48o 



NATURE 



\Oct. 15, 1874 



M. Van der Waals makes his'molecules elastic spheres, 

 which, when not in contact, attract each other. His 

 treatment of the " molecular pressure " arising from their 

 attraction seems ingenious, and on the whole satisfactory, 

 though he has not attempted a complete calculation of 

 the attractive virial in terms of the law of force. 



His treatment of the repulsive virial, however, shows a 

 departure from the principles on which his investigation 

 is founded. He considers the effect of the size of the 

 molecules in diminishing the length of their " free paths," 

 and he shows that this effect, in the case of very rare gases, 

 is the same as if the volume of the space in which the 

 molecules are free to move had been diminished by four 

 times the sum of the volumes of the molecules themselves. 

 He then substitutes for V, the volume of the vessel in 

 Clausius' formula, this volume diminished by four times 

 the molecular volume, and thus obtains ths equation — 



where / is the externally applied pressure, -' , is the 



molecular pressure arising from attraction between the 

 molecules, which varies as the square of the density, or 

 inversely as the square of the volume. The first factor is 

 thus what he considers the total effective pressure. V is 

 the volume of the vessel, and b is four limes the voluaic 

 of the molecules. The second factor is therefore the 

 "effective volume" within which the molecules are free 

 to move. 



The light hand member expresses the kinetic en;rgy, 

 represe.ited by tiro absolute temperature', mulliplicd by a 

 quantity. A', constant for each gas. 



The results obtained by M. Van der Waals by a 

 compArison of this equation with the determinations of 

 Regnauk and Andrews are very striking, and would 

 almost persuade us that the equation represents the true 

 stale of the case. But though this agi cement would be 

 strong evidence in favour of the accuracy of an empirical 

 formula devised to represent the experimental results, 

 the equation of M. Van der Waals, professing as it does 

 to be derived from the dynamical theory, must be sub- 

 jected to a much more severe criticism. 



It appears to me that the equation does nut agree with 

 the theorem of Clausius on which it is founded. 



In that theorem p is the pressure o( the sides of the 

 vessel, and /' is the volume of the vessel. Neither of 

 these quantities is subject to correciion. 



The assumption that the kinetic energy is determined 

 by the t;mperature is true for perfect gases, and we have 

 no evidence that any other law holds for gises, even near 

 iheir liqucf)ing point. 



The only source of deviation from Boyle's law is 

 therefore to be looked for in the term i22(/i.V), which 

 expresses the virial. 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 

 repulsiou. 



Neglecting the effect of attraction, I iind that the efftct 

 of the impulsive repulsion reduces the equation of Clausius 

 to the form — 



8 +.7 



vS;c.) 



pV=^\^{mv-)Y - 2log.(i - "^ , -/ „. — , ( 

 where o- is the density of the molecules and [> the mean 

 density of the medium. 



The form of this equation is quite different from that of 

 M. Van der Waals, though it indicates the effect of the 

 impulsive force in increasing the pressure. It takes no 

 account of the attractive force, a full discussion of which 

 would carry us into considerable difficulties. 



At a constant temperature the effect of the attractive 

 virial is to diminish the pressure by a quantity varying as 

 the square of the density, as long as the encounters 

 of the molecules are, on the whole, between two at a 

 time, and not between three or more. The effect of 

 the attraction in deflecting the paths of the molecules 

 is to make the number of molecules which at any given 

 instant are at distances between r and ;■ + dr of each 

 other greater than the number in an equal volume at a 

 greater distance in the proportion of the velocities corre- 

 sponding to these distances. As the temperature rises, 

 the volume being constant, the ratio of these velocities 

 approaches to unity, so that the distribution of molecules 

 according to distance becomes more uniform, and the 

 virial is thus diminished. 



If there is a virial arising from repulsive forces acting 

 through a finite distance, a rise of temperature will in- 

 crease the amount of this kind of virial. 



Hence a rise of temperature at constant volume will 

 produce a greater increase of pressure than that given by 

 the law of Charles. 



The isothermal lines at higher temperatures will exhibit 

 less of the diminution of pressure due to attraction, and 

 as the density increases will show more of the increase of 

 pressure due to repulsion. 



1 must not, however, while taking exception to part of 

 the work of M. Van der Waals, forget to add that to him 

 alone are due the suggestions which led me to examme 

 the theory of virial more carefully in order to explore the 

 continuity of the liquid and the gaseous states. 



1 cannot now enter into the comparison of his tlieoreti- 

 cal results with the experiments of Andrews, but 1 would 

 call attention to the able manner in which he expounds 

 the theory of capillarity, and to the remarkable phenomjnon 

 of the surface tension of gases which he tells (p. 3S) has 

 been observed by Bosscha in tobacco smoke. As tobacco 

 smoke is simply warm air witli a slight excess of carbonic 

 acid, carrying solid particles alpng with it, the change of 

 properties at the surface of the cloud must be very slight 

 compared with that at the surface where two really dilfe- 

 rent gases first come together. If, therefore, the pheno- 

 menon observed by Bosscha is a true instance of surface- 

 tension, we may expect to discover much more striking 

 phenomena at the meeting-place of different gases, if we 

 can make our observations before the surface of discon- 

 tinuity has been obliterated by the inter-diffusion of the 

 gases. J. Clerk-Maxwell 



LETTERS TO THE EDITOR 



[ The EJltjr iloa not hold liimsdf responsible for ofntions expressed 

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 No notice is taken of auonyinous comniunications.'\ 



An Anagram 



TllE practice of enclosing discoveries in sealed packets and 

 sending them to Academies, seems so inferior to the old one of 



