Properties of the Equation of State. 399 



function conforming to the law of corresponding states, but 

 its form need not be the same for each substance. In a 

 subsequent paper I will show that w is not zero (as it 

 might be), but a function of T whose form depends on the 

 nature of the substance. 



It will be easily seen that the law of corresponding states 

 can be deduced if we assume that the functions U. u, and Z 

 have the forms given by the above equations. 



Special Forms oj the Equation of State. 



The equation of state is often written, as we have already 

 remarked, in the form _p + P»=<£(T, v), where F n denotes 

 the intrinsic and p the external pressure, and $(T, v) the 

 effect of the motion of translation of the molecules which 

 balances the intrinsic and external pressure. Van der Waals 



•pm 



writes — j- for the right-hand side of the equation, 



m(v — b) fo * 



where b is equal to four times the volume actually occupied 

 by the molecules of a gram of matter. The equation would 

 then strictly hold if the average velocity of a molecule and 

 its chance of colliding with another molecule is not affected 

 by the attraction of the molecules upon one another. But 

 this is by no means the case, as I have shown in a previous 

 paper*. It was shown that when a molecule passes through 

 a point in a substance at which the forces of attraction 

 of the surrounding molecules neutralize one another, its 

 velocity is equal to that it has in the gaseous state at the same 

 temperature. But the average velocity increases with the 

 density of the substance when it does not obey the laws of 

 a perfect gas, and in the case of a liquid is from five to ten 

 times the foregoing velocity. Further evidence of this will 

 be brought forward in a subsequent paper. 



It is necessary therefore to change van der Waals 5 ex- 

 pression into a form taking this fact into account. The 

 pressure exerted by the molecules per cm. 2 in a substance 

 (balancing the intrinsic and external pressure) is propor- 

 tional to the number of molecules crossing a cm. 2 per sec. in 

 one direction f. This number is proportional to the average 

 velocity other things remaining the same. It is evident 



then that we must multiply the expression by W*, where 



Y a is the average velocity of a molecule and V its velocity 

 in the gaseous state at the same temperature. 



* Loc, cit. f Loc. cit. 



