784 THE MECHANISM OF AGGLUTINATION 



that this motion was due to the bombardment of the particles by the molecules of 

 the solvent. The motion, therefore, becomes strictly analogous to the kinetic motion 

 of the molecules themselves. A quantitative theory for this motion was worked out 

 independently by Einstein and von Smoluchowski and verified experimentally by 

 Perrin. The part of the theory which is of interest in this connection is the prediction 

 regarding the final distribution of the particles at equilibrium. If the Brownian move- 

 ment is really analogous to the kinetic motion of gases, then the distribution of the 

 particles at equilibrium should be determined by the same law that regulates the 

 density of a gas at different levels. Equilibrium will be established when the effect of 

 gravity exactly equals the osmotic pressure (in this case the Brownian movement) of 

 the particles or molecules. In the case of gases this formula is 



gM p 



where h is the height, po the pressure at the bottom of the column, p the pressure at 

 height /;, g the acceleration due to gravity, and M the molecular weight.' Since the 

 osmotic pressure is proportional to the number of particles per unit of volume, the 

 formula, as appHed by Perrin to suspensions, becomes 



, RT , Ho 



gN-Tf^D-d) " 



in which iV is Avogadro's number, D is the density of the particle, and d the density 

 of the liquid. 



Perrin's experiments leave little doubt that the relation between the size of the 

 particles, the rate of settling, and the final distribution is accurately expressed by 

 formulas (i) and (2). If the necessary data regarding the size of the particles, the vis- 

 cosity of the solution, etc., are known, it is, therefore, possible to calculate both the 

 rate of fall of the particles and the final state of equilibrium. Briefly, it may be said 

 that if the size alone is varied, the rate at which the particles fall will increase as the 

 square of the radius and that at equilibrium the distance from the bottom, at which 

 the concentration of particles will be halved, will be inversely proportional to the 

 mass. 



Perrin's equation and results were confined to the region near the surface of the 

 suspension. Porter and Hedges' have measured at deeper levels, and find the particles 

 distributed in accordance with the equation 



^ = K„(i-bny 



in which n is the number of particles, y the depth, and K and b are constants. 



The rate of settling and the final distribution therefore appear to be on a firm ex- 

 perimental and theoretical basis. Application to bacterial suspensions is diflicult, 



' Freundlich, H.: loc. cit. 



» Porter, and Hedges: Phil. Mag., pp. 641-51. 1922; Tr. Far. Soc. 1923. 



