# 15.2: Gases

**At Grade**Created by: CK-12

## Lesson Objectives

The student will:

- describe the relationship between molecular motion and temperature in Kelvin.
- describe random motion of gaseous molecules and explain how their collisions with surfaces cause pressure on the surface.
- recognize that zero kinetic energy of molecules corresponds to \begin{align*}0 \ \mathrm{K}\end{align*}.

## Vocabulary

- joule

## Introduction

Gases are tremendously compressible, can exert massive pressures, expand nearly instantaneously into a vacuum, and fill every container they are placed in regardless of size. All of these properties of gases are due to their molecular arrangement and constant molecular motion.

## Gases Readily Change Volume

When describing gases, the word “full” loses some of its original meaning. A glass of water may be \begin{align*}1/4\end{align*} full, \begin{align*}1/2\end{align*} full, or completely full, but a container containing a gaseous substance is always full. The same amount of gas will fill a quart jar, a gallon jug, a barrel, or even a house. Gas molecules are able to separate further away from each other and spread out uniformly until they fill whatever container they are in. On the other hand, gases can also be compressed to fractions of their original volume. If gas molecules are pushed together to the point that they touch, the substance would no longer be in the gas form and will become a liquid. As a result, one method of converting a gas to a liquid is to cool it, while another method is to compress the gas.

## Gases Exert Pressure

The constant random motion of gas molecules causes them to collide with each other and with the walls of their container. These collisions of gas molecules with their surroundings exert a pressure on the surroundings. When you blow up a balloon, the air particles inside the balloon push against the elastic sides of the balloon, causing the balloon to expand. This pressure is produced by air molecules pounding on the inside walls of the balloon.

When you look at the surface of a blown-up balloon, the balloon wall appears to be firm with no visible vibration or movement in its position. It is not apparent that the wall is actually being held in position by billions of collisions with tiny particles. If you place a book on its edge and tilt it over slightly so that it would fall, you can prevent the book from falling by tapping it very rapidly with your finger on the underside. Unlike the balloon surface, the book doesn’t stay steady because you can’t tap it fast enough to keep it exactly in one position. If you can imagine being able to tap it millions of times per second, you can see how the balloon wall maintains a steady position.

## Gas Temperature and Kinetic Energy

Kinetic energy is the energy of motion, so therefore all moving objects contain kinetic energy. The mathematical formula for calculating the kinetic energy of an object is: \begin{align*}KE = \frac{1} {2} mv^2\end{align*}. This formula applies to all objects, regardless of whether we are talking about the moon moving in its orbit, a baseball flying toward home plate, or a gas molecule banging around in a bottle. As you can see from the formula, kinetic energy is dependent on both the mass of the object and the velocity of the object. For example, the kinetic energy of a \begin{align*}0.20 \ \mathrm{kg}\end{align*} ball moving at \begin{align*}20. \ \mathrm{m/s}\end{align*} would be \begin{align*}KE = \frac{1} {2} (0.20 \ \mathrm{kg}) \cdot (20. \ \mathrm{m/s})^2 = 40.\ \frac {\mathrm{kg} \cdot \mathrm{m}^2} {\mathrm{s}^2}\end{align*}. The units \begin{align*}\frac {\mathrm{kg} \cdot \mathrm{m}^2} {\mathrm{s}^2}\end{align*} is also known as joules. The **joule** is the SI unit for energy. The kinetic energy of a molecule would also be calculated in this exact same way. You should note that if the mass of an object is doubled while its velocity remains the same, the kinetic energy of the object would also be doubled. If, on the other hand, the velocity is doubled while the mass remains the same, the kinetic energy would be quadrupled due to the square in the formula.

The molecular motion of molecules is related to their temperature. Recall from the chapter “Measurement in Chemistry” that the thermometer reflects the temperature of the surrounding because the molecules of material surrounding the thermometer will collide with the tube and transfer heat during the process. When you measure the temperature of a group of molecules, what you are actually measuring is their average kinetic energy. The relationship between the average kinetic energy of a group of molecules and the temperature is: \begin{align*}KE_{avg} = \frac {3} {2} RT\end{align*}, where \begin{align*}R\end{align*} is a constant of proportionality and \begin{align*}T\end{align*} is the absolute temperature (Kelvin). As a result, when a substance is heated, the average kinetic energy of the molecules increases. Since the mass of the molecules cannot be increased by heating, the velocity of the molecules must be increasing. The relationship between the kinetic energy of an object and its velocity, however, is not linear. Because the velocity is squared in the formula for kinetic energy, the average kinetic energy is doubled when the absolute temperature is doubled, but the velocity is increased only by a factor of \begin{align*}1.4\end{align*}.

It is absolutely vital that you keep in mind that the mathematical relationship between the temperature and the average kinetic energy of molecules only exists when the temperature is expressed in the Kelvin scale. In order for the direct proportion to exist, the molecules must have zero kinetic energy when the temperature is zero. Molecules do *not* have zero kinetic energy at \begin{align*}0^\circ \text{C}\end{align*} - balloons and automobile tires do not go flat when the outside temperature reaches \begin{align*}0^\circ \text{C}\end{align*}. Instead, the temperature at which molecular motion stops is \begin{align*}0 \ \mathrm{K} \ (-273^\circ \text{C})\end{align*}. If temperature is measured in Kelvin, then the average kinetic energy of a substance at \begin{align*}100 \ \mathrm{K}\end{align*} is exactly double the average kinetic energy of a substance at \begin{align*}50 \ \mathrm{K}\end{align*}. Make sure any work you do with the kinetic energy of molecules is done with Kelvin temperatures.

These two videos contain a discussion of the relationship between absolute zero and kinetic energy **(4a, 4f, 4g)**: http://www.youtube.com/watch?v=K4sOfGKEaxs (4:03), http://www.youtube.com/watch?v=Mgyp94TZdqQ (5:55).

Flaws in videos:

- statement that temperature is related to average molecular velocity rather than average molecular kinetic energy.

- statement that the carbon dioxide molecule has an angular shape rather than linear shape.

## Lesson Summary

- Gases readily change volume, as they are both expandable and compressible.
- Collisions of gas molecules with their surroundings exert a pressure on the surroundings.
- The relationship between temperature and the average kinetic energy of molecules can be expressed as \begin{align*}KE = \frac {1} {2} mv^2 = \frac {3} {2} RT\end{align*}.
- Zero molecular kinetic energy corresponds to 0 K.

## Review Questions

- Ball \begin{align*} A\end{align*} has a mass of \begin{align*}4 \ \mathrm{daltons}\end{align*} and a speed of \begin{align*}16 \ \mathrm{meters}\end{align*} per second. Ball \begin{align*}B\end{align*} has a mass of \begin{align*}16 \ \mathrm{daltons}\end{align*}. What velocity is necessary for ball \begin{align*}B\end{align*} to have the same kinetic energy as ball \begin{align*}A\end{align*}?
- Suppose you blow up a balloon, tie off the opening, and place the balloon in a freezer for one hour. When you take the balloon out of the freezer, what will be the most significant difference in its appearance? What do you think will happen as the balloon sits out in the room for a while?
- Suppose you drive home from school on a hot day and check the pressure in your automobile tires when you get home. You find the tire pressure is over the manufacturer's recommended pressure, so you let some air out of the tires until the pressure is appropriate. What will the tire pressure be in the morning when you go out to go to school?
- Weather balloons are large balloons that are used to carry meteorological instruments up through the atmosphere and radio back measurements on weather conditions, such as temperature, pressure, and humidity, as it passes through many different altitudes. When these balloons are filled with helium before they are released from earth, they are only a little more than \begin{align*}10\%\end{align*} the maximum capacity of the balloon. This provides enough lift to carry the instruments, but the balloon would have more lift if it were filled completely. Why don't they fill the weather balloons to maximum capacity?
- If molecules of \begin{align*}\text{H}_2\end{align*} (\begin{align*}\mathrm{molar \ mass} = 2)\end{align*}, \begin{align*}\text{O}_2\end{align*} \begin{align*}(\mathrm{molar \ mass} = 32)\end{align*}, and \begin{align*}\text{N}_2\end{align*} \begin{align*}(\mathrm{molar \ mass} = 28)\end{align*} are all placed in the same container at the same temperature, which molecules will have the greatest average kinetic energy?
- If molecules of \begin{align*}\text{H}_2\end{align*} (\begin{align*}\mathrm{molar \ mass} = 2)\end{align*}, \begin{align*}\text{O}_2\end{align*} \begin{align*}(\mathrm{molar \ mass} = 32)\end{align*}, and \begin{align*}\text{N}_2\end{align*} \begin{align*}(\mathrm{molar \ mass} = 28)\end{align*} are all placed in the same container at the same temperature, which molecules will have the greatest velocity?

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