Physical Chemistry Third Edition

5.5 Surfaces in One-Component Systems 227

tension force is downward, and a mercury meniscus is depressed in a glass capillary

tube, as shown in Figure 5.17c.

Exercise 5.15

Assuming a contact angle of 180◦, calculate the distance to which a mercury meniscus is

depressed in a glass capillary tube of radius 0.500 mm at 20◦C.

The Laplace Equation

If a liquid droplet is small enough, it will remain suspended in a vapor phase almost

indefinitely. Figure 5.18 depicts a system that consists of a small droplet of liquid

suspended in a gaseous phase of the same substance. The system is contained in a

cylinder with a movable piston. The piston is displaced reversibly so that an amount

of work is done if the volume changes bydV:

dwrev−P(g)dV−P(g)(dV(g)+dV(l)) (5.5-10)

whereP(g)is the pressure of the gas phase,V(g)is the volume of the gas phase, and

V(l)is the volume of the liquid phase. The surface tension exerts an additional pressure

inside a small liquid droplet, so thatP(l), the pressure inside the droplet, differs from

P(g). OnlyP(g)enters in the expression fordwrevbecause only the gas phase is in

contact with the piston.

Liquid droplet

Vapor phase

Figure 5.18 System to Illustrate

the Pressure Difference in a

Spherical Droplet.The droplet is

surrounded by the vapor phase, and

is assumed to be small enough that

it does not immediately settle to the

bottom of the vessel.

We can write a different expression for the reversible work by considering the phases

separately. The surface work term in Eq. (5.5-7) is added to the usual expression for

dwrevfor the liquid phase,

dwrev−P(g)dV(g)−P(l)dV(l)+γdA (5.5-11)

We assume that the vapor phase has negligible surface tension with the cylinder and

piston, so no such term is added for the vapor phase. Equating the two expressions for

dwrevand canceling the termP(g)dV(g)from both sides, we obtain

(P(l)−P(g))dV(l)γdA (5.5-12)

The droplet is assumed to be spherical, so

dV(l)d

(

4

3

πr^3

)

 4 πr^2 dr (5.5-13a)

dAd(4πr^2 ) 8 πrdr (5.5-13b)

Substituting these relations into Eq. (5.5-12) and canceling the common factordr,we

obtain theLaplace equation:

P(l)−P(g)

2 γ

r

(Laplace equation) (5.5-14)

The Laplace equationisnamedfor
PierreSimon,Marquisde Laplace,
1749–1827,a great French
mathematician,physicist,and
astronomer who also proposedthat the
solar system condensedfrom a rotating
gas cloud.