9.5 The Pressure of a Dilute Gas 411
a.Explain why you think this is true.
b.How do you think that the speed of sound in air would
depend on the temperature?
c.How do you think that the speed of sound in helium gas
would compare with its speed in air? Why do you
think a person’s voice sounds different after a breath of
helium?9.26 Find the fraction of molecules in a gas that has speeds
greater than 80.0% of the mean speed.
9.27 It is shown in the theory of hydrodynamics that the speed
of sound,vs, is given by
v^2 sVmCP
MκTCV
whereVmis the molar volume,Mis the molar mass,κTis
the isothermal compressibility,CPis the heat capacity at
constant pressure, andCVis the heat capacity at constant
volume.
a.Find the speed of sound in N 2 gas at 298.15 K and
1.000 atm. Look up the value ofCP,min the appendix.
Assume thatVmandκTcan be approximated by
assuming that N 2 is an ideal gas. Look up the speed of
sound in air and compare it with your answer.
b.Assume thatCP,mof a gas can be approximated by^72 R.
Find a formula for the speed of sound in a gas as a
function of temperature and find the speed of sound in
N 2 gas at 298.15 K and 1.000 atm.
c.Find the speed of sound in helium at 2981.15 K and
1.000 atm.
d.Find the ratio of the speed of sound to the mean speed
of N 2 molecules at 298.15 K.
e.Find the fraction of N 2 molecules at 298.15 K that have
speeds greater than the speed of sound.9.28 Find the value of each of the ratios:
a.The mean speed of helium atoms at 298.15 K divided
by the mean speed of oxygen molecules at 298.15 K.
b.The root-mean-square speed of helium atoms at
373.15 K divided by the mean speed of helium atoms at
373.15 K.
c.The mean speed of nitrogen molecules at 400 K
divided by the mean speed of nitrogen molecules at
200 K.
d.The mean speed of nitrogen molecules at 400 K divided
by the most probable speed of nitrogen molecules at
400 K.
9.29 a.The standard deviation of an arbitrary probability
distribution for the variableuis defined by Eq. (9.3-34).
Find the expression for the standard deviation for the
distribution of speeds of gas molecules.
b.Find the value of this standard deviation for oxygen
molecules at 298 K.
9.30 Find the ratio of the mean speeds of O 2 and O 3 molecules:
a. at 298 K
b. at 1000 K
9.31 If oxygen molecules had mass 32 g instead of 32 amu,
what would their mean speed be at 298 K?
9.32 a.Find a general expression for the median speed of
molecules in a dilute gas.Hint:See the identity in
Problem 9.20. Your expressions will contain a constant
that you will have to evaluate numerically.
b.Find the median speed for oxygen molecules at
298 K.9.5 The Pressure of a Dilute Gas
We now want to show that our model gas of point-mass particles obeys the ideal gas
law. We assume that the box confining our model system is rectangular with walls that
are perpendicular to the coordinate axes and that the walls are smooth, slick, flat, and
impenetrable. A collision of a molecule with such a wall is called aspecular collision,
which means: (1) It is elastic. That is, the kinetic energy of the molecule is the same
before and after the collision. (2) The only force exerted on the particle is perpendicular
to the wall.