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This is a Sequential Activity
Group Activity: Gas Laws,
Problem Set 1
Introduction
In this activity you will be introduced to the concepts
of the Kinetic Molecular Theory of Gases. So far you know
a little about how gases act when subjected to changes in
temperature, pressure and volume. Now you will have a
chance to learn a little about why they act that
way. You will also get a more detailed picture of just
what pressure and temperature really are.
Also, you will learn about Dalton’s Law of Partial
Pressures. This law says that the pressure of a mixture
of gases is simply the sum of the pressures that each gas
would exert if it were alone in the container. For
instance, in Earth’s atmosphere the major
components are oxygen (O2 ) and nitrogen
(N2 ). Ignoring
the other minor components for a moment we can say that
O2 (22% of the
atmosphere) has a partial pressure of 0.22 atm and
N2 has a partial
pressure of 0.78 atm. Adding these together you can see
that they equal the total pressure of 1 atm.
used in the lab
a column of mercury in a glass tube makes a simple
barometer; 1 mm of that column is 1 mmHg or 1 torr
kiloPascals
kPa
101.325 kPa
= 1 atm
the SI unit of P
You may be wondering why you are looking at still more
units of pressure. Well, there happen to be a lot of
units for pressure (bar, lb/in2,
lb/ft2, ton/ft2, millimeters of
mercury, etc.) It isn’t helpful to know them all
but these two are important because of what they can tell
us about the nature of pressure.
First, a word about kPa: The study of physics led to the
development of units of force. The unit of force is
called the newton (N), named after Isaac Newton, a
scientist of some note. Pressure is defined as force
divided by the area over which the force acts: P =
(force)/(area). Using the standard unit of force, the
units of pressure are newtons per square meter
(N/m2). These units are commonly called
pascals (Pa) after Blaise Pascal, a French mathematician
and scientist. That is the origin of the kilopascal
(kPa), which is equal to 1,000 pascals. Your text uses
the kPa unit for pressure.
Pascals point to a concept of pressure that has some real
value. What is causing the force on an area when we talk
about gas pressure? That force is due to the collisions
between the molecules and atoms of a gas and the sides of
their container. As the gas particles strike the sides of
the container they exert a force on it depending on how
fast they are moving and how often they hit it. The more
times they hit it, in a given amount of time, and the
harder they hit, the higher the pressure.
Second, the ‘new’ unit introduced above is
not really all that new. The mmHg stands for millimeters
of mercury (Hg is the
chemical symbol for mercury). That is in fact the
definition of the torr: 1 torr equals 1 mmHg. This unit
was named for the first person (supposedly) to realize
that the atmosphere exerts a pressure on everything on
Earth: Evangelista Torricelli. More important than that,
though, is the fact that he invented the mercury
barometer (at right). The barometer is simple but
dangerous because mercury is chemically very reactive and
is in fact a very strong poison. It works like this: you
fill a long glass tube with mercury and cover the open
end. Place the open end in a pool of mercury and let the
mercury in the tube flow into the pool. On an average day
you will find that the height of the column of mercury is
about 760 mm (76 cm or about 29.92 in.) A vacuum forms at
the top of the closed column of mercury because the
mercury is being pulled by gravity out of the tube.
What keeps any mercury in the tube? Atmospheric pressure. The atmosphere
pushes down on the pool of mercury with a force per unit area of 101.3 N/m2,
or 14.7 pounds per square inch. This is exactly enough pressure to hold up
a 76-cm tall column of Hg . Consider the power of atmospheric pressure as seen
in the can-crushing demonstration and the
upside-down glass of water.
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The Kinetic Theory of Gases
The kinetic theory of gases is based on the following
assumptions:
Gases are made of particles that are a lot smaller
than the distances between the particles: about 40 times
smaller under usual conditions. They can be treated as
mathematical “points” with mass but no
volume.
Gas molecules are always moving around, colliding
with one another constantly. Collisions between molecules
are perfectly elastic: that is, energy from one molecule
can be transferred to another through collisions. The
total energy of a collection of gas particles remains the
same no matter how much the particles exchange energy
with each other.
Gas particles do not have any power to attract other
particles to themselves or to repel other particles.
The average kinetic energy of a collection of gas
particles is directly proportional to the temperature of
the gas in kelvins.
Pressure is a result of the collision of gas
particles with the walls of their container.
The kinetic energy of any object is directly proportional
to the mass and velocity of the object (in case you are
interested, KE = ½mv2). So the faster
something moves and the more massive it is, the more
kinetic energy it has. (Kinetic comes from the
Greek for movement.) Gas particles are not very massive
but they move incredibly fast at normal temperatures
(~1,100 mph). The motion
of gas particles can be measured with a thermometer because
temperature is proportional to the average kinetic energy of a collection
of molecules. So the faster the molecules are flying around,
the higher their temperature. Naturally, the faster they fly, the harder they
hit things they collide with. It is these collisions which gives rise to
pressure as the particles of a gas collide with each
other and with other objects. The constant pushing from
trillions and trillions of gas particles against objects
is what we call gas pressure.
I know after all this reading you are anxious to get on
to some problems. So have at it:
A
sample of gas has an average kinetic energy of 3.4
kilojoules (kJ, a unit of energy). Another sample has an average kinetic energy
of 4.5 kJ. Which has a higher temperature?
If both of the samples of gas
from the previous problem were in a container of the same
volume, which do you think would have a higher pressure?
Why? (Think in terms of moving particles when you answer
the second question in this problem).
Explain in your own words how
molecular motion gives rise to pressure.
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Why is it that the pressure of a
gas in a constant-volume container goes up when the
temperature is raised? What is happening at the molecular
level?
Put the idea of higher
temperature in terms of what you now know about
gases. What does a higher temperature mean from the point
of view of the particles of a gas?
Does adding more gas to a
container (say by inflating a balloon) have an effect on
the temperature? What effect does it have on T?
How about on the pressure? For
this question imagine not a balloon (which changes volume
as you add gas) but a steel gas bottle. What effect does
it have on P if V is constant?
Explain your answers to the
previous question in terms of the Kinetic Molecular
Theory of gases.
In the picture at right in which
the volume is reduced and the pressure increases at
constant temperature (as according to Boyle’s Law)
what is causing the increase in pressure?
The previous question mentions
constant temperature. If the temperature is constant,
then the average kinetic energy must be constant. Why is
it wrong to say that the molecules hit harder when
the pressure increases at constant temperature? What does
it take to make them hit harder?
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Your teacher has shown you how atmospheric pressure
can be used to crush a soda can.
Describe the procedure used to make this happen. Explain what happens in
terms of what you know about the gas laws. Also (and finally) explain what
happens in terms of the Kinetic Molecular Theory of Gases. (This question
should have three parts in the answer!)
Dalton’s Law of Partial Pressures
When two or more gases mix, and do not chemically react,
then the total pressure of the mixture equals the sum of
the pressure of each component if it were present alone.
In your imagination fill a steel vessel with 1 mol of
He. Say the total pressure
in the vessel is 10 atm. Add another mole of He but keep the volume and temperature
constant. According to PV=nRT the total pressure should
now be 20 atm because twice as many moles of gas are in
the vessel. Now, how would the pressure have changed if
instead of He, you added
O2? Just as when
you added more He, the
pressure doubles.
Mathematically, the idea behind this law can be stated
this way:
Ptotal = P1 + P2 +
P3 + … + Pn
This idea has an application that you will use in a lab later in this course.
When a gas is collected over water the gas that is being collected accounts
for most of the pressure. The rest is due to water vapor. Water molecules
are always evaporating and condensing at the surface of water. When the water
is in a closed container the pressure
of water vapor depends only on the temperature. Here is an excerpt from
a data table showing the vapor pressure of water at various temperatures:
The pressures listed are the partial pressures of
water that you would have at the temperature shown in a
closed container. Say you were to generate O2 according to the following
chemical equation: 2KClO3 --> 2KCl + 3O2. This equation describes the
decomposition of potassium chlorate into oxygen and
potassium chloride. If you set up some lab apparatus as
shown in the illustration then you can collect the
O2 to use for
something else. When you collect the gas over water, the
pressure inside a 1-L capacity jar is equal to the
external atmospheric pressure.
That is, the total pressure. So if atmospheric
pressure is 101.3 kPa today then that is also the total
pressure inside the jar. But what is the pressure of the
oxygen? Use Dalton’s Law of Partial Pressures:
PT = P1 + P2 or
PT – P2 = P1. If
the temperature of the system is 25°C then
P2, the partial pressure of H2O is 3.169 kPa. So the
pressure of O2
alone is 98.16 kPa (which is 0.967 atm). And why did we
want to know? We wanted to know how muchO2 was generated
in the chemical reaction. So how can you find that out?
Use PV=nRT. For the volume use the total amount of oxygen
generated, say it was 5.5 L.
PV
n = ——
RT
(0.967 atm)(5.5 L)
substituting n = ———————————————————————— = 0.217 mol
values: (0.0821 L atm/K mol)(298 K)
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Dalton’s Law of Partial Pressures, continued
And what is a mole? A mole is a unit of chemical
measurement for amount of substance. It is directly
proportional to the number of atoms or molecules. For now,
we will just be using the unit without any discussion of
what it means: we will leave that for our discussion of the
periodic table.
The mole is useful in the discussion of partial pressures
because of the idea of a mole fraction. Say a container has
equal amounts of two gasses, say He and Ar.
The amount of He in the
container is 50% of the total by volume. Its mole
fraction is 0.5. Similarly, the mole fraction of
Ar is 0.5. Mole fraction can
be calculated using equation 1.
naXa = ——————————————
na + nb + … nn
equation 1
In eq. 1 Xa is the mole fraction of
gas ‘a’ and nx is the number of moles
of gas ‘x’. The sum in the denominator of this
fraction is the total number of moles of gas. Say you know
that a mixture of gases is 20% N2. Its mole fraction is then
0.20 and if you have the total number of moles of gas then
you can find the number of moles of N2.
Pa = XaPT or Pb = XbPT
equation 2
Also, if you know the mole fraction of a gas that is
part of a mixture and you know the total pressure, you can
find the partial pressure of that gas using equation 2. In
equation 2, Pa and Pb are the partial
pressures of gases a and b, X is the mole fraction,
and PT is the total pressure.
Now for some problems:
A container holds three gases:
oxygen, carbon dioxide, and helium. The partial pressures
of the three gases are 2.00 atm, 3.00 atm, and 4.00 atm,
respectively. What is the total pressure inside the
container?
What is the mole fraction of
each of the gases in the previous problem?
Check the answer to the previous
problem by adding up all the mole fractions. What should
the sum of the mole fractions be?
Thirty-five liters of hydrogen
gas (H2) are
collected over water at 99 kPa and 24°C. What is the
partial pressure of H2?
How many moles of H2 were collected?
What was the mole fraction of
H2 in the
previous two problems? Express the number as a decimal
and as a percent.
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A container with two gases,
helium and argon, is 30.0% by volume helium. What is the
mole fraction of He? And of
Ar?
Calculate the partial pressure
of helium and argon if the total pressure inside the
container is 4.00 atm.
If 60.0 L of nitrogen is collected over water at 25.0°C when the atmospheric pressure is 760.0 mmHg, what is the partial pressure of the nitrogen? How many moles is that?
General Gas Law Problems
The universal gas constant
R = 0.0821 L· atm/K· mol
A gas has a volume of 800.0 mL at -23.00°C and 300.0 torr. What would the volume of the
gas be at 227.0°C and 600.0 torr of pressure?
If a gas is cooled from 323.0 K to 273.15 K and the volume is kept constant what final
pressure would result if the original pressure was 750.0 mmHg?
The temperature of a 4.00 L sample of gas is changed from 10.0°C to 20.0°C. What will the
volume of this gas be at the new temperature if the pressure is held constant?
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What volume will 1.27 moles of helium gas occupy at STP? (STP means Standard Temperature and Pressure which is 1 atm and 0°C).
V
P
T
V at STP
20.3 L
98.3 kPa
42.0°C
21.2 L
97.8 kPa
40.4°C
19.7 L
100.2 kPa
39.5°C
20.8 L
99.0 kPa
40.1°C
The table shows volumes of O2 that were collected at the temperatures and pressures shown. In order to compare experimental results it is necessary to know the volume collected at STP. Please find the volume of each sample at STP.
45.0 mL of water-vapor-saturated argon gas is collected at 729.3 mmHg and 25.0°C. What volume of dry gas does this represent at STP?
At what pressure would 0.150 moles of N2 at 23.0°C occupy 8.90 L?
In order to keep the lungs from collapsing a scuba diver needs to have air delivered at a pressure that corresponds to the pressure due to the depth under water. No matter what the total pressure, oxygen must have a partial pressure of 0.22 atm in order to avoid dangerously low or high levels. Say the air pressure delivered by the scuba diver’s regulator needs to be 350 kPa. In the pressurized air delivered to the diver at the given pressure, what is the mole fraction of oxygen necessary to maintain the proper partial pressure? What percentage of the total air delivered is oxygen? How does that compare to normal atmospheric oxygen levels of 22%?
