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

Himenz P.C. Polymer Chemistry. The Basic Concepts - Copyright, 1984. - 736 p.
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2. Equation (1.34) is a normalized expression which means that,
integrated
over all values of x, it equals unity. Accordingly,
P(x,t)dx = (2 7Ã Kt l2) 1/2 exp
630
Frictional Properties of Polymers in Solution
concentration of solute in a series of slices through the apparatus-
that is, / ñ dx = / c0P(x, t) dx-always accounts for all of the solute,
whatever its distribution.
3. The combined consideration of items (1) and (2) indicates that the
profile of ñ throughout the apparatus gets flatter as it gets broader. At
t = 0 the initial concentration is sharply peaked at c0 at x = 0; at t =
°°, ñ is uniform from one end of the apparatus to the other.
All of these points are entirely reasonable. What is confusing about
Eq. (9.83) is the step length and the undetermined constant K.
Fortunately, they can both be eliminated in a single step. If Eq. (9.83)
is a solution to Eq. (9.74), then both sides of the latter must be equal
when the indicated operations are performed on Eq. (9.83). Carrying out
these operations yields the following:
1. The first derivative of ñ with respect to t is
(Ý.
= c0 (27Ã Kt 12)~1/2 e-x2/2Kt,:
2 Kt'
1
2t
2. The second derivative of ñ with respect to x is obtained from
(S)t = ^/2Kt|2 (- ø)
followed by
Kt
Kt
3. Combining items (1) and (2) as required by Fick's second law yields
(9c/9t) x2/2 Kt2!2 - l/2t K2t2l4 Kl2
D =
(Ý2 ñ/Ýõ2 )t x2 /Ê212 Ã - 1 /Ktl2 2 Kt2 r
These manipulations show that Eq. (9.83) satisfies Fick's second law with
The Diffusion Coefficient: Experimental Aspects
631
Ê =
2D
(9.84)
so the solution becomes
c(x, t)dx - c0(47rDt) 1/2 exp
dx
(9.85)
Figure 9.11 is a plot of this function at two different times for a
solute with a diffusion coefficient arbitrarily selected to be 5 X 10-11
m2 sec-1.
We can imagine measuring experimental curves equivalent to those in
Fig. 9.11 by, say, scanning the length of the diffusion apparatus by some
optical method for analysis after a known diffusion time. Such results
are then interpreted by rewriting Eq. (9.85) in the form of the normal
distribution function, P(z) dz. This is accomplished by defining a
parameter z such that
in terms of which Eq. (9.85) becomes
The inflection point of this function-where the second derivative changes
sign-occurs at z = 1; hence the experimental analogs of Fig. 9.11 are
examined for the location of their inflection points (subscript infl).
The distance through which the material has diffused at this point is
therefore given by
from which D can be evaluated if t is known. If Fig. 9.11 had been
determined experimentally, such a procedure would be used by observing
that xinfl = 10-2 m for the curve at t = 106 sec. From this information,
D = (10~2 m)2/2(106 sec)= 5 X 10-11 m2 sec-1.
The experiment we have just described is not very satisfactory from a
practical point of view, since it is very difficult to deposit a thin
layer of solution between two bulk portions of solvent without some
mixing. An experimentally more convenient method consists of layering
equal volumes of solvent and solution so that a sharp boundary exists
between them at x = 0, with ñ = c0 for

(9.87)

(9.88)
632
Frictional Properties of Polymers in Solution
x x 102 (m)
Figure 9.11 Variation of c/c0 with x for one-dimensional diffusion
[calculated from Eq. (9.85) with D = 5 X 1ÑÃ11 m2 sec-1 ].
x < 0 and ñ = 0 for x > 0. This procedure also depends on layering
liquids, but it is easier to do this with bulk quantities of material
than to sandwich a thin layer between two bulk portions of solvent as
required above.
The Diffusion Coefficient: Experimental Aspects
633
At the outset of such an experiment, the gradient of concentration is
zero on either side of the boundary and sharply peaked near x = 0. As
diffusion occurs, the boundary becomes progressively less sharp. For x <
0, ñ < c0, and for x > 0, ñ > 0, with both the magnitude and spatial
extension of the effect increasing with time. Figure 9.12a shows how the
overall concentration profile varies with time, and Fig. 9.12b shows the
gradient dc/dx at corresponding times. Formal solution of Fick's second
law with these boundary conditions proves that the shape of the gradient
curves in this situation are identical to the curves in Fig. 9.11.
Measuring the concentration gradient poses no serious experimental
difficulty, so this method can also be analyzed by Eq. (9.87).
We are aware of the bending of a light ray when light passes from a
medium of one refractive index to another. If light passes through a
layer of optically homogeneous material and then reenters the original
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