December 17, 1891] 



NATURE 



53 



In ih^ first case, in substituting p + alv'- for A. and 

 R(i -I- a/) for ^Imu^, the well-known formula of Prof. 

 Van der Waals is arrived at. In the second case, the 

 same substitution leads to a quite worthless formula, 

 unfit to explain even qualitatively the conduct of gases 

 under compression. 



The first form is the one which presents itself most 

 naturally when, as was done by Van der Waals, the 

 extension of the molecules is considered as a diminution 

 of the volume in which they are moving ; the second is 

 obtained as a first approximation, when the virial equa- 

 tion is extended to the repulsive forces which come into 

 play at the collisions. Of course, both methods, if they 

 could be worked out with absolute rigour, would give the 

 same result ; but, this being impossible for both of them, 

 the question as to which gives the better approximation 

 is not at all an unreal one. 



Now, it is extremely improbable that this question 

 should have to be answered in a different way for 

 linear and for three-dimensional space ; yet for linear 

 space the first method leads to a quite easy and abso- 

 lutelv rigorous solution, and the equation thus obtained 

 is analogous to ^\i& first form. 



14 -Vz V3 



In order to prove this, let A b (Fig. i) represent a linear 

 space of length /, bounded by two rigid walls, A and B, 

 and let there be moving in this space some perfectly 

 elastic particles, all of the same mass, w, and length, X, 

 but having different velocities, t/j, v^, . . . 7/„. At every 

 encounter of these particles there will be simply an 

 exchange between their velocities ; therefore at every 

 moment one of the particles will have the velocity 7^1. 

 On this particle we fix our attention, following it on its 

 way till the next collision. After this collision we leave it 

 to its fate, directing our attention to the other particle, 

 which has now acquired the velocity z'j. Proceeding in 

 this manner, it is obvious that at every collision a distance 

 X is economized, which has not to be travelled over by 

 the centres of the molecules. Starting, then, from the 

 wall A and passing over to the wall B and back again, 

 the number of collisions is i{n - i), and the distance 

 economized 2«X, (adding 2X for the collisions against 

 the walls). The distance travelled over by the centres 

 consequently being 2/ - itiK, it is clear that the number 

 of collisions with velocity v^ against one of the walls 



amounts to -- — ' — - in one unit of time, and the corre- 

 2(L - «X) 



sponding change of momentum to -. ^^^ . , so that the 



«X' 



pressure on the wall is measured by 

 h - n\' ' 



P 



.(3) 



Of course, for space of two and three dimensions, the 

 problem is much more complicated. Yet in 1877 I gave 

 a solution ' of it for spherical particles which, according 

 to my opinion, is rigorous so far as the several encounters 

 between the molecules may be looked at as independent 

 of one another. For a short time after each collision 

 the possibilities of fresh collisions are considerably in- 

 fluenced by the proximity of the departing molecule. 

 This influence, certainly of very difficult mathematical 

 treatment, is disregarded in my calculations. 



The outcome of these calculations - is that of every 



' Verslagen en Mededeelingeti der Kon. Ak. u. Amsterdam, 2" Reeks, 

 Deel X. ; Archives Neerlandaises, t. xii. 



* I am bound to acknowledge that the same correction, which is indicated 

 further on for Lorentz's calculations, has to be applied to the number of 

 collisions given in my paper. 



unit of distance only the (i - 4<5i/7/)th part has to be 

 travelled over by the centres of the molecules, the ifi-^jv th 

 part being economized at the coHisions. Therefore the 

 number of the encounters with the bounding walls is 

 augmented in the proportion i : I - \hh-, and the formula 

 pv — h^mu- is to be changed to 



/>{v - 4<5i) = ^2ww- (4) 



In 1 881, my friend H. A. Lorentz applied the virial 

 equation to the calculation of the influence of the size of 

 the particles on the pressure. In this manner he obtained 

 the formula— 



/>v = l-^iini- • (i + — ) 



(5) 



His paper, published in Wiedemann's Annalen, Bd. 

 xii. p. 127, was inserted in the German and English 

 versions of Van der Waals's pamphlet " On the Continuity 

 of the Liquid and Gaseous States of Matter," at the end 

 of the sixth chapter. 



Considered as a determination of the factor, with 

 which the total volume ^j of the particles, when intro- 

 duced \wv — b, is to be multiplied, our results were iden- 

 tical, and confirmed the opinion expressed by Van der 

 Waals about the value of this coefficient. Mr. Lorentz 

 viewed his results in no other light, and had no inten- 

 tion at all to substitute his formula (5) for that given by 

 Van der Waals. Indeed, in the passage of his paper 

 which I quote here, he clearly indicates the weak point 

 of his calculation : — " Strictly speaking, a correction ought 

 to be made here, indicated by Mr. Van der Waals ; in 

 calculating the number of encounters, the extension^ 

 of the molecules should have been taken into account. 

 The matter is simplified, however, if the influence of the 

 virial arising from the repulsive forces, or the size of the 

 molecules, is small ; and if a correction to the first order 

 is sufficient, then the uncorrected value of the number of 

 encounters maybe used in calculating the small repulsive 

 virial." 



Now it is not impossible to apply to Lorentz's formula 

 the correction alluded to in this passage. In 1875 I cal- 

 culated for the first time,- by a more rigorous method, 

 the shortening of the mean free path of spherical par- 

 ticles, in consequence of their extension in the direction 

 of motion. Some months later, Mr. Van der Vaals suc- 

 ceeded in the same calculations by a somewhat different 

 method, extending it to the case of two sets of particles 

 of different diameters Both calculations lead to the 

 same result, viz. that the mean free path is shortened in 

 the proportion v w — \bx\ therefore the number of the 

 collisions, and the term in the virial equation dependent 

 upon these collisions, must be augmented in the reciprocal 

 proportion ; but then this equation takes the form — 



pv 





Apy 



(6) 



NO. I 155, VOL. 45] 



and becomes identical with the equation (3) of Van der 

 Waals.3 



In this manner the true formula is obtained by means 

 of the virial equation, as it has been by the method of 

 economized distances, and these verifications of the equa- 

 tion derived by Van der Waals are not without import- 

 ance. Indeed, I always held the opinion that it is not 

 quite allowable to conclude directly from the diminution 

 of the free path of the molecule to a proportional aug- 

 mentation of the pressure on the bounding walls. The 

 number of the mutual encounters of the molecules, 

 and the number of their collisions with the walls (or, 

 rather, their passages through an ideal plane), are not 



» The extension in the direction of the motion is meant here. I have 

 translated the first phrase from the original paper in IVied. Ann., where it 

 runs: " Streng genommen miisste man also hier eine Correction anbringen, 

 wie sie von Herrn van der Waals angegeben wiirde ; man hatte niimlich bei 

 der Stosszahl die Grosse der Molecule zu beriicksichtigen." 



^ yerslagen en ^tedcdcelingen, 2 Reeks, Deri x. ; A rehires Xeerlandaises, 

 t. xii. 



3 I owe this remark to a verbal communication by Van der Waals. 



