9G REPORT — 1894. 



The Connection between the Virial Equation, the Second Law, and the 

 Boltzmannr Maxwell Law. 



57. The virial equation cannot be used to jyrove the Second Law. 

 When applied to a perfect gas it leads to the equation 



pv=nO (59) 



From this equation, combined with the Second Law, certain properties of 

 pei'fect gases, with which we are all familiar, are deduced in treatises on 

 thermodynamics. These are that the latent heat of expansion is equal to 

 the pressure, that the difference of specific heats is equal to R, and so on. 

 Hence corresponding conditions must hold good in molecular thermo- 

 dynamics in order that the Second Law may be consistent with the virial 

 equation. 



A general condition under which the Second Law is true was originally 

 found by R. C. Nichols,^ and has recently been put into a somewhat 

 different form by Burbury.- If ^ or U denote the potential energy, this 

 condition may be written 



T\dv^-dv)-~dtdv 



(GO) 



where the bar denotes average values. 



Burbury •* has rightly pointed out that the methods of Clausius and 

 Szily ■• require such a condition to be satisfied in order that they may give 

 the Second Law. The condition comes in when we try to assign a mean- 

 ing to the ' quasi-period i,' without which meaning the result is unin- 

 telligible. Burbury shows that if we assume 



i=i;*T'- (i.e., a definite time), 

 tlien x must be a function of v only, so that 



^X=^X (Gi) 



dv dv 



and this is a special case of Nichols' condition. 



It is also to be observed that in the Clausius-Szily method the 

 averages are time-averages. This, if not an objection, is at least a 

 disadvantage, since it does not show the relation between the Second 

 Law and the Boltzmann-Maxwell Law, in which averages are taken over a 

 large number of molecules in a ' special state ' of permanent distribution. 



58. That relation forms the subject of a very recent paper by Burbury,'' 

 which is a development of his second letter to ' Nature.' ^ 



The proof of the Second Law on the assumption of the Boltzmann- 

 Maxwell Law has long been known, and was given by Boltzmann as early 



' 'On the Proof of the Second Law of Thermodynamics,' Pliil. Mag., 1876 (i.), 

 p. .S69. 



^ Natiire, December 14, 1893; January 11, 1894. 

 ' Ibid., December 14, 1893. 



* ' Report on Thermodynamics,' Part I. Section 1, Cardiff Report, 1891, p. 88. 



* 'The Second Law of Thermodynamics,' Phil. Mag., June 1894, p. 574. 

 « Nature, Jan. 11, 1894. 



