The Kinetic-Molecular Theory

The ratio corresponds to the coefficients in the balanced equation for the reaction. Hundreds of examples based on experimental observations could be cited. Here are a few. 

One volume of nitrogen can react with three volumes of hydrogen to form two volumes of ammonia 

One volume of methane reacts with (bums in) two volumes of oxygen to give one volume of carbon dioxide and two volumes of steam 

Sulfur (a solid) reacts with one volume of oxygen to form one volume of sulfur dioxide 

Four volumes of ammonia bum in five volumes of oxygen to produce four volumes of nitric oxide and six volumes of steam 

Careful scientific studies of the behavior and properties of gases were begun in the seventeenth century by Robert Boyle (1627-1691). Flis work was carried forward by many investigators, and the accumulated data were used in the second half of the nineteenth century to formulate a general theory to explain the behavior and properties of gases. This theory is called the kinetic-molecular theory (KMT). The KMT has since been extended to cover, in part, the behavior of liquids and solids. 

The KMT is based on the motion of particles, particularly gas molecules. A gas that behaves exactly as outlined by the theory is known as an ideal gas. No ideal gases exist, but under certain conditions of temperature and pressure, real gases approach ideal behavior or at least show only small deviations from it. Under extreme conditions, such as very high pressure and low temperature, real gases deviate greatly from ideal behavior. For example, at low temperature and high pressure many gases become liquids. 

The principal assumptions of the kinetic-molecular theory are as follows  

The distance between particles is large compared with the size of the particles 

Gas particles move in straight lines in all directions, colliding frequently with one another and with the walls of the container. 

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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 —... 

In Chap. 11 the laws governing the behavior of gases were presented. The fact that gases exert pressure was stated, but no reasons why gases should exhibit such behavior were given. The kinetic molecular theory explains all the gas laws that we have studied and some additional ones also. It describes gases in terms of the behavior of the molecules that make them up. 

Graham s law may be explained in terms of the kinetic molecular theory as follows Since the two gases are at the same temperature, their average kinetic energies are the same ... 

Does the kinetic molecular theory state that all the molecules of a given sample of gas have the same velocity since they are all at one temperature ... 

Ans. No. The kinetic molecular theory states that the average kinetic energy is related to the temperature, not the velocity or kinetic energy of any one molecule. The velocity of each individual molecule changes as it strikes other molecules or the walls. 

Suppose that we double the length of each side of a rectangular box containing a gas. (a) What would happen to the volume (b) What would happen to the pressure (c) Explain the effect on the pressure on the basis of the kinetic molecular theory. 

Ans. The gas laws work for unbonded atoms as well as for multiatom molecules, and so it is convenient to classify the unbonded atoms as molecules. If these atoms were not classified as molecules, it would be harder to state the postulates of the kinetic molecular theory. For example, postulate 1 would have to be stated "Molecules or unbonded atoms are in constant random motion. ... 

According to the Kinetic-Molecular Theory, all gas molecules have the same average kinetic energy (= M2 mu2) at the same temperature, where u is the average velocity. Hence, at the same T ... 

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

The kinetic-molecular theory (KMT) represents the properties of gases by modeling the gas particles themselves at the microscopic level. The KMT assumes that ... 

Too little attention is generally paid to the concentrations of the reactants in preparative organic work. With the exception of rare cases (e.g. in intramolecular rearrangements) we are concerned with reactions of orders higher than the first, and in these several kinds of molecules—usually two—are involved. Since, according to the kinetic molecular theory, the velocity of bimolecular reactions is proportional to the number of collisions between the various dissolved molecules and therefore to the product of the concentrations,... 

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. 

The Kinetic Molecular Theory attempts to represent the properties of gases by modeling... 

A gas that obeys these five postulates is an ideal gas. However, just as there are no ideal students, there are no ideal gases only gases that approach ideal behavior. We know that real gas particles do occupy a certain finite volume, and we know that there are interactions between real gas particles. These factors cause real gases to deviate a little from the ideal behavior of the Kinetic Molecular Theory. But a non-polar gas at a low pressure and high temperature would come pretty close to ideal behavior. Later in this chapter, we ll show how to modify our equations to account for non-ideal behavior. 

Before we leave the Kinetic Molecular Theory (KMT) and start examining the gas law relationships, let s quantify a couple of the postulates of the KMT. Postulate 3 qualitatively describes the motion of the gas particles. The average velocity of the gas particles is called the root mean square speed and is given the symbol rms. This is a special type of average speed. 

We can understand how atmospheric pressure is just the weight of the atmosphere pushing down on the Earth s surface, but how can we apply the basic definition of pressure to a confined gas To expand our concept of pressure and provide a basic framework for understanding the behavior of gases, we use a simple model for a confined gas. This model is known as the kinetic molecular theory. The kinetic molecular theory states ... 

The kinetic molecular theory is used throughout the discussion of gases and should become clearer as examples are used that illustrate this theory. 

Continuing to use a syringe as a container, the basic gas laws can be explained. These laws apply to what is referred to as an ideal or perfect gas. An ideal or perfect gas can be thought of as a gas that conforms to the kinetic molecular theory. In reality, gas molecules do have volume and exert forces on each other. Under normal conditions of temperature and pressure, though, the kinetic molecular theory explains the behavior of gases quite well. It is only when a gas is at very low temperatures and/or under extremely high pressure that a gas no longer behaves ideally. 

Explaining the kinetic molecular theory Moving between phases... 

Kinetic energy is the energy of motion. Gas particles have a lot of kinetic energy and constantly zip about, colliding with one another or with other objects. The picture is complicated, but scientists simplified things by making several assumptions about the behavior of gas pcirticles. These assumptions are called the postulates of the kinetic molecular theory. They apply to a theoretical ideal gas ... 

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 ... 

All gas particles have some volume. All gas particles have some degree of interparticle attraction or repulsion. No collision of gas particles is perfectly elastic. But imperfection is no reason to remain unemployed or lonely. Neither is it a reason to abandon the kinetic molecular theory of ideal gases. In this chapter, you re introduced to a wide variety of applications of kinetic theory, which come in the form of the so-called gas laws. ... 

This shows that Equation (73) occupies a position for the random coil that is analogous to the position of average kinetic energy in the kinetic molecular theory. This is not just a fortuitous similarity, but a reflection of the statistical basis of both. As a little self-test Did you recognize the similarity between the formalisms of the last few sections and kinetic theory as we went along ... 

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. 

EXAM PLE 9.6 Rate of Atomic Collisions as a Function of Pressure. Assuming 1019 atoms per square meter as a reasonable estimate of the density of atoms at a solid surface, estimate the time that elapses between collisions of gas molecules at 10 6 torr and 25°C with surface atoms. Use the kinetic molecular theory result that relates collision frequency to gas pressure through the relationship Z = 1/4 vNIV, for which the mean velocity of the molecules v = (BRTI-kM) 12 and NIV is the number density of molecules in the gas phase and equals pNJRT. Repeat the calculation at 10 8 and 10 10 torr. 

The kinetic molecular theory (KMT see Sidebar 2.7) of Bernoulli, Maxwell, and others provides deep insight into the molecular origin of thermodynamic gas properties. From the KMT viewpoint, pressure P arises merely from the innumerable molecular collisions with the walls of a container, whereas temperature T is proportional to the average kinetic energy of random molecular motions in the container of volume V. KMT starts from an ultrasimplified picture of each molecule as a mathematical point particle (i.e., with no volume ) with mass m and average velocity v, but no potential energy of interaction with other particles. From this purely kinetic picture of chaotic molecular motions and wall collisions, one deduces that the PVT relationships must be those of an ideal gas, (2.2). Hence, the inaccuracies of the ideal gas approximation can be attributed to the unrealistically oversimplified noninteracting point mass picture of molecules that underlies the KMT description. 

Some aspects of the kinetic molecular theory (KMT) of ideal gases were outlined in Sidebar 2.7. The simplest form of KMT refers to monatomic ideal gases, for which the internal energy U and enthalpy H=U + PV = U + nRT can be written explicitly as... 

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