or, alternatively, as 



THE SEDIMENTATION VELOCITY METHOD 91 



m P {\ ■ - d L /d p ) 



co 2 r / 



The left-hand side of this expression is defined as the sedimentation 

 coefficient s. It is in terras of s that many biological materials are de- 

 scribed, since it is equivalent to stating the mass or molecular weight of 

 the particle. Since s is a ratio between the velocity attained by a particle 

 and the acceleration a) 2 r given it by the particular machine used, it will 

 be the same for all centrifugation machines. Also, the right-hand term 

 must be independent of the particular machine used, and it can be seen 

 that it depends only on the properties of the particle and the solvent 

 liquid. 



To proceed further, we must, in effect, measure /. The usual procedure 

 is to utilize a relation first derived by Einstein: 



, kT 



where T is the absolute temperature, k is the molecular gas constant, and 

 D is the so-called diffusion coefficient, which can be determined from the 

 distance particles diffuse (by Brownian motion) in a measured time 

 interval. Thus, by using Einstein's expression for /, we obtain an expres- 

 sion for s which depends on measurable quantities: 



m p (l — d L /d p ) D Nm p (l — d L /d p )D 

 S ~ ' ~kf~ NkT 



where we have multiplied numerator and denominator by N, Avogadro's 

 number. Here Nm p is the molecular weight M of the particle and Nk is 

 the more usual form of the (molar) gas constant R, as used in the gas 

 law, pV — RT. Thus the final expression for the sedimentation coefficient 

 is 



M(l - d L /d p )D 

 S ~ RT 



In practice, s, T, and d L are obtained in one experiment and D in a 

 separate experiment. The particle density d p may be obtained by cen- 

 trifuging the particles in liquids of various densities to determine the 

 density at which the particles do not move ; this liquid density must then 

 equal the particle density. Thus, finally, we can obtain the molecular 

 weight, which is the object of all this work and analysis. 



For spherical particles, we have indicated that the diffusion coefficient, 

 D, is inversely proportional to the cube root of the molecular weight. 



