Page 66 - Essentials of physical chemistry

28 Essentials of Physical Chemistry

In itself this is only a formula that tells us the velocity of the fluid at a specific radius r in terms of the
pressure P, the viscosity coefficient h, the length of the tube l, and the outer radius R of the pipe. In
order to obtain a formula that can be used to measure something, we ask how much volume flows
through the tube in a given time, the bulk flow rate as (volume=time) by integrating the function of
r from zero to the outer radius R:

ð R ð R
2 2 ð R 2 3
V P(R r ) PpR r Ppr
dr:
t ¼ (v)(2prdr) ¼ 4hl (2prdr) ¼ 2hl 2hl
0 0 0
So, we have
R
2 2 4 2 2 2 4
V pPR r r pPR R R pPR
:
¼
¼
¼
t 2hl 2 4R 2 2hl 2 4 8lh

0
While this derivation is in two parts and requires some careful thinking, we have shown the power
of using differential calculus to obtain a relatively simple final formula. The method illustrates the
ideas of using small differential quantities with physical reasoning and then integrating to obtain the
macroscopic formula. Note especially that the bulk rate of flow depends on the fourth power of
the radius of the tube! Doubling the radius of the tube will increase the flow by a factor of 16!
Finally, we obtain a way to measure the viscosity coefficient of a fluid by measuring the time it takes
for a fixed volume to flow through a pipe of known dimensions:
4
pPR t
:
h ¼
8lV
We are supposed to be learning an essential form of physical chemistry, and once you are aware of
laminar viscosity you may notice it in a number of applications. Note that this leads to a simple
relationship if we have a standard fluid=liquid with a standardized viscosity such as water. Then, the
time for a standard liquid can be compared with the flow time for the same amount of an unknown

2 4 3
pPR
h x 8lV t x t x
fluid in the same device to obtain the viscosity of the unknown liquid: ¼ 5 ¼ ,so
4
h std pPR 4 t std t std
8lV

t x
that h ¼ h std .
x
t std
Pure water can be used as a standard for viscosity measurements, although a correction should be
applied for changes in density with the temperature:

r t x
x
h ¼ h .
x std
r t std
std
3
For H 2 O, r ffi 1 g=cm .
MEASUREMENT OF VISCOSITY
There are several types of devices which measure viscosity but we will only show the most common
type here, the Ostwald viscometer (Figure 2.3).
The Ostwald type can be purchased from a variety of supply houses with different diameter
bores. Small bores are used for a low-viscosity range near that of water and larger bores for more
viscous liquids. In order to measure the viscosity coefficient of liquids such as motor oil or molasses,