I once defined chemistry for a freshman as “what chemists do, and how they do it,” a better definition, I think, than one describing it as a stockpile of information. What it is that physical scientists try to do was aptly stated by Max Born in his book Experiment and Theory in Physics:
The problem in physics is how the actual phenomena, as observed with the help of our sense organs aided by instruments, can be reduced to simple notions which are suited for precise measurement and used for the formulation of quantitative laws.
How they do it is by ingenuity and imagination, by patience and persistence, by trial and error, following no rules or principles except to shun all deception of either oneself or others. There is no right method for constructing a work of art, whether it be a painting or a scientific theory. There are certain devices much used by physical scientists that are broadly called models. I will discuss them under five categories: Scale Models; Idealized Systems or Reference States; Analogs; Models of Space Relations; and Models Embodying Theoretical Concepts, together with errors to beware of in each case.
I do not presume to criticize models used outside of my own field, but members of an audience such as this may detect possible applications of what I say to such models as: “the exceptional child,” “the business cycle,” “the economic man,” “the American way of life,” and “the conservative.”
Scale Models
Scale models are used extensively in engineering design to predict the behavior of large-scale constructs. Examples are: models of dams, devised to test stresses and strains; airfoils, for testing in wind tunnels; “pilot plants” in chemical industry; and models of tidal harbors to study the most advantageous placement of jetties, etc.
In order to predict the behavior of a full-scale construct from that of a model, “scale factors” must be used. Judgment is called for in the selection of scale factors, because predictions based upon even a sophisticated scale factor are subject to more or less error, as lengths, areas, and volumes do not “scale up” in the same ratio.
J. B. S. Haldane, in a charming essay “On Being the Right Size,” pointed out that Giant Pagan, depicted in his child’s edition of Pilgrim’s Progress, whose height and breadth of limb were ten times those of Christian, would have weighed a thousand times as much, and since his weight per unit cross-section on his leg bones was but ten times that on Christian’s bones, he would have broken his leg on his first step. Had Christian known about scale factors he would have had nothing to fear.
Reference States
Reference states are illustrated by the “ideal gas,” the “ideal solution,” and the “regular solution.” An ideal gas is one that would conform to the “ideal gas law,” ( PV = RT ). The letters denote, respectively, pressure, volume per mole, a “gas constant” (whose numerical value depends upon the units used), and temperature on the “absolute,” Kelvin scale. The molecules of such an imaginary gas would have mass and velocity but no attractive forces or volumes. Real gases deviate more or less from this simple model. But since molecular volumes and attractions play smaller and smaller roles as volume increases, deviations become correspondingly less and may often be neglected with sufficient accuracy for practical purposes.
The ideal solution is a well-known reference state. It is a solution in which the forces between the molecules of like and unlike species are all the same. Non-ideal solutions are described, in theory and practice, by activity coefficients that express deviations from the ideal solution.
The regular solution is a further reference state. It serves to differentiate non-ideal solutions, in which selective, “chemical” forces operate, from those in which forces are non-specific, general, “physical.”
Analogs
Analogs are freely used by teachers, preachers, lecturers, and writers, as evident in parable, metaphor, and simile. Skillful teachers invent analogies in order to make new ideas clear and vivid to their students. A teacher of elementary physics uses the analogy between the flow of water in pipes and the flow of electricity in wires in order to assist his students in gaining a working knowledge of electrical units.
The model of the atom shown in Figure 2 was designed by Niels Bohr to account for the fact that negative electrons do not fall into a positively charged nucleus. The electrons were assumed to revolve around the nucleus in discrete circular or elliptical orbits by analogy with the solar system. This “planetary atom” model had to be abandoned with the advent of the Heisenberg “uncertainty principle,” which made clear the impossibility of determining orbits of undisturbed electrons. The model now used represents the probabilities of electrons being at various positions. This is an “electron cloud” model, illustrated in Figure 3.
Valuable and nearly indispensable as analogies are for the communication of ideas, one must resist the temptation to mistake an analogy as evidence. It may be suggestive, for example, to assert that the relation of the individual to society is analogous to that of the cell to the individual, but it would not follow that the individual can be equated to the square root of the product of the cell by society.
Models of Space Relations
My next category consists of models constructed to assist in dealing with space relations. Although our brains are three-dimensional, only sculptors and machinists have minds able to visualize in three dimensions.
Figure 4 shows a model of the arrangement of the atoms of hydrogen and oxygen in a crystal of ice. Every hydrogen atom acts as a bond between two oxygen atoms. In order to satisfy this structure, the lattice is expanded over the more closely packed structure that can be imposed by high pressures. When the ice melts, the atoms gain thermal motion; the number of hydrogen bonds diminishes, and water is more dense than ice, which, unlike most crystalline substances, floats upon its liquid phase.
This breaking of hydrogen bonds continues as the temperature rises, and the density reaches a maximum at 4° C. After that, the normal expansion exceeds the effect of the further breaking of hydrogen bonds.
Models Embodying Theoretical Concepts
I turn, finally, to models designed to represent theoretical concepts and to serve as bases for devising quantitative relations and predicting properties. Historic examples include: the Ptolemaic model of the solar system; the “caloric fluid,” adopted to explain the flow of heat; successive models of the atom, including the Bohr “planetary atom” and the modern “electron cloud” model.
The molecular kinetic theory is a conceptual complex that serves to formulate quantitative relations for a variety of phenomena, including pressure-temperature-volume relations, diffusion, viscosity, heat capacity, heat conductivity, and the liquefaction of gases.
The pitfall in the use of a conceptual model is to regard it as “true” if it can be made to fit some of the properties of a system by tinkering with adjustable parameters. A prudent investigator seeks evidence designed to put maximum strain upon his model or theory before throwing it out to be tested by competitors.
In 1907 I attended lectures in Berlin by the distinguished Geheimrat Professor Doktor Walther Nernst, the arbiter of the period in all questions in physical chemistry. He began by listing seven principles or laws that he regarded as so well established as to be no longer subject to doubt. Two of these were the law of conservation of mass and the law of conservation of energy. These two conservation laws still survive, but not separately; they are combined into one. A third item on the list was the model of an elastic ether, postulated to serve as a medium for the transmission of light through space. This had to be discarded because of the discovery, then already made, that the speed of light is independent of the direction of the motion of the earth through space.
There is at present no consensus regarding a significant model of the structure of simple liquids. One model envisages molecules in complete disorder, as in a highly compressed gas; other published models represent a liquid as retaining some of the order of its solid crystal form, as seen in Figures 7 and 8. Certain authors use such terms as “liquid lattice,” “quasi-crystalline,” “holes,” “cells,” “solid-like molecules.”
But there is now considerable evidence that a liquid composed of approximately spherical molecules contains none of the long-range order present in the crystal. Morrell and I, in 1934, constructed a model to represent this concept. Our simulated molecules were small spheres of uniform size made by drops of hot gelatin solution falling through a tall column of cold oil. These balls were put into a solution of gelatin of the same composition but which had been boiled to prevent jelling. The density and refractive index of balls and solution were almost identical, hence all the balls, except a few that contained lampblack, were nearly invisible. Balls and suspending solution were put into a cubic plate glass cell that had coordinate axes ruled on two faces. The cell was vibrated to agitate the contents, then stopped, and a photograph was quickly taken normal to two faces by the aid of a mirror, by the light of a spark.
Figure 9 shows one of the exposures. Morrell measured the coordinates of the visible balls and calculated their radial distance from one another. The results of hundreds of observations are seen in Figure 10, showing the probability of finding molecules at various distances around a central molecule, measured in diameters. There is, of course, a high probability that some pairs of balls will be about one diameter apart, but as distance increases, the position of a ball has little influence upon the position of balls even two diameters removed.
Four years earlier, Debye and Menke had determined the distribution of molecules in liquid mercury by x-ray scattering. The correspondence with the gelatin ball model is striking, as illustrated in Figure 10 in juxtaposition with the results of our model.
There are certain properties of liquids that one can deal with rather well on the basis of a lattice model. The mean distances between the molecules in a liquid and its crystal expanded to the same volume are not very different; hence, the energy involved in vaporizing liquid and solid can be correlated by reference to either model. But it is the entropy, a measure of molecular disorder, that furnishes the critical tests. It is not possible to introduce into such a model as is represented in Figure 8 enough disorder in the form of holes to match the enormous energy required for their production.
Quasi-solid theories of liquid structure seem quite inconsistent with the following properties:
- Supercooling: Many liquids can be “supercooled” far below their freezing points. White phosphorus, whose molecules are rapidly rotating tetrahedra of four atoms, and which melts at 44° C, has remained liquid to -70° C. If micro-crystals were present above 44°, they would surely grow to a mass of crystalline solid when subjected to such great undercooling.
- Hole Theory: If there were holes in liquids like those depicted in Figure 8, they would be larger, the larger the molecules, and one would expect that small molecules of another substance could fit into the large holes but not into the smaller. The octamethyl cyclotetrasiloxane, referred to earlier, has molecules five times the size of iodine molecules; those of cyclohexane are less than twice as large, but when iodine is introduced it expands each by the same amount; it finds no ready-made, hospitable holes in the former and must make its own holes.
- Crystallization and Viscosity: In Figure 11 are seen two-dimensional formulas of the flat molecules of meta-xylene and para-xylene. These substances differ only in the relative positions of the two methyl groups, -CH₃. Para-xylene melts at 13.2° C, and from there on up to their boiling points, which differ by only 0.8°, the two xylenes have almost identical viscosities and volumes. The latter are plotted against the reciprocal of the Kelvin temperature in Figure 12. If cooled below 13.2°, the para-xylene will freeze if seeded by a tiny crystal, with a contraction of 16.7%; the meta-xylene must be cooled to -47° before it can be frozen. If there were micro-crystals in these liquids, surely there would be more of them in the para-xylene at, say, 14°, than in the meta-, and it would have begun both to contract and to become more viscous. The fact is that the para-xylene has no inkling even at 15° of what could happen to it at 13.2°. It evidently contains no “solid-like” aggregates eager to grow. Crystallization is a completely discontinuous process.
- Regular Solution Theory: The theory of regular solutions, which has served, for example, to correlate solubilities of iodine in 15 solvents over a 300-fold range, as shown in Figure 1, is a theory containing no “holes”; it is altogether incompatible with a quasi-lattice model of liquid structure.
The moral to be drawn from this conflict is that a model should be regarded as suspect if it yields inferences in serious conflict with any of the pertinent properties of a system, regardless of how closely it can be made to agree with some, especially if there are adjustable parameters. A model that is consistent with all the pertinent properties, even if only approximately, probably can be refined to become more precise; but if it is in irreconcilable conflict with any part of the evidence, it is destined to be discarded, and in the meantime predictions and extrapolations based upon it should be regarded as unreliable.
Concluding Remarks
In summary, models in physical science serve as indispensable tools for understanding, predicting, and describing complex phenomena. The five categories of models discussed—scale models, reference states, analogs, models of space relations, and models embodying theoretical concepts—all have their own strengths and pitfalls.
Each of these categories provides insights into the physical world, but they are not without limitations. The scale models, while useful for engineering applications, often face challenges due to scaling issues in different dimensions. Reference states like the ideal gas or ideal solution are immensely helpful for understanding general behavior, but real-world deviations from these idealized models must be accounted for.
Analogs play an important role in teaching and conveying ideas to audiences that may not be familiar with the underlying science, but caution must be exercised to avoid taking these analogies literally. Space relation models, such as those used to understand crystal structures, help visualize complex atomic arrangements but may lack the details needed to convey all physical properties.
Models embodying theoretical concepts, such as those used for atomic structure, often evolve as new evidence comes to light. The shift from the Bohr planetary model to the modern quantum model of the atom illustrates how scientific models must adapt when faced with new empirical data.
Ultimately, models are not meant to be absolute representations of reality but rather tools to help scientists formulate hypotheses, understand relationships, and predict outcomes. They must be flexible, adaptable, and always subject to scrutiny and reevaluation in light of new evidence. This dynamic nature of modeling is what drives the progress of physical science, as new discoveries refine and sometimes completely change our understanding of the natural world.
The use of models, therefore, should always be tempered with a clear understanding of their limitations and the contexts in which they are valid. They are, at best, approximations of a complex reality, and while they can provide incredible insights, we must always be cautious about over-relying on them or treating them as complete descriptions of the systems they represent.
Joel H. Hildebrand
Professor Emeritus of Chemistry, University of California, Berkeley
(Read April 24, 1964)