4 o SCIENCE PROGRESS 



15 At, and taking B = o*i and hence 7r=r5 atmospheres, then 

 = 6*3 1 which makes T = 189*3 K. From this to about = 2 the 

 inversion occurs at nearly the same values of 7r<f>/0, the pressures 

 rising to 77 = 8*95, which with hydrogen = 134 At. This is not very 

 different from the results found by Olszewski at Cracow using 

 a method which is not strictly carried out on the principles 

 on which the Kelvin equation is deduced. Also it is known 

 from practical experience that hydrogen experiences a sensible 

 cooling when expanded through a fine jet at pressures of about 

 100 atmospheres at the temperature of liquid air, which is 

 about 83 K., as this has been used to effect the liquefaction of 

 hydrogen in combination with the regenerative process as used 

 by Linde originally for air. 



This limiting value for helium, with a Tc = 5*1 and pc = 2*3 

 about, will be T = 32*2 K. with & = o*i. This result is again to 

 some extent substantiated by experiment, as the isothermal 

 determinations of H. K. Onnes at Leiden showed that the 

 minimum value />v/T would be at about 18° K. for very small 

 densities, and, as has been pointed out above, the relations 

 expressed by equations (10) and (n) make this temperature just 

 half that of the inversion point for the same density. 



By using a temperature of 15 K. obtained by means of liquid 

 hydrogen boiling under reduced pressure, H. K. Onnes was 

 able to liquefy helium with ease. As a contrast is the case of 

 oxygen, in which Tc = 1 55° K. and pc = 50 At. at a density of 

 0*02, which would be equivalent to a pressure of about 1 

 atmosphere = 6'6, whence T = 1023 K. = 750 C, while, where 

 8 = 1*5; 7T = 5 '9 ; so that at a temperature of — 40 C, the pressure 

 at which inversion would occur would be about 300 kg. and 

 hence quite within measurable limits. 



It should again be emphasised that the results obtained by 

 the use of equation (7), or indeed any other theoretical equation, 

 are not to be taken as numerically accurate, but only as indicat- 

 ing the probable course of the relation. If anything were 

 wanted to make this clear, it would be a consideration of the 

 limiting temperatures found by the use of the various equations 

 of state and equation (8). Some of the more important are 



Clausius 3*182 V 1 + — tc, Berthelot 4*24 tc, Reinganum 5*36 tc 



in place of the 6*75 tc found with the v.d. Waals equation. On 

 the other hand, the empirical equation (8) gives a value just 



