Kinetic Theory of Gases. 797 



that is to say by e 2 ^^. Making this correction the value 

 of b given by (20) follows at once by the usual arguments. 

 Equations (19) and (20) used in the ordinary formula 

 NkTb — a must give the exact form for the second virial 

 coefficient. 



We may note here the differences between the usual values 

 of a and b and the values given by (19) and (20). The- 

 usual values are really obtained on the assumption that 

 2jU(a) is small so that e^ 11 ^ is very nearly unity, and the 

 effect of this factor on a and b negligible. The value of a 

 so obtained is of course 



a = 27rN 2 ( xm(x)dx. 



Jo 



It is most important to recognize that increasing the 

 dilution has no effect whatever on the accuracy or inaccuracy of 

 this approximation, for it does not affect 2/II(cr). The ap- 

 proximation can only be justified if the temperature is 

 sufficiently high. But by equation (4) 2/TI(a-) is of the order 

 5S/T. For an ordinary monatomic or diatomic gas, whose 

 viscosity is analysed by Sutherland's formula, S>100. 

 [The only exceptions are He, H 2 , and Ne.] Analysed by a 

 more exact formula, taking account of all powers of 1/T, a 

 considerably smaller value of S (perhaps 5-10 times smaller) 

 might fit the observations, and agree with the compressibility 

 data (see § 2). But even so, 2jH(a) is about unity at a 

 temperature of 100° absolute, and can hardly be considered 

 small at temperatures less than 1000° absolute. It is not 

 till we reach these temperatures that the clustering effect can 

 be ignored, and the gas regarded as genuinely of uniform 

 density throughout. 



§ (10). The direct calculation of the Virial of Clausius. — 

 It is possible to calculate independently by this method the 

 exact value of the coefficient of l\v in the expression for pv, 

 and so confirm the results of the last section. The virial 

 method is, however, unsuitable for making allowance, when 

 the need arises, for the effect of the boundary field. 



We take a molecular model for which the law of impulsion 

 f(r) between a pair of molecules is 



/(*') = Q, (<7-6<r<cr), 



/W = -<Kr), (r>ar). 



We obtain the elastic sphere model (diameter a) with the- 



