Physical Chemistry Third Edition

410 9 Gas Kinetic Theory: The Molecular Theory of Dilute Gases at Equilibrium

Exercise 9.15

Find the mean speed and the root-mean-square speed of helium atoms at 298 K.

Exercise 9.16

Show that the two ratios〈v〉/vpandvrms/vphave the same two values for all gases at all

temperatures and find those values.

Exercise 9.17

Find the ratio of the probability densities for each of the following:

a.vpand〈v〉for argon gas at 298.15 K.

b.vpand〈v〉for argon gas at 500.0 K.

c.vpand〈v〉for helium gas at 298.15 K.

d.vpandvrmsfor argon gas at 298.15 K.

e.vpandvrmsfor argon gas at 500.0 K.

f.vpandvrmsfor helium gas at 298.15 K.

PROBLEMS

Section 9.4: The Distribution of Molecular
Speeds

9.20 The escape velocity at the earth’s surface is

1. 12 × 104 ms−^1  2. 5 × 104 miles per hour.
   a.Find the fraction of helium atoms at 298 K
   having a speed exceedingvesc. You can use the
   identity
   ∫x

0

t^2 e−at

2

dt

√

π

4 a^3 /^2

erf

(√

ax

)

−

x

2 a

e−ax

2

b.Find the temperature at which the mean speed of

helium atoms equals the escape velocity.

9.21 a.Find the fraction of molecules that has speeds greater
than

√

kBT/m. Explain the relationship of this fraction

to the fraction computed in Problem 9.17. You can use

the identity in Problem 9.20.

b.Find the fraction of molecules that has kinetic energies

greater thankBT/2. Explain the relationship of this

fraction to that of part a.

9.22 For N 2 molecules at 298.15 K, calculate the probability
thatvexceeds the speed of light, 2. 997 × 108 ms−^1
according to the Maxwell–Boltzmann probability
distribution.

9.23 a.Find the temperature at which hydrogen atoms

have an average translational energy equal to

1. 0 × 109 J mol−^1 (perhaps enough energy to initiate a

fusion reaction).

b.What is the mean speed of hydrogen atoms at this

temperature?

c.Find the temperature at which deuterium atoms

have an average translational energy equal to

1. 0 × 109 J mol−^1.

9.24 Find the fraction of molecules in a gas that has:

a.speeds less than the most probable speed.

b.speeds between the most probable speed and the mean

speed.

c.speeds between the most probable speed and the

root-mean-square speed.

d.speeds greater than the root-mean-square speed.

e.Explain why these fractions are independent of the

temperature and of the mass of the molecules.

Note: You can use the identity in Problem 9.20.

9.25 The speed of sound in air and in other gases is somewhat

less than the mean speed of the gas molecules.