1904.] Catalysis by Colloidal Metals and similar Substances. 363 



This equation, taken in conjunction with the inequality 3, and with 

 Equation 11, leads to the conclusion that the greatest conceivable value 

 for the difference between C and D is JD (it + #!). 



Now, in all the cases we have to consider v is smaller than lO" 6 , so 

 that we may, without appreciable error, make D equal to C. Remem- 

 bering, also, that the decrease - dC of C in the time dt equals NFdt, 

 and expressing the radii of the spheres by their diameter L, we obtain 

 from Equations 9 and 11 



~F 



from which, by utilising the limiting condition C = C for t = 0, we 

 find by integration 



l log = K ( 12 ) 



the constant K being given by the equation 



K = ^?r (13), 



and having the minimum value 



K = T? (I*)- 



Examples to Show that Equations 9, 12, and 14, are a Close Approximation 

 to the Correct Result. 



(a) A Sphere in Infinite Space. In deducing the foregoing equations, 

 we assumed the permanent state to be always produced instantaneously, 

 and we have made it clear that the result thus obtained for the rate of 

 change of concentration will certainly not be too large. In reality, 

 when the spheres are so small as those we have to deal with, and 

 the volume they draw solute from more than a millionfold as great 

 as their own, no appreciable error is made by our assumption. We 

 can readily prove this by integrating Equation 5 completely for simple 

 cases, and showing that the results approximate very quickly to those 

 we have calculated. 



As a first example, we determine the amount of solute which would 

 be removed by each sphere after the time t if it were placed in an 

 infinite amount of solution at the beginning of the experiment. For 

 this purpose we calculate an expression for the concentration throughout 

 the liquid at any time t by integrating Equation 5 under the limiting 

 condition c = D between x = r and x = QO for t = 0, and the con- 

 dition expressed by Equation 6. 



VOL. LXXIV. 2 E 



