Physical Chemistry , 1st ed.

Example 22.4

What is the change in pressure across the surface of a droplet of water having

a radius of 0.100 mm? What if the droplet had a radius of 0.001 mm (that is,

1 m)? The surface tension of water is 72.75 erg/cm^2.

Solution

Mathematically, this example is simple from a numerical perspective, but we

must watch our units. Using equation 22.14:

p

2

r





We need to revise the units so that they are consistent and work out to units

of pressure. First, convert the mm unit in the denominator to centimeters:

p

1

1

0

c

m

m

m

1.455  104 

c

e

m

rg

 3

Notice how the complex fraction involving the units simplifies into a simpler

fraction. Additionally, we recognize that 1 dyne, a unit of force, equals

1 erg/cm. (An erg is a unit of energy.) We substitute:

p1.455  104 

c

d

m

yn

 2

Force per unit area (that is, dyn/cm^2 ) is defined as pressure, but what pres-

sure unit is this? If you consider the units dyne and centimeter, you can show

that 1 dyn/cm^2 is one-millionth of a bar, the standard SI unit of pressure.

Converting:

p1.455  104 

c

d

m

yn

 2 

p0.01455 bar

This might not seem like a large pressure difference, less than 2% of an atmo-

sphere. But keep in mind that this is for a drop that’s only 0.100 mm—a tenth

of a millimeter—in radius! This is a substantial pressure difference for such

a small droplet. Using 0.001 mm as a radius and repeating the substitution

and conversions, we can show that

p1.455 bar

This pressure difference is greater than atmospheric pressure!

The above example illustrates how large pressure differentials can be across

an interface. Keeping in mind that pressure differentials will act to force liquid

molecules to vaporize, one can see the advantage to vaporizing a liquid by

separating it into as small a droplet as possible (a process misleadingly called

atomization).

Interfaces also exist between liquid and solid phases. Under certain condi-

tions their behavior is governed by surface tension effects. Consider a small

droplet of a liquid on a surface. The way the droplet behaves depends on the

1  10 ^6 bar



1 

c

d

m

yn

 2

(^2) 72.75 
c
e
m
rg
 (^2) 

0.100 mm
(^2) 72.75 
c
e
m
rg
 (^2) 

0.100 mm
774 CHAPTER 22 Surfaces