Gases and the Equation of Virial. 495 



pressure and temperature at constant volume is in fact 

 linear, or 



r=TfW + xW; (2) 



and it is of interest to inquire whether such a form is to be 

 expected on theoretical grounds, when (f>(p) can no longer 

 be neglected. It has indeed been maintained* that (2) is a 

 rigorous consequence of the general laws of thermodynamics 

 and of the hypothesis that the forces between molecules are 

 functions of the distance only. The argument proceeded 

 upon the assumption that the distances of the particles, and 

 therefore the mutual forces between them, remain constant 

 when the temperature changes, provided only that the volume 

 of the body is maintained unaltered. According to this the 

 virial term in ^1) is a function of volume only, so that (1) 

 reduces to (2), wither) proportional to r _1 . But, as Boltz- 

 mann pointed out, the assumption is unfounded, and in fact 

 inconsistent with the fundamental principles of the molecular 

 theory. The molecules are not at rest but in motion ; and 

 when the temperature varies there is nothing to hinder the 

 virial from varying with it. 



The readiest proof of this assertion is by reference to the 

 case where the molecules are treated as " hard elastic spheres," 

 that is where the force is zero so long as p exceeds a certain 

 value (the diameter of the spheres) and then becomes in- 

 finite. From the researches of Van der Waais, Lorentz. 

 and Tait it is known that in that case 



itp<f>(p) = -itmY*.\ .... (3) 



where b, denoting four times the total volume of the spheres, 

 is supposed to be small in relation to v. So far from the 

 virial being necessarily independent of temperature, it is 

 here directly proportional to temperature. The introduction 

 of the special value (3) into (1) gives the well-known form 



p(v-£)=£gmV 2 =ET, .... (4) 



in which l> is still regarded as small in comparison with v. 

 It is worthy of note that this particular case, although of 

 course sufficient to upset the general argument that the 

 virial is independent of temperature, nevertheless itself con- 

 forms to (2), proportionality to T being for this purpose as 

 good as independence of T. 



Not only is the linear relation maintained in spite of the 

 forces of collision of elastic spheres when no other forces 



* M. Levy, C. 11. t. lxx.wii. pp. 449, 188, 554, 049, 070, 826 (ls78). 



