Forces on the Osmotic Pressure of Electrolytes. 745 



w is the most probable, or two-thirds of the average, kinetic 

 energy of translation possessed by an ion or a molecule at 

 the temperature. T, 



7 /4tt 2NX* 2 / , • 



vT'Ty r /™ W 



h is a pure number, being the ratio of a potential energy to 

 the kinetic energy w ; the potential energy is that of two 

 ions at a distance apart equal to the radius of the sphere in 

 which on a random distribution the average number of ions 

 contained would be one. <j){h) is a function of Ji the value 

 of which is theoretically determinate, and of which approxi- 

 mate numerical values for a few different values of h were 

 calculated. 



In equation (1) the interionic forces are reckoned 4- when 

 repulsive, and E is a negative quantity, 4> (h) being always 

 negative. It will be more convenient for our present pur- 

 pose to deal with a positive quantity for the virial. Put 

 W=-E and /(A) = -0(70, then 



W=NWi/(/t) 



is positive and may be called the attractive virial. If the 

 volume V contains 1 gm. mol. of the associated ions, as we 

 shall henceforward assume, then N = v, and w = RT/N, 

 whence 



W = RTA/(/0 (3) 



A curve showing the approximate course of f(Ji) with h is 

 here appended (tig. 1, p. 746). 



Strictly, the result (3) only holds if the ratio h is a small 

 number, the chief reason being that the assumption is made 

 in the calculation that association of the ions due to elec- 

 trical forces is negligible, and this would become less and 

 less true as li increases. Also to simplify the expressions 

 the virial was calculated for an assumed gaseous mixture, 

 the forces between a pair of ions being put # 2 /r 2 . Under 

 these conditions, however, Jt comes out to be quite a large 

 number, thus the result is evidently not applicable to an 

 actual gaseous mixture of ions which have the ordinary ionic 

 charges. This conforms with the fact that a gaseous mixture 

 of ions initially dissociated is not in equilibrium, but undergoes 

 spontaneously an almost complete association. The case is, 

 however, different when we apply the result to electrolytes. 

 We do not know what will be the exact effect on the in- 

 terionic forces of the water molecules between the electro- 

 lytic ions ; it is doubtless a very complicated one, but it will 

 certainly diminish the forces very considerably. In the 



