BW#5 Ch. 12: CQ2, 4-7, 9a, 10 P. 7, 11, 12.
Q12.2. Reason: Hydrogen and helium can escape the atmosphere while heavier elements such as oxygen and nitrogen
cannot. This is because hydrogen and helium, being lighter, move faster. From Chapter 11, we know that the root-meansquare speed of a particle is given by the following:
vrms 
3kBT
m
It follows that the lighter a particle is, the greater is its rms speed.
Assess: We can be thankful that our Earth’s gravity holds on to the vital nutrients oxygen and nitrogen. A smaller body
than the earth might not be able to hold on to these elements. For example, the moon’s gravity is too weak for it to have an
atmosphere of oxygen or nitrogen.
Q12.4. Reason: From the ideal gas law, the lower the pressure of a container of gas, the higher the volume. The
pressure inside the bag equals the pressure outside. So when the atmospheric pressure goes down, the pressure in the bag
goes down and the volume goes up.
Q12.5. Reason: From Equation 12.7, we know that the temperature of a gas is directly proportional to the square of
the rms speed of the molecules. Thus, doubling the typical speed of molecules in a gas increases the temperature by a
factor of four. We also know, from Equation 12.11, that the pressure of a gas is directly proportional to its temperature, so
doubling the typical speed of the molecules must also increase the pressure by a factor of four. Increasing the speed of
molecules in a gas increases the amount of force a molecule exerts on the wall of the container and the rate of collisions
with the walls.
Assess: It makes sense that the pressure depends on the square of the average velocity of molecules in a gas since the
force and collision rate are proportional to the velocity of a molecule.
Q12.6. Reason: Note that N /V and vrms are the same for both gases.
(a) In the process of deriving the ideal gas law we saw that
1N
2
mvrms
3V
or that p  m, so given the conditions above, the gas with the more massive molecules (gas 2) will have the higher pressure.
p2 > p1
p
(b) The ideal gas law can be rearranged as
N
kBT
V
which shows that p  T , so given the answer to part (a) the temperature of gas 2 must be greater than the temperature of gas
1.
T2 > T1
p=
Assess: We conclude that, other things being equal, the gas with the more massive molecules will have a greater
pressure and a greater temperature.
Q12.7. Reason: (a) As defined in the chapter, a mole is 6.02 1023 basic particles, regardless of which chemical
element we have. So there are equal numbers of particles in a mole of helium gas and a mole of
oxygen gas.
(b) A mole of helium gas has a mass of 4 g, while a mole of oxygen gas has a mass of 32 g, so one gram of helium gas has
1
 6.02 1023 particles, while one gram of oxygen gas has 321  6.02 1023 particles. Therefore, the gram of helium gas has
4
more particles than the gram of oxygen gas.
Assess: We note that the basic particles for the helium gas are helium atoms while the basic particles for the oxygen gas
are diatomic oxygen molecules. That is why the O2 molecule has a mass of 32 u.
Q12.9. Reason: (a) The thermal energy of the gas does change. In Section 11.4 we are given
Eth 
1
2
Nmvrms
2
so if the speed of every molecule is doubled then the thermal energy is increased by a factor of 22  4.
Q12.10. Reason: Since the helium atoms are five times lighter than the neon atoms, they are traveling, on average,
5 times faster. For the sake of argument, we will compare a sample neon atom to a sample helium atom which is
traveling faster by exactly 5 and in the same direction. You might think the helium atom would exert greater pressure
since it is faster. However, since momentum depends on velocity and mass, its momentum is actually less by a factor of
5. This means that every time the helium atom collides with the walls of the container, it imparts a force which is less
than that exerted by the neon atom by
5. Finally, since the helium atoms travel faster, they collide more often, again by
5. The combined effect of a force which is weaker by 5 with collisions which occur 5 times more often is that the
time-averaged force is the same for both atoms. Consequently, the pressures exerted by the two different gases are the
same.
Assess: It is interesting to see how a number of effects have canceled so that the two gases have the same pressure.
Similarly, helium and neon atoms have the same average kinetic energy even though the helium atoms are faster because
they are correspondingly lighter.
P12.7. Prepare: The absolute pressure is the gauge pressure plus one atmosphere at sea level. 1atm  14.7psi.
Solve:
p  pg  1atm  35.0 psi  14.7 psi  49.7 psi
Assess: The difference between p and pg is due to the fact that your tire gauge measures pressure differences.
P12.11. Prepare: In order to use the ideal gas law (Equation 12.11) we need to know the number of helium atoms in the
gas.
N  nNA  (7.5 mol)(6.02 1023 mol1)  4.52 1024
 1m3 
3
V  15L 
  0.015m
1000L


As a further preliminary calculation add 1 atm to the gauge pressure to give the absolute pressure and convert the pressure to SI
units.
 1 atm  101.3 kPa 
p  pg  1 atm  65 psi  14.7 psi  79.7 psi 

  549 kPa
 14.7 psi  1 atm 
549Kpa = 5.49 x 105 Pa Equation below needs this in it since SI unit for pressure is the Pascal
Solve: (a) Solve Equation 12.11 for T.
T
pV
(549 kPa)(0.015 m3 )

 132 K  105 C
NkB (4.52 1024 )(1.38 1023 J/K)
(b) Now use Equation 12.5 for K ave.
Kave 
3
3
kBT  (1.38 1023 J/K)(132 K)  2.7 1021 J
2
2
Assess: The answer to part (a) is a cold temperature, but it needs to be to get that much gas in that volume.
P12.12.
Prepare: We need the mass of a mole of CO2. Since carbon has an atomic mass of 12 and oxygen has an atomic
mass of 16, the molecular mass of CO2 is 44. Hence a mole of CO2 has a mass of 44 g or 0.044 kg.
Solve: We use Equation 12.10 for rms speed but modify it by multiplying numerator and denominator by Avogadro’s
number.
vrms  3kT / m  3RT / (mmol )  3(8.315 J/(mol  K))(210 K)/(44 10 3 kg/mol)  350 m/s
Assess: This is a typical speed for a gas molecule with a temperature in the hundreds of Kelvins. For example, the rms
speed of an O2 molecule at room temperature is about 480 m/s.