Chapter 13: Gases 1

Chapter 13: Gases 1

chapter 13

Gases

Introduction

We live in a gaseous environment. The air we breathe is a mixture of gases. Yet gases are not as conspicuous as liquids and solids are, and it is easy to overlook their significance. Understanding why gases behave as they do can help us understand everyday occurrences such as “low pressure systems” in our weather and the apparent decrease of the amount of air in car tires in cold weather. This chapter focuses on how gases behave under various circumstances and why they behave as they do.

Chapter Discussion

One of the best byproducts of studying gas laws is achieving a greater understanding of the concept of models and getting a better feel for scientific thinking. As with most scientific endeavors, one of the goals is to be able to explain real phenomena; for example, explaining how a hot-air balloon works or why our ears “pop” when flying in an airplane. Observations are made, and when they appear to be consistent, they are termed laws (for example, Boyle’s law). It is important to understand that these laws tell us “what” (for example, the pressure of a gas is inversely proportional to its volume if the amount of gas and temperature are held constant), but do not tell us “why.” Models (or theories) are developed to explain the “whys.” In the case of gases, the kinetic molecular theory is offered as an explanation. Remember, though, that models are not reality; that is, they include simplifications.

Sections 13.2, 13.3, and 13.4 give us observations that, for the most part, consist of facts. For example, place a balloon filled with air in the freezer, and it will shrink. Thus temperature and the volume of a gas are directly related as long as pressure and the amount of gas remain constant (as the temperature decreases, the volume decreases). These observations should not be surprising to you, but they are presented in these sections as mathematical expressions.

There is no need to memorize all of these relationships because we bring them all together in the ideal gas equation:

PV = nRT

This equation brings together all of the gas laws discussed in Chapter 13. There are quite a few example problems in Sections 13.1-13.5. After reading these sections and working on the examples, you should be able to show how all the other gas laws (Boyle’s law, etc.) can be derived from the ideal gas law equation. Pay attention to what is constant and why. This way you need to know only this one equation.

For example, consider Charles’s law. This law states that the volume of a gas is directly dependent on the temperature of the gas (in Kelvin) provided that the amount of gas and the pressure are held constant. Thus V and T are changing, and P and n are constant (as is R, of course, as always). Let’s rearrange the ideal gas law equation so that the factors that change are on one side, and the factors that are constant are on the other:

Because R, P, and n are all constant, the factor “” must be constant as well. Therefore, we can write this as

= constant

This shows us that the ratio “” is a constant, as long as P and n are held constant (and the temperature is in Kelvin). We can rewrite this as

which is the mathematical representation of Charles’s law you will find in Section 13.3 of your text. You should spend some time and derive the other laws using this same approach.

These relationships explain what happens, but not why. To explain why, we need to consider a model of gases. Section 13.6 in your text discusses Dalton’s law of partial pressures. Read through this section and understand that this law is an observation; that is, it is a fact. The text also says that this law tells us two important things about gases that are important enough to repeat here:

1.The volume of the individual gas particle must not be very important.

2.The forces among the particles must not be very important.

Make sure to understand how Dalton’s law of partial pressures tells us this. Discuss this with friends or an instructor. These conclusions lead us to our model, the kinetic molecular theory (KMT).

The KMT is discussed in Section 13.8. Notice that two of the premises come from the above conclusions, which come from observations. This is how model development works. That is, we make observations and use the significance of these observations in developing a model. For example, consider two of the assumptions in the KMT:

1.The volume (size) of the individual particles can be assumed to be zero.

2.The particles are assumed not to attract or to repel each other.

Note how these two premises are actually “ideal” cases that come from Dalton’s law of partial pressures. That is, Dalton’s law of partial pressures leads us to believe that the volume of gas particles is not important, nor are the forces among these particles. In the KMT, we assume that not only are these factors not important, they are non-existent. Why do we do this? Because it is easier; that is, it makes for a simpler model. And recall, our goal is to make the model as simple as we can as long as it explains the observations we want it to explain.

You should strive to understand the significance of the KMT. Read through sections 13.8 and 13.9 and make sure you can explain the relationships among pressure, volume, amount of gas, and temperature.

Learning Review

1.Gas in compressed gas cylinders is usually under a great deal of pressure. If the gas in a particular cylinder has a pressure of 2500 psi, how many torr is this?

2.Convert each of the units of pressure below.

a.0.408 atm to torr

b.68,471 Pa to mm Hg

c.50.9 psi to atm

3.The relationship observed by Boyle between volume and pressure is

a.linear

b.proportional

c.inversely proportional

d.no relationship

4.Examine the cylinder with a moveable piston. If the piston moves downward, causing the volume of the gas to decrease, will the pressure of the gas become larger or smaller?

5.A sample of nitrogen gas at 2.4 atm has a volume of 50.3 L. If the pressure is decreased to 1.9 atm, will the new volume be greater or smaller? What is the new volume?

6.Absolute zero is the temperature at which gases have zero volume. But zero volume has never been measured in the laboratory. So how do we know at what temperature the volume of a gas equals zero? Be as specific as possible.

7.Which of the following equations cannot be derived from Charles’s law, V = bT?

8.A container with a moveable piston contains 0.89 L methane gas at 100.5 °C. If the temperature of the gas rises by 11.3 °C, what is the new volume of the gas?

9.Avogadro’s law describes the relationship between the amount (number of moles) of a gas and the volume of the gas. Under what conditions of temperature and pressure is Avogadro’s law true?

10.Where does the universal gas constant, R, come from?

11.Use the ideal gas law to solve the problems below.

a.A sample of chlorine gas at 543 torr has a volume of 21.6 L. If the temperature of the chlorine is 0 °C, how many moles of chlorine gas are present?

b.Poisonous carbon monoxide gas is a product of the internal combustion engine. If 1.2 mol of CO at 11°C and 102 mm Hg are present in a container, what will be the volume of the CO gas?

c.0.45 mol of a gas has a pressure of 299 torr at 300 °C and a volume of 53.8 L. At the same temperature and pressure, the volume of the gas is decreased to 39.7 L. How many moles of gas are present after the volume has changed?

12.Which statements about Dalton’s law of partial pressure are true and which are false?

a.The total pressure (PTOTAL) of a mixture of gases is independent of the sizes of the gas particles.

b.Attractive forces between gas particles are important in determining PTOTAL.

c.For ideal gases, PTOTAL depends solely on the total number of moles of gas for any temperature and volume.

13.Assume that a sample of humid air contains only nitrogen gas, oxygen gas and water vapor. If the atmospheric pressure is 745 mm Hg and the partial pressure of N2 is 566 mm Hg and of oxygen is 140. mm Hg, what is the partial pressure of water vapor in the air?

14.A 7.5-L mixture of gases is produced by mixing 4.0 L of N2 at 450 torr, 3.5 L of O2 at 252 torr, and 0.21 L of CO2 at 150 torr. If the temperature is held constant at 65°C, what is the total pressure of the mixture?

15.Explain the difference between a law and a model.

16.Which statements about the Kinetic Molecular Theory are true, and which are false?

a.The postulates of the Kinetic Molecular Theory are true for all gases at all temperatures and pressures.

b.Gas particles are assumed to either attract or repel each other.

c.The distance between individual gas particles is much greater than the volume of an individual gas particle.

d.As the Kelvin temperature of a gas increases, the average kinetic energy increases.

17.According to the Kinetic Molecular Theory, what are we measuring when we measure the temperature of a gas?

18.Carbon dioxide is produced during the combustion of liquid propane fuel.

C3H8 + 5O2 3CO2 + 4H2O

If 5.0 kg of propane are burned at 1.000 atm pressure and 400.°C, what volume of CO2 gas is produced?

19.A sample of fluorine gas has a volume of 19.9 L at STP. How many moles of fluorine are in the sample?

Answers to Learning Review

1.There is no conversion factor directly between psi and torr. However, we can convert psi to atm and atm to torr.

a.0.408 atm × = 310. torr

b.68471 Pa × = 513.6 mm Hg

c.50.9 psi × = 3.46 atm

3.Boyle observed that as the volume increased, the pressure decreased. Pressure and volume are inversely proportional. The correct answer is c.

4.If the volume of gas inside the cylinder becomes smaller, then the pressure of the gas will become larger.

5.This problem provides pressure and volume data. Boyle’s law relates pressure to volume. We can use P1V1 = P2V2, which is one way of writing Boyle’s law, to solve this problem. In this equation, P1 and V1 represent initial, or starting, conditions. P2 and V2 represent final or changed conditions. 2.4 atm is the initial pressure, P1,and 50.3 L is the initial volume, V1. Pressure has changed so P2 is 1.9 atm, and we are asked for the new volume, V2. First, will the new volume be larger or smaller? The pressure decreases from 2.4 atm to 1.9 atm. There is an inverse relationship between temperature and pressure, so if the pressure decreases, we would expect the volume to increase.

Rearrange the equation to isolate V2 on one side.

P1V1 = P2V2

Now substitute values into the equation.

V2 = 50.3 L ×

V2 = 64 L N2

This answer makes sense. The pressure has decreased from 2.4 atm to 1.9 atm, so we would expect the volume to increase. The volume has in fact increased from 50.3 L to 64 L. There is an inverse relationship between pressure and volume.

6.We can measure the volume of a gas at various temperatures, some of which are very cold but not quite absolute zero. We can then plot on a graph each temperature and volume pair.

Although we cannot measure the volume of a gas at −273.2 °C, we can determine at what temperature the volume would be zero by drawing a continuation of the straight line of the graph and noting where the temperature axis hits the line. This has been done for many different gases, and the result is always the same; the temperature at which the volume of a gas would be zero is −273.2°C.

a.Charles’s law can be written as V = bT, where V is volume, T is temperature and b is a constant. We can rearrange the equation by dividing both sides by T.

So, can be derived from Charles’s law.

b.If and , then it is also true that , so this relationship can also be derived from V = bT.

c.We know that and that . We also know that , but we cannot add and together to equal the constant b. So this relationship cannot be derived from V = bT.

d.By multiplying temperature and volume by the same value, we are not changing the equation, because and are both equal to the constant b.

so this relationship can also be derived from .

8.This problem provides temperature and volume data and asks about the volume of methane gas when the temperature changes. We can use Charles’s law to solve the problem. One form of Charles’s law is

The initial volume of methane is 0.89 L, and the initial temperature is 100.5 °C. The rise in temperature is 11.3 °C, so the final temperature is the sum of 100.5 °C plus 11.3 °C, or 111.8 °C. The unknown quantity is the final volume, V2. Rearrange the equation to isolate V2 on one side.

Because both the initial and final temperatures are given in °C, we must convert to Kelvin.

initial temperature (T1)

TK = T°C + 273

TK = 100.5 + 273

TK = 374

final temperature (T2)

TK = 111.8 + 273

TK = 385

Now substitute values into the equation.

V2 = 0.89 L methane ×

V2 = 0.92 L methane

This answer makes sense. The temperature has increased, so the volume should also increase. There is a direct relationship between the temperature in Kelvin of a gas and its volume.

9.Avogadro’s law, which expresses a relationship between the volume of a gas and the number of moles of that gas, is true only when the gas temperature and pressure are constant.

10.When the three laws that relate volume to pressure, to temperature and to the number of moles are combined, the constants that are present in each individual law are combined into one constant, called R, the universal gas constant. When volume is expressed in liters, pressure in atmospheres, and temperature in Kelvin, R has units of L·atm/K·mol and a value of 0.08206 L atm/K mol.

11.The ideal gas law can be expressed as PV = nRT. Remember that temperature must be expressed in kelvins, pressure in atmospheres, and volume in liters. If any of the quantities given in the problem are expressed in other units, we must convert them before we can use the ideal gas law.

a.This problem provides the pressure of chlorine gas in torr and the temperature in °C. Before we proceed let’s convert torr to atmosphere and °C to K.

TK = T°C + 273

TK = 0 + 273

TK = 273

We are provided with pressure, volume, and temperature data, and we are asked for the number of moles. Rearrange the ideal gas law equation to isolate n on one side.

PV = nRT

Now substitute values into the equation.

n = 0.688 mol Cl2

b.This problem asks us to calculate the volume of carbon monoxide gas, CO, given temperature, pressure and the number of moles of gas. The pressure of CO is given in mm Hg and temperature in °C. Before we proceed, we need to convert mm Hg to atmospheres and °C to K.

102 mm Hg × = 0.134 atm

TK = T°C + 273

TK = 11 + 273

TK = 284

We can use the ideal gas law to solve this problem because we are provided with temperature, pressure and the number of moles of gas. Rearrange the ideal gas law to isolate V on one side.

PV = nRT

Now substitute values into the equation.

V = 210 L CO

c.In this problem we are provided with information about two different gases. The information about the first gas is complete. That is, we know the number of moles, the temperature, the volume and the pressure of the gas. We know the temperature, the volume and pressure of the second gas and are asked to calculate the number of moles, n. We could use Avogadro’s law, which relates volume and moles, to solve this problem, but we are asked to use the ideal gas law. Because we know the temperature and the volume and the pressure of the second gas, we can use the ideal gas law to calculate the number of moles. First convert pressure, which is given in torr, to atm and temperature, which is given in °C, to K.

299 torr ×

TK = TC + 273

TK = 300. + 273

TK = 573

Rearrange the ideal gas law to isolate n on one side.

PV = nRT

Now substitute values into the equation.

n =

n = 0.332 mol

12.

a.The total pressure (PTOTAL) depends on only the total quantity of gas, not on the kind and size of the particles. So this statement is true.

b.Because PTOTAL does not depend on the identity of the gas particles but only on the quantity of particles, this statement is false.

c.The pressure of a single gas or a mixture of gases depends on the number of moles, and also the temperature and volume. A mixture of gases follows the ideal gas law, just as a single gas does. So this statement is false.

13.PTOTAL represents the pressure exerted by the gases in the atmosphere, which in this case is equal to 745 mm Hg. PTOTAL equals POXYGEN + PNITROGEN + PWATER VAPOR. The two major components of air are nitrogen and oxygen. If PNITROGEN equals 566 mm Hg, and POXYGEN equals 140. mm Hg, then we can use the following equation to calculate PWATER VAPOR.

PTOTAL = POXYGEN + PNITROGEN + PWATER VAPOR

Rearrange the equation to isolate PWATER VAPOR on one side.

PWATER VAPOR = PTOTAL − (POXYGEN + PNITROGEN)

Now substitute values into the equation.

PWATER VAPOR = 745 mm Hg − (140. mm Hg + 566 Hg)

PWATER VAPOR = 39 mm Hg.

14.We are asked to calculate PTOTAL for a mixture of three gases. To determine PTOTAL, we need to know the partial pressure of each individual gas. Calculating the partial pressure of any gas by using the ideal gas law requires us to know the number of moles, n, of that gas. So we need to find a way to calculate n. We are given the initial volume of each gas, the initial pressure, and the initial temperature, so we can use the ideal gas law in the form to find the number of moles. Because the number of moles of each gas does not change when the gases are mixed together, we can use the value of n we calculated for each gas before they are mixed together to calculate the partial pressure of each gas under the conditions present when the gases are mixed. Because pressure is given in torr and temperature in °C, convert to atmospheres and Kelvin before proceeding.