128 Physical Properties [CH. vi 



molecules is vanishingly small. Thus the impacts upon the boundary, of the 

 molecules nearest to the boundary, will be infinitely frequent, and the pressure 

 will be infinite. The effective density of molecules in collision with the 

 boundary is infinite, for all the molecules nearest to the boundary are in 

 position in. of fig. 5. Hence when v = v , we have p = oc , p b = oo . It 

 follows that a system of values of p, v, T for which v < v is impossible ; and 

 that in the corrected form of fig. 6 the isothermals are limited to the region 



v >V Q . 



147. In general we have by differentiation of equation (303), treating ty 

 and cr c as constants for the sake of simplicity, 



Hence it appears that = 0, (i) when p = 0, (ii) when T = (except 

 when p b oo ), and (iii) when 



The right-hand member of this equation is inversely proportional to 

 d log pb/dp. Now when p is small, we have by equation (304) 



27T<7 3 



whence log p b = log p + -= /?+..., 



and therefore by differentiation 



dig = i + 2pV. (309) 



dp p 3m 



Since we are dealing with a unit mass of gas, the number of molecules 

 will be 1/m, and as we are supposing these for the present purpose to be 

 spheres of diameter cr, the smallest volume into which they can be packed 

 without overlapping will be given by 



27TO- 3 2 V2-7T 



whence _= -^tj, 



The relation just found enables us to express equation (309) in the form 



cHogp 6 2\/27T . . 



or = v H -- s v + terms in inverse powers ot v ...... (311). 



dp 



This equation enables us to determine the asymptotic value of d log p^/dp 

 when v is great. We also know that d log p^/dp becomes infinite and positive 

 as v approaches the value v = v from the side v > v . 



