Page 64 - Essentials of physical chemistry

26 Essentials of Physical Chemistry

A
v

A d

FIGURE 2.1 Sliding layers to derive Poiseuille’s law.

materials will have some sort of rough hills and valleys on the surfaces. We can also expect that
adding weight to the upper sheet will increase the friction, but really that only makes the surfaces
squeeze together more tightly and we have already included the inverse dependence on mean
distance d between the layers. We can anticipate that the friction will depend on the applied load
on the top sheet, but that will not affect the unit analysis of the friction:

cm

cm 2 2
Av s cm
2
f ¼ ma ¼ gcm=s ¼ h ¼ h ¼ h :
d cm s
This leads to the phenomenological units of the coefficient of viscosity in the cgs system as
h ¼ g=cm s 1 poise:

While this unit is easy to derive using reasoning from everyday experience, the poise (pwaz) is
an ancient unit and viscosity is now measured in (pascal seconds), so that 1 poise ¼ 0.1 Pa s in SI
units:

!
2
gcm=s g
2
0:1 Pa s ¼ [(10 dyne=cm )=10]s ¼ 1 s ¼ 1 ¼ 1 P:
cm 2 cm s
More properly called the Hagen–Poiseuille law, it was developed independently by Gotthilf
Heinrich Ludwig Hagen (1797–1884) and Jean Louis Marie Poiseuille. Poiseuille’s law was
experimentally derived in 1838 and formulated and published in 1840 and 1846 by Jean Louis
Marie Poiseuille (1797–1869). Hagen also carried out experiments in 1839. While there are a
number of derivations, we follow a simple one here from Physical Chemistry by Castellan [5].
Consider a pipe with some fluid forced through it by a pressure difference (P 1 P 2 ) where
P 1 > P 2 . Although we will eventually consider the phenomenon from a molecular view, we stress
the power of calculus here to represent a macroscopic effect in terms of infinitesimals. Assume the
fluid (gas, liquid, or slurry) is flowing down, but there is some sort of friction between thin layers as
cylinders sliding within each other like concentric rings of pipe or tree rings (Figure 2.2). We can
see that in the limit as one goes out to the outer wall, the velocity of the layers must be zero while the
velocity is greatest in the center of the tube. Note the total area of the friction is the surface of the
outer shell of a cylinder whose radius varies from zero at the center to R at the wall of the pipe, and
the variation of the velocity can be described as a derivative (dv=dr), so we can write the frictional
force on any given cylinder as

dv
f ¼ hA :
dr