the Theory of Solutions. 603 



Xevertheless Van Calcar and De Bruyn * have succeeded in 

 showing that solutions really can be concentrated by centri- 

 fugalisation. 



From the term for the concentration gradient we see, 

 that — independent of the intensity of the force — it is equal 



to zero, when ~- is equal to zero at the concentration 

 0<- 



present. In such a solution dc will consequently be equal 



to zero, and the concentration will be the same throughout 



the solution. We shall suppose that we only have to do with 



the influence of gravity, and we shall notice the expression 



(11 (7) more minutely. If ^— is positive, the densitv will 



(X 

 increase with the concentration, and the concentration will 



decrease with the height. When ~ is negative, the con- 

 centration increases upwards. From this it follows, as we 

 have already pointed out at the beginning of this article, 

 that in both cases the centre of gravity of the solution will 

 be lowered by the action of the force. 



For each solution we can consider the density as a function 

 of the concentration, provided that the temperature and 

 pressure are kept constant. This function can be represented 

 as is well known, by putting p proportional to the distance 

 from the point to a fixed line A — B ? and c is represented by 

 the ratios of the distances from the point to two lines 

 perpendicular to A — B (fig. 2). Each solution gives a curve. 



At a point where ^ is equal to zero, the tangent of the curve 



is parallel to A — B and the density has for this concentration an 

 intermediate maximum (or minimum). If p is a rational non- 

 linear function of c, it would always be possible to find one or 



more values of the concentration, for which—- is equal to zero. 



dc 



If. however, these values are to have a physical interpretation 



they must be positive and real, and also be within the 



domain of solubility. In fact we shall find, that to most 



solutions there will be no mixture proportion for which the 



concentration is constant throughout the solution. As the 



contraction of the solution is quite slight, we can take it for 



granted that the solution for which the gradient vanishes 



must consist of components whose densities differ very little 



from each other. 



* Rec. Trav. Chem. Leiden, vol. xxiii. pp. 219-223 (1903). 



