REPRESENTATION of STATISTICAL DISTRIBUTIONS by CONTINUOUS SPECTRA* T. H. HAZLEHURST, JR.
AND
WALTER H. KELLEY, JR.
Lehigh University, Bethlehem, Pennsylvania
A method of representing statistical distributions has been elaborated and applied to the representation of Maxwellian distribution of molecular nrelocities, "electron density" in the hydrogen atom, black-body radiation, phase space for a unidimensional gas, and "particle density" in the classical and the quantized linear harmonic oscillator. Other applications are suggested.
The method consists in plotting the curves in the usual way and blackening the entire area outside the curve so that the area between the curve and the horizontal axis shows up white against a black background (see Figure 5 ) . The diagram is then wrapped around a wooden cylinder with the axis of abscissas in the figure parallel to the axis of the cylinder. The cylinder is mounted on an axle so that it may be rotated a t 1000 to 1200 R.P.M. by means of a small motor. The eye is
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TATISTICAL magnitudes are most commonly encountered by the student of chemistry through his contacts with the kinetic theory. The statistical treatment of the gas laws, diffusion, conduction of heat, viscosity, and the second law of thermodynamics has been elaborated and standardized since the days of Maxwell and Boltzmann and has become a regular part of instruction in advanced courses, and no college student majoring in chemistry receives his degree without having some knowledge of Maxwell's law of distribution of molecular velocities. This law takes a somewhat different form according as we deal with one, two, or all three dimensions of the gas. Problems of evaporation or of solution, which naturally focus attention upon the component of velocity perpendicular to the interface, involve consideratibn of the distribution of velocities along that one dimension snly. Problems in surface chemistry require consideration of the two dimensions in the plane of the surface, the dimension perpendicular to i t being irrelevant. Finally, the usual problems of gas pressures necessitate the consideration of all three physical dimensions. The three different functions in question, namely the three distribution laws referring respectively to one, two, and three dimensions, may be represented by curves (as they usually are), by tables of figures, which lack the picturesqueness of curves, or by a third method about to be described which makes use of the ability of the eye to compare relative brightness. This third method, while not entirely new, has been applied here for the first time to this and a few related problems. The distribution of velocities becomes visible as a distribution of light intensity.
* Presented before the Division of Chemical Education at the 88th meeting of the American Chemical Society, Cleveland. Ohio, SeDt. 11. 1934.
a, b, and c are spectra of the distribution of molecular velocities along one, two, or three dimensions of the gar, respectively. d. The ordinates of the original curve are values of E/eT, where E is the energy for any wave-length, T is the absolute temperature, and a is a constant. The abscissas are values of bh, where b is a constant, and A the wave-length.
unable, of course, to follow the motion of the diagram, and what is observed is the conglomerate image of the black and white portions of the diagram along what was originally an ordinate in the curve. The higher the ordihate in the curve as originally drawn, the greater is the proportion of white and the brighter that portion of the diagram appears to the eye. When viewed through a slot the whole thing appears just like a continuous spectrum, the intensity being everywhere proportional to the fraction of molecules having the energy represented by the abscissa a t that point. It is obvious that the scale of the diagram must be so chosen that no ordinate is greater than the circumference of the cylinder. Figure 1 a, b, c shows the appearance of such spectra for the distribution of resultant molecular velocities for one, two, and three dimensions of a gas. The curves of
It is sometimes of value to distinguish between the distribution of resultant molecular velocities in, say, a two-dimensional gas and the distribution along each component velocity. What is required is a diagram so shaded that the intensity along any line through the origin shall fall off exponentially in the manner of the "spectrum" for a one-dimensional gas.' Such representations are readily obtained by rotating the proper diagram fastened to the face of a disc instead of to the surface of a cylmder. The abscissas of the curve for the distribution of velocities for the one-dimensional gas (Table 1, column 2) are plotted on polar coordinate paper as radii and the ordinates as angles, being read as degrees either directly or after having been multiplied by some convenient numerical factor. It is evident that if such a diagram is placed concentrically upon a rotating disc the intensity of reflected illumination a t any distance from the center will be proportional to the ordinate in the curve a t that value of the abscissa (radius), that is, to the fraction of molecules having velocities whose components are the coordinates of the point in question, referred to Cartesian axes with origin a t the center. It is as though the velocity of each molecule were represented by a microscopic white dot located a t a point whose x coordinate is equal to the x component of the velocity of the molecule and whose y coordinate is equal to the y component of the velocity of the molecule. This point is called the representative point and i t is customary to refer to the "density" of such points, meaning the number per unit area. On account of the vast number of molecules and thesmall scale of the diagram the density is always huge and the eye sees, not each dot separately, but a gray of lighter or darker shade. The magnitude of the resultant velocity is, of course, measured by the radius vector to the point in question and will be identical for all points on a circle with center a t the origin. By superposing upon such a picture first Cartesian and then polar coordinates i t is made quite obvious to the student (1) that the distribution along each axis in Cartesian coordinates is independent of the distribution along the other; (2) that no particular direction is numbers involved, making it obvious that the ordinates naturally selected as the x- or y-axis, that is, that the have become almost too small to plot by the time w has orientation of the cartesian system makes no differattained the value 3. For any particular gas and any ence; (3) that the division of the plane into cells by be transparticular temperature the values of ,tangular axes, where each cell naturally has the lated into velocities by multiplication by ~ ~ R T / M . same area, gives rise to a distribution with a maximum Other statistical distributions which may be demon- densitv , (number of reoresentative ooints oer cell) a t the strated in the same manner will readily oc.cur to mind. origin, whereas subdivision of the plane into cells by Figure id shows the distribution of energy in the means of radii and circles (polar coordinates), a subspectrum of radiation emitted from a black body. division in which the cells are by no means uniform but The curve used in obtaining this picture is the familiar are much smaller near the origin than elsewhere, gives one demanded by the Planck formula for black-body rise to a distribution with a maximum density not a t the radiation. It would be a simple matter to plot curves origin but farther out. for the spectral distribution of radiation from any sort This last demonstration serves to clear up a point of radiator such as a tungsten filament, a neon tube, which is sometimes troublesome to a student. He fails or a mercury arc lamp. In fact, colored whirling dia- to see why, if the most probable velocity for a onegrams may be profitably employed for direct visual observation by. the class, although naturally they are of HERZPELD, K. F., "Kinetische Theorie der Warme." Vieweg no help in making photographs. und Sohn, Braunschweig, 1925, p. 31. the original diagrams have been so drawn that the areas under them are the same for all; that is to say, the distribution functions have been multiplied by their normalizing factors. This refinement is really superfluous because it would be of use only if we wished to compare quantitatively the intensities in two different pictures, and, aside from the fact that the information so derived could be obtained much more easily in another way, no particular precautions were taken to make the photographs identical in illumination, exposure, or development. The abscissa scale is in units of w = cl-where 6 is velocity, M is the molecular weight of the gas, R the gas constant, and T the absolute temperature. Table 1 shows the actual
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dimensional gas is zero, the most probable resultant velocity for a two- or three-dimensional gas should be other than zero. The demonstration shows that when we consider the resultant magnitude of the velocity in two or three dimensions we are essentially using the plane (or space) polar coordinate system with its cells of varying size. Exactly the same diagram may be used to demonstrate the distribution of representative points in phase space for a one-dimensional gas, one axis repre-
rotating diagram. The diagram is illuminated from the front and observed visually by the class or is photographed for a permanent record. The axle is turned by means of a pulley driven by a small stirring motor. The disc diagrams are mounted on a copper disc (diam. 7") fastened to the end of the axle which bears the cylinder. The brass bushing used to fasten the disc is shown in Figure 2. The diagrams are centered by making use of the tiny projection left on the bushing when it is turned in the lathe, and are fastened to the disc by bolts near the periphery. Naturally, the exact dimensions and set-up are a matter of convenience and it would be feasible to construct an apparatus suitable for demonstrating such figures to a large class in a lecture hall. Anv statistical distribution is amenable to this treatment. For example, i t is possible to reproduce the diagrams published by Ruark and Urey2 or those published by Dusbma+ by constructing circular diagrams of the proper kind. A table (Table 2) of values of the required functions multiplied by factors permitting 2
~
b C FIGURE3 Electron "density" in the hydrogen atom in the states 100 (a), 200 (b), and 300 (c), respectively Q
senting position measured from a fixed point of minimum potential energy and the other ~ o m e n t u m . Strictly speaking, although there will be an'.exponential decrease in density of representative points along both axes, the rate at which the density decreases may not be the same along the two axes, so that contours of equal density would in general be ellipses and not circles. However, in view of the difference in physical nature of the quantities measured along the two axes, we may so select the two scales, which are entirely independent of one another, as to make the rates of decrease identical without loss of generality. A brief description of the apparatus used by the authors in demonstrating such things follows. The wooden cylinder was turned on the lathe from white oak and made exactly 9"in circumference (diam. 2.86"). A pair of brass bushings and a set screw serve to fasten the cylinder to an axle made of 1/4" drill rod. The cylinder is 4" long. The diagrams are conveniently fastened to it by thumbtacks. A slotted piece of black cardboard is arranged so that the slot permits a view of the
accurate comparison is included here for the convenience of those who might wish to construct the figures. The illustrations (Figure 3) show pictures of rotating diagrams on this apparatus demonstrating the electron density in the hydrogen atom in the three quantum states 100, 200, 300. Scales reading in Angstrom units may be photographed simultaneously in order to convey some notion of what one might mean by the term "radius of a hydroxen atom" in each of the states. As a final i l k r k i o n of the use of this apparatus consider the representation of the simple harmonic oscillator as pictured by the wave or quantum mechanics. The classical vibrator with frequency v has a very definite amplitude for each value of the energy. All values of the energy are possible, but for any one value there is a maximum displacement beyond which the vibrating particle will never go. It is possible to plot the probability of finding (at some random mop
a RUARKAND UREY, "Atoms, molecules, and quanta," McGraw-Hill Book Co., Inc , New Yark City, 1930, p. 565. 3 DUSHMAN. S., J. CHEM.EDUC.,8, 1106 (June, 1931).
TABLE 2 PROBABILITY F U N C T I O S 90R ELBETRON DISTRIBUTION IN TRB
HYDRODBN
ATOM 2 1.WOO 0.8187 0.6703 0 5488 0.4493 0.3670 0.3012 0.2466 0.2010
n
0
7-3
Explnnorion: Column I lists values of x = 2rIan. where r is the radius, the radius of the first Bohr orbif, and n the quantum number. Colllmnr
ing factors
ment) the particle a t any chosen point of its path by plotting the reciprocal of the velocity of the particle a t that point against the position of the point itself (the displacement). The reciprocal of the velocity is of course a measure of the time spent in that region and so of the probability of finding the particle there. This is shown in Figure 45, which was made by photo-
b
a
c FIGURE4
d
e
a. Particle "density" for a classical harmonic vibrator of frequency u and energy '/r hv. b, c, d , e. The corresponding quantity for the quantized simple harmonic vibrator of frequency and energy '/x hv, 3 / ~ hv, 5/s hu, '/% hu, respectively.
+
maohing a whirling diagram bearing the curve produced by plotting tKe reciprocal of the velocity against t-..h e Aisnlm-ement The r.---..-- .......... ........intpnsitv reaches a maximum a t the maximum amplitude (because the particle is there moving most slowly) and beyond that point there is com~leteblackness because the particle is not permitted to transgress that limit. The quantized vibrator is not confined to a particular amplitude but may pass even to infinity upon certain rare occasions. The wave mechanics gives a method of calculating the probability of finding the particle at any chosen displacement depending upon the quan"
A
u
L
FIGURE~ . - T S P I C A L DIAGRAMS When rotated at high speed upon the cylinder and disc, respectively, they exhibit the appearance shown in Figures 4c, 3e, and 3c, respectively.
tum number of the e n e r n state in which it may happen to be. Not all energystates are permitted, but only l/p)hv, where h those for which the energy equals (n is Planck's constant, u is the frequency of the vibrator, and n is zero or some positive integer. Table 3 shows the values of the probability functions for the oscillator in the fundamental and the first three excited states (n = 0, 1, 2, 3). These values are plotted against x, which is the ratio of the actual displacement to the amplitude which a classical oscilla-
+
-
-
-
&9lonolion: x r 1.7, where r actual displacement and a amplitude of e l w i c a l harmonic meillator of identical frequency r and enerm ' h hv. the lowest walm of t h e e n e r n oermitted bv . the ouaotum thcorv. The other columns represent the probability of finding the vibrating particle at poGtion x in the four c a s s n 0. 1 . 2 . 3 rcawetivel~. The" are ealeulated from the
-..
.
tor would have in the lowest permitted energy state, in the same manner as the diagrams for the distribution of molecular velocities. Figures 4b, c, d, e show photographs of the resulting appearance when the diagrams are placed upon the cylinder and rotated. Another method of showing the probability function for the classical vibrator is even more direct,.although somewhat less accurate. The path of the particle is plotted (a sine curve) so that the total nineinch length of the diagram just suffices for one complete vibration. The path is drawn with a width of about l/s" and shows white on a black background. If this diagram is fastened to the roller and rotated slowly while being viewed through the slotted cardboard the "particle" may be seen to vibrate back and forth
with a frequency dependent, of course, upon the speed with which the cylinder is rotated. If the cylinder is rotated rapidly a "spectrum" is observed which is nearly identical with that of Figure 4a. The identity would be absolute if the path of the particle were infinitely narrow instead of being '/8"wide. One of our most onerous and important tasks as teachers is that of attempting not only to keep up with recent advances in science ourselves, but to transmit those advances to our pupils in a form which is within their range of comprehension but which does not obliterate the significance of the discoveries. I t is of course out of the question to impart to undergraduates any extremely clear notion of the recent advances in subatomics. It takes 15 to 20 years for even a highly diluted version to find its way into the elementary textbooks. But there is no reason why we should not do what little we can to point out the sort of thing that is being done on the boundaries of science. If i t is useless to attempt to expound quantum mechanics and the philosophical consequences of the uncertainty principle to a class in elementary chemistry, it is quite possible and even desirable to demonstrate the facl that the Bohr picture of electronic orbits of definite shape and size (propounded in 1914 and now a t least mentioned in all texts) must be replaced by pictures such as those in Figure 3, and the notion of oscillators with definite amplitudes enlarged to take in vibrating particles which may move to infinity. The general idea should get across and, while the mental reaction and comment of the student may be crude and naWe in the extreme, it will none the less be a germ of an idea which will develop gradually in his scientificprogress. The authors wish to express theii appreciation of the help of Mr. E. M. Musselman of Lehigh University in the construction of the apparatus described.
