Page 70 - Essentials of physical chemistry

32 Essentials of Physical Chemistry

viscosity is a pure number but it is sometimes reported as a reciprocal concentration such as
deciliters=gram due to the denominator of the definition:

h ln h
h i
sp r
and :
c c!0 c c!0
[h] ¼ [h] ¼
In 1930 Staudinger [8] proposed to adapt Einstein’s formalism to solutes of polymers, which may or
may not be spherical, even rod-like, or plate-like in a simple formula known as Staudinger’s rule:
[h] ¼ KM:

The interesting thing about this technique of relating viscosity to molecular weight is that if one can
measure the viscosity of various concentrations of polymer in the solvent and plot, both the
h ln h
h i
sp r
c c
values of [h] ¼ and [h] ¼ on the y-axis of a graph and the concentration
c!0
c ¼ grams=100 ml on the x-axis (100 mL ¼ 1dL ¼ 0.1 L) the two lines should=will meet at
the same value of the intrinsic viscosity [h]. Thus, both the methods of plotting the graph
yield the same intrinsic viscosity value. Sometimes other units are reported such as

g 10 10 g 1000 10 kg

.
¼
dL 10 L 1000 m 3
¼
This type of work requires careful laboratory technique, but it is very satisfying to see both lines
have the same intercept. Often other types of viscometers are used for this work but the Ostwald
viscometer can be used for dilute solutions. Further work by Staudinger and his associates was
carried out to find the value of K for various types of polymers, and the relationship was later refined
to use two parameters (K, a)as in
a
[h] ¼ KM :
In each case, there had to be calibrations of the molecular weight using other techniques for absolute
molecular weights such as melting point methods, osmotic pressure, and light scattering, but the
ease of using the viscosity measurements then allows the determination of the molecular weight of
an unknown. Today, this is a standard laboratory procedure in industries where polymer properties
are measured. Thus, while ‘‘viscosity is a drag,’’ its measurement is of great practical importance in
industry and of some use as a diagnosis technique in hematology, the study and science of blood.

Example
Castellan [5] gives three data points for polystyrene dissolved in benzene at 258Cas
3
[C(kg=m ), h(mPa s)]: (21.4, 1.35), (10.7, 0.932), and (5.35, 0.757). Plot these data using both
definitions of the intrinsic viscosity and extrapolate to zero concentration. h for benzene is 0.606
0
mPa s (Figure 2.5). The two intercepts should be close but take the average of the two intercepts as
a
the best value of the intrinsic viscosity. Castellan suggests using the expression [h] ¼ KM with
3
K ¼ 1:71 10 3 m =kg and a ¼ 0:74 (slightly different from the 308C data in Table 2.2) along
with the viscosity to calculate M, the effective molecular weight of this sample:
1

ln ([h]) ln K 1 [h] [h] a ðÞ
a
[h] ¼ KM , ln ([h]) ¼ ln K þ a ln M, ln M ¼ ¼ ln ¼ ln ,
a a K K
1

[h] a ðÞ h sp ln h r
so that M ¼ , now set up tables of [h] ¼ and [h] ¼ for
K c c
c!0
3
[C(kg=m ), h (mPa s)]: (21.4, 1.35), (10.7, 0.932), and (5.35, 0.757).