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## 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.

### Some New Units

 Pressure Unit Symbol Convert Example torr = mmHg torr or mmHg 1/760 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:

1. 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.
2. 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.
3. Gas particles do not have any power to attract other particles to themselves or to repel other particles.
4. The average kinetic energy of a collection of gas particles is directly proportional to the temperature of the gas in kelvins.
5. 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:

1. 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?
2. 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).
3. Explain in your own words how molecular motion gives rise to pressure.

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1. 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?
2. 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?
1. 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?
2. 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?
3. Explain your answers to the previous question in terms of the Kinetic Molecular Theory of gases.
1. 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?
2. 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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1. 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:

```°C  kPa
19  2.1978
20  2.3388
21  2.4877
22  2.6447
23  2.8104
24  2.985
25  3.169
26  3.3629
```

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 much O2 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.

```           na
Xa = ——————————————
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:
1. 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?
2. What is the mole fraction of each of the gases in the previous problem?
3. Check the answer to the previous problem by adding up all the mole fractions. What should the sum of the mole fractions be?
4. Thirty-five liters of hydrogen gas (H2) are collected over water at 99 kPa and 24°C. What is the partial pressure of H2?
5. How many moles of H2 were collected?
6. 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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1. A container with two gases, helium and argon, is 30.0% by volume helium. What is the mole fraction of He? And of Ar?
2. Calculate the partial pressure of helium and argon if the total pressure inside the container is 4.00 atm.
3. 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
1. 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?
2. 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?
3. 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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1. 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
1. 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.
2. 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?
3. At what pressure would 0.150 moles of N2 at 23.0°C occupy 8.90 L?
1. 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%?
Last updated: Jun 30, 2006             Go Back    |     Home