The Kinetic-Molecular Theory of Gases

How many moles of KCIO are required to produce this amount of O2  

2 What is the mole ratio between KCIO and O2 in the balanced equation  

4 What is the mass of KCIO (molar mass 122.6 g/mol) in the original sample  

We have so far considered the behavior of gases from an experimental point of view. Based on observations from different types of experiments, we know that at pressures of less than 1 atm most gases closely approach the behavior described by the ideal gas law. Now we want to construct a model to explain this behavior. 

Before we do this, let s briefly review the scientific method. Recall that a law is a way of generalizing behavior that has been observed in many experiments. Laws are very useful, since they allow us to predict the behavior of similar systems. For example, if a chemist prepares a new gaseous compound, a measurement of the gas density at known pressure and temperature can provide a reliable value for the compound s molar mass. 

Solution From Dalton s law of partial pressures we know that 

Check The density of the oxygen gas is (0.164 g/0.128 L), or 1.28 g/L, which is a reasonable value for gases under atmospheric conditions (see Example 5.8). 

Each of the color spheres represents a different gas molecule. Calculate the partial pressures of the gases if the total pressure is 2.6 atm. 

The gas laws help us to predict the behavior of gases, but they do not explain what happens at the molecular level to cause the changes we observe in the macroscopic world. For example, why does a gas expand on heating  

A typical dive might be 40 to 65 ft, but dives to 90 ft are not uncommon. Because seawater has a slightly higher density than fresh water—about 1.03 g/mL, compared with 1.00 g/mL—the pressure exerted by a column of 33 ft of seawater is equivalent to 1 atm pressure. Pressure increases with increasing depth, so at a depth of 66 ft the pressure of the water will be 2 atm, and so on. 

3 What is the partial pressure of oxygen in a gas mixture that 

917 mole of carbon dioxide and that has a total pressure of 

The gas laws were derived empirically, and they enable us to predict the macroscopic behavior of gases. They do not explain, however, why gases behave as they do. The kinetic molecular theory, which was put forth in the nineteenth century by a number of physicists, notably Ludwig Boltzmann and James Maxwell, explains how the molecular nature of gases gives rise to their macroscopic properties. The basic assumptions of the kinetic molecular theory are as follows  

A gas is composed of particles that are separated by relatively large distances. The volume occupied by individual molecules is negligible. 

Gas molecules are constantly in random motion, moving in straight paths, colliding with the walls of their container and with one another in perfectly elastic collisions. (Energy is transferredhuX not lost in the collisions.) 

The kinetic-molecular theory is primarily a mathematical model of a gas based on the following postulates  

Comment 1.00 mole of helium occupies 22.4 L at STP, a volume about 20 percent larger than a 5-gallon gas can. 1.00 mole of helium atoms (6.022 x 1023 atoms) has a volume less than 0.03 percent of the volume of the gas. At STP, 99.97 percent of the volume of the gas is empty space. The fraction of its volume taken up by atoms decreases still further as its volume increases. Gases are mostly empty space  

Comment The straight-line motion and collision behavior of molecules is similar to the way billiard balls move and collide with one another on a billiard table, with an important exception Billiard balls eventually will slow to a stop because of friction the particles in a gas do not they continue to move. 

Pressure is caused by molecular collisions with the container walls. At constant temperature, a gas exerts greater pressure as its volume is compressed, which increases the concentration of particles and the number of collisions for each instant. The relationship between the pressure applied to a gas and its volume is related in Boyle s law. 

Comment The particles of real gases do attract one another when they are squeezed close together by very high pressures. Low temperatures enhance this attraction. If this were not so, it would not be possible to liquefy gases. The fact that attractive forces are exhibited in real gases at high pressures and low temperatures causes the gas laws to be less accurate under these conditions. 

However, although laws summarize observed behavior, they do nor tell us why nature behaves in the observed fashion. This is the central question for scientists. To try to answer this question, we construct theories (build models). The models in chemistry consist of speculations about what the individual atoms or molecules (microscopic particles) might be doing to cause the observed behavior of the macroscopic systems (collections of very large numbers of atoms and molecules). 

A model is considered successful if it explains the observed behavior in question and predicts correctly the results of future experiments. Note that a model can never be proved to be absolutely true. In fact, any model is an approximation by its very nature and is bound to fail at some point. Models range from the simple to the extraordinarily complex. We use simple models to predict approximate behavior and more complicated models to account very precisely for observed quantitative behavior. In this text we will stress simple models that provide an approximate picture of what might be happening and that fit the most important experimental results. 

An example of this type of model is the kinetic molecular theory, a simple model that attempts to explain the properties of an ideal gas. This model is based on speculations about the behavior of the individual gas particles (atoms or molecules). The postulates of the kinetic molecular theory can be stated as follows  

A gas is made up of a large number of particles (molecules or atoms) that are small in comparison with both the distance between them and the size of the container. 

Collisions between the molecules/atoms themselves and between the molecules/atoms and the walls of the container are perfectly elastic. 

FIGURE 3.1 A single particle in a cubical box undergoing elastic collision with a wall. 

Even though collisions will occur on more than one face, we can calculate how long it takes before another collision will occur at this face in terms of the velocity and the distance. This is based on the familiar equation (distance) = (rate)(time), so that the time is the distance/rate in the denominator, 

The following step is an approximation usually used in a freshman presentation in that we note that the motion of the atom is random and the velocity is a vector with three components in general. The motion is random so all three components are favored equally, but at the moment we only need v. If we use the dot product, we can get the square of the velocity containing all three equally weighted components as 

OBJECTIVE To understand the basic postulates of the kinetic molecular theory. 

Module 16 Gas Law and the Kinetic Molecular Theory covers concepts in this section, 

Gases consist of tiny particles (atoms or molecules). 

These particles are so small, compared with the distances between them, that the volume (size) of the individual particles can be assumed to be negligible (zero). 

The particles are in constant random motion, colliding with the walls of the container. These collisions with the walls cause the pressure exerted by the gas. 

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

Gas molecules do not exert attractive or repulsive forces on one another. 

The average kinetic energy, Ey, of gas molecules in a sample is proportional to the absolute temperature  

The ideal-gas equation describes how gases behave but not why they behave as they do. Why does a gas expand when heated at constant pressure Or why does its pressure increase when the gas is compressed at constant temperature To understand the physical properties of gases, we need a model that helps us picture what happens to gas particles when conditions such as pressure or temperature change. Such a model, known as the kinetic-molecular theory of gases, was developed over a period of about 100 years, culminating in 1857 when Rudolf Clausius (1822-1888) pubhshed a complete and satisfactory form of the theory. 

The kinetic-molecular theory (the theory of moving molecules) is smnmarized by the following statements  

The combined volume of all the molecules of the gas is negligible relative to the total volimae in which the gas is contained. 

Attractive and repulsive forces between gas molecules are negligible. 

Energy can be transferred between molecules during collisions but, as long as temperature remains constant, the average kinetic energy of the molecules does not change with time. 

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In addition to and r nis ai other way of characterizing coil dimensions is to consider which end-to-end distance has the greatest probability of occurring for specified n and 1 values. Derive an expression for this most probable value of r, r, from Eq. (1.44). Compare the ratio r ms/ m the ratio from the kinetic molecular theory of gases (consult, say,... 

The kinetic-molecular theory of gases explains the behavior of gases at the molecular level. All of the following are part of this theory EXCEPT —... 

According to the kinetic-molecular theory of gases, the internal energy of an ideal gas, U,... 

While studying gases in this chapter you will consider four main physical properties—volume, pressure, temperature, and amount—and their interrelationships. These relationships, commonly called gas laws, show up quite often on the AP exam, so you will spend quite a bit of time working problems in this chapter. But before we start looking at the gas laws, let s look at the Kinetic Molecular Theory of Gases, the extremely useful model that scientists use to represent the gaseous state. 

Chapter 10 sets down the basic assumptions of the kinetic molecular theory of gases, a set of ideas that explains gas properties in terms of the motions of gas particles. In summary, kinetic molecular theory describes the properties of ideal gases, ones that conform to the following criteria ... 

Instead, we must turn to the kinetic molecular theory of gases for an estimate of the frequency with which molecules collide with a solid surface. We shall not be misled, however, if we anticipate that this pressure is low. Example 9.6 is a numerical examination of gas collisions with walls. 

The temperature of an ideal gas is proportional to the average kinetic energy of the particles in a sample of the gas. The relationship between kinetic energy and temperature is derived from the Kinetic Molecular Theory of Gases and is given by ... 

The Kinetic Molecular Theory of Gases is based on four basic tenets. These tenets exactly describe an ideal gas. However, as we shall see, this theory does not describe real gases under all conditions. 

List the basic tenets of the Kinetic Molecular Theory of gases. 

The kinetic molecular theory of gases makes the following... 

When we refer to the volume of a gas, we are in fact talking about the volume of the container. The definition of the volume of a gas is the space available for gas molecules to move around in. The kinetic molecular theory of gases assumes that the volume of each gas molecule is essentially zero. Thus, the amount of space for them to move around in is the volume of the container. For all gases, Vgas = the volume of the container holding the gas. (Do not confuse this with the molar volume of gases. You will learn about molar volume in Chapter 12.)... 

Section 12.1 introduces the concept of pressure and describes a simple way of measuring gas pressures, as well as the customary units used for pressure. Section 12.2 discusses Boyle s law, which describes the effect of the pressure of a gas on its volume. Section 12.3 examines the effect of temperature on volume and introduces a new temperature scale that makes the effect easy to understand. Section 12.4 covers the combined gas law, which describes the effect of changes in both temperature and pressure on the volume of a gas. The ideal gas law, introduced in Section 12.5, describes how to calculate the number of moles in a sample of gas from its temperature, volume, and pressure. Dalton s law, presented in Section 12.6, enables the calculation of the pressure of an individual gas—for example, water vapor— in a mixture of gases. The number of moles present in any gas can be used in related calculations—for example, to obtain the molar mass of the gas (Section 12.7). Section 12.8 extends the concept of the number of moles of a gas to the stoichiometry of reactions in which at least one gas is involved. Section 12.9 enables us to calculate the volume of any gas in a chemical reaction from the volume of any other separate gas (not in a mixture of gases) in the reaction if their temperatures as well as their pressures are the same. Section 12.10 presents the kinetic molecular theory of gases, the accepted explanation of why gases behave as they do, which is based on the behavior of their individual molecules. 

Several gas laws have been introduced in this chapter, but no explanation as to why those laws apply to all gases has been proposed. This section introduces the kinetic molecular theory of gases, which explains the gas laws and when extended, also explains some properties of liquids and solids. Five postulates explain why gases behave as they do ... 

The kinetic molecular theory of gases predicts that an increase in temperature increases molecular velocities and so increases the frequency of in-termolecular collisions. This agrees with the observation that reaction rates are greater at higher temperatures. Thus there is qualitative agreement between the collision model and experimental observations. However, it is found that the rate of reaction is much smaller than the calculated collision frequency in a given collection of gas particles. This must mean that only a small fraction of the collisions produces a reaction. Why ... 

The kinetic molecular theory of gases relates the macroscopic properties of a gas to the structure of the constituent molecules, the forces between them, and their motions. Because the number... 

Apply the kinetic-molecular theory of gases and describe how this theory is consistent with the observed gas laws... 

The final topic will briefly present the postulates of the kinetic-molecular theory of gases that describe the behavior of gases on the molecular level. Let s begin with one of the most important properties of a gas, its pressure. 

If you answered incorrectly, review The Kinetic-Molecular Theory of Gases, page 319. 

LIQUIDS OR SOLIDS IN MANY WAYS. MOLECULAR MOTION IN GASES IS TOTALLY RANDOM, AND THE FORCES OF ATTRACTION BETWEEN GAS MOLECULES ARE SO SMALL THAT EACH MOLECULE MOVES EREELY AND ESSENTIALLY INDEPENDENTLY OF OTHER MOLECULES. SUBJECTED TO CHANGES IN TEMPERATURE AND PRESSURE, GASES BEHAVE MUCH MORE PREDICTABLY THAN DO SOLIDS AND LIQUIDS. ThE LAWS THAT GOVERN THIS BEHAVIOR HAVE PLAYED AN IMPORTANT ROLE IN THE DEVELOPMENT OF THE ATOMIC THEORY OF MATTER AND THE KINETIC MOLECULAR THEORY OF GASES. 

The findings of Maxwell, Boltzmann, and others resulted in a number of generalizations about gas behavior that have since been known as the kinetic molecular theory of gases, or simply the kinetic theory of gases. Central to the kinetic theory are the following assumptions ... 

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