# Polymer Chemistry. The Basic Concepts - Himenz P.C.

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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

accuracy.

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

639

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

(9-99)

2. The flux due to diffusion is given by Fick's first law:

Jdiff = - D ?

(9.100)

3. Under conditions of sedimentation equilibrium, the sum of these two

fluxes equals zero, or

D - = cco2sr

(9.101)

dr

which can be integrated to give

Inc = ^ co2r2 + const.

(9.102)

Using Eqs. (9.96) and (9.79) to replace s and f, respectively, we

obtain M(1 - p/p2) . .

In ñ = ------------- ñî r + const.

(9.103)

2NAkT

7

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.

640

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

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