Physical Chemistry Third Edition

9.9 The Molecular Structure of Liquids 437

The probability distribution for molecular speeds is

fv(v) 4 π

(

m

2 πkBT

) 3 / 2

v^2 e−mv

(^2) / 2 kBT
The mean speed of molecules in our model gas is given by
〈v〉

√

8 πkBT

3 m



√

8 πRT

3 M

Equations were derived for the most probable speed and the root-mean-square speed.

The rate of wall collisions per unit area per unit time is given by

v

1

4

N

V

〈v〉

1

4

N〈v〉

The model system obeys the ideal gas law and Dalton’s law of partial pressures.

A second model system was the hard-sphere gas. We derived an approximate equa-

tion of state for this system and discussed molecular collisions using this model system.

We obtained formulas for the mean free paths between collisions and for collision rates,

for both one-component and multicomponent systems. An important result was that the

total rate of collisions in a one-component gas was proportional to the square of the num-

ber density and to the square root of the temperature. In a multicomponent gas, the rate of

collisions between molecules of two different substances was found to be proportional

to the number densities of both substances and to the square root of the temperature.

A few elementary ideas about the molecular structure of liquids were presented. In

a liquid, the shell of nearest neighbors contains voids, so that fewer nearest neighbors

are present than in the solid. In a typical liquid, a molecule undergoes roughly 100

times as many collisions per second as does a molecule in a typical gas.

ADDITIONAL PROBLEMS

9.77Assume that a certain sample of air at 25◦C contains dust
particles with diameter 5.0μm and density 2500 kg m−^3.
a.Find the most probable speed, the mean speed, and the
root-mean-square speed of the dust particles, treating
them as though they were molecules.

b.Assuming that the dust particles are described by a

Boltzmann distribution, find the ratio of the

concentration of dust particles at a height of 1.000 m to

the concentration at a height of 0.00 m.

c.Find the rate of collisions of one dust particle with other

dust particles if their number density is 1. 0 × 109 m−^3.

d.Find the total rate of collisions per cubic meter of pairs

of dust particles.

e.Find the rate of collisions of one dust particle with

nitrogen molecules.

f.Assume that a dust particle is stationary and calculate

the rate at which nitrogen molecules strike its surface.

Compare your answer with that of part e, and explain

any difference.

9.78At 25◦C and 1.000 atm, methane gas has a mean free path

of 5. 33 × 10 −^8 m.

a.Calculate the effective hard-sphere diameter of methane

molecules.

b.Calculate the mean number of collisions per second

undergone by one methane molecule at this temperature

and pressure.

c.Calculate the total number of collisions per cubic meter

per second in methane gas at this temperature and

pressure.

d.Keeping the volume of the container and the

temperature fixed, enough oxygen gas is added to make