Page 69 - Essentials of physical chemistry

Viscosity of Laminar Flow 31

V
So, we find ¼ 2:5548 gal=min. This is too high even with the duty factor of 0.1, so let us
t
standardize the effective duty factor for an average adult human of about 4900 mL=min. Then we have
4900 mL
ffi 1:29493 gal=min:
(4 qt=gal)(946 mL=qt)

Thus, if we wish to use the formula from a rigid pipe for a flexible aortic wall we need to adjust the
duty factor:

Duty x
¼ (2:5548 gal=min ) ¼ 1:29493 gal=min,
Cycle 0:1
so, x ffi 0:0507 and we have an effective formula which takes into account the elastic wall of a large
artery like an aorta and the pulse nature of the heartbeat as

4
P h P l 6 2 2:54 cm 1 gal 60 s
p (1:01325 10 dyne=cm atm) (0:0507)

V 760 R in: 3784 cm 3 min
:
¼
t 8(L in:)(2:54 cm=in:)(h g=cm s)
We note that in some cases, we could rearrange the equation to calculate the effective radius of the
pipe and use the fourth root (square root of the square root) of the rearranged formula if we know the
(V=t) bulk flow rate. This formula also teaches us a lot about using different units.

STAUDINGER’S RULE FOR POLYMER MOLECULAR WEIGHT

Although Albert Einstein is most well known for his work in the theory of relativity and for his
analysis of the photoelectron effect, he also developed a foundation [3,4] for the theory of solution
viscosity. His initial work was on solutions of colloidal spheres and sugar solutions and that work
was limited in application. However, as a result of plastics and synthetic rubber being developed
during WWII, the field of Polymer Science emerged with great significance to chemical industry.
While chemists developed new synthetic methods for formation of polymers, the inevitable
question arose as to the values of molecular weights. The question is complicated by the fact that
often the products of polymerizations are ‘‘polydisperse,’’ that is, there is a mixture of various
molecular weights (n-mers) of similar compounds after the reaction. We can only give a glimpse of
polymer science here, but the measurement of viscosity is now a standard technique in determining
average molecular weight of large polymer molecules.
Initially, the application of viscosity measurements to polymer solutions extended the relation-
ships derived by Einstein for colloid solutions. While colloids might be assumed to be roughly
spherical, polymer molecules can be flexible, rod-like, or plate-like, so adjustments had to be made.
Einstein [4] defined some useful terms. Let h be the viscosity of the solvent alone and h be the
0
viscosity of the solution in question. As the solution is diluted, the viscosity will approach the h
0

h
. A further
r
h 0
value, but at other concentrations we can define the relative viscosity as h
quantity was defined as the ‘‘specific viscosity’’ as the amount by which the viscosity of a given
solution differed from that of the solvent as h h 1. Finally, yet another quantity was defined
sp
r
as the ‘‘intrinsic viscosity,’’ which has an interesting graphic property and is believed to be an
intrinsic property of the polymer solute. In Einstein’s original work [3,4], the intrinsic viscosity for
hard spheres is 2.5, but we expect lower values for quasi-linear polymers. Theoretically, the intrinsic