Physical Chemistry Third Edition

422 9 Gas Kinetic Theory: The Molecular Theory of Dilute Gases at Equilibrium

PROBLEMS

Section 9.7: The Model System with Potential Energy

9.51 a.Construct an accurate graph of the Lennard-Jones 6–12

potential function for two argon atoms.

b.Find the value ofrfor whichuLJ(r)^32 kBTfor

273 K, 373 K, and 473 K. Compare your values with

the effective hard-sphere diameter of argon at these

temperatures from Table A.15.

9.52 Find the value of the Lennard-Jones representation of the

interatomic potential function of argon at interatomic

distances equal to each of the effective hard-sphere

diameters of argon at different temperatures in Table A.15

of Appendix A. Explain the temperature dependence of

your values.

9.53 a.Estimate the average distance between

nearest-neighbor atoms in gaseous argon at 1.000 atm

and 87.5 K, its normal boiling temperature. Do this by

calculating the side of a cube containing one argon

atom.

b. Calculate the value of the Lennard-Jones potential

function for two argon atoms at this separation.

c.Calculate the minimum value of the Lennard-Jones

potential function for two argon atoms. Calculate the

ratio of the value in part b to this value.

9.54Another approximate representation for

intermolecular pair potentials is theexponential-6

potential function

u(r)be−ar−cr−^6

wherea,b, andcare parameters to be chosen to fit data for

each substance. The function has a nonphysical maximum

and approaches negative infinity asrapproaches zero. For

values ofrsmaller than the value at the maximum, this

expression must be replaced by a different representation.

The usual procedure is to defineuto be positively infinite

in this region. Find the values ofa,b, andcin the

6-exponential representation of the interatomic potential

of argon that matches the Lennard-Jones representation

such thatc 4 εσ^6 and such that the minimum is at the

same value ofras the minimum of the Lennard-Jones

representation.

9.55Assume that air is 80% nitrogen and 20% oxygen, by

moles, at sea level. Calculate the percentages and the total

pressure at an altitude of 20 km, assuming a temperature

of− 20 ◦C at all altitudes. Calculate the percent error in the

total pressure introduced by assuming that air is a single

substance with molar mass 0.029 kg mol−^1.

9.56 Calculate the difference in the density of air at the top and

bottom of a vessel 1.00 m tall at 273.15 K at sea level.

State any assumptions.

9.57 Estimate the difference in barometric pressure between the

ground floor of a building and the 41st floor, assumed to

be 400 feet higher. State any assumptions.

9.58A helium-filled balloon is filled with helium at sea level

and at a temperature of 20◦C. The design of the balloon is

such that the pressure inside the balloon remains equal to

the external atmospheric pressure (the volume can

change).

a.If the volume of the balloon at sea level is 1000 m^3 find

the mass of helium required to fill the balloon and the

mass that can be lifted (including the mass of the

balloon).

b.Assuming that the atmosphere has a uniform

temperature, find the volume of the balloon at an

altitude of 10.0 km and find the mass that it can lift,

assuming that the same amount of helium is in the

balloon as in part a.

9.8 The Hard-Sphere Gas

The simplest representation of the pair potential function is thehard-sphere potential:

u(r)

{

∞ (0<r<d)

0(r>d)

(9.8-1)

which is depicted in Figure 9.17. The parameterdis the distance of closest approach

of the centers of the two spheres and is thus equal to the diameter of one molecule.