Physical Chemistry Third Edition

9.9 The Molecular Structure of Liquids 435

has a nearest-neighbor distance equal to 372 pm. This distance is slightly smaller

than the distance at the minimum of the Lennard-Jones pair potential, 380 pm, as we

might expect due to the attractions of molecules beyond the shell of nearest neighbors.

Although the density of liquid argon is smaller than that of the solid, the nearest neigh-

bors are at very nearly the same average distance as in the solid. On the average there

are fewer nearest neighbors, because the nearest neighbors are disordered and voids

exist between them.

Exercise 9.26

Estimate the number of nearest neighbors around an argon atom in the liquid by multiplying

12, the number of nearest neighbors in the solid, by the ratio of the density of the liquid to the

density of the solid. The density of solid argon is equal to 1.82 g mL−^1 , and that of liquid argon

is equal to 1.40 g mL−^1.

Since molecules in the liquid are surrounded by other molecules their motions are

very different from those of gaseous molecules. The nearest neighbors form a “cage” in

which a given molecule is confined. Instead of moving considerable distances between

occasional collisions, a molecule of a liquid is involved in frequent collisions as it

undergoes a kind of rattling motion. Intermolecular forces do not depend on velocities,

so the velocity distribution of Section 9.4 is valid for a liquid or solid as well as for a

gas. The mean speed is given by Eq. (9.4-6):

〈v〉

√

8 kBT

πm



√

8 RT

πM

(9.9-1)

For example, the mean speed of water molecules in liquid water at 100◦C is the same

as in water vapor at the same temperature. The input of energy required to vaporize the

liquid changes the potential energy, not the kinetic energy.

Since the molecules in a liquid are much closer together than in a gas, and since

they are moving just as rapidly on the average as in a gas at the same temperature, the

rate of collisions in a liquid is much greater than in a gas. There is some ambiguity

in defining a collision between two molecules in a liquid, because the molecules are

not exactly like hard spheres and there is no unique instant of contact between them.

However, if some definition of a collision is adopted, the rate of collisions between

liquid molecules can be estimated.^7

EXAMPLE9.21

Estimate the collision frequency of an argon atom in liquid argon at its normal boiling tem-

perature, 85 K. Assume that the atom is moving at the mean speed for this temperature.

Estimate the time between collisions as the time required to travel twice the distance from the

minimum in the potential function of Figure 9.15 to a value ofrsuch that the potential energy

is equal to the kinetic energy of a particle moving at the mean relative speed.

(^7) P. K. Davis,J. Chem. Phys., 57 , 517 (1972).