( 315 ) 



Physics. — "On the law of molecular attraction for electrical 

 double points." By Prof. J. I), van der Waals Jr. (Com- 

 municated by Prof. J. D. van deh Waals). 



Though Prof. M. Reinganum has sometimes expressed the opinion 



that from the supposition thai the molecules are electrical double 



points would follow that the molecular attract ion decreases proportio- 



1 

 nallv to — , he pointed out to me in a private letter that his deduc- 



tion of the equation of stale does not depend upon the accuracy of 

 (his opinion. In fact for every pair of molecules we have for the 

 value of the potential energy 



— — - y/3 (.•os* # -|- 1 . cos <p 



and so for the virial i. e. for \rR — if r represents the line joining 

 the molecules and R the component of the molecular force in the 

 direction of this line — 



m 



— 1/3 cos 7 i> -f- 1 . cos <p, 



2r 4 



. . virial 3 



so the ratio — — , =— . 



potential energy 2 



As this ratio has the same value for each pair of molecules, it is 

 also the ratio between the total virial of the attracting forces of all 

 the molecules and the total potential energy. And it is this ratio which 

 Reinganum uses for his deduction of the equation of state. 



So we see that this ratio has the same value as if the attraction 

 decreased proportionally to i/V 4 , though in reality the mean attraction 

 decreases much more rapidly. If we wished to calculate this ratio by 

 differentiating the value for the mean potential energy 1 ) with respect 

 to r, in order to find the force, and by multiplying this value by ! ', /• in 

 order to find the virial, we should find a value different from ' ,. 

 This is because we should not differentiate the average potential 

 energy in order to find the average force, as I have I.e. erroneously 

 written. In order to find the average force we must first find the 

 force between two molecules by differentiating the potential energy 

 with respect to r. Then we must multiply the result by : 



ui'l '3 cos* &-j- I . cos f 

 e 3r ' 1 l J A sin <f> d<f sind-tlïï. 



!) These proceedings XI, p. 132. 



21 



Proceedings Koyal Acad. Amsterdam. Vol. XI. 



