68 PROCEEDINGS OF THE AMERICAN ACADEMY. 



L N N' L' minus the area M M' N' N. The former area is equal to 



P 



R T\n~, and if the latter be designated by A we have the equation, 



RT\nt = ET\n^,-A. (29) 



Now if the point M' is moved in the direction of greater volume, equa- 

 tion (29) holds true continuously, and therefore is true if M' is taken at 

 infinite volume. But at infinite volume 



^' --^ P', 

 and therefore 



R T\n^ = PT\DP-A^, (30) 



if A^ represents the total area bounded by the line M N and the curves 

 M M' and N N', each produced to infinity. This equation may be 

 written, 



i?7'ln^ = -^^,orln^ = --^,or ^^Pe^^, (31a) 



where e is the base of natural logarithms. 



The deviation of the fugacity from the gas pressure is, therefore, 

 dependent upon the area A^. The case that has been chosen in which 

 the curve M M' lies within N N' is of course the common one. For 

 gases of the opposite type, hydrogen and helium, the formulae will be, 



i? T'ln-^ = + ^^ and i/. = Pe^ • ' (31b) 



"We see at once that for all known gases and vapors except hydrogen 

 and helium the escaping tendency is less than the gas pressure ; for 

 these two, greater. The determination of the value of the fugacity at 

 any pressure involves the estimation of the area A^. This must be 

 done by integrating the most exact empirical equation of the isotherm of 

 a gas between the pressure in question and the pressure zero. This 

 method has the disadvantage of all extrapolation, but the value thus 

 obtained may be checked by using a second empirical equation of 

 another form and recalculating A^. If the two results coincide the 

 value obtained will in all probability be very near the true value of A^. 

 In conclusion it may be remarked that equation (29) applies to the 

 isothermal of all substances, not merely to gases, and can be frequently 

 of use. For example, if it is possible to pass continuously from vapor to 

 liquid along an isothermal, it is evident that in passing from a saturated 

 vapor to its liquid, 



