Physical Chemistry Third Edition

5.5 Surfaces in One-Component Systems 229

The surface of a cavity inside a liquid acts like the surface of a droplet, exerting

force that tries to reduce the size of the cavity. It reduces the pressure in the liquid

at the surface of the cavity while increasing the pressure inside the cavity. The vapor

pressure of the liquid is decreased below that of a planar surface.

EXAMPLE5.14

Show that the vapor pressure inside a spherical cavity is given by

ln

⎛

⎝Pvap

Pvap(p)

⎞

⎠−^2 V

(l)

mγ

rRT

(5.5-17)

Solution

We can repeat the derivation of Eq. (5.5-14), the Laplace equation, with the role of liquid and

gas reversed. The result is that the pressure of the gas (inside the cavity) is greater than that

of the liquid (outside the cavity):

P(g)−P(l)P(inside)−P(outside)

2 γ

r

We have already obtained a relation between the total pressure and the vapor pressure,

Eq. (5.3-18):

P 2 P 1 exp

(

Vm(P 2 ′−P 1 ′)

RT

)

whereP′represents the total pressure andPrepresents the vapor pressure. Since the pressure

in the liquid is smaller, and vapor pressure of the liquid is obtained by substitutingP(g)−P(l)

forP 2 ′−P′ 1 , giving

PvapPvap(p)exp

(

−Vm(P(g)−P(l))

RT

)

P(p)vapexp

(

−

2 Vmγ

rRT

)

Exercise 5.16

Find the vapor pressure of water at 298.15 K inside a spherical cavity with diameter 0.0200 mm.

Because the lessening of the vapor pressure at the surface of a cavity is greater for a

smaller cavity, the formation of a cavity requires a higher temperature than vaporizing

the liquid from a planar surface. For this reason, a liquid can often be superheated

well above its normal boiling temperature if its container is smooth and there are no

particles in the liquid at which cavities can begin to form. The superheated liquid can

suddenly boil (“bump”) when the metastablesuperheated liquidfinally begins to form

cavities. This effect has been observed when heating water in a microwave oven if the

water is contained in a vessel with a smooth inner surface. In an organic chemistry

laboratory “boiling stones” with rough surfaces are often placed in a liquid to avoid

superheating.