Physical Chemistry Third Edition

436 9 Gas Kinetic Theory: The Molecular Theory of Dilute Gases at Equilibrium

Solution

〈v〉

√

8 RT

πM



√

8(8.3145 J K−^1 mol−^1 )(85 K)

π(0.039948 kg mol−^1 )

212 m s−^1

The kinetic energy of an argon atom moving at this speed is 1. 49 × 10 −^21 J. To find the value

ofrthat corresponds to this value of the potential energy function, we set

1. 49 × 10 −^21 J 4 ε

[

(σ/r′)^12 −(σ/r′)^6

]

wherer′is the value ofrthat we seek. This equation must be solved by numerical approx-

imation, using a spreadsheet or a simple computer program. With values ofεandσfrom

Table A.14, we find thatr′ 3. 3 × 10 −^10 m. The minimum of the potential energy is at

3. 8 × 10 −^10 m. Twice the distance fromr′to this value ofris 1. 0 × 10 −^10 m. The time

required to traverse this distance at 212 m s−^1 is equal to 4. 7 × 10 −^13 s, giving an estimate

of the collision rate equal to 2. 1 × 1012 s−^1.

As seen in the previous example, collision rates of a molecule in a typical liquid are

several hundred times larger than in a typical gas.

PROBLEMS

Section 9.9: The Molecular Structure of Liquids

9.74 The density of ice is 0.917 g mL−^1 at 0◦C, and that of
liquid water is 1.000 g mL−^1. Ice is completely
hydrogen-bonded with four nearest neighbors for each
water molecule. Estimate the average number of nearest
neighbors in liquid water at 0◦C. Explain your answer, and
explain why liquid water is denser than ice.

9.75 a.Calculate the density of solid xenon from the
Lennard-Jones parameters, assuming that the
interatomic distance is 2% smaller than the minimum
in the pair potential function.

b.The density of liquid xenon is 3.52 g mL−^1. Estimate

the number of nearest-neighbor atoms in the liquid.

9.76 a.If 1.000 mol of argon atoms is in a perfect crystal lattice

such that each atom has 12 nearest neighbors at the

interatomic distance equal to the distance at the

minimum in the Lennard-Jones potential function,

calculate the energy required to turn the crystal into a

gas, neglecting all interactions except those of nearest

neighbors. Assign half of the interaction energy of a

pair to each member of the pair, so that each atom has

to break six attractions in the sublimation process.

Compare this energy with the actual energy of

sublimation at 0 K, 8.49 kJ mol−^1.

b.Calculate the energy of vaporization of argon, assuming

that each argon atom has approximately 10.5 nearest

neighbors.

c.Calculate the energy of fusion of argon.

Summary of the Chapter

The first model system designed to represent a dilute gas consists of noninteract-

ing point-mass molecules that obey classical mechanics. We obtained the Maxwell–

Boltzmann probability distribution for molecular velocities:

g(v)

(

m

2 πkBT

) 3 / 2

e−mv

(^2) / 2 kBT