Physical Chemistry Third Edition

222 5 Phase Equilibrium

5.34 a.Write a computer program to carry out the Maxwell
equal-area construction, assuming the van der Waals
equation of state. It is probably best to choose a trial
value of the coexistence pressure and to calculate
the two areas, and then to carry out successive
approximations until the areas are as nearly equal as
you desire.
b.Using the equal-area construction, find the vapor
pressure of water at 100. 0 ◦C according to the van der
Waals equation of state.
c.Find the molar volumes of the coexisting liquid and
vapor phases of water at 100. 0 ◦C according to the
van der Waals equation of state and the equal-area
construction.

d.Find the value of the compression factor, Z, for water

vapor in coexistence with liquid water at 100. 0 ◦C.

5.35 Find the difference in the slopes of the two tangent lines at

the cusp in each of the following graphs for pure water:

a.A graph ofGmas a function ofTat 1.000 atm in the

vicinity of 0. 0 ◦C.

b.A graph ofGmas a function ofPat 273.15 K in the

vicinity of 1.000 atm.

c.A graph ofGmas a function ofTat 1.000 atm in the

vicinity of 100. 0 ◦C.

d.A graph ofGmas a function ofPat 373.15 K in the

vicinity of 1.000 atm.

5.5 Surfaces in One-Component Systems

Many of the thermodynamic equations that we have presented are valid only in the case

that surface contributions to the energy can be neglected. For example, we assumed

that the thermodynamic energy of a one-component one-phase fluid system depended

onT,V, andn, but not on the surface area. Although this is ordinarily an excellent

approximation, there is a significant surface contribution to the energy of a liquid in

the case of a small droplet or a liquid in a small capillary tube.

The Energy Attributed to a Surface

The surface contribution to the energy of a liquid is primarily due to intermolecular

attractions. Since molecules at the surface of the liquid have fewer nearest neighbors

than molecules in the bulk (interior) of the liquid, they have a different average potential

energy than molecules in the bulk.

EXAMPLE5.10

For liquid carbon tetrachloride in contact with its vapor, estimate the surface energy per unit

area, using the enthalpy change of vaporization to estimate the net attractive energy of the

molecules.

Solution

In a solid lattice, carbon tetrachloride molecules can pack together like spheres and are

surrounded by 12 nearest-neighbor molecules. Since the liquid is somewhat disordered and

is less dense than the solid, we assume that each molecule in the interior of the liquid has

approximately 10 nearest neighbors. Molecules at the surface of the liquid have no nearest

neighbors on the vapor side, and we assume that a molecule at the surface has approximately

seven nearest neighbors.