CHAPTER VIIBROWNIAN MOVEMENTS IN GASES
I. HISTORICAL SUMMARY
In 1827 the English botanist, Robert Brown, first made mention of the fact that minute particles of dead matter in suspension in liquids can be seen in a high-power microscope to be endowed with irregular wiggling motions which strongly suggest “life.”[70]Although this phenomenon was studied by numerous observers and became known as the phenomenon of the Brownian movements, it remained wholly unexplained for just fifty years. The first man to suggest that these motions were due to the continual bombardment to which these particles are subjected because of the motion of thermal agitation of the molecules of the surrounding medium was the Belgian Carbonelle, whose work was first published by his collaborator, Thirion, in 1880,[71]although three years earlier Delsaulx[72]had given expression to views of this sort but had credited Carbonelle with priority in adopting them. In 1881 Bodoszewski[73]studied the Brownian movements of smoke particles and other suspensions in air and saw in them “an approximate image of the movements of the gas molecules as postulated by the kinetic theory of gases.” Others, notably Gouy,[74]urged during the next twenty yearsthe same interpretation, but it was not until 1905 that a way was found to subject the hypothesis to a quantitative test. Such a test became possible through the brilliant theoretical work of Einstein[75]of Bern, Switzerland, who, starting merely with the assumption that the mean kinetic energy of agitation of a particle suspended in a fluid medium must be the same as the mean kinetic energy of agitation of a gas molecule at the same temperature, developed by unimpeachable analysis an expression for the mean distance through which such a particle should drift in a given time through a given medium because of this motion of agitation. This distance could be directly observed and compared with the theoretical value. Thus, suppose one of the wiggling particles is observed in a microscope and its position noted on a scale in the eyepiece at a particular instant, then noted again at the end of(for example, 10) seconds, and the displacementin that time along one particular axis recorded. Suppose a large number of such displacementsin intervals all of lengthare observed, each one of them squared, and the mean of these squares taken and denoted by:Einstein showed that the theoretical value ofshould bein whichis the universal gas constant per gram molecule, namely,,the temperature on the absolute scale,the number of molecules in one gram molecule, anda resistance factor depending uponthe viscosity of the medium and the size of the drop, and representing the ratio between the force applied to the particle in any way and the velocity produced by that force. If Stokes’s Law, namely,,held for the motion of the particle through the medium, thenwould have the value,so that Einstein’s formula would becomeThis was the form which Einstein originally gave to his equation, a very simple derivation of which has been given by Langevin.[76]The essential elements of this derivation will be found inAppendix C.
The first careful test of this equation was made on suspensions in liquids by Perrin,[77]who used it for findingthe number of molecules in a gram molecule. He obtained the mean value,which, in view of the uncertainties in the measurement of bothand,may be considered as proving the correctness of Einstein’s equation within the limits of error of Perrin’s measurements, which differ among themselves by as much as 30 per cent.
II. QUANTITATIVE MEASUREMENTS IN GASES
Up to 1909 there had been no quantitative work whatever on Brownian movements in gases. Bodoszewski had described them fully and interpreted them correctly in 1881. In 1906 Smoluchowski[78]hadcomputed how large the mean displacements in air for particles of radiusought to be, and in 1907 Ehrenhaft[79]had recorded displacements of the computed order with particles the sizes of which he made, however, no attempt to measure, so that he knew nothing at all about the resistance factor.There was then nothing essentially quantitative about this work.
In March, 1908, De Broglie, in Paris,[80]made the following significant advance. He drew the metallic dust arising from the condensation of the vapors coming from an electric arc or spark between metal electrodes (a phenomenon discovered by Hemsalech and De Watteville[81]) into a glass box and looked down into it through a microscope upon the particles rendered visible by a beam of light passing horizontally through the box and illuminating thus the Brownian particles in the focal plane of the objective. His addition consisted in placing two parallel metal plates in vertical planes, one on either side of the particles, and in noting that upon applying a potential difference to these plates some of the particles moved under the influence of the field toward one plate, some remained at rest, while others moved toward the other plate, thus showing that a part of these particles were positively electrically charged and a part negatively. In this paper he promised a study of the charges on these particles. In May, 1909, in fulfilling this promise[82]he made the firstquantitative study of Brownian movements in gases. The particles used were minute droplets of water condensed upon tobacco smoke. The average rate at which these droplets moved in Broglie’s horizontal electric field was determined. The equation for this motion wasThe meanwas next measured for a great many particles and introduced into Einstein’s equation:From these two equationswas eliminated andobtained in terms of.Introducing Perrin’s value of,De Broglie obtained from one series of measurements;from another series on larger particles he got a mean value several times larger—a result which he interpreted as indicating multiple charges on the larger particles. Although these results represent merely mean values for many drops which are not necessarily all alike, either in radius or charge, yet they may be considered as the first experimental evidence that Einstein’s equation holds approximately, in gases, and they are the more significant because nothing has to be assumed about the size of the particles, if they are all alike in charge and radius, or about the validity of Stokes’s Law in gases, the-factor being eliminated.
The development of the oil-drop method made it possible to subject the Brownian-movement theory to a more accurate and convincing experimental test than had heretofore been attainable, and that for the following reasons:
1. It made it possible to hold, with the aid of the vertical electrical field, one particular particle under observation for hours at a time and to measure as many displacements as desired on it alone instead of assuming the identity of a great number of particles, as had been done in the case of suspensions in liquids and in De Broglie’s experiments in gases.
2. Liquids are very much less suited than are gases to convincing tests of any kinetic hypothesis, for the reason that prior to Brownian-movement work we had no satisfactory kinetic theory of liquids at all.
3. The absolute amounts of the displacements of a given particle in air are 8 times greater and in hydrogen 15 times greater than in water.
4. By reducing the pressure to low values the displacements can easily be made from 50 to 200 times greater in gases than in liquids.
5. The measurements can be made independently of the most troublesome and uncertain factor involved in Brownian-movement work in liquids, namely, the factor,which contains the radius of the particle and the law governing its motion through the liquid.
Accordingly, there was begun in the Ryerson Laboratory, in 1910, a series of very careful experiments in Brownian movements in gases. Svedberg,[83]in reviewing this subject in 1913, considers this “the only exact investigation of quantitative Brownian movements in gases.” A brief summary of the method and results was published by the author.[84]A full account was published by Mr. Harvey Fletcher inMay, 1911,[85]and further work on the variation of Brownian movements with pressure was presented by the author the year following.[86]The essential contribution of this work as regards method consisted in the two following particulars:
1. By combining the characteristic and fully tested equation of the oil-drop method, namely,with the Einstein Brownian-movement equation, namely,it was possible to obtain the productwithout any reference to the size of the particle or the resistance of the medium to its motion. This quantity could then be compared with the same product obtained with great precision from electrolysis. The experimental procedure consists in balancing a given droplet and measuring, as in any Brownian-movement work, the quantity,then unbalancing it and measuring,and (;the combination of (24) and (25) then givesSince it is awkward to square each displacementbefore averaging, it is preferable to modify by substituting from the Maxwell distribution law, which holds for Brownian displacements as well asfor molecular velocities, the relationWe obtain thusor
The possibility of thus eliminating the size of the particle and with it the resistance of the medium to its motion can be seen at once from the simple consideration thatso long as we are dealing with one and the same particlethe ratiobetween the force acting and the velocity produced by it must be the same, whether the acting force is due to gravity or an electrical held, as in the oil-drop experiments, or to molecular impacts as in Brownian-movement work. De Broglie might have made such an elimination and calculation ofin his work, had his Brownian displacements and mobilities in electric fields been made on one and the same particle, but when the two sets of measurements are made on different particles, such elimination would involve the very uncertain assumption of identity of the particles in both charge and size. Although De Broglie did actually make this assumption, he did not treat his data in the manner indicated, and the first publication of this method of measuringas well as the first actual determination was made in the papers mentioned above.
Some time later E. Weiss reported similar work to the Vienna Academy.[87]
2. Although it is possible to make the test ofin just the method described and although it was so made in the case of one or two drops, Mr. Fletcher worked out a more convenient method, which involves expressing the displacementsin terms of the fluctuations in the time required by the particle to fall a given distance and thus dispenses with the necessity of balancing the drop at all. I shall present another derivation which is very simple and yet of unquestionable validity.
In equation (28) letbe the time required by the particle, if there were no Brownian movements, to fall between a series of equally spaced cross-hairs whose distance apart is.In view of such movements the particle will have moved up or down a distancein the time.Let us suppose this distance to be up. Then the actual time of fall will be,in whichis now the time it takes the particle to fall the distance.If nowis small in comparison with,that is, ifis small in comparison with(say ⅒ or less), then we shall introduce a negligible error (of the order ¹⁄₁₀₀ at the most) if we assume thatin whichis the mean velocity under gravity. Replacing then in (28) (,byin whichis the square of the average difference between an observed time of fall and the mean time of fall,that is, the square of the average fluctuation in the time of fall through the distance,we obtain after replacing the ideal timeby the mean time
In any actual workwill be kept considerably less than ⅒ the mean timeif the irregularities due to the observer’s errors are not to mask the irregularities due to the Brownian movements, so that (29) is sufficient for practically all working conditions.[88]
The work of Mr. Fletcher and of the author was done by both of the methods represented in equations (28) and (29). The 9 drops reported upon in Mr. Fletcher’s paper in 1911[89]yielded the results shown below in whichis the number of displacements used in each case in determiningor.
When weights are assigned proportional to the number of observations taken, as shown in the last column ofTable XIV, there resultsfor the weighted mean value which represents an average of 1,735 displacements,or,as against,thevalue found in electrolysis. The agreement between theory and experiment is then in this case about as good as one-half of 1 per cent, which is well within the limits of observational error.
This work seemed to demonstrate, with considerably greater precision than had been attained in earlier Brownian-movement work and with a minimum of assumptions, the correctness of the Einstein equation, which is in essence merely the assumption that a particle in a gas, no matter how big or how little it is or out of what it is made, is moving about with a mean translatory kinetic energy which is a universal constant dependent only on temperature. To show how well this conclusion has been established I shall refer briefly to a few later researches.
In 1914 Dr. Fletcher, assuming the value ofwhich I had published[90]for oil drops moving through air, made new and improved Brownian-movement measurements in this medium and solved forthe original Einstein equation, which, when modified precisely as above by replacingbyandbecomesHe took, all told, as many as 18,837’s, not less than 5,900 on a single drop, and obtained.This cannot be regarded as an altogether independent determination of,since it involves my value. Agreeing, however,ofas well as it does with my value of,it does show with much conclusiveness that both Einstein’s equation and my corrected form of Stokes’s equation apply accurately to the motion of oil drops of the size here used, namely, those of radius fromcm. tocm..
In 1915 Mr. Carl Eyring tested by equation (29) the value ofon oil drops, of about the same size, in hydrogen and came out within .6 per cent of the value found in electrolysis, the probable error being, however, some 2 per cent.
Precisely similar tests on substances other than oils were made by Dr. E. Weiss[91]and Dr. Karl Przibram.[92]The former worked with silver particles only half as large as the oil particles mentioned above, namely, of radii between 1 and.and obtainedinstead of 9,650, as in electrolysis. This is indeed 11 per cent too high, but the limits of error in Weiss’s experiments were in his judgment quite as large as this. K. Przibram worked on suspensions in air of five or six different substances, the radii varying from 200to 600,and though his results varied among themselves by as much as 100 per cent, his mean value came within 6 per cent of 9,650. Both of the last two observers took too few displacements on a given drop to obtain a reliable mean displacement, but they used so many drops that their meanstill has some significance.
It would seem, therefore, that the validity of Einstein’s Brownian-movement equation had been pretty thoroughly establishedin gases. In liquids too it has recently been subjected to much more precise test than had formerly been attained. Nordlund,[93]in 1914, using minute mercury particles in water and assuming Stokes’s Law of fall and Einstein’s equations, obtained.While in 1915 Westgren at Stockholm[94]by a very large number of measurements on colloidal gold, silver, and selenium particles, of diameter from 65to 130(), obtained a result which he thinks is correct to one-half of 1 per cent, this value is,which agrees perfectly with the value which I obtained from the measurements on the isolation and measurement of the electron.
It has been because of such agreements as the foregoing that the last trace of opposition to the kinetic and atomic hypotheses of matter has disappeared from the scientific world, and that even Ostwald has been willing to make such a statement as that quoted onp. 10.