Physical Chemistry Third Edition

5.6 Surfaces in Multicomponent Systems 235

The Clausius–Clapeyron equation gives the vapor pressure of a liquid or solid phase:

ln(P)

−∆Hm

RT

+constant

where∆Hmis assumed to be temperature-independent.

Inclusion of surface effects leads to the expression fordU

dUTdS−PdV+γdA+μdn

whereγis the surface tension and whereA is the interfacial surface area of the

system. In most systems the effects of the surface energy are negligible, but the effects

are significant in systems with large surface areas per unit mass, such as small droplets,

which are found to have a larger vapor pressure than larger droplets.

In a multicomponent system, some substances tend to accumulate at a surface, and

others tend to avoid the surface. These phenomena were related to the surface Gibbs

energy.

ADDITIONAL PROBLEMS

5.50 The molar enthalpy change of vaporization of water is
equal to 44.01 kJ mol−^1 at 298.15 K, and the vapor
pressure of water at this temperature is equal to
23 .756 torr.
a.Use the Clausius–Clapeyron equation to estimate the
vapor pressure of water at 100◦C. Compare your result
with the actual value, 760.0 torr.

b.Use the modified Clausius–Clapeyron equation derived

in Problem 5.27 to revise your estimate of the vapor

pressure at 100◦C, assuming that the heat capacities are

constant. Comment on your result.

c.Assume that the heat capacities of the liquid and vapor

phases are constant and equal to their values at

298 .15 K. Find the value of∆Hm,vapat 100. 0 ◦C, and

compare your value with the correct value,

40 .66 kJ mol−^1.

d.Use the Clausius–Clapeyron equation to estimate the

vapor pressure of water at the critical temperature,

647 .4 K. The actual critical pressure is equal to

218 .3 atm. Explain any discrepancy.

e.Use the modified Clausius–Clapeyron equation derived

in Problem 5.27 to estimate the vapor pressure of water

at 647.4 K. Comment on your result.

5.51 a.Consider a coexistence curve in a pressure–temperature
phase diagram of a single pure substance. Show that the
phase on the high-temperature side of the curve is
the phase of higher molar entropy and that the phase on

the high-pressure side of the curve is the phase of

smaller molar volume.

b. Interpret the three curves in the water phase diagram in

terms of this result.

c.Interpret the curves in the^3 He phase diagram in terms

of this result.

5.52 a.The vapor pressure of mercury at 260◦C is equal to

100 torr, and at 330◦C it is equal to 400 torr.

Find the enthalpy change of vaporization of

mercury.

b.Find the normal boiling temperature of mercury and

compare it with the experimental value, 356. 9 ◦C.

5.53 Label each of the following statements as either true or

false. If a statement requires some special condition to

make it true, label it as false.

a.At equilibrium, a substance that occurs in two phases

will have the same concentration in both phases.

b.Two phases at equilibrium must have the same pressure.

c.The Clapeyron equation applies only to a phase

transition involving a vapor phase.

d.The Clapeyron equation is an exact thermodynamic

equation.

e.The Clausius–Clapeyron equation is an exact

thermodynamic equation.