Physics
Kinetic Theory of Gases
The Kinetic Theory of Gases describes the behavior of gases in terms of the motion of their particles. It states that gas particles are in constant, random motion and that their kinetic energy is directly proportional to the temperature of the gas. This theory helps explain the macroscopic properties of gases, such as pressure and temperature, in terms of the behavior of their individual particles.
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12 Key excerpts on "Kinetic Theory of Gases"
eBook - PDF- R. Prasad(Author)
- 2016(Publication Date)
- Cambridge University Press(Publisher)
1.1 Kinetic Theory of Gases Out of the three prominent states of matter, solid, liquid and gas, the Kinetic Theory of Gases is perhaps the most developed and complete. Further, under suitable boundary conditions, the Kinetic Theory of Gases may be applied to the solid and the liquid states as well. 1.1.1 What do we expect from a good theory of gases? Any good theory of gases must be able to explain satisfactorily all experimentally observed facts about gases, like the Boyle’s law, Charles’ law, Gay-Lussac’s law, Graham’s law of effusion, Daltons law of partial pressure, magnitudes of specific thermal capacities and their ratios for different gases, their temperature dependence, thermal conductivity and viscosities of gases, etc. The simple classical theory, called the Kinetic Theory of Gases, detailed below, brings out the main properties of gases remarkably well. It, however, fails to explain some finer points as regards to the temperature dependence of some gas properties. It is also not expected that such a simple theory will explain all details of complex gas systems. Laws of quantum statistics, also called statistical mechanics applied to gaseous systems, explain most of the properties that remained unexplained by the kinetic theory. The first step toward the development of the Kinetic Theory of Gases is to define an ideal gas. An ideal gas is a hypothetical or imaginary gas, properties of which are defined through the following assumptions of the Kinetic Theory of Gases. 1.1.2 Assumptions or postulates of the kinetic theory 1. Gases are made up of molecules. In any given measurable or macroscopic volume there is large number of molecules of the gas. 2. The molecules of a gas are in state of continuous motion and the relative separation between the molecules is much larger than their own dimensions. 3. Molecules exert no force on one another except when they collide.
eBook - PDF- Mehran Kardar(Author)
- 2007(Publication Date)
- Cambridge University Press(Publisher)
3 Kinetic Theory of Gases 3.1 General definitions Kinetic theory studies the macroscopic properties of large numbers of particles, starting from their (classical) equations of motion . Thermodynamics describes the equilibrium behavior of macroscopic objects in terms of concepts such as work, heat, and entropy. The phenomenological laws of thermodynamics tell us how these quantities are constrained as a system approaches its equilibrium. At the microscopic level , we know that these systems are composed of particles (atoms, molecules), whose interactions and dynamics are reasonably well understood in terms of more fundamental theories. If these microscopic descriptions are complete, we should be able to account for the macroscopic behavior, that is, derive the laws governing the macroscopic state functions in equilibrium. Kinetic theory attempts to achieve this objective. In particular, we shall try to answer the following questions: 1. How can we define “equilibrium” for a system of moving particles? 2. Do all systems naturally evolve towards an equilibrium state? 3. What is the time evolution of a system that is not quite in equilibrium? The simplest system to study, the veritable workhorse of thermodynamics, is the dilute (nearly ideal) gas. A typical volume of gas contains of the order of 10 23 particles, and in kinetic theory we try to deduce the macroscopic properties of the gas from the time evolution of the set of atomic coordinates. At any time t , the microstate of a system of N particles is described by specifying the positions q i t , and momenta p i t , of all particles. The microstate thus corresponds to a point t , in the 6 N -dimensional phase space = N i = 1 q i p i . The time evolution of this point is governed by the canonical equations ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ d q i d t = p i d p i d t = − q i (3.1) where the Hamiltonian p q describes the total energy in terms of the set of coordinates q ≡ q 1 q 2 ··· q N , and momenta p ≡ p 1 p 2 ··· p N . 57
- 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.
- Iain D. Boyd, Thomas E. Schwartzentruber(Authors)
- 2017(Publication Date)
- Cambridge University Press(Publisher)
Part I Theory 1 Kinetic Theory 1.1 Introduction The primary aim of kinetic theory is to relate molecular level behavior to macroscopic gas dynamics. This is achieved by consideration of the behavior of individual particles, and integrating their collective properties up to the macroscopic level. Consider the simple case of a gas at rest as illustrated in Fig. 1.1. At the macroscopic level, this is an uninteresting situation because all the gas properties, such as density (ρ ), pressure ( p), and temperature (T ), are constants. However, at the molecular level, there is a great deal of activ- ity with particles traveling individually at relatively high speed, and under- going collisions with other particles. When one considers the behavior of particles at the molecular level, they really only undergo two processes: trans- lational motion in space due to their velocity, and intermolecular collisions with other particles in the gas. While kinetic theory analysis has to consider these two physical phenomena, we will see that it is a complex process. For example, the motions of particles will be divided into consideration of bulk, directed motion, and random, thermal motion. Collisions of particles involve a nonlinear process that includes elastic events where only the particle veloc- ities change, and inelastic processes involving energy exchange with internal modes and even chemical reactions. 1.2 Fundamental Concepts In this section, we first provide an introduction to some basic concepts and definitions that will be needed to achieve our goal of relating molecular behavior to macroscopic gas dynamics. We then employ these concepts later in the chapter to analyze a number of different gas flow situations. 3 4 Kinetic Theory Molecular Macroscopic no gradients ρ, p, T Figure 1.1 Macroscopic and molecular views of a gas at rest.
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- 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 - 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.
- 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- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2020(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 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. 472 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- Robert J. Silbey, Robert A. Alberty, Moungi G. Bawendi(Authors)
- 2012(Publication Date)
- Wiley(Publisher)
P A R T T H R E E K inetic theory introduces the calculation of the rates of certain processes by use of a simple model of atoms and molecules in the gas phase. The probabilities of molecular speeds and the values of average speeds depend on the molecular mass and temperature for noninteracting gas molecules. The frequency of collisions and the transport properties (viscosity, diffusion, and heat conduction) for gases of rigid spherical molecules can be calculated. However, the behavior of real gases is more complicated, again because of intermolecular interactions. The prediction of rates of chemical reactions is much more difficult, so we will first consider the experimental aspects of gas reactions and the use of this information to obtain mechanisms of reactions. Then we turn to chemical dynamics to learn about the role of the transition state and to photochemistry to learn about the various processes that can occur after a molecule has absorbed a photon. The last chapter in this part of the book deals with the kinetics of reactions in the liquid state. The study of viscosity, diffusion, and electrical transport of ions provides information that is useful in understanding the rates of reactions in liquids. Relaxation methods are useful for studying very fast reactions in the liquid phase, and the theory of diffusion-controlled reactions yields an upper limit for the rate constants of bimolecular reactions. This will help us to better understand the acid–base catalysis, enzyme catalysis, and the rates of electrochemical reactions. Kinetics This page intentionally left blank
eBook - PDF- Robert J. Silbey, Robert A. Alberty, George A. Papadantonakis, Moungi G. Bawendi(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
PART 3 Kinetics K inetic theory introduces the calculation of the rates of certain processes by use of a simple model of atoms and molecules in the gas phase. The probabilities of molecular speeds and the values of average speeds depend on the molecular mass and temperature for noninteracting gas molecules. The frequency of collisions and the transport properties (viscosity, diffusion, and heat conduction) for gases of rigid spherical molecules can be calculated. However, the behavior of real gases is more complicated, again because of intermolecular interactions. The prediction of rates of chemical reactions is much more difficult, so we will first consider the experimental aspects of gas reactions and the use of this information to obtain mechanisms of reactions. Then we turn to chemical dynamics to learn about the role of the transition state and to photochemistry to learn about the various processes that can occur after a molecule has absorbed a photon. The last chapter in this part of the book deals with the kinetics of reactions in the liquid state. The study of viscosity, diffusion, and electrical transport of ions provides information that is useful in understanding the rates of reactions in liquids. Relaxation methods are useful for studying very fast reactions in the liquid phase, and the theory of diffusion–controlled reactions yields an upper limit for the rate constants of bimolecular reactions. This will help us to better understand the acid–base catalysis, enzyme catalysis, and the rates of electrochemical reactions. 601
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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