76 METHODS FOR DETERMINING MOLECULAR SIZE AND SHAPE 



where x 2 is the average of the square of the displacement of the particles 

 from their starting point, k is the molecular gas constant, t is the time in 

 seconds at which the displacement is x, T is the absolute temperature, r) 

 is the viscosity of the medium (to be measured or looked up in hand- 

 books) , and r is the radius of the particle. 



Sedimentation. If a gravitational or centrifugal force is applied 

 to a particle in solution, the particle will accelerate until its velocity is 

 such that the accelerating force is just balanced by the frictional force. 

 A measurement of this terminal velocity then gives a relationship be- 

 tween the mass, size, and shape of the particle and the frictional force. 

 If the shape is spherical, it is possible to obtain a direct measure of the 

 diameter and the mass of the particle. If the shape is not spherical, one 

 obtains only a relation between the mass and the diameter, and informa- 

 tion must be added from other methods to permit the knowledge of the 

 mass m and diameter separately. The relationship for spherical particles 

 is 



mv m{\ — dL/dp) 

 F Girrir 



where the symbols are the same as in the relation for the diffusion 

 method, F is the magnitude of the applied force, d L is the density of the 

 liquid solution, d v is the density of the particle, and v is the velocity of 

 the particle in cm/sec. 



In both the diffusion and sedimentation methods, it is necessary to 

 measure the position of the particle at various times. Since the particles 

 of interest are normally invisible even in the light microscope, it is 

 necessary to use some clever method for determining these properties. 

 This is normally done for diffusion by carefully putting two solutions 

 together at a sharp, flat interface. One solution contains the particles of 

 interest, the other is identical except that these particles are missing. 

 Under these conditions, as the particles move they advance like the front 

 line of a parade of soldiers, as indicated in Fig. 35. By optical methods 

 which will not be discussed here, this boundary can be made visible and 

 its progress measured, thereby giving an average displacement for the 

 particles. 



Electrophoresis. If an electric force is applied to a particle in solution, 

 the particle will accelerate until it reaches a constant velocity, as in sedi- 

 mentation. A measurement of this terminal velocity then gives a rela- 

 tionship between the size and shape of the particle, the electric driving 

 force, and the resisting force. Since the electric force is proportional to 

 the electric charge on the particle, and since that charge is unknown, this 

 relationship docs not permit deductions concerning the size and shape of 



