9.5 The Pressure of a Dilute Gas 415
Exercise 9.18
Find the number of oxygen molecules withvy444 m s−^1 that must strike an area of 1.000 m^2
in 1.000 s in order for a force of 101325 N to be exerted on the area.The Pressure of a Mixture of Gases
Now consider a model system that contains a mixture of several gaseous substances,
each one represented by point-mass particles. We denote the number of substances byc.
We continue to assume that the molecules have zero size and do not exert forces on each
other, so the molecules of each substance will move just as though the other substances
were not present. Because the pressure is the sum of the effects of individual molecules,
the total pressure is the sum of the pressures exerted by each set of molecules:PP 1 +P 2 +P 3 + ··· +Pc (9.5-14)whereP 1 is thepartial pressureof substance 1, defined to be the pressure that this
substance would exert if it were alone in the container, and similarly for the other
substances. Note that the partial pressure does not depend on the mass of the particles.
Equation (9.5-14) isDalton’s law of partial pressures. Because each gas obeys the
ideal gas law, the mixture of gases also obeys the ideal gas law,P
n 1 RT
V+
n 2 RT
V+ ··· +
ncRT
V(n 1 +n 2 + ··· +nc)RT
V
nRT
V(9.5-15)
wherenis the total amount of all gases:nn 1 +n 2 + ··· +nc (9.5-16)PROBLEMS
Section 9.5: The Pressure of a Dilute Gas
9.33 In the discussion of this section, it was assumed that the
system was contained in a rectangular box. Explain why
the pressure of a gas is independent of the shape of the
container.
9.34 For a helium atom moving withvy1255 m s−^1 , find
the average impulse (force multiplied by the time over
which the force is exerted) on a wall perpendicular to
theyaxis if the particle collides with the wall. Compare
your value with that of Example 9.12. Comment on the
fact that 444 m s−^1 is the mean speed of O 2 molecules at
298.15 K while 1255 m s−^1 is the mean speed of He
atoms at 298.15 K.
9.35 Give a verbal explanation of the fact that the pressure
of an ideal gas does not depend on the mass of the
molecules.
9.36 Show that in a specular wall collision on a flat wall, the
angle of incidence and the angle of reflection are equal.
The angle of incidence is the angle between the line
perpendicular to the surface and the initial velocity,
and the angle of reflection is the angle between the
line perpendicular to the surface and the final
velocity.
9.37 a.Estimate the total force of the atmosphere on the outer
surface of your body.
b.Estimate the mass (in pounds and in kilograms) such
that its gravitational force at the surface of the earth is
equal to the force in part a.
9.38 Derive the ideal gas law without using the velocity
distribution function.Hint:Start with Eqs. (9.3-28) and
(9.5-11).
9.39 Estimate the mass of the earth’s atmosphere. Calculate the
area of the earth. Make an estimate of the mean barometric
pressure and the mean temperature of the atmosphere.
Look up the mass of the earth and calculate the ratio of the
atmosphere’s mass to the earth’s mass.