62 PRINCIPLES OF GENERAL PHYSIOLOGY 



a multiplied by nonir. root of 2, or less than twice; this root, expressed as 

 exponent ( -\ usually lies between the values of 0*1 and 0'5. In the latter case 



\M/ 



it is, of course, the simple relation of the square root, but, as we shall have many 

 opportunities of seeing in succeeding pages, it is very rarely that it is precisely 

 of this value. A table of values for a number of typical cases will be found 

 on pp. 150 and 151 of Freundlich's work (1909). In other words, the more 

 dilute the solution, the greater is the proportion of its contents that is adsorbed. 

 The equation expressing this relationship is given by Freundlich (1909, p. 146) 

 in the following form : 



x 1 



m ~ 

 where a; is the amount adsorbed by the surface m, from a solution whose final 



concentration is C, a and - being constants for a particular surface and solution. 



N 



The temperature is supposed constant, so that the expression is that of the 

 adsorption isotherm, a may be defined as the quantity adsorbed by unit surface 

 from a solution which is of unit concentration when in equilibrium with the 

 amount adsorbed by the surface. Its value varies considerably in different 

 instances, according to surface tension, electric charge, and so on. The range 



of its values is very much greater than that of - 



The relation of this formula to that correlating diminution of surface tension 

 with concentration, as given on p 52 above, will be evident. If we consider 

 the effect of successive deposits on a surface, it will be clear that the first one 

 will cause greater diminution of surface energy than succeeding ones, and each 

 of these less than its predecessor. Each successive deposit occurs on a surface 

 whose energy is already lessened by the previous deposit. Finally, a state of 

 saturation is reached. 



The curve expressed by Freundlich's equation is usually, but incorrectly, called an 

 "exponential" one. Properly speaking, an exponential curve is one whose equation has 

 one of the variables as an exponent: y = a.e. tx . Our curve is one of the forms of the general 



equation to the family of parabolas : y ax" ; when t = 2, or - =0'5, the curve is the ordinary 

 parabola, when n = 3, it is called a cubic parabola. 



In order to determine the values of - and a for a series of experimental results, the simplest 



n 



way is to plot out the values on logarithmic paper. Freundlich's formula may be written thus, 

 by taking logarithms throughout : 



log 5 :. log a + _ log C. 



in n 



This formula is that of a straight line inclined to the axes. If the values of log be 



m 



represented as ordinates, and those of log as abscissa*, ., is the tangent of the angle made 



71 



by the straight line joining the series of points with the axis of absciss*. This line cuts the 

 axis of ordinates at a point above the origin ; the distance of this point from the origin is the 

 value of log a. 



Although this formula satisfies adsorption processes through a wide range, 

 it has been shown by G. C. Schmidt (1911, p. 660) that a more complex one 

 is needed to satisfy extremes of concentration, and he gives the following : 



A (8-jr) 



\ = Kxe 8 



where x is the amount adsorbed, a the amount of substance originally present, 



and v the volume in which a was dissolved. a ~ x j s then the concentration 



v 



of the solution in equilibrium. S is the amount at the maximum, i.e., the 

 amount adsorbed when in equilibrium with a saturated solution, and, therefore, 



S-x 

 o is the proportion of the amount adsorbed at a given concentration to 



