Physical Chemistry Third Edition

5.5 Surfaces in One-Component Systems 223

The molar enthalpy change of vaporization of CCl 4 at 20◦C is equal to 33.77 kJ mol−^1 ,

so that the molar energy change of vaporization is

∆vapUm∆vapHm−∆(PV)≈∆vapHm−RT

33770 J mol−^1 −(8.3145 J K−^1 mol−^1 )(293 K)31330 J mol−^1

The energy change of vaporization per molecule is

31330 J mol−^1

6. 022 × 1023 mol−^1

 5. 203 × 10 −^20 J

We assume that this energy is the amount by which the potential energy of the molecule is

increased when it breaks free from its neighbors and enters the vapor phase, where attractions

to other molecules are negligible. When a molecule is brought to the surface, it gains energy

roughly equal to 30% of this value, since it on the average loses approximately 3 of its 10

nearest neighbors. The surface energy per molecule is thus

(surface energy per molecule)≈(6. 203 × 10 −^20 J)(0.30) 1. 6 × 10 −^20 J

The density of CCl 4 at 20◦C is equal to 1.594 g cm−^3 , and its molar mass is 153.82 g mol−^1.

This gives a molar volume of 96.5cm^3 mol−^1 , and a volume per molecule of

(volume per molecule)

96. 5 × 10 −^6 m^3 mol−^1

6. 022 × 1023 mol−^1

 1. 60 × 10 −^28 m^3

A sphere with this volume has radius equal to 3. 4 × 10 −^10 m. We assume that on the average

each molecule on the surface occupies an area equal to the area of a square 6. 8 × 10 −^10 m

on a side:

area per molecule(6. 8 × 10 −^10 m)^2  4. 6 × 10 −^19 m^2

The surface energy per square meter is denoted byγand is approximately equal to

γ≈

1. 56 × 10 −^20 J

3. 6 × 10 −^19 m^2

 0 .043 J m−^2

This value agrees only roughly with the experimental value at 20◦C, 0.02695 J m−^2.

Exercise 5.13

a.Estimate the surface energy per square meter for liquid water, assuming that the princi-

pal intermolecular force is hydrogen bonding, with a bond energy of 20 kJ mol−^1 for each

hydrogen bond. Assume that a molecule in the interior of a sample of liquid water has four

hydrogen-bonded nearest neighbors and that a molecule in the surface has three. Remember

that each hydrogen bond involves two atoms. Compare your result with the experimental

value at 25◦C, 0.072 J m−^2.

b.For 1.00 mol of liquid water contained in a beaker with diameter 3.00 cm, find the ratio of

the surface energy of the upper surface to the energy required to vaporize 1.00 mol of water.

c.Explain why gases have negligible surface energy.