Physical Chemistry Third Edition

9.8 The Hard-Sphere Gas 433

be easy to remember because all four factors are things to which the rate should be

proportional. The most important physical fact shown in Eq. (9.8-31) is this:The total

rate of collisions between molecules of two substances is proportional to the number

density of each substance.

Exercise 9.25

Assume that 0.800 mol of nitrogen (substance 1) and 0.200 mol of oxygen (substance 2) are

contained in 24.45 L at 298 K.

a.Findλ1(2),λ2(1),λ1(1), andλ2(2).

b.Findz1(2),z2(1),z1(1), andz2(2).

c.FindZ 11 ,Z 22 , andZ 12.

d.Find the total number of collisions per second.

PROBLEMS

Section 9.8: The Hard-Sphere Gas

9.59 a.Write an expression for the excluded volume of a pair
of hard spheres of different sizes.
b.Obtain an equation of state for a mixture of two
different hard-sphere substances.
c.Compare your equation of state with the van der Waals
equation of state using the mixing rules in Problem
1.38.

9.60 Calculate the pressure of carbon dioxide gas at 298.15 K
and a molar volume of 24.00 L mol−^1 , assuming (a) the
ideal gas law, (b) the van der Waals equation of state, and
(c) Eq. (9.8-6), taking the same value of the parameterbas
for the van der Waals equation of state.

9.61 What would the pressure of nitrogen gas have to be at
25 ◦C for a nitrogen molecule to undergo 1. 00 × 106
collisions per second?

9.62 For N 2 gas at 298.15 K and 1.000 atm, calculate the ratio
of the total volume of the molecules to the volume of the
gas, (a) using the value of the hard-sphere diameter from
Table A.15 of Appendix A, and (b) using the value of
molecular diameter calculated from the value of the van
der Waals parameterb.

9.63 Calculate the coefficient of ther−^6 term in the
Lennard-Jones potential, 4εσ^6 , for He, Ne, N 2 ,O 2 , Ar, and
CO 2. Make a graph of this quantity versus the number of
electrons in the atom or molecule. Comment on your result
in view of the interpretation that the London attraction is
due to synchronized fluctuating dipoles in the electrons of
the two attracting atoms or molecules.

9.64An approximate equation of state for the hard-sphere fluid

is due to Carnahan and Starling:

PV

NkBT



1 +y+y^2 −y^3

( 1 −y)^3

where

y

πNd^3

6 V



πNd^3

6

Manipulate this equation of state into the virial equation of

state and find the expressions for the second and third

virial coefficients. Compare your formula for the second

virial coefficient with that of Exercise 9.22.

9.65 Evaluate the second virial coefficient of helium, using the

value of the hard-sphere diameter in Table A.15 of the

appendix. Compare your value with those in Table A.4 of

Appendix A.

9.66 For a mixture of 2.000 mol of CO and 1.000 mol of O 2 at

292 K and 1.000 atm, calculate:

a.the number of collisions with O 2 molecules suffered by

one CO molecule in 1.000 s, taking

d 2. 94 × 10 −^10 m for CO;

b.the number of collisions with CO molecules suffered by

one O 2 molecule in 1.000 s;

c.the length of time required for one CO molecule to

have as many collisions with O 2 molecules as there are

O 2 molecules in the system.

9.67A mixture of 1.000 mol of methane and 2.000 mol of

oxygen is held at a total pressure of 1.000 bar and a