Physical Chemistry , 1st ed.

22.13 is a fundamental equation for the behavior of an interface and is called

the Laplace-Young equation,after Pierre-Simon Marquis de Laplace, a French

mathematician, and Thomas Young, an English scientist whose work was also

considered in Chapter 9.

How can we use equation 22.13? First, consider a droplet of liquid as shown

in Figure 22.6. Regions I and II are marked for reference. Just by considering

the figure and equation 22.13, there is an implication that the pressure in re-

gion I is greater than the pressure in region II. Consider why this is so. Surface

tension is always positive, and the derivative ( A/ VI) is also positive, always:

as the volume of a region increases, so does its area. Therefore, the right side

of equation 22.13 is always positive, so the difference (pIpII) must always be

positive. The only way for this to be so is for pIto be greater than pII: the pres-

sure inside the liquid is greater than the pressure outside the liquid.

If the droplet of liquid is in fact spherical, we can use expressions for the

surface area and the volume of a sphere from geometry:

A 4 r^2 V^43  r^3

where ris the radius of the droplet. Because area and volume are dependent on

another variable, the radius, we can take differentials ofAand Vin terms ofr

and substitute those expressions into the Laplace-Young equation. We find that

dA 8 rdr

for the differential of area in terms of radius. For the change in volume,

dV 4 r^2 dr

Plugging these expressions into the Laplace-Young equation, we find

(pIpII) 

4

8

r

r

2





d

d

r

r



By defining pas the change in pressure across the interface, the above equa-

tion simplifies to

p

2

r



 (22.14)

If the system under consideration were a bubble (that is, a film with an inner

and an outer surface) instead of a droplet, then both surfaces would contribute

a surface energy (that is, surface tension) and equation 22.14 would be

p

4

r



 (22.15)

which is twice that for a droplet.

Equations 22.14 and 22.15 have some interesting applications. If there are

unbalanced pressures in a system, then something usually happens. For exam-

ple, in the case of a simple gas-in-a-piston system, unbalanced internal and ex-

ternal pressures cause irreversible expansions or contractions of the system’s

volume. In the case of liquid droplets, equation 22.14 implies that the smaller

the droplet, the greater the vapor pressure on the gas-phase side of the liquid-

gas interface. This suggests that smaller droplets evaporate faster. This fact has

implications for such differing topics as perfume spraying and gasoline engine

performance. (There are other equations we can use to understand the behav-

ior of liquid droplets further. We won’t consider them here, but see the exer-

cises at the end of the chapter for another example.)

22.3 Interface Effects 773

Region I

Region II

Interface

Figure 22.6 Region I is the liquid and region
II is vacuum or some gas. The interface separates
the two regions. The Laplace-Young equation pre-
dicts some of the properties of the liquid droplet.