Chemistry
Kinetic Molecular Theory
The Kinetic Molecular Theory explains the behavior of gases based on the motion of their particles. It states that gas particles are in constant, random motion and that their kinetic energy is directly proportional to their temperature. Additionally, the theory describes the relationship between pressure, volume, and temperature in ideal gases.
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12 Key excerpts on "Kinetic Molecular Theory"
eBook - PDF- Don Shillady(Author)
- 2011(Publication Date)
- CRC Press(Publisher)
3 The Kinetic Molecular Theory of Gases INTRODUCTION We remind ourselves we are trying to present the essential aspects of physical chemistry and we consider this one of the most essential topics. In our treatment of the van der Waals gas, we have already mentioned the ideas of the collisions of small atoms, which have a lot of space between them as in Dalton ’ s law. Here, we go into further detail regarding the behavior of gas molecules using the ideas of Ludwig Boltzmann (1844 – 1906), who was one of the intellectual giants of the late nineteenth century and whose ‘‘ Boltzmann principle ’’ of energy distribution is one of the pillars of modern science. The breakthrough here was due mainly to Boltzmann ’ s PhD thesis on the theory of gases. Here, we will fi rst review the freshman chemistry derivation of part of Kinetic Molecular Theory of gases (KMTG) and then introduce Boltzmann ’ s amazing energy principle. KINETIC ASSUMPTIONS OF THE THEORY OF GASES 1. 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. 2. The molecules = atoms are in continuous random motion. 3. Collisions between the molecules = atoms themselves and between the molecules = atoms and the walls of the container are perfectly elastic . Let us consider the idea that gas pressure is caused by impacts of atoms = molecules with the wall of a container (or the diaphragm of a pressure gauge). We know a gas will fi ll any shaped container, but to make the derivation simpler, we assume a cubical container of dimension L L L where each side is of length L (Figure 3.1). Thus, each inner face of the container has area A ¼ L L . Looking ahead to the idea that pressure is force = area, we put just one atom in an empty cubical box and analyze the force on one face of the box.
eBook - PDF- Arther Adamson(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
CHAPTER TWO Kinetic Molecular Theory OF GASES 2-1 Introduction The treatment of ideal and nonideal gases in Chapter 1 was carried out largely from a phenomenological point of view. Behavior was described in terms of the macroscopic variables Ρ, V, and Γ, although some molecular interpretation was included in the discussion of the a and b parameters of the van der Waals equation (Section 1-9) and in the Special Topics section. We take up here the detailed model of a gas, that is, the Kinetic Molecular Theory of gases. In this model, a gas is considered to be made up of individual molecules, each having kinetic energy in the form of a random motion. The pressure and the temperature of a gas are treated as manifestations of this kinetic energy. In its simplest form, kinetic theory assumes that the molecules experience no mutual attractions. The elementary picture is the familiar one of a molecule having an average velocity u and bouncing back and forth between opposite walls of a cubical container. With each wall collision a change in momentum 2mu occurs, where m is the mass of a molecule. If the side of the container is /, the frequency of such collisions is u/2l, and the momentum change per second imparted to the wall, that is, the force on it, is mu 2 /l. The pressure, or force per unit area, becomes mu 2 /l 3 or mu 2 /. The quantity u refers to the velocity component in some one direction, and the total velocity squared, c 2 , is c 2 = u 2 + v 2 + w 2 , where ν and w are the components in the other two directions; on the average these components should be equal, and so we conclude that u 2 = c 2 /3, and obtain the final equation Pv = mc 2 , (2-1) where ν denotes the volume per molecule. Per mole, this becomes PV = Mc 2 , (2-2) where Μ is the molecular weight of the gas. One now takes note that the simple picture corresponds, for real gases, to the 39
eBook - PDFIntroduction to Physics
Mechanics, Hydrodynamics Thermodynamics
- P. Frauenfelder, P. Huber(Authors)
- 2013(Publication Date)
- Pergamon(Publisher)
C H A P T E R 8 KINETIC THEORY O N E goal of kinetic theory is to provide a deeper understanding of at least some of the thermodynamic properties of matter. The basic postulates of kinetic theory are the molecular structure of matter and the principle of molecular chaos. In order to treat the problem quantitatively, it is necessary to make various assumptions, most of which have arisen from some of the early concepts of the structure of matter, or idealizations of that structure. In this sense, kinetic theory is quite inferior to thermodynamics, which is based on only a very few general principles. However, kinetic theory offers us something which thermodynamics lacks, due to the very nature of its formulation, and that is an insight into the behavior of matter on a micro-scopic level. To explain the behavior of gases at low pressures, it is sufficient to ascribe to their molecules only a few very general properties. The more closely the molecules are packed, the more specific must our assumptions be. Eventually, in order to explain the properties of liquids and solids, we shall have to take into account the structure of molecules and atoms themselves. These considerations point the way in which our study of kinetic theory should proceed. We will consider first the kinetic theory of an ideal gas. From the discrepancies with experiment, the theory of real gases will follow natur-ally, and this will lead us to an understanding of the liquid and solid states 73. THERMAL MOTION The concepts of temperature and quantity of heat were introduced earlier in a purely formal way. Both of these quantities have a physical interpreta-tion from the molecular point of view. Temperature and quantity of heat can be clearly explained in terms of other concepts with which we are already familiar. In terms of the molecular picture of matter, solids, liquids, and gases are all composed of atoms and molecules.
eBook - ePubIntroduction to Plasmas and Plasma Dynamics
With Reviews of Applications in Space Propulsion, Magnetic Fusion and Space Physics
- Hai-Bin Tang, Thomas M. York, Haibin Tang(Authors)
- 2015(Publication Date)
- Academic Press(Publisher)
Chapter 2Kinetic Theory of Gases
Abstract
This chapter begins with a definition of idealized molecular behavior that incorporates classical momentum and energy conservation. The gas laws at standard conditions are derived and explained within this formalism. The transport phenomena of viscosity, conduction, and diffusion are similarly treated. Statistical concepts are introduced to establish a mathematical basis for deriving macroscopic properties. The velocity (Maxwellian) distribution function is derived from physical laws and with the introduction of entropy as an important descriptive variable of state. Average values of molecular speeds are derived. The extension of the ideal molecular model as the basis for describing real gases is discussed.Keywords
Equilibrium distribution function; Kinetic theory; Pressure; temperature, and energy relationships; Transport processes (viscosity, conduction, and diffusion)Introduction
In the study of the mechanics and energetics of fluid flow, normally the fluid is considered to be a continuous medium (continuum), describable by properties such as density, temperature, pressure, and viscosity. For example, energy is defined as C V T 0 . Since the basic problem is that of the interchange of a large amount of energy in and out and fluid systems, we must look at what a fluid is “in the small” (microscopically) as well as “in the large” (macroscopically) so that we can understand what energy “is” (what its forms are), and how it can change when added to or removed from a fluid. The energy exchange is central, and the effects of the energy exchange are secondary.Kinetic theory originated in an attempt to explain and correlate the familiar physical properties of gases on the basis of molecule behavior (perfect gas law as stated for imperfect gases, viscosity, conduction, and diffusion).Basic Hypotheses of Kinetic Theory
Basic Hypotheses (Present, 1958 )
1. Molecule hypothesis—“matter is composed of small discrete units known as molecules: that the molecule is the smallest quantity of substance that retains its chemical properties, that all molecules of a given substance are alike, and there are three states of matter which differ in the arrangement and state of motion of molecules.”
eBook - PDF- R. Prasad(Author)
- 2016(Publication Date)
- Cambridge University Press(Publisher)
The Kinetic Theory of Gases 1.0 Kinetic Theory, Classical and Quantum Thermodynamics Two important components of the universe are: the matter and the energy. Interplay between them creates a variety of processes and phenomenon. In order to understand and appreciate the vast spectrum of happenings around us, it is required to know more intimately the properties of the different forms of matter and their interactions with energy. This may be approached in two different ways. In the first approach, often called the microscopic approach, some assumptions about the nature of the matter present in the universe is made and then the well-known and well-established laws of interaction are applied between the assumed entities of the matter to explain the observed natural phenomenon. The kinetic theory of matter and the statistical mechanics (or quantum statics ) are the examples of the microscopic approach. In kinetic theory of matter it is assumed that matter is made of elements, which in turn are made of molecules that are in motion. Molecules of an element are all alike, while molecules of different elements are different. Molecules are themselves assumed to be made of atoms. Having made assumptions about the constitution of the matter, the kinetic theory applies the laws of Newtonian mechanics, like the law of conservation of energy, law of conservation of liner momentum, the law that states that the rate of change of momentum is equal to force, etc. to the molecules and obtain expressions for average properties of the system, like the pressure exerted by a gas, etc. In the case of quantum statics or statistical mechanics, it is assumed that matter is made of different kinds of identical particles or entities; the number of each type of entity in a given piece of matter is very large and, therefore, the entities follow the laws of quantum statistics.
- Frank R. Foulkes(Author)
- 2012(Publication Date)
- CRC Press(Publisher)
CHAPTE R FOU R THE KINETIC THEORY OF GASES The theory is called a “kinetic” theory because it assumes that all the macroscopic properties of a gas result from molecular motion . To be considered “macroscopic” requires * 10 6 particles, which corresponds to only about 10 –17 L of gas! 4.1 POSTULATES (1) A gas consists of a large number of small, identical particles, which are relatively far apart. The particles must be identical because gases form uniform solutions with uniform properties. The particles must be small and relatively far apart because gases are transparent, easily compressed, and quickly diffuse through each other. (2) The molecules are in continuous, random, very rapid motion, colliding frequently with each other and with the retaining walls of the container. The molecules must be in continuous motion because they don’t settle with time. The motion must be random because pressure is exerted in all directions; furthermore, the molecules move in all directions to occupy the container. The motion must be rapid because gases quickly move in straight line motion to very quickly fill a vacuum. There must be frequent collisions between particles because diffusion in other gases is much slower than in a vacuum. Finally, there must be frequent bombardment with the walls in order to account for the pressure of the gas. (3) The collisions are perfectly elastic, with no loss in total energy. If energy were gradually dissi-pated, there would be less bombardment with time, resulting in a gradual decrease in the pressure of the gas. This is not observed. (4) The only type of energy is kinetic energy ( KE ), and the average KE of translation is directly proportional to the absolute temperature; the same constant of proportionality holds for all gases. For a constant volume of gas, increasing the temperature increases the pressure, which means in-creased bombardment.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
Air behaves approximately as an ideal gas, for which the internal energy per mole is proportional to the Kelvin temperature. Mustafa Quraishi/©AP/Wide World Photos 392 CHAPTER 14 The Ideal Gas Law and Kinetic Theory Check Your Understanding (The answers are given at the end of the book.) 11. The kinetic theory of gases assumes that, for a given collision time, a gas molecule rebounds with the same speed after colliding with the wall of a container. If the speed after the collision were less than the speed before the collision, the duration of the collision remaining the same, would the pressure of the gas be greater than, equal to, or less than the pressure predicted by kinetic theory? 12. If the temperature of an ideal gas were doubled from 50 to 100 °C, would the average translational kinetic energy of the gas particles also double? 13. The pressure of a monatomic ideal gas doubles, while the volume decreases to one-half its initial value. Does the internal energy of the gas increase, decrease, or remain unchanged? 14. The atoms in a container of helium (He) have the same translational rms speed as the atoms in a container of argon (Ar). Treating each gas as an ideal gas, decide which, if either, has the greater temperature. 15. The pressure of a monatomic ideal gas is doubled, while its volume is reduced by a factor of four. What is the ratio of the new rms speed of the atoms to the initial rms speed? 14.4 *Diffusion You can smell the fragrance of a perfume at some distance from an open bottle because perfume molecules leave the space above the liquid in the bottle, where they are relatively con- centrated, and spread out into the air, where they are less concentrated. During their journey, they collide with other molecules, so their paths resemble the zigzag paths characteristic of Brownian motion. The process in which molecules move from a region of higher concentration to one of lower concentration is called diffusion.
- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
These three variables associated with a gas—volume, pressure, and temperature—are all a consequence of the motion of the atoms. The volume is a result of the freedom the atoms have to spread throughout the container, the pressure is a result of the collisions of the atoms with the container’s walls, and the temperature has to do with the kinetic energy of the atoms. The kinetic theory of gases, the focus of this chapter, relates the motion of the atoms to the volume, pressure, and temperature of the gas. Applications of the kinetic theory of gases are countless. Automobile engi- neers are concerned with the combustion of vaporized fuel (a gas) in the auto- mobile engines. Food engineers are concerned with the production rate of the fermentation gas that causes bread to rise as it bakes. Beverage engineers are concerned with how gas can produce the head in a glass of beer or shoot a cork The Kinetic Theory of Gases 544 CHAPTER 19 The Kinetic Theory of Gases from a champagne bottle. Medical engineers and physiologists are concerned with calculating how long a scuba diver must pause during ascent to eliminate nitrogen gas from the bloodstream (to avoid the bends). Environmental scientists are concerned with how heat exchanges between the oceans and the atmosphere can affect weather conditions. The first step in our discussion of the kinetic theory of gases deals with measur- ing the amount of a gas present in a sample, for which we use Avogadro’s number. Avogadro’s Number When our thinking is slanted toward atoms and molecules, it makes sense to measure the sizes of our samples in moles. If we do so, we can be certain that we are comparing samples that contain the same number of atoms or molecules.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
In fact, it can be shown that the internal energy is proportional to the Kelvin temperature for any type of ideal gas (e.g., monatomic, diatomic, etc.). For example, when hot-air balloonists turn on the burner, they increase the temperature, and hence the internal energy per mole, of the air inside the balloon (see Figure 14.11). in the gas. Thus, temperature is a property that characterizes the gas as a whole, a fact that is inherent in the relation 1 2 mv 2 rms 5 3 2 kT. The term v rms is a kind of average particle speed. There- fore, 1 2 mv 2 rms is the average kinetic energy per particle and is characteristic of the gas as a whole. Since the Kelvin temperature is proportional to 1 2 mv 2 rms , it is also a characteristic of the gas as a whole and cannot be ascribed to each gas particle individually. Thus, a single gas particle does not have a temperature. Figure 14.11 When the burner is turned on to heat the air within a hot-air balloon, the temperature of the air rises. Air behaves approximately as an ideal gas, for which the internal energy per mole is proportional to the Kelvin temperature. Mustafa Quraishi/©AP/Wide World Photos Check Your Understanding (The answers are given at the end of the book.) 11. The kinetic theory of gases assumes that, for a given collision time, a gas molecule rebounds with the same speed after colliding with the wall of a container. If the speed after the collision were less 338 Chapter 14 | The Ideal Gas Law and Kinetic Theory 14.4 | *Diffusion You can smell the fragrance of a perfume at some distance from an open bottle because perfume molecules leave the space above the liquid in the bottle, where they are rela- tively concentrated, and spread out into the air, where they are less concentrated. During their journey, they collide with other molecules, so their paths resemble the zigzag paths characteristic of Brownian motion.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
A form of the ideal gas law that applies when the number of moles and the pressure are constant is called Charles’ law and is given by Equation 14.4. P i V i = P f V f (14.3) V i _ T i = V f _ T f (14.4) 14.3 Kinetic Theory of Gases The distribution of particle speeds in an ideal gas at constant temperature is the Maxwell speed distribu- tion (see Figure 14.9). The kinetic theory of gases indicates that the Kelvin temperature T of an ideal gas is related to the average trans- lational kinetic energy ¯ KE of a particle, according to Equation 14.6, where υ rms is the root-mean-square speed of the particles. ¯ KE = 1 _ 2 mυ rms 2 = 3 _ 2 kT (14.6) The internal energy U of n moles of a monatomic ideal gas is given by Equation 14.7. The internal energy of any type of ideal gas (e.g., monatomic, diatomic) is proportional to its Kelvin temperature. U = 3 _ 2 nRT (14.7) 14.4 Diffusion Diffusion is the process whereby solute molecules move through a solvent from a region of higher solute concentration to a region of lower solute concentration. Fick’s law of diffusion states that the mass m of solute that diffuses in a time t through the solvent in a channel of length L and cross-sectional area A is given by Equa- tion 14.8, where ΔC is the solute concentration difference between the ends of the channel and D is the diffusion constant. m = (DA ΔC)t _ L (14.8) 434 CHAPTER 14 The Ideal Gas Law and Kinetic Theory Focus on Concepts Additional questions are available for assignment in WileyPLUS. Section 14.1 Molecular Mass, the Mole, and Avogadro’s Number 1. All but one of the following statements are true. Which one is not true? (a) A mass (in grams) equal to the molecular mass (in atomic mass units) of a pure substance contains the same number of mol- ecules, no matter what the substance is. (b) One mole of any pure substance contains the same number of molecules. (c) Ten grams of a pure substance contains twice as many molecules as five grams of the substance.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
The mass per mole (in g/mol) of a substance has the same numerical value as the atomic or molecular mass of one of its particles (in atomic mass units). The mass m particle of a particle (in grams) can be obtained by dividing the mass per mole (in g/mol) by Avogadro’s number, according to Equation 3. 14.2 The Ideal Gas Law The ideal gas law relates the absolute pressure P, the volume V, the number n of moles, and the Kelvin temperature T of an ideal gas, according to Equation 14.1, where R 5 8.31 J/(mol ? K) is the universal gas constant. An alternative form of the ideal gas law is given by Equation 14.2, where N is the number of particles and k 5 R N A is Boltzmann’s constant. A real gas behaves as an ideal gas when its density is low enough that its particles do not interact, except via elastic collisions. A form of the ideal gas law that applies when the number of moles and the temperature are constant is known as Boyle’s law. Using the subscripts “i” and “f” to denote, respectively, initial and final conditions, we can write Boyle’s law as in Equation 14.3. A form of the ideal gas law that applies when the number of moles and the pressure are constant is called Charles’ law and is given by Equation 14.4. 14.3 Kinetic Theory of Gases The distribution of particle speeds in an ideal gas at constant tem- perature is the Maxwell speed distribution (see Figure 14.8). The kinetic theory of gases indicates that the Kelvin temperature T of an ideal gas is related to the average translational kinetic energy KE of a particle, according to Equation 14.6, where v rms is the root-mean-square speed of the particles. The internal energy U of n moles of a monatomic ideal gas is given by Equation 14.7. The internal energy of any type of ideal gas (e.g., monatomic, diatomic) is proportional to its Kelvin temperature.
eBook - PDFCourse of Theoretical Physics
Physical Kinetics
- L. P. Pitaevskii, E.M. Lifshitz, J. B. Sykes(Authors)
- 2017(Publication Date)
- Pergamon(Publisher)
C H A P T E R I KINETIC THEORY OF GASES §1. The distribution function T H I S chapter deals with the kinetic theory of ordinary gases consisting of electric-ally neutral atoms or molecules. The theory is concerned with non-equilibrium states and processes in an ideal gas. An ideal gas, it will be recalled, is one so rarefied that each molecule in it moves freely at almost all times, interacting with other molecules only during close encounters with them. That is to say, the mean distance between molecules, f ~ N _ 1 / 3 (where Ν is the number of molecules per unit volume), is assumed large in comparison with their size, or rather in com-parison with the range d of the intermolecular forces; the small quantity N d 3 ~ (d/r) 3 is sometimes called the gaseousness parameter. The statistical description of the gas is given by the distribution function /(i, q, p) of the gas molecules in their phase space. It is, in general, a function of the generalized coordinates (chosen in some manner, and denoted jointly by q) and the corresponding generalized momenta (denoted jointly by p), and in a non-steady state also of the time t. Let άτ - dq dp denote a volume element in the phase space of the molecule; dq and dp conventionally denote the products of the differentials of all the coordinates and all the momenta respectively. The product fdr is the mean number of molecules in a given element dr which have values of q and ρ in given ranges dq and dp. We shall return later to this definition of the mean. Although the function / will be everywhere understood as the distribution density in phase space, there is advantage in expressing it in terms of suitably chosen variables, which need not be canonically conjugate coordinates and momenta. Let us first of all decide on the choice to be made.
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