Physics. — ''On the Equation of State for Arbitrary Temperatures 

 and Volumes. Analog ij with Plan(;k'3 For)nnla\ Bj Dr. J. J. 

 VAN Laar. (Coiniuiuiicated by Prof. H. A. IjOrentz). 



(Communicated in tlie meeting of January 25. 1919). 



^ 1. Introduction. 



In foiii- papers') I tried more closely to study the depeiuleiire on 

 the teinperatiue of the quantities a and 6 of van dkr Waai.s's e?iiiation 

 of state on the ground of kinetic considerations. I then came to the 

 conclusion that the quantity a must steadily increase with descending 

 temperature to a maximum value in the neighboui-hood of the absolute 

 zero point, after which it again decreases to the value at 7'=0. 

 All this with very large volume. 



Also with respect to /> I carried oul similar computations, but 

 the mathematical ditiliculties become greater and greater, and the 

 results obtained become very complicated. And for small volumes 

 such a treatment of the problem is still less suitable. I, therefore, 

 gave np the idea of ]>ublisliing what was still found in connection 

 with the said paper, and tried to solve the question by another and 

 simpler way. 



The thought had already occurred to me before, to substitute for 

 the three-dimensional problem an analogous problem of one dimen- 

 sion, and then to transform the result in the well-known way to 

 one which would hold for three dimensions. It is clear that so 

 doing the nature of the sought dependence on the temperature and 

 the volume of the constants occurring in the equation of state will 

 not be moditied; there can oidy arise some difference in a few 

 numerical factors. But these are aflei' all immaterial, when in the 

 result some groups of quantities, the said factors included, are joined 

 to one (U' more constants. 



This method has besides also the advantage that it cannot only 

 be used for large volumes, but also for small volumes, and results 

 are, therefore, obtained which are universally valid, not only for 

 arbitrary temperatures, but also for arbitrary volumes, from v = (X) 

 up to ?' = h. 



1) These Proceedings XX, p. 750 and 1105; XXI, p 2 and 16. 



