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the more diffuse the boundary between solvent and solution becomes owing
to this effect. Figure 9.13b schematically illustrates the effect of this
broadening in the Schlieren traces of the concentration profiles shown in
Fig. 9.13a. The widths of these peaks depend on D, although correction
must be made for the acceleration, which distorts the shape of the peaks
from the ideal form indicated in Fig. 9.13b. Nevertheless, in principle,
a study of sedimentation velocity permits the sedimentation coefficient
to be evaluated unambiguously and the diffusion coefficient to be at
least estimated. The proviso "in principle" is added, since s and D vary
differently with M, and it may be only for an optimum range of M's that
both s and D can be determined from a single experiment with sufficient
Once a sedimentation coefficient has been measured, there are several
ways in which it can be used:
1. As noted above, ss° can be used directly to characterize a solute.
This practice is widely followed with biopolymers.
2. The particle can be assumed to be spherical, in which case M/NA can
be replaced by (4/3)7rR3p2, and f by 6ïò]0Ê. In this case the radius can
be evaluated from the sedimentation coefficient: s = 2R2(p2 - ð)/9ò?0.
Then, working in reverse, we can evaluate M and f from R. These
quantities are called, respectively, the mass, friction factor, and
radius of an equivalent sphere, a hypothetical spherical particle which
settles at the same rate as the actual molecule.
3. If we assume that the densities of the solute and solution are known
from separate experiments, the ratio M/f can be evaluated: M/f = NAs/(l -
p/p2). By Eq. (9.79), f can be replaced by kT/D, with no assumptions
regarding the shape of the particle. Therefore, if D is measured from the
Schlieren traces of the sedimentation or in an independent experiment, s
can be interpreted to give a value for M:
NAkT s (1 - P/Pa) D
Sedimentation Velocity and Sedimentation Equilibrium
4. For polydisperse systems the value of M obtained from the values of
s° and D°-or, better yet, the value of the s/D ratio extrapolated to ñ =
0-is an average value. Different kinds of average are obtained, depending
on the method used to define the "average" location of the boundary. The
weight average is the type obtained in the usual analysis.
If a sedimentation experiment is carried out long enough, a state of
equilibrium is eventually reached between sedimentation and diffusion.
Under these conditions material will pass through a cross section
perpendicular to the radius in both directions at equal rates: "downward"
owing to the centrifugal field, and "upward" owing to the concentration
gradient. It is easy to write expressions for the two fluxes which
describe this situation:
1. The flux due to sedimentation is equal to the concentration of
solute times vs as given by Eq. (9.96):
Jse d = CVs = CC°2rS
2. The flux due to diffusion is given by Fick's first law:
Jdiff = - D ?
3. Under conditions of sedimentation equilibrium, the sum of these two
fluxes equals zero, or
D - = cco2sr
which can be integrated to give
Inc = ^ co2r2 + const.
Using Eqs. (9.96) and (9.79) to replace s and f, respectively, we
obtain M(1 - p/p2) . .
In ñ = ------------- ñî r + const.
Note that this expression is equivalent to the barometric formula which
gives the variation of atmospheric pressure (°c c) with elevation (°c r).
A first-order dependence on the distance variable holds in the barometric
equation, since the acceleration is constant in this case.
Frictional Properties of Polymers in Solution
If the solute concentration is measured at different values of r in a
sedimentation equilibrium experiment, Eq. (9.103) predicts that a plot of
ln ñ versus r2 is linear with a slope equal to M(1 - p/p2)co2/2RT. This
type of experiment, therefore, allows M to be determined without either
measuring D or assuming anything about the shape of the particle. For a
polydisperse sample with a continuous distribution of molecular weights,
several different kinds of average values of M-starting with Mw and
including Mz [Eq. (1.19)] and still higher averages-can be extracted from
this kind of data, particularly under 0 conditions. Protein preparations
which are more commonly studied by this method are likely to contain