Concise Physical Chemistry

c11 JWBS043-Rogers September 13, 2010 11:26 Printer Name: Yet to Come

170 LIQUIDS AND SOLIDS

question including all sleeves within it. One can think of a pencil of liquid driven

by a force,fgrav, flowing coaxially down the tube. The gravitational force on each

sleeve isfgrav=pA=pπr^2 , whereris the radius of the sleeve. Under conditions

of constant flow rate, these forces are equal and opposite:

fviscous=−fgrav

η 2 πrl

dv

dr

=−pπr^2

dv=−

p

η 2 l

rdr

Integrating the left-hand side of the equation from velocity of flow of 0 at the inner

surface of the tube tovthe velocity at radius r, we get

∫v

0

dv=−

p

η 2 l

∫r

R

rdr

Recall thatRis the radius of the tube wherev=0 andr<Ris the radius of a moving

sleeve of liquid:

v=−

p

η 4 l

(r^2 −R^2 )=

p

η 4 l

(R^2 −r^2 )

A little further analysis along the same lines leads to the Poiseulle equation for the

volume of flowVper unit time:

V=

πpR^4

8 ηl

or

η=

πpR^4

8 Vl

permitting determination of the viscosity coefficientηby measuring the volume of

flow of a fluid through a tube of knownRin unit time.

11.4 CRYSTALS

In many crystals, atoms are arranged in a very regular three-dimensional pattern of

rows and columns, echelons deep. Not surprisingly, when radiation strikes a crystal, it

is reflected fromplanesof atoms, ions, or molecules in the way that light is reflected

from a mirror. The difference is that experimentalists use penetrating X rays to detect

planeswithinthe crystal. The appearance of planes within a crystal is a consequence