they are pure imaginary, the reaction will orbit about the equilibrium point indefinitely without converging to equilibrium. If they have positive real parts, the amounts of the characteristic species grow without limit instead of decaying to zero. Since thermodynamics applies to chemical kinetics and since the existence of the thermodynamic Liapounov functions (entropy, Gibbs free energy, etc.) tells us that the system is dissipative, we know that pure imaginary solutions are not possible for monomolecular systems in chemical kinetics. Also we know from the law of conservation of mass and from the nonexistence of negative amounts of the various components that positive real parts are not possible. More subtle physical principles are required, however, to eliminate the possibility of complex values with negative real parts. It is shown in Appendix I that the principle of detailed balancing is sufficient to insure that the parameters - X i are nonpositive real numbers. The general proof, given in Appendix I, that the parameters --Xi are nonpositive real numbers makes use of the transformation of the rate constant matrix K into a symmetric matrix K by the similarity transformation

D-%KD%= This proof, first given by Hearon (55) in 1953,* is apparently little known to investigators in chemical kinetics. Renson (6) appeals to physical intuition and thermodynamics in an attempt to establish that the constants - X i are neither positive nor pure imaginary and Bak (56) uses a very lengthy and involved argument that requires the partitioning of the rate constant matrix into a number of tridimensional matrices to prove that constants - X i are real. It is interesting that Wigner (57),in a paper concerning the reciprocal relations of Onsager, independently pointed out the correct way to establish the reality of the parameters --Xi by means of a transformation equivalent to D - ~ K D ~ . Our development makes use of the straight line reaction paths as new

* Note added in proof: The authors are indebted to Prof. Michel Boudart for calling attention to an earlier publication containing this proof [Jost, W., 2. Naturforsch. as, 159 (1947)l.

360

JAMES WE1 AND CHARLES D. PRATER

coordinate axes. The existence of straight line reaction paths in complex monomolecular reaction systems has been noted by Allen et al. (58) and by Piret and Bilous (59). They did not point out, however, the true significance of these reaction paths; Allen et al. suggested that they might be used to establish the location of the equilibrium point. We have interpreted these axes as representing new hypothetical species and we have seen how these hypothetical species may be interpreted as a special package of Ai molecules that transfer as a unit in the reaction. The possible existence of such transference units has been suggested by Koefoed (80) and Rice (61). Matsen and Franklin (62) have suggested that a transformation to a new coordinate axes should lead to useful results. Since their investigation is the work most closely related to our formulation and solution for monomolecular systems,* we shall discuss in what ways their development falls short of being an adequate treatment and in what ways their development is actually incorrect. In this discussion we shall transform their notation into our notation although the length of the vectors used in the transformation matrix X will not be the same as the one used by us; this is not important for the discussion to follow. Matsen and Franklin make use of an analogy between the formulation of monomolecular systems and normal mode analysis of the vibrations of polyatomic molecules. The too strict use of this analogy leads them to make an assumption that is wrong for monomolecular systems in general. Normal mode analysis of vibration spectra treats symmetric matrices; except in very special cases, the rate constant matrix for the reactions of the various species Ai is asymmetric when the amounts are expressed in the A system of coordinates. The heart of their formulation is expressed by Eqs. (2), (3), and (4) of their paper, which will be designated (MF2), (MF3), and (MF4) when expressed in our notation. Matsen and Franklin begin by assuming that a transformation matrix X exists such that @ = a =

XTa X@

(MF2) (MF3)

and

where A is a diagonal matrix analogous to the matrix given in Eq. (26). From Eqs. (MF2) and (MF3), we have

XT

=

X-1

Hence, the matrix X must be composed of orthogonal vectors of unit length (see Appendix 1,D). Equation (MF4) tells us, however, that the rate con-

* Note added in proof: The formulation of Jost is actually the one most closely related.

ANALYSIS OF COMPLEX REACTION SYSTEMS

361

stant matrix K is diagonalized by the similarity transformation XTKX. Consequently, the rate constant matrix K must be symmetrical since the matrix X is composed of orthogonal column vectors (see Appendix 1,D). This also implies that the amounts of all the various components are equal when equilibrium is reached; this is the only system that can be treated by this formulation. But we know already that, in general, the rate constant matrix K is not symmetrical for monomolecular reaction systems. Matsen and Franklin state explicitly: “The development has been based on the assumption of the existence of a set of orthogonal eigenconcentrations” (orthogonal characteristic directions in our notation). We see that such an assumption is not justified when the composition a! is used as is explicitly stated in their paper. It is true that we can transform the rate constant matrix K into a symmetric matrix g, but there is no orthogonal transformation available to change the composition a! to a new coordinate system that will simultaneously diagonalize the matrix K as required by equation (MF4). The transformation of the matrix K into the symmetric matrix K also requires the transformation of the composition vector a! into the new composition vector a whose ith element is equal to ai/dai*. Hence, the equations that Matsen and Franklin should have written are @ =

a

=

1% 10

and

The vector @ and the matrix A in Eq. (460) have elements of different numerical value from the corresponding quantities used in the main part of the text because of the shift in the length of the unit vectors in the orthogonal B system of coordinates. This difference is not essential to the discussion here. The other differences between (MF3) and (MF4) and the set of Eqs. (460) are not trivial, but contain the essential character of the monomolecular reaction system. In addition, Matsen and Franklin make no attempt to relate the eigenconcentration (our characteristic directions) to experimentally measured quantities. Hence, even for the special case of symmetric rate constant matrices (equal amounts at equilibrium), their development represents only another method for obtaining the formal solution to the set of rate equations for monomolecular systems and it is not, as they have formulated it, well adapted to passing from experimental data to the values of the rate constant. Their approach, however, is cer-

362

JAMES WE1 AND CHARLES D. PRATER

tainly in the right direction and represents the nearest approach to our treatment of monomolecular reaction systems found in the literature. * We shall now turn to a discussion of the pseudomonomolecular system. The treatment of the pseudomonomolecular system given in Section VI represents a general extension of the polynomial fraction rate equations often obtained in the treatment of heterogeneous reactions [see for example Benson (6), Laidler (6.91. There appears to have been no discussion in the literature of the conditions under which a common term 4 may be factored from all individual rate expressions for a general complex heterogeneous reaction system and of the consequences of such a factorization. I n Section VI, we have only given some sufficiency conditions for such a factorization to arise; the establishment of the necessary conditions should be sought because of the usefulness of such properties in the testing of reaction models. The papers of Manes et al. (64), Horiuti (65), Gadsby et al. (66), and Prigogine et al. (67), in which they discuss the relations of the over-all forward and backward rate constants for each over-all mass action step of the complex reaction to the thermodynamic equilibrium constant for this mass action step, are concerned with this factorization although the fact is not explicitly stated. Manes, Hofer, and Weller use thermodynamic reasonings near the equilibrium point and Horiuti uses absolute reaction rate theory to arrive a t their theorems concerning this relation. An examination of the nature of the arguments required leads one to suspect that such theorems are primarily algebraic theorems arising from the possibility of decomposing a complex reaction ultimately into a series of mass action steps as given by condition (1) in Section VI,A,2,a. The possibility of making such a decomposition seems to be an implicit assumption in the treatments given by Horiuti and by Manes, Hofer, and Weller. This point will be discussed more fully by us in a subsequent paper. As an additional example of the usefulness of the factorization of pseudomonomolecular systems into a 4 term and a monomolecular mass action term, we may mention the advantage of studying the c j term along the straight line reaction path. Along such a path the amounts of the various species are linearly related to each other; hence, the amount of each component may be expressed as a linear function of the amount of one of them, say aj. Therefore, the 4 term along the straight line reaction path may be written

* Note added in proof: The formulation of Jost is actually the one most, closely related, and is free from the objections arising from Eqs. (MF2), (MF3), and (MF4). Jost also gave the correct form of the “normal coordinates” and the equation for the reaction paths in the B system of coordinates [our Eq. (58)].

ANALYSIS OF COMPLEX REACTION SYSTEMS

363

which is the form of the equation for a simple Langmuir adsorption isotherm involving only the species A j . By measuring the values of the constants Ci for each species and using the known linear relations between the amounts of the various components we may easily obtain the values of the coefficients lG(l)ilk,i and the coefficient /GI in Eq. (330). Our discussion in this article has been confined for the most part to monomolecular and pseudomonomolecular systems for the obvious reason that the difficulty increases enormously on passing to highly coupled nonlinear systems. Some solutions exist for certain simple special cases [see for examLotka (69)]but the mathematical tools do not ple Benson (6), Chien (69, exist for giving general explicit solutions. Search should, however, be made for any structural features of the various classes of nonlinear systems by use of the tools a t hand even though the final solution must be obtained by numerical methods. Such a procedure is useful because the insight obtained enables the investigator to reduce error magnification and to plan more efficient and better computer programs for the testing of particular models. An especially interesting nonlinear couples reaction system is the autocatalytic system discussed by Lotka (69, 70) and by Denbigh et al. (71). This type of system leads to periodic solutions and the amounts of certain components oscilate permanently when the system is operated a t a steady state away from equilibrium. Except for the implicit discussion of steady state conditions given in Section VI, we have not discussed the conditions for the origin of, the characteristics of, and the transitions between such states in complex reaction systems because of lack of space. Denbigh et al. (71) have given such a discussion for certain special cases. The reader should have little difficulty in treating the steady state for general monomolecular systems in terms of the formulation given in Sections II and IV. The techniques presented may also be used to treat many systems involving physical processes such as the performance of distillation columns and the conduction of heat in laminar structures. It is also not difficult to include diffusion steps and distribution of residence times in the discussion of highly coupled monomolecular systems. These will be the subjects of future publications. In closing, again we would like to remind the reader that the discussion given is not to be thought of as merely providing methods for obtaining numerical values of rate parameters in complex reaction systems but as providing methods for testing proposed mechanisms and for providing insight into the behavior of such systems. With the present effortsin science to understand the behavior of complex biological and physical systems such insight is badly needed. In technology, chemical processes are becoming more complex and the need for the understanding of the behavior of complex systems is great in the design of processes and in the prediction of the behavior of the processes when process variables are changed.

Appendix I. The Orthogonal Characteristic System A. TRANSFORMATION OF THE RATE CONSTANT MATRIXINTO A SYMMETRIC MATRIX I n this appendix, we shall prove, for n-component reversible monomolecular reaction systems, that (1) the characteristic roots are real numbers and (2) there are n independent characteristic directions (vectors). It is sufficient to show that such reversible monomolecular reaction systems can always be transformed into an equivalent new reaction system (to be mathematically precise, a similar system) with new coordinate axes such that

where the new rate constant matrix is a symmetric matrix and ti is the composition vector in the new coordinate system. A symmetric matrix is a matrix that is equal to its transpose;

G=

GT

where T signifies the transpose. A symmetric matrix has two important properties: (1)all of its characteristic roots are real and (2) it has n independent orthogonal characteristic directions (vectors) (72). Thus, the desired conclusions follow immediately when the proof is obtained that the reaction system can be transformed into a similar system with a symmetrical rate constant matrix. This proof also provides the tools for proving the characteristic roots are nonpositive (negative or zero). Furthermore, it also provides the transformation to an orthogonal characteristic system of coordinates useful for consistency checks and for obtaining the inverse of the matrix X. Clearly, if kij = kji for all values of i and j , the matrix K is already symmetrical and the above properties follow immediately. This is a very special case, however, where each forward rate constant equals its corresponding backwards rate constant ; the equilibrium composition will then be equimolar. The principle of detailed balancing from statistical mechanics provides a means for converting any asymmetric rate constant matrix K into a sym364

365

ANALYSIS OF COMPLEX REACTION SYSTEMS

metric matrix S. For all steps in a monomolecular reaction systems, this principle states ( I , 73, 74) that a t equilibrium

k..a.* 3% = k ija.* 3

641)

where a,* is the equilibrium amount of A;. Then, if the matrix plied on the right by the diagonal matrix

"-\%

a3*

0 0

0

)

...

.. .. .. ...

K is multi-

(A21

an*

we obtain the matrix S. h n

al* 0

h n

0

...

0

n

az* . . . 0

... ... .... ..

... ... ... knl

knz

-

$1

kj,

0

0

. . . an*

j =1

.... .. Lai*

...

knzaz*

-

kinan* j=l

or

KD

=

S

Since Eq. (Al) applies, the matrix S is symmetrical. This, however, is not the transformation of K needed. At equilibrium all a;* > 0 since all steps are reversible; consequently, the matrix D must be nonsingular and has an inverse D-l. Then, from Eq. (11) of text, Eq. (A4), and DD-l = I, we have

366

JAMES WE1 AND CHARLES D. PRATER

The vector a on the left is not the same as the vector (D-la) on which the matrix S acts. Hence, the action of S on a vector is not equivalent to taking the derivative of this vector as required by Eq. (11) of text. The above matrices S and D, however, are used in establishing the required transformation. Let us search for the required transformation by a different route. As has been seen, the operation of forming the derivative of a vector is equivalent to a transformation of this vector into a new vector and that K is a matrix representation of this transformation. As one might expect, the n X n matrix K is not the only matrix that transforms vectors with n elements into their derivatives. Multiplying each side of Eq. (11) of text from the left by an arbitrary n x n matrix P, which has an inverse P-' (nonsingular), and using the fact that the unit matrix I = PP-' may be placed a t any point in the equation without changing its value, we obtain

Thus, all transformations of the n X n matrix K of the form P-'KP (similarity transforms, see Section IV) by arbitrary nonsingular n X n matrices P yield matrices that transform vectors into their derivatives (75). Another important characteristic of this type of transformation is shown by multiplying both sides of Eq. (59) of text from the left by P-' and using PP-l = I

(P-'KP) (P-'x~)= --Xi(P-'xi)

(A7)

Thus, the matrix P-IKP has the same characteristic roots - X i as K even though the characteristic vectors differ. We shall now show that there i s a symmetric matrix similar to K, and that the elements of the transform required for its calculation are known quantities related to the matrix D. Multiplying both sides of Eq. (A4) from the left by D-l, we obtain

K

=

SD-'

Let Df6 and D-$$ be the diagonal matrices

(AS)

ANALYSIS OF COMPLEX REACTION SYSTEMS

367

and

0

0

-

1

da3,

...

0

... ...

Multiplying both sides of Eq. (A8) from the left by D-" and from the right by D36,we have

D-taKDH

=

D-%(SD-')DJ'i= D-%S(D-'D+%)

(All)

Substituting Eq. (A12) into Eq. ( A l l ) , we have

where the rule has been used that the transpose of the matrix formed by the products of matrices is equal to the product of the transpose of each matrix in the product taken in reverse order (76);that is (PG)T= GTPT. Clearly, the only matrices that are equal to their transpose are symmetrical matrices. Hence, D-%SD-%is a symmetrical matrix and by Eq. (A13) SO is D-%KDj6.Thus, the required similarity transform has been found in terms of known quantities, namely, the equilibrium concentration of the reacting species. The matrix D-36KD36will be designated g.

368

JAMES WE1 AND CHARLES D. PRATER

B. TRANSFORMATION TO THE ORTHOGONAL CHARACTERISTIC COORDINATE SYSTEM Since

t is symmetrical, it

will have orthogonal characteristic vectors

Zi that, with P-l = D->* and P = DH, are related to the characteristic vectors xi of the K matrix by [Eq. (A7)] Zi = D-Wx;

(A171

The transformation Zi to xi is obtained from Eq. (A17) by multiplying both sides from the left by DS6;this gives

xi = Df6Zi According to Eq. (A6) the

(Y

(A18)

vector is transformed into it by the matrix

D-X; D-36, (A19) Multiplying Eq. (A19) on the right by DJ6gives the transform from & to 5 =

a;we have

D ~ E (-420) Multiplying Eq. (62) of text on the left by D-" and using D3*DD-% = I, (Y

=

we have

(D-)*KDJs)(D-MX)

=

D+SX A

Thus,

it = D-%X and

Kit = xn The inverse of Eq. (A21) is it-1

=

X-lD%

since the inverse of the product of two matrices is equal to the product of their inverses taken in the reverse order (77), that is, (PG)-' = G-lP-'. Multiplying Eq. (A23) on the right by D-" gives

X-1

=

Z-1D-x

Since @ = X-'a

(A241

ANALYSIS OF COMPLEX REACTION SYSTEMS

369

Thus, the vector Q does not have the values of its coordinates changed by the transformation to the orthogonal B system of coordinates. Multiplying both sides of Eq. (A25) from the left by gives

Figure 37 shows the transformation for a two component system. A grid for the B system before and after the transformation is shown in Fig.

,OMPOSl TlON VECTOR

FIG.37. A two component system illustrating the transformation of the orthogonal characteristics system. Fig. (37a) shows a composition vector in both the A and B systems of coordinates. The A system of coordinates is considered to be fixed to the background. The composition vector and the B system of coordinates are considered to be attached to a rubber sheet to which a stretch and shear is applied t o obtain the orthogonal B system shown in Fig. (37b).

37(a) and (b) respectively. Fig. 37(a) shows the same composition vector in both the A and B coordinate system. In the A system, the composition vector is

and in the B system

@=[-:I

370

JAMES WE1 AND CHARLES D. PRATER

The characteristic vectors x i are

and x1 =

(-0.310 0.310)

Imagine that the A coordinate system is fixed to the background and that the B coordinate system and the composition vector are attached to a rubber sheet. Stretch and shear are applied to the sheet until the axes of the B coordinate system are orthogonal as shown in Fig. 37(b). Looking back into the A coordinate system in the background, the composition vector has changed to

'

1.140 = (0.448)

and therefore] is not the same as the original. Its behavior, however, will be the same since the rate constants for the reaction have also been changed to %. Meanwhile, the composition in the B system is still unchanged since the vector is attached to the rubber sheet, that is,

as shown by Eq. (A25). The new characteristic vectors are now 0.546 'O

= (0.835)

and =

(-0.514 0.317)

C. PROOF THAT THE CHARACTERISTIC ROOTSOF THE RATE CONSTANTS MATRIXK ARE NONPOSITIVE REALNUMBERS Since the matrix K is similar to a symmetrical matrix R, it follows immediately from the above discussion that the characteristic roots are real numbers. The proof that the characteristic roots are nonpositive depends on the theorem that, if yTGy 6 0 for any vector y, then the symmetric matrix Gc has only nonpositive characteristic roots (78). Hence, we need only show that yTgy 6 0 for any vector y to prove that the roots are

ANALYSIS OF COMPLEX REACTION SYSTEMS

371

nonpositive. First we note that the off diagonal elements of the matrix are given by

g

and the diagonal elements are

Therefore,

Using the values of (K)ij given above and collecting terms, we have

Hence, the rate constant matrix has only nonpositive characteristic roots and since the rate constant matrix K is similar to the matrix it,it also has only nonpositive characteristic roots.

D. THE CALCULATION OF THE INVERSE MATRIXX-’ The inverse of the matrix X can be computed by transforming to the orthogonal characteristic system and using the fact that the inverse of a matrix composed of orthogonal column vectors of unit length is simply the transpose of the matrix (79). Hence, after the matrix X has been transformed to the orthogonal system by use of Eq. (A17), we need only to adjust the length of its column vectors to unit length in the A system of coordinates. Equations (92) and (93) of the text give the adjustment for each vector of the matrix; hence, for the entire matrix, we have where L-J* is the diagonal matrix

rh

L-?4

=

0

...

l o z.. ......... I i

I

0

1

...

0 0

*

0

...

1

dl2

372 Since the matrix

JAMES WE1 AND CHARLES D. PRATER

f is composed of column vectors of unit length, we have f-1

=

(A291

fT

From Eq. (A27) we have

IT=

( j t ~ - j . ; )= ~

L--N~~T

(A301

where the rule has been used that the transpose of the product of two matrices is equal to the product of their transposes taken in reverse order. Since L-f6 is a diagonal matrix, the interchanging of the rows and columns to form the transpose leaves it unchanged. Multiplying Eq. (A27) from the right by matrix L3.;,we obtain ji: = f L W

(A311

Hence, jt-1

=

(fL9i-l = L-tax-1

where the rule has been used that the inverse of the product of two matrices is the product of the inverses of the individual matrices taken in reverse order. Using Eqs. (A30) and (A29) in Eq. (A32), we have

z-i =

L-I~T

(-433)

Equation (A23) shows that

X-lD!d

=

L-1x.T

or x-1

=~ - i j t ~ ~ - t 6

But from Eq. (A21), we have

hence, X-1

=

L-1XTD-1

II. Explicit Solution for the General Three Component System A convenient way to derive the explicit expression for the time course of the reaction in the general three component system [Eq. (3)j is first to reduce the matrix K [Eq. (lo)] to a 2 X 2 matrix. This is possible because the constraint given by the law of conservation of mass tells us that the system is over determined with regard to amounts, i.e., only two of the three amounts al, a2, and a3 need be specified to determine the system. First, let us shift the origin of the natural coordinate system to the equi-

373

ANALYSIS OF COMPLEX REACTION SYSTEMS

librium point; this is accomplished by subtracting the vector a* from the vector a. Then, d(a -

dt

= -da =

dt

Ka

=

K(a

(A371

a*>

since

da* = K a * = dt

0

Let the matrices V and V-' be given by

v=

(i y H)

and

v-'=

1 0 -1

(

0 1 -1

Multiplying Eq. (A37) from the left by V a d usin W-'

=

I, we have

Evaluating the vector V(a - a*) and the matrix VKV-l, we obtain

(A411

and

Because the last entry in the vector V(a - a*) is zero and all the elements of the last row of the matrix VKV-' are zero, the last column of the matrix VKV-1 does not contribute to the transformation. Hence, Eq. (A40) is an equation in two dimensional space instead of three. Consequently, let us define

374 analogous to

JAMES WE1 AND CHARLES D. PRATER

of Fig. 8, and

Thus, Eq. (A40) becomes dii = g, -

dt

Then

It will be convenient to determine the left rather than the right characteristic vectors. Except for a possible discrepancy in length, which determines the size of unit amounts of the various species Bi, these vectors form the rows of the inverse matrix X-1 used to transform compositions from the A to the B system of coordinates (see footnote Section IV1A,4,a).These characteristic vectors are /?

21 = 1, u1

L448)

W

and

e,

n = 1, u2

(A49)

W

where

and

I n Eqs. (A50) and (A51),

A =

(ka

- ka)'

+4kA

The characteristic roots corresponding to 21 and

9i2

are

ANALYSIS OF COMPLEX REACTION SYSTEMS

375

and

respectively. The matrix

2 formed from the row vectors is

which has the inverse

These matrices 2 and 1Z-l transform compositions between the natural and characteristic systems in the two dimensional spaces; and where

Q is the vector (A59)*

Furthermore, corresponding to Eq. (75) for the vector @, Q(t) =

(""" 0 O ) Q(0) e-ht

I

and, corresponding to Eq. (77) for a,

Hence, writing the vectors and matrices in terms of their components, we see that a2 - a2* u2 - u1 * The elements bi in Eq.

(A59) do not necessarily have the same unit amounts as the elements bj defined for the vector 0 in the remainder of the text because the matrix 2 has not been adjusted in length to correspond to X-l. However, this does not matter for the discussion given in this appendix.

376

J A M E S WE1 AND CHARLES D. PRATER

where a2 is the initial amount of the species A i . Thus, writing the components of Eq. (A62), we obtain a1

=

+ u2 -1 u1 {[(a? - al*) + u,(a2 - [(a? - al*) + uz(a2 - ~ ~ * ) ] u ~ e - ~ z ~ }

a1*

c~~*)]u~e-~l~

~

(A63)

and a2 = a2*

+ u2 -1

{ - [(a? - al*)

~

u1

+ ul(azO- ~ , * ) ] e + ~

+ [(alo- al*) + u2(az0- c ~ ~ * ) ] e - ~ n ~

(A641

Of course a3 = 1 - a1 - a2

(A651

Equations (A63), (A64), and (A65) with the defining equations (A46), (A50), (A51), (A52), (A53), and (A54) give the explicit solution for the general three component system. It is obviously complex and not easy to use.

Ill. A Convenient Method for Computing the Characteristic Vectors and Roots of the Rate Constant Matrix K When the matrix K is known, the constants Xi and cji in Eq. (6) may be determined by first solving a n algebraic equation of degree n - 1 to obtain the value of the constants X i and then a set of simultaneous linear algebraic equations to obtain the constants ci,. This method is discussed in many standard works on chemical kinetics (6) and will not be presented here. There is, however, a method (80-82) that is convenient and relatively easy to use, which will be given with its geometrical interpretation. We shall begin with the geometrical presentation of the principle of the method as illustrated by a 2 X 2 matrix that is not a rate constant matrix. This change makes the geometry of the illustration simpler. The rate constant matrix always leads to a reflection of each characteristic vector since the characteristic roots are negative numbers; we shall choose a matrix with positive characteristic roots to illustrate the determination of the characteristic vector with the largest characteristic root and return to the use of the rate constant matrix K for the discussion of the determination of the remaining vectors and roots. Let G be the 2 x 2 matrix,

(38 3) 4 - (2.667

a=

2

10

3

3

-

1.333)

0.667 3.333

ANALYSIS OF COMPLEX REACTION SYSTEMS

377

that transforms vectors in the two dimensional coordinate system (z, y) into new vectors in this coordinate system. For this matrix the characteristic directions G1 and G2, with characteristic roots w1 = 4 and w2 = 2 respectively, are shown in Fig. 38. Throughout the calculation the characteristic

FIG.38. The rotation of the vector n into the characteristic direction GIby repeated apphations of the transformation matrix G as discussed in the text.

directions and roots are to be considered as unknown except in discussing the theory of the method and for comparison to see our progress towards their determination. The problem is: given the matrix G, determine a vector g1 in the characteristic direction with the largest root, in this case the direction G1. Also we need to know the value of this characteristic root. Let us apply the matrix G to transform the vector n into a new vector ii’. The vector n can be decomposed into components along G1 and G2; n = clg,

+ c2g2

(A661

where gl and g2 are vectors in the characteristic directions GI and G2 respectively and where c1 and c2 are scalar constants. Then, we have

Gn

= ii’ =

clGg1

+ czGg2 = olclgl + wZc2g2 = 4C1g1 + 2c2g2

Hence, the vector n is lengthened and rotated in the direction of the characteristic coordinate with the largest root. Let the vector n be given by

378

JAMES WE1 AND CHARLES D. PRATER

Then 2.667 1.333)(1) = (2.667) 0.667 = fi’ 3.333 0

= (0.667

as shown in Fig. 38. Let us return the element in the vector fi’ corresponding to the s-coordinate to the value it had in n, namely, unity. This gives

Applying the matrix G to the vector n’, we have 1.000

3.000

= o(0.250) = (1.500) =

”‘

The vector fit’ is rotated still further towards the characteristic direction GI as shown in Fig. 38. Again returning the first element to unity, we have

When the multiplying factor is dropped, this gives a vector which terminates on the line - - - (Fig. 38) that passes through the end of n and is parallel to the y-axis. This adjustment determines the length of the characteristic vector that we obtain in our convergence process. This procedure is repeated until the vector coincides with the direction GIwithin the degree of accuracy desired. Let us select accuracy in the third decimal place; then the sequence is 1 .ooo

* 2.667 (0.250) 1 .ooo (0.250) -* 1 .ooo (o.500) +

1.000 3‘000 (0.500) 1 .ooo 3.333 (0.700)

1 .ooo ------t3*600 (0.833) 1 .ooo 3*778(0.912) 1.000 3*883(0.955) 1 .ooo 1.000 (0.955) -+3‘940 (0.977)

(i;::) (kE) -* ):h(!

ANALYSIS OF COMPLEX REACTION SYSTEMS

379

1.ooo (0.989) 1.ooo -----f ‘985 (0.994) 1.000 3’992 (0.997) 1.ooo -+3.996 (0.999) 1.ooo 1.ooo (1.000) (0.999) --+3.999

1.ooo (0.977) 1.ooo (0.989) 1.000 (0.994)

-+3’967

-----f

(i:::;)

The vector g1 only changes its length under the action of the matrix G within the accuracy limits set and is the characteristic vector sought. Since cfe1

=

M

l

the constant multiplier 4.000 is the characteristic root sought. The rate of convergence may be increased by using the results of one or two calculations to make guesses as to the location of the direction to which the process is leading. One major difficulty with the method is that, when the values of the characteristic roots are close together, the convergence will be slow, and the choice of starting vectors becomes very important. The method given above always converges to the characteristic vector with the largest decay constant. Hence, to determine the characteristic vector with the second largest decay constant, a matrix must be determined for which the effects of the vectors with the largest roots are removed. To do this, we shall return to the use of the rate constant matrix K.Furthermore, we shall use the rate constant matrix K for the orthogonal system, which is related to the rate constant matrix K by (Appendix 1,A)

K

=

D-%KD%

The characteristic vector corresponding to the largest characteristic root will be considered as having been determined by the above method and as having its length adjusted to unit length-it will be designated L-1. Let the matrix we seek be designated Kn-z;

-

Kn-z = K

- Xn--ljin-lXn-iT

(A671

Let the vector ii be decomposed into components along the various characteristic directions so that ii = bJo

+

bJ1.

. . bn-Jn-l

380

JAMES WE1 AND CHARLES D. PRAlXR

Multiplying the vector

e by Kn-z, we have

(t- X,-lii,-J,-lT)(b$O. . . + b,-1iin-1) + bJtiil. . . + b n - 2 r n , - 2 + b n - 1 i b n - 1 - Xn-1bJn-Jn-1TfO - Xn-lbliin-ljin-lTjil . . . - Xn-lbn-2iin-ljin-1Tln-2 - Xn-lbn-Jn-Jn-lTjin-l = blXlii1 + . . . bn-2Xn--Jn--2

Zn-2E = =

bogiio

since fn-lTjii = 0 (i # n - 1) and iin-lTf,-l = 1. Let X1 6 Xz 6 X < '* - * * 6 A,-z 6 X,-1. Thus, for the matrix RnF2,the characteristic vector X,-2 has the largest characteristic root. This vector and its corresponding root may now be determined by the convergence method given above. The matrix Kn-3may then be calculated and the next characteristic vector obtained and so on unt.il all have been evaluated. In calculating the various matrices Kn4, Kn+ . . . , t1, care must be taken to minimize and evaluate errors accumulated in the purge process given by Eq. (A67).

-

IV. Canonical Forms The Jordan canonical form of a matrix is best defined in terms of elementary matrices J, which are matrices with entries --Xi along the principal diagonal, entries 1 on the diagonal next below the principal diagonal and all other entries zero. For example,

is an elementary Jordan matrix. A matrix is the Jordan canonical form when it is composed of elementary Jordan matrices arranged in blocks along the principal diagonal and all other entries zero. For example, (-h) 0

J=

0 0 0 0 0 0 0 0 0 0

0 (--A2) 0 0

0 0 0 0 0 0 0 0

0 0

0 0 0

(-A3)

0

/-A4

0 0

0 0

0 0 0 0':

O ( l - - A 4 O I

0 0 0 0 0 0 0

', 0 0

0 0 0 0 0

1 0 0 0 0 0 0

-A4;

0 0 0 0 0

0

0 0 0 0 0 0

(-k)

0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

O(-A,) 0 0 ,/-A7 0 0 o', 0 0 ' 1 - A 7 0 0 : 0 o l o 1--x70: 0 0 ',o 0 l-A7;

is the Jordan canonical form for a 12 X 12 matrix with threefold degeneracy in one characteristic vector and fourfold degeneracy in another. The ele-

381

ANALYSIS OF COMPLEX REACTION SYSTEMS

<

mentary Jordan matrices are indicated by j. Any matrix composed of real (or complex) numbers is similar to a matrix in the Jordan canonical form (83). Let us now show that the Jordan canonical form is similar to a matrix in the canonical form N [Eq. (183)]. The elementary Jordan matrices are transformed into the required form by

(Rem)-'JemRem = Nem where 0

Rem=

xjm-2

0

Xy-1

0

' I (Rem)w1=

0

...

xi

0

0

0

...

i 0

0

... ... ...

1 ~

Xim-2

...

... 0...

0

.... ..

(A721 1

0

0

...

, o

0

... 0

0

0 0

-

xi

and '--xi

xi

Nem

=

0

--xi

Xi

... 0 ... 0

... .... .... .... 0...

--Xi

0' 0 0

(A73)

382

JAMES WE1 A N D CHARLES D. PRATER

ai* The equilibrium amount of the ith molecular species (ai) Average value of at

uio The initial value of ai ai The weight of the ith molecular species a Composition vector with elements ai Zo Active catalytic sites containing no adsorbed species Zi Adsorption complex between the species Ai and the active catalytic site Bo ?,Ifi An adsorbed species with no corresponding free species a i The amount of the adsorbed species Z i a0 The amount of free catalytic sites ZO a'i The amount of adsorbed species %'i Zc An adsorption complex corresponding to more than one free species a, The amount of adsorbed species Zc a Composition vector for species adsorbed on active sites ai* The equilibrium amount of the adsorbed species Zi Bi The ith characteristic species in the characteristic coordinate system bi The amount of the characteristic species Biin the characteristic coordinate system Bo The characteristic species corresponding to the equilibrium composition bo The amount of the characteristic species BO bjo The initial amount of the characteristic species Bj 2% A second type of active catalytic site containing no adsorbed species Bi The adsorption complex between the species Ai and the active catalytic site a0 c,i A constant parameter in the general solution to the reversible monomolecular rate equations Ci A constant ci A constant D A diagonal matrix with diagonal elements equal to the equilibrium amounts ai* D-l A diagonal matrix with diagonal elements equal to the reciprocal of the equilibrium amounts ai* Dx A diagonal matrix with diagonal elements equal to D-$* A diagonal matrix with diagonal elements equal to 1/&? D A diagonal matrix with diagonal elements equal to a;* and a i*

ANALYSIS OF COMPLEX REACTION SYSTEMS

383

% A diagonal matrix with diagonal elements equal to ai" E The equilibrium point in composition space E , The equilibrium point corresponding to the rth choice of the characteristic vector xo(r)

ei The unit row vector 0 . . . 1 . . . 0 A constant A function A driving force; a force A matrix The element in the i t h row and the j t h column of the matrix

G The element in the ith row and the j t h column of the matrix G The transpose of the matrix G The constant bio/(bjfl)X"Xi The steady state matrix for pseudomonomolecular systems The element in the ith row and j t h column of the matrix G The matrix formed by replacing the j t h row of the matrix by a row of ones The matrix formed by replacing the j t h row of the matrix by the ith row of the matrix U The determinant of the matrix G The determinant of the matrix G ( ' ) j The determinant of the matrix Gij Gibbs free energy Gibbs free energy at equilibrium Characteristic directions of the matrix G Characteristic vectors of the matrix G The constant - (Ajx&O)/ (Alzilbl") [HZl The partial pressure of hydrogen I The identity matrix Invariants arising from species interaction The flux corresponding to the driving force, grad S , in irreversible thermodynamics IT The transpose of the flux vector j The ith component of the flux vector j Ji The flux vector determined in the laboratory The ith component of the flux vector j" The transpose of the flux vector j An elementary Jordan matrix of degeneracy m Jem

384

JAMES WE1 AND CHARLES D. PRATER

J A matrix in the Jordan canonical form The rate constant for the reaction of the ith to the jth species in monomolecular systems K The rate constant matrix for the natural system of coordinates K’ A relative rate constant matrix in the natural system of coordinates k‘ij A relative rate constant in the matrix K‘ K The relative rate constant matrix in the natural system of coordinates with one element normalized to unity kij A relative rate constant in the matrix K k ; The ith rate constant in the system kji

-kl

lc,,

kb,

lc,, kd

& Q kdi

lcai Kji K

K,,Ka, K,, Kd (K)ii (K)ij K

g K,-i L li

L-’ L% L-’6 L’, 1” (L’)ii

kl

A1 A2 A3 Constant parameters in the general solution for the general three component system Unperturbed rate constant matrix Perturbation rate constant matrix The desorption constant for the ith species The absorption constant for the ith species The thermodynamic equilibrium constant for the reaction from the ith to the jth species The grand transition rate constant matrix Partitions of the grand transition rate constant matrix The diagonal elements of the symmetric rate constant matrix ti; The off diagonal elements of the symmetric rate constant g The rate constant matrix in the orthogonal system The collapsed rate constant matrix The purged rate constant matrix with A,-i as the largest characteristic root A diagonal matrix with diagonal elements equal to the squares of the lengths of the characteristic vectors x i used to form the matrix X The jth diagonal element of the matrix L The inverse of the matrix L The diagonal matrix with diagonal elements equal to the lengths of the characteristic vectors x i The inverse of the matrix L’* Diagonal matrices in the expansion of the rate constant matrix A The diagonal elements of the matrix 1’

ANALYSIS OF COMPLEX REACTION SYSTEMS

385

1 The relative rate constant matrix in the B system of coordinates equivalent to the relative rate constant matrix t The unperturbed matrix 1 L The transformation matrix required to change the driving force, grad S , into the flux vector LT The transpose of the matrix L The elements of the matrix L The transformation matrix required to change the laboratory driving force 2 into the laboratory flux 3 The transpose of the matrix 1 The elements of the matrix L A constant The minor formed by deleting the jth row and the ith column of the matrix A The determinant of the matrix Mji I Mjil n The number of components in monomolecular reaction systems N A matrix in a particular canonical form N Number of active sites Number of active sites of type 3 NgI Number of active sites of type 23 Nd An elementary matrix with m-fold degeneracy in the Nem canonical form N 0 The null matrix P A nonsingular matrix P-1 The inverse of the matrix P P T The transpose of the matrix P P The matrix for transforming the vector a into the vector a P The number of reactions in a complex reaction system The ith parameter that characterizes a system Qi Ti

Xj/Xi

r The fraction of the distance between two vectors R The gas constant Rem The diagonal matrix with diagonal elements equal to Am- i

The inverse of the matrix Rem The matrix formed from the elementary matrices Rem The inverse of the matrix R The entropy of a system The entropy at equilibrium A symmetric matrix Time

386

JAMES WE1 AND CHARLES D. PRATER

Tt The stochastic matrix X (exp At) X-I T The absolute temperature u1,u 2 Parameters in the general solution for the general three component system U The rate constant matrix for the adsorbed species uij Elements of the rate constant matrix U V A Liapounov function V A transformation matrix V-’ The inverse of the matrix V wZ A constraint vector arising from the interaction between species WZT The transpose of the vector Wl wli The ith element of the vector W Z W An antisymmetric matrix WT The transpose of the matrix xi The unit vector in the j t h characteristic direction written in the natural or A system of coordinates xij The ith element of the j t h unit characteristic vector in the natural or A system of coordinates x0 The equilibrium characteristic vector X The n X n square matrix formed by writing the n unit characteristic vectors xj side by side as column matrices X-1 The inverse of the matrix X XT The transpose of the matrix X xri The vector formed by displacing the unit characteristic vector xj to the equilibrium point x ” ~ The displaced characteristic vector in the opposite direction from x’i fj The j t h orthogonal unit characteristic vector expressed in the natural or A system of coordinates xjT The transpose of the vector xj fjT The transpose of the vector fj jijl An inaccurate vector fj fj The j t h orthogonal characteristic vector of unit length in the natural or A system of coordinates IiT The transpose of the vector f j xij Elements of the vector Ii xo(r) A vector terminating rth of the distance between two characteristic vectors with zero characteristic roots X The characteristic matrix for the unperturbed rate constant matrix k

ANALYSIS OF COMPLEX REACTION SYSTEMS

387

E The characteristic matrix for the perturbed rate constant matrix K

2-1 %I,

The inverse of the matrix X

I-' The inverse of the matrix E I" Matrices in the expansoin of E

(I')ij

2 X

Xi f XT

xi

ii:

it-1

ZT

ii iiT

1-1 Yi Y

The element in the ith row and j t h column of the matrix I' A coordinate axis The driving force grad S The ith component of the vector x A laboratory force satisfying the condition (dS/dt) = pj The transpose of the vector x The ith component of the vector 2 The characteristic matrix in the orthogonal coordinate system The inverse of the matrix 2 The transpose of the matrix 2 The normalized characteristic matrix in the orthogonal system The transpose of the matrix The inverse of the matrix 1 The vector used to complete the set of vectors when degeneracy occurs in the characteristic directions The transformation matrix for degenerate irreversible systems The inverse of the matrix Y The vector yi displaced to the equilibrium point A coordinate axis The left characteristic vector of the matrix K A coordinate axis A left characteristic vector of the collapsed matrix &: The transformation matrix formed from the characteristic vectors & The inverse of the matrix 2 The composition vector in composition space expressed in the natural or A system of coordinates The value of the composition vector a t the time t The value of the composition vector (Y a t the time t = 0 The equilibrium composition vector expressed in the natural or A system of coordinates The composition vector into which the vector (Y is changed by the action of the dimensionless transformation matrix K

388

JAMES WE1 AND CHARLES D. PRATER

t

Qj

QZ;

iE: L

e P

E

n ?i

ii‘

n‘ %f’

n“

A vector in the j t h characteristic direction expressed in the natural or A coordinate system The characteristic composition vector corresponding to the characteristic vector xi The value of the vector aZiat time t = 0 The value of the vector uzia t the time t = 0 The composition vector that is rth of the distance from the vectors ~ ( tto) a2(t) The composition vector expressed in the orthogonal system The composition vector in (n - 1)-dimensional composition space The composition vector expressed in the characteristic or B system of coordinates The composition vector in the (n - 1)-dimensional space defined by the straight line reaction paths The composition vector 0 at time t The composition vector 8 at time t = 0 An arbitrary vector The j t h element of the vector y A vector in the orthogonal system of coordinate The transpose of the vector y A small interval of time An interval of time The discriminant of the three component system A matrix for the pseudomonomolecular system The determinant of the matrix A The scalar quantity equal to the inner product between two vectors A small number An arbitrary vector An element of the vector n A new vector derived from the vector n The vector formed from the vector ii’ by adjusting its length A new vector derived from the vector n‘ The vector formed from the vector ii” by adjusting its length The pseudo-rate-constant for the reaction from the j t h to the ith species The rate constant matrix for the pseudomonomolecular system

ANALYSIS OF COMPLEX REACTION SYSTEMS

0‘

A A‘

exp At

389

The pseudo-rate-constants in the relative system with one constant normalized to unity The relative rate constant matrix with one element normalized to unity The forward and backwards pseudo-rate-constants for a second order system A rate constant in the pseudo-monomolecular system The ith characteristic decay time for a monomolecular system The rate constant matrix in the characteristic or B system The relative rate constant matrix in the characteristic or B system The diagonal matrix

A The unperturbed rate constant matrix in the characteristic

U

or B system A rate constant in the pseudo-mass-action system A relative rate constant in a pseudo-mass-action system The non-negative forward and backward stoichiometric coefficients The net stoichiometric coefficient vi,” - vipf The matrix of the net stoichiometric coefficients A degree of freedom; the degree of advancement A degree of freedom or degree of advancement in different units Standard deviation

2’kji

The sum of the constants kii for all values of j from 1 to

yip V

t‘i FP

Tk

j =I

2

n except j =

i

The sum of all entries from j

=

1 to j

=n

j=l 7

+

A new time scale for pseudo-monomolecular systems An unspecified function of the amounts of the various species and time in pseudo-mass-action systems

390

JAMES WE1 AND CHARLES D. PRATER

c $ ~ t,

4’ A modified 4 function # ~ The ~ functions 4 for the active sites % and 8 respectively

\Ej

w

bj/bjo

A characteristic root

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4. Kaplan, W., “Ordinary Differential Equations.” Addison-Wesley, Reading, Massachusetts, 1958. 6. Frost, A. A., and Pearson, P. G., “Kinetics and Mechanism,” p. 160. Wiley, New

York, 1953. 6. Benson, S. W., “The Foundations of Chemical Kinetics.” McGraw-Hill, New York, 1960. 7. Lanczos, C., “Applied Analysis.” Prentice-Hall, Englewood Cliffs, New Jersey, 1956. 8. Haag, W. O., and Pines, H., J. A m . Chem. SOC.82, 387, 2488 (1960). 9. Lago, R. M., and Haag, W. O., to be published. 20. Wiggill, J. B., to be published.

11. Scarborough, J. B., “Numerical Mathematical Analysis.” Johns Hopkins Press, Baltimore, Maryland, 1950. 12. Birkhoff, G., and MacLane, S., “A Survey of Modern Algebra,” p. 334. Macmillan, New York, 1953. IS. Hamilton, W. M., and Burwell, R. L., Jr., Proc. 2nd Intern. Congr. on Catalysis, Paris, 1960 Paper 44, p. 987. 14. Rossini, F. D., Pitzer, K. S., Arnett, R. L., Braun, R. M., Pimentel, G. C., “Selected

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ANALYSIS O F COMPLEX HEACTION SYSTEMS

39 1

26. Jacobson, N., “Lectures in Abstract Algebra,” p. 5. Van Nostrand, Princeton, New Jersey, 1951. 87. Slater, J. C., “Introduction to Chemical Physics,” pp. 124-155. McGraw-Hill, New York, 1939. 88. Denbigh, K . G., A.Z.Ch.E. Journal 6, 20 (1959). 89. DeGroot, S. R., “Thermodynamics of Irreversible Processes,” pp. 169-171. Wiley (Interscience), New York, 1952. 30. Coleman, B. D., and Truesdell, C., J. Chem. Phys. 33, 28 (1960). 31. Laidler, I(.J., “Chemical Kinetics.” McGraw-Hill, New York, 1950. 38. Einstein, A,, “The Theory of the Brownian Movement.” Dover, New York, 1956. 33. Dirac, P. A. M., Proc. Roy. Soc. (London) All2, 661 (1926); A114, 243 (1927). 34. Pauli, W., “Sommerfeld Festschrift,” p. 30. Leipeig, 1928. 36. Tolman, R. C., “The Principles of Statistical Mechanics,” p. 463. Oxford Univ. Press, London and New York, 1938. 36. Cox, R. T., “Statistical Mechanics of Irreversible Change.” Johns Hopkins Press, Baltimore, Maryland, 1955. 37. Thomsen, J. S., Phys. Rev. 91, 1263 (1953). 38. Feller, W., “An Introduction to Probability Theory and Its Applications,” p. 420. Wiley, New York, 1957. 39. Feller, W., ‘‘An Introduction to Probability, Theory and Its Applications.” Wiley, New York, 1957. 40. Bharucha-Reid, A. T., “Elements of the Theory of Markov Processes and Their Applications.” McGraw-Hill, New York, 1960. 41. Bellman, R., “Introduction to Matrix Analysis.” McGraw-Hill, New York, 1960. 48. Chandrasekhar, S., Revs. Modern Phys. 16, 1 (1943). 43. Rakowski, A., 2. physik. Chem. 67, 321 (1907). 44. Alberty, R. A., and Miller, W. G., J . Chem. Phys. 26, 1231 (1957). 46. de Boer, J. H., and van der Borg, R. J. A. M., Proc. 2nd Intern. Congr. of Catalysis, Paris, 1960 Paper 40, p. 919. 46. Jungers, J. C., Balaceanu, J. C., Coussemant, F., Eschard, F., Giraud, A., Hellin,

47. 48. 49. 50. 51. 68. 63.

64. 66. 66.

67. 68.

M., Leprince, P., and Limido, G. E., “Cinetique chimique appliquee.” Soc. Editions Tech., Paris, 1958. Zwolinski, N. B., and Eyring, H., J . Am. Chem. SOC.69, 2702 (1947). Carrington, T., J . Ckem. Phys. 36, 807 (1961). Montroll, E. W., and Shuler, K. E., Advances i n Clem. Phys. 1, 371 (1959). Arrivos, A., and Amundson, N. R., Ind. Eng. Chem. 47, 1533 (1955). Acrivos, A., and Amundson, N. R., CI,eni.Eng. Sci. 4, 29, 68, 141, 159, 206, and 249 (1955). Dunford, N., and Schwarte, J. T., “Linear Operators,” (Part 1. Pure and Appl. Math. Ser., Vol. 7), p. 607. Wiley (Interscience), New York, 1958. Booth, G. W., and Peterson, T. I., “A.1.Ch.E. Computer Program Manual No. 3.” Am. Inst. Chem. Engs., New York, 1960. Peterson, T. I., Chem. Eng. Process Symposium Series No. 31 66 (1960). Hearon, J. Z., Bull. Math. Biophys. 16, 121 (1953). Bak, T. A., Acta Chem. Scand. 12, 999 (1958); Bull. classe sci. Acad. Toy. Belg. [5] 46, 116 (1959); “Contributions to the Theory of Chemical Kinetics.” Munksgaard, Copenhagen, 1959. Wigner, E. P., J. Chem. Phys. 22, 1912 (1954). Allen, R. H., Turner, A., Jr., and Yats, L. D., J . Am. Chem. SOC.81,42,5289 (1959).

392

JAMES WE1 AND CHARLES D. PRATER

Piret, E. L., and Bilous, O., A.1.Ch.E. Journal 1, 480 (1955). Koefoed, J., J. Colloid. Sci. 12, 131 (1957). Rice, 0. K., J. Phys. Chem. 61, 622 (1957). Matsen, F. A., and Franklin, J. L., J . Am. Chem. SOC. 72, 3337 (1950). Laidler, K. J., in “Catalysis” (P. H. Emmett, ed.), Vol. 1, p. 75. Reinhold, New York, 1955. 64. Manes, M., Hofer, L. J. E., and Weller, S., J. Chem. Phys. 18, 1355 (1950). 65. Horiuti, J., Advances in Catalysis 9, 339 (1957). 66. Gadsby, J., Hinshelwood, C. N., and Sykes, K. W., Proc. Roy. SOC.8187, 129 (1946). 67. Prigogine, I., Outer, P., and Herbo, C., J. Phys. & Colloid Chem. 62, 321 (1948). 68. Chien, J., J . Am. Chem. SOC.70, 2256 (1948). 69. Lotka, A,, “Elements of Mathematical Biology.” Dover, New York, 1956. 70. Lotka, A., J. Am. Chem. SOC.42, 1595 (1920). 71. Denbigh, K. G., Hicks, M., and Page, F. M., Trans. Faraday SOC.44, 479 (1948). 79. Birkhoff, G., and MacLane, S., “A Survey of Modern Algebra,” pp. 277 and 314. Macmillan, New York, 1953. 73. Fowler, R. H., “Statistical Mechanics,” p. 660. Cambridge Univ. Press, Londoii and New York, 1936. 74. Tolman, R. C., “The Principles of Statistical Mechanics,” p. 522. Oxford Univ. Press, London and New York, 1938. 76. Birkhoff, G., and MacLane, S., “A Survey of Modern Algebra,” p. 248. Macmillan, New York, 1953. 76. Birkhoff, G., and MacLane, S., “A Survey of Modern Algebra,” p. 217. Macmillan, New York, 1953. 77. Birkhoff, G., and MacLane, S., “A Survey of Modern Algebra,” p. 223. Macmillan, New York, 1953. 78. Birkhoff, G., and MacLane, S., “A Survey of Modern Algebra,” p. 273. Macmillan, New York, 1953. 79. Birkhoff, G., and MacLane, S., “A Survey of Modern Algebra,” p. 258. Macmillan, New York, 1953. 80. Buckingham, R. A., “Numerical Methods.” Pitman, London, 1957. 81. Faddeeva, V. N., “Computational Methods of Linear Algebra.” Dover, New York, 1959. 82. Margenau, H., and Murphy, G. M., “The Mathematics of Physics and Chemistry.” Van Nostrand, Princeton, New Jersey, 1943. 83. Birkhoff, G., and MacLane, S., “A Survey of Modern Algebra,” p. 334. Macmillan New York, 1953. 69. 60. 61. 6%. 63.

Catalytic Effects in Isocyanate Reactions A. FARKAS

AND

G. A. MILLS

Houdry Process and Chemical Company, Marcus Hook, Pennsylvania

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Polymerization of Isocyanates. . . . . . . . . . . . . . . A. Dimerization. . . . . . . . . . . . . . . . . . . . . B. Trimerization. . . . . . . . . . . . . . . . . . . . . . . . . . . ........................... C. Linear Polymerization. . . . . . . . . . . . 111. Reactions of Isocyanates with Compou A. General . . . . . . . . . . . . . . . . . . . . . . . . . B. Reaction of Isocyanates with Hydroxyl Compounds.. . . . . . . . . . . . . . . . . . . C. Reaction of Isocyanates with Water., . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Reaction of Isocyanates with Amines., . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Reaction of Isocyanates with Thiols. . . . ........................... F. Miscellaneous Reactions of Isocyanates ........................... IV. Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . ........................... References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........

Page 393

399

403 427 431 434 439 441 443

1. Introduction The chemistry of organic isocyanates has gained tremendous technical importance in recent years, since through certain isocyanate reactions a vast variety of plastics, coatings, elastomers, and especially foamed articles have become available. In spite of the relatively high reactivity of the isocyanate group, reactions involving isocyanates are carried out in almost all cases under the directive influence of catalysts. Catalysis in isocyanate chemistry is of broad interest from a theoretical viewpoint. It is possible to vary the structure of each of the reactants and of the catalyst and to observe profound effects. As for the catalyst, it has been established that both “basicity” and steric properties are of fundamental importance. Kinetic measurements have proved most useful in investigations, carried out for the most part since 1945. Knowledge gained has permitted consideration of the steric and electronic factors of the active reactant-catalyst complex, key to the catalytic mechanisms. One interesting feature of these reactions ‘is the relatively high intrinsic activity of the catalysts; the other is the requirement that in certain cases two or more concurrent reactions must be catalyzed at carefully balanced relative rates. 393

394

A. FARKAS AND G. A. MILLS

The rapid industrial growth of polyurethanes and related polymers in the last 20 years has provided a surge of research dealing with both practical and theoretical aspects. The chemistry of isocyanates has been reviewed by Saunders and Slocombe ( I ) and by Arnold et al. ( 2 ) . The extensive developments in isocyanate chemistry will be again brought up to date in a forthcoming book by Saunders and Frisch ( 3 ) .The study of the mechanism of reaction has been emphasized particularly by Baker and co-workers (4)and by Dyer and co-workers ( 5 ) .Detailed aspects of polyurethane foam formations are given comprehensive treatment by Dombrow (8)and by Saunders (7). In contrast to the general references given above, this chapter is concerned specifically with catalysis of isocyanate reactions. Reactions of isocyanates provide an example of classical catalysis in that a catalyst-reactant complex is first formed which is then able to react with a second reactant molecule with an over-all high reaction velocity and specificity. Factors affecting rate and amount of complex formation, provision of paths of low activation energy, as well as steric and electronic effects, are all important. It cannot be said that these factors are well known. However, the catalysts most studied-the tertiary amines-have shown enormous variations in catalytic properties as their structure was varied. It will be recalled that the list of catalytic acids-HF, H2S04, H20-Si02-A1203, etc.-is both relatively small and difficult to interrelate. On the other hand, the possibility for systematic variations in organic base catalysts presents a very great advantage. Probably the most significant finding is the great importance of the steric factor; that is, the molecular structure of the amine catalyst greatly modifies its activity. While the steric effect is not a new concept, yet a study of isocyanate catalysis provides an exceptionally good example whose application can be of widespread application. It should be an objective to place the steric factor on a quantitative theoretical basis. The elucidation of the catalytic effects in isocyanate reactions thus is a promising field not only from its own scientific and technical importance but also from the solution of catalytic problems in general. The chemical reaction of greatest present interest is the addition of molecules containing an active hydrogen, particularly alcohols : R-N=C=O

catalyst + HOR’ --+ R-NH-

8

-OR’ Urethane

It is this catalyzed reaction, involving the N=C bond, which, when carried out with diisocyanates and glycols, leads to polymers of great interest. The relationship of such polymers to the polyamides, proteins and nylons, is quickly grasped from a comparison of their structures:

CATALYTIC EFFECTS I N ISOCYANATE REACTIONS

395

Polyurethane glycol) (diisocyanate

+

Polyamide (dicarboxylic acid diamine)

+

Isocyanates react with compounds which give a positive Zerewitinoff test. These include alcohols, water, phenols, amino groups (primary and secondary), carboxylic acids, as well as certain compounds having activated methylene groups such as acetoacetic ester, malonic ester, etc. With water, the reaction product is a carbamic acid which decomposes to form an amine and carbon dioxide, the latter being responsible for foam generation. Secondary reactions are of special significance since they lead to chain branching. Important in this regard are reactions of isocyanates with amines (generated from the reaction with water) to form ureas, and their further reaction to produce biurets. Also, urethanes themselves react further to form allophanates. Actually, reactions of isocyanates with themselves, as distinguished from reactions with compounds having active hydrogen, lead to interesting polymers and this field is discussed in the next section.

I I . Polymerization of Isocyanates One of the major fields of isocyanate catalysis is polymerization. The formation of cyclic dimers and trimers from aryl isocyanates was established over 100 years ago by Hofmann (8). This early work has been reviewed by Saunders (1). Dimers are formed by aryl isocyanates a t room temperature in the presence of certain amines or phosphines. Trimerization occurs in the presence of bases such as potassium acetate. Linear polymerization has been recently reported by Shashoua and co-workers a t lower temperatures in polar solvents with an anionic catalyst such as metallic sodium (9, 10). The general mechanism proposed for polymerization consists of formation of an active complex by attachment of a nucleophilic catalyst, X, onto the carbon of the isocyanate group with displacement of an electron pair to the nitrogen atom, followed by addition of a second isocyanate molecule: X C=N

8k

+ X -+

@

C-N

/!)A

X

8 + C=N

dk

€B C-N-C-N

I I I I I

J

R

O

R

8

396

A. FARKAS AND G . A. MILLS

Which polymer is formed depends upon the relative rates of subsequent reactions. If chain termination then occurs with loss of X, a cyclic dimer is produced; if a third isocyanate molecule is added, followed by loss of X, a cyclic trimer occurs; if chain termination is relatively slow, addition of further monomers takes place with formation of a linear polymer. Conditions such as temperature, catalyst concentration, and character contribute to the reaction pattern. The tendency to cyclize no doubt plays a specially large part in isocyanate polymerization. A. DIMERIZATION Catalysts which have been found to promote dimerization of phenyl isocyanate include: pyridine (11), methylpyridine ( I d ) , triethylamine ( I S ) , N-methyl- (or ethyl-)morpholine, triethylphosphine, and other alkyl or alkyl-arylphosphines (14, 15). Alkylphosphines bring about a very violent polymerization since they act as active catalysts and the polymerization is quite exothermic. Triphenylphosphine is inactive. Alkyl-arylphosphines are not as active as alkylphosphines and permit better control of the reaction. Another convenient method (14, 16) for control of phosphinecatalyzed dimerization involves the addition of an alkylating agent such as benzyl chloride in an amount stoichiometrically equivalent to the substituted phosphine present. Complete deactivation of the catalyst results. By this means the reaction may be mitigated or even quenched and then activated by the addition of more catalyst. Isocyanate dimers are solids having a fairly high melting point. For example, the dimer of phenyl isocyanate melts a t 175". As expected, diisocyanates can polymerize to form resins. Much attention has been given to the physical properties of dimers including mixed dimers (2, 17), but such properties are not of primary concern here. The structure, a t least in the solid state, has been established (18) from X-ray measurements to be symmetrical as shown below:

Phenyl isocyanate dimer, also uretidinedione

Dimers, in contrast to trimers, are in dynamic equilibrium with monomer. Toluene diisocyanate dimerizes to a greater extent the lower the temperature, 90% a t 10" compared to 73% at 25" (19). I n the absence of a catalyst,

CATALYTIC EFFECTS I N ISOCYANATE REACTIONS

397

dissociation occurs rapidly only at elevated temperatures. The dimer of toluene 2,4-diisocyanate is almost completely converted to monomer at 175" and has been reported to dissociate at 150" (20). However, in the presence of a phosphine catalyst, dimers dissociate to equilibrium amounts even at room temperature (19). The structure of the isocyanate affects its tendency to polymerize. While diisocyanates can polymerize to form resins due to cross-linking, the dimer of toluene 2,4-diisocyanate is readily formed as a separate compound. This is because the isocyanate group ortho to the methyl is relatively much less active than the para-isocyanate group:

0

The general mechanism of dimerization by nucleophilic catalysts has been described above. As Arnold et al. (2) remark, the availability of a pair of unshared electrons on the catalyst is not in itself sufficient, however. Obviously, structural or other factors must be involved. For example, dimethylphenylphosphine and dimethylaniline are about equal in base strength (in 50% alcohol), but the phosphine is an extremely active catalyst and the amine is completely inactive.

B. TRIMERIZATION Treatment of isocyanates with a number of catalysts has been found to yield cyclic trimers, or isocyanurates :

Such catalysts include potassium acetate, sodium benzoate, certain basic inorganic salts (15, 21, 22), lead oleate and other lead salts (23), alkali soaps (23), and metal naphthenates (24). Also at 125O, trimers of phenyl isocyanate form in high yield catalyzed by N-methylmorpholine in the presence of ethyl alcohol (26). Epoxides have been reported to be active for trimerization in the presence of a small amount of amine (26, 27). Ethylene carbonate also is effective in production of trimer (68).Recently

398

A. FARKAS AND G . A. MILLS

Shashoua and co-workers (9, 10) described cyclic trimer formation with a sodium catalyst under certain conditions which were not ideal for linear polymerization. While metal salts catalyze trimer formation a t lower temperature, at 135' Dyer and Read (29) have found they are selective in promoting carbodiimide formation: A

2ArNCO -4

catalyst

COZ

+ ArN=C-NAr.

A special technique of trimerization has been described by Kogon (24, 25). Phenyl isocyanate reacts with ethyl alcohol to form a urethane (ethyl carbanilate). At 125' a substantial yield of ethyl a,y-diphenyl allophanate is observed as well as a small amount of phenyl isocyanate dimer. However, when N-methylmorpholine (NMM) is added as a catalyst, the reaction is altered and the product is triphenylisocyanurate (isocyanate trimer) in high yield. The reaction sequence is believed to be:

+

+N=C=O C2H50H -+ +NH-COOCzH5 +NCO ---* +N-C=O +-NHCOOCzH, H \ N-C=O

+

'

(1) (2)

d NMM

6NCO -----+

(dNC0)z

0 II (GNCO), + 4 NH-C-y-

(3)

COOC,H,

Thus, as shown in detailed studies of the trimer, two isocyanate molecules are supplied by the dimer, and the third by the isocyanate portion of the allophanate. This mechanism applies specifically for the trimerization reaction of an isocyanate carried out in the presence of a tertiary amine catalyst and either an alcohol or a urethane. For reaction (2), electron-donating groups in the aromatic ring of the urethane (carbanilate) and electron-withdrawing groups in the phenyl isocyanate appear to aid in the reaction (25).More study was given to the reaction of phenyl isocyanate and ethyl carbanilate (24).

CATALYTIC EFFECTS I N ISOCYANATE REACTIONS

399

Jones and Savill (26) have described the very rapid catalytic effect upon trimerization of isocyanates by ethylene oxide, propylene oxide, styrene oxide, or epichlorhydrin when a trace of pyridine is present. No reaction occurs in the absence of pyridine. They express the opinion that the function of the tertiary base is to open the epoxide ring and that the catalytic action is believed due to initial formation of @N-

AI &I -

-08.

No further features of the mechanism of this catalysis were given. When toluene diisocyanate is employed with propylene oxide and pyridine, a highly cross-linked resin is obtained. In another investigation, Tsuzuki et al. (28) found that trimerization of phenyl isocyanate occurs in the presence of ethylene carbonate and catalytic amounts of a tertiary amine. In this case, the trimer forms a complex with the carbonate which can react further at elevated temperatures by reverting to monomeric isocyanate. As mentioned earlier, metal carboxylates are trimerization catalysts (23,Zd). They also catalyze the reaction of aryl isocyanates and substituted ethyl carbanilates to give ethyl a,y-diarylallophanates even at room temperature ($4). C. LINEARPOLYMERIZATION An extremely interesting investigation (9, 10) concerns the discovery of the polymerization of monoisocyanates to linear high molecular weight polymers. This was achieved by the use of anionic catalysts a t low temperatures (- 20" to - 100'). This procedure is applicable to both alkyl and aryl isocyanates. The polymers are

where n may be up to 2000. These polymers can be classified as N-substituted 1-nylons. As an example, where R is n-propyl, the softening point is 180" and decomposition temperature 250". Polymerization was carried out by treating solutions of the monoisocyanates, generally in N,N-dimethylformamide, with catalysts such as sodium cyanide, sodium benzophenone ketyl, sodium naphthalene, or sodium metal. Certain classical anionic reagents such as metal alkyls,

400

A. FARKAS AND G . A. MILLS

Grignard reagent, and sodamide were unsuccessful. The authors ascribed this lack of activity to either side reactions or to insolubility in the reaction medium. It was shown that the catalyst must be in solution to be an initiator. Free radical catalysts, such as potassium persulfate and azo compounds, are ineffective in initiating this type of polymerization of isocyanates. Also, catalysts, such as triethylphosphine and triethylamine, which are known to catalyze dimer and trimer formation at higher temperatures, are not effective in initiating the polymerization to linear 1-nylons. The polymerization of isocyanates to 1-nylons is thought to be a new example of anionically polymerizable systems requiring basic catalysis and low temperatures. Thus, the initiation step can be considered as the attack of the anion (X-) at the isocyanate group as shown below. Initiation step:

This is analogous to the reaction of nucleophilic reagents such as water, alcohols, etc.,with isocyanates. The propagation step is believed to be continued by the nitrogen anion until the linear trimer stage. Then, there are two courses of reaction open: (1) cyclization with release of the initiating anion to give a trimer, or (2) reaction with more monomer to give a linear polymer. Propagation step:

linear trimer

J n Linear polym.er Cyclic trimer

CATALYTIC EFFECTS IN ISOCYANATE REACTIONS

401

From a study of the polymerization conditions, it was observed that both these reactions do occur. The trimerization reaction is favored at high concentrations of initiator and high temperature. The polymerization reaction is favored by low temperatures, high monomer concentrations, and low concentrations of initiator. These facts are consistent with the postulate that the amount of polymer formation is governed by the relative rate of cyclization. By decreasing the temperature the cyclization rate can be rendered small, to favor the linear polymer formation. The termination step most probably occurs during the isolation of the polymer and/or by reaction with adventitious water present in the reaction medium in a similar manner to the reaction. Termination step:

In this regard, it is of interest to note that the addition of 5% ethyl succinate, 10% acetonitrile, or 5% methylene chloride does not markedly influence the polymerization. However, the addition of 1% formamide completely inhibit's the polymerization. In general, for a reaction mechanism which involves the formation of an equilibrium amount of a catalyst-reactant complex, followed by reaction between complex and another reactant molecule, the reaction rate is proportional to (steric collision factors) (complex concentration) (activation energy term). Thus, increased catalytic activity can depend upon any of these terms. Although extensive kinetic measurements have been carried out for reactions with isocyanates, the elucidation of which of these factors is important in any particular case remains a challenge.

111. Reactions of Isocyanates with Compounds Containing Active Hydrogen

A. GENERAL The reactivity of a particular isocyanate toward a given alcohol or amine increases with increasing electrophilicity (electron attracting power) of the radical attached to the isocyanate group. The phenyl group pulls electrons away from the isocyanate group and make it more positive while alkyl groups repel electrons and increase the negative charge on the isocyanate group. Thus, in general, aromatic isocyanates are more reactive than aliphatic isocyanates. The introduction of electron withdrawing

402

A. FARKAS AND G . A. MILLS

groups, e.g., NOz, into the phenyl nucleus increases the reactivity while substitution with electron repelling methyl or methoxy group lowers the reactivity. On the other hand, the reactivity of alcohols and amines toward isocyanates is increased by the introduction of electron repellent substituents and is higher for the aliphatic than for the aromatic compounds. Thus, one has the following series of decreasing tendency to release electrons and react with isocyanates (SO) : R

\A/H

Ri"

H

R

\o/

H

> .. > RNHz

H

H

> \o/

..

>

NHa

H

> \o/

CO-R

..

>

+NHz

In the series amine-urea-amide-urethane R :NH

4

R :NH

>

A0

I NH I

R :NH

R :NH

A

OR

' A0 > AI0

R

electron availability decreases progressively and lower reactivity results. In the urea the carbonyl group pulls the electrons from one of the nitrogen atoms and causes the existence of the resonance forms: R :NH

co

R :NH

:NH R

I R In the case of the amide the reduction of the electron availability on the nitrogen is even more in evidence because there is only one nitrogen atom in the molecule. Still lower reactivity results for the urethane. It appears that the nucleophilic character of the active hydrogen compounds enables them to act also as their own catalysts in their reaction with isocyanates. On the whole their catalytic activity increases with the availability of the free electron pair. Morton and Deisa (31) measured the reaction rate of phenyl isocyanate in dioxane at 80"with a variety of active hydrogen compounds and obtained the relative rates as shown by Table I. The high reactivity of carbanilide (leading to the formation of biuret) is worth noting.

403

CATALYTIC EFFECTS IN ISOCYANATE REACTIONS

TABLE I Reaction of Phenyl Isocyanate with Active Hydrogen Compounds in Dioxane at 80" for Reactant Ratio 1 :1 Reactant

Initial [+NCO], M.

k X lo4 liters/mole/second ~

n-Butyl phenyl carbamate Acetanilide n-Butyric acid Carbanilide Water n-Butanol

0.25 0.50 0.50 0.11 0.50 0.50

~~~~

Relative rate

~

0.06 0.99 1.56 4.78 5.89 27.50

1 16 26 80 98 460

Somewhat different relative reactivities for n-butanol, water, and carbanilide with phenyl isocyanate are given by Hostettler and Cox (Sb), with and without various catalysts (Table 11). TABLE I1 Relative Activity of Various Active Hydrogen Compounds at '70"in Dioxane at Stoichiometric Concentration5 Catalyst

n-Butanol

None Et3N Triethylenediamine BusSnOAc BuzSn(0Ac)a

1 90 1200 80,000 600,000

a

Water 1.2 50 400 14,000 100,000

Carbanilide 2 4 90 8000 12,000

Initial phenyl isocyanate concentration 0.25 M ; catalyst concentration 0.025 M.

The differences in the relative rate of the uncatalyzed reactions are due to the fact that the reactions involving alcohols, water, and carbanilide follow different kinetics and therefore are influenced in a different manner by reactant concentrations.

B. REACTION OF ISOCYANATES WITH HYDROXYL COMPOUNDS The reaction of isocyanates with hydroxyl compounds is probably the most widely studied field of isocyanate chemistry. The reaction rate in this system depends on the nature of the isocyanate, of the alcohol, the solvent used and, of course, on the catalyst. In the following sections first the studies concerned with the reaction of isocyanates and alcohols in the absence of catalysts, i.e., the uncatalyzed or spontaneous reaction, will be discussed. These discussions will then be followed by a section on the catalyzed reaction.

404

A. FARKAS AND G . A. MILLS

1. The Spontaneous Reaction of Isocyanates with Hydroxyl Compounds

The first systematic study of the kinetics of the uncatalyzed or rather spontaneous reaction between phenyl isocyanates and alcohols was carried out by Baker and Gaunt (4c,e). Interestingly, this study was done after the work on the catalyzed reaction had been carried out and the theory for the catalysis had been developed (4a,b). Baker and Gaunt found that the reaction between various alcohols and phenyl isocyanates in di-n-butyl ether and in benzene followed essentially second-order kinetics (first order with respect to the alcohol and to isocyanate concentration) as long as the reactants were in equimolar concentration. The second-order velocity constant calculated on the basis of such kinetics, however, increased with increasing (excess) alcohol concentration, indicating that the alcohol acted as its own catalyst. Baker and Gaunt postulated that the first step of the reaction was the formation of a complex (C) between the alcohol and the isocyanate and that in the second step this complex then reacted with a second molecule of alcohol to form the urethane and a free alcohol molecule according to

+ ROH krki PhNC-0-

PhNCO

RdH

+

(complex)

PhNy-0-

+ ROH 2 PhNHCOOR + ROH

(2)

If one denotes with [I], [A] and [C] the isocyanate, alcohol, and complex concentrations, respectively, the following relations are obtained : -=

kl[II[AI - kz[Cl - kdCI[AI

dt = ka[C][A].

(3) (4)

If it is now assumed that a t any given concentration of the reactants a steady state concentration of [C] is reached, i.e., d[C]/dt = 0, then

and

CATALYTIC EFFECTS I N ISOCYANATE REACTIONS

405

A comparison of this formula with the simple second-order rate equation

9 = ko[I][A] dt

(7)

shows directly that the experimentally determined ko calculated from Equation (7) is actually defined by

This relation readily explains the increase of the second-order velocity constant ko with increasing alcohol concentration. The limiting value of ko will be kl which will be reached when k,[A] >> Icz. Rearrangement of Equation (8) yields

indicating that a plot of A/ko vs. A should give a straight line-% relation that is in agreement with the experimental results obtained in dibutyl ether and to a lesser extent with the results in benzene. A strict interpretation of these data should allow the determination of both kl and k2/k3 from the slope and intercept of the [A]/ko vs. [A] plot. The temperature coefficients for k1 and k3/k2 will then give the corresponding energies of activation or differences in the energies of activation, ( E ) . From the results given in Table 111,it is apparent that while the reaction TABLE I11 Rate Constants and Activation Energies for the Spontaneous Reactim of Phenyl Isacyanate with Various Alcohols in Dibutyl Ether and Benzene Solution EtOH

MeOH BuzO k1,O.

kalkzzo-

El, kcal/mole E8-E2,kcal/mole

0.0217 0.28 6.5 6.4

0.024 3.79 11.8 -14.2

i-PrOH

BuzO

C6H6

BuzO

0.0168 1.08 11.6 -1.1

0.0201 4.32 11.4 -11.7

0.0052 1.56 10.5 -0.7

velocity is similar in both solvents, the reaction is faster in benzene. According to Baker and Gaunt, this effect is due mainly to the large increase of the value k3/k2 which is attributed to the circumstance that in benzene E3is much smaller than E2. This is thought to be caused by a large decrease in the activation energy for the reaction of the complex with the alcohol in benzene solution rather than by any significant increase in activation energy for the decomposition of the complex, This explanation is based on

406

A. FARKAS AND G. A. MILLS

the observation that the infrared spectra show a considerable concentration of monomeric alcohol in benzene solution, whereas no monomeric alcohol exists in the dibutyl ether in which the alcohol is associated or complexed with the ether ( 4 4 . According to Baker and Gaunt, in the uncatalyzed reaction in benzene the difficult step is the formation of the intermediate polar alcohol-isocyanate complex, but once this is formed the whole reaction has a very great tendency “to run downhill” by a reaction with the second molecule of alcohol to give the uncharged product. On the other hand, in butyl ether there are two possibilities; namely, the dissociation of the intermediate complex into isocyanate and alcohol or its further reaction with alcohol to give the urethane. These possibilities appear to be more evenly balanced in the benzene solution. There are a number of questionable and unexplained points in Baker’s theory, some of which were indicated by Baker himself and some of which were raised in subsequent studies. In dibutyl ether no monomeric alcohol molecules are present. Therefore, the use of the stoichiometric alcohol concentration in the kinetic equations is not justified. Baker does not make allowance for the change in the association equilibrium on changing the temperature, and therefore the interpretation of the temperature coefficients, the slope, and intercept of the [A]/ko vs. [A] plot is not quite correct. If free monomeric alcohol molecules participate in the isocyanate reactions as reactants and/or catalysts, the concentration of unassociated alcohol molecules has to be taken into account even in benzene solution a t higher alcohol concentrations although this solvent itself does not associate with alcohol. Another shortcoming of the Baker theory is the fact that a t low alcohol concentrations systematic deviations appear in the [A]/ko vs. [A] plots since the [A]/ko values tend to be constant or increase rather than decrease with decreasing alcohol concentration. Actually, Baker’s mechanism requires that a t low alcohol concentrations a t which ka[A]<< ICZ the kinetics follow strictly third-order kinetics, second order with respect to alcohol concentration and first order with respect to the isocyanate concentration. Therefore, the proper way to handle Baker’s data would be to calculate the third-order velocity constants for each initial alcohol concentration. A plot of the reciprocal of the third-order constant vs. the alcohol concentration could then be expected to give a straight line relationship. Another questionable point in Baker’s derivation is his neglecting the effect of formation of the complex on the concentrations of the free alcohol and of the isocyanate.

CATALYTIC EFFECTS IN ISOCYANATE REACTIONS

407

The validity of Baker's mechanism for the spontaneous isocyanate alcohol reaction was also challenged by Ephraim et al. (33). These authors studied the effect of solvents on the reaction of phenyl isocyanate with methanol a t 20". If second-order kinetics is applicable to this reaction (i.e., first order with respect to the isocyanate concentration and first order with respect to the alcohol concentration), then plotting the reciprocal isocyanate concentration for equal reactant concentrations or plotting the logarithm of the alcohol to isocyanate ratio for unequal reactant concentrations against time should give a straight line. It was observed by these authors that two effects occur upon changing the solvent, one being deviation from the second-order rate plot (positive and negative deviation) and the other being a change in the reaction rate. The plots for the reaction with methylethyl ketone (MEK) and acetonitrile were found to indicate good second-order kinetics over the complete range. With toluene and nitrobenzene, downward curvature of the plot was apparent, indicating slowing down of the reaction. For the runs in n-butyl acetate and dioxane the plots show upward curvature. The apparent second-order rate constants obtained from the initial slope of the second-order plots in the various solvents were found to decrease in the series: benzene > toluene > nitrobenzene > di-n-butyl ether > n-butyl acetate > MEK > dioxane > acetonitrile. In view of the fact that the methanol-phenyl isocyanate reaction is known to be an auto-catalytic reaction due to the weakly basic character of the phenylurethane formed, one would expect an upward drift of the secondorder plot if the reaction were truly of the second order. This is apparently the case in n-butyl acetate and dioxane. The straight line plot in MEK and acetonitrile and the downward curvature found in toluene and nitrobenzene were taken to indicate an order greater than 2 with respect to the reactants. It appeared that the deviations from second-order kinetics toward higher order in the sequence toluene > nitrobenzene > acetonitrile > M E K > n-butyl acetate > dioxane increased with an increase in the hydrogen bonding capacity of the solvent. On the other hand, there was little correlation between the initial rate of the phenyl isocyanate-methanol reaction and the hydrogen bonding capacity of the solvent. With the exception of nitrobenzene and dioxane, the logarithms of the apparent second-order rate constant were found to decrease linearly with an increase in the quantity D / ( j 5 D l), where D is the dielectric constant of the solvent. On the basis of these results, Ephraim and co-workers expressed serious doubt as to the validity of the Baker mechanism. According to their views, there is no evidence for a complex formation between alcohol and phenyl isocyanate, On the other hand, a t high concentrations methanol has been

+

408

A. FARKAS AND G . A. MILLS

shown to complex with itself to a small extent in benzene. In dibutyl ether and other solvents of higher dielectric constant, it complexes with both itself and with the solvent. In contrast to Baker's mechanism, Ephraim and co-workers assume the formation of three complexes: (1) between two alcohol molecules, (2) between alcohol and the solvent, and (3) between the alcohol and the urethane. Each of these complexes then is thought of being capable of reacting with the isocyanate leading to the formation of the urethane and the complex forming reaction partner according to the following reaction steps:

* complex 1

Kla

2ROH

(10)

Kib

+ solvent complex 2, Klb = Kh[SOlV.] complex 3 ROH + urethane Complex 1 + RNCO urethane + ROH Complex 2 + RNCO urethane + solvent Complex 3 + RNCO 2 urethane.

RON

(11)

Kic

(12)

kto

-+

kzb

---f

klc

-+

KI,, KG,, and K1, are the respective equilibrium constants while kZu,k 2 b , and kzC are the velocity constants. The rate of isocyanate disappearance is then found to be -d[RNCol = (kz,[complex I] dt

+ ka[complex 21 + kz,[complex 3]][RNCO].

(16)

If it is assumed that the reaction of the isocyanate with the complexes is slow compared with the decomposition rate of the complexes, then the following reaction equation results: -d'RNCol dt

=

{

[RNCO]

1a

+ kfb [ROH] + K z o [ROH][urethane]t .

[ROHI2

Kib

It is now evident that the order of the initial reaction rate constant will lie between 2 and 3, depending on the relative importance of the first two terms on the right-hand side of Equation (17). I n solvents which form hydrogen bonds with the alcohol, the second term will predominate, and a t low extent of reaction and low alcohol concentration a second-order reaction will be found. I n either case beyond the initial stage the reaction rate is increased by auto-catalysis due to the urethane complex catalysis. 2. The Catalyzed Reaction of Isocyanates with H ydroxyl Compounds a. General. The catalytic effects of bases and, in certain cases, of aluminum chloride on the formation of urethanes from isocyanates and alcohols,

CATALYTIC EFFECTS IN ISOCYANATE REACTIONS

409

and especially from phenols, were discovered early and were utilized in the preparation of derivatives of alcohols for their characterization (34). French and Wirtel (35) prepared some 30 naphthyl-urethanes from phenols and 1-naphthylisocyanate using triethylamine as the catalyst. The catalytic activity of hydrogen chloride was found by Tarbell and Kincaid (34g).Continuing this work, Tarbell et al. (36) showed that certain acids and bases affect the yields of urethanes obtained from a-naphthyl isocyanate and phenols. Thus, catalytic effects were observed with sodium carbonate and acetate, pyridine, triethylamine, acetic acid, trichloroacetic acid, zinc chloride, hydrogen chloride, and boron fluoride etherate. The latter catalyst and triethylamine were found to be the most effective acidic and basic catalysts, respectively. The discovery of the catalytic effect of metal compounds on the reaction of isocyanates with alcohols came relatively recently (37)and led eventually to finding of the most potent catalysts of this kind in certain organometallic derivatives of tin (32). b. T h e Amine-Catalyzed Reaction. (i) Mechanism and kinetics. The kinetic analysis of the amine-catalyzed reaction between isocyanates and alcohols by Baker and Holdsworth (4a) was the subject of the first study aimed at the elucidation of the mechanism of this reaction. The initial series of runs were carried out by these authors in dibutyl ether solution with phenyl isocyanate and methanol in the presence and absence of triethylamine and of other tertiary amines. The reaction was found to follow second-order kinetics, first order with respect to the isocyanate and first order with respect to the alcohol, the second-order rate constant kz being given by the relation kz = ko

+ k, [amine]

where ko is the rate constant for the uncatalyzed reaction and k, is the catalytic constant characteristic of the activity of the catalyst. At 20' this rate constant expressed in liters/mole minute was k2 = 0.0017 7.45[NEt3] for triethylamine and was found to be reasonably constant for isocyanate concentrations and methanol to isocyanate ratios of 0.5:l to 4 : l (see Table IV). On varying the base catalyst there was some correlation between the constant k, and the base strength of the bases tested with the exception of dimethyl- and diethylaniline which were found inactive. These data are summarized in Table V. The first obvious reaction mechanism involving the formation of an alkoxy ion from the alcohol by the abstraction of a proton by the bases was rejected by Baker and Holdsworth primarily because of the inactivity of the anilines. This type of mechanism should show catalytic activity

+

410

A. FARKAS AND G . A. MILLS

dependent mainly on the base strength, and consequently the inactivity of the anilines cannot be explained. On the other hand, if a mechanism involving the complex formation between the isocyanate and the base is accepted, the inactivity of the anilines can be explained by steric consideraTABLE IV Reaction of PhNCO with MeOH in Dibutyl Ether at 20'. Effect of the Reactant Concentration [NEII]= 0.0306 M [PhNCO] moles/liter

[MeOH] moles/liter

[MeOHl/[PhNCO]

10' kz

0.2315 0.2380 0.1190 0.2380 0.1241

0.2315 0.2380 0.2380 0.1190 0.4941

1 1 2 0.5 4.0

22.6 22.8 22.0 22.0 19.4

tion. The anilines are essentially planar with the alkyl groups in the plane of the benzene ring. This structure can then approach the isocyanate group so to speak sideways only, whereas the trialkylamines or pyridine can be more favorably oriented toward the isocyanate group. TABLE V Catalysis of the PhNCO-MeOH Reaction, Both 0.2419 M in di-n-Butyl Ether, by Various Tertiary Amines

Quinoline Pyridine or-Picoline Triethylamine Diethylaniline Dimethylaniline

0.29 0.56 0.61 7.45 0 0

6.3 X 2 . 3 x 10-9 3 x 10-8 5.65 x 10-4 4 . 5 x 10-8 1 x 10-9

On the basis of these observatioqs and considerations, Baker and Holdsworth postulate that the catalyzed reaction depends on the reaction of the alcohol molecule with the complex formed from the isocyanate and the base. The formation of the isocyanate-base complex is a reversible reaction and its rate has to be taken into consideration in developing the over-all rate equation. These authors assume then the following reaction steps: ArNCO

ki + NRI * ArNCO k? I

NRa

CATALYTIC EFFECTS IN ISOCYANATE REACTIONS

ArNCO

I

kr + R'OH -+ ArNHCOOR' + RaN.

411 (2)

NR3

Using the stationary state condition the concentration of the complex is given by the expression [complex] =

ki[hNCOI[NRal kz ks[R'OHl

+

(3)

and hence the rate of the catalyzed reaction can be expressed by -d[ArNCO] dt

- kika[ArNCOl[NRal[R'OHI. k2 + ka[R'OH]

(4)

The over-all rate constant is then

where k, is the rate constant for the spontaneous reaction and k, is the previously mentioned catalytic constant. An inspection of the formula indicates that for large values of kz (kz > k3 X [R'OH]), Equation ( 5 ) reduces t o kbi =

ko

+[NRsJ = ko + Kka[NRaI k2 klk3

(6)

where K is the equilibrium constant of complex formation. Another consequence of Equation (5) is the dependence of k,,i on the alcohol concentration. It is apparent that neglecting ko with increasing alcohol concentration the rate constant for the catalyzed reaction should decrease according to NRs/kbi= Ilk, = kf/klk3

+ (ROH)/ki.

(7)

In further studies, Baker and Gaunt (4b1 e ) investigated the effect of the alcohol concentration on the triethylamine and pyridine-catalyzed reaction with phenyl isocyanate, and were able to confirm fully the validity of Equation (7) at 20" and 30" for methanol, ethanol, and isopropanol. The data are shown in Fig. 1. The intercept on the ordinate is the value k2/klk3 whiIe the slope of the line is equal to kl. From these values, k3/k2 were calculated. As shown by Table VI, the kl decreases in the series methanol to ethanol to isopropanol while the values for k3/k2 increase. The product klk3/kz decreases in the same series. Comparative values for El, kl, E8-E2,and ka/kz for the catalyzed reaction in benzene and in dibutyl ether are shown in Table VII. The main cause of the increased reaction velocity in benzene evidently lies in a much greater

412

A. FARKAS AND G . A. MILLS

'

0

2 0.4

0.8

Alcohol

1.2

1.6

2.0

2.4

conc. (moles/liter)

FIG.1. Base-catalyzed reaction between phenyl isocyanate and alcohols in di-n-butyl ether a t 20'C. Curve 1, methanol with 0.03 M NEta; curve 2, ethanol with 0.03 M NEt3; curve 3, i-propanol with 0.03 M NEt3; curve 4, methanol with 0.3 M pyridine; curve 5 , ethanol with 0.3 M pyridine; curve 6, i-propanol with 0.3 M pyridine.

TABLE VI Values of kt and ka/kz for the Base-Catalyzed Reaction of Alcohols with 0.24 M PhNCO in Dibutyl Ether Methanol Alcohol : k , a t 20' k3/kz a t 20" a

Ethanol

Isopropanol

a

6

a

b

a

b

5.45 1.71

1.02 0.60

1.60 2.27

0.48 0.79

0.096 2.26

0.095 1.32

0.0307 M triethylamine. 0.304 M pyridine.

CATALYTIC EFFECTS I N ISOCYANATE REACTIONS

413

value of k 3 / h in the benzene. Also, the negative values of E8-E, are very much larger in benzene than in ether. Again, it appears very likely that this difference is due to a large decrease in E, rather than to any significant change in E2. Thus, the attack by monomeric alcohol on the amine-isocyanate complex is greatly facilitated. The general character of the difference between the reactions in benzene and butyl ether is thus of the same kind as exhibited by the spontaneous reaction. TABLE VII Reaction between Phenyl Isocyanate and Methanol and Ethanol Catalyzed by 0.01 M NEt3 in ( a ) dibutyl Ether and (b) Benzene

h @ 20" (a) (b)

El, kcal./mole (a) (b) k 3 / h @ 20" (a) (b) E3-EZ, kcal./mole (a) (b)

Methanol

Ethanol

5.45 6.97 14.0 7.9 1.60 47.8 -9.4 -25.2

1.60 2.66 13.7 8.8 2.27 10.77 -8.5 -18.4

While the experimental confirmation of Equation (7) was considered by Baker and co-workers as supporting the proposed mechanism, the question was immediately raised as to why the kl's, the rate constants for the formation of isocyanate-base complex, should be dependent on the nature of the alcohol. A mechanism based solely on the complex between the base and the alcohol providing the effective catalyst was not capable of accounting for the straight line relationship indicated by the experiment. The mechanism involving the reaction NEts

+ ROH

(NHEt,)+(OR)-

(8)

is also excluded because of the inactivity of the anilines (the basicity toward hydrogen ions should not depend on steric effects). It was not thought that in the relatively low alcohol concentration the medium consisting of the solvent and the alcohol should have an effect on the rate of the complex formation between the base and the isocyanate. Another puzzling feature of this reaction is the dependence of the temperature coefficient of the reaction on the alcohol concentration indicating that the over-all activation energy increases with the alcohol concentration. Subsequent work on the infrared absorption of the system alcohol-dibutyl

414

A. FARKAS AND G. A. MILLS

ether-base ( 4 4 showed that no unassociated alcohol molecules existed even in high dilution. On this basis two possible explanations were offered by the authors. According to the first hypothesis, the amine complexes with the alcohol and both the amine and this complex can function as catalysts for the reaction, the same isocyanate-base complex resulting from attack by either catalytic species. Therefore, the original mechanism is really a composite one of the following two steps: PhNCO

+ NRs klk¶ PhNC-0I

+NRs PhNCO

ki‘ + NRs . . . H . . OR’ S PhNC-0- + R’OH. I

(10)

12’

NRa

+

From such a mechanism a simple straight line relationship between the reciprocal velocity constant and the alcohol concentration would not follow but the relation would obtain [B]/kbi

- K[B] = kz/kiks

+ [R’OH]/ki

(11)

where B is the base concentration and K is a composite constant composed of the velocity coefficients involved in the various reaction stages. An alternative hypothesis was based on the role of solvation and on the

+

polar nature of the intermediate complex, BNC(NR,)-O-. It is thought that this complex is solvated by the alcohol and that the greater the solvation of the complex, the more readily the forward reaction would take place and the greater the value of kl would be. On both polar and steric grounds, the energy of solvation for such a complex would be expected to decrease in the order methanol, ethanol, isopropanol. Consequently, the values for the rate constants for the formation of the complex should also decrease in this order, a fact which was found experimentally. In a closer study of the amine-catalyzed reaction of phenyl isocyanate with alcohols in toluene solution, Burkus (38) found that contrary to Baker’s results observed in dibutyl ether solution, the second-order rate constant was not a linear function of the triethylamine concentration. In order to explain the linearity in dibutyl ether and the nonlinearity in toluene of the second-order rate constant as a function of the triethylamine concentration, Burkus considers the role of the solvent. In dibutyl ether the solvent-alcohol interaction causes the absence of monomeric alcohol. The addition of the base does not affect the nature of the alcohol

CATALYTIC EFFECTS IN ISOCYANATE REACTIONS

415

species present and produces only a small base-alcohol interaction and base-alcohol complexes of low stability. Thus, in dibutyI ether, the base interacts mainly with the isocyanate. The weak base-alcohol complex may also attack the isocyanate to form the base-isocyanate complex. The concentration of the base-isocyanate complex is then proportional to the base concentration, and the linear relation between the rate and the base concentration follows. On the other hand, in toluene the base-alcohol interaction is appreciable, and the stability of this complex is much greater than in dibutyl ether. The molecular species of alcohol present in toluene is a function of the base concentration. The strong interaction between the base and alcohol prevents appreciable reaction with the isocyanate, and it alters the nature of the species of alcohol present. At low base concentrations, the base-alcohol complex forms mainly by reaction with the polymeric alcohol molecules and only to a lesser extent with monomeric alcohol. At higher base concentrations, the complex between the base and the alcohol is formed a t the expense of monomeric alcohol. Thus, the nonlinearity between the rate and the base concentration in toluene is not unexpected. Burkus also concludes that because of the interactions there would be no linear relation between the reciprocal value of the catalytic constant, k,, and the alcohol concentration in benzene in Baker’s experiments. Replotting the data of Baker and Gaunt (4e>, deviations from linearity are indeed observed. These arguments also predict that in benzene a t very high alcohol concentrations the strong interaction between the base and the alcohol causes a typical uncatalyzed reaction; that is, the rate of reaction should increase with increasing alcohol concentration. This conclusion is in agreement with the data of Farkas and Flynn (39). According to Burkus, the lower basicity and the higher nucleophilicity of 1,4-diazabicyclooctanefavor the formation of the complex between the base and the isocyanate and minimize the interaction between the base and the alcohol. Thus, at low base concentrations, a direct proportionality exists between the base concentration and the base-isocyanate complex concentration. According to Pestemer and Lauerer (401, the existence of addition complexes between isocyanates and tertiary amines as postulated by Baker and co-workers is shown by the appearance of infrared bands at expected frequencies. The assumed structure of the complex

416

A. FARKAS AND G. A. MILLS

is similar to that of certain "krypto" isocyanates

which have infrared absorption bands at 1750 to 1765 cm.-l (41). Mixtures of phenyl isocyanate with tertiary amines such as triethylamine, cyclohexyldimethylamine, and pyridine dissolved in hydrophobic solvents (paraffin oil, hexachlorobutadiene) show several new bands which are absent in the spectra of the components. In each of these systems bands are present in the region 1635-1652 cm.-l which are assigned to the structure

The complex bands did not appear instantaneously on mixing the components but their development took several minutes. If the complex thus formed is indeed the one postulated by Baker, it should be possible to establish a correlation between the rate of the catalyzed reaction and between the rate of the complex formation. In the phenyl isocyanate-triethylenediamine system, no bands were observed in the 1700 cm.-l region (42). (iij The effect of the structure of the arnine. As mentioned previously, Baker and Holdsworth recognized that the catalytic activity of amines while dependent on the basicity was also affected by steric relations. The low activity of N-dimethylaniline compared to that of pyridine, an amine of equal basicity, was ascribed to the planar configuration of the dimethylaniline which prevents close approach of the isocyanate molecule. As an example of an amine of unusually high catalytic activity, Farkas et al. studied the 1,4-diazabicyclooctane catalyzed reaction of phenyl isocyanate with 2-ethylhexanol (39). This amine is a di-tertiary base, N(C2H&N, with the N atoms at the bridge heads. The reaction was found to follow the second-order kinetics, and the rate of reaction was proportional to the diazabicyclooctane concentration. The temperature dependence of the uncatalyzed and the catalyzed reaction between the 23' and 47" corresponds to an energy activation of 11.1 and 5.5 kcal./mole for the uncatalyzed reactions, respectively. A comparison of the catalytic constants as defined by Baker for diazabicyclooctane and the structurally related triethylamine, 1,4-dimethyl-

417

CATALYTIC EFFECTS IN ISOCYANATE REACTIONS

piperazine, and N-ethylmorpholine shows the unusually high activity of the bicyclic amine and that there is no correlation between the basicity and the catalytic constants (see Table VIII). Under the assumption that Baker’s mechanism is valid, Farkas and Flynn draw the following conclusions from their results. The rate of the catalyzed reaction depends on the concentration of the amine complex and on the specific velocity constant for the reaction which involves this complex and the alcohol. Since the diazabicyclooctane is free from steric hindrance and the nitrogen atoms are readily accessible to the reactants, the formation of the complex between this molecule and the isocyanate molecules takes place more readily than with an amine having a carbon-nitrogen bond capable of free rotation around the nitrogen. Related to the postulate of higher TABLE VIII Reaction between PnNCO and 3-Ethylhexanal-1 (both 0.073 M ) at 93’ in Benzenea

1,4-Diazabicyclooctane Triethylamine 1,4Dimethylpiperazine N-Ethylmorpholine

pKA

kz, liters/mole hour

ko

8.6 10.64 8.39 7.51

7.8 1.9 1.56 0.58

5480 1260 1020 315 -

Q

Catalyst concentration:0.0014 M .

stability of the diazabicyclooctane-isocyanate complex as compared with a similar complex of tertiary amine is the observation of Brown and Eldred (43) on the formation of quaternary alkyl derivatives of quinuclidine, CH (CZH&N, a compound similar to diazabicyclooctane but containing only one bridgehead nitrogen atom. Brown and Eldred found that certain quaternary salts of quinuclidine formed 50-700 times faster than the corresponding derivatives of triethylamine and thought that the difference in reactivity was attributable to the absence of steric hindrance. Farkas and Flynn point out that the affinity of an amine for a proton is not necessarily a measure of that amine’s ability to complex with a molecule such as phenyl isocyanate because the proton being much smaller than the isocyanate is not greatly influenced by steric factors. Thus, the higher stability of the unhindered amine-isocyanate complexes will not be shown by measurements of the basicity of the amine toward the proton. An increased catalytic activity of the sterically unhindered diazabicyclooctane could also be expected if the concentration of its complex were not larger than the concentration of the hindered amine complex, provided the velocity constant for the reaction involving this complex is larger than that involving the complex formed from the hindered amine. Burkus (38) determined the activity for 23 different amines in the

418

A. FARKAS AND 0. A. MILLS

catalysis of the reaction of phenyl isocyanate with butanol-1 in toluene solution. He found (see Table IX) that the catalytic activity of the amines varied from 0.075 to 23.9, the highest activity being shown by 1,4-diazabicyclo(2,2,2)octane even though its basicity was almost the lowest among the amines examined. It was noted that another amine of very high activity, TABLE IX Tertiary Amine Catalytic Activities in the Reaction of Phenyl Isocyanate with 1-Butanol in Toluene at 59.69"" Catalyst N-Methylmorpholine N-Ethylmorpholine Ethylmorpholinoacetate Dimorpholinomethane N-(3-Dimethylaminopropyl)-morpholine Trieth ylamine N-Methylpiperidine

N,N,Nf,N'-Tetramethy1-1,3-propanediamine

N,N-Dimethyl-N',N'-diethyl-1 ,3-propanediamine N,N,N',N',N'-Pentamethyldiethylenediamine N,N,N',N'-Tetraethylme thanediamine Bis (2-diethylaminoethyl)-adipate Bis-(2-diethylaminoethyl)-adipate N,N-Dimethylcyclohexylamine N,N-Diethylcyclohexylamine N-Methyl-N-octylcyclohexylamine N-Methyl-N-dodecylcyclohexylamine N-Me thyl-N-(2-ethylhexyl)-cyclohexylamine N-Methyldicyclohexylamine 1,4-Diazabicycl0-(2.2.2)-octane 1,2-Dimethylimidazole Quinine Pyridine

Catalytic activity 1.00 0.68 0.21 0.075 2.16 3.32 6.00 4.15 3.10 3.47 0.085 1.oo 1.92 6.00 0.70 2.00 1.90 0.16 0.16 23.9 13.9 11.3 0.25

PKa 7.41 7.70 5.2 7.4 10.65 10.08 9.8

-

9.4 10.6 8.6 8.8 10.1 10.0 9.8

-

9.6 8.6 7.8 5.29

0 Catalytic activity of N-methylmorpholine is taken as 1.00. The amines were compared at equal amine equivalents which waa about 0.0300 N . The isocyanate and alcohol concentrations were about 0.100 M .

quinine, contained bridgehead nitrogen atoms. The methylamines showed higher activities than the corresponding ethylamines. Substituted cyclohexylamines, although of about equal basicities, varied greatly in the catalytic activity. N,N-diethylcyclohexylamine showed very low activity, nor did N,N,N',N'-tetraethylmethanediamine and dimorpholinomethane catalyze the reaction significantly. Burkus's conclusion is that the catalytic mechanism consistent with the experimental data involves a base-isocyanate complex formed by the direct

CATALYTIC E F F E C T S I N ISOCYANATE REACTIONS

419

attack of the isocyanate by the free base but not a base-isocyanate complex formed according to the equation RaN . . . HOR”

+ R’NCO + R’NCOe + R”0H. I

@NRa

The dependence of the catalytic activity on the steric requirement of the base does not support a mechanism which involves a base-alcohol interaction resulting in the formation of an alkoxide ion, since such a mechanism would not be sensitive to steric factors because of the small space requirement of the proton. The basicity of an amine is not connected with the steric requirement of the base. The mechanism involving ion formation cannot account for the high activity of 1,4-diazabicyclooctane, quinine, or 1,2-dimethylimidazole. The low activity of the tetraethylmethanediamine and dimorpholinomethane are also inconsistent with the ionic mechanism. I n connection with the selection of amine catalysts for the production of polyurethanes, Britain and Gemeinhardt (23) compared the catalytic activity of a large number of amines by measuring the gelation times of certain isocyanate-glycol mixtures. Since the conditions of these tests do not allow quantitative conclusions to be drawn, the results are summarized in Table X only for orientation purposes. In these tests tolylene diisoTABLE X Catalyst Tests of Tertiary Amines for the Isocyanate-Hydroxyl Reaction Gelation a t 70”, minutes

1,4-Diasa-(2.2.2)-bicyclooctane N-Ethylethyleneimine 2,4,6-Tri (dimethylaminomethy1)-phenol N,N,N’, N’-Tetramethylethylenediamine 1-Methyl-4- (dimethylaminoethy1)-piperasine Triethylamine N-Ethylmorpholine 2-Methylpyrasine Dimethylaniline

4 32 50 60 90 120 180 >240 >240

cyanate (mixtures of the 2,4- and 2,6-isomers) and a secondary hydroxylcontaining polyoxypropylenetriol were reacted a t 70” in the presence of the catalyst. c. Catalysis by Metal Compounds. There is no doubt that metal compounds excel over all other known catalysts in their diversity and activity. In recent reports (23, 32, 44) compounds of some 20 metals covering all

420

A. FARKAS AND G . A. MILLS

eight groups of the periodic systems were mentioned as catalysts. These included salts of organic and inorganic acids, halides, organo metallic compounds, a carbonyl, and alcoholates. Among these certain tin compounds stand out as the most effective catalysts known for the hydroxyl-isocyanate reaction. In spite of a bewildering richness of material, unfortunately there has been very little quantitative work of kinetic nature carried out that would allow some definite conclusions concerning the mode of action of these catalysts. In order to explain the high catalytic effect of cobalt naphthenate, Bailey et al. (45) suggested that the cobalt atom forms coordination complexes with the nitrogen and oxygen atoms in two isocyanate groups whereby the positive charges on the carbon atoms are increased and the isocyanate groups are activated for the reaction with the hydroxyl compound :

+ Ar-N=C-Q ‘\

I,

I

:c 0 . \

0’-C

=N-Ar 2.

O’Brien and Pagano (46) determined the rate constants for the metalsalt-catalyzed reactions of t-octyl isocyanate CH3 - C(CH3)z - CH2 - C(CHs)zNCO

with a large excess of ethanol and p-menthane diisocyanate

7% I

CH,-C-NCO

I

CH,

with a large excess of 1- and 2-butanol. Both of these tertiary isocyanates have unusually low reactivity. In all cases the reactions were first order with respect to the isocyanate concentration. This result makes it unlikely that the reactive complex contains two molecules of the isocyanate. The rate constants of the cupric nitrate- or ferric chloride-catalyzed reaction of t-octyl isocyanate with ethanol were found to be linearly dependent on the square root of the metal salt concentration. This relation necessitates the assumption that these metal salts are present in the alcohol solution in

CATALYTIC EFFECTS I N ISOCYANATE REACTIONS

42 1

an inactive dimeric form and that the dimer is in equilibrium with the active monomeric form which complexes with the isocyanate according to (metal sa1t)z 2 metal salt isocyanate metal salt $ complex.

+

The complex then reacts with the alcohol in a manner similar to that postulated by the Baker mechanism for the base-catalyzed reaction. The kinetics involving this square root law is not valid for the cupric acetateor zinc naphthenate-catalyzed reaction of these tertiary isocyanates. It seems that metal salts of strong acids and of weak acids conform to different mechanisms. For the reactant system p-menthane diisocyanate-alcohol the relative effectiveness of the metal naphthenates was found to be in the order Cu

> Pb > Zn > Co > Ni > Mn.

The ratio of the rate constants for the reaction of p-menthane diisocyanate with 1- and with 2-butanol at 25' was 3.5 with copper naphthenate as the catalyst but rose to 26 with lead naphthenate. By following the reaction to high conversion with 1-butanol, the ratio of the reactivities of the two isocyanate groups was found to depend on the catalyst used as shown by the values Naphthenate

Relative reactivity

CU

1.5 1.7 5.8

Zn Pb

It is interesting to observe the higher sensitivity of the lead catalyst to steric effects involving either the alcohol or the isocyanate. Hostettler and Cox (32)found that the catalysis of the phenyl isocyanatemethanol reaction by di-n-butyltin diacetate in dibutyl ether or dioxane follows second-order kinetics giving the following velocity constants : Catalyst

None Triethylamine Di-n-butyltin diacetate

Mole % catalyst 1

0.0088

k2 x 104 liters/mole second 0.51 5.7 118

The high rate observed with the tin compound even at the relatively low concentration is worth noting. This compound is 2400 times more active than triethylamine. The effect of the concentration of this catalyst on the

422

A. FARKAS AND G . A. MILLS

reaction was studied for the phenylisocyanate-1-butanolreaction and was found to correspond to the relation kz = ko

+ k , (catalyst)"

where n = 0.89 for di-n-butyltin diacetate and 1.04 for triethylamine. For practical purposes, n may be taken as unity, a value that would indicate that the catalytic is directly proportional to the catalyst concentration. A comparison of the activity of various tin compounds and of other isocyanate catalysts is shown in Table XI. The wide variation in the activity TABLE XI Phenyl Isocyanate-Alcohol Reaction (52) Relative activity a t yo catalyst concentration

1 mole

Catalyst None Triethylamine Cobalt naphthenate Benzyltrimethylammonium hydroxide Ferric acetonylacetonate Tetra-n-but yltin Tri-n-butyltin acetate Di-n-butyltin diacetate Di-n-butyltin dilaurate n-Butyltin trichloride Di-n-butyltin dichloride Dimethyltin dichloride Stannic chloride Trimethyltin hydroxide Stannous chloride Tetraphenyltin Di-n-butyltin sulfide 2-Ethylhexylstannoic acid

Methanol

Butanol

1

11 23 60

-

82 500 26,000 37,000 830

-

2100

99 1800

-

-

1 8

3 100

160 31,000 56,000 56,000

-

57,000 78,000 2600

-

2200 9 20,000 30,000

is worth noting. It will be also recognized that the basic tin derivative, trimethyltin hydroxide, is far more active than the much more basic benzyl trimethylammonium hydroxide. A similar conclusion is valid for acidic pair dimethyltin dichloride-stannic chloride in which case the latter is the stronger acid but weaker catalyst. On the basis of these facts, Hostettler and Cox suggest that a weak complex with one or both of the reactants is responsible for the catalytic activity. A more detailed mechanism for metal catalysis has been suggested by Britain and Gemeinhardt (23) based on the differences in the catalytic

423

CATALYTIC E F F E C T S I N ISOCYANATE REACTIONS

activation of the reactions of aromatic and aliphatic isocyanates. These authors used the gelation times of a secondary hydroxyl-containing polyoxypropylene triol-diisocyanate mixture at 70” as the measure for reactioii rates and compared the reactivities of toluene diisocyanate, m,xylylene diisocyanate, and hexamethylene diisocyanate in the presence of various metal compounds and other catalysts. In the absence of catalysts, the three diisocyanates differ very widely in their reactivity as shown by Table XII. In the presence of catalysts TABLE XI1

Reaction Rates of Three Diferent Diisocyanates -

Abbreviated name

a

Formula

TDI

Nco 80 %

XDI

4

OCN-R-NCO

b

353

32

23

21

20 %

CH~NCO I

‘CH~NCO OCN (CH2)sNCO

HMDI

Relative rate

1

0.5

+ R‘OH = OCN-R-NHCOOR’. + R’OH = R’OCONH-R-NHCOOR’.

* OCN-R-NHCOOR’

their reactivity is increased to a different extent depending on the nature of the catalyst. While tertiary amine catalysts do not change greatly the relative reactivities of the different types of diisocyanates, the metal compounds activate the aliphatic isocyanates more than the tolylene diisocyanate. Certain compounds of Zn, Co, Fe, Sn, Sb, and Ti have so much larger effect on the aliphatic dusocyanates that their reactions become faster than that of the aromatic diisocyanates. The gelations times for the three diisocyanates observed with a variety of catalysts are listed in Table XIII. Britain and Gemeinhardt suggest that the metal-catalyzed isocyanate reaction involves a complex formation between the metal compound, the isocyanate, and the hydroxyl compound as shown below. The complex formation can occur in two steps with either the isocyanate or the hydroxyl compound reacting first. In the double complex, the isocyanate and

424

A. FARKAS AND G. A. MILLS

TABLE XI11 Cuatalyst Tests with Aliphatic and Aromatic Diisocyanates Gelation time (minutes) at 70" Tolylene diisocyanate ~

m-Xylylene diisocyanate

Hexamethylene diisocyanate

~~

Blank Triethylamine 1,4-Diaaabicyclooctane Stannous octoate Dibutyltin di (2-ethylhexoate) Lead 2-ethylhexoate Sodium o-phenylphenate Potassium oleate Bismuth nitrate Tetra-(2ethylhexyl) titanate Ferric 2-ethylhexoate Cobalt 2-ethylhexoate Zinc naphthenate Antimony trichloride Stannic chloride Ferric chloride

>240 120 4 4 6 2 4 10 1 5 16 12 60 13 3 6

>240 >240

>240 >240

80 3 3 1 6 8

>240 4 3 2 3 3

M

35

2 5 4 6 3

2 4 4

10 6

w

$5 34

34

hydroxyl compound are positioned on the same side of the metal compound in close proximity to each other in a manner which allows very high rates for the catalytic reaction. This mechanism is shown for the case in which first the isocyanate complex forms first. It would readily explain the especially high activation for the aliphatic isocyanates which, because of lack of steric hindrance, would allow them to participate particularly easily in the indicated complex formation.

L

. R-N=C-O

@ I H-0-MX,

I RI

@

L

R-N-C=O

I

I

H O

I

R'

+

MX,

425

CATALYTIC EFFECTS I N ISOCYANATE REACTIONS

Weisfeld (44) compared the catalytic activity of various metal acetylacetonates by measuring the time required for a certain polyethylene adipate ester-diphenylmethane-diisocyanatemixture to reach a predetermined viscosity in the presence of the catalyst and also used the same technique for determining the catalytic order n defined by k = ko

+ k, (catalyst)".

The catalytic order was 1.0 for the acetonylacetonates of Cr, Cu, Co, V, and Fe. Only for the Mn compound was n = 1.75. This result indicates that in the concentration range studied more than one manganese atom is necessary to provide the intermediate complex necessary for the catalysis of the isocyanate reaction. While no specific picture is made as to the nature of the metal catalysis, it is suggested that this type of catalysis may be associated with the formation of the triplet state and related to the paramagnetic properties of the metal. TABLE XIV Catalytic Effect of HCl and BF3-ETnO on the Reaction of Phenyl Isocyanate and Butanol in Toluene at 86' Isocyanate conc.

OH

NCO

BFa-EtnO conc. moles/liter

k

x

104

liters/mole/second

~

0.396

0.386

1:l

5: 1

0.0059 0.021 0.032 0.048 HC1 0.018 0.031 0.040 BFpEtnO 0 0.016 0.32 HC1 0 0.018 0.03

2.87 4.9-4.6 5.4 6.95 4.95 7.2 8.4 4.73, 4.74, 5 . 0 , 4 . 9 6 4.96 5.0 4.83 5.5 6.0

d. Catalysis by Acids and Bases. There is little information on the acid-catalyzed reaction of isocyanates and alcohols. Tarbell et al. (36) reported first on acid catalysis, but other authors claimed that acids cause inhibition (45, 47'). According to Tazuma and Latourette (@), hydrogen chloride and boron fluoride-etherate, two examples of proton and Lewis acids, have a relatively weak catalytic effect on the reaction of phenyl isocyanate with butanol-1 in toluene as shown by Table XIV at

426

A. FARKAS AND G . A. MILLS

an OH/NCO ratio of 1/1. The catalytic constants defined by

kexp = ko

+ k, (catalyst)

for HC1 and for BFsEtzOand are 10-15 times smaller than the corresponding constant for triethylamine for the same system. At a 5/1 OH/NCO ratio the catalytic effect of the acid became very small for HC1 and practically disappeared for BFrEtzO. For explaining the catalytic effect of acids (A), two mechanisms were considered one of which involves a complex formation with the isocyanate and the other an alcohol-acid complex. Mechanism I : 0 + A * Ph-N=C-0-A@ @ 8 Ph-N=CO-A + ROH -+ PhNHCOOR + A

PhN=C=O

Mechanism ZI:

ROH

+A

@ 8 RO-A

I

H @

R-0-A I

8

+ PhNCO -+

PhNHCOOR

+A

k Both of these mechanisms are compatible with the experimental results. At low alcohol/isocyanate ratios both give a catalytic effect which is proportional to the acid concentration. Also, both predict that at high alcohol excess the catalytic effect disappears. According to mechanism I this would be due to the formation of the complex becoming the rate determining step at high alcohol excess. On the other hand, in the case of mechanism 11, TABLE XV Catalyst Tests of Various Acidic Metal Compounds for the Isocyanate-Hydroxyl Reaction Compound tested Blank Stannic chloride Ferric chloride Antimony trichloride Antimony pentachloride Vanadium trichloride Arsenic trichloride Boron trifluoride-ether complex

Gelation time a t 70"' minutes

>240 3 6 13 90 90 240 >240

CATALYTIC E F F E C T S I N ISOCYANATE REACTIONS

427

the auto-catalysis by the formation of active complex between the isocyanate and alcohol leading to increased reaction rates could obscure the relatively small catalytic effect of the acid. The relative activities of some acidic catalysts are shown in Table XV in terms of gelation times according to Britain and Gemeinhardt (23). For explaining the activity of strong bases in catalyzing the isocyanatehydroxyl reaction Britain and Gemeinhardt (23) suggest the following mechanism patterned after that of Baker:

H 0-R” [R-Ik-f~~]

+

[

I

H-0-R”

H--0-R” I

,

R-N-b---Oe A-R’

I 1 H

I

1

eO-R”

H 0-R”

R-N-C=O b R ‘

80-R’

C. REACTION OF ISOCYANATES WITH WATER This reaction has great significance in the preparation of polyurethane products as it causes chain extension and branching, provides the carbon dioxide necessary for foaming and forms the basis of air curing of polyurethane coatings. It has been generally accepted that the primary reaction between isocyanates and water leads to the formation of carbamic acid which then decomposes either directly or after undergoing reactions involving the release of carbon dioxide. Naegeli et al. (49, 50) developed the following scheme for the interaction of isocyanates and water:

+Hz0

I. RNCO --4

+RNCO

-GO2

RNHCOOH ---t RNHz - RNHCONHR ---f

+RNCO

-coz

11. RNHCOOH ___ --+ (RNHC0)ZO_ _ -+ RNHCONHR

111. RNHCOOH-

+RNHz

-H*O

+ [RNHCOO]-[NHaR]++ RNHCONHR.

428

A. FARKAS AND G. A. MILLS

According to this picture, the first step in the reaction between the isocyanate and water is indeed the formation of the carbamic acid which, however, has the choice of three reaction paths. It can either decompose to the amine and COz, react with isocyanate to form the carbamic acid anhydride or combine with the amine to form a carbamate salt. In subsequent reaction steps, then, the anhydride and salt are converted to the urea by loss of COZ and water respectively. The relative rates of these reaction sequences were found to depend 011 the nature of the medium (homogeneous or heterogeneous), rate of addition of water, temperature, reactivity of the amine (formed by the decomposition of the carbamic acid) with the isocyanate, concentration, and other factors. For example, in the reaction with phenyl isocyanate, cold water and heterogeneous medium favored the formation of diarylurea while with boiling water the main product was aniline. Dilution also favored aniline formation. With the introduction of nitro groups in phenyl isocyanate, the yield of the amines increases a t the expense of the ureas. 2,4-Dinitro- and 2,4,6-trinitrophenyl isocyanates gives only amines. The introduction of a methoxy or methyl group in para position decreases the amirie yield. The amine yield increases in the following series: 4 M e O < 4-Me < H < 3-Me0 < < 3,5-(NOz)z < 2-NO2 < 2,4-(N02)

< 4-NO2 < 2,4,6-(N02)a.

A morerdetailed quantitative study of the water-o-tolyl isocyanate reaction by Shkapenko et aZ. (51) showed that at 80" in dioxane solution and in the presence of triethylamine or other catalysts the consumption of the isocyanate was complete within a short period when only approximately half of the theoretical amount of carbon dioxide was released. The evolution of carbon dioxide proceeded from this point on at a slow rate. It was also demonstrated that by heating the reaction mixture to lOO", 30-35% of theoretical COZ was released, and that this portion of the COz was given off by the decomposition of the carbamic acid anhydride formed from the acid and a second molecule of isocyanate. Additional tests showed that 4-5% of the isocyanate formed o-tolyl ammonium-N-o-tolyl carbamate, 18.7% of the water added remained unreacted, and that a trace of the free o-tolyl amine was also present. I n addition, the presence of di-o-tolyl urea was proven. Since according to these results 18% of water remained unreacted at a time when all of the isocyanate had been consumed, it is apparent that the isocyanate must have reacted with the diphenyl urea-a reaction that was actually demonstrated by these authors to occur a t 80" under the experi-

429

CATALYTIC EFFECTS IN ISOCYANATE REACTIONS

mental conditions of the water reaction a t approximately the same rate as that of the water-isocyanate reaction. A much simpler picture is obtained for the reaction between water and isocyanate if only the disappearance of the isocyanate is followed. This state of affairs is not necessarily expected on the basis of complex nature of the reactions discussed above even if the first step leading to the formation of carbamic acid is the slowest step. According to Morton and Deisz (31), the kinetics of the spontaneous reaction between water and phenyl isocyanate is very similar to the corresponding rcaction for n-butanol. The reaction in dioxane follows secondorder kinetics and the second-order rate constant k depends on the initial water concentration in accordance with the Baker mechanism : _ [HOHI _=-

[HOW

k

kt

+ &*

kik3

The observed rate constants are tabulated in Table XVI and are compared with the constants for the butanol reaction a t several reactant concentrations and temperatures. Thus, it appears that the spontaneous reaction between isocyanates and water follows the same mechanism as the alcohol reaction and that the water serves as its own catalyst. Similar conclusions regarding the mechanisms of the water and the alcohol reactions were reached by Farkas and Flynn (39)who studied the base catalyzed reaction between phenyl isocyanate and water. TABLE XVI Reaction of Phenyl Isocyanate ([NCO] = 0.6 M ) with Water and n-Butanol in Dioxane Initial reactant/NCO ratio 4: 1 2:I 1:I 1:I 1:l 1:l

Activation energy, kcal./mole

k x 10‘ liters/mole second Temp., “C

water

n-butanol

25 25 25 35 50 80 -

1.42 0.77 0.41 0.73 1.53 5.89 11 .o

2 2.53 4.47 7.57 27.50 9.3

The progress of the reaction with time in dioxane a t 23’ in the presence of diazabicyclooctane as the catalyst did not conform to second-order kinetics as the reaction tended to proceed faster than required by secondorder kinetics after half of the isocyanate had been consumed. This effect

430

A. FARKAS AND G . A. MILLS

3 Hours

FIG.2. Reaction of +NCO with HzOin dioxane at 23°C in the presence of triethylene diamine (0.0014 M). Effect of reactant concentration. [HzO] = jg[+NCO].

is shown by the upward curvature of the plot of the reciprocal isocyanate concentration vs. time (see Fig. 2) and is probably caused by the ureide formed in this reaction. The initial second-order rate constants for various concentrations of isocyanate and water, summarized in Table XVII, show a satisfactory conTABLE XVII Reaction of Phengl Zsocvanate with Water at

a

M'a

+NCO moles/liter

HzO moles/liter

kZ liters/mole hour

0.036 0.144 0.073 0.073

0.018 0.072 0.072 0.144

0.72 0.64 0.86 0.62

Catalyst: 0.0014 M diazabicyclooctane; solvent: dioxane.

43 1

CATALYTIC EFFECTS IN ISOCYANATE REACTIONS

stancy indicating that the second-order kinetics is valid at least at the initial stages of the reaction. Triethylamine, N,N-dimethylpiperazine, and N-ethylmorpholine showed 3-6 times lower catalytic activity than triethylenediamine. In benzene both the catalyzed and uncatalyzed water reactions were faster than in dioxane. In either of the two solvents the rate of the catalyzed water reaction was about one-third of that of the corresponding reaction between phenyl isocyanate and 2-ethylhexanol. The relative rates of phenyl isocyanate with stoichiometric concentrations of n-butanol and water (0.25 N ) in dioxane at 70"in presence of various catalysts as determined by Hostettler and Cox (32) are summarized in Table XVIII. For sake of comparison the relative rates with diphenylurea are also included. TABLE XVIII Relative Rates of Urethane Reactions Relative rates

Catalyst None N-Methylmorpholine Triethylamine

Catalyst conc., M

-

0.025 0.025 N,N,N',N'-Tetramethyl-l,3-butanediamine 0.025 1,4-DiazabicycIo(2,2,2)octane 0.025 Tri-n-butyltin acetate 0.00025 Di-n-butyltin diacetate 0.00025

Butanol

Water

Diphenylurea

1 .o 40 86 260 1200 800 5600

1.1 25 47 100 380 140 980

2.2 10 4 12 90 80 120

In the absence of catalysts the reactivity follows the order urea > water > butanol. In the presence of catalysts the order of reactivity is reversed and the alcohol exhibits the highest reactivity. It is interesting to note that the preferential acceleration of the alcohol reaction by diazabicyclooctane is more pronounced than that by N-methylmorpholine or triethylamine. The high ratios of relative rates for the alcohol and water reactions observed with certain catalysts make these useful in the so-called one-shot foaming process in which for good foam stability a rapid polymerization process involving a reaction between diisocyanate and bifunctional hydroxy compounds and a slower CO, evolution are desirable.

D. REACTIONOF ISOCYANATE WITH AMINES According to Naegeli el al. (6U), the reactivity of substituted phenyl isocyanates toward m i n e s increases in the series: 4MeO < &Me < H < 3-Me0 < 3-NO2 < 4-NO2 < 3,5-(NOz)z < 2-NO2 < 2,4-(NOz)z < 2,4,6-(Noz)a.

432

A. FARKAS AND G . A. MILLS

A similar substitution on anilines causes the reverse effect. Nitro groups in ortho position either in the isocyanate or the aniline lower the reactivity by steric hindrance. These authors also reported that the reaction is subject to catalysis by pyridine, tertiary bases, and certain carboxylic acids but is unaffected by water, inorganic acids, bases, or salts. Relative rates for the reactions of some primary aliphatic amines with phenyl isocyanates have been determined by Davis and Ebersole (5%'). Systematic studies of the reaction rates of isocyanates with amines were carried out by Craven (53) and Baker and Bailey (4j). Craven studied the reactions of phenyl and o- and p-tolyl isocyanates with aniline, 0- and p-toluidine and o-chloroaniline in dioxane solution maiiily in the absence of catalysts. The reactivity in these systems agreed with the classical electronic picture according to which electron donating group that increase the nucleophilicity or base strength increase the reactivity of the amine. Substituents in ortho position, particularly on the isocyanate, cause steric hindrance and reduce the reactivity (see Table XIX). TABLE XIX Half-Lives of Reactions of Isocyanate8 (0.1N ) with Amines (0.1 N ) in Dioxane at 31" Isocyanate

Amines

Phenyl

Aniline o-Toluidine p-Toluidine o-Chloroaniline Aniline o-Toluidine Aniline p-Toluidine

O - T d yl

p-TOlyl

Rase strength, 4.6 3.3 20.0 0.023 4.6 3.3 4.6 3.3

Half -1if e, minutes 43 60 5 > 1200 202 > 1000 54 25-30

While bases, like pyridine and diethylcyclohexylamine, or sulfuric acid and water showed little catalytic action, ureas were found to accelerate the reaction. This action of ureas is reponsible for the autocatalysis observed in the interaction of isocyanates and amines. The initial rate is more dependent on the concentration of the amine than on that of the isocyanate, the order with respect to the amine being close to 2. On the basis of these observations, the following mechanism was suggested : Spontaneous reaction: kl

+ ArNH2 1% complex I ka Complex I + ArNHs (ArNH)&O + ArNHz ArNCO

+

(1) (2)

CATALYTIC E F F E C T S I N ISOCYANATE REACTIONS

433

Product-catalyzed reaction: Complex I

+ (ArNH)tCO

ka’ --+

2(ArNH)2.

(3)

For the initial stages of the reaction, the rate is given by

As long as k3(ArNH2) < kz, this expression leads to an order of 2 for the amine concentration, while this order becomes unity when k3(ArNH) >> k2. As the reaction proceeds, the urea concentration increases, the amine concentration decreases, and the product-catalyzed reaction becomes more prominent. The over-all reaction can be represented by

_ _ia_ = _ k2. di/dt

1

klkl u

1

+ k3/k13a+ k

(5)

where i, a, and u are the isocyanate, amine, and urea concentrations, respectively. The kinetic data were found to be in accord with this scheme and led to the conclusion that the complex formation was fast and that productinduced reaction occurred with higher efficiency than the amine-induced reaction ( I c ~ / I C ’ ~ < 1). A somewhat different mechanism was deduced by Baker and Bailey (4f)from their own studies of the systems phenyl isocyanate and ethyl paminobenaoate (benzocaine) ; cyclohexyl isocyanate and aniline; phenyl isocyanate and aniline; p-methoxyphenyl isocyanate and aniline. While Craven postulates one complex, Baker and Bailey assume two complexes of the isocyanate: one with the amine, and one with the product urea. The reactions formulated for the uncatalyzed reaction are as follows: kr

ArNCO

+ NHzR ks

Complex I

+ NH2R

complex I k6 -+

ArNHCONHR

+ NHzR

ki

ArNCO

+ ArNHCONHR k2 complex I1

Complex I1

--f

+ NHZR -+ka 2ArNHCONHR.

(9)

Thus, the truly spontaneous reaction involves complex I while the product catalyzed reaction involves complex 11. Applying the stationary state method for the formation and decomrosition of these two complexes, the following velocity constants are obtained :

434

A. FARKAS AND G . A. MILLS

k. and k’, being the velocity constant for the truly spontaneous and the product-catalyzed reactions. These constants are related to the over-all velocity constant k = k,

+ k’, (ureide).

(12)

In the presence of a tertiary base (B), there is a base-catalyzed reaction between the amines and isocyanate superimposed on the spontaneous and the product-catalyzed reactions. According to Baker and Bailey, the following steps are involved in the base-catalyzed reaction. First there is the reaction involving the isocyanate-base complex and the amine :

+ +

ArNCO B complex Complex NHZR + ArNHCONHR

+ B.

(13) (14)

In addition, there is interaction between the isocyanate, the product urea, and the base leading to a ternary complex which in turn reacts with an amine molecule forming two product ureas and reforming the base:

+

+

ArNCO ArNHCONHR B ternary complex B. Ternary complex NHzR -+ 2ArNHCONHR

+

+

(15) 06)

The over-all velocity constant for the base catalyzed reaction k, can then be represented by k, = k.

+ k,[ureide] + k,B + k”,[B][ureide].

(17)

E. REACTION OF ISOCYANATES WITH THIOLS The reaction of thiols with isocyanates leading to the formation of thiolcarbamates is very similar to that of alcohols but has received much less attention. According to Dyer and Glenn (6), the reaction between phenyl isocyanate and 1-butanethiol can be described by the following equations: Spontaneous reaction: +NCO

+ BuSH

-+

+NHCOSBu

Base-catalyzed reaction:

-a + EtaN + +N-C-0

-a

@NCO

(complex I)

I NEt3

Complex I Product-catalyzed reaction: +NCO

+ BuSH

+

+ +NHCOSBu

+ Et8N

+a - a + +NHCOSBu * +N=C-0 . . . H-N-+ &OSBu (complex 11)

(34

435

CATALYTIC EFFECTS I N ISOCYANATE REACTIONS

Complex I1

+ BuSH

+ 26NHCOSBu

(3b)

Base-product-cata’yzed reaction: Complex I

+ 6NHCOSBu S +NC-0

. . . H-N-4

(44

&Eta LOSBu (complex 111) Complex I11

+ BuSH

--t

2+NHCOSBu

+ Et3N.

(4b)

The first of these equations represents the spontaneous reaction which is considerably slower than the corresponding reaction involving the alcohol. The slowness of this reaction is caused by the lower nucleophilicity of the thiols. The base-catalyzed reaction in Equations (2a) and (2b) is very similar to the corresponding alcohol reaction with this difference. Since the mercaptide ion is more nucleophilic than the alkoxide ion one will expect that smaller differences in the charge density around the carbonyl carbon of the isocyanate will affect the reactivity of the mercaptides than that of the alkoxides. This would explain the higher sensitivity of thiols to basecatalysis and the higher activity of bases for the catalysis of the thiol reaction. The third group of equations describes a relatively weak effect while the last group of equations accounts for the apparent acceleration of the basecatalyzed reaction as the reaction proceeds. The formation of complex I11 involves the amide hydrogen in the thiol carbanilate since the fully substituted compound, n-butyl N,N’-diphenylthiolcarbamate was found to be without catalytic effect. The second-order rate constant calculated for the initial stages of the TABLE XX Catalytic Rate Constants and Base Strength of Various Amines k, for -

Amine

PKh

1-BUSH

1,2,2,6,6-Pentamethyl piperidine Tri-n-but ylamine Triethylamine Tri-n-propylamine N-ethylpiperidine N-methylpiperidine Benzyldimethylamine 1,4Diaaa-(2,2,2)-bicyclooctane N-ethylmorpholine Diethylaniline Pyridine

2.75 3.11 3.26 3.35 3.60 3.87 5.07 5.40 6.30 7.44 8.85

0.37 0.22 1.03 0.19 0.65 0.55 0.37 2.17 0.0147 0 0.00016

2-BuSH

0.028 0.13

0.000051

2-OctSH

436

A. FARKAS AND G. A. MILLS

base-catalyzed reaction was found to be proportional to the base concentration and independent of the thiol to isocyanate ratio. The catalytic constants for the base-catalyzed reaction in toluene calculated according to k, = k,/[cat] and the base strengths of the amines are summarized in Table XX and Fig. 3. It is immediately apparent that base 5

o 1.4 Diaza-Bicyclaoctane

I

E t-Piperidine 0.E 0

p

0

Eenzyl-Oi-Me-Arnine

0.1

0 0

-I

0.0E

0.01

0.00:

I

I

I

I

I

I

L

3

4

5

6

7

8

9

PKb

FIG.3. Reaction of phenyl isocyanate with butanethiol-1. The relationship between the catalytic constants of various amines and their base strength.

strength is not the only factor responsible for catalytic activity and that steric hindrance plays an important role. The low activity of the highly hindred but strongly basic penta-methylpiperidine and the high activity of the weakly basic diazabicyclooctane possessing readily accessible tertiary nitrogen atoms give two striking examples for the two extremes.

CATALYTIC EFFECTS I N ISOCYANATE REACTIONS

437

‘These authors note that “the fact that a true Bronsted plot is not dbserved in the base-catalyzed reaction of phenyl isocyanate with l-butaneIthiol indicates that the function of the amine does not depend on proton transfer and argues strongly for the isocyanate amine complex proposed by Baker and Holdsworth (4a) and by Dyer and co-workers (5).” The lower reactivity of the secondary thiols when compared with that of the primary thiols is similar to the behavior of the corresponding alcohols. The effect of the solvents on the triethylamine-catalyzed reaction 1-butanethiol with phenyl isocyanate is shown in Table XXI. TABLE XXI Eflect of Solvents on the Triethylamine-Catalyzed Reaction of Butanethiol with Phenyl Isoc?yanate ~~~~~~

Dielectric constant Solvent Toluene Butyl acetate Nitrobenzene Acetonitrile

20”

k, for triethylamine

2.39 5.01 36.1 38.8

1.1 2-2.4 200 0.001

In either dimethyl-formamide or dimethyl sulfoxide, the reaction rates became too fast to measure even in the absence of a catalyst. It thus appears that while the ionizing power of the solvent as indicated by the dielectric constant is an important factor for the solvent effect, it is not the only one. The slow reaction in the case of acetonitrile may have been caused by the nitrile competing with the isocyanate for the electrons of the base catalyst and thereby “neutralizing” the catalyst by complexing. For the energies and entropies of activation (AE and AS) of the triethylamine-catalyzed reaction of butanethiols with phenyl isocyanate, the following values were obtained from the temperature dependence of the reaction between 20” and 25”:

I-BUSH 2-BuSH

AE

As

(kcal./mole)

(e.u.)

3.9

-64

4.6

-58

Iwakura and Okada (64) interpret the mechanism and kinetics of the tert-amine catalyzed reactions of isocyanates with Gbutanethiol and n-dodecanethiol somewhat differently. They found that the catalyzed reaction was strictly first order with respect to the thiol, isocyanate, and

438

A. FARKAS AND G . A. MILLS

the tertiary amine. The catalytic coefficient for strongly basic tert-amines calculated from the equation: k = k~

+ k,[catalyst]

were larger for the phenyl isocyanate-butanethiol reaction than for the phenyl isocyanate-ethanol reaction while for the catalytic coefficient of the weak bases the reverse was true. Thus diazabicyclooctane, while still more effectivecatalytically than expected from its basicity, showed only slightly higher catalytic constant than trimethylamine. The rate constants and catalytic constants are given in Table XXII. The catalytic coefficient of TABLE XXII Reaction of Phenyl and Ethyl Isocyanate with I-Butanethiol (both 0.116 M ) in Toluene Solution at SO"

Tertiary amine

pK,

Conc. of t-amine molee/liter

Phenyl isocyanate 1,4-Diazabicyclooctane Triethylamine Diethylcyclohexylamine Tributylamine N-Methylmorpholine Pyridine N,N-Dimethylaniline

8.60 10.78 0 9.93 9.29 5.23 5 .OO

0.000468 0.005 0.00497 0.0050 0.010 0.10 0.10

0.0365 0.345 0.178 0.0754 0.0108 0.00314

77.9 68.6 39.6 15.6 1.08 0.0314 0

0.05

0.0422

0.86

Ethyl isocyanate Trimethylamine

8.60

Second-order rate constant, liters/mole minute

Catalytic coefficient

-

kc

triethylamine is about 80 times smaller for ethylisocyanate than for phenyl isocyanate. The rate of the triethylamine-catalyzed reaction of phenyl isocyanate with n-butanethiol depends on the solvent used and increases in the order benzene, toluene, dibutyl ether, dioxane, methyl ethyl ketone, and nitrobenzene. This solvent effect increased with the ionizing power of the solvent and was the reverse of that observed in the reaction of isocyanates with alcohol. The reactivity of various thiols is shown in Table XXIII. The lack of reactivity in the case of thiophenol is worth noting. While Baker and Gaunt postulate that the mechanism of the basecatalyzed reaction of isocyanates and alcohols involves the attack of the isocyanate-base complex by the alcohol, according to Iwakura and Okada the higher acidity of the active hydrogen containing compounds changes

CATALYTIC EFFECTS I N ISOCYANATE REACTIONS

439

the mechanism in the case of the thiol-isocyanate reaction in as much as the catalyst is complexed with the thiol (55). The catalyst-thiol complex then forms a ternary complex with the isocyanate which eventually reacts with free thiol and is then converted to the product and the catalyst. TABLE XXIII The Reaction of Phenyl Isocyanate and Ethyl Isocyanate with Thiols (both Reactants at 0.1250 M ) in Toluene Solution at 30”in the Presence of 0.05 M EtsN Thiol Phenyl isocyanate I-Dodecanethiol 1-Butanethiol Phenylme thanethiol Thiophenol

PK,

Second order rate constant, liters/mole minute

13.8 12.4 11.8 8.3

0.306 0.345 3.1 Too slow

The relative reactivity of the HS and OH groups was determined in an elegant way by Smith and Friedrich (56) by allowing 1 mole of phenyl isocyanate to react with 1 mole of 2-mercaptoethanol. In agreement with expectations, the uncatalyzed or the acid-catalyzed reaction lead to O-urethanes (I) while the base-catalyzed reaction yielded the S-urethanes (11). RNCO

+ HOCzH4SH
zH4SH (I)

RNHCOSC2H40H (11).

The synthesis of polythiolcarbamates from methylene-bis-(4-phenylisocyanate) and 1,6-hexanediisocyanate and from various dithiols in the presence of tri-n-propylamine and 1,4-diazabicyclo(2,2,2)octane was described by Dyer and Osborne (57).

F. MISCELLANEOUS REACTIONS OF ISOCYANATES Substituted ethyl carbanilates and aryl isocyanates react at elevated temperatures (above 130”) to form an equilibrium mixture of the starting materials and a,y-diarylallophanates according to (24, 25) RCaHiNCO

+ R’C~HINHCOOR‘’

RCaH4NHCON(CaH4R’)COOR’’.

At lower temperature the equilibrium is shifted in favor of the allophanate but the rate of equilibrium decreases and a t room temperature no reaction occurs in 2 weeks. Metal carboxylates (naphthenates, 2-ethylhexanoates, linoresinates of Pb, Co, Cu, Mn, Fe, Cd, V, Zn) were found to accelerate this reaction very appreciably. Particularly active are the Iead and cobalt salts which caused complete conversion at room temperature in a few hours. All the metal salts with the exception of the zinc salts also catalyze the

440

A. FARKAS AND G . A. MILLS

trimerization of the isocyanates to isocyanurates. However, this latter reaction is very much slower than the allophanate formation. Diary1 allophantes as indicated by the above equation have been prepared with: R = H, O-CH3, p-C1, p-CH,; R' = H, p-CHI, m-OCH,; R" = C2H6, CaH6. The allophanate formation fails to occur if the carbanilate or the aryl isocyanate is replaced by ethyl carbamate or by ethyl isocyanate, respectively. In connection with their study of the water-isocyanate reaction, Shkapenko et al. (51) found that at 80" o-tolyl isocyanate reacts with symditolylurea in dioxane at 0.2 M concentration and forms biuret a t approximately the same rate as that of the reaction between the isocyanate and water takes place. According to Saunders (58), the reaction between isocyanates and ureas follows second-order kinetics. The reactions with urethanes are much slower and do not comply with the second-order law. In view of Kogon's work (24, 25), the slowing down of the reaction with urethanes can be ascribed to the backward reaction leading to the establishment of the equilibrium : RNCO

+ R'NHCOOR" S RNHCONCOOR". I

R'

N-Ethylmorpholine, a relatively weak isocyanate catalyst, was found to be ineffective for these reactions. Other t-amines appear to have little effect. The reaction rates of two isocyanates with two ureas and with ethyl phenyl carbanilate (59) are shown in Table XXIV. TABLE XXIV Relative Rates of Reactions of Aromatic Isocyanates and Substituted Ureas and Urethanes5 k X lO4,liters/mole second

Isocyanate Phenyl 0-Tolyl Phenyl o-ToIYI (1

Urea

CEHSNHCONH~ (CEH~NH)ZCO CeH6NHCONHz (CEH~NH)ZCO

-

-

Urethane

100"

140"

-

32 9.9

23

CaHsNHCOOEt CaH6NHCOOEt

11

3.6 -

48 18 8.2 0.2

0.1

Approximately 0.01 M in each reactant in dichlorobenzene.

The first step of the reaction of isocyanates with acids is the formation of a mixed anhydride (60) which can only be isolated occasionally and usually

CATALYTIC EFFECTS I N ISOCYANATE REACTIONS

441

decomposes either to the amide and COzor to the symmetrical urea and the anhydride of the acid (61).On further reaction, the latter two products give again the amide and urea (62):

+

RNCO R’COOH -+ RNHCOOCOR’ RNHCOOCOR’ + RNHCOR’ COz 2RNHCOOCOR’ -+ RNHCONHR R’COOCOR’.

+

(A)

(B)

+

(C)

An unusual example of this type of reaction is that between benzoic acid and phenyl isocyanate in the presence of dimethyl sulfoxide, the products of which are sym-diphenylurea, carbon dioxide, and a-benzoyldimethyl sulfide (63). Hydrazines and hydrazides have been known for some time to react with aromatic and aliphatic isocyanates to give the expected products. Recently this type of reaction was used for the preparation of condensation polymers from diisocyanates and dihydrazides and hydrazine (64). Oximes react with isocyanates according to R=NOH OCNR’ + R=NOCONHR’. In the case of hydroxylamine, the reaction takes place preferentially with the amino group and then with the hydroxyl group. Campbell (65) used the reaction between diisocyanates and dioximes for the preparation of polymers. Of the three catalysts tried, triethylamine proved to be generally applicable and the most effective one.

+

IV. Applications The discussion in previous sections deals with the chemistry of individual reactions which are important in isocyanate catalysis, In commercial practice, particularly in polyurethane foam manufacture, the balance of these reactions as well as their absolute velocities becomes especially important. Gas formation and polymer growth must be balanced or matched so that the gas is trapped efficiently and the polymer has the right strength at the end of the gas evolution period to maintain its volume without collapse. Also, reactions which occur in “curing” are important since these serve to complete linkages in the foam which are necessary to develop maximum strength and minimize compression set. Other significant factors are odor, vapor pressure, solubility, toxicity, and cost. These considerations, as well as the colloid chemistry of nucleation phenomena, bubble stability and the rheology of the polymer system have been given excellent treatment by Saunders (7’) and Dombrow (6). Studies (66, 67) have been directed to improvement of foam properties by selective choice of type and amount of catalyst. In practice, toluene diisocyanate is used universally and the resin is usually a polypropylene polyol. For low density foams, the equivalent ratio of diisocyanate to polyol is normally higher than 1:l. Cross-linking is

442

A. FARKAS AND G . A. MILLS

introduced for the most part by use of branched polyols. The average weight per branch point is in the order of 400-700 for “rigid” foams and 2500-20,000for “flexible” foams (7, 68). Branching may also be developed by allophanate or biuret formation. One of the most significant advances in commercial practice occurred through the development of highly active catalysts about 1958. Prior to that time, it has been the custom to react the isocyanate with all or part of the polyol to form a “prepolymer.” Subsequently, this prepolymer was reacted with water and the balance of polyol. The advent of the very active catalysts-1,4-diaza-(2,2,2)-bicyclooctane (69, 70) and tin compounds (32, 68)-provided a means to practice the so-called “one-shot” technique whereby the prepolymer step is avoided (71).Thus, all the reactants and catalyst can be mixed and formed into desired polyurethane in one step with substantial economies. As may be expected, many factors affect polyurethane properties which are important in practical applications. For example, films, adhesives, and foamed resins can be obtained by suitable formulation, particularly by variation in the nature of the polyol (72). Naturally, the effect of catalysts on polyurethane properties has been recognized to be particularly significant. Alzner and Frisch (66) reported in 1957 on improvement of foam properties by the selective choice of type and amount of catalyst. Rates of curing were measured at 70” and 120”. These were determined by measurement of tensile, modulus, elongation, compression set, deflection load, dewiity, open cells, and resistance to aging a t high humidity. Optimum catalyst concentrations were established for the six amines tested in a standard formulation employing a polyether prepolymer. The activities of these catalysts was found to be proportional to their base constant (Kb).From a consideration of the kinetics of the reactions for liberation of COz and chain extension, a method was proposed for estimating, from kinetic measurements, the optimum catalyst concentration for a given system. Similarly, Gmitter et al. (67) reported significant improvements in cushioning properties can be attained by proper choice and use of catalysts. An adipate polyester was used and special attention was given to resiliency, effect of flexing, and resistance to aging. Certain of the 40 amines tested displayed wide variation in the ability to impart desirable properties to the foam so that the catalysts could be classified into groups. The resiliency of the foams produced with one group of catalysts decreased markedly with flexing or with aging and not with others. It was not possible to relate catalyst structure to class exactly although the three beta amino ethanol catalysts tested all showed a marked improvement in humid aging stability. Considerable interest has been shown in performing laboratory tests

CATALYTIC EFFECTS IN ISOCYANATE REACTIONS

443

which will indicate catalyst activity in actual commercial foam production. Bailey et al. (73) measured catalytic activity by determining the per cent unreacted diisocyanate as a function of time for the reaction of toluene diisocyanate with diethylene glycol adipate polyester. They showed that the rate curves can be used to predict the activity of the catalyst in urethane applications. With the exception of one catalyst (dimethylethanolamine) showing unexpectedly high activity in coating applications, only one other real anomaly occurred. This was the attempt to use cobalt naphthenate as a catalyst and probably can be explained by solvent effects. The technique of measuring maximum heat rise during reaction has been utilized as a measure of catalytic activity. In contrast to many kinetic measurements made in dilute solution, relatively concentrated solutions can be employed which more nearly represent practical foam manufacture. The isocyanate-water and isocyanate-hydroxyl reaction can be studied separately. This has been used extensively under the designation of the Wolfe test (74).The results of these tests demonstrate the very high activity of triethylenediamine for the isocyanate-water reaction. Likewise, this catalyst is the most active amine catalyst for the isocyanate-hydroxyl reaction, although less active than tin compounds. This test can also be used effectively for testing mixtures of catalysts. This is in accord with present commercial practice of using an amine-usually triethylenediamine -and a tin compound to achieve optimum results. While the fundamental chemistry of isocyanates was well established by Hofmann over a century ago, it was only recently that isocyanates have assumed industrial importance. The properties of these polymers of special excellence are abrasion resistance and, important in coatings, resistance to weathering. The most extensive development, however, has been in foam manufacture, the greatest growth being in the U.S.A. in the later 1950’s. In this application, the high strength to bulk density has been a key economic factor. A second key factor has been the relative ease and simple equipment for manufacture of polyurethane foams which is particularly adaptable to mass production methods. The use of foams has therefore grown very rapidly in insulation, automotive and furniture cushioning, packaging, filters, clothing interliners, and even smokes. The widespread use of rigid polyurethane foams in building construction offers an exceptionally large field for future application. Likewise, improvements being made in polyurethane elastomers and coatings promises substantial future growth in that direction.

REFERENCES 1. Saunders, J. H., and Slocombe, R. J., Chem. Rev. 43, 203 (1948). 2. Arnold, R. G., Nelson, J. A., and Verbanc, J. J., Chem. Rev. 67, 47 (1957).

444

A. FARKAS AND G . A. MILLS

3. Saunders, J. H., and Frisch, K., “Polyurethanes.” Interscience, New York (in press). 4u. Baker, J. W., and Holdsworth, J. B., J. Ckem. SOC.p. 713 (1947). 4b. Baker, J. W., and Gaunt, J., J . Chem. SOC.p. 9 (1949). 4c. Baker, J. W., and Gaunt, J., J . Chem. SOC.p. 19 (1949). 4d. Baker, J. W., Davies, M. M., and Gaunt, J., J . Chem. Soc. p. 24 (1949). 4e. Baker, J. W., and Gaunt, J., J . Chem. SOC.p. 27 (1949). 4f. Baker, J. W., and Bailey, D. N., J . Chem. SOC.pp. 4649, 4663 (1957). 5. Dyer, E., Taylor, H. A., Mason, S. J., and Samson, J., J. Am. Chem. SOC.71, 4106 (1949); Dyer, E., and Glenn, J. F., ibid. 79, 366 (1957); Dyer, E., Glenn, J. F., and Lendrat, E. G., J . Org. Chem. 26, 2919 (1961). 6. Dombrow, B. A., “Polyurethane.” Reinhold, New York, 1957. Y. Saunders, J. H., Rubber Chem. and Technol. 33, 1259, 1293 (1960). 8. Hofmann, A. W., Jahresber, 1868, 349. 9. Shashoua, V. E., J. Am. Chem. SOC.81, 3156 (1959). 10. Shashoua, V. E., Sweeney, W., and Tieta, R. F., J. Am. Chem. SOC.82, 866 (1960). 11. Snape, H . L., J. Chem. SOC.49, 254 (1886). 12. Lockwood, W. H., “Supplemental Report on Application of Diisocyanates,” Fiat Final Rept. No. 1301 (September 15, 1947). 13. Lyons, J. M., and Thomson, R. H., J . Chem. SOC.p. 1971 (1950). 14. Balon, W., and Stallman, O., U. S. Patent 2,683,144. 16. Blair, J. S., and Smith, G. E. P., Jr., J . Am. Chem. SOC.66, 907 (1934). 16. Balon, W. J., Laugerak, E. O., Simons, D. M., and Stallman, O., Am. Chem. SOC. Div. of Paint, Plastics, and Printing Ink Chem., preprints, Am. Chem. Soc. meeting, September, 1956, Symposium on Isocyanate Polymers, Atlantic City, New Jersey, p. 67. 17. Raiford, L. C., and Freyermuth, H. B., J. Org. Chem. 8, 230 (1943). 18. Brown, C. J., J. Chem. SOC.p. 2931 (1955). 19. Barthel, E., Kehr, C. L., Langerak, E. O., Pelley, R. L., and Smelte, K. C., Am. Div. of Paint, Plastics, and Printing Ink Chem., preprints, Am. Chem. Chem. SOC. Soc. meeting, September, 1956, Symposium on Isocyanate Polymers, Atlantic City, New Jersey, p. 78. 20. Bayer, O., BIOS Final Rept. No. 719 (July, 1946). 21. Hofmann, A. W., Ber. 3, 765 (1870); 4, 246 (1871). 22. Frentzel, W., Ber. 21, 411 (1888). 23. Britain, J. W., and Gemeinhardt, P. G., J. Appl. Polvmer Sci. 4, 207 (1960). 24. Kogon, I. C., J . Org. Chem. 24, 83 (1959). 25. Kogon, I. C., J . Am. Chem. SOC.78, 4911 (1956). 26. Jones, J. E., and Savill, N. G., J . Chem. SOC.p. 4392 (1957). 27. Burkus, J., U. S. Patent 2,979,485. 28. Tsueuki, R., Ichikawa, K., and Kase, M., J . Org. Chem. 26, 1009 (1960). 29. Dyer, E., and Read, E., in press. 30. Reilley, C. B., M.S. Thesis, Univ. of Cincinnati, Cincinnati, Ohio, 1955. 31. Morton, M., and Deisz, M. A., Am, Chem. SOC.Div. Paint, Plastics, and Printing Ink Chem., paper No. 34, Am. Chem. SOC.meeting, Atlantic City, New Jersey, September, 1956. 32. Hostettler, F., and Cox, E. F., Ind. Eng. Chem. 62, 609 (1960). $3. Ephraim, S., Woodward, A. E., and Mesrobian, R. B., J. Am. Chem. SOC.80, 1326 (1958).

CATALYTIC EFFECTS I N ISOCYANATE REACTIONS

445

S4a. Leukart, R., Ber. 18, 873 (1885). 34b. Vallee, C., Ann. chim. phys. [7] 16, 331 (1908). S4c. Michael, A., and Cobb, P. H., Ann. 64, 363 (1908). 34d. Claisen, L., Ann. 82, 418 (1919). S4e. Farinholt, L. H., Harden, W. C., and Twiss, D., J . Am. Chem. SOC.66,3383 (1933). 34f. Dieckmann, W., Hoppe, J., and Stein, R., Ber. 37, 4627 (1904). 349. Tarbell, D. S., and Kincaid, J. F., J . Am. Chem. Soc. 62, 728 (1940). 35. French, H . E., and Wirtel, A. F., J . Am. Chem. SOC.48, 1736 (1926). 36. Tarbell, D. S., Mallatt, R. C., and Wilson, J. W., J . Am. Chem. SOC.64, 2229 (1942). Rothrock, H. S., U. S. Patent 2,374,136. Burkus, J., J. Org. Chem. 26, 779 (1961). Farkas, A,, and Flynn, K. G., J . Am. Chem. Sac. 82, 642 (1960). Pestemer, M., and Lauerer, D., Angew. Chem. 73, 612 (1960). Kuehle, E., and Wegler, R., Ann. 616, 183 (1958). 48. Farkas, A., and Mills, G. A., unpublished results. 43. Brown, H. C., and Eldred, N. R., J . Am. Chem. SOC.71, 445 (1949). 44. Weisfeld, L. B., J. Appl. Polymer Sci. 6, 424 (1961). 46. Bailey, M. E., McGinn, C. E., and Spaunbaugh, R. G., Am. Chem. SOC.Div. Paint, Plastics, and Printing Ink Chem., paper No. 36, Am. Chem. SOC.meeting, Atlantic City, New Jersey, September, 1956. 46. O’Brien, J. L., and Pagano, A. S., Second Delaware Valley Regional Meeting, Am. Chem. SOC.,Philadelphia, Pennsylvania, 1958. 47. Slocombe, R. J., U. S. Patent 2,620,349. 48. Tazuma, J. J., and Latourette, H. K., Am. Chem. SOC.,Div. Paint, Plastics, and Printing Ink Chem., paper No. 35, Am. Chem. Soc. meeting, Atlantic City, New Jersey, September, 1956. 49. Naegeli, C., Tyabji, A., Conrad, L., and Litwan, F., Helv. Chim. Acta 21, 1100 (1938). 60. Naegeli, C., Tyabji, A., and Conrad, L., Helv. Chim. Acta 21, 1127 (1938). 51. Shkapenko, G., Gmitter, G. T., and Gruber, E. E., Ind. Eng. Chem. 62,605 (1960). 68. Davis, T. L., and Ebersole, F., J . Am. Chem. Sac. 66, 885 (1934). 53. Craven, R. L., Am. Chem. SOC.,Div. Paint, Plastics, and Printing Ink Chem., paper No. 33, Am. Chem. SOC.meeting, Atlantic City, New Jersey, September, 1956. 54. Iwakura, Y., and Okada, H., Can. J . Research 38, 2418 (1960). 55. Gordy, W., and Stanford, S. C., J . Am. Chem. SOC.62, 497 (1940). 56. Smith, J. F., and Friedrich, E. C., J . Am. Chem. SOC.81, 161 (1959). 57. Dyer, E., and Osborne, D. W., J . Polymer Sci. 47, 361 (1960). 58. Saunders, J. H., Rubber Chem. and Technol. 32, 337 (1959). 59. Bennet,, W. B., Saunders, J. H., and Hardy, E. C., Annual Meeting of the Alabama Academy of Sciences, Tuscaloosa, Alabama, April 2, 1954. 60. Dieckmann, W., and Breest, F., Ber. 30, 3052 (1906). 61. Naegeli, C., and Tyabji, A., Helv. Chim. Acta 17, 931 (1934); 18, 142 (1935). 68. Agre, C. L., Dinga, G., and Pflaum, R., J . Org. Chem. 20, 695 (1955). 63. Sorenson, W. R., J . Org. Chem. 24, 978 (1959). 64. Campbell, T. W., Foldi, V. S., and Farago, J., J . A p p l . Polymer Sci. 2, 155 (1959). 66. Campbell, T. W., Foldi, V. S., and Parrish, R. G., J . Appl. Polymer Sci. 2, 81 (1959). 66. Alzner, B. G., and Frisch, K. C., Ind. Eng. Chem. 61, 715 (1959). 37. 38. 39. 40. 41.

446

A. FARKAS AND G . A. MILLS

67. Gmitter, G. T., Gruber, E. E., and Joseph, R. D., SPE Journal 16, 957 (1959). 68. Bolin, R. E., Saabat, J. F., Cote, R. J., Peters, E., Gemeinhardt, P. G., Morecroft, A. S., Hardy, E. E., and Saunders, J. H., J . Chem. & Eng. Data 4, 261 (1959). 69. Farkas, A., Mills, G. A., Erner, W. E., and Maerker, J. B., Ind. Eng. Chem. 51, 1299 (1959); J . Chem. & Eng. Data 4, 334 (1959). 70. Orchin, M., U. S. Patent 2,939,851. 71. Erner, W. E., Farkas, A., and Hill, P. W., Modern Plastics 37, 107 (1960). 72. Heiss, H. L., Saunders, J. H., Morris, M. R., Davis, B. R., and Hardy, E. E., Ind. Eng. Chem. 46, 1498 (1954). 73. Bailey, M. E., Khawam, A., and Toone, G. C., Am. Chem. Soc., Div. Paint, Plastics, and Printing Ink Chem., paper No. 37, Am. Chem. SOC.meeting, Atlantic City, New Jersey, September, 1956.

74. Wolfe, H. W., Jr., “Catalyst Activity in One-Shot Urethane Foam,” du Pont Company Bull. (March 16, 1960).

Author Index Numbers in parentheses are reference numbers and are included to assist in locating references when the authors’ names are not mentioned in the text. Numbers in italic refer to the page on which the reference is listed. Bellman, R., 356, 391 Bennet, W. B., 440(59), 446 Benson, S. W., 209(6), 210, 355, 356, 359, 362, 363, 376(6), 390 Benton, A. F., 2, 60 Bertocci, U., 123(80), 136 Bertolacini, R. J., 179(35), 190 Bethe, H. A., 81(47), 85(47), 86(47), 134, 136 90(47), 136 Allen, R. H., 360, 391 Bharucha-Reid, A. T., 356, 391 Allison, S. K., 98(62), 136 Bielanski, A., 30(61,63), 39, 61, 66, 63 Allsopp, H. J., 25, 61 Bigeleisen, J., 126(81), 136 Alzner, B. G., 441(66), 442, 446 Bilous, O., 360, 392 Amphlett, C. B., 32(72), 62 Birkhoff, G., 276(12), 316, 323(16), 364 Amundson, N. R., 357, 391 (72), 366(75), 367(76), 368(77), 370 Anderson, J. S., 23, 61 (781, 371(79), 381(83), 390, 396 Apker, L., 101(66), 136 Blair, J. S., 396(15), 397(15), 444 Arnett, R. L., 286(14), 390 Blench, E. A., 2(2), 60 Arnold, R. G., 394, 396, 397, 443 Bloch, H. S., 158(20), 163(20), 170, 189 Aschenbrand, L. M., 127(86), 136 Ashkin, J., 81(47), 85(47), 86(47), 90 Block, J., 36, 37, 39, 66 Bonnet, J. B., 75(43), 136 (471, 136 Booth, G. W., 358, 391 B Boreskov, G. K., 34, 66 Bailey, D. N., 394(4f), 432, 433, 444 Borissov, M., 44, 63 Bailey, M. E., 420, 425(45), 443, 446, 446 Boudart, M., 36(93), 66 Bak, T. A., 359, 391 Braun, R. M., 286(14), 390 Baker, J. W., 394, 404, 406, 409, 411, Breest, F., 440(60), 446 414(4d), 415, 432, 433, 437, 444 Brennan, D., 22(38), 34(82), 61, 66 Baker, R. W., 175(31), 178(31), 179(31), Brennan, H. M., 179(35), 190 Britain, J. W., 397(23), 399(23), 419, 190 Balaceanu, J. C., 356(46), 391 422, 427, 444 Ballandine, A. A,, 56(15,16), 77, 134 Brooks, H., 86(52), 91(52), 92(52), 136 Balon, W. J., 396(14,16), 444 Brown, C. J., 396(18), 444 Barry, T. I., 24(48,49), 25, 31(48), 41 Brown, H. C., 417, 446 (48), 45, 46(48), 47, 48(127), 49 Brownell, G., 81(49), 83(49), 85(49), 88 (491, 96(49), 136 (1281, 61, 63, 56(20), 76, 119, 134 Buckingham, R. A., 376(80), 396 Barthel, E., 396(19), 397(19), 444 Baumbach, H. H., 22(39), 27(39), 61 Bull, H. I., 2(2), 60 Burlein, T. K., 94, 136 Bayer, O., 397(20), 444 Burhop, E. H., 90(57), 136 Beebe, R. A., 24, 61 447

A Acrivos, A., 357, 391 Agre, C. L., 441(62), 4.66 Aigrain, P., 30(65), 38(65), 66 Alberman, K. B., 23(43), 61 Alberty, R. A., 356, 391 Allen, A. O., 56(13,14), 75(13,45), 76,

448

AUTHOR INDEX

Burkus, J., 397(27), 414, 417, 444, 446 Burtt, B. P., 62(33), 126(33), 136 Burwell, R. L., Jr., 286, 287, 289, 390

C Cabrera, N., 22(34), 61 Caffrey, J. M., 56(13), 75(13), 76, 134 Calvert, J. G., 47, 63 Cameron, J. F., 89, 136 Campbell, T. W., 441(64,65), 445 Caratheodory, C., 345, 390 Carrington, T., 357, 358, 3.91 Caylord, N. C., 74(40), 136 Chandrasekhar, S., 356, 391 Chessick, J. J., 22(37), 61 Chien, J , 363, 398 Chrenko, R. M., 49(130), 63 Ciapetta, F. G., 157, 158(12), 175, 178 (31), 179(31), 180(39), 189, 190 Cimino, A., 38, 39, 63 Cippolini, E., 39, 63 Claisen, L., 409(34d), 446 Cobb, P. H., 409(34c), 446 Coekelbergs, R. F., 56(8,9,10), 57(9), 58(24), 60(25), 62(8,9), 66(8), 67 (9), 68(9,24), 70(10), 111(25), 114 (24), 134 Cohn, G., 181, 182, 190 Coleman, B. D., 353, 391 Compton, A. H., 98(62), 136 Connor, J. E., Jr., 179(37), 190 Conrad, L., 427(49,50), 431(50), 446 Courant, R., 344(24), 990 Courtois, M., 21, 61 Coussemant, F., 356(46), 39f Cote, R. J., 442(68), 446 Cox, E. F., 403, 409(32), 419(32), 421, 422(32), 442(32), 444 Cox, R. T., 356, 991 Craven, R. L., 432, 446 Crucq, A., 58(24), 61(28), 114(24), 134

D Dalmai, G., 114(73), 136 Damkohler, G., 147(6), 189 Darby, P. W., 29(58), 30(58), 61 Davies, M. M., 394(4d), 406(4d), 414 (4d), 444 Davis, B. R., 442(72), 446 Davis, T. L., 432, 4-45

Dawson, J. K., 66(35), 136 De, K. S., 34(81), 62 de Boer, J. H., 34, 66, 140, 189, 356, 391 Decot, J., 56(10), 70(10), 134 DeDonder, T., 341, 390 Degols, L., 61(28), 134 DeGroot, S. R., 340, 352, 353, 390, 391 Deisz, M. A,, 402, 429, 444 Dell, R. M., 5(16), 8(16), 16, 20(16), 22 (361, 32(70), 33(70), 60, 61, 62 Denbigh, K. G., 352, 363, 391, 396 Deren, J., 30(61,63), 39(101), 61, 62, 63 Derry, R., 56(57), 61 Devries, A. E., 75(45), 135 Dieckmann, W., 409(34f), 440(60), 446 Dienes, G. J., 87(55), 88(55), 91(55), 95(55), 136 Dinga, G., 441(62), 446 Dirac, P. A. M., 355, 391 Dobres, R. M., 175(31), 178(31), 179 (31), 190 D’Olieslager, J . F., 60(27), 61 (271, 194 Dolle, I,., 62(30), 126(30), 184 Dombrow, B. A., 394, 441, 444 Donaldson, G. R., 152(10), 157, 175(10), 189

Dondes, S., 62(31), 66, 126(31,34), 194, 136

Dorling, T. A., 55(7), 115(7), 134 Dorroselskaia, N. P , 56(15,16), 77(15, 161, 134 Dowden, D. A,, 4, 5(8), 24, 26(8), 30 (62), 33, 35, 60, 61, 62 Drake, L., 57(22,23), 134 Drikos, A,, 121(76), 136 Drushel, H. V., 128(89), 136 Dry, E. M., 36, 37, 39(96), 63 Dubar, L., 27, 61 Dunwald, H., 22(39), 34(39), 61 Duflo, M., 56(18), 79, 194 Dugas, C., 30(65), 38(65), 62 Dunford, N., 357(52), 391 Dyer, E., 394, 398, 434, 437(5), 439, 444, 446

E Ebersole, F., 432, 446 Edwards, C . G., 66, 196 Egloff, G., 158, 163(20), 189 Ehrenberg, W., 192, 801

449

AUTHOR INDEX

Eibner, A., 40, 63 Einstein, J., 355, 391 Eischens, R. P., 17, 21, 61 Eldred, N. R., 417, 446 Emmett, P. H., 168(24), 189 Engell, H. J., 23, 30(65), 32, 37(71), 38 (651, 39, 61, 62 Ephraim, S., 407, 444 Erner, W. E., 442(69,71), 446 Eschard, F., 356(46), 391 Evans, R. D., 82, 83 Evans, R. S., 85(51), 136 Eyring, H., 356, 391

F Faddeeva, V. N., 376(81), 392 Fan, H. Y., 107(70), 136 Farago, J., 441 (64), 446 Farinholt, L. H., 409(34e), 445 Farkas, A,, 415, 416, 429, 442(69,71), 446, 446 Farnsworth, H. E., 191(1), 201 Feeler, M., 75(44), 136 Feller, W., 356, 391 Fensham, P. J., 24(46), 61 Ferguson, I. F., 24, 61 Field, E., 75(44), 136 Filey, E. F., 121(79), 136 Fischer, F., 138(1), 189 Flynn, K. G., 415, 416(39), 429, 446 Foldi, V. S., 441(64,65), 446 Fowler, R. H., 365(73), 391 Franklin, J. L., 360, 392 French, H. E., 409, 446 Frenckel, J., 101(651, 136 Frennet, A., 56(10), 70(10), 134 Frentzel, W., 397(22), 444 Freyermuth, H. B., 396(17), 444 Friedlander, H. N., 74(41), 136 Friedman, L., 126(81), 136 Friedrich, E. C., 439, 446 Frisch, K. C., 394, 441(66), 442, 444, 446 Frost, A. A,, 209(5), 355, 356, 390 Fruton, J. S., 186(42), 190 Fujita, Y., 41, 42, 43, 47, 63

G Gadsby, J., 362, 392 Gale, R. L., 5(18), 20(18), 23(18), 26 (18), 27(18), 38(100), 60, 63

Garcia de la Banda, J. F., 30(62), 62 Garner, W. E., 2, 3, 4, 5 , 6, 7, 8, 17(15), 24, 26(5,8,15), 27(13), 29(57), 30 (62), 45, 60, 61, 52, 121(77,78,79), 136

Gaunt, J., 394(4b, 4c, 4e), 404, 406, 411, 414(4d), 415, 444 Gemeinhardt, P. G., 397(23), 399(23), 419, 422, 427, 442(68), 444, 446 Germer, L. H., 192(3), 194(5,6), 197(6), 200(6), 201 Gevantman, L. H., 128(87), 136 Gibson, E . J., 55(7), 115(7), 134 Giraud, A., 356(46), 391 Glang, R., 32(71), 37(71), 39, 62 Glazounov, P., 56(15), 77(15), 134 Glenn, J. F., 394(5), 434, 437(5), 444 Gmitter, G. T., 428(51), 440(51), 441 (677, 442, 446, 446 Goodeve, C . F., 49(129), 63 Gordy, W., 439(55), 446 Gosselain, P. A., 56(8,9), 57(9), 58(24), 60(25), 62(8,9), 66(8), 67(9), 68(9, 241, 111(25), 114(24), 134 Grant, F. A., 40(105), 63 Gray, T., 121(78), 136 Gray, T. J., 5(13,14), 6(13), 24(14,46, 47), 27, 29(56,57,58,59), 30(58), 60, 61

Greaves, J. C., 33, 62 Greensfelder, B. S., 158, 163(21), 170, 189

Gruber, E . E., 428(51), 440(51), 441 (67), 442(67), 446, 446 Gruver, J. T., 75(42), 136 Gundermann, J., 22(39), 27 (39), 61

H Haag, W. O., 247, 259, 390 Haayman, P. W., 36(88), 61 Haber, J., 30(61,63), 39(101), 49(131), 61, 62, 63 Haensel, V., 152(10), 157, 175(10), 189 Hagedoorn, A. L., 98(63), 99, 136 Haissinsky, M., %(I), 56(18), 79, 134 Hall, W. H., 2(2), 60 Hallon, J. V., 128(89), 136 Hamilton, W. M., 286, 287, 289, 390 Harden, W. C., 409(34e), 446

450

AUTHOR INDEX

Hardy, E. C., 440(59), 442(68,72), 446, 446

Harper, E. A., 23, 61 Harteck, P., 62(31), 66, 126(31,34), 134, 136

Hartman, C. D., 192(3), 194(5,6), 197 (61, 200(6), Harwood, J. I., 93(60), 136 Hattrnan, J. B., 157(11), 162(11), 175

(Ill, 189 Hauffe, K., 22(39), 23, 27(39), 30(65), 32, 35, 36, 37(71), 38(65), 39, 45 (119), 61, 62, 63, 126(85), 136 Haussner, H., 93(60), 136 Hayward, D. O., 22(38), 61 Hearon, J., 359, 391 Heiland, G., 40(104), 41(104, log), 46 (104), 63 Heinemann, H., 157, 158(13), 162, 175 ( l l ) , 189 Heinrich, G., 75(43), 136 Heiss, H. L., 442(72), 446 Hellin, M., 356(46), 391 Hendee, E. D., 186, 187, 190 Herbo, C., 362(67), 392 Herington, E. F. G., 139(3), 169, 189 Hicks, M., 363(71), 396 Hill, P. W., 442(71), 446 Hindin, S. G., 158, 171, 189 Hine, G. J., 81(49), 83(49), 85(49), 88 (491, 96(49), 136 Hinshelwood, C. N., 362(66), 392 Hofer, L. J. E., 362(64), 392 Hofmann, A. W., 395, 397(21), 444 Hogeboom, G. H., 187(44), 190 Holdsworth, J. B., 394(4a), 404(4a), 409, 437, 444 Hoppe, J., 409(34f), 446 Horiuti, J., 362, 392 Hostettler, F., 403, 409(32), 419(32), 421, 422(32), 442(32), 444 Hunter, J . B., 157, 158(12), 189 Hurwitz, H., 175(30), 189

I Ichikawa, K., 397(28), 399(28), 444 Imelik, B., 114(73), 136 Iwakura, Y., 437, 446

J Jacobi, R. B., 123(80), 136 Jacobson, N., 345(26), 391 James, H. M., 106, 136 Jennings, T. J., 23(40), 61 Johnston, H. S., 126(SZ), 136 Jones, H. A., 17, 61 Jones, J. E., 397(26), 399, 444 Joseph, R. D., 441(67), 442(67), 446 Jost, W., 369 Juliens, J., 56(9), 57(9), 62(9), 67(9), m(9), 134 Jungers, J. C., 60(27), 61(27), 134, 356, 391

K Kanev, S., 44, 63 Kaplan, W., 209(4), 390 Kase, M., 397(28), 399(28), 444 Kehr, C. L., 396(19), 397(19), 444 Keier, N. P., 36, 38, 63 Kennedy, D. R., 48, 49(128), 63 Kennedy, P. J., 108(71), 136 Keulemans, A. I. M., 175, 182, 189, 190 Khawam, A., 443(73), 446 Kincaid, J. F., 409, 446 Kinchin, G. H., 81(48), 85(48), 86(48), 90(48), 91(48), 96(48), 136 Kingman, F. E. T., 2, 60 Kircher, J. F., 62(33), 126(33), 136 Kirsch, F. W., 157(11), 162(11), 175(11), 189

Kitchener, J. A., 49(129), 63 Klarke, R. W., 55(7), 115(7), 134 Klemperer, D. F., 22(36), 61 Klier, K., 16, 60 Klingman, W. H., 45, 47, 63 Koch, E., 22(39), 27(39), 61 Koefoed, J., 360, 392 Koehler, J. S., 87(54), 88(54), 90(54), 91(54), 92(54), 94(54), 196 Koster, W., 22, 61 Kogan, S. M., 42, 63 Kogon, I. C., 397(24,25), 398, 399(24, 25), 439(24,25), 440, 444 Kohn, H. W., 55(2,3,4), 115(2,3,4), 116, 134

Kokes, R. J., 26, 61, 168, 189

AUTHOR INDEX

Komuro, I., 47, 63 Kraus, G., 75(42), 136 Krusemeyer, H. J., 30(65), 38(65), 62 Krustinsons, J., 5, 60 Kubokawa, Y., 30(60), 45(122), 61, 63 Kuehle, E., 416(41), 446 Kutseva, L. N., 38, 63 Kwan, T., 41, 42, 43, 47, 63

45 1

Maktkishima, S., 33(76), 62, 55(6), 115(6), 134

Mallatt, R. C., 409(36), 425(36), 446 Manes, M., 362, 392 Marcellini, R. P., 15(24,25), 16(24,25), 20(25), 60 Margenau, H., 376(82), 392 Mark, H. F., 74(40), 136 Mason, S. J., 394(5), 437(5), &$ 1 Massey, H. S., 90(57), 136 Lago, R. M., 183(28a), 189, 247, 259, 390 Mastel, B., 94, 136 Matsen, F. A., 360, 392 Laidler, K. J., 355, 362, 391, 392 Mechelynck-David, C., 56(11,12), 72, Lampe, F. W., 68, 128(39), 136 128, 134 Lanczoe, C., 210,390 Medved, D. B., 41(111), 63 Lang, W. H., 179(34), 190 Langerak, E. O., 396(16,19), 397(19), Melnick, D. A,, 41(110), 63 Mesrobian, R. B., 407(33), &$ 44.4 Michael, A., 409(34c), 446 Lanyon, M. A. H., 22(35), 61 Mikkalenko, I. E., 56(16), 77(16), 134 Lark-Horowitz, K., 106, 107(70), 136 Miller, W. G., 356, 391 Latourette, H. K., 425, 446 Milliken, T. H., 157(13), 158(13), 189 Lauerer, D., 415, 446 Mills, G. A,, 157, 158, 162(11), 171(18), Law, J. T., 193, 201 175(11), 189, 416(42), 442(69), 446, Lefschetz, S., 209(3), 344, 390 446 Lendrat, E. G., 394(5), 437(5), 444 Molinari, E., 38, 39, 63 Leprince, P., 356(46), 391 Mollwo, E., 40(104), 41(104), 43(116), Leukart, R., 409(34a), 446 46(104), 63 Leum, L. N., 179(37), 190 Montarnal, R., 56(19), 79, 134 Leverenz, H. W., 103(67), 136 Montroll, E. W., 357, 391 Lien, A. P., 179(36), 190 Morecroft, A. S., 442(68), 446 Limido, G. E., 356(46), 391 Morin, F. J., 34, 62 Linnett, J. W., 33, 62 Morrell, J. C., 158(20), 163(20), 189 Litwan, F., 427(49), 446 Morris, M. R., 442(72), 446 Lockwood, W. H., 396(12), 444 Morrison, J. A., 15(23), 16, 60 Long, G., 66(35), 136 Morrison, S. R., 43, 63 Lotka, A,, 363, 392 Morse, J. G., 93(60), 136 Lyons, J. M., 396(13), 444 Morton, M., 402, 429, 444 M Moseley, F., 62, 66, 126(32), 136 Mott, N. F., 22(34), 61 McConnell, J. D. M., 24, 61 Munns, G. W., Jr., 162, 165, 166, 178 McGinn, C. E., 420(45), 425(45), 446 (23), 189 McHenry, K. W., 179, 190 Murphy, G. M., 376(82), 392 Mackende, J., 48, 49(128), 63 Myasnikov, I. A., 40, 44, 47, 63 Mackenzie, N., 33, 62 Myers, C. G., 162, 165, 166, 178(23), 179, McKie, D., 2(2), 60 180, 189, 190 MacLane, S., 276(12), 316, 323(16), 364 (72), 366(75), 367(76), 368(77), 370 N (78), 371(79), 381(83), 390, 392 Naegeli, C., 427, 431, 441(61), 446 Maerker, J. B., 442(69), 446 Natta, G., 138, 189 Maggs, J., 3(6), 45, 60

452

AUTHOR INDEX

Nelson, J. A,, 394(2), 396(2), 397(2), 443 Newman, R., 49(130), 63

Pshezhetskii, S. Y., 40, 44, 63 Putseiko, E. K., 41(108), 53

R

0

Raiford, L. C., 396(17), 444 Rakowski, A,, 140(9), 189, 356, 391 Rallman, K. W., 75(42), 135 Ranc, R. E., 15(25), 16(25), 20(25), 60 Rauch, W. G ,93(60), 135 Read, E., 398, 444 Reilley, C. B., 402(30), 444 Rheaume, L., 33, 62, 126(83), 1S6 Rhodes, J. R., 89, 135 Rice, 0. K., 360,392 Rideal, E. K., 139(3), 169, 189 P Rigamonti, R., 138, 189 Pagano, A. S., 420, 445 Ritchey, W. M., 46,47(124), 53 Page, F. M., 363(71), 392 Ritchie, M., 48, 49(128), 53 Pallade, G. E., 187(44), 190 Ritter, H., 57(23), 134 Pappenheimer, A. M., Jr., 186, 187, 190 Robbins, H , 344(24), 390 Parravano, G., 33, 36, 38, 39, 52, 63, 126 Roberts, J. P., 25, 61 (831, 136 Roberts, L. E. J., 23, 61 Parrish, R . G., 441(65), 445 Roberts, R., 48, 63, 56(20), 76, 119, 134 Pauli, W., 355, 391 Roginskii, S. Z., 20, 21, 36, 38(95), 61, 63 Pearson, A. D., 35, 6.2 Roginsky, S. Z. (see Roginskii, S. Z.) Pearson, P. G., 209(5), 355, 356, 390 Rohrer, J. C., 175(30), 189 Pease, R . S., 81(48), 85(48), 86(48), 90 Romeijn, F. C ,36(88), 52 (48), 91(48), 96(48), 136 Romeo, G., 38, 63 Pelley, R. L., 396(19), 397(19), 444 Romero-Rossi, F., 44(118), 45(118), 46 Pestemer, M., 415, 445 (123), 48, 49(118), 53, 118, 121(75), Peters, E., 442(68), 44G 128, 136 Peterson, T. I., 358, 391 Rossini, F. D., 286(14), 390 Pflaum, R., 441(62), 446 Rothrock, H. S., 409(37), 446 Pierce, C., 57(21), 134 Rudham, R., 5(17,18), 8(17), 11(17), 20 Pimentel, G. C ,286(14), 3BO (18), 22(36), 23(18), 26(18), 27(18), Pines, H., 180(38), 190, 247, 390 60, 61 Piret, E. L., 360, 39.2 RUB, P., 15(24), 60 P i t h , P. M., Jr., 179, 190 Rylander, P. N., 181, 182, 190 Pitzer, K. S., 286(14), 390 Pliskin, W. A,, 17, 21, 61 Polin, R. E., 442(68), 446 Saito, Y., 33(76), 68, 55(6), 115(6), 134 Ponsaerts, E., 60(26), 71(26), fS4 Samson, J., 394(5), 437(5), 444 Pope, D., 55(7), 115(7), 134 Saunders, J. H., 394, 395, 440, 441, 442 Pooovakii. V. V.. 34. 63 (7,68,72), 449, 444, 446, 446 Prater, C. D , 140(5), 147(6), 160, 169, 175, 176, 177(33), 178, 179, 183, 189, Savage, S. D., 29(56,59), 61 Savill, N. G.,397(26), 399, 444 190, 334(18), 336(18), 390 Sazonova, I. S., 36, 38(95), 63 Preve, J., 56(19), 79, 134 Prigogine, I., 205, 352, 362(67), 390, 392 Scarborough, J. B., 275(11), 390 Provoost, F., 56(11), 72(11), 128(11), 134 Scharowsky, E., 45, 63

Oblad, A. G., 157(13), 158(13), 189 O’Brien, J. L., 420, 445 Okada, H., 437, 445 O’Keeffe, M , 6(20), 24(20), 60 Onsager, I,., 205, 351, 365(1), 390 Orchin, M., 442(70), 446 Osborne, D. W., 439, 446 Outer, P., 362(67), 392 Ozeroff, J., 93(59), 135

s

453

AUTHOR INDEX

24(14,15,48), 26(15,18), 27(13,18), Scheibner, E. J., 194(5), 201 30(64), 31(48,66), 32(70), 33(70), 34 Schlier, R.E., 191(1), 201 (80,81),36, 37, 38(100), 39(96), 41 Schneider, W.C., 187(44), 190 (48), 44(118), 45(118), 46(48,123), Schotsmans, L. J., 56(8,9), 57(9), 58 48(118), 49(118,131), 60, 61, 52, 53, (24), 62(8,9),66(8), 67(9), 68(9,24), 105(68), 118,121(75,78,79),128,136 114(24), 134 Swallow, A. J., 128(88), 136 Schuit, G. C. A,, 182, 190 Schwab, G.M., 36, 37, 39, 5d, 121(76), Sweeney, W., 395(10), 398(10), 399(10), 136

Schwartz, J. T., 357(52), 391 Seelig, H.S., 139(35), 190 SegrB, E., 84 Seguin, M., 114(73), 136 Seitz, F.,86(53), 87,88(54), 90(54), 91, 92(53,54), 94(54), 135 Shashoua, V. E., 395, 398, 399(9,10),

444 Shkapenko, G., 428,440,445, 446 Shaw, A. W., 180(38), lli0 Shockley, W., 101(64), 108(64), IS5 Shuler, K.E , 357,391 Sibbett, D.J., 180(39), 190 Simmonds, S.,186(42), 190 Simons, D. M., 396(16), 444 Sinfelt, J. H., 175,189 Slater, J. C., 346,391 Slocombe, R. J., 394, 395(1), 425(47), 443, 445 Sloczynski, J., 30(61,63), 39(101), 51,52, 63 Smelts, K. C., 396(19), 397(19), 444 Smith, G.E. P., Jr., 396(15), 397(15),

444

Smith, J. F., 439,446 Smith, R.L., 169(26), 189, 334,336,390 Snape, H.L., 396(11), 444 Solonitzin, Yu., 41, 42, 46, 63, 114(74),

444

Swegler, E. W., 55(5), 115(5), 184, 158, 169,189 Sutherland, J. W., 56(14), 134 Szabat, J. F., 442(68), 446 Sykes, K.W., 362(66), 399

T Tnft, E., lOl(66), 136 Takaishi, T., 30(65), 38(65), 52 Tammann, G., 22, 51 Tarbell, D.S., 409,425(36), 445 Taylor, E. H., 55(2,3,4), 115(2,3,4), 116,134 Taylor, H. A., 394(5), 43761,444 Taylor, H.S.,2, 17,50, 61 Tazuma, J. J., 425,445 Teichner, S.J., 15, 16,20, 21,50, 51 Terenin, A., 41,42,46, 63, 114(74), 136 Thiele, E. W., 147(6), 189 Thomas, C. L., 158(20), 163(20), 170, 189

Thomas, D. G., 25,30(65), 38(65), 61, 62 Thommen, K. Z., 90(58), 135 Thomsen, J. S.,356,391 Thomson, R.H., 396(13), 444 Tietz, R. F., 395(10), 398(10), 399(10),

444

Tiley, P. F., 5(15), 7, 8(15), 17(15), 22 (36), 24(15), 26(15), 32(70), 33(70), 136 50, 51, 62 Sorenson, W. R., 441(63), 445 Tobin, H., 168(24), 189 Sowden, R. G., 66(35), 135 Spaunbaugh, R. G., 420(45), 425(45), 4i5 Tolman, R. C., 355,365(74), 391, 399 Spiteyne, V. I., 56(15,16), 77(15,16),134 Toone, G. C., 443(73), 446 Toyama, O., 30(60), 51 Stallman, O., 396 (14,16),444 Trapnell, B.M. W., 22(35,38),33,61,68 Stanford, S. C., 439(55), 445 Tropsch, N., 138(1), 189 Stein, R.,409(34f), 445 StGckmann, F., 40(104), 41(104), 46(104), Truesdell, C., 353,391 Truswell, A. E., 62, 126(32), 135 63 Stone, F. S., 5(12,13,14,15,16,17,18),Tselinskaya, T.F., 20,61 397(28), 399, 444 6(13), 7, 8(15,16,17),11(17), 16, 17 Tsueuki, R., (15), 20(16,18), 22(36), 23(18,40), Turner, A,, Jr., 360(58), 391

454

AUTHOR INDEX

Twiss, D., 409(34e), 446 Tyabji, A., 427(49,50), 431(50), 441(61),

Verechtchinski, I., 56(15), 77(15), 134 Verwey, E. J. W., 34, 36, 62 Vesselovsky, V. I., 56(17), 79, 134 Vierk, A. L., 45(119), 63 Vineyard, G. H., 87(55), 88(55), 91(55), 95(55), 136 Voge, H. H., 158, 163(21), 170, 175, 189

Weller, S. W., 158(18), 171(18), 189, 362 (64), 392 Wells, D., 35, 6.9 Wheeler, A., 147(6), 189 White, G. R., 81(50), 94(50), 136 White, T. A., 2, 60 Wiggill, J. B., 259, 390 Wigner, E. P., 359, 891 Wilkowa, T., 30(61), 61 William, P. R., 128(87), 136 Wilson, J. L., 179(35), 190 Wilson, J. W., 409(36), 425(36), 446 Winter, E. R. S., 11, 12, 16, 18, 20, 24 (211, 33, 37, 38, 39, 60, 62, 63, 126 (841, 136 Wirtel, A. F., 409, 446 Wolfe, H. W., Jr., 443, 446 Wolkenstein, Th. (Volkenstein, F. F.), 32(68), 35, 36(68), 42, 62, 63, 113 (721, 117(72), 136 Woodward, A. E., 407(33), 444 Wourtzel, E., 62(29), 134

W

Y

44 V Vallee, C., 409(34b), 446 van der Borg, R. J. A. M., 140, 189, 356, 391

Van Der Venne, M., 56(9), 57(9), 60 (251, 62(9), 67(9), W 9 ) , 111(25), 134

Van Rysselberghe, P., 341(20), 390 Veal, F. J., 3(5), 4, 5(5), 6, 26(5), 60 Verbanc, J. J., 394(2), 396(2), 397(2), 443

Wagner, C., 22, 27, 32(69), 34, 37, 61, 62 Walton, G. N., 123, 136 Wapstra, A. H., 98(63), 99, 136 Ward, T., 4, 5(9,10), 60 Wegler, R., 416(41), 446 Wei, J., 140(5), 169(26), 189, 334(18), 336(18), 343, 344, 348, 890 Weisfeld, L. B., 419(44), 425, 446 Weisz, P. B., 30(65), 38(65), 62, 55(5), 115(5), 134, 147(6,7,8), 148(7,8), 158, 160, 163(19), 169, 171(19), 175, 176, 177(32,33), 178, 179, 183, 189, 190, 334, 390

Yats, L. D., 360(58), 391 Yoneda, Y., 33(76), 62, 55(6), 115(6), 134

Yu, Y. F., 22(37), 61

Z Zeldovich, J. (see Zeldowitsch, J. B.) Zeldowitsch, J. B., 21, 61, 147(6), 189 Zelikoff, M., 127(86), 136 Zettlemoyer, A. C., 22(37), 61 Zwolinski, N. B., 356, 391

Subject Index A Adsorption, heat of, of CO on metallic oxides, 5 of CO on metallic oxides, 6 of oxygen on nickel, 193 Aging mechanisms in reforming, 178 Alphas, 90 Alumina, butene isomerisation over, 247 Amine catalyzed isocyanate reactions, 409 Amines, reaction of isocyanate with, 431 Ammonia synthesis, 70 Aromatic-octane number relationship, 176 Aromatization, of alkylcyclopentanes, 170 of cyclohexane, 169 of methylcyclopentane, 171 Apparatus, electron diffraction, 192 Atom displacements, 86

B Beta irradiation of Germanium, 88 rays, 86 Biochemical reactions, cell dimensions in, 185 Boundary-Layer theory of chemisorption, 30 Bremsstrahlung, 87 Butene, isomerization, 247

C

Catalytic reactions, activity patterns in, 32 Cell dimensions in biochemical reactions, 185 Characteristic vectors, orthogonality relations between, 239 Chemisorption, boundary layer theory of, 30 of CO on oxides, 3 on metallic oxides, 1-50 semiconductivity changes during, 27 Chromia, adsorption of CO on, 3 Chromia-alumina in dehydrogenation of cyclohexane, 139, 169 CO adsorption on metallic oxides, 5 oxidation at low temperatures, 17 oxidation of, 5 Cobalt naphthenate, catalytic activity in isocyanate reactions, 420 oxide, adsorption of CO on, 8 catalytic oxidation of CO on, 17 Complex formation, 6 reaction systems, structure and analysis of, 203-389 equilibrium point in, 343 Conductors, fission fragments produced from, 93 “Coupling” through a side product, 152 Crystal surface study by electron diffraction, 191 Cumene cracking, 183 CuzO, adsorption of CO on, 6 catalytic oxidation of CO on, 17 exchange of heavy oxygen on, 12 semiconductivity changes in oxygen adsorption on, 27 Cyclohexane reactions over Pt-ALO, catalysts, 334 Cyclohexanol dehydration, 77

Catalysis of acids and bases in isocyanate reactions, 425 on metallic oxides, 1-50 pseudo-mass-action systems of, 313 Catalyst activation by preliminary irradiation, 115 under irradiation, 117 D components, physically mixed, 156 for phenylisocyanate dimerisation, Degradation, of radiation energy in 396 solids, 60 scheme of radiation energy in solids, systems, model of multifunctional porous, 145 80 455

456

SUBJECT INDEX

n-Hexane isomerization on catalyst Deutons, 90 mixtures, 159 Diffusion criteria, 153 Hexane dehydrogenation equilibrium, Dodecane hydrocracking over mixed 143 catalysts, 164 n-Dodecene conversion over SiO-ALO3, Homogeneous irradiation in nitrogen fixation, 66 163 N20 radiolysis, 62 E H2-D2exchange in paraffins, 180 Hydrocracking, 165 Equilibria hexane dehydrogenation, 143 of n-decane and n-hexadecane, 163 Equilibrium point in complex reaction of paraffins, 162 systems, 343 selectivity, 169 Electron diffraction, low energy, 191Hydrogenation-dehydrogenation of C6 201 cyclics, 34 new technique in, 191 Hydrogenolysis, 165 Electronic factor, 27 Hydroxyl compounds, reaction of isoimperfections, 107 cyanates with, 404 Ethylene polymerization, 72 Energy transfer from irradiated solids,

I

119

by excited electronic states, 121 by temperature “spikes,” 123 by photons, 125 in radiation catalysis, 119 Extrinsic semiconductors, 118

F Fission fragments, produced from conductors, 83 from insulators, 93 Fixation of nitrogen, 66

G Gamma rays, 81 in ethylene polymerization, 72 methanol synthesis by, 76 pentane radiolysis by, 75 Germanium, beta irradiation of, 88 irradiation displaced atoms in, 91 H Heat of adsorption, of CO on CulO, 7 of CO on nickel and cobalt oxides, 8 Heterogeneous ethylene polymerization, 72 catalysis, polyfunctional, 137-189 irradiation in nitrogen fixation, 66 radiolysis in liquid phase, 79 N1O radiolysis, 62 n-Heptane isomerization and cracking over mixed catalysts, 106

Imperfections, electronic, 107 lattice, 104 Impurity atoms from nuclear reactions, 105 Insulators, fission fragments produced from, 93 Interaction with electron cloud, 89 Interception of reaction path, 149 Intermediates in heterogeneous catalysis, 139 Intimacy requirement, nature of, 160 Ions in radiation catalysis, 98 Irradiated heterogeneous systems, 56 Irradiation, catalyst activation under, 117 Isocyanate dimerization, 396 linear polymerization of, 399 trimerization, 397 reactions, catalytic effects in, 395-443 effect of amine structure on catalyzed, 416 with amines, 431 with compounds containing active hydrogen, 401 with hydroxyl compounds, 403 with thiols, 434 with water, 427 Isocyanurates, see isocyanate trimers Isomerization of butenes over alumina, 247 of n-heptane, 160

457

SUBJECT INDEX

of n-hexane, 159 of paraffins, 158 selectivity in paraffin reactions, 167

K Kinetics, of oxygen uptake by metals and metallic oxides, 21 reaction, of phenylisocyanates and alcohols, 404, 413

L Liapounov functions in complex reaction systems, 344 relation to direction of reaction paths, 349 Low energy electron diffraction, 191-201

0 Orthoyonal characteristic system, 364372 relations between vectors, 239 Oxidation of CO at low temperatures, 17 of CO on metallic oxides, 5, 36 Oxides, adsorption and catalysis on, 35 catalytic studies with, 39 doped, oxidation of CO on, 36 Oxygen adsorbed on nickel, electron diffraction of, 193 different forms of chemisorbed, 23 reactivity of adsorbed, 26 uptake by metals and metal oxides, 21

M

P

Mechanisms, of aging in reforming, 178 of isocyanate reactions with hydroxyl compounds, 404, 409, 413 Metal acetylacetonates, activity in isocyanate reactions, 425 naphthenates, activity in catalytic isocyanate reactions, 421 Metalhc oxides, chemisorption and catalysis on, 1-53 Methane radiolysis, 68 Methanol synthesis, 76 Methylcyclopentane dehydrogenation, 173 Microporous solids, properties of, 56 Mixed catalyst technique, 156 Monomolecular systems, analysis of reversible, 208-244 of irreversible, 270-294 determination of rate constants for typical, 244-285 geometry of, 213, 270 nature of new analysis of, 210 miscellaneous topics concerning, 295 rate equation for, 208

Paraffin hydrocracking, 162 Pattern, diffraction, of O? adsorbed on nickel, 194 Pentane radiolysis, 75 Petroleum naphtha “reforming” reaction, 175 Phenyl isocyanate dimer, 396 trimer, 397 reaction, catalyzed by metal compounds, 419 with active hydrogen compounds, 403 with butanethiol, 437 with alcohols, 413 Perturbations of rate constant matrix, 302 Photoadsorption on metallic oxides, 40 Photocatalysis on metallic oxides, 40 Photons, 90, 102 Physically mixed catalyst components, 156 Polymerization of ethylene, 72 of isocyanates, 395 Polyfunctional catalyst selectivity, 149 Polystep reaction, general criterion for, 148 in enzymatic processes, 184 inorganic, 188 mass transport in, 144 of hydrocarbons, 157 principles of, 138 systems, criterion for, 149

N Neutrons, fast, effect of, 94 Nickel oxide, adsorption of CO on, 8 catalytic oxidation of CO on, 20, 36 exchange of heavy oxygen on, 12 photoeffects with, 49 Nonstoichiometry, 21

458

SUBJECT INDEX

trivial and non-trivial, 142 Polyurethane foam manufacture, 441 Pseudo-mass-action systems in heterogeneous catalysis, 313

Q Quasi-intermediates in polystep reactions, 182 role of, 141

R Radiation catalysis, 55-133 induced imperfections, influence of, 104

Radiolysis, of NsO, 62 of methane, 68 Rate constants, experimental procedures for determining, 285 Reaction, in enzymatic processes, 184 (see also polystep reactions) organic reduction, 181 sequence, physical meaning of, 139

S Selectivity of alkylcyclopentane aromatization, 173 of polystep paraffin reactions, 167 Semiconductivity, 27 Semiconductors, extrinsic, 118

Synthesis, of methanol, 76

T Temperature “spikes,” 87 Titanium dioxide, photo effects with, 48 Thermodynamics of polystep rate process, 153 Three component system, explicit solution for, 372 Transition metal oxides, d-electron configuration and catalytic activity of, 33 “Turnover numbers,” 187

U Uretidenedione, see phenylisocyanate

V Vectors, orthogonality relations between, 239

X Xylene-ethylbenzene interconversion, 179

Z Zinc oxide, catalytic reactions photosensitized by, 45 photoactivity of, 40