438 9 Gas Kinetic Theory: The Molecular Theory of Dilute Gases at Equilibriumthe total pressure 2.000 atm. Find the total number of
collisions of each kind per cubic meter per second. The
effective hard-sphere diameter of oxygen is 0.361 nm.9.79 Consider a spherical water droplet in a cloud at 25◦C. The
radius of the droplet is 10.0μm. The equilibrium vapor
pressure of water at this temperature is 23.756 torr.
a.Calculate the rate at which water molecules strike the
surface of the droplet, assuming that the air is saturated
with water vapor (the partial pressure equals the
equilibrium vapor pressure).
b.Assume that the air is supersaturated (the water vapor
is supercooled) with a water partial pressure of
30.0 torr. Find the rate at which water molecules strike
the surface of the droplet.c.Calculate the rate at which the mass of the droplet in
part b is growing. State any assumptions.9.80 The number of three-body collisions is far smaller than the
number of two-body collisions in a dilute gas. Consider
three-body collisions in a sample of pure argon at 1.000 bar
and 300 K. Assume that a three-body collision occurs when
a third body collides with a pair of molecules in the act of
colliding,
a.Estimate the number density of colliding pairs by
estimating the time during which two colliding
molecules are close enough together to be struck by a
third body. Take this time as the time for a molecule
moving at the mean relative speed to travel a distance
equal to the collision diameter.
b. Estimate the rate of three-body collisions by estimating
the rate of collisions between colliding pairs and third
bodies. Take an effective hard-sphere diameter of the
colliding pair to be twice that of a single molecule.9.81 Uranium hexafluoride has a vapor pressure at 56◦C equal
to 765 torr, so UF 6 is a gas at 60◦C and 1.000 atm. Various
diffusion and thermal diffusion processes were used in the
Manhattan Project of the United States in World War II to
separate gaseous^235 UF 6 molecules from^238 UF 6
molecules.
a.Find the mean speed of^235 UF 6 molecules at 60. 0 ◦C.
Round off atomic masses in amu to the nearest integer.b. Find the mean speed of^238 UF 6 molecules at 60. 0 ◦C.
Round off atomic masses in amu to the nearest integer.
c.The effective hard-sphere diameter of UF 6 molecules
is 570 pm. Find the mean free path of UF 6 molecules at
60 ◦C and 1.000 atm.d.Find the number of collisions that one UF 6 molecule
undergoes in 1.000 s at 60. 0 ◦C and 1.000 atm.
9.82 a.Calculate the most probable speed and the mean speed
for ozone (O 3 ) molecules at 298.15 K.
b.On a smoggy day in Los Angeles, ozone can reach
0.5 parts per million by moles (mole fraction
0. 5 × 10 −^6 ). Estimate the number of ozone molecules
that would strike your face during 8 hours of exposure
to this air. State any assumptions.
c.Estimate the number of collisions per second with
other ozone molecules undergone by an ozone
molecule at this concentration. Estimate the mean free
path between such collisions. Assume what you think
is a reasonable effective hard-sphere diameter for
ozone and state any other assumptions.
9.83 Label each of the following statements as either true or
false. If a statement is true only under special
circumstances, label it as false.
a.If a given sample of a pure gas is isothermally
expanded to twice its original volume, the total rate of
collisions in the entire sample drops to one-fourth of its
original value.
b.If a given sample of a pure gas is isothermally expanded
to twice its original volume, the rate of collisions per
unit volume drops to one-fourth of its original value.
c.The mean speed of water molecules at 100◦C has the
same value in the liquid as in the vapor.
d.The ratio of the most probable speed to the mean
speed has the same value for all gases at all
temperatures.
e.The ratio of the mean speed to the root-mean-square
speed has the same value for all gases at all
temperatures.
f.Ordinary gases behave nearly like ideal gases because
the molecules are far enough part on the average that
the intermolecular forces are small.
g.In a typical gas under ordinary conditions, the average
distance between neighboring molecules is roughly 10
times as great as the distance between neighboring
molecules in the liquid.
h.The mean free path in an ordinary gas is roughly equal
to the average distance between neighboring molecules.
i.Because the temperature on the Kelvin scale cannot be
negative, a state of higher energy cannot have a greater
population than a state of lower energy.