Advances in Catalysis and Related Subjects, Volume 13

ADVANCES IN CATALYSIS AND RELATED SUBJECTS

VOLUME 13

This Page Intentionally Left Blank

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

NO PART O F THIS BOOK MAY BE REPRODUCED I N ANY FORM

BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION F R O M THE PUBLISHEliS

ACADEMIC PRESS INC. 111 FIFTHAVENUE NEWYORK3, N. Y.

United Kingdom Edition Published by

ACADEMIC PRESS

INC.

(LONDON) LTD.

BERKELEY SQUARE HOUSE, LONDONW. 1

Library of Congress Catalog Card Number 49-Y76,5

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

This Page Intentionally Left Blank

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 .

.

.

.

.

.

.

.

.

.

.

.

V

PREFACE .

.

.

.

.

.

.

.

.

.

.

.

vii

PROFESSOR W . E . GARNER.

.

.

.

.

.

.

.

1

.

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

.

.

.

.

.

.

.

.

.

.

.

.

. .

. .

. .

.

.

.

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

. .

. .

. .

. . .

. . .

. . .

. . .

. . .

. . .

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

. .

. .

. .

. .

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

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) $$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 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

REFERENCES 1. Haissinsky, M,, Second International Congress on Catalysis, Paris, Comm. No. 69 (1960), Vol. 11, p. 1429. Technip., Paris, 1961. 2. Kohn, H. W., and Taylor, E. H., J. Phys. Chem. 63,966 (1959). 3. Kohn, H. W., and Taylor, E. H., J . Phys. Chem. 63, 500 (1959). 4. Kohn, H. W., and Taylor, E. H., Second International Congress on Catalysis, Paris, Comm. No. 71 (1960), Vol. 11,p. 1461. Technip., Paris, 1961. 6. Weise, P. B., and Swegler, E. W., J. Chem. Phys. 23, 1567 (1955). 6. Saito, Y., Yoneda, Y., and Makishimba, S., Nature 183,388 (1959). 7. Klarke, R. W., Gibson, E. J., Dorling, T. A., and Pope, D., Proc. 2nd Intern. Conf. Peaceful Uses Atomic Energy, Geneva, 1968 2B, 312, (1959). 8. Coekelbergs, R. F., Gosselain, P. A., and Schotsmans, L. J., PTOC. 2nd Intern. Conf. Peaceful Uses Atomic Energy, Geneva, 1968 29, 429 (1959). 0. Coekelbergs, R. F., Gosselain, P. A., Juliens, J., Schotsmans, L. J., and Van Der Venne, M., in “Large Radiation Sources in Industry, Warsaw, 1959,” Vol. I, p. 191. Intern. Atomic Energy Agency, Vienna, 1960. 10. Coekelbergs, R. F., Decot, J., and Frennet, A., unpublished data (1961). 11. Mechelynck-David, C., and Provoost, F., Intern. J. Appl. Radiation and Isotopes 10, 191 (1961). 12. Mechelynck-David, C., unpublished data (1960). 13. Caffrey, J. M., and Allen, A. O., J . Phys. Chem. 63,879 (1959). 1.6. Sutherland, J. W., and Allen, A. O., in “Large Radiation Sources in Industry, Warsaw, 1959,” Vol. 11, p. 3. Intern. Atomic Energy Agency, Vienna, 1960. 16. Balandine, A. A., Spiteyne, V. I., Dorosselskaia, N. P., Glaeounov, P., and Verechtchinski, I., Second International Congress on Catalysis, Paris, Comm. No. 68 (1960), Vol. 11, p. 1415. Technip., Paris, 1961. 16. Balandine, A. A., Spitzyne, V. I., Dorosselskaia, N. P., and Mikkalenko, I. E., Dolclady Akad. Naulc S.S.S.R. 121,496 (1958). 17. Vesselovsky, V. I., pro^. 1st Intern. Conf. Peaceful Uses Atomic Energy, Geneva, 1966 7, 678 (1956). 18. Haissinsky, M., and Duflo, M., Proc. 2nd. Intern. Conf.Peaceful Uses Atomic Energy, Geneva, 1968 29, 4 (1959). 19. Preve, J., and Montarnal, R., Compt. rend. acud. sci. 249, 1667 (1959). 20. Barry, T. I., and Roberts, R., A.E.R.E. 1/R 2746 (1958). 21. Pierce, C., J. Phys. Chem. 67, 149 (1953). 22. Drake, L., I d . Eng. Chem. 41, 780 (1949). 23. Ritter, H., and Drake, L., Ind. Eng. Chem. Anal. Ed. 17, 787 (1945). $4. Coekelbergs, B. F., Crucq, A., Gosselain, P. A., and Schotsmans, L., Intern. Conf. on the Chem. E$ects of Nuclear Transformations,Prague Comm. CENT/38 (1960). 26. Coekelbergs, R. F., Gosselain, P. A., and Van Der Venne, M., Ind. chim. belge 22, 153 (1957). 26. Ponsaerts, E., Bull. SOC. chim. Belges 38, 110 (1939). 27. D’Olieslager, J. F., and Jungers, J. G., Bull. SOC. chim. Belges 40, 75 (1931). 28. Crucq, A., and Degols, L., unpublished data (1961). 29. Wourteel, E., Le Radium 11, 289, 332 (1919). 30. Dolle, L., Proa. 2nd Intern. Con$ Peaceful Uses Atomic Energy, Geneva, 1968 29, 367 (1959). 31. Harteck, P., and Dondes, S., Nucleonics 14, No. 3, 66 (1956).

RADIATION CATALYSIS

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.

136

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


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).

This Page Intentionally Left Blank

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

206

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.

208

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




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 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


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], 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