September 24, 1891] NATURE 



499 



contain a large number of Chinese novelties. One part 

 of the last volume is devoted to the Stapelia^ of South 

 Africa. The seventeenth volume is wholly devoted to 

 new ferns ; and the first volume of what it is intended 

 to call the fourth series will consist entirely of orchids. 

 Three parts of this have already appeared. 



ON VAN DER PVAALS'S TREATMENT OF 

 LAPLACE'S PRESSURE IN THE VIRIAL 

 EQUATION: A LETTER TO PROF. TAIT. 



MY DEAR PROF. TAIT— In Part IV. of your 

 " Foundations of the Kinetic Theory of Gases," * 

 you take exception to the manner in which Van der 

 Waals has introduced Laplace's intrinsic pressure K into 

 the equation of virial. " I do not profess to be able fully 

 to comprehend the arguments by which Van der Waals 

 attempts to justify the mode in which he obtains the 

 above equation. Their nature is somewhat as follows : — 

 He repeats a good deal of Laplace's capillary work, in 

 which the existence of a large, but unknown, internal 

 molecular pressure is established, entirely from a statical 

 point of view. He then gives reasons (which seem, on 

 the whole, satisfactory from this point of view) for assum- 

 ing that the magnitude of this force is as the square of the 

 density of the aggregate of particles considered. But 

 his justification of the introduction of the term aiv"^ into 

 an account already closed, as it were, escapes me. He 

 seems to treat the surface-skin of the group of particles 

 as if it were an additional bounding-surface, exerting an 

 additional and enormous pressure on the contents. Even 

 were this justifiable, nothing could justify the multiplying 

 of this term by {v — /3) instead of by v alone. But the 

 whole procedure is erroneous. If one begins with the 

 virial equation, one must keep strictly to the assumptions 

 made in obtaining it, and consequently everything con- 

 nected with molecular force, whether of attraction or of 

 elastic resistance, must be extracted from the term 

 2(Rr).'' 



With the last sentence all will agree ; but it seemed 

 to me when I first read Van der Waals's essay that his 

 treatment of Laplace's pressure was satisfactory, and on 

 reperusal it still appears to me to conform to the require- 

 ments above laid down. As the point is of importance, it 

 may be well to examine it somewhat closely. The ques- 

 tion is as to the effect in the virial equation of a mutual 

 attraction between the parts of the fluid, whose range is 

 small compared with the dimensions of bodies, but large 

 in comparison with molecular distances. 



The problem thus presented may be attacked in two 

 ways. The first, to which I will recur, is that followed 

 by Van der Waals ; but the second is more immediately 

 connected with that form of the equation which you had 

 in view in the passage above quoted. 



In the notation of Van der Waals (equation 8) 



|2wV- = \lfp - ^SRrcos (R, r), 



where V denotes the velocity of a particle tn, which 

 is situated at a distance r from the origin, and is 

 acted upon by a force R, while (R, r) denotes the angle 

 between the directions of R and r. The intermediate 

 term is to be omitted if R be the total force acting upon 

 711. It represents the effect of such forces, / as act 

 mutually between two particles at distances from one 

 another equal to p. In the summation the force between 

 two particles is to be reckoned once only, and the forces 

 accounted for in the second term are, of course, to be 

 excluded in the third term. 



In the present application we will suppose all the 

 mutual forces accounted for in the second term, and that 

 the only external forces operative are due to the pressure 



of the containing vessel. No one disputes that the effect 

 of the external pressure is given by 



- i2Rr cos (R, r) - :)pv ; 

 so that 



nmV^ = %pv -\- i2p0(p), 



if with Laplace we represent by <^(p) the force between 

 two particles at distance p. The last term is now easily 

 reckoned upon Laplace's principles. For one particle in 

 the interior we have 



h • 41- 



j ^ (t>{p)p^dp, 



and this, as Laplace showed,^ is equal to 3K. The 

 second summation over the volume gives ^Kv, but this 

 must be halved. Otherwise each force would be reckoned 

 twice. Hence 



^2wV2 = %pv + ^Kv 

 = |z/(/ + K), 



showing that the effect of such forces as Laplace sup- 

 posed to operate is represented by the addition to />, the 

 pressure exerted by the walls of the vessel, of the intrinsic 

 pressure K. In the above process the particles situated 

 near the surface are legitimately neglected in comparison 

 with those in the interior. 

 I Van der Waals's own process starts from the original 

 form of the virial equation — 



i i2wV- = - pRr cos (R, r), 



] where R now refers to the w/wle force operative upon 

 any particle ; and it appears to me equally legitimate. 

 For all particles in the interior of the fluid R vanishes in 

 virtue of the symmetry, so that the reckoning is limited 

 to a surface stratum whose thickness is equal to the 

 I range of the forces. Upon this stratum act normally 

 I both the pressure of the vessel and the attraction of the 

 j interior fluid. The integrated effect of the latter through- 

 I out the stratum is equal to the intrinsic pressure, and, on 

 I account of the thinness of the stratum, it enters into the 

 I equations in precisely the same way as the external pres- 

 sure exerted by the vessel. The effect of Laplace's forces 

 is thus represented by adding K to p, in accordance with 

 the assertion of Van der Waals. 



I am in hopes that, upon reconsideration, you will be 

 able to admit that this conclusion is correct. Other- 

 wise, I shall wish to hear more fully the nature of your 

 objection, as the matter is of such importance that it 

 ought not longer to remain in doubt. 



Believe me yours very truly, 



Rayleigh. 

 L'Abbaye de St. Jacut-de-la-Mer, September 7. 



NOTES. 



The French Association for the Advancement of Science met 

 at Marseilles on September 17, under the presidency of M. P. 

 P. Deherain, who chose as the subject of his address the part 

 played by chemistry and physiology in agriculture. The meet- 

 ing comes to an end to-day. There were general excursions on 

 Sunday to Aries, and on Tuesday to Aix ; and it is proposed that 

 to-morrow, the 25th, there shall be a final excursion to the 

 Mediterranean coast. 



The Congress of German Naturalists and Physicians was 

 opened at Halle on Monday by Prof. His, of Leipzig. The 

 meeting was attended by 1215 persons, including many dis- 

 tinguished foreign physicians and men of science and 280 

 ladies. 



The Helmholtz celebration, deferred from August 31, is now 

 fixed for November 2. After the ceremony the delegates and 

 others will dine together at the Hotel Kaiserhof. 



Ed. Tr.^ns., vol. 



Part 2, p. 261 



NO. I 143. VOL. 44] 



See also PAil, Mag., October 1890, p. 292. 



