Physical Chemistry Third Edition

9.8 The Hard-Sphere Gas 425

EXAMPLE9.17

Obtain a formula for the second virial coefficient of a hard-sphere gas.

Solution

For the hard-sphere gas with molecular diameterd,

u(r)∞ ifr<d

u(r)0ifr>d

e−u(r)/kBT− 1 −1ifr<d

e−u(r)/kBT− 1 0ifr>d

B 2 −

NAv

2

∫d

0

− 14 πr^2 dr

2 πNAvd^3

3

Exercise 9.22

Write Eq. (9.8-6) in the form

PVm

RT



1

1 −b/Vm

and use the identity

1

1 −x

 1 +x+x^2 +x^3 + ··· (|x|< 1 )

to transform Eq. (9.8-6) into the form of the virial equation of state. Show that the same formula

results for the second virial coefficient as in the previous example.

Exercise 9.23

a.For a hard-sphere gas with molecules of radius 1. 8 × 10 −^10 m find the pressure of the gas

at a temperature of 298.15 K and a molar volume of 0.0244 m^3 mol−^1.

b.Find the pressure of the gas using the ideal gas law under the same conditions. Express the

deviation of the pressure from the ideal pressure as a percentage of the ideal pressure.

The equation of state of a hard-sphere fluid has been the subject of considerable

research, and far better approximate equations than Eq. (9.8-4) have been obtained.^5

One such equation of state is found in Problem 9.64. Much additional research on gases

and liquids has used the technique ofmolecular dynamics, in which solutions to the

classical equations of motion for a system of several hundred particles are numerically

simulated by a computer program. Energies, pressures, etc., are then calculated by aver-

aging over the particles’ positions and velocities. As we would expect, the molecular

dynamics calculations indicate that there is no gas–liquid condensation in the hard-

sphere system. However, there is considerable evidence from these calculations that

a gas–solid phase transition occurs.^6 This result was originally somewhat surprising

because of the absence of attractive forces.

(^5) R. Hoste and W. Van Dael,J. Chem. Soc. Faraday Trans., 2, 80 , 477 (1984).
(^6) H. Reiss and A. D. Hammerich,J. Phys. Chem., 90 , 6252 (1986).