Physical Chemistry Third Edition

420 9 Gas Kinetic Theory: The Molecular Theory of Dilute Gases at Equilibrium

Exercise 9.20

The altitude of Pike’s Peak is 14,110 ft. Estimate the barometric pressure on Pike’s Peak on a

winter day when the temperature is− 15 ◦F.

Intermolecular Forces

We now modify our model to include two-body intermolecular forces. That is, the force

on particle 1 due to particle 2 is unaffected by the positions of particle 3, particle 4,

and so on. This is a good approximation for gases, and a fair approximation for liq-

uids.^3 We assume also that the intermolecular forces are independent of the molecules’

velocities, so that they can be derived from a potential energy. With our assumption

the intermolecular potential energyVis a sum of two-body contributions.

V

N∑− 1

i 1

∑N

ji+ 1

u

(

rij

)

(9.7-4)

The functionu(rij) is called thepair potential energy functionof particlesiandj.We

consider only monatomic gases, for whichudepends only onrij, the distance between

the centers of particlesiandj. The limits on the double sum in Eq. (9.7-4) are chosen

so that the contribution of a single pair of particles is counted only once.

The Lennard-Jones potential is named
for J. E. Lennard-Jones, 1894–1954, a
prominent British theoretical chemist.

A common approximate representation for the pair potential function is the

Lennard-Jones 6–12 potential function:

uLJ(r) 4 ε

[(

σ

r

) 12

−

(σ

r

) 6 ]

(9.7-5)

The parameterεis equal to the depth of the minimum in the curve and the parameter

σis equal to the intermolecular separation at which the potential energy is equal to

zero. The designation 6–12 denotes the choice of the exponents in the formula. This

function is sometimes referred to simply as the Lennard-Jones potential function and

is depicted in Figure 9.15. Table A.14 of Appendix A gives values ofσandεfor a

few substances. The minimum in the function occurs atr 21 /^6 σ.Ifris greater than

this value, there is an attraction, and ifris smaller than this value, there is a repulsion.

More accurate potential energy functions have been obtained, but the Lennard-Jones

potential function remains widely used.^4

0

–2

u 2

/10

21

J 4

6

8

1234567

r 31010 /m

Minimum at r= 21/6s= 3.82 310 –^10 m

u = –1.66 310 –^21 J

r= = 3.40 310 –^10 m

Figure 9.15 The Lennard-Jones Rep-
resentation of the Intermolecular Poten-
tial of a Pair of Argon Atoms.

EXAMPLE9.15

a.Show that for the Lennard-Jones potential,

uLJ(σ) 0 (9.7-6)

b.Show that the value ofrat the minimum in the Lennard-Jones potential is

rmin 21 /^6 σ(1.12246)σ (9.7-7)

(^3) D. R. Williams and L. J. Schaad,J. Chem. Phys., 47 , 4916 (1967). See C. A. Parish and C. E. Dykstra,
J. Chem. Phys., 101 , 7618 (1994) for a three-body potential for helium atoms.
(^4) See for example D. E. Moon,J. Chem. Phys., 100 , 2838 (1994).