434 9 Gas Kinetic Theory: The Molecular Theory of Dilute Gases at Equilibriumtemperature of 298.15 K. The effective hard-sphere
diameter of methane molecules is 414 pm and that of
oxygen molecules is 361 pm. Assume that no reaction has
taken place.
a.Calculate the mean number of collisions per second that
a methane molecule undergoes with oxygen molecules.
b.Calculate the mean number of collisions per second that
an oxygen molecule undergoes with methane
molecules. Explain why this quantity is not equal to
your result for part a.
c.Calculate the total number of methane–oxygen
collisions per second in the system.9.68 Assume that in a certain automobile the exhaust gas has a
carbon monoxide mole fraction of 0.0140 and a
temperature of 500 K when it reaches the catalytic
converter.
a.Find the mean free path between CO–CO collisions of
one CO molecule in the exhaust gas. (Collisions with
other types of molecules can intervene.)
b.Findz1(1), the mean collision rate of CO–CO collisions
for one CO molecule in the exhaust gas of this
automobile.
9.69 Assume that the mole fraction of carbon dioxide in the
earth’s atmosphere is 0.000306.
a.Estimate the mean free path between CO 2 −CO 2
collisions in the atmosphere at sea level if the
temperature is 298 K (collisions with molecules of
other substances can intervene, so that the free path in
question can have bends in it).
b. Estimate the number of collisions with other CO 2
molecules per second undergone by a CO 2 molecule
under the conditions in part a.
9.70Assume that a certain region of interstellar space contains
1 hydrogen atom (not molecule) per cubic meter at a
temperature of 5 K. Estimate the mean time between
collisions for a hydrogen atom. You will have to estimate
a value for the atomic diameter.9.71A stoichiometric mixture for the combustion of carbon
monoxide with oxygen is prepared at 25◦C and a total
pressure of 1.000 atm.a.Calculate the total number of oxygen–carbon monoxide
collisions per cubic meter per second, assuming that no
reaction takes place.b. If every such collision led to reaction, how long would
it take for half of the material to react? How do you
think this compares with the actual half-life of the
reaction at this temperature?9.72 a.Find the mean speed of atoms in gaseous argon at 85 K,
the normal boiling temperature of argon.b. Find the rate of collisions of one argon atom with other
argon atoms in pure gaseous argon at 85 K and
1.00 atm. The effective hard-sphere diameter of argon
atoms is 3. 16 × 10 −^10 m.c.Find the mean free path of argon under the conditions
of part b.9.73 Estimate the mean free path for hydrogen atoms under the
conditions of Problem 9.709.9 The Molecular Structure of Liquids
A great deal of research has been carried out on the properties of liquids, but a single
comprehensive theory of the liquid state does not exist. The theories that do exist are
much more complicated than gas kinetic theory, and we present only a few elementary
comments. We note a simple but important fact:The structure and properties of a
liquid are primarily determined by its potential energy, whereas the properties of a
gas are primarily determined by its kinetic energy.For example, consider liquid argon.
The Lennard-Jones pair potential function for argon was shown in Figure 9.15. The
minimum is at 380 pm (3.8 Å). The potential energy of a pair of atoms at this distance
must be increased either to separate them or to push them closer together.
Substances such as argon consist of spherical atoms. In a solid lattice of spheres, 12
nearest-neighbor spheres can surround a given sphere. Each of these nearest neighbors
touches the central sphere and touches four of the other nearest neighbors. Solid argon