i007Fig. 7
Fig. 7
Fig. 7
The experimental arrangements are shown in Fig. 7. The brass vesselwas built for work at all pressures up to 15 atmospheres, but since the present observations have to do only with pressures from 76 cm. down, these were measured with a very carefully made mercury manometer,which at atmospheric pressure gave precisely thesame reading as a standard barometer. Complete stagnancy of the air between the condenser platesandwas attained, first, by absorbing all of the heat rays from the arcby means of a water cell,80 cm. long, and a cupric chloride cell,and, secondly, by immersing the whole vesselin a constant temperature bathof gas-engine oil (40 liters), which permitted, in general, fluctuations of not more than .02° C. during an observation. This constant-temperature bath was found essential if such consistency of measurement as is shown here was to be obtained. A long search for causes of slight irregularity revealed nothing so important as this, and after the bath was installed all of the irregularities vanished. The atomizerwas blown by means of a puff of carefully dried and dust-free air introduced through cock.The air about the dropwas ionized when desired, or electrons discharged directly from the drop, by means of Röntgen rays from X, which readily passed through the glass window.To the three windows(two only are shown) in the brass vesselcorrespond, of course, three windows in the ebonite strip,which encircles the condenser platesand.Through the third of these windows, set at an angle of about 28° from the lineand in the same horizontal plane, the oil drop is observed through a short-focus telescope having a scale in the eyepiece to make possible the exact measurement of the speeds of the droplet-star.
In plotting the actual observations I have used the reciprocal of the pressurein place of,for the reason thatis a theoretical quantity which is necessarily proportional to,whileis the quantity actually measured.This amounts to writing the correction-term to Stokes’s Law in the forminstead of in the formand consideringthe undetermined constant which is to be evaluated, as wasbefore, by dividing the slope of our line by its-intercept.
Nevertheless, in view of the greater ease of visualization ofall the values of this quantity corresponding to successive values ofare given inTable X.Fig. 5shows the graph obtained by plotting the values of,againstfor the first 51 drops ofTable X, andFig. 6shows the extension of this graph to twice as large values ofand.It will be seen that there is not the slightest indication of a departure from a linear relation betweenandup to the value,which corresponds to a value ofof .4439 (see drop No. 58,Table X). Furthermore, the scale used in the plotting is such that a point which is one division above or below the line inFig. 5represents in the mean an error of 2 in 700.It will be seen from Figs.5and6that there is but one drop in the 58 whose departure from the line amounts to as much as 0.3 per cent. It is to be remarked, too, that this is not a selected group of drops, but represents all of the drops experimented upon during 60 consecutive days, during which time the apparatus was taken down several times and set up anew. It is certain, then, that an equation of the form (15) holds very accurately up to.The last drop ofFig. 6seems to indicate the beginning of a departure from this linear relationship. Since such departure has no bearing upon the evaluation of,discussion of it will not be entered into here, although it is a matter of great interest for the molecular theory.
Attention may also be called to the completeness of the answers furnished by Figs.5and6to the question raised inchap. IVas to a possible dependence of the drag which the medium exerts on the drop upon the amount of the latter’s charge; also, as to a possible variation of the density of the drop with its radius. Thus drops Nos. 27 and 28 have practically identical values of,but while No. 28 carries, during part of the time, but 1 unit of charge (seeTable X), drop No. 27 carries 29 times as much and it has about 7 times as large a diameter. Now, if the small drop were denser than the large one, or if the drag of the medium upon the heavily charged drop were greater than its drag upon the one lightly charged, then for both these reasons drop No. 27 would move more slowly relatively to drop No. 28 than would otherwise be the case, and hencefor drop No. 27 would fall belowfor drop No. 28. Instead of this the twofall so nearly together that it is impossible to represent them on the present scale by two separate dots. Drops Nos. 52 and 56 furnish an even more striking confirmation of the same conclusion, for both drops have about the same value forand both are exactly on the line, though drop No. 56 carries at one time 68 times as heavy a charge as drop No. 52 and has three times as large a radius. In general, the fact that Figs.5and6show no tendency whatever on thepart of either the very small or the very large drops to fall above or below the line is experimental proof of the joint correctness of the assumptions of constancy of drop-density and independence of drag of the medium on the charge on the drop.
The values ofandobtained graphically from the-intercept and the slope inFig. 5areand,being measured, for the purposes of Fig. 5 and of this computation in centimeters of Hg at 23° C. andbeing measured in centimeters. The value ofin equations 15 and 16 (p. 101) corresponding to this value ofis .874.
Instead, however, of taking the result of this graphical evaluation of,it is more accurate to reduce each of the observations ontoby means of the foregoing value ofand the equationThe results of this reduction are contained in the last column ofTable X. These results illustrate very clearly the sort of consistency obtained in these observations.The largest departure from the mean value found anywhere in the table amounts to 0.5 per cent and “the probable error” of the final mean value computed in the usual way is 16 in 61,000.
Instead, however, of using this final mean value as the most reliable evaluation of,it was thought preferable to make a considerable number of observations at atmospheric pressure on drops small enough to makedeterminable with great accuracy and yet large enough so that the whole correction term to Stokes’s Law amounted to but asmall percentage, since in this case, even though there might be a considerable error in the correction-term constant,such error would influence the final value ofby an inappreciable amount. The first 23 drops ofTable Xrepresent such observations. It will be seen that they show slightly greater consistency than do the remaining drops in the table and that the correction-term reductions for these drops all lie between 1.3 per cent (drop No. 1) and 5.6 per cent (drop No. 23), so that even thoughwere in error by as much as 3 per cent (its error is actually not more than 1.5 per cent),would be influenced by that fact to the extent of but 0.1 per cent. The mean value ofobtained from the first 23 drops is,a number which differs by 1 part in 3,400 from the mean obtained from all the drops.
When correction is made for the fact that the numbers inTable Xwere obtained on the basis of the assumption,instead of(seeSection II), which was the value ofchosen in 1913 when this work was first published, the final mean value ofobtained from the first 23 drops is.This corresponds to
I have already indicated that as soon asis known it becomes possible to find with the same precision which has been attained in its determination the exact number of molecules in a given weight of any substance, the absolute weight of any atom or molecule, the average kinetic energy of agitation of an atom or molecule at any temperature,and a considerable number of other important molecular and radioactive constants. In addition, it has recently been found that practically all of the important radiation constants like the wave-lengths of X-rays, Planck’s,the Stefan-Boltzmann constant,the Wien constant,etc., depend for their most reliable evaluation upon the value of.In a word,is increasingly coming to be regarded,not only as the most fundamental of physical or chemical constants, but also the one of most supreme importance for the solution of the numerical problems of modern physics. It seemed worth while, therefore, to drive the method herewith developed for its determination to the limit of its possible precision. Accordingly, in 1914 I built a new condenser having surfaces which were polished optically and made flat to within two wave-lengths of sodium light. These were 22 cm. in diameter and were separated by 3 pieces of echelon plates, 14.9174 mm. thick, and all having optically perfect plane-parallel surfaces. The dimensions of the condenser, therefore, no longer introduced an uncertainty of more than about 1 part in 10,000. The volts were determined after each reading in terms of a Weston standard cell and are uncertain by no more than 1 part in 3,000. The times were obtained from an exceptionally fine printing chronograph built by William Gaertner & Co. It is controlled by a standard astronomical clock and prints directly the time to hundredths of a second. All the other elements of the problem were looked to with a care which was the outgrowth of five years of experience with measurements of this kind. The present form of the apparatus is shown in diagram inFig. 8, and inFig. 9is shown a photograph taken before the enclosing oil tank had been added.
i008Fig. 8—,atomizer through which the oil spray is blown into the cylindrical vessel.oil tank to keep the temperature constant.and,circular brass plates, electric field produced by throwing om 10.009-volt battery.Light from arc lampafter heat rays are removed by passage throughand,enters chamber through glass windowand illuminates dropletbetween platesandthrough the pinhole in.Additional ions are produced aboveby X-rays from the bulb.
Fig. 8—,atomizer through which the oil spray is blown into the cylindrical vessel.oil tank to keep the temperature constant.and,circular brass plates, electric field produced by throwing om 10.009-volt battery.Light from arc lampafter heat rays are removed by passage throughand,enters chamber through glass windowand illuminates dropletbetween platesandthrough the pinhole in.Additional ions are produced aboveby X-rays from the bulb.
Fig. 8—,atomizer through which the oil spray is blown into the cylindrical vessel.oil tank to keep the temperature constant.and,circular brass plates, electric field produced by throwing om 10.009-volt battery.Light from arc lampafter heat rays are removed by passage throughand,enters chamber through glass windowand illuminates dropletbetween platesandthrough the pinhole in.Additional ions are produced aboveby X-rays from the bulb.
i009Fig. 9
Fig. 9
Fig. 9
This work was concluded in August, 1916, and occupied the better part of two years of time. The final table of results and the corresponding graph are given inTable XIand inFig. 10. The final value ofcomputed on the basisis seen to be nowinstead of 61.085, or .07 per cent higher than the value found in 1913. But Dr. Harrington’s new value of,namely, .00018226, is more reliable than the old value and is lower than it by .07 per cent. Sinceappears in the first power in,it will be seen that the new value[54]of,determined with new apparatus and with a completely new determination of all the factors involved, comes out to the fourth place exactly the same as the value published in 1913, namely,The corresponding values ofandare now .000617 and .863, respectively.
Since the value of the Faraday constant has now been fixed virtually by international agreement[55]at 9,649.4 absolute electromagnetic units, and since this is the numberof molecules in a gram molecule times the elementary electrical charge, we haveAlthough the probable error in this number computed by the method of least squares fromTable XIis but one part in 4,000, it would be erroneous to infer thatandare now known with that degree of precision, for there are four constant factors entering into allof the results inTable Xand introducing uncertainties as follows: The coefficient of viscositywhich appears in the ³⁄₂ power introduces intoanda maximum possible uncertainty of less than 0.1 per cent, say 0.07 per cent. The cross-hair distance which is uniformly duplicatable to one part in two thousand appears in the ³⁄₂ power and introduces an uncertainty of no more than 0.07 per cent. All the other factors, such as the volts and the distance between the condenser plates, introduce errors which are negligible in comparison. The uncertainty inandis then that due to two factors, each of which introduces a maximum possible uncertainty of about 0.07 per cent. Following the usual procedure, we may estimate the uncertainty inandas the square root of the sum of the squares of these two uncertainties, that is, as about one part in 1000. We have then:
Perhaps these numbers have little significance to the general reader who is familiar with no electrical units save those in which his monthly light bills are rendered. If these latter seem excessive, it may be cheering to reflect that the number of electrons contained in the quantity of electricity which courses every second through a common sixteen-candle-power electric-lamp filament, and for which we pay ¹⁄₁₀₀₀₀₀ of 1 cent, is so large that if all the two and one-half million inhabitants of Chicago were to begin to count out these electrons and were to keep on counting them out each at the rate of two a second, and if no one of them were ever to stop to eat, sleep, or die, it would take them just twenty thousand years to finish the task.
Mean = 61.126.
i010Fig. 10
Fig. 10
Fig. 10
Let us now review, with Figs.5and10before us, the essential elements in the measurement of.We discover, first, that electricity is atomic, and we measure the electron in terms of a characteristic speed for each droplet. To reduce these speed units to electrical terms, and thus obtain an absolute value of,it is necessary to know how in a given medium and in a given field the speed due to a given charge on a drop is related to the size of the drop. This we know accurately from Stokes’s theory and Arnold’s experiments when the holes in the medium, that is, when the values ofare negligibly small, but whenis large we know nothing about it.Consequently there is but one possible way to evaluate e, namely, to find experimentally how the apparent value of,namely,,varies withor,and from the graph of this relation to find what value,approaches asorapproaches zero. So as to get a linear relation we find by analysis that we must plotinstead ofagainstor.We then getfrom the intercept of an experimentally determined straight line on the-axis of our diagram. This whole procedure amounts simply to reducing our drop-velocities to what they would be if the pressure were so large orso small that the holes in the medium were all closed up.For this case and for this case alone we know both from Stokes’s and Arnold’s work exactly the law of motion of the droplet.