TEMPERATURE AND THE PROPERTIES OF GASES 39 



two special points, A where i/<j> = o and -k$\Q — 8/3, at which all 

 the isotherms converge, and B which is the critical point. 



This diagram of 7r<£/0 is particularly interesting and con- 

 venient for showing the whole range in consequence of these 

 limits. It is to be noticed that the change of 7n/>/0 obtained 

 in passing along an isotherm is a change of entropy with 

 change of density (— dty/dv), which is very important in many 

 theoretical discussions. 



By the usual process of finding minimum values it will be 

 found that the minima of irfyjd are given by 



(10) <t>\27 - 80) - 1 80 + 3 = o 



for various values of 6, the limiting values to give real solutions 

 being 6 = - 2 f and - s \ where irfyjQ — % and zero respectively (fig. 1, 

 curve c). We shall see later also that values of 7r</>/0 = pv/T\ 

 are of considerable interest in the treatment of the variability 

 of certain quantities such as the specific heats. 



It has been mentioned above that one of the criteria of a 

 perfect gas is that it shall not be heated or cooled in expanding 

 through a small orifice under a small difference of pressure. 

 Now it is found in practice that nearly all gases are cooled on 

 expansion and that at the ordinary temperature only helium 

 and hydrogen will be heated among the known gases. The 

 effect with helium has not yet been observed directly, that with 

 hydrogen being measured with some uncertainty by Joule and 

 Kelvin in their famous experiments. All that one can justly 

 deduce from their results with hydrogen is that the change was 

 very small, but towards a heating rather than a cooling effect. 



The general equation given by Lord Kelvin reduces when 

 there is no heating to (3), and if this is applied to the reduced 

 equation (7) the following relation is obtained 



("5 2 (27-40)-i8<H-3 = o, 



which is the same equation as (10) if 6 has twice the value it 

 has there. 



The values obtained from this are shown in curve d, fig. 1. 

 Hence at all values inside the curve there will be cooling and 

 at all values outside heating. The maximum temperature 

 according to equation (1 1) at which the inversion will take place 

 will be 6 = 675, at which it will occur at zero density. Con- 

 sidering the case of hydrogen and assuming Tc = 30 K., pc = 



