DIMENSIONAL ANALYSIS OF CHEMICAL LAWS AND THEORIES 0. THEODOR BENFEYL Harvard University, Cambridge, Massachusetts
Dmrm the past decades scientists have been working in ever increasing numbers on problems a t the borderlines between different sciences. For a time the sciences developed in isolation from each other, developing their own concepts and their own language. But we are now discovering not only the interrelation between the skiences, hut also the large measure of agreement in the approaches and concepts of the different sciences. The stage has been reached when it becomes possible to develop a comparative study of the sciences in order to discover what asperts are common to all of nature, from the fundamental particles to complex molecules and organisms, and what aspects are specific t,o a particular level of complexit,y. For this purpose a symbolism needs to be developed, applying to those dimensions which different sciences or scientific theories utilize in their descriptions of natural phenomena. This type of dimensional analysis does not analyze a particular quantity such as force or energy into its constituent dimensions, but rather seeks to discover the totality of independent dimensions incorporated in a given law or theory. When such an analysis is complete, it should become possible to state the dimensional struct,ure of the law or theory in the form of a set of symbols. Thereby its relation to other fields of study in the same science or in other sciences would become apparent. In this paper some chemical lawe and theories are analyzed, using the following symbolism, and the implications of the results are discussed. Symbol N
Dimension repmaenled Number
D d
Diatanee distance
-
time
t
t
C A
Codomhic interaotion Association
Meaning Independent units, e.g., atoms, molecules, bonds, organisms. Space dimension. Internal space dimension leading to finite volume of the units. Time. Direction of time ("Time's Arrow") introduced, for instance, in the second Law of Thermodynamios. Attraction and repulsion between electrically charged entities. Attraction between like entities such as are explained in waveby "exchange mechanics forces." Such attraction often leads to new units made up by the association of units of lower complexity (ex.. molecules from atoms). The new units are not mere aggregates of the constituent parts.
THE DEVELOPMENT O F THE KINETIC THEORY
Considering first the classical kinetir theory of gases, the units of the theory are independent point particles having no attractive or repulsive forces arting between them. This particle nature is designated by the symbol N. The particles move in three dimensional spare (3D), their movement being measured by a time measure (t) such as seconds or minutes. The kinetic theory approximately accounted for gaseous behavior a t high temperature and low pressure but seriously failed under other conditions. An analysis of the failures pointed to the necessity of introduring further parameters. The van der Waals b constaut mas related to the space occupied by the molerules ( 3 4 , while the constant a of the van der Waals equation was necessitated by attractive forces between the particles, which we designate as A (for associative tendency). In the limit, the particles associate to form a liquid, a state completely unexplainable by the original kinetic theory. The theory as thus developed, including the van der Waals parameters, has the dimensional structure (N, 30, 3d, t, A). This more complex theory still cannot account for order-disorder phenomena, dealt with by statistical theory which introduces "time's arrow," that is, the direction of time +
(I). Further, the theory fails to account for the
behavior of gaseous ions for which coulombic attractive and repulsive forces are required (C). Had the kinetic theory been tested under conditions where the gas was largely dissociated into atoms, the associative parameter A would have been necessitated by the tendency of the atoms to form covalent bonds with each other. A theory including all these paramethrs has the struc+
ture (N, 30, 3d, t, t , A , C). I t is apparent that such a theory can be arrived a t in several different mays. One may begin, as the kinetic theory did, with two or three of the parameters, and discover the others through failures of the original model. Alternatively, one may begin with a model containing all these parameters. or all except those which can be derived from the chosen ones. Thus the finite size of atoms and molecules is derivable in quantum mechanics from coulombic interaction and the wave properties of the constituent entities. For this theory, particle behavior corresponds to N, wave properties to A, and the theory +
includes 3 0 , t , t (from statistical theory) and C, as A
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286
before. Its structure is (N, 3 0 , t, t, A, C) from which JOURNAL OF CHEMICAL EDUCATION
it derives, a t least in principle, 3 4 i.e., the average covalent, ionic and van der Waals radii. I t will therefore discover no failures corresponding to those encountered by the kinetic theory. The two approaches in their final form may be spoken of as dimensionally rommensnrate. ORGANIC STRUCTURAL THEORY
Applying the dimensional approach to the historical development of organic structural theory and considering only the dimensions already cited, Dalton's atomir t,heory would rorrespond to (N), the nature of the linking between atoms to form molecules not being specified. Berzelius' electrochemical theory, on the other hand, had the structure (N, C), and because he could only conceive of association between oppositely charged entities, he could not accept diatomic molecules of oxygen, hydrogen, and so on. For the same reason, he ronld not accept simple substitution of H by C1 in organic compounds, and this led to the famous controversy with Dumas about the structure of the chloracetic acids. The substitution theories of Laurent and Dumas assumed non-electrochemical binding forces though they had 110 idea of the nature of surh forces. With the development of valence theory, this problem was pushed into the distant future, by making the bond the unit of the new theory rather than the constituent atoms. These bonds constituted the fundamental entities in the theories of organic compounds advanced Ly Kekul6 and A. S.Couper in 1858. The structure of t,heir theories was simply (N), for as G. W. Wbeland points out,2 the theories did not demand any spatial representation (though Kekul6 certainly thought in structural terms). The simple requirement of a fixed number of ualences for each type of atom leads to a distinguishable formula for every chemical species. These formulas can be represented in the form of numerical tables rat,her than as linked atoms.2 The failures of this simple theory fall into five classes: (1) Van't Hoff and Le Bel, to account for the fact, that several different substances often have the same classical formula, demanded the direction of valence bonds in space (30). ( 2 ) In order to account for optical isomerism in the biphenyl series, the requirement of a finite volume ( 3 4 of the constituent parts was introduced. (3) To account for the properties of benzene and the absence of two ortho-disubstituted benzenes such as oxylene, Kekul6 proposed a dynamical theory (N, t ) , a rapid succession of L'classical" structures. This, however, proved untenableQnd was replaced by the t,heory of resonance, which permitted "polycentrir" as opposed to dicentric bonds, that is, the association of t,hese bonds which were considered independent units in the original theory. Resonance theory therefore introduces the parameter A , the mathematics for arriving a t the hybrid structure of benzene by the valence bond met,hod (whirh takes the Kekul6 structures as starting points) being identical in form with that used in the desrription of covaleut bond formation
between atoms. In both cases, "exchange forces" account for the greater stability of the associated entity. The above discussion makes clear that resonance theory was necessitated by the lark of a parameter in the Kekult? theory. It appears as a separate postulate if we start out with independent dicentric classical bonds. Molecular orbital theory, on the other hand, leads, though admittedly by considerable mathematical approximations, to a single acetone formula and a single benzene formula with no conceptually different basis for the two calrnlations. It can do this because it utilizes the wave nature of the electrons ( A ) ,and hence does not need to introduce a new assoriative property to explain polycentric bonds. (4) The Kekul&Couper theory predicts substances such as vinyl alcohol CHFCHOH which have not, in fact, been isolated. Comparison with similar compounds leads to the conclusion that if formed they would transform or decompose into other substanres before being detected. The incorporation of this failure into structural theory demands both the direr+
tion of time and a measure of time ( t and t ) . The direction of time is demanded because there must be other structures of lower free energy iuto which the postulated substance would spontaneously transform; a measure of time is needed because isolability depends on the rate of transformation of the substance and the rapidity with which the experimenter is able to detect it. (5) Finally there exist organic onium compounds, falling outside classical structural theory and introducing the parameter C. The complete theory, as developed above is commensurate with statistical quantum mechanics and with the developed gas theory. On the other hand, the definition of a molecule as a three dimensional spatial structure is not commensurate with the wave-particle nature of the fundamental particles (photons, protons, neutrons, electrons), nor is it commensurate with biological organisms. Yet modern cosmological theories4 demand evolution from the fundamental particles through atoms and molecules to living organisms. The definition of a molecule in spatial terms, that is, as (N, 3L), 3 4 A , C), fails to account for the non-isolability of vinyl alcohol. A more satisfactory definition of a molecule must, therefore, include time. A molecule is not a spatial entity, but a pattern of activity, persisting long enough in the particular pattern to he observable before changing into a uew pattern. A proton, separated a t a given instant from an electron by the Bohr radius, is not necessarily a hydrogen atom. The electron may merely be moving past the proton a t high speed. A minimum time is required for the system of proton and electron to execute the pattern (i.e., the probability distribution) that constitutes the hydrogen atom. An atom or molecule thus has a life history from the time of its formation, through its characteristic patterns, to its final destruction or its absorption in a larger pattern. This view of a molecule brings chemistry closer to the view of nature as constituted of ,‘organisms," whirh A. iY.Whitehead has suggested as
WHELAND,G. W., "4dvanced Organic Chemistry," 2nd
cd., John Wiley & Sons, Inc., New York, 1949, p. 87. T o r a discussion of the so-called resonance frequency, see WHELAND,G. W., "Resonance in Organic Chemistry," John Wiley & Sons, In?., New York, 1955, p. 608.
VOLUME 34, NO. 6, IUNE, 1957
See, for instance, GAHOW, G., "The C~.mt,ionof the Universe," Viking Press, Xew York, 1952.
necessary for both physical and biological ~ciences.~ This article has confined itself to space and time dimensions, wave and particle nature, and conlonlbic WHITEHEAD, A,
N-, iiSeience and the Modern World,22The
Mxcmillan Co., New York. 1925.
attraction and repulsion. KO account has been taken of other parameters, nor have relations between dimensions, such as thespace-time relation of relativity theory, been discussed. Yet the application to the relatively simple theories here considered points up the value of a symbolic representation of lams and theories.
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