Molecular Motions One of the aims of molecular physics is to account for the bulk
properties of nlatter in terms of the beha()ior of its particles. High-speed computers are helping physicists realize this goal
by B. 1. Alder and Thuma, E. Wainwright
D uring the 19th century, as evidence in favor of the atomic theory mounted, an ancient hope of science began to bear fruit. As long ago as the first century B.C. Lucretius had proposed not only that matter is composed of tiny particles called atoms, but also that the behavior of these par-
ticles is the key to understanding the properties of bulk matter. For centuries this idea remained simply an interesting hypothesis. Then Isaac Newton set forth laws of motion from which the behavior of atoms might be calculated. At the same time a number of investigators were making quantitative observations
on the gross properties of matter. The stage was set for an attempt to realize Lucretius' dream.
The earliest tries were mostly unsuccessful. But in 1739 Daniel Bernoulli succeeded in proving that the product of the pressure and the volume of a gas is proportional to the average kinetic
PHYSICAL MODEL of a molecular Eystem consists of gelatin balls suspended in a tank of liquid. The tank is shaken to generate
typical patterns of separations between balls. In the experiment depicted here separations were measured for the seven black balls.
1959 SCIENTIFIC AMERICAN, INC
No.6 of a series
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energy of its atoms, provided that the atoms do not interact. His result is in fact valid for gases at very low density, where interactions between atoms or molecules are rare. It was not until the 19th century, however, that the program got under way on a grand scale, with the work of such men as Rudolf Clausius, James Clerk Maxwell and Ludwig Boltzmann. The task they faced was immense. Their predecessors had dealt with such problems as computing the orderly motions of a few planets under the gravitational attraction of the sun. These men were concerned with millions of particles, colliding with one another and darting about in all directions. To follow the trajectory of any individual atom in a piece of matter it would be necessary to know the position at all times of every other particle close enough to exert a force on it. The total force on the atom could then be computed from instant to instant, and, assuming that Newton's laws applied, its motion could be calculated.
In practice such a detailed calculation was hopelessly complicated for a dozen particles, let alone millions. Fortunately it was not necessary. The bulk properties of matter depend upon the average behavior of many atoms, and not upon the detailed motion of each. Therefore statistical methods could be used and a new branch of physics known as statistical mechanics grew up. The statistical approach fulfilled many of its founders' hopes. In some problems, however,
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o 1.2 1.4 1.6 1.8
even this type of mechanics bogged down in cumbersome mathematics.
T oday we are in a position to overcome some of the practical difficulties. U sing high-speed computers we can perform calculations that were hitherto impossible. With the help of these machines we are moving a little closer to the ideal of understanding the properties of matter in terms of the mechanical behavior of its constituent particles.
In many applications of statistical mechanics it is possible to proceed with no knowledge of the velocities of individual atoms. We ask only how the atoms are distributed on the average in space. From this distribution alone many of the properties of the material can be calculated.
To appreciate what is meant by the spatial distributions, imagine that we have a microscope powerful enough to see the individual atoms or molecules in a sample of matter, and a camera fast enough to stop them in their rapid flight. A stereoscopic picture made with this arrangement shows how the particles are distributed at a given instant. We examine the particles in the snapshot in turn, measuring the distances between each one and all the others. The information is summarized on a graph, with distance of separation plotted along the horizontal axis, and the number of pairs at each distance aiong the vertical [see illustration below J.
Suppose we make a number of snap-
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DISTANCEOFSEPARATION PLOT for a crystalline solid is characterized by peaks and valleys. Peaks represent preferred distances of separation between the molecules of the crystal. Horizontal axis measures distance in molecular diameters. Vertical axis measures the relative probability that pairs of particles will lie at each distance. From this curve it is possible to calculate the actual numbers of pairs that are separated by each distance.
1959 SCIENTIFIC AMERICAN, INC
shots in rapid succession, say 1,000 in SYLVANIA ELECTRONIC SYSTEMS IN RECONNAISSANCE a second. We plot the distances of sep-aration in each picture and then obtain the average of all the curves. This graph represents the average distribution of pair separations for that second. If the system is in equilibrium, then the average distribution remains constant for all time.
Between each pair of molecules in a piece of matter there is a force that depends on the distance between them. Assuming that we know how the force varies with distance, we can, for example, use our average plot to compute the forces exerted on a typical molecule by its neighbors. This gives us the pressure of the system.
How do we actually find the distances of separation in a system of invisible particles darting about erratically at tremendous speeds? In some cases we can obtain the information experimentally. A beam of X-rays sent through a solid or a liquid is diffracted in a way that depends on the spacing between the particles of the substance. From the size and intensity of spots in the diffraction pattern we can deduce the distances of separation and the number of pairs at each separation. For a crystalline solid the pair-separation graph shows a number of distinct peaks and valleys: as in the illustration on the opposite page. The peaks indicate that the molecules tend to lie preferentially at certain specific distances from one another. This is what we should expect if the molecules are arranged in an orderly grid or lattice. As the temperature of the solid is lowered, the peaks in the plot become sharper, showing that the molecules vibrate less widely about their central positions in the lattice. ''''ith increasing temperature, on the other hand, the peaks grow broader. If the material is heated above its melting temperature, the peaks are still present at distances as small as a few molecular diameters. They disappear at large distances because the lattice structure has disintegrated, and there is no longer an ordering force between molecules at longer ranges.
F'rom the point of view of statistical mechanics we should like to be able to find the distribution of distance between molecules theoretically and to explain how physical conditions such as temperature and density influence the distribution. This would enable us, for example, to predict the pressure or energy of a substance from its temperature and density.
One way of approaching the problem
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