422
9 Gas Kinetic Theory: The Molecular Theory of Dilute Gases at Equilibrium
P R O B L E M S
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 of r for which u LJ (r)
3
2 k B T for
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.
9.54 Another approximate representation for
intermolecular pair potentials is the exponential-6
potential function
u(r) be
−ar
− cr
−6
where a, b, and c are parameters to be chosen to fit data for
each substance. The function has a nonphysical maximum
and approaches negative infinity as r approaches zero. For
values of r smaller than the value at the maximum, this
expression must be replaced by a different representation.
The usual procedure is to define u to be positively infinite
in this region. Find the values of a, b, and c in the
6-exponential representation of the interatomic potential
of argon that matches the Lennard-Jones representation
such that c 4εσ 6 and such that the minimum is at the
same value of r as the minimum of the Lennard-Jones
representation.
9.55 Assume 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.
9.58 A 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 the hard-sphere potential:
u(r)
∞ (0 < r < d)
0
(r > d)
(9.8-1)
which is depicted in Figure 9.17. The parameter d is the distance of closest approach
of the centers of the two spheres and is thus equal to the diameter of one molecule.
9 Gas Kinetic Theory: The Molecular Theory of Dilute Gases at Equilibrium
P R O B L E M S
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 of r for which u LJ (r)
3
2 k B T for
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.
9.54 Another approximate representation for
intermolecular pair potentials is the exponential-6
potential function
u(r) be
−ar
− cr
−6
where a, b, and c are parameters to be chosen to fit data for
each substance. The function has a nonphysical maximum
and approaches negative infinity as r approaches zero. For
values of r smaller than the value at the maximum, this
expression must be replaced by a different representation.
The usual procedure is to define u to be positively infinite
in this region. Find the values of a, b, and c in the
6-exponential representation of the interatomic potential
of argon that matches the Lennard-Jones representation
such that c 4εσ 6 and such that the minimum is at the
same value of r as the minimum of the Lennard-Jones
representation.
9.55 Assume 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.
9.58 A 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 the hard-sphere potential:
u(r)
∞ (0 < r < d)
0
(r > d)
(9.8-1)
which is depicted in Figure 9.17. The parameter d is the distance of closest approach
of the centers of the two spheres and is thus equal to the diameter of one molecule.