November 26, 1891] 



NATURE 



81 



If the volume of the molecules be very small in com- 

 parison with the space they occupy, the virial of the im- 

 pulsive forces may be neglected, and the equation may 

 be written 



pv = l^ZmN-, (I 



where p is the pressure exerted upon the walls of the 

 inclosure, v the volume, m the mass, and V the velocity of 

 a molecule. 



In his essay of 1873 Van der Waals took approximate 

 account of the finite size of the molecules, using a peculiar 

 process to which exception has been taken by Maxwell 

 and other subsequent writers. It must be said, however, 

 that this method has not been proved to be illegitimate, 

 and that at any rate it led Van der Waals to the correct 

 conclusion — 



p{v - b)^ ^2;//V-, ....... (2) 



in which b denotes four times the total volume of the 

 spheres. In calling (2) correct, I have regard to its 

 character as an approximation, which was sufficiently in- 

 dicated by Van der Waals in the original investigation, 

 though perhaps a little overlooked in some of the 

 applications. 



In his (upon the whole highly appreciative) review of 

 Van der Waals's essay, Maxwell (Nature, vol. x. p. 477, 

 1874) comments unfavourably upon the above equation, 

 remarking that in the virial equation v is the volume of 

 the vessel and is not subject to correction. ^ " The effect of 

 the repulsion of the molecules causing them to act like 

 elastic spheres is therefore to be found by calculating the 

 virial of this repulsion." As the result of the calculation 

 he gives 



/-.' = i5wv2Ji-2iog('i - 8^ + 'Xe: _ . . j j, . (3) 



where o- is the density of the molecules, and p the mean 

 density of the medium, so that pja- = b':\v. If we expand 

 the logarithm in (3), we obtain as the approximate 

 expression, when pier is small, 



(4) 



(5) 



pv - i2wV-(i + a,blv), . . . 

 or, as equally approximate, 



p{v - 4/;) = ^5;//V" 



which does not agree with (2). 



The details of the calculation of (3) have not been pub- 

 lished, but there can be no doubt that the equation itself 

 is erroneous. In his paper of 1881 {Wied. Ann., xii. 

 p. 127), Lorentz, adopting Maxwell's suggestion, inves- 

 tigated afresh the virial of the impulsive forces, and 

 arrived at a conclusion which, to the order of approxima- 

 tion in question, is identical with (2). A like result has 

 been obtained by Prof. Tait (Edin. Trans., xxxiii. p. 90, 

 1886). 



It appears that, while the method has been improved, 

 no one has succeeded in carrying the approximation 

 beyond the point already attained by Van der Waals in 

 1873. But a suggestion of great importance is contained 

 in Maxwell's equation (3), numerically erroneous though 

 it certainly is. For, apart from all details, it is there 

 implied that the virial of the impacts is represented by 

 ^2 w V-, multiplied by some function of p'j(T, so that, if the 

 volume be maintained constant, the pressure as a func- 

 tion of V is proportional to 2 m V-. The truth of this 

 proposition is evident, because we may suppose the velo- 

 cities of all the spheres altered in any constant ratio, 

 without altering the motion in any respect except the 

 scale of time, and then the pressure will necessarily be 

 altered in the square of that ratio. 



It will be interesting to inquire how far this conclusion 

 is limited to the suppositions laid down at the commence- 



' In connectio:j with this it may be worth notice that for motion in one 

 dimension the form (2) is exact. 



ment. It is necessary that the collisions be instantaneous, 

 in relation, of course, to the free time. Otherwise, the 

 similarity of the motion could not be preserved, the 

 duration of a collision, for example, bearing a variable 

 ratio to the free time. On the same ground, vibrations 

 within a molecule are not admissible. On the other 

 hand, the limitation to the spherical form is unnecessary, 

 and the theorem remains true whatever be the shape of 

 the colliding bodies. Again, it is not necessary that the 

 shapes and sizes of the bodies be the same, so that 

 application may be made to mixtures. 



In the theory of gases, 2 wV- is proportional to the 

 absolute temperature ; and whatever doubts may be felt 

 in the general theory can scarcely apply here, where the 

 potential energy does not come into question. So far, 

 then, as a gas may be compared to our colliding bodies, 

 the relation between pressure, volume, and temperature is 



/ = T.^('4 



(6> 



NO. I 152, VOL. 45] 



where <^(z') is some function of the volume. When v is 

 large, the first approximation to the form of <^ is 



V 



In the case of spheres, the second approximation is 



V v^ 



where b is four times the volume of the spheres. 



Thus far we have supposed that there are no forces 

 between the bodies but the impulses on collision. Many 

 and various phenomena require us to attribute to actual 

 molecules an attractive force operative to much greater 

 distances than the forces of collision, and the simplest 

 supposition is a cohesive force such as was imagined by 

 Young and Laplace to explain capillarity. We are thus 

 led to examine the effect of forces whose range, though 

 small in comparison with the dimensions of sensible 

 bodies, is large in comparison with molecular distances. 

 In the extreme case, the influence of the discontinuous 

 distribution of the attractive centres disappears, and the 

 problem may be treated by the methods of Laplace. 

 The modification then required in the virial equation is 

 simply to add ^ to/ a term inversely proportional to z/,-' 

 as was proved by Van der Waals ; so that (6) becomes 



/!= T<^(z;) - av-- ■ • • (7) 



According to (7), the relation between pressure and 

 temperature is linear — a law verified by comparison with 

 observations by Van der Waals, and more recently and 

 extensively by Ramsay and Young. It is not probable, 

 however, that it is more than an approximation. To 

 such cases as the behaviour of water in the neighbour- 

 hood of the freezing-point it is obviously inapplicable. 



In their discussions, Ramsay and Young employ the 

 more general form — 



T<^(Z-) + X(Z'): 



(8) 



and the question arises, whether we can specify any 

 generalization of the theoretical conditions which shall 

 correspond to the substitution of x (■^) for <^ ''''"^- It 

 would seem that, as long as the only forces in operation 

 are of the kinds, impulsive and cohesive, above defined, 

 the result is expressed by (7) ; and that if we attempt to 

 include forces of an intermediate character, such as may 

 very probably exist in real liquids, and must certainly 

 exist in solids, we travel beyond the field of (8), as well 

 as of (7). It may be remarked that the equation sug- 

 gested by Clausius, as an improvement on that of Van 

 der Waals, is not included in (8). 



' It thus appears that, contrary to the assertion of Maxwell,/ is subject 

 to correction. It is pretty clear that he had in view an attraction of much 

 smaller range than that considered by Van der Waals. 



