The Periodic System and Atomic Structure I.
An Elementary Physical Approach WILLIAM J. WISWESSER Cooper Union, New York City
INTRODUCTION
T
HE excellent article by De Vault (I) on an elementary presentation of quantum mechanics draws attention to a major dilemma in teaching college chemistry. Important as these atomic structure fundamentals are, there is a general belief that wave mechanics is far too difficultto introduce in the elementary codrses. Thus only one elementary text (Z), written for students not majoring in chemistry, was found which contained a section on "Wave Mechanics." The writer believes that a major simplification of wave mechanics can be accomplished by a careful coordination of three teaching aids: 1. Emphasis on the nodes of the matter-wave patterns, rather than the resulting electron distribution functions. 2. Interpretation of the simpler wave functions by diagrams colored to show the sign and intensity of the function. 3. Employment of similarly colored melhematical curves in dynamic models which are so simple that the students can construct some themselves.
A simplified approach employing these aids has given some encouraging results, especially with students who were sufficiently interested to hold a brief discussion outside class. (This extracurricular time is necessary in order to introduce the physical and mathematical concepts such as standing waves, nodes, and polar coordinate plots.) The details given in this article are considerably more complete than those which were given the students. It must be thoroughly understood that this approach makes no pretense to be theoretically complete, nor perfectly exact, nor certainly as ideal as broader experience and discussion can make it. Further development and perfection of these teaching aids should make i t possible to introduce the basic notion of wave mechanics very early in the chemistry curriculum. This is desirable because wave mechanics is an important basic tool for the further understanding of matter. It gives the only logical explanation for: (1) matter-energy interactions; (2) atomic structure; (3) the Periodic System; (4) valence and molecular structure; (5) the cause of chemical action, stability or instability; and (6) the nature of chemical change. It further serves as an ideal common ground in the three basic subjects of mathematics, physics, and chemistry, and i t is a powerful intellectual stimulant. THE IMPORTANCE OF WAVE MECHANICS
Popular Recognition Most of the recent popular books on science (3,4, 5, 6,7,8) make some mention of the quantum theory and
wave mechanics, in recognition of their tremendous implications. These studies have brought about some of the most revolutionary changes in scientific thought since Galileo refuted Aristotle--that is, since science began. For example, Trattner's (9) account is typical: "As a result of wave-mechanics, we are stepping into a new and different kind of an order, quite unlike that which we were taught to believe under the discipline of the older concepts of physics. hut nonetheless s rational order capable of mathematical formulation. . . . And it is a revolutionary one at that. To deny the universal validity of the principle of causality is to strike at the very roots of science as humanity has known it since the days of Galileo and Newton."
The "revolutionary" degree of this change is not a popular exaggeration, but is fully acknowledged by expert physicists. A year ago Margenau and Wightman (10) began their review of "Atomic and Molecular Theory Since Bohr" with this statement: "The revolution in man's conception of the physical universe which has accurred during the last two decades is comparable, both in magnitude of philosophic conception and in pragmatic fertility, to the upheaval that took place during the sixteenth and seventeenth centuries."
This matter-energy, particle-wave fusion with its inherent shattering of determinism is a major topic in almost every popular account of the physical world published since 1930 (11,12,13,14,15,16). Eddington (8) summarizes its importance fairly bluntly: "In present-day physics the most fundamental equation is the wave equation of an electron."
How mnch longer can chemistry ignore these heavy reverberations in its basic physical doctrines? The student who reads these popular accounts rightly expects a few further details in his chemistry course, which is supposed to deal with the structure and atomic behavior of matter. Moreover, imaginative and mechanically minded students deserve a mnch better substitute for the obsolete Bohr-Sommerfeld model than the discouraging remarks that "the new structures cannot be visualized." After all, the structures can be visualized, and these notions are not new, but older than heavy hydrogen, neutrons, positrons, or artificial radioactivity.
Teaching Value A carefully simplified wave-mechanical approach to atomic structure, the Periodic System, and chemical change, etc., can be presented as marvelously inspiring evidence of a penetrating-almost religious--cosmic unity. This "moral lift" is quickly sensed by the
young student who may be meeting the discipline of tellectual honesty, many of the disturbing vagaries scientific thinking for the first time in his life. In con- and complications can he avoided in a simplified aptrast, very poor teaching psychology is employed in the proach, such as the one that follows. historical or empirical "atomic weight" approach to the Periodic System, with its errors, defects, and ex- Practical Value The practical value of matter-waves is even more ceptions; and in the elaboration of an obsolete "solar evident: Within a few years after de Broglie (20) and system" atom. The inquiring mind, a t any age, craves a pictorial Schrodinger (21) predicted atomic matter waves, representation for almost any kind of structure, and Davisson and Germer (22) in the Bell Laboratories the pictures suggested by atomic wave patterns are produced them; others (23, 24, 25, 26) soon confirmed unequalled in their intricate symmetry, striking beauty, the theory with beams of hydrogen and helium atoms and perfect harmony of finite discreteness with infinite and protons, as well as electrons. The electron difcontinuity. James Clerk Maxwell thought enough of fraction technique (27) which immediately followed hespherical harmonic charge distribution patterns to in- came a new and singularly powerful tool which deterclude several contour-lined views in the appendix of mines molecular structures in the gaseous state, much his famous treatise (17). It seems strange that so few as Laue's use of X-rays determines atomic structures modern texts or popular accounts illustrate these newly in the crystalline state. This new method became more discovered infinitesimal spherical harmonics within prominent after Pauling invented an improved interthe atom-as though the pictures somehow were preting technique in 1935. The enormously powerful wrong although the mathematical equations describing electron microscope represents a still greater applicathem still were right. "Electron cloud" pictures are tion. This instrument, in commercial production since illustrated in Ruark & Urey's treatise (18) while addi- 1940, uses magnetic lenses to take direct vision pictures tional patterns are shown on p. 135 of Darwin's popular with electron matter waves. The skeptic who does not account (6), and in De Vault's article (1). The most "believe in" these wave notions will have a hard time extensive collection of these pictures is given in White's explaining how and why this instrument works. This treatise (19) on pp. 71 and 146. Few of these references practical result of highly theoretical speculations is the greatest tool for extending man's vision since Galileo (5) contain similar-indeed almost identical-pictures of the corresponding waoe patterns, which have a more and Leeuwenhoek put glass lenses together to invent understandable mathematical origin with their positive the telescope and microscope. With this new elecand negative regions and meridian nodes. Such are tronic tool, "the scientist today stands a t the threshold of a vast new world of chemistry . . ." (28). the illustrations presented in this article. Finally, if the objection is raised that wave meIntellectual Value chanics is "physical" rather than "chemical," then i t The intellectual value of these new matter-wave must be noted that so are the extensive discussions on fundamentals has been shown in their proper emphasis gas measurements and properties, kinetic theory, moon statistically correct probability analysis rather than lecular weights, liquids (humidity, etc.), solutions, "hit-and-miss" spot-testing. Einstein and Infeld's colloidal dispersions, the solid state, and related popular account (13) closes with these remarks: "physical preliminaries" which collectively take up fully a third of the general course (29). The course "Quantum physics formulates laws governing crowds and not individuals. Not properties but probabilities are described, not still concerns one basic science, organized for conlaws disclosing the future of systems are formulated, but laws venience into two overlapping divisions. governing the changes in time of the probabilities and relating to great congregations of individuals."
SUMMARY O F A SUGGESTED APPROACH
For a simplified wave-mechanical approach to This new philosophy truly cuts to the heart of our science. No longer is the physical world based merely atomic structure and the Periodic System, the student on matter and energy and perfect predictability; the may he directed along the stepwise sequence of iuonly constancy in the universe is change itself, and the quiries summarized below. Further details will be only certainty is an ultimate limit to certainty. This found in the subsequent sections of later papers, third ultimate physical reality is embodied in the unit . numbered to correspond to these integrated steps. of change or action which a t the same time is the "mini- In the first two steps and sections, the impressive accumulation of evidence leading to the unified theory mum uncertainty" known as Planck's constant. These strange new aspects of matter waves and en- is cited (recognizing the principle that science should ergy particles give a greater unification in the apparent begin with the facts and not with the speculations). behavior of the physical universe. Whatever its re- The next four steps describe the wave patterns and maining shortcomings, wave mechanics gives a far more show how these alone completely determine the pattern accurate, more completely satisfying and experimen- of the Periodic System. The mathematical section tally valid picture of the structure of matter than any VII and all that follows is for advanced students and other theory or model yet proposed. Therefore a cer- those who are interested in using the equations to make tain amount of simplification is justified in the intro- working models, etc. Here also the Dirac refinements .ductory interpretations; and without sacrificing in- of the simplified Schrodinger equation are mentioned.
These elementary interpretations should be illustrated richly, not only with multiple views of the wave patterns but with a few simple dynamic models which employ spinning mathematical curves to simulate, for example, the cross-cut view of the atom. I.
COMPLEXITY OF ATOMIC STRUCTURE
The complexity of atomic structure is suggested by spectroscopic evidencc and by Rutherford's &scription of a nuclear model. The contributions and deficiencies of the chemist's Lewis-Langmuir model, the physicist's Bohr-Sommerfeld model, and the Hund-Land6 vector model are reviewed. All of these are shown to be fundamentally incapable of explaining quantum restrictions, fixed charge distributions, and matter-energy interactions. Three quantum numbers are introduced in the Bohr model to describe the restricted size (n), shape ( l ) , and tilt (m)of the electron's orbit. A relativity variation of mass causes the virtual orbit to precess in its orbital plane, as shown by Sommerfeld's "rosettes," and the atom's magnetic field causes this tilted plane in turn to precess around the magnetic axis. These motions are cleverly described by Max Born's marginal flip pictures (5). Finally, the coupling of the electron's spin-motion with this doubly precessing orbital motion completely blurs the complex three-dimensional traverse of even a single electron. Atomic systems with more than one electron involve hopelessly complex dynamics. Wave-mechanical patterns thus are introduced as a necessary simplification rather than an auxiliary complication.
like the particle nature of radiant energy, was revealed in a cumulative series of recent discoveriessuch as the atomic emission and absorption of light, the generation of X-rays, the photoelectric effect, the Compton effect, the Raman effect, the positron annihilation, electron diffraction, and electron microscopy (all briefly described in section 11). These various interactions between particles and waves required a more plausible explanation than existing theories could give. The quantum theory and wave mechanics offered this interpretation. Energy as well as matter and electricity is composed of ultimate discrete particles; however, the interactions among radiant photons, material protons, and electromagnetic electrons are no mere mechanical collisions, but rather interference and "beat" phenomena among their attendant behavior waves. The recent success of the electron microscope leaves little remaining doubt about the physical existence of these matter waves, and in accordance with the same interpretation, Maxwell's elegantly developed laws for electromagnetic waves "are to be regarded as probability laws for photons or particles" (10).
The electron's attendant matter wave is a mathematically exact, statistically precise pattern of its behavior. The wave equations tell nothing about what an electron is, nor what the wave itself is, but-what is more important practically-how an electron behaves. The wave intensity & a t any point in space (or the point-value of the wave function &) determines the probable occurrence of the electron a t that point. 11. WAVE NOTION OR MATTER More precisely, the product of the complex expression d\/ and its symmetrical conjuThe wave notion of matter was initiated, not by & (containing theoretical speculations, but by accumulated experi- gate &* (containing gives the electric charge mental evidence, so i t need not be introduced apolo- distribribution, which means the electron's total domain. getically. The unsuspected wave nature of matter, The electron's probable path through space is deter-
+
LIMITING SPHERE
...........1....... .....J ..)' ..' I a
l . - T r m ~ ~TWESOF NODESm THB BEHAVIOR PATTENOP ATOM-BOUNDELECTRONS FIGURE Spherical Nodes Spheres of constant radius Regions where #V = 0
Paired Conical Nodes Cones of constant colatitude Regions where fie = 0
Equidistant Meridian Nodes Planes of constant longitude Regions where #I)= 0
DOWEL-PIN
PIVOT
COLOR AND POSITION O F SECOND CURVE GIVING OP/VSYm P M S E
CARDBOARDWHEEL (DIAM.=A) CEMENTED T O AXLE (RADIVS*
F I G U R E 2.-MECHANICAL MODEL GENERATES A SURFACE PATTERN PROM A LINEAR "VIBEATION" When viewed from the top, the spinning and rotating paper curve shows origin of linear nodes as edge views of a paper curve which is colored black on one side and red on the other. Linear standing nodes arise only when the axle radius is an integral multiple of the wheel radius, thus synchronizing the two angular motions.
mined by its train of "pilot-waves" just as a photon of light is determined by its electromagnetic waves. For atom-bound electrons these wave-trains are required to overlap in harmony, so that the wave crests synchronize or "mesh in." This two-directional cyclic requirement produces the stationary or standing waves long known in spherical harmonics. Similar requirements for these standing waves cause a dampened radical oscillation, rapidly falling to zero with increasing distance from the nucleus. Thus the three-dimensional pattern is described completely in terms of twb direction angles and the radius.
The progressive development of these various kinds of nodes from the original point node can be demonstrated with mechanical-analogy models and diagrams. (See Figure 2.) For example, circular nodes arise from the rotation of a point node; linear nodes develop when the rotating and vibrating frequencies are integrally related (synchronized) because all points in the vibration pass through the equilibrum position at the same instant. (See Figure 3.) In an analogous way, conical nodes arise from the rotation of a linear node, and meridian planes develop when this rotation is synchronized with the over-all surface vibration. (See Figure 4.) N. WAVE PATTERNS AND NODES An electron's behavior pattern is completely defined These wave patterns of atom-bound electrons, being in all its radial and angular values by specifying the centered on the nucleus and the magnetic axis, are most number of each kind of node, because these same numeasily described by the nodes, regions where the wave bers enter the mathematical equations defining the intensity is zero. This approach is used by Darwin wave function or intensity $ a t any point. An addi( 6 ) , Born (5). and Einstein and Infeld (13). A vi- tional reason for putting emphasis on the nodes, rather brating string has poiat nodes, one for each successive than the intensity maxima, rests in a simple correharmonic. A vibrating drum or membrane has two spondence with the three quantum numbers n, I, and kinds, both being lines: circular nodes and diagonal m, introduced in the Bohr-Sommerfeld model. Thus nodes. Hence a vibrating volume can have three kinds, m is the number of meridian planar nodes, 1is the numall being surfaces: spherical nodes, conical nodes, and ber of linear nodes (i. e., meridian planes and co-latitude planar nodes. Figure 1 shows these three kinds of cones, both generated by the straight lines defming the nodal surfaces which together define three-dimensional respective angles), and n is the total number of nodes standing wave patterns. including the limiting sphere and other finite spheres
JOURNAL OF
CHEMICAL EDUCATION
ROTATINR POINT N m f GENEWITES A CIRCULAR NODE
SUCCESSIVE PROJECTIONS O f THE SPINNING AND ROTATING CURVE -CURE
3.-PAF'ER
CURVE ~~ATES A
VIEZRATINGSTRINGWEEN SPUN,AND A VIBRATINGSWRYACE WITH NEWLINEAR NODEWHEN ROTATED IN HARMONY WITH ITSSPIN
The equally clear mathematical significance of these three nodal numbers is discussed in section VII.
of constant radius. Therefore the three nodal quantum numbers are defined from n, 1, and m as follows: = no. o f m e r i d i a n nodes. including - sine and cosine altern a t e orientations, for convenience indicated by *m c = no. of conical nodes = 1 - nz p = no. of finite spherical nodes = n - 1 - 1
V.
m
SIN
30
SIN
RESULTANT SUWACE PATTERN
Z$
SIN
Q
COMPLETE DESCRIPTION OR ATOM-BOUND ELECTRONS
The fourth or spin quantum number s initially is introduced merely as the mathematical sign of the over-
em 0
1
COE
TOP OR POLAR VIEWS SHOW T H E RADUL A N 0 MERIDIAN VARIATIONS MERIDIAN NODES ARE SHOWN DY DOTTED LINES
SECTORAL VIEWS
SHOW THE RADIAL AND
CONICAL
NODES ARE
SHOWN
LATERAL VARIATIONS BY
SOLID
20
CoS
30
(glb)
('YeO)
LINES,
E S U L T A N I THREE-DIMENSIONAL VARIATIONS IN WAVE INTENSITY (W) SIDE VIEWS OF THE Q THIS ROW (n.3 1.3) COMPLETES THE WRiE PATTERN SYNOPSIS SHOWN ON FI6URE 5
FIG-
~.-PICTORIAL ANALYSISOP
AND SEVEN IDENTICAL PATTERNS WITH $r,8,$ PATTERNS w m THREE ~ L~mm NODES. THESE REVERSED COLORS ACCOUNT noa THE FOURTEEN IL\RE E ~ T H S
all wave equation (neglecting a t this point Dirac's rdnement). This interpretation, also given by Born (30), is considerably simpler than the vector model consequences of the spin; and according to Dirac (31), the most important consequence of the spin is not its mechanical effect (which is small) but its introduction of a simple fourth variable. Pauli's exclusion principle (32) is introduced as a common-sense statement: In any one atom, no two electrons can be exactly alike in their total behavior, or else they would be an identity. Since the four quantum numbers, n , 1, m , s completely define the behavior pattern, i t follows that each atom-bound electron must have a unique set of quantum numbers: s may be only plus or minus, m may be a plus or minus integer to designate the sine or cosine orientations, while n and 1 must be positive integers. At this point i t is convenient to tabulate the twofold aspects (wave meaning and particle meaning) of each quantum number, showing the mathematical definition of each number, and incidentally noting the suggestive meaning of the letter-symbols.
Table. The two groups of active metals on the left side, the s i r groups of regular elements on the right side, the ten groups of irregular or transition metals, and the fourteen rare earths, all reflect the number of patterns permitted for a given number of linear nodes, these being 0, 1, 2, and 3, respectively. For example, all of the rare earths are characterized fundamentally by the fact that in their atoms, the last-added electron has a pattern with three linear nodes, permitting these : variations (as indicated by the mathematical definitions in Table 1) :
A similar accounting for the ten, six, and two should be obvious from a study of Table 1. Figure 4 illustrates the last seven of the above fourteen patterns; the other seven are identical except that the black and red colors are reversed. The bottom SlGNlsrCaNCB 0. TIIB FOUP QUANTUM NUMBBPS TABLE 1. row of this figure completes Figure 5 (Frontispiece) which shows similar rows of five, three, and one, each wave pattern representing two electrons differing only 1, 2, 3, 4, 5. . . . Principal Q.N. n Total no. of nodes Sire 01 orbit in "spin" or sign. In both Figures 4 and 5, half of the "Lesser" Q.N. I No. of linear nodes Shape of orbit 0 up to (n - 1) (-1) to ( + I ) Magnetic Q.- m No. of meridian Tilt of orbit wave patterns with meridian nodes are shown "magN. nodes netically resolved," by plotting the real part (e. g., cos Spin Q.N. s Over-all math. sign Spin direction Minus or plus 24) of the complex meridian function (e. g., J.O = c*"* The wave-particle correspondence is a reminder = cos 24 + i sin 24); the other half of the wave that both the Bohr-Sommerfeld dynamic model and patterns are "merged," thus representing the square the Lewis-Langmuir static model (discussed in section root of the corresponding electric charge pattern (W*) I) can be partially reconciled in the wave theory. The which has no meridian variation (e. g.. cosZ 24 discontinuous change from one state to another, pro- il sin2 24 is unity). After determining the number and picturing the hibiting uoninteger quantum numbers, is a reflection of the obvious physical fact that fractions of nodes in a types of patterns permitted for any n , 1 energy value, standing wave cannot exist. In these important re- the next step is to determine the proper sequence of spects this simplified interpretation is understandable these n , 1 values. Those patterns having the greatest binding energy are those which permit the closest apand impressive. proach of the electron to the attracting positive nuVI. THE PLAN OF THE PERIODIC SYSTEM cleus, and these certainly will be occupied first. Now The general plan of the Periodic System can be deter- all linear nodes, again regardless of their kind, slice mined completely from the filling sequence of the wave through the nucleus and thereby deny the electron to patterns. Since each added electron must be unique, it this strong-binding region. Spherical nodes, on the follows that each element can be identified by the pattern other hand, allow a high probable occurrence in this of its last-added electron. Therefore any predetermined nuclear region. Consequently, when inner groups of natural sequence of the wave patterns also will be a electrons and high nuclear charges are present, linear nodes keep the electron almost mice as far from the nunatural sequence of the corresponding elements. Successively added electrons will fall spontaueously cleus as spherical ones. This difference is especially into the unoccupied patterns having the greatest bind- marked when the electron might penetrate inner layers ing energy, just as water falls spontaneously in the of electrons and thereby experience a higher nuclear earth's field. The electron's binding is affected only charge which these inner electrons otherwise would by its distance from the nucleus, hence in the absence neutralize. Except for this difference between linear of a strong magnetic field, conical and meridian nodes and spherical nodes, patterns with the least number of hawe the same binding energy. Thus all the patterns for nodes (prohibited regions) will be occupied first. a given number of linear nodes, and the same number of Counting linear nodes twice for their relative value, 1 spherical nodes, will be occupied successively. Here the patterns will be filled i n increasing order of n is the underlying reason for the vertical subdivisions (rather than n alone). A better approximation, com1). in the extended (metallurgical) form of the Periodic pletely specific,is given by n 1 - 1 ( / 1
+
+
+
TWO GROUPS WITH 2-0
EXPLANATORY NOTES
SIX GROUPS
WlTH
1. Each picture is a side view of the three-dimensional standing wave or "behavior pattern" for an atom-bound electron. 2. The electron's distribution. or electric charge density, is determined by squaring the plus (black) or minus (red) wave intensities. 3. A second or "paired" electron is described by thesame pattern with the colors reversed, keeping the north magnetic pole a t the top. 4. Each element is characterized by the wave pattern of its last-added electron. 5. The pattern number, in red, is theatomic number of the element. . 6. The rare earth patterns are shown in Figure 4. 7. The trigonometric expressions shown in red describe the angular variations (surface hmonics) of the wave intensity +o,+ 8. The meridian variations. shown where m = -1 and +2. constitute onlv the real part of a camplei imaginary (++ = c""). The electron distributfon is the product of this function and its complex conjugate. e-irnm. Hence the distribution patterns have no meridian variations and are symmetrical about the are shown where polar axis. The corresponding "merged" wave patterns n = -2 and f l .
I=1
m=*l SIN $
a
10
GROWS OF
m--2 SIN
20
TRANSITION METALSWlTH 1-2
n--1 SIN+.COS~
1-0 JCOS.~-I
Wave Pattern Synopsis of the Periodic System (Figure 5.
See page 319)
n =+1 COS+.COS~
m-+l! cos 2+
Thus the patterns with (n, 1) equal to (1, 0) will be occupied first; then (2, O), (2, 1). (3, O), and (3, 1). However (4, 0) will be filled before (3, 2) where the n plus 1sum is five, since the two linear nodes in the latter patterns will keep the electron further from the nucleus than the extra spherical node in the former. For the same reason, the (5, 0) patterns will be filled before (4, 2), and (6, 0) before (4, 3) or (5, 2). The wave pattern synopsis of the Periodic System (Figure 5) summarizes the successive order of the ho&ontal (n, 1) rows of patterns according to the relatively simple measure n 1 1/(1 1). The vertical groups are plotted according to increasing 1 value from left to right. This condensed "spectroscopic" or wavemechanical Periodic Chart preserves the adjacent sequence of the first three periods, but breaks the atomic number continuity in the long periods. In compensation, however, the irregular transition metals-with their incomplete inner shells, colored ions, variable and irregular valences, horizontal similarities, etc.-are separated from the regular elements which do not have these peculiar characteristics. Hence arguments about the "ideal" chart get nowhere, and the illustrated synopsis is but one of many possible compromises. Those who recognize the deeply significant meaning of the Periodic Sjstem are bound to be impressed by this purely mathematical and physical method of determining its general plan from genuinely basic principles, without requiring the memorization of a single chemical fact.
thereby indicating a fractional number of roots. Such polynomials would be so "improper" that they could not be plotted; for example (-x)' is negative when # is odd, and positive when p is even, but it could haveno physical meaning when # is fractional. Further details of the mathematical equations and their relation to the illustrated wave patterns are given later in section VII. VIII.
CHEMICAL ACTION OR STABIJATT
+- +
Chemical action or stability is explained in a fundamental yet simple way with the mathematical equations. When all of the electric charge patterns (+$*) in any (n, 1) subshell are added together, the corresponding equations show that the directional inequalities cancel out, producing a spherically symmetrical electron distribution. For example, the meridian variations disappear because the product of the complex conjugates yields dm'.e-im+, which is unity Similarly, all the lateral for any value of m or equations in a subshell add to give the terms of the expansion (sin2 8 cos%)', which also remains unity for any value of 8 or 1. Therefore when groups of patterns are completed, all the electrons cancel one another's directional peculiarities so that they are completely inert toward directed bond formation, i. e., any directed interaction with otlier atoms. This gives a plausible explanation why the inert gas configurations are inert, why "symmetry means stability" in analogous ionic configurations, and why only the unbalenced outer or valence electrons form atom-toM. THE MATEEMATICAL EQUATIONS AND CURVES The mathematical equations and curves for the wave atom bonds and are involved in chemical changes. The inner electrons, except for their symmetrical refunctions further clarify the meaning of the three nodal pulsion, behave as though they are not there. 1 quantum numbers m, c (= 1 m), and p (= n Pauling (33) has gone' still further on empirical 1). The total wave equation is the product of three independent Y vs x equations, one x for each spherical grounds, and has shown that these wave pattern equations (atomic orbitals) can be additively combined polar coordinate 8, and r, respectively. That is, the wave intensity $ = $+&.$,, where the real part of to produce the tetrahedral bond pattern of carbon-like the meridian function $+ = sin m+ or cos m+, and atoms, the octahedral bond pattern of iron-like atoms, either of these can be expanded as polynomials of and other variations. However, the methods of applying these atomic patterns to molecules are too uncer(sin 6)'" or (cos 4)'"; the lateral function GOcontains a tain to verify Pauling's results. related oscillating polynomial of cos 8 to the degree c; While speculations on linear combinations of atomic and the radial function rC; contains a polynomial of the orbitals ("LCAO") await more convincing proofs and radius r to the degree p. Thus the three kinds of nodes justifications, much progress has been made in develare merely the roots or x-intercepts in these oscillating oping the competing hypothesis of molecular orbitals. curves; or the number of times each curve cuts the xFajans (34) recently has cited considerable evidence to axis, giving values of 4, 8, or r, respectively, where the show that molecules in themselves may have wave corresponding intensity $ is zero. It should be recogpatterns like those in atoms; e. g., the subshells in the nized that the nodes always are preserved when equaN 2 molecule are strongly suggestive of over-all CO or tions are multiplied; e. g., if YI = x -2 and Y2= x those in the neon atom. Similar ionic analogies exist - 3, then Yl.Y2 will have nodes a t 2 and 3. This in the CN- and NO+ ions. The most active frontiers would be evident on plotting the polynomial Y = x2 in atomic studies lie in these directions. 5x 6, equally well expressed by the root factors (x - 2) (x 13); IX. SPECTROSCOPIC STUDIES Again the reason for the strange quantum or wholenumber restrictions becomes obvious: these oscillatinn Snectrosconic studies such as the internretation of curves may cut the axis a t any fractional x-value, but complex atomic spectra, the resolution of molecular and they cannot cut it a fractional number of times. These nuclear structures, and the determination of basic polynomials cannot contain fractional powers (degrees), chemical affinities in terms of binding energies, are
-
- -
+,
+
-
+.
+
closely related to atomic structure studies. The wave-mechanical model emphasizes particularly well the inherent difficulties in these problems. It clearly shows that the different electrons within the same atoms or molecules cannot he treated as independent particles. Through their wave patterns, these electrons perturb or influence each other in a complex manner. The separate patterns blend into single configuration patterns, each combination being uniquely different from all others. Thus the enormous number of mathematically permitted pattern combinations alone explains why a very few changing electrons can give rise to thousands of energy level differences, or spectrum lines, for a single element. These complex atomic spectra reveal many simultaneous multiple-electron transitions. These occur as naturally as single-electron "jumps" because the combination wave patterns (configurations) change as a whole. The wave-mechanical model, besides explaining the naturalness of perturbations and multiple-electron transitions, also gives (1) a plausible time-continuous mechanism for the "jumps," (2) a mathematical accounting for the line intensities as transition probabilities, and (3) mathematical reasons for the selection (transition) rules. X.
SPECULATIONS
or intermolecular forces, (5) metallic nature, (6) magnetism, (7) nuclear structure and nuclear phenomena. Margenau and Setlow (36) give an excellent summary of these successes. The theory is supported by the soberly awesome "experimental miracles," as Margenau calls them, of the cyclotron, betatron, and electron microscope. ACKNOWLEDGMENTS
During the development of this article, the writer received a great deal of encouragement and constructive criticism on physical, mathematical, theoretical, and grammatical details from Dr. John J. Grebe of the Dow Chemical Company; from Dr. Theodore Berlin of the University of Michigan; and, a t The Cooper Union, from Professors Arthur H. Radasch, Frederic H. Miller, Albert Reis, Clarence S. Sherman, J. K. L. MacDonald, Donald N. Read, and Dr. Leon Goldmau. Messrs. David Saletan and William Summers, while sophomore students, contributed helpful points of view, and Mrs. Joseph Allerton contributed many hours of secretarial work on the manuscript. LITERATURE CITED
( 1 ) DEVAULT, D., J. CHEM.EDUC., 21, 526 (1944). ( 2 ) DEMING,H. G., "Fundamental Chkmistry," John Wiley & Sons, Inc., New York, 1940, pp. 315-19. ( 3 ) ALLEN, J. S., ET AL., "Atoms, Rocks and Galaxies." 2nd Ed., Harper & Bros., New York, 1942, pp. 369, 500. ( 4 ) BARNES, E. W., "Scientific Theory and Religion," The Macmillan Company, New York, 1933, pp. 199,261-310. M., "The Restless Universe," Harper & Bros., New ( 5 ) BORN, York, 1936, pp;, 139-65, 186-97. ( 6 ) DARWIN, C. G., New Conceptions of Matter." The Macmillan Company, New York, 193:: pp. 1 2 9 4 1 . (7) DIETZ,D., "The Story of Science, 4th Ed., New Home Library, New York, 1942, pp. 2 4 - 5 4 , (8)EDDINGTON, A., "New Pathways in Science." The Macmillan Company, New York. 1935, pp. 41-7.226, ( 9 ) TRATTWR, E. R., "The Story of the World's Great Thinkers, Wew Home Llbrary, New York, 1942, pp. 158-62. A m WIGHTMAN, Am. J. Physics, 12, 119, 121 (10) MARGENAU
Finally, it is appropriate to conclude with an honest reminder that many pressing questions remain unanswered, and many more comprehensive generalizations await discovery. A recent announcement (35) raises the speculation that biological processes may he remotely related to atomic wave patterns. Some collaborating biologists and electronics experts have perfected a high impedance electronic voltmeter which consumes less than a micro-micro ampere of current. With this ultra-sensitive probing instrument, they have proved conclusively that every living organism--egg, plant, or animal-has a characteristic electrical field /1044\ pattern. This may have far-reaching significance, and (11) DARROW. K . K.. "The Renaissance of Physics," The Maca t least ought to draw attention to these far more millan Company, New York, t936. BROGLIE.L.. "Matter & Lieht." W. W. Norton & Cornminute atomic patterns. For a long time i t was sus- (12) . . DEpany, N& ~ b r k1939, , p. 1g7. ' pected that the strange, orderly "dance of the chromo- (13) EINSTEIN, A,, AND L. INFELD, "The Evolution of Physics." somes" in the nuclear process of cell division might be Simon & Schuster, New York, 1938, pp. 288, 313. M.. "The Universe in the Light of Modern Physdirected by some kind of electric field radiating from the (14) PLANCK, ics." W. W. Norton & Comoanv. New York. 1931. D.32. polar bodies through their asteroids, but the mecha- (15) SCHR&DINGER, E., "Science an'd t c L ~ u m a nTe&pera&nt." nism is still a mystery. Yet, without doubt, all our W. W. Norton & Company, New York, 1935, pp. 52f, e , ? . LU",. sensory perceptions and physiological "directives" (16) Cox, R. T . , "Time, Space and Atoms," Williams & Wilkins are transmitted by intricate impulses which are purely Company, Baltimore, 1933, pp. 12k5-4; MILLIKAN.R. A., "Time, Matter and Values," University of North electrical in nature. Perhaps a certain universal Carolina Press, Chapel Hill, 1932, pp. 61-7. electrical pattern rules these biological processes, and (17) MAXWELL, J. C., "Treatise of Electricity and Magnetism." there may be some remote connection between these 3rd Ed., Clarendon Press, Oxford, 1891, Vol. I, p. 216. AND UREY,"Atoms, Molecules and Quanta," Mcbiochemical patterns and the infinitely smaller atomic (18) RUARK Graw-Hill Book Company, Inc., New York, 1930, p. 565. patterns, just as visible crystalline forms reflect the (19) WHITE, H. E., "Introduction to Atomic Spectra," McGrawsymmetry of their component atoms. Hill Book Company, Inc., New York, 1934, pp. 71, 146. These are mere speculations: i t is far more important (20) DE BROGLIE,L., Phil. Mag., 47, 446 (1924); Ann. Phys. ( 1 0 ) . 3, 22 (1925). to remember that the wave-mechanical interpreta- (21) ScHRon~~cen, E., Phys. Rev., 28, 1049 (1926); Ann. tions alone are successful in interpreting (1) atomic Phys., 79, 361. 489; 80, 437; 81, 109 (1926). . . Davrsso~AND GERMBR.Phvs. Rev... 30.. 705 (1927): . .. -T. spectra and the Periodic System, (2) molecular spectra, (22) CHEM.EDUC., 5, 1041 (1928). (3) valence or intramolecular forces, (4) van der Waals (23) TXOMS~N, G . P., Proc. Roy. Soc. (London), 119, 651 (1928); ,A"~=z,.
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