19.2 Postulates and Pressure
19.1.Define “postulate” in the scientific sense. You may have
to consult a good dictionary that can define scientific terms
appropriately. Why don’t we try to prove the postulates of the
kinetic theory of gases?
19.2.What is the kinetic energy of a single atom of mercury
that has a speed of 200 m/s? (This is a good speed for Hg
atoms at room temperature.) What is the kinetic energy of a
mole of Hg atoms having that speed?
19.3.Show geometrically that the following vector relation-
ship v^2 vx^2 vy^2 vz^2 is correct and, by analogy, v^2 avg
v^2 avg,xv^2 avg,yv^2 avg,z.
19.4.At a temperature of 273.15 K and pressure of 1 atm or
1.01325 bar, many gases have an approximate volume of
22.4 L. (This is a very useful approximation.) What are the av-
erage speeds of (a)He atoms and (b)Kr atoms under these
conditions? Compare your answer for part a with the answer
in Example 19.1.
19.5.Use equation 19.8 and the classical definition of kinetic
energy to argue that the average kinetic energy of a gas is the
same for all gases at the same absolute temperature.
19.6.Interstellar space can be considered as having 10 mol-
ecules of hydrogen per cubic centimeter and an average tem-
perature (far away from stars!) of 2.7 K. Determine (a)the
pressure of hydrogen in interstellar space and (b)the average
speed of the hydrogen molecules. Compare these answers
with values under normal Earth conditions.
19.3 Velocities
19.7.Compare the temperatures required to have an rms-
average speed of 200, 400, 600, 800, and 1000 m/s for Cs
atoms. Note that the average speeds form a pattern. What is
the pattern of the calculated temperatures?
19.8.If relativistic effects were ignored, what temperature is
required for hydrogen atoms to have an rms-average speed of
3.00 108 m/s? What do you think is the potential for actu-
ally achieving this temperature?
19.9.Verify equation 19.25. You will need to consult the table
of integrals in Appendix 1, and use the idea that K(K).
19.10.Distinguish between the definitions of g, G, and as
the three probability functions defined in the derivation of the
Maxwell-Boltzmann distribution.
19.11.Show that the constant K, as defined by 1/vx
gx (vx)/gx(vx) and 1/v (v)/ (v), has the same value for
1/vygy (vy)/gy(vy) and 1/vzgz (vz)/gz(vz).
19.12.Derive equation 19.34.
19.13.What is the ratio of vrms/vmost probfor any gas at a
given temperature?
19.14.Use the Maxwell-Boltzmann distribution function to
numerically estimate (that is, do not evaluate the integral) the
percentage of O 2 molecules at 300 K moving (a)between 10
and 20 m/s; (b)between 100 and 110 m/s; (c)between
1000 and 1010 m/s; (d)between 5000 and 5010 m/s; and
(e)between 10,000 and 10,010 m/s. Each interval has the
same absolute value. What do your answers tell you about the
distribution of velocities among the gas molecules?
19.15.Current research that focuses on low temperatures
uses crossed laser beams to slow down gas atoms (the phrase
“optical molasses” is a good analogy) so much that their “tem-
perature” is close to absolute zero.
(a)If atoms of Rb are moving at 1 cm/s, what is the approx-
imate “temperature” of the Rb gas? (You can use any defini-
tion of “average temperature” for this estimate.)
(b)How relevant is the word “temperature” to systems like
the one described? Develop arguments for and against the
applicability of the term to gas atoms trapped in optical
molasses.
19.16.Use the form of G(v) to find v^2 , then take the square
root of the answer you get and show that you get the defini-
tion of vrms, the root-mean-square speed.
19.17.Compare relative values of vrms, vmost prob, and v. Will
they always have the same relative values, or can variations in
conditions like temperature or molar mass change their rela-
tive magnitudes?19.4 Collisions
19.18.Vacuum systems use some gauges that measure pres-
sures in millitorr (where 760 torr 1 atm). Express the answer
from exercise 19.6 in units of millitorr.
19.19.Derive equation 19.41.
19.20.Use the conditions of exercise 19.6 to determine the
mean free path between hydrogen molecules in interstellar
space if d1.10 Å for hydrogen.
19.21.Explain why the molecular diameter for argon, at
2.6 Å, is about the same as that for molecular hydrogen, at
2.4 Å, even though hydrogen is a much smaller atom than
argon.
19.22.Tanks of nitrogen gas are often pressurized to 2400 psi
(pounds per square inch) at room temperature. What is the
mean free path of a nitrogen molecule, d3.20 Å, under
these conditions? There are 14.7 psi in 1.00 atm.
19.23.For a given sample of gas (which has a certain molar
mass, collision diameter, and so on), what variable(s) does the
average collision frequency depend on?678 Exercises for Chapter 19
EXERCISES FOR CHAPTER 19