Physical Chemistry Third Edition

9.8 The Hard-Sphere Gas 423

Table A.15 in the appendix gives the values of the effective hard-sphere diameters of

several substances.

u

r = d r

0

`

Figure 9.17 The Hard-Sphere
Representation of the Intermo-
lecular Potential of a Pair of Atoms.

Consider a model system consisting ofNidentical hard spheres in a container of

volumeV. We now obtain an approximate equation of state for this model system. We

first pretend that all of the particles in the system are stationary except particle number

1. This moving particle has access to a volume that is smaller thanVbecause of the
   presence of the other particles. Figure 9.18 shows the volume due to particle 2 into
   which the center of particle 1 cannot penetrate. The radius of thisexcluded volume
   is the sum of the radii of the two molecules, equal to the diameter of one molecule.
   Its volume is 8 times as large as the volume of one particle. If the gas containsN
   particles the total volume from which particle 1 is excluded isN−1 times the volume
   in Figure 9.18:

Vexc(N−1)

(

4

3

πd^3

)

≈N

4

3

πd^3 (9.8-2a)

where we neglect unity compared withN.

If all of the particles are moving, the excluded volume in Figure 9.18 represents not

only the volume from which particle 1 is excluded because of particle 2, but also the

volume from which particle 2 is excluded because of particle 1. We assign half of this

volume to each particle and write for the net excluded volume for a single particle

Vexc

N

2

(

4

3

πd^3

)

N

2

3

πd^3 (9.8-2b)

We take as our approximate equation of state that of an ideal gas with a volume

equal to the volume in which each particle can actually move,

P(V−Vexc)NkBTnRT (9.8-3)

wherekBis Boltzmann’s constant,nis the amount of the gas measured in moles, and

Ris the ideal gas constant. We divide Eq. (9.8-3) bynand write

P(Vm−b)RT (9.8-4)

Excluded volume

for particle 1

d

r

r

1

2

Figure 9.18 The Excluded Volume of a Pair of Hard Spheres.