222 RUCRER ON THE XXII.-On the Range of Molecular Forces. By A. W. R~~CKER M.A. F.R.S. THE subject on which I have been asked to address the Chemical Society is the Range of Molecular Forces and it will perhaps be well that I should by way of prelude explain the meaning which I myself attach to that term. The investigation of the movements of a group of atoms or molecules is-though far more complex-in some respects similar to the study of the solar system. Newton proved that the Sun th RANGE OF MOLECULAR FORCES. 223 planets and their satellites behave as if a mutual action at ti distance taking place between them modified their motions in accordance with a very simple rule. The wonderful impetus which this rule-the law of gravitation-gave to astronomy has led to many attempts to apply a similar method t o molecular dynamics.Newton’s law is thus frequentlyregarded as being only the first term of a more complex expression which if complete would-on the hypothesis of action a t a distance-give the true law of mutual force between the ultimate particles of matt,er. The first term expresses all the results of experiment when the distances between the particles are con-siderable but is insufficient when they are near together. The investigation of the other terms which then become important may be properly spoken of as the study of molecular forces. Sir William Thomson (Proc. Roy. Institution 11 Pt. 111 483) has indeed expressed the opinion that it is possible that the phenomena of cohesion and others which arc! ordinarily ascribed to a departure a t small distances from the law of gravitation may not be incon-sistent with it.In that case the additional terms are introduced by the attempt to apply a formula founded on the assumed continuity of matter to phenomena which are caused by its “ coarse grainedness.” Interesting as this suggestion is it has not been worked out suffi-ciently tjo make it easy to translate all that we know of molecular action into language consistent with it. I shall therefore adhere to the ordinary usage and assume that it is probable that a somewhat complex expression is required for the full statement of the law of force between two molecules. If this formula were fully known the physical interpretation to be given to it would still be open to discussion Formerly it would no doubt have been considered sufficient to state as an ultimate fact that the law of the force in play between two molecules varies with the distance t!hat it is for instance attractive when they are far apart, repulsive when they are near together.Now such a supposition is branded as artificial but I venture to think that the artificiality is due rather to the acceptance of the hypothesis of action a t a distance than to the assumed complication of the law. There are two closely related yet really distinct ways in which an apparent repulsive force may be (in the ordinary sense of the word) explained. It may as has been the case with centrifugal force be shown to be an effect of motion and inertia without any abandonment of the theory of action at a distance in the case of the other forces involved.Or it may be explained as a result of the properties of a medium by which mat,ter is surrounded or of which each atom is a epecialised part If action a t a distance is thus reduced to action in proximity if machinery is imagined adequate to account for tlie effects which distant bodie 224 RUCKER ON THE produce upon each other it should explain not only the repulsions but also the attractions not only molecular elasticity but gravita-tion. It is I believe sometimes thoughtX that the next step in the progress of the theory of the constitution of matter will be the assumption of an unexplained attraction only between its ultimate particles while their elasticity is okherwise accounted for. This explanation of elasticity may or may not involve the hypothesis of a medium extending between the molecules.If we dispense with it we must not be content with vague analogies to account for the behaviour of two molecules during an encounter. It is true that a comet coming out of space toward the solar system might and probably would travel round the Sun without a collision but in meteor swarms and in gases with non-repelling molecules collisions must take place and Sir William Thomson (Rep. Brit. ASSOC. 1884 616) has insisted on the fact that the result of such collisions in a gas must be the transformation of energy of trans-lation into energy of vibration with the spontaneous cooling of the gas as a result. It is precisely because no such effect is observed that the theory of elastic molecules is abandoned.We are thus driven to suppose that elasticity is due to a repulsion, and if we refuse to accept the theory of action a t ;;L distance to introduce a medium by which the effect of a repulsion acting a t :t distance may be produced. It is however absurd to accept attraction a t a distance and to refuse to conceive of a repulsion acting nnder similar circumstances to admit the one without explanation and to invent a medium to account for the other. The most pregnant suggestions as to the constitution of matter which have hitherto been made do not proceed on these lines. An unexplained attraction is not assumed between vortex atoms in addition to the effects which follow from the laws of hydrodynamics. If it were necessary to adopt such an hypothesis the vortex atom theory would evidently be as artificial as that embodied in the bald statement that the law of force changes with the distance from attraction to repulsion.It is perhaps possible that some such hybrid theory might serve as a useful basis for calculation but from the philosophical standpoint it would not be a whit more conceivable than any other which involves action a t a distance. If then we are to retain the language of the latter theory in any part of our discnssion it will be convenient and not less accurate to retain it throughout 011 the distinct understanding t h a t it is a conventional mode of representing facts which we do not fully understand and which it does not suffice to explain completely.* “ On the Law of Molecular Force,” by W. Sutherland Phi7. Mag. July 1887, p. 127 RANGE OF JIOLECULAR FORCES. 285 A better method of expression may be found when suggestions like the vortex-atom theory of Sir William Thomson and the granular theory of Professor Osborne Reynolds (Phil. Mag. December 1885) are worked out. They no doubt will present grave difficulties as to the true nature of the action in proximity which takes place between contiguous granules or between neighbouring layers of the ideal fluid in which the vortices are formed but they will be justified as working hypotheses if they reduce the di5cul ties connected with the explana-tion of a large number of physical phenomena under a few heads. If then we use provisionally the language of action at a distance in the expectation that it will ultimately be replaced by a theory of action in proximity I think we ought from the first to admit that the law of force between molecules may be very complicated.We must not dismiss any idea which experiment suggests-such for instance as that there are several alternations of attraction and repulsion between two molecules as the distance between them diminishes-merely because it appears arbitrar-y and lacking in simplicity. It may be admitted for the sake of argument that we naturally look for simplicity in our fundamental assumptions but if the machinery by which distant bodies affect each other if the medium by which force is transmitted is simple it by no means follows that its effects on matter can be expressed on the action at a distance hypothesis by an easy formula.Even in the case of a single ball moving through a perfect liquid bounded by an infinite plane it will be apparently attracted to or repelled by the boundary according as it is projected parallel to or towards it. In the vortex-atom theory the behaviour of two molccules during an encounter would depend entirely upon the circumstances of the collision and cannot be very shortly de-scribed. I n the important case of a single vortex ring passing by a large number of others uniformly distributed it will experience a repulsion.* In Professor Osborne Reynolds’s grauular theory two molecules would exhibit mutual attraction and repulsion at different distances. I n none of these cases can the fundamental assumptions be regarded as complicated yet they all give rise to repulsions as well as attractions.They do not lead to simple expressions for the forces in play between molecules separated by distances of the same order of magnitude as their diameters. It would be perhaps too much to say that a simple result could only be produced by a medium of com-plicated constitution but it is certainly true that we have ci priori no more right to expect simplicity in the results of its action than simplicity in its constitution and that the two are not necessarily obtained together. f ‘‘ Motion of Vortex Rings,” J. J. Thomson. Mncmillan 1883 p. 55 226 RUCEER ON THE Thus much it has been needful t o say in order to explain the point of view from which I wish to regard molecular forces in this lecture.I shall use the language of the action a t a distance theory throughout, not-as I hope I have made clear-because I believe in it but because, i n so far as it can express them at all it affords a self-consistent method of describing facts the causes of which are as yet imperfectly under-stood. I shall not discuss the question of the representation of the forces by an algebraical formula. I cannot in the short time a t my disposal exhanst tho more limited subject to which I intend to confine myself. I desire only t o lay before you an outline of the results of the principal experiments which have been made with the view of determining the distances through which a law of force apparently different from that of gravitation obtains.The greatest distance from a molecule at which this deviation is sensible is called " the radius of molecular action." It constitutes the superior limit to the range of molecular forces. The inferior limit is what is ordinarily called the radius of a molecule but which if we regard the molecules as exer-cising when in close proximity a mutual repulsion is a length related to half the average minimum distance between their centres during an encounter. This distance may depend on the temperature and on the physical state of the body so that the diameter of a molecule may be different according as it is determined from experiments on gases or liquids. While retaining it as a convenient phrase it will emphasise the con-ventional sense in which it is used if we speak of the diameter of a gaseous or liquid molecule as the case may be.Between the limits thus defined the law of force is unknown, though interesting suggestions have been made by Maxwell and others, but apart from this question which as I have said I do not now discuss, the limits themselves may be determined very differently by different methods. The question as to whether a molecular force is " sensible " a t a given distance from the molecule depends partly upon the sensitiveness of the means used to detect it and partly upon the nature of the phenomenon-electrical optical or otherwise-studied. It is impossible therefore to group the results of various observa-tions into a connected whole but it may nevertheless be useful to give a short ~e'surne' of the conclusions to which different observers have been led and to attempt to arrange them as far as may be in order.The largest values which have been obtained for the magnitude of the radius of molecular action have been deduced from observations on the condensation of films of gases and vapours on the surfaces of solids. Quincke (Pogg. Ann. 108 326 1859) in 1859 argued that if it be assumed that the law of molecular force is the sam RAKU'GE OF BIOLECULAR FORCES. 227 for molecules in the gaseous liquid and solid states the superior specific gravity of a solid would enable it to condense a gas upon its surface. It is evident however from recent observations that the nature of the solid is of eren greater importance than its density. Among the more remarkable investigations on this point is that of Bunsen ( V i e d .Ann. 20 552 1883). A bundle of glass threads the total surface of which was determined by preliminary observations and calculations was enclosed in a chamber connected with a long tube, the lower end of which was dipped in mercury. The gradual rise of the mercurial column showed that an apparent absorption of carbonic acid by the glass was still going on a t the end of three years. Later observations ( W i e d . Ann. 24 1885 322) proved that although the glass had been carefully dried it is impossible to get rid of all the adhering moisture unless the temperature is raised to a point not far short of the critical temperature of water. If this precaution Bas been omitted carbonic acid if present will according to Bunsen be dissolved in the water-film and since the inner layers of the liquid are subjected by molecular attraction to a pressure which is measured by hundreds of atmospheres they are capable of absorbing enormous quantities of the gas.The strong acid thus formed appears to attack the glass and it was found that nearly 6 per cent. of the total mass of glass threads employed had been disintegrated ( W i e d . Ann., 29 1886 161). The long-continued apparent condensation was, therefore really slow chemical action. Nay more when the glass had been dried a t a high temperature no appreciable condensation of carbonic acid on the surface took place in eight days ( W i e d . Ann., 24 1885 335). A small quantity of water was then introduced and absorbed by the glass threads with a rapidity which showed that when really dry they acted as a more powerful desiccator than calcium chloride.Immediately after the introduction of the water the absorption of the C02 began as before which proved that moisture was necessary to produce the phenomenon or that carbonic acid does not condense to a measurable amount on dry glass. By exposing glass threads to a series of constant teniperatures until in each case no more moisture could be exhacted by the passage of a current of dry air over them and by measuring the successive quantities of water thus obtained Burlsen waa able to calculate the total thickness of the water film which at each of these temperatures is irremovable by dry air. Under ordinary conditions water does not eva,porate when its vapour exerts upon the surface a particular pressure the magnitude of which varies with the temperature.Any internal layer parallel to the surface is subjected to an additional molecular pi-essure which increases rapidly with the depth until th 228 RUCKER ON THE 6 *YO 5'47 3 -63 1'32 0 *4,2 0 -00 boundary of the superficial portion of the liquid is reached after which it becomes constant throughout the interior. If the interior mass of water be replaced by a solid which exerts, ceteris paribus a greater attraction on vater than that of water itself, the molecular pressure would be increased and thus the vapour-tension might be diminished without evaporation taking place. The defect of the external pressure would be balanced by the increased molecular attraction.If then we assume that the thickness of the water film which cannot a t any given temperature be removed by dry air is such that the pressure due to molecular attraction a t the surface of the film is equal to the pressure of aqueous vapour at the temperature a t which the experiment is made it is possible when the vapour-tension is known to calcnlat-e the molecular pressure f o r given thicknesses of the film. The following table expresses Bunsen's results. The temperature is expressed in degrees centigrade. The thickness neglecting some minor corrections is indicated by D and expressed in terms of micromillimetres (p.p.)* The pressure is ex-pressed in atmospheres :-1 *278 20 .791 ----t. D. 23' 107 21 5 329 415 468 503 If the desiccation with dry air was incomplete the thickness of the films very much exceeded the above limits.Thus in one experiment in which the drying was purposely imperfect the water layer was 232.4 p.p. thick. The interpretation to be placed on these results has however been again rendered doubtful by the experiments of Warburg and Ihmori (Wied. Ann. 27 481 1886). These observers constructed a small balance of extraordinary delicacy which was enclosed iii an exhausted receiver which could be connected at pleasure with vessels containing strong sulphuric acid or water. When it had been dried by frequent evacuation water vapour was admitted, and the weight of the films deposited on a thin glass bulb suspended from the balance was determined.They found that if the glass was washed with boiling water before the experiment the deposited film was very much thinner than if this * The micromillimetre is the millionth part of a millimetre RANGE OF MOLECULAR FORCES. 229 precaution had been omitted. The thickness diminished in two experi-ments in the proportion of 48 to 4 and 23 t o 2. I n a third case no film could be detected even when the temperature of the receiver was only 0.2" above the dew point. Glass rods which have been boiled for a few minutes will not discharge an electroscope even when they have for long been in a relatively damp atmosphere under circumstances such that rods of the same glass which have not been similarly treated conduct readily. The film of moisture adherent to glsss may thus be divided into two parts distinguished as the permanent and temporary respectively, of which the latter disappears under the influence of a long-con-tinued current of dry air while the former can only be removed by raising the temperature.Warburg and Ihmori conclude that the temporary film (with which they alone deal) is not produced by tlie molecular at,traction of the glass as a whole on water vaponr. They refer to experiments which prove that if glass powder be boiled in water measurable quantities of alkali are dissolved. They therefore, assume that there is a certain quantity of free or loosely combined alkali on the surface of the glass and that it absorbs water until a solution is formed the vapour-tension of which corresponds to thc hygrometric state of the air in the neighbourhood.If carbonic acid is then absorbed by the solution the glass may be attacked and the process continued. Further experiments have been made by Ihmori (Wied. Ann. 31, 1006 1887). He finds that the water films on clean unvarnished metal surfaces are extremely thin varying from 10 to 3p.p. On oxidised metal they may be twice as thick and he inclines to tlie view that in all cases the phenomenon is due to oxidation. Varnished metal may in 20". absorb enough t o produce a layer 286 p.p. thick, and sealing-wax also absorbs large quantities. Nineteen experiments on quartz gave a mean thickness of 22 ,u.p. with a maximum of 62 p.p. Six observations made when the crystal had been previously washed gave a mean of 4 p.p.and a maximum of 6 p.p. only. The mean result of 11 experiments on platinum was under 3 p.pu. the maximum being 122 p p . I n the case of a piece which was specially cleaned by heating no condensation could be detected. Agate absorbs large quantities of water. Films the thickness of which varied between 562 and 1640 pp. are stated to be the result of an hour's exposure to a moist atmosphere. It is however well known though Herr Ihmori does not' refer to the fact that agate consists of alternate layers of quartz and a porous form of silica allied to opal. Professor Judd has kindly furnished me with specimens which have been immersed in coloured solutions. These have been absorbed by the porou 230 RUCKER ON THE layers and thus coloured bands are formed.I n this way a good imitation of an onyx may be produced. There can be no doubt that tlhe surface exposed by the agate to the water-vapour includes that of the interior of a vast number of capillary tubes and is enormously greater than the mere external surface. The quantity of water absorbed does not therefore give any indication of the thickness of the water film and no deduction as to the radius of molecular action can be drawn from it. The net result of these experiments is to render it doubtful whether in the case of substances which are not dissolved or chemically acted on by water any measurable temporary film is formed a t temperatures above the dew point. If such a film is formed in these cases its thickness is according to Warburg and Ihmori in general very much less than the radius of molecular action as determined by Quincke.PfeifEer (Wied. Bei. 8 635 1884) who published some experi-ments on the absorption of gases by solids a t high pressures arrived a t the conclusion that layers of ammonia and carbonic acid of the thick-ness of 450 p.p. and 240 p.p, are formed on charcoal made from firwood. As however the result is based on calculations made from box-wood charcoal in which it is assnmed that it condenses SO exactly in the same way as glass does but little reliance can be placed on it. It is probable that water films play as important a part in the ap-parent condensation of SO on the surface of glass as they do in that of co,. Another group of experiments has been made with iron oxide, alumina and silica which readily absorb water and carbon bisulphide.Thus Muller-Erzbach (Ezner’s Rep. 21,409,1885) measured the grains of finely powdered oxide of iron under the microscope and concluded that a certain area was greater than that of a given weight of the powder. He then deduced from this datum and the weight of CS absorbed by the oxide the thickness of the film. Assuming that the specific gravity of the absorbed CS was increased from 1-27 to 1.60 by causes similar to those which affect the specific gravity of water of crystallisation he concluded that the film was a t least 1000 p p . in thickness. Similar calculations (Exner’s Rep. 21 553 1885) gave 1700 +,u. for the thick-ness of a film of CS adherent to alumina. He finally concludes that the radius of molecular action is at least 1500 p.p.(Wied. Ann., 28 696 1886). Kayser (Wied. Ann. 14 450 ISSl) as the result of experiments on the condensation of gases on glass thread was of opinion that the quantity condensed depended on t’he closeness with which the fibres were packed and that the radius of molecular action was of the same order of magnitude as the diameter of the threads. He fixes its magnitude a t from 2000 to 3000 p.p. As these observations were made before the importance of getting rid of the water film b RASGE OF MOLECULAR FURCES. 231 heating had been demonstrated they cannot be accepted as support-ing this enormoils value. Passing next to the condensation of the more permanent gases, I may refer to a calculation made by Callendar," on the assnmp-tion that the differences between the coefficients of expansion of air between 0" and 100" C.given by various air thermometeru, depend on the quantity of air condensed when they are cooled and thus upon the ratio between the surface and volume of the bulbs, which of course varies with their shapes. He shows that the values for the coefficients of expansion of air at constant volume between 0" and 100" C. obtained by Regnault Bidfour Stewart and himself, would agree if the weight of air condensed between those tempera-tures is 10-6 grams per sq. em. From this we deduce that the thick-ness of the film removed by heating from 0" to 100" C. would be 10 p.p. if the density be assumed to be the same as that of water. Schumann (Wied. Am,. 27 91 1886) has also recently pointed out that if the layer of air condensed on glass reaches con-siderable dimensions the length of a mercurial thyead in a capil-lary tube would be appreciably different according as the film was or was not present.He therefore connected a long bent capillary tiibe with a bulb and when the positions of the ends of a thread of mercury had been determined it was transferred to the bulb. The tube was then exhausted and heated to 312" C. which could be accomplished without heating the mercury. When the apparatus had become cold the mercury was returned to the capillary tube. Its length was found to be precisely the same as before. The thickness of the air film removed by the heating could not therefore according to Schumann have been greater than 70 pp.His method of course involves the assumption which seems legitimate that the mercury would not remove the fiIm from the glass as it moved along the tube. An argument to the same effect may be drawn from some observa-tions made by Bottomley (Chem. News 51 85 1885) and published in 1885. He exhausted a vessel containing glass fibres by means of ft mercury pump until the pressure as indicated by a McLeod gauge was 0.3 M.? He then heated the vessel and its contents until some of the glass fibres began to soften and collected the gas which was given off. It amounted in all to 0.45 c.c. at 15" C. and 760 mm. and when analysed was found to contain 8-24! per cent. GOz 248 per cent. Oz, and 75.2 per cent. N?. The total surface of the fibres was 14-48 sq. cm. The gas as it left the vessel was dried and it is not stated that the * " On the Practical Measurement of Temperature," Phil.Trans. vol. 178 (1887)) A. p. 161. t M = 1 millionth of an atmosphere 232 RUCKER ON THE glass fibres had been previously washed so that there probably was a water film of the magnitude of which no estimate could be formed. It is therefore probable that the gases were partly dissolved but a t all events the observations lent no support to the idea that the gas film on glass dried only by contact with dry air is very thick. If we assume that the mixture when collected was of the same density as air and that when in contact with the glass it had the same density as water the thickness of the film of condensed gas was 4 ,u.p. which is somewhat less than the number deduced on the same hypothesis from Callendar’s suggestion.On the whole looking only a t the very contradictory results attained by different researches and without regard to arguments which T shall presently adduce I must confess that I do not think we can a t present draw any certain conclusion as to the magnitude of the radius of molecular action from observations on the condensa-tion of vapours or gases. TO justify this view I cannot perhaps do better than quote from two of the gentlemen who have studied the phenomena most closely. I n 1885 Miiller-Erzbach remarks (Exner’s Rep. 21 1885 407) :-“Ich habe nun . . . ein Mittel gefunden durch welches ich auf einfache Weise glaube beweisen zu konnen dass die i n Betracht konimenden Molecularkrafte nicht nur bei unmittelbarer Beriihrung wirksam sind sondern selbst noch in einem griisseren Abstand als ihn Hr Quincke nach seinen Versuchen bestimmt hat.” In 1886 Warburg and Ihmori sum up these results as follows (Wied.Ann. 27 507 1886) :-“ Es liegt uns fern die Richtigkeit der Schlusse Quincke’s auzueweifeln . . . . Allein in den Messungen, welche wir uber das Gewicht der Wasserhaut bei Glas und anderen Kor pern anges tellt ha ben ist uns nic hts en t’gege n ge t reten woraus eine Wirking der Molecularkrafte auf niessbare Distanzen hin zu erschliessen wiire.” The first important attempt to measure the radius of molecular action was made by Plateau (Stntique des Liguides 1873 1, 210). Arguing that the surface-tensions would decrease if the thickness of a soap film became less than twice the radius he made experiments to determine whether the pressure exerted on the enclosed air depended on the thickness.His method was open to criticism. The soap bubble was made of a mixture of soap water, and glycerine and thus its constituticn would alter unless it were surrounded by aqueous vapour of a determinate tension. No pre-cautions were taken to secure this condition. The bubble was pro-duced a t the end of a tube bent so as t o form a manometer. The liquid used to measure the pressure was water while the rate of thinning of the bubble was accelerated by enclosing it in a covere RASGE OF MOLECULAR FORCES. 233 beaker in which were placed some sticks of caustic potash. Under these circumstances it is impossible to say what the final constitution of the liquid might be.By measuring the specific electrical resistance of various mixtures of glycerine and soap and water and the resist-ance of cylindrical soap films formed of the same substances Pro-fessor Reinold and I have been able to measure the changes i u the constitution of films when subjected to variations in the temperature or hygrometric state of the surrounding air. We found it very diffi-cult to secure constant conditions and that under circumstances far more favourable than those of Plateau’s experiments the films lost one of the 57.7 volumes of water originally contained in every 100 of solution in times which varied between four and eight minutes (Phil. Trans. Part 11 1881 486). As Plateau’s film lasted two days it is evident a change of compo-sition sufficient to have caused a marked or considerable change in surface- tension due to thinning might have occurred.Platleau observed no change in the pressure when the colours of the bubble proved that its thickness was 118 p.p and thence concluded that the radius of molecular action is < 59 p.p. This inference is not so certain as he appears to have thought ii; to be. Maxwell (Art. “ Capillary Action,” Enc. Brit. ed. ix) has shown that if we neglect the change of density in the surface of the liquid and the thermal phenomena which accompany the thinning of a film the surface-tension will remain nnchanged until the thickness is equal to the radius of molecular action. It is difficult to estimate the extent to which this result might be affected by a theory which took cognisance of the motion of molecules and the change of surface-density.Per-h a p therefore all that we are entitled to say is that if no change is observed in the tension of a film of given thickness the radius of molecular action must be less than that thickness but that if a change is observed it must be greater than half that thickness. Thus the superior limit fixed by Plateau’s experiment would be t’wice as great as it has been generally assumed to be. Quincke (Pogg. Ann. 1869 137 402) attacked the problem in another way. He placed a layer of Martin’s silvering solutiorl between a glass cylinder of 120 mm. radius and a plane sheet of glass. A double wedge of silver which was thinnest in the centre, was deposited on the surface.Two sheets of glass thus prepared were fastened together with a small interval between them with their silver sides inwards and adjusted so that silver layers of equal thickness were as nearly as possible opposite to each other. A glass cell open at the top and bottom having thus been formed the lower part was immersed in distilled water. The cell being vertical the water rose highest in the centre where it might be considered to be VOL. LIII. 234 RUCKER ON THE in contact with the gIass. On each side as the silver sheet became thicker the capillary elevation diminished. It was measured at known distances from the centre and the angle between the solid and the liquid surface calculated. This would became constant when the thickness of the silver layer was such that the attraction of the glass on the water was negligible.The silver was afterwards converted into iodide of silver and the thickness of the layer a t different parts deduced from the colour. Similar experiments were made wihh other substances. The results may be summed up as follows if we write p for the radius of molecular action in terms of micromilli-metres :-p > 54-2 for watw silver and glass, - 43.3 , mercury sulphide of silver aud gIass, - 59.0 , mercury iodide of silver and glass, < 80.0 , mercury collodion and glass. The quantity p as given by these experiments strictly speaking measures not the radius of molecular action but the distance a t which the difference between the molecular forces exerted by glass and silver becomes inappreciable.This would probably be somewhat less than the true radius but nevertheless the net result is to show that the radius of molecular actiop is approximately = 50 p.p. It is much to be desired that this conchion should be in every way tested, and that similar observations should be undertaken by other physicists. I hope tjo show in the course of this lecture that it does receive im-portant confirmation from the behaviour of thinning soap films. Plateau’s experiment has been repeated and modified in various ways. Ludtge ( P o g g . Ann. 139 1870 SSO) inshead of directly measuring the pressure exerted by bubbles compared the pressures due to thick and thin films by balancing them against each other. A soap film having been formed a t the end of a tube it was allowed to thin and the other end was then closed by another film.Air was forced in the films assumed the form of spherical segments and tl]eii* curvatures were compared. If p is the pressure exerted by a soap bubble of which T and R are the surface-tension and radius respec-tively p = 4T/R. Hence if the tensions of the thick and thin films were different their radii would be different also. He concluded that the radius of molecular action was much larger than Plateau and Quincke’s observations would have led us to suppose and that con-trary to expectation the thicker film had the less surface-tension. His experiments were repeated and extended by Van der hlensbruyghe (BruxeZZes Acad. Sci. BUZZ. 30 1870 322) who was unable to detect the alleged change of tension.Afterwards howerer he suggested and adduced experiments to prove that the phenonienoi RAKGE OF MOLECULAR FORCES 235 was probably real and due to the cold produced by the continu;al evolution of fresh liquid surFaces as the films thiuned (Bruxelles Acad. Sci. Mern. 43 1882 No. 4 18). Lastly Professor Reinold and I (Phil. Trans. 177 Part 11 627, 1886) have employed similar methods. We balanced two cylindrical films the one against the other ; one of them was kept thick by passing up it an electyic current which we have shown carries the matter of a thin film with it (Phil. Mag., 19 94 1885). The other was allowed to thin and the ten-sions were deduced from the cuwatures. The above figurerepre-sents diagrammatically the essential parts of the apparatus.The cylinders were formed between platinum rings and their interiors could be put in connection with each other or with the external air by stopcocks. The apparatus actually used was somewhat complicated. The films were formed in a closed glass box surrounded by water. They could be made and adjusted without opening the box so that the temperature and hygrometric state of the enclosed space were constant. A difference of surface-tension was indicated by a bulging of one film and a contraction of the other. Several possible causes of error were investigated and as the distorted films were unduloids, formulse were devised by means of which we could at once calculate the difference of the tensions of the two films when their lengths and maximum or minimum diameters were known.We found as has indeed been noticed by others that the surface-tension of a newly-formed film diminishes and that from 10 to 15 minutes must elapse before it acquires an approximately constant valne. By measuring the magnitude of the changes of surface-tension thus developed we proved that they were far too great to be accountled for as Van der Mensbrugghe supposed by cooling due t o thinning. The calculated R 236 RUCKER ON THE change was 0.0016 per cent. while we observed changes of 9 per cent. The effect is probably only a striking instance of the difficulty of preserving a liquid surface pure. If however two films of very different thicknesses but neither of which had been very recently formed were compared the difference of tension (if any) was very small and was not constant either as to sign or amount.We concluded that no evidence of a change in surface-tension dependent on the thickness of the film is furnished by a direct comparison of the tensions of thin and thick films over a range of thickness extending from 1350 milliooths of a millimetre down to the stage of extreme tenuity when the film shows the black of the first order of Newton’s scale of colours. Had any such dif-ference as large as one-half per cent. of the value of the tension existed we must have detected it. The magnitude of the lower limit when the film appears black was given bp the results of a previous research (Pld. Trans. Part 11 1813, 645). We had determined the thickness of black soap-films by measnreaents based on two independent methods the one electrical, the other optical.I n the first we measured the resistance of cylin-drical films and deduced the thickness on the assumption that the specific resistance was the same as that of a thick layer of the same liquid. The apparatus used is shown in the figure. The film was formed The lower part of the glass between the platinum rings A and B RANGE OF MOLECULAR FORCES. 2 37 Method. vessel was flooded with the solution and C is an endless linen band which dipped in the liquid and could be rotated from the outside. I t was thus kept moist and the hygrometric state of the air was main-tained at a constant point. The current flowed from the binding screw D to A thence through the film to B and E.At F a pair of insulated gold wires penetrated the film. They were concected with the opposite quadrants of an electrometer and the differences of potential between them and between the extremities of a known resistance inserted in the circuit were measured alternately. Prom these the resistance of the film between the needles was deduced. I n the second method we passed the two rays of light used in an apparatus* for the production of the phenomenon of interference by means of thick plates through two tubes in which a number of plant: films had been formed. A known number of films was then broken in each tube in turn. The thickness was deduced from thedisplacement of the interference bands on the assumption that the mean I-efractive index of the thin films was the same as that obtained by the ordinary methods from experiments on the liquid in mass.The results may be summarised as follows :-No. of films observed. Liquid. Electrical . . Electrical . . Optical . . ,. Optical . . . . Liquide glycQrique . . . . Soap solution, out glycerine .ft.?- { 99 )) 5 7 13 9 Mean thick-ness in terms of 10-6 mm. 11.9 10 -7 11 -7 12 -1 Probable error of a single observation. -f 0.2 f 0’6 f 1’4 f 0.8 The close agreement between these numbers obtained by different methods and by calculations based upon different assumptions proves conclusively that the thickness of a black film is generally about, 12 p . ~ . We found that the thickness of different films might vary within several millionths of a millimetre but that in any given film the thickness of the black part remains constant-at all events from a short time after its first formation.At first sight then it appears as though our result was in direct opposition to that obtained by Quincke and proved that the radius of molecular action is <12 pp. This is not the interpretation we ourselves put upon it. The black and coloured parts of a film are separated by a sharp line, which shows that there is a discontinuity in the thickness. Thus in extreme cases the rest of the film may be 250 times thicker than the black part with which it is apparently in contact. * Sometimes called a Jamin’6 Interferential Ref ractometer 238 RUCKER ON THE The accompanying figures represent the life history of a film which Professor Reinold and I watched for several hours.They are sections deduced from the colours observed at intervals. The thickness is magnified 5000 times more than the length. The upper part of the film. was black and the enormously rapid change in thicknem at the edge of the black is well shown. Sir William Thornson (Proc. Roy. Instit. 11 Part 111 485 1887) and Professor Reinold and myself (Phil. Trrans. 177 Part 11 679 and 684 1886) independently arrived at the conclusion that our observations on the uniform thickness of the black part of a film and on the discontinuity in the thickness at its edge prove that when the film reaches a certain degree of tenuity the snrface-tension diminishes to a minimum and begins to increase again when the thickness is somewhat greater than 12 p,p.The relation between the surface-tension and thickness may thus be represented by a curve like that shown in the accompanying figure. When t,he thickness is great the surface-tension is constant. When it reaches the value which corresponds to P the tension begins t RANGE OF MOLECULAR FORCES. 239 diminish. The thicker parts of the film now tear the thinner parts asunder. Rupture would inevitably follom were it not for the fact that when a certain degree of tenuity is reached the surface-ten-sion again increases and when the thickness is 12 p . ~ . becomes equal to that of a thick film as indicated by the equality of the ordi-nates at P and Q. Equilibrium is thus possible between two parts of a film of which the one has the thickness corresponding to Q and the other any thickness greater than that corresponding to P.It is also stable for any further decrease in the thickness of the film below Q would cause a further increase of tension; the thinner parts would therefore contract and become thicker. In other words the film could not under ordinary circumstances thin to below that thickness for which the surface-tension regains its normal value. The discontinuity at the edge of the black and the uniform thickness G f a black film are thus both accounted for. Our failure to detect any measurable difference of surface-t'ension between thick and thin films means not that the radius of molecular action is less than 12 p.p., but that the changes of tension which produce the sharp edge of the black are certainly <W5 per cent.of its whole value. Let us then examine this remarkable phenomenon a little more closely. It is a result of ordinary observation that in a thinning film there is a range of unstable thickness which is always missing between the black and coloured parts. The instability is very strikingly shown by an experiment which Professor Beinold and I bave often performed. If an electric current be sent up a cylin-drical film the upper part of which is black the sharp edge is obliterated. The current carries liquid up with it smoothes oi€' the discontinuity and the colours pass into the black by a gradual tran-sition through grey. As soon however as the current is broken the old state of things is re-established.The grey disappears and the black is again bounded by a definite sharp edge. The change takes place in from 10 to 16 seconds. The colours which thus vanish correspond to the range of unstable thickness. Its lower limit is fixed by the experi-ments of Professor Reinold and myself as being nearly 12 .u.p. The upper limit is more difficult to determine as the colour by which the black part of the film is bordered varies and is probably largely determined by accident. This however may certainly be said that when the film thins in the normal way the discontinuity in the khick-ness never occurs within the grey region. The colour next to the black may rise into the second or higher orders ; it never sinks below a full white of the first order. It is therefore probable that the decrease i n surface-tension begins at a thickness less than that which corre-sponds to the middle of the white and greater than that whic 240 RUCEER ON THE corresponds to the beginning of the black or faint blue which surrounds it.According to Newton these thicknesses are 96 and 45 ,u.p. respectively the mean being 70 ,u.p Let us now assume that Quincke’s value of the radius of molecular action is correct. The greatest possible thickness a,t which the surface-tension of a film could begin to diminish is then 100 p.p. Any film thicker than this would have two complete surface layers and a layer of “ interior ” liquid separating them. Its surface-tension could not therefore depend on the thickness. On the other hand if Maxwell’s theory were correct, the tension would remain unaltered until the thickness was equal to the radius of molecular action that is 50 p p .It is not I think, probable that any improvement in the theory would reduce this limit, though it might increase it. Hence we arrive a t the conclusion that t h e limits of thickness f i w d b y observation as those between which t h e sugnce-tension of a f i l m begins t o diminish (96 and 45 ,u.p.> are yrutc-tically i d e n f i c a l with the l i m i t s deduced by theory from Quiszcke’s expe-riment (100 a n d 50 p.,u.> as those within which such decrease o q h t $ r s t to be observed. Curious and important as I venture to think this conclusion is T do not wish to press it too far. The fact that the limits of doubt imposed by two independent lines of argument are a t the present moment the same,is a more or less accidental coincidence.The vital point is that the value of the radius of molecular action as determined by Quincke is certainly of the same order as and cannot possibly differ much in mag-nitude from that which may be deduced from the properties of soap films. Quincke’s result is therefore not an isolated fact. It receives the strongest possible confirmation from a totally different line of research. The radius of molecular action cannot if Maxwell’s theory be accepted, be greater than 96 pp. which is the superior limit to the tllickness a t which the surface-tension begins to decrease. If the ordinary view be correct it cannot be less than one-half of 45 p.,u. which is the lower limit to that thickness.Hence the true value of the radius of molecular action lies between 96 and ‘23 p.p and the value found by Quiiicke (50 ,u.,u.) is inter-mediate between these. However therefore we combine the figures we deduce from the two observations the same result viz. that 50 p.p. is of the same order of magnitude as the radius of molecular action a conclusion which i t is not too much to say has now strong claims to rank as an ascertained fact. Van der WaalsX deduced from his theory distances between 0.15 and * “ Die Continuitat des ga*foimigen und fliissigen Zustandes.” Van der Waals. The matter may also be presented in another way. Trtinslat,ed by F. Roth Leipzig 1881 p. 107 RANGE OF MOLECULAR FORCES. 2-21 0.29 p . ~ which are less but as he thinks not very much less than the radius of molecular action and he expresses the opinion that Quincke's value is larger than our knowledge of capillary phenomena will allow.As the numbers he himself obtains are from 0.01 to 0.02 of the thickness of a black soap-film i t is evident from the above discus-sion that they are very much too small. Passing next to the lower limit of the unstable thickness ( l a ,p.p.), we must enquire what is the cause of the increase of surface-tension to which the uniform thickness of the black film is due. On this point it may be well to speak with a certain amount of reserve until the theory of the constitution of liquids is more fully developed. If however, we accept equations obtained by Maxwell in which the movements of the molecules and the surface change of density are neglected the phenomenon can be at once explained if we suppose that the increase of surface-tension corresponds to a change from attraction to repul-sion in the intermolecular forces.If the force exerted by a liquid mass on a particle is repulsive when the distance of the particle from the surface lies between certain limits then the tension of films the thick-ness of which is comprised between the same limits will increase instead of decreasing as the thickness diminishes. From this point of view therefore the explanation of the sharp edge of the black part of a soap film would be that when the molecular force between a liquid bounded by a plane surface and a molecule in its neigh-bourhood first becomes sensible i t is an sattraction but that a t some lesser distance which is nevertheless greater than 1 2 x 10-6 mm., it becomes a repulsion.It must however be distinctly understood that the explanation that the increase in surface-tension is due to the action of a repulsive force is only put forward as suggested by Max-well's theory. I think that this conclusion is very much more doubt-ful than that which determines the thickness at which the surface-tension would begin to diminish b u t the further discussion of this p i n t would involve a mathematical argument with which I will not a t present trouble you. If however apart from the question as to how it may be mechani-cally explained the view be accepted that the surface-tension falls t,o a minimum and is again increasing when the thickness is 12 pp.the veiy interesting question arises whether there is any experimental evidence that at some thickness less than 12 k.p. it again diminishes. In answer it may be remarked that in general the black spreads slowly and quietly over the film and may take an hour or more in travelling from the top to the bottom of a cylindrical film 26 mm. long. All the statements I have hitherto made refer to cases i n which the mode of formation was thus normal (Phil. Trans. 177 [ a ] 677, 1886). " At times however the black is formed with something like 212 RUCKER ON THE convulsion. Not only does it spread with extraordinary rapidity but the edge is violently disturbed and large patches rise through the coloured part of the film.Whenever this occurs the film breaks before long but in four cases we were able to obtain measurements before rupture. We are not able to produce this phenomenon at will, but the few observations we have been able to make on it are in agreement among themselves.” I n all cases the cylinder which thinned most rapidly bulged the other contracted. The differences thus produced between the diameters varied from 0.35 to 0.75 mm., and could not be accounted for by the sudden renewal of the surface of the thinning film (which would have produced a change in the other direction) or by any other cause known to us. The measure-ment of the thickness of such films would probably settle the question as to whether the black when formed in this abnormal way corre-sponds to the stateof unstable equilibrium which would exist if after increasing the surface-tension again diminished as the film became thinner or to a second state stable within narrow limits of thickness.Such experiments would however be attended with extraordinary difficulties as they would involve measurements on films which are practically always short-lived and which are possibly theoretically unstable. Another method of investigating the magnitude of the radius of molecular action is based on the phenomenon of electrolytic polarisa-tion. If we immerse in acidulated water two similar metal plates which are not attacked by the acid they will be a t the same potential. When a current is passed from the one to the other they will if the metal and acid have been properly chosen become covered with films of oxygen and hydrogen respectively.The sum of the differ-ences of potential due to metal I gas I liqiiid is not the same as that due to the single metal I liquid contact and varies with the nature of the gas. Hence the coated plates assume different potentials but the full difference is not established until the surface-density of the deposited gas exceeds a certain value. If then we suppose that the film is uniform and that the metal and liquid cannot be regarded as completely separated until the thickness of the film exceeds t,he radius of molecular action we may by plausible assumptions as to the density of the gas estimate its magnitude. Thus F. Kohlrausch ( Y o g g . Ann. 148 153 1873) concluded that if the gases are eupposed to be a t their ordinary densities the polarisa-tion of a platinum electrode is complete when it is coated with a layer of oxygen 20 pp.in thickness. It is evident that this assump-tion as to the density of the gas is totally a t variance with the views ordinarily pnt forward in discussions on the condensation of gases 011 solids as to the great molecular pressure to which the condensed fil RAXGE OF MOLECULAR FORCES. 243 is subjected and that doubt on this point deprives such observations of all value for our present purpose. To reduce the uncertainty as to the density of the polarising layer it is evidently better to substlitute another metal for a gas. This has recently been done by Oberbeck (Wied. Ann. 31 337 1887). The films were deposited on platinum electrodes and the liquids used were solutions of ZnSO, CdS04 and CuSO4.Three platinum plates were immersed in the solution contained in a rectangular cell. Two plates of the metal of which the sulphate was used (zinc say) were interposed between the central platinum plate and the other two and were used as electrodes by means of which a Iziyer of zinc was deposited on both sides of the central plate. Electrolysis was continued until the difference of potential between the coated plate and the external platiiiums which were not affected by the current was the same as that between Zn and Pt (1.13 Daniell). The current was then stopped and for a time the electromotive force slowly diminished after which a very rapid decrease was observed.The film was spontaneously re-dissolved and the sudden change in the rate of the fall of the electromotive force was regarded as indicating that the thickness of the metallic layer had become less than the radius of molecular action. If a is the quantity of metal deposited on each sq. cm. (which could be calculated from the current strength &c.) ; and e the time whicb elapsed after the completion of the eiectro-lysis before the rapid fall of E.M.F., i t was found t.hat these quantities were connected by a relation of the form-a = A + Be, where A and B are constants. Of these A is the quantity of zinc on each sq. cm. when its thickness is just less than the radius of molecular action and by means of two experiments in which a and 8 have different values it can be calculated.Oberbeck concludes that if the specific gravities of the electrolytic layers are the same as those of the metals under ordinary circumstances the thicknesses necessary to establish the full difference of potential are between 2 and 3 ,up. for zinc between 1 and 2 p.p. for cadmium and rather less than 1 p+. for copper. Interesting as these results are they are as Oberbeck himself points out open to criticism. The rapid decrease in the E.M.F. might be explained by supposing that when the zinc layer becomes very thin, parts of the platinum plate are uncovered and that local action takes place which rapidly dissolves the zinc. ThiB is certain to occur unles 2-14 RUCKER ON THE the metallic film is uniform. To test its uniformity the experiment was repeated with a solution of acetate of lead as the electrolytic liquid.The colour of the platinum electrode showed that the deposit was uniform over the greater part of the plate but was slightly thicker towards the edges. No data as to the colours displayed are given but unless the difference of the tints was very slight it would correspoud to a variation of tliickness greater than that assigned to the radius of molecular action. The next method which I propose to describe aims at a measure-ment of the distance between two consecutive layers of molecules. If plates of Zn and Cu are connected by a metallic wire they assume different potentials (P and p ) the Zn becomes positively the Cu negatively electrified. When the plates are parallel to each other, and separated by a distance t centimetres they form a condenser and if + e is the charge upon 1 sq.cm. of the Zn plate e = (P -p>/47t. Hence st is a constant which depends only on the nature of the metals and is independent of the distance between them. When the metals are in contact the potential difference remains unaltered and we may regard the surface molecules as being oppositely charged and separated by a very small interval. The two charges are said by v. Helmholtz to constitute an electric double layer ( P o g g . Ann., 89 211 2853 ; Wied. Ann. 7 337 1879). The mutual action of two metals or of a metal and liquid when in contact is t'herefore the resultant of the molecular forces and the electrical attractions and repulsions which are in play between the different parts of the double layer.Thus v. Helmholtz (BerZin Wissenschnft. Abh. 925 1882 ; see also Wied. Ann. 16 31 1882) has proved theoretically that the surface-tension depends on the electrical charge and is a maxi-mum when it vanishes and as is well known Lippmann (Annales de Chemie 5 4<94 1875j has shown that the surface-tension of mercury in contact with dilute acid is a function of the difference of potential between them arid that every motion of the common surface changes the potential difference in such a way as to produce an alteration in the surface-tension which checks the motion. By means of a theory which it is unnecessary to reproduce here he (Compt. rend. 95 687 1882) drew from his experiments the con-clusion that for such differences of potential as he employed the capacity of a given area of a Hg 1 H,O surface is constant.Hence the distance between the two electrified surfaces is constant and can be deduced from the theory. Oberbeck ('CVied. Avzn. 21 157 1884) and Falck have measured the electro-motive force of polariss tion produced by alternating currents on metals immersed in solutions of Ki,SOa KCl KBr and KI. They The value found is 0.03 p.p RANGE OF MOLECULAR FORCES. 245 conclude that the capacity of the double layer is not constant but is a function of the charge so that its t,hickness must be regarded as variable. They deduce however its limiting value when the charge is zero i.e. the distance between the nearest layers of molecules under normal conditions when no current is passing.The magni-tude of this initial value depends more on the metal than on the liquid. The following table holds for solutions of KC1 or KBr and gives the thickness of the double layer deduced from the forrnula-t = l/4n-C, where C is the initial capacity. t in t,erms of Metal. 1 P*P-I-- --Nickel Aluminum Gold . Silver 1-04 0 *67 0 *06 0 *02 When dimensions so small as these are reached the validity of the method of representing the phenomenon as due to two uniform layers of electricity is very doubtful. The values of t in the case of gold and silver are comparable with the diameters of the molecules them-selves and thus t can only be regarded as a conventional length re-presenting the thickness of an artificial condenser by which the real molecular arrangements may be approximately imitated.L. Lorenz (Pogg. Ann. 140 644 1870) has also based upon electrical theory an estimate of the distance between neighbouring water molecules. He concludes that it is < 0.1 p.p. A very interesting paper has lately been published by 0. Wiener (Wied. Ann. 31 629 1887) in which he attacks the problem of the determination of the thickness of the thinnest metallic plate which affects reflected light in the same way as a thick plate of the same metal. It is well known that in general when light passes from a less dense to a more dense transparent medium the phase of the reflected ray is altered by half a wave-length. This is proved by the fact that the centre of Newton’s rings as seen by reflected light is black.For the difference in the phases of the rays reflected from the front and back surfaces of the film respectively depends partly on the thickness of the film and partly on any change of phase which the rays may undergo on reflection or refraction. As the film becomes very thin the difference in the paths of the two rays due to its thicknes 246 RUCKER ON THE becomes negligible and thus if the phase were not affected by reflection or refraction the rays reflected from the centre of the rings where the film is thinnest would be nearly in accord or the centre would be bright when viewed by reflected light. The fact that the centre is dark is explained by the assumption that when a ray of light, passes from a less dense to a more dense transparent medium the phase of the reflected ray is altered by half a wave-length.When light is reflected at a metallic surface an alteration of phase also takes place but it is not necessarily half a wave-length and it is on this peculiarity that Wiener’s method is based. I n the light reflected from a thin film of air enclosed between the two glass plates those rays will be wanting for which the difference of phase produced (1) by the difference in the lengths of the paths of the rays reflected at the first and second surfaces and (2) by the change of half 5t wave-length produced on reflection at the air-glass surface is an odd multiple of half a wave-length. In the spectrum of such light dark interference bands will be visible corresponding to the missing rays.If the second surface had been silvered (the thickness of the air film remaining unaltered) the effect of the first of the above two causes would be the same as before but that of the second would be different. Hence the particular kind of light for which the total retardation was previously an odd multiple of the half wave-length would no longer satisfy that condition and the interference bands in the spectrum would occupy new positions. If the second surface had been partly silvered two contiguous spectra could be obtained in which the interference bands appeared broken. I n the experiments with which we are specially concerned Wiener proceeded as follows :-A thin film of mica was partly covered by R second with a straight edge. Silver was deposited on it by discharge from a silver electrode (Wied.AWL 29 353 1886). The layer thus formed was thickest in the centre and thinned away gradually. When the covering mica was removed the silver film was bounded on one side by a straight line. Thus when light was reflected from th RANGE OF MOLECULAR FORCES. 217 film on to the slit of a spectroscope (the silvered side being furthest from the instrument) two spectra were seen side by side as is shown in the figure (p. 246). The displacement of the interference bands varied with the thickness of the film but became constant when the thickness exceeded a certain value. It was measured for certain parts of the spectrum at a number of points the positions of which on the mica were determined. The silver was then converted into silver iodide, and the displacement of the interference bands was again determined at the selected points.From this latter measurement the thickness of the iodide and therefore of the original silver film could be deduced by formulE for the discussion of which I must refer to the original paper (Zoc. cit. p. 664). Curves were then drawn showing the rela-tion between the thickness of the silver and the change of phase pro-duced by it. Curves I 11 and These are reproduced in the figure. I11 were obtained by the same mirror but by observations in different parts of the spectrum. Curve I corresponds to the orange (X = t;47), I1 to the green (X = 534). and I11 to the blue (X = 455). The re-tardation increases very rapidly for the blue less rapidly for the other colours till a thickness of about 4 p.p.is attained. Afterwards it alters more slowly and is nearly constant at the greatest thickness for which the measurements were made viz. 12 p.p. Observations made with another mirror confirmed the result t,hat the change of phase reached its maximum value for a thickness of about 12 p+ but indicated a more uniform rate of increase. Herr Wiener ascribes this difference to a slight oxidation of the silver films. The smallest thickness for which any displacement of the bands could be observed is estimated as rather less than 0.2 p.p. We have now reached the point at which we may investigate the inferior limit to the range of molecular forces viz. the so-calle 248 RUCKER ON THE radius of the molecules. This part of my subject has been so fully discussed by Sir William Thomson (hTatumZ Philosophy Thomson and Tait Pt.11 495 1883; Proc. Roy. Instit. 1883 ; Exnw's Rep., 21 182 1885) and 0. Meyer ( D i e Kir~etiscke Theorie der Gase, 225 1877) that it will be unnecessary f o r me to reproduce their arguments in full. I shall therefore content myself with shortly stating their results and describing at greater length a more recent method developed by Dorn and Exiier. Sir William Thomson (Natural Pldosophy 502) concludes that the diameter of the gaseous molecule cannot be less than 0.02 p.,u., and that the distance from centre to nearest centre in solids and liquids may be estimated a t from 0.07 to 0.02 ,K.,u. He points out that when plates of zinc and copper which are con-nected by a metal approach each other work is done in virtue of the attraction caused by their assuming different electrical potentials.If the plates are split up into an increasing number of thin layers and arranged Zn and Cu alternately so that the thicknesses of t.he plates and of the intervening spaces are equal the work done will vary as the square of the number of plates. I f the thickness i n question were 0.1 p p . the heat-equivalent of the work done would be sufficient to raise the temperature of the metals by 62" C. if it were 0.025 p.p. the heat would suffice to raise the mass through 992" C. The conclusion is drawn that the molecules of Zn and Cu are pro-bably a t least 0.1 p+. and certainly more than 0.025 p . ~ . in diameter. Again when a liquid film is stretched work is done upon it and it.is also cooled. To keep its temperature constant heat must be supplied, and if the thickness were reduced to 0.05 p.p. the heat-equivalent of the total amount of energy imparted to the film would be about twice the latent heat of steam. As it is incredible that the film could absorb so large a quantity of energy and yet remain in the liquid state it is certain that if it could be reduced to this extreme tenuity, the work done in stretching it would cceterisparibus be less when it was very thin than when it was relatively thick. Hence the surface tension must diminish before the thickness of the film is 0.05 p.p., and Sir William Thomson thinks t]hat there cannot ''be any consider-able falling off in the contractile force as long as there are several molecules in the thickness.It is therefore probable that there are not several molecules in a thickness of " 0.05 p p . From a consideration of the transmission of light t'hrongh transparent bodies he also con-cludes that the distance between the centres of contiguous molecules i n solids and liynids is greater than 0.05 pp. The fourth method used by Sir William Thomson is based on the theory of gases. An important formula has been deduced by Clausius, and in a slightly different form by Maxwell which establishes RANGE OF RIOLECULAR FORCES. 249 relation between the diameter of the molecule (d) the mean free path (L) and the ratio of the total volume of the molecules to the volume of the gas (21). It may be written d = 6J%L.The value of v which is called by Loschmidt the condensation coefficient has been obtained in various ways. Sir William Thomson concludes from the general results of experiments on the condensa-tion of gases that a gas could not be made 40,000 times denser than it is under ordinary atmospheric pressure and at ordinary temperatures. Loschmidt (Sitxungsber. Wien. Akad. m a t h Classe 52 Abt. 2 404, 1866) made use of Kopp’s formula-specific volume = molecular weight divided by the density at the boiling point-to calculate the densities in the liquid state of gases which had not then been liquefied. He assigned to the various elements specific volumes somewhat different from those selected by Kopp. Thus assuming those of oxygen and nitrogen to be 11 and 12 respectively the calculated densities are 16/11 = 1.4545 and 14/12 = 1-1666.Hence taking air as a mixture of four parts of nitrogen and one of oxygen he calculated the density in the liquid state to be 1.224. If the molecules are spheres they will when packed as closely as possible occupy a space which bears to the sum of their volumes the ratio 1-17 1. He assumes that in a liquid they are closely packed and deduces as an approximation to the true density 1.224 x 1-17 = 1.5 say. Hence v = 0-001293/1.5 = 0.00086. He takes as the value of the mean free path 140 p.p., whence d = 1 p.p. If however we use the value of L given by Meyer ( D i e kinetische Theorie der Gass l40) viz. 95 p.,u. we get d = 0.68 p.p. 0. Meyer (Thenrie der Gase 225) employing a similar method for nine substances the density of which is known both in the liquid and gaseous states by direct experiment found Talues for the molecular diameters which lie between 1.18 p.p.for N,O and 0.44 p p . for H,O. Dorn ( W i d Arm. 13 378 1881) and more recently Exner (Rep. der Phjysik 21 425 1885) have obtained the value of the so-called condensa tion-coefficien t v in another way. Clausius ( D i e mechanische Behaizdlmy der Electricitat I11 Abschnitt) has given a formula which connects K the specific inductive capacity of a dielectric and ZI as above defined on the assumption that the molecules of the dielectric are conductors and are surrounded by a non-conducting medium. This formula is-Hence v > 25 x According t o Maxwell’s electromagnetic theory of light if n is the TOL.LXII. 250 RGCICER ON THE Air H2 . GO,. . co N,O refractive index of the dielectric for r a p of infinite wave-length K = u2 a t all events to a first approximation. This equation is not satisfactorily fulfilled in the case of liquids or easily condensible gases, partly perhaps because our knowledge of the law of dispersion is insufficient to enable us to calculate the value of PZ. from the refractive indices of the comparatively short luminous and dark waves which have been studied experimentally. In the case of gases in which the dispersion is very small this difficulty is not met with. As the specific inductive capacity is also very nearly unity the experimental difficul-ties which attend its determination are great.The first measurements of this kind were made by Boltzmann and Professors Ayrton and Perry. More recent observations of Klernen@i6 are in good accord with the results obtained by Boltzmann. The agreement between the values of /Kand of n as determined by Mascart is not satisfactory f o r vapours but is very close in the case of the morc perfect gases. The following table is abstracted from that given by Klemen6i6 (Exner’s Rep. 21 611 1885) :-1 *000255 1 *000132 1 -000473 1 -00034.5 1 *000 4.97 1 d K . Boltzrnann. --- I KlemenEiE. I -I---1 -000347 1 -000579 n. 1 ‘000293 1*000139 1 ’00e454 1 -000338 1 a000516 This table shows that the value of 2 may be approximately deter-mined in the case of gases for which we know either K or n. As TI is the ratio of the space occupied by the molecules to the t’otal volume of the body of which the former is a const,ant and the latter varies iiiversely as the density (8) of the substance it is evideIit that for each substance 1118 should be a constant.Hence (n’ - l)/S(n2 + 2) shonld be the same a t all temperatures and for all physical states of the same substance. This result has been obtained independently from optical consiclera-tions by H. A. Lorentz ( W i e d . Ann. 9 641) and L. Lorentz ( W i e d . Aszn. 11 70) and has been tested experimentally by the latter and Prytz (Wied. Ann. 11 104) in a large number of cases. Although the refractirc indices are those for D and not for waves of iufinite length the agreement is very close. 1 give in the following table as samples the first three substances mentioned in the final tables of these two observers.The number RASGE OF MOLECULAR FO LLCES. 251 Substance, compared are the values of (a2 - l)/8(n2 + 2) for the same sub-stance in the liquid and gaseous states :-Molecular Tolurne, 10-5 x Substance. I- --E t)liyl ether . Ethyl alcohol. . Water . Methyl acetate. Ethyl forrnate . Methyl alcohol Observer. Liquid. -~ 0 *30264 0 -28042 0’ 20615 0 ‘2567 0 ‘2375 0 ‘2437 Vapour. 0 * 3068 0.2825 0 * 2068 0 -2559 0 2399 0.2419 The following table contains the values of z1 for some of the ele-ments. From Avogadro’s law it follows that these numbers are pro-portional to the volumes of the molecules ; and if we divide them by the number of atoms in the molecule we obtain numbers proportional to the atomic volumes.In the case of H, ZI was determined from the specific inductive capacity in all other cases from the refi-active index. I n this and the next table I quote from Exner. I I-- -Hz. N2 . 0 2 . c1 . 84. . P‘j . . . . . . . . . . . . . Hg C (from CO - 0) 8 . 8 20 18 51 108 91 37 -Atomic volume, 10-5 x 4 -4 10 9 25 27 23 37 14 From these atomic volumes it is of course possible to calculate the molecular volume of any compound of these substances. Thus the molecalar volume of water = 8.8 + 9 = 18 nearly. obtained from K in the cases of the first five substances and from n in that of the others, together with the calculated values deduced from the above atomic volumes :-The following table gives the values of s 252 RUCKER ON THE Substance.Air CO NzO CHj . C2Hj . NH . Roo . NO H2S HC1 C,N . so v (observed), 10-5 x 17*[20] 31 33 31 44 26 17 20 43 30 56 44 v (calculated), 10-5 x 19 32 34 32 45 23 18 19 36 29 48 45 These results on the whole confirm the accuracy of the physical meaning of the expression (nz - l ) / ( n z + 2) and tend to show that the diameter of the molecule is the same in the liquid and gaseous states. It is important to note however that from the theoretical point of view there is a good deal of confusion. The meaning of the expression (K - l)/(K + 2) is deduced from an electrical theory put forward by Clausius.It should only be equivalent to (w2 - l)/(nz -+ 2) when v is calculated for waves of infinite length and as a matter of fact K and n2 are not eqml for most vapours when n has a value proper to any of the visible rays. If then (I( - 1)/8(K + 2) is really the same for a liquid and its vapour for neither of which n, is known we should not prim& facie expect that (nZD - l ) / S ( ~ 2 ~ + 2) would be the same for both. Nevertheless experiment shows that the variations produced in this expression by the passage from the liquid t o the vaporous state are less than the discrepancies due to the imper-fect agreement between the values of K and w2 in the case of most vapours for which both have been determined. I n cases where K is not = nz the value of v deduced from n2 is to be preferred.Thus the specific inductive capacity of flint glass as determined by Dr. Hopkin-son (PTOC. Roy. Xoc. 43 161) is 9.5 which makes 21 nearly = 0.8. I f we assume the refractive index to have been 1.7 we get 21 rather less than 0.3 which is in far better agreement with the results obtained from gases. I n the case of conductors the values of K are very high. The annexed table gives the value of v for several liquids calcu-lated directly from K and ,n2 as determined by Dr. Hopkinson (Zoc. cit.). The value is also given deduced from the atomic volumes of the gases, * This number appears to be incorrect. Boltzmann’s value for K is l%OOti90, which gives = 20 x This will be used hereafter RANGE OF MOLECULAR FORCES.253 CSHlo C,jH,j C,Hlo CloH12 C,H, viz. 14 x 10” for C and 4.4 x the chemical formula of which is C,H, we have-for H. Thus for the substance 2.05 1.9044 0.260 2-38 2’2614 0.325 2.42 2.2470 0.321 2.39 2’2238 0.317 2.25 2’2254 0.294 D 2 = (14n + 4 . 4 ~ ) 1 0 - ~ 12n x 0-00008961 L1 where D is the density of the liquid. Substance. -Amylene . Benzol . . . Toluol . . Xylol . . . . Cymol . . . . I- I-I-4P”-0 -232 0 -296 0 -291 0 -290 0 -290 v calculated. --0 ‘237 0 *278 0 -286 0 *284 0 *295 In these cases then all three methods of calculating v indicate that from one-fourth to one-third of the volume of the liquid is filled with matter. Another interesting point is that this method of regarding the formula (mz-l)/S(n2 + 2) enables us to assign a physical meaning to the specific refraction of a substance.I n the above calculations it has been assumed that the atomic volume of a substance is the same whatever the nature of its union with the other atoms may be. Landolt however (Liebig’s Annnlen, 213 1882 75) has undertaken a careful comparison of specific refractions calculated by the ordinary formula (n-l)/8 and by (nz-1)/ij(n2 + 2). He finds that the latter is more constant when the values obtained for the liquid and gaseous states are compared and he calculates the specific atomic refractions by means of it. He finds it necessary to assign different values to 0’ and O” which from the point of view we are discussing indicates a dieerenee of atomic volume.It must also be remarked that a very low value of the specific inductive capacity of air when the pressure was 0.001 mm. has been obtained by Professors Ayrton and Perry which might if confirmed by future experiment affect the questions we have been discussing. The fol-lowing table of their results is abridged from Ayrton’s Practical Elec-tricity p. 310. The letter k indicates that the specific inductive capacity of air at $60 mm. is taken as unity. The numbers which refer to air are alone extracted 254 RUCKER ON THE Approximatre pressure in mm. 0 *001 5 760 k. 0 *994* (about) 0.9985 1 ~0000 They have in a pamphlet " On Certain Modifications that must be Introduced in the E'undamental Notions of the Mathematical Theory of Electricity," p.5 proved that if 6 be the density referred to air a t 760" mm. and 0" C. as unity if 1.000294 be the refractive index of air under the same standard condition referred to that of a vacuum as unity and if k be defined as above-0.0005888 + 1 1.000388 " k = This expression is obtained by Biot and Arago's formula (nz-l)/a = constant but a practically identical result will be attained if we use instead the expression employed by Exner. It follows that k has a limiting value when 6 = 0 such that if k is the specific inductive capacity of a vacuum referred to t h a t of air at 760 mm. as unity, k = 1/1*000588 = 0.999412. At 5 mm. k = 0.999416 as givenby the formula. This agrees with the value obtained by Boltzmann viz., 0.99941 (Practical Electricity loc. cit.) but is not in such close agreement with that obtained by Ayrton and Perry themselves.The difference might easily be ascribed to errors of experiment but the value 0.994 when the pressure was 0.001 mm. was obtained in a later research (Rep. IBril. Ass. 1880.) Its accuracy is inde-pendent of that of the measurements a t a pressure of 5 mm. and as far as I am aware no other observers have carried out experiments in gases of such extreme tenuity. It is therefore much to be desired that further observations should be made on the specific inductive capacity of air a t low pressures. The importance of such a research would be enhanced from the fact that it has been pointed out by Professor Fitzgerald (Xep. Brit. ASS., 1880 Zoc. cit.) that the values obtained for the capacity of an air condenser between " about 0.02 and 0.2 mm.pressure bear a general resemblance to those obtained for the Crookes' force." For my present purpose however it is snficient to remark that if the ratio of the specific inductive capacities of air at pressures of 0.001 and 760 mm. is about 0.994 1 then either Maxwell's theory fails when Professor Ayrton informs me that that giren in Practical Electrirify viz. 0.94 is a misprint. * This number is correct RAKGE OF MOLECULAR FORCES. 255 applied to rare gases or the refractive indices of air a t these pressures are in the ratio JO-994 1 that is 0.997 1. Now the fact that the re-fractive index from a vacuum to air a t atmospheric pressure is about 1*000294 is proved iiot only by direct experiments on air of different densities but also by the agreement between the observed and calcu-lated results of the effect of atmospheric refraction on the apparent positions of stars.Hence if we admit the validity both of tbe expe-rimental determination of the specific inductive capacity of air a t a pressure of 0.001 mm. and of the application of Maxwell’s theory of this case we must conclude that the refractive index of highly rarefied air referred to that of a vacuum as unity is 0.997 x 1.000294 = 0.997293 and that it is about 0.27 per cent. less than that of a vacuum. The alteration in the ordinary refractive index of air which would be required to make this quantity > 1 would make the calculated atmospheric refraction nearly ten times greater than that which is actually observed.It is therefore evident that either the experimental result is affected with error or that Maxwell’s theory does not apply to a rare gas. I n either case the conclusions arrived a t need not affect the application of Maxwell’s theory to sub-stances for which K = n2 and f o r which therefore it appears to be a t all events an approximation t o the truth. obtained by Exner are remarkably confirmed by those deduced from the theory of Van der Waals. In the general relation between pressure volume and temperature given by him a constant b occurs which is according to Van der Waals = 471 and, according to 0. Meyer = $ 4 2 ~ . Meyer deduces it from a comparison between the terms of Van der Waals’ formula and the constants of a similar empirical formula of Regnault’s.The values so obtained are of the same order as those given by other methods, but I do not think that hleyer’s plan is as satisfactory as those adopted by Van der Waals himself as it largely depends on the valnc of a very small constant in the empirical formula. Van der Waals makes two comparisons-the one with Regnault’s observations (Die Continuitut &c. 73 to 79) and the other with Cailletet’s results (Zoc. cit. 98 and 99) and he shows that the constants obtained by the former method produce a fair agreement with the observations of Andrews (Zoc. cit. 80). The table is obtained by taking these results (comp. Meyer Kine-tische Theorie 75) and reducing all values to the standard pressure of one atmosphere. The values of The value of b may be obtained in several ways 256 RUCKER ON THE Mean 1 Value of b from Value of v according to I Regnault.0*00195 1 0.00049 0 *00050 0 *00012 0 *0023 0 -00057 I- -0*00035 0.000088 0 *00041 Air .E€z . . . . . . COa . . . . , Cailletet. --0.00050 -0*00195 0 -00049 0 -0023 0. Meyer, The value of b for SO2 deduced from experimenh by Cagniard de la Tour is 0.0032 which leads to v = 0.0006 according to Meyer’s formnla. As these early observations were probably not so accurate as the others we are discussing I shall not make use of his results, also as Meyer’s relation between b and v appears to be the best I shall hereafter employ it only. Before discussing this table further I must point out that there are some slight errors in the books which deal with the subject and which account for discrepancies between the figures as given by others and hy myself.The numerical value of b which is proportional to the ratio of the volume of the molecules to the volume of the gas under standard conditions increases with the standard pressure and may be taken as proportional to it. The values obtained by Van der Waals from Regnault and Caille-tet’s results are referred to pressures of 1 m. and ‘760 mm. respectively, Ruhlmann (Xechanz’sche Warmetheorie 2 244 Vieweg und Sohn, 1885) reduces them all to a pressure of 1 metre but in so doing he multiplies the numbers which are referred to an atmosphere by 0.76 instead of dividing. Thus as corresponding to 0.00198 he gives 0.0015 and so on.By this mistake he conceal8 the practical identity of the values obtained by Van der Waals. The value of b for CO, referred to a pressnre of 1 m. is 0.003 (Die Continuitat &c. 74). Van der Waals calculates it for 1 atmcisphere on p. 80 and finds b = 0.0023. Yet Meyer (Kinetische Theorie 231) treats this value as though it referred to the larger pressure and finds that for COz d = 0.18 p.p. ilistead of 0.23 p.p. which is the correct value deduced from his other data. I point out these errors only to prevent confusion on the part of those who might happen to compare the different tables of values. In the next table are the values of v obtained from the specific inductive capacity the refractive index and the theory of V#n &r Waals :-0. Meyer himself makes a similar mistake RASGE OF MOLECULAR FORCES.--Air . Table of Values of v. Calculated from K as deter-mined by Calculated from n, Mascart. Boltzmann KlemendiE 10-5 x 10-5 x 10-5 x ----20 20 20 H2 8.8 8 -8 coz . 33 I 32 257 9 . 3 30 Calculated from b Van der Waals and 0. Meyer, 10-5 x 35 41 8.8 Without laying stress on the extraordinary similarity between the values of 21 obtained in the case of H from the dynamical theory of gases and from electrical and optical formulae I think that the agree-ment as to the order of the magnitudes of v calculated by such various methods is very strong evidence that they are approximately correct. It will also be observed that they are intermediate in value between the superior and inferior limits given by Loschmidt and Sir William Thomson respectively.Loschmidt regarded the liquid as formed of molecules in contact an assumption which could not give too small a value for v ; Sir W. Thomson selects a condensation coefficient which he is sure is not large enough. We thus get in the case of air-Superior limit (Loschmidt) 86 x lod5 Actual value calculated from b 35 x lom5 79 9 , n . . 20 x 7 9 , K 20 x 10-5 2.5 x Inferior limit (Sir W. Thomson) We may therefore conclude that the space occupied in the sense previously defined by the molecules in air at 0" C. and 760 mm. is about one five-thousandth (0.0002) of t,he volume of the gas. As Loschmidt's calculation is based on the assumption that in a liquid v= 1/1.17= 0.85, and as Exner's value in the above table is 2O/% of his it follows that in liquid air the value of v would be about 0.2.Very similar values are obtained for substances which can be liquefied easily. Thus the observed value of v for water vapour is 0.00017. This is referred to 0" C. and 760. But at 0" C. and 4.6 mm. (the maximum tension of aqueous vapour at that temperature) the volume of saturated steam is 210,660 times the volume of the liquid. Under standard conditions this would be reduced to 210,660 x 4*6/760 = 1275 258 RUCKER ON THE Substance. Viscosity. Hence for liquid water 21 = 0.00017 x 1275 = 0.22. We conclude that in liquids about one-fifth only of the total volume is filled with matter which is in fair accord with the numbers before obtained from Hopkinson’s results.To calculate the diameter of a molecule we must know not only v but also L. This may also be determined by three independent methods of experiment viz. by the determination of the coefficients of viscosity diffusion and thermal conductivity. As the values of the coefficient of viscosity given by Meyer are not reduced to 0” C. I take the values of L deduced from these given by Ruhlmann (Me-c7~zanische Wai-rnetheoyie 227). For those which depend on diffusion, I quote Stefan’s deductions from Loschmidt’s experiments as given by Exner (loc. cit. 450). Taking the coefficients of thermal con-ductivity determined by experiment by Kundt and Warburg, Winklemann and Stefan as given by 0. Meyer (Zoc. cit. 194) I have calculated back to the coefficients of viscosity by the formula f = 1*53qc where 5 and q are the coefficients of conductivity and viscosity and c is the specific heat a t constant volume and have then deduced L from the coefficient of viscosity thus calculated.The results are given in the following table and prove that there is a t all events no doubt as to the order of the magnitude of L. Thermal conductivity. Diffusion. Values of L. I L in p.p. calculated from l--l--I-- --Air R3 co . co . N,O 99 194 66 97 66 ~ 71. 139 50 65 42 110 186 55 106 59 We are now if the various theoretical assumptions are allowed in a position t o calculate the diameter of a molecule of air H,. o r Con, by three absolutely independent met,hods. We may combine the values of 8 obtained from the specific inductive capacity the refrac-tive index and the theory of Van der Waals with the values of L deduced from the coefficients of viscosity diffusion and thermal con-ductivity respectively.The results are showu in the following table : RANGE OF MOLECULAR FORCES. 259 Substance' Diameter of gaseous molecule in p.p. calculated from Specific inductive Refractire index Expansion and thermal and diffusion. conductivity. capacity and viscosity. Air. . . . . . . . . . . . . . Hz coz . . . . . . . . . . . . . 0 -17 0 *14l 0 .18 0 -12 0.11 0 *13 0 -33 0.14 0 -19 Without insisting too much on an agreement which can be exem-plified in the case of a few gases only and which would probably not be exhibited by the results of experiments on vapours it is not too much to say that it cannot possibly be fortuitous and that it leaves very little doubt that the so-called diameter of a gaseous molecule is of the same order of magnitude as 0.2 ~ .p . I n conclusion I think i t may be well to attempt to class the phe-nomena which have been observed in very thin layers of matter and the results of calculations on the size of molecules in the order of the magnitudes involved. It is probable that such a statement will have to undergo much correction in the futnre but i t may be useful and suggestive in the present. A t all events I think it will show that the time has passed when any estimate however rough as to the magnitude of' molecules or of the radius of molecular action is to be welcomed.We know now what the order of these magnitudes is, and observations are wanted based on reliable methods and leading to definite results. I n drawing up such a table I shall therefore reject measurements which appear to me to be open to very grave doubt. In the first place all results as to the condensation of liquid films 011 solids which lead to values of the radius of molecular action of several hundred or even several thousand micromillimetres must be rejected until they are confirmed by other methods. The onus of proring that the bodies used are not porous not absorbent and not affected with impurities which can unite chemically with or dissolve in, water lies with the invest,igntors who adopt this method. Not only have the observations on agate varnished metals and glass shown that these are grave and probable sources of error but Ihmori has proved that when they are as far as possible got rid of the thickness of the condensed film is very small.The fact that soap films exhibit no trace of change in their surface-tension or other properties till a thickness of about 50 p.p. is reached makes it absolutely incredible that the radius of molecular attraction should have a magnitude o 260 RUCKER ON THE from 500 to 3000 p.p. I n this view I am supported by the opinion of Sir W. Thomson who has laid it down as “ quite certain that the molecular attraction does not become sensible until the distance is much less than 250 micromillimetres ” (Proc. Roy. Inst. 11 Part 111, 415 1887). Important too as the observations of Ihmori and Warburg are, I do not think that they can be used for our present purpose.They refer only to the temporary film and therefore do not afford direct information as to the distance between the surface of the glass and the outermost water layer. This is given by Bunsen’s observations, though it is doubtful what the nature of the attraction by which the water is held may be. I shall also reject calculations based on the polarisation produced by gases. The uncertainty as to the density of the films deprives these estimates of all value. The same objection does not however, apply to measurements of the thickness of the electrical double layer. I shall therefore include these and the results of observations on the polarisation of metal by metal subject of course to t,he criti-cisms which I have already made.I include Plateau’s results on account of their historical interest. Table of Properties of Thin Films and of Molecular Mag?iitudes. 118 p.p. Superior limit to the radius of molecular action deduced from Plateau’s experiments on the pressure of a soap bubble by Maxwell’s theory that the surface-tension first diminishes when the thickness of the film = p. 96-45 p.p. Between these limits the thickness of a film begins to be unstable, Hence the radius of that is the surface-tension begins to diminish. molecular action must be < 96 p.p. and > 22 p.p 59 p.p. Superior limit to p deduced by Plateau on the assumption that the surface-tension first diminishes when the thickness = 2p.50 p.p. Value of p deduced by Quincke from experiments on capillary elevation. Hence the thickness should begin to be unstable when it is 100 p.p. or 50 p.p. according as we adopt Plateau’s or Maxwell’s views. There is therefore a remarkable accord between Quincke’s result and the superior limit Probably the truth lies between the two RANGE OF MOLECULAR FORCES. 261 t o the unstable thickness (96-45) obtained by Reinold and Rucker from experiment. 12 p.p. Average thickness of black soap-films measured by two independent methods. As the tension of a black film is equal to that of a thick film the surface-tension which begins to diminish at 50 p.p. must increase again and reach its original value at 12 p . ~ The fact that each black film is of uniform thickness proves that the surface-tension is still increasing at 12 p.p.which is the lower h i i t to the range of unstable thickness. This is also about the thickness below which, according to 0. Wiener a thin silver plate will no longer produce the same effect on the phase of reflected light as a thick silver plate would do. 10.5 1A.p. Thickness of t'he permanent water film observed by Bunsen on unwashed glass at a temperature (23" C.) at which the vapour pressure of water is small. 4 p.p. to 3 p.p. Average distance from centre to nearest centre of molecules in gases under standard conditions calculated by Meyer. If Exner's values of v be accepted the distance would be more nearly 2 p.p. 3 p.p. to 1 p.p. Thickness of metal films required to polarise platinum completely according to Oberbeck.1 p.p. to 0.02 pp. Thickness of electric double layer according to Obei-beck and Falck. Lippmanzl found 0.3 p . p . 0.2 p.p. Smallest thickness of silver which affects the phase of reflected light. 0.14 to 0.11 p.p. Diameter of gaseous hydrogen molecule as given by combining-(1.) The specific inductive capacity and coefficient of viscosity. (2.) The refractive index and coefficient of diffusion. (3.) The law of expansion and the thermal conductivity. 0.07 t o 0.02 p.p. Average distance bet ween centres of molecules snpposed arrange 2 62 SCHUNCTL ON Ti-IE SUPPOSED IDENTITY uniformly in liquids and solids according to Thornson. limit found by L. Lorenz wils 0.1 p.p. A superior 0.02 p,p. Inferior limit to the diameter of a gaseous molecule according to These results may be shortly summed up as follows:-Thomson. 118 96-45 59 50 12 12 10-5 4-3 3 -1 1-0'02 0 *2 0'14-0 '11 0 * 07-0 * 02 0'02 Superior limit to p i I 1 } { 1 Range of unstable thickness begins Superior limit to p . . Range of unstable thickness ends { Action of silver plate on phase of reflected Thickness of permanent water film on glass at 23°C Mean distance between centres molecules in gases at '760 mm. Thickness of metal films which polarise plati-Thickness of electric double layer Smallest appreciable thickness of silver film Diameter of gaseous hydrogen molecule Mean distance between centres of nearest Inferior limit to diameter of gaseous molecule . . Magnitude of p light alters r!?.t } num i liquid molecules,. Plateau. (!Maxwel!). R,einold and Riicker. Plateau. Quincke. Rcinold and Itiicker. Wiener. Bunsen. 0. Meyer. Oberbeck. Lippmann and Oberbeck. Wiener. Exner. 0. Meyer. Van der Waals. W. Thornson. W. Thomson