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Kinetic theory of gases

mathematical model explaining macroscopic properties of gases in microscopic terms

The kinetic theory of gases is a simple, historically significant classical model of the thermodynamic behavior of gases, with which many principal concepts of thermodynamics were established. The model describes a gas as a large number of identical submicroscopic atoms or molecules, all of which are in rapid, random motion undergoing random elastic collisions between themselves and with the enclosing walls of the container. The basic model describes the ideal gas, and considers no other interactions between the particles. The theory explains macroscopic properties of gases, such as volume, pressure, and temperature, as well as transport phenomena such as viscosity, thermal conductivity and mass diffusivity. The model also accounts for Brownian motion. The kinetic theory of gases was the first explicit exercise of the ideas of statistical mechanics.

    Random particle motion
with elastic collisions, in 2D

Quotes

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  • Hari Seldon devised psychohistory by modeling it upon the kinetic theory of gases. Each atom or molecule in a gas moves randomly so that we can't know the position or velocity of any one of them. Nevertheless, using statistics, we can work out the rules governing their overall behavior with great precision. In the same way, Seldon intended to work out the overall behavior of human societies even though the solutions would not apply to the behavior of the individual human beings.
  • In this book I have tried... to make clearly comprehensible the path-breaking works of Clausius and Maxwell. The reader may not think badly of me for finding also a place for my own contributions. These were cited respectfully in Kirchhoff's lectures [on Maxwell's kinetic theory] and in Poincare’s Thermodynamique at the end, but were not utilized where they would have been relevant. From this I concluded that a brief presentation, as easily understood as possible, of some of the principal results of my efforts might not be superfluous. Of great influence on the content and presentation was what I have learned at the unforgettable meeting of the British Association in Oxford and the subsequent letters of numerous English scientists, some private and some published in Nature.
    I intend to follow Part I by a second part, where I will treat the van der Waals theory, gases with polyatomic molecules, and dissociation. ...Unfortunately it was often impossible to avoid the use of long formulas to express complicated trains of thought, and... to many who do not read over the whole work, the results will perhaps not seem to justify the effort expended. Aside from many results of pure mathematics which, though likewise apparently fruitless at first, later become useful in practical science as soon as our mental horizon has been broadened, even the complicated formulas of Maxwell’s theory of electromagnetism were often considered useless before Hertz’s experiments. I hope this will not also be the general opinion concerning gas theory!
  • Boltzmann decided to publish his lectures, in which the most important parts of the theory, including his ...contributions, were carefully explained. ...[H]e included his mature reflections and speculations on such questions as the nature of irreversibility and the justification for using statistical methods in physics. His Vorlesungen über Gastheorie was... the standard reference... for advanced researchers, ...[and] a popular textbook ...for the first quarter of the [20th] century ...The reason why the classical theory works is that, while the internal structure of molecules must be described by quantum mechanics, the interaction between two molecules can be fairly well described by a classical model which ignores this structure and simply uses a postulated force law whose parameters can be chosen to fit experimental data. ...Aside from phenomena at very high densities or very low temperatures, the only property that the classical theory fails to account for is the ratio of specific heats. ...Boltzmann ...simply concludes that for some unknown reason all the possible internal motions of a molecule do not have an equal share in the total energy, and takes this into account as an empirical fact.
  • Boyle... proposed a theoretical explanation for the elasticity of air... "a heap of little bodies, lying upon one another"... The atoms are said to behave like springs... Boyle also tried the "crucial experiment" which was to help overthrow his own theory in favor of the kinetic theory two centuries later, though he did not realize its significance... Experiment No. 26... places a pendulum in the evacuated chamber... [A]bsence of air makes hardly any difference to the period of the swings or the time... to come to rest. In 1859, James Clerk Maxwell deduced from the kinetic theory that the viscosity of a gas should be independent of its density... which would be very hard to explain on the basis of Boyle's theory. ...Neither Boyle nor Newton claimed that the hypothesis of repulsive forces between atoms is the only correct explanation for gas pressure; both were willing to leave the question open. Boyle mentions the Descartes theory of vortices (1644)... somewhat closer in spirit to the kinetic theory since it relies more heavily on the rapid motion of the parts of the atom as a cause of repulsion. (Though Descartes did not believe in "atoms" in the classical sense.) Nevertheless, the Boyle-Newton theory of gases was apparently accepted by most scientists until about the middle of the 19th century, when the kinetic theory finally managed to overcome Newton's authority.
    • Stephen G. Brush, Introduction to Kinetic Theory Of Gases: An Anthology Of Classic Papers With Historical Commentary (2003) History of Modern Physical Sciences, Vol. 1.
  • It is difficult to understand the relative lack of progress in gas theory during the 18th century ...[T]here was little interest in the properties of freely moving atoms. The atoms in gas were... conceived as... suspended in the ether, although they could vibrate or rotate enough to keep other atoms from coming too close. This model was... awkward... mathematically, as... seen from an... attempt by Leonhard Euler in 1727. ...[O]ne contribution from this period has been... recognized as the first kinetic theory of gases. This is Daniel Bernoulli's derivation of the gas laws from a "billiard ball" model—in 1738... [H]is kinetic theory is... a small part of a treatise [Hydrodynamica (1738)] on hydrodynamics... Bernoulli's formulation and... applications of the principle of conservation of mechanical energy (...vis viva ..."living force" ...) were ...more important than the fact that he proposed a kinetic theory ...a century ahead of its time ...Heat was still regarded as a substance ...Bernoulli's assumption that heat was nothing but atomic motion was unacceptable, especially to scientists interested in... radiant heat. The assumption that atoms could move freely through space until they collided like billiard balls... neglected the drag of the ether and oversimplified the interaction between atoms. ...When physics reached the stage of development at which the kinetic theory no longer conflicted with established principles, ...[it] had almost been forgotten and had to be rediscovered. ...In a very real sense, the man who persuades the world to adopt a new idea has accomplished as much as the man who conceived that idea.
    • Stephen G. Brush, Introduction to Kinetic Theory Of Gases: An Anthology Of Classic Papers With Historical Commentary (2003) History of Modern Physical Sciences, Vol. 1.
  • The influence of Quetelet's ideas spread throughout the sciences, even to the physical sciences. The two primary founders of the modern kinetic theory of gases, based on considerations of probability, were James Clerk Maxwell and Ludwig Boltzmann. Both acknowledged their debt to Quetelet. ...[H]istorians generally consider the influence of the natural sciences on the social sciences, whereas in the case of Maxwell and Boltzmann, there is an influence of the social sciences on the natural sciences, as Theodore Porter has shown.
    • I. Bernard Cohen, The Triumph of Numbers: How Counting Shaped Modern Life (2005)
  • It will be shown in this paper that, according to the molecular-kinetic theory of heat, bodies of microscopically visible size suspended in liquids must, as a result of thermal molecular motions, perform motions of such magnitude that these motions can easily be detected by a microscope. It is possible that the motions to be discussed here are identical with the so-called "Brownian molecular motion"; however, the data available to me on the latter are so imprecise that I could not form a definite opinion on this matter.
    If it is really possible to observe the motion to be discussed here, along with the laws it is expected to obey, then classical thermodynamics can no longer be viewed as strictly valid even for microscopically distinguishable spaces, and an exact determination of the real size of atoms becomes possible. Conversely, if the prediction of this motion were to be proved wrong, this fact would provide a weighty argument against the molecular-kinetic conception of heat.
    • Albert Einstein, "On the Motion of Small Particles Suspended in Liquids at Rest Required by the Molecular-Kinetic Theory of Heat" Annalen der Physik 17 (1905) pp. 549-560.
  • In 1909 Perrin suspended particles of gamboge in a liquid of slightly lower density, and found that the heavy particles did not sink to the bottom of the lighter liquid; they were prevented from doing so by their own Brownian movements. If the liquid had been infinitely fine-grained, with molecules of infinitesimal size and weight, every solid particle would have had as many impacts from above as below; these impacts, coming in a continuous stream, would have just cancelled one another out, so that each particle would have been free to fall to the bottom under its own weight. But when they were bombarded by molecules of finite size and weight, the solid particles were hit, now in one direction and now in another, and so could not lie inertly on the bottom of the vessel. From the extent to which they failed to do this, Perrin was able to form an estimate of the weights of the molecules of the liquid... and this agreed so well with other estimates that there could be but little doubt felt as to the truth either of the kinetic theory of liquids, or of the associated explanation of the Brownian movements.
    • James Jeans, An Introduction to the Kinetic Theory of Gases (1940) Introduction, p. 8.
  • Now, although the plans of the edifice of the electromagnetic theory of light were laid in 1880 by H. A. Lorentz, and even indicated much earlier by W. Weber, a full 10 years were required before the discoveries of Heinrich Hertz gave the impetus to collect the building stones and work them into shape. In the years 1890-93 a number of works appeared by F. Richarz, H. Ebert and G. Johnstone Stoney, mostly dealing with the mechanism of the emission of luminous vapours, and in which attempts are made, on the basis of the kinetic theory of gases, to determine the magnitude of the elementary electrical quantity, called by Stoney by the now universally accepted name of electron. ...H. Ebert proved that the amplitude of an electron in luminous sodium vapour need only be a small fraction of a molecular diameter in order to excite a radiation of the absolute intensity determined by E. Wiedemann. The way of determining the amount of electricity contained in the electron is very simple. The quantity of electricity required for the electrolytic evolution of 1 cubic cm. of any monatomic gas is divided by Loschmidt's number—i.e., the number of gas molecules contained in 1 cubic cm.
    • Walter Kaufmann, "The Development of the Electron Idea" (Nov. 8, 1901) The Electrician Vol. 48 pp. 95-97. Lecture delivered before the 73rd Naturforscher Versammlung at Hamburg. From the Physikalische Zeitshrift, of October 1, 1901.
  • One of the most important and interesting aims of physical chemistry is to explain the properties of matter in terms of the motions and spatial arrangements of atoms and molecules. This aim has been more nearly achieved in the physical chemical study of gases at low pressures [below a few atmospheres at ordinary temperatures] than the study of matter in any other conditions. ... The structure of gases at these pressures is particularly simple: such gases are collections of molecules which move randomly in space and which collide with each other relatively infrequently—that is, the molecules are so far apart that much of the time they exert little influence on each other. ...[T]he properties of the gaseous state play a role in many important practical processes, such as [in]... the internal combustion engine, the function of the lungs, the motions of the winds across the earth and the flight of airplanes. Gases... provide a useful and pedagogically attractive starting point for the introduction of students to physical chemistry.
  • The kinetic theory of gases is a small branch of physics which has passed from the stage of excitement and novelty into staid maturity. ...Formerly it was hoped that the subject of gases would ultimately merge into a general kinetic theory of matter; but the theory of condensed phases... today, involves an elaborate and technical use of wave mechanics, and for this reason it is best treated as a subject in itself.
    The scope of the present book is, therefore, the traditional kinetic theory of gases. ...[A]n account has been included of the wave-mechanical theory, and especially of the degenerate Fermi-Dirac case... There is also a concise chapter on statistical mechanics, which... may be of use as an introduction... [T]he discussion of electrical phenomena has been abbreviated... the latter voluminous subject is best treated separately. ...[F]undamental parts have been explained... [as] to be within the reach of college juniors and seniors. The... wave mechanics and statistical mechanics... are of graduate grade. ...[A] number of carefully worded theorems have been inserted in the guise of problems, without proof... to give... a chance to apply... lines of attack exemplified in the text.
    To facilitate use as a reference book, definitions have been repeated freely, I hope not ad nauseam. ...Ideas have been drawn freely from ...books such as ...of Jeans and Loeb...
    • Earle Hesse Kennard, Kinetic Theory of Gases With an Introduction to Statistical Mechanics (1938) Preface.
  • The object of... this book is to formulate a Kinetic Theory... which shall apply equally well to matter in any state. ...The difficulty hitherto experienced in applying ...the Kinetic Theory of Gases ...to liquids and intermediary states of matter has been primarily due to the difficulty of properly interpretating molecular interaction. In the case of gases this difficulty is in most part overcome by the... assumption that a molecule consists of a perfectly elastic sphere not surrounded by any field of force. But since such... does not exist, the results obtained in the case of gases hold only in a general way, and the numerical constants involved are therefore of an indefinite nature, while in the case of dense gases and liquids this procedure does not lead to anything that is of use in explaining the facts.
    Instead of an atom, or molecule, consisting of a perfectly elastic sphere, it is more likely that each may be regarded simply as a center of forces of attraction and repulsion. If the exact nature of the field of force surrounding atoms and molecules were known... [one could calculate] the resulting properties of matter. But our knowledge... is at present not sufficiently extensive to permit... development... along these lines. ...[F]undamental progress will have been made only if molecular interaction is not, as is usually the case, represented by the collision of elastic spheres. ...[T]he subject may be developed... along sound mathematical lines... without knowing the exact nature and immediate result of molecular interaction. Thus it will be found... that the definition of the free path of a molecule in connection with viscosity, conduction of heat, diffusion, etc., may be given a form... not involving the exact nature of molecular interaction, which is mathematically quite definite, and which therefore applies equally well to the liquid and gaseous states. Since in the gaseous state each kind of path is proportional to the volume... its interest is then mainly associated with the characteristic factor of the volume which makes the product numerically equal to the path. A direct physical meaning may be given to this factor.
    In constructing a general Kinetic Theory the problem... is the dependence of the velocity of... a molecule in a substance on... density and temperature. It is often assumed that this velocity is the same in the liquid as in the gaseous state at the same temperature. ...[H]owever ...this holds only for each molecule at the instant it passes through a point in the substance at which the forces of the surrounding molecules neutralize each other. The total average velocity corresponding to the whole path of a molecule is usually much greater than... in a liquid and dense gas on account of the effect of the molecular forces of attraction and repulsion.
    ...The expansion pressure of a substance is evidently balanced by the sum of the external pressure and the intrinsic or negative pressure due to the attraction of the molecules upon each other, the equation expressing this relationship being a form of the equation of state.
    In the case of a mixture it is convenient to introduce the terms "partial expansion pressure," "partial intrinsic pressure," and "partial external pressure," in connection with each constituent. ...The property of osmotic pressure can ...be shown to be a function of the corresponding partial intrinsic pressure of the mixture, and the number of molecules giving rise to the osmotic pressure which cross a square centimeter from one side to the other per second. ...With the foregoing results ...and the modified definitions of the free paths ...the foundation of a general Kinetic Theory can be laid which applies to matter in any state, and which furnishes a number of important formulæ. ...If the formulæ obtained in connection with viscosity, conduction of heat, and diffusion, be applied to a perfect gas, they assume the well-known forms ...The development presented is perfectly sound without involving difficult mathematics. ...As a physicist I have often failed to see the usefulness from a physical standpoint of the extremely intricate mathematical investigations purporting to work out to the utmost limit the results of certain assumptions (usually in connection with molecular collision), and which have usually led to results whose usefulness seems incommensurate with the labor involved, seeing that the assumptions are usually not likely to be true. ...The development of a general Kinetic Theory of matter will be of service in the study of chemical action, principally in connection with the constant of mass-action and the reaction velocity constants. ...[T]he kinetic aspect of the chemical interaction of molecules apart from the thermodynamical aspect ...can only proceed along lines having such a Kinetic Theory as a basis. Thus... the rapidity of a chemical reaction in a gaseous or liquid mixture must be intimately connected with the number of molecules crossing a square centimeter per second ...[for] each constituent. The constant of mass-action must evidently be intimately connected with the free diffusion path of a molecule, etc.
    The study of viscosity, conduction of heat, diffusion, etc., has usually been confined to pure substances and to mixtures whose constituents do not interact chemically. It would... be of great interest to study these effects in the case of substances partly dissociated, since the interacting between molecules is then influenced by chemical affinity. Further light may be thrown on the nature of this property by the application of the formulae obtained.
    ...The properties of matter treated in this book... depend mainly on the dynamical properties of molecules modified in most cases by the molecular forces of attraction and repulsion. There is evidently, therefore, another side to the subject of the properties of matter, namely that which deals with those properties which depend in the main on the molecular forces modified in some cases by the dynamical properties of the molecules. Thus... the internal heat of evaporation and the intrinsic pressure probably do not depend directly on molecular motion. This part of the physico-chemical properties of matter will be dealt with in a separate book... "Molecular Forces." The present book may serve... as an introduction to the study of the purely thermodynamical aspect of material properties.
  • The last thirty years have seen the beginning and development of a new period in physics and chemistry, namely the atomic period. In contrast to the period preceding it where nature's processes were described in terms of continua, recent developments have emphasized the discrete structure of the submicroscopic universe. Thus, today one hears of the atoms of matter, the atoms of electricity, and even the atoms of energy, the quanta. ...[T]he atomic theory of matter is the oldest and perhaps the most complete. ...[B]ecause of its relative simplicity the problem of the atomic theory of gases, in the form of the kinetic theory of gases, has attained the highest degree of perfection in this field. Its admirable methods of analysis are therefore indispensable... This book... endeavors to develop the various concepts... independently...Besides a simple introduction of each concept it gives derivations... elementary ones, using little or no calculus; more advanced classical derivations; and in some cases the most recent developments available. It also contains the comparison of the theoretical deductions with modern experiment and a critique of the theories.
  • The science of Thermodynamics, founded by the labors of these three illustrious men [Nicolas Léonard Sadi Carnot, William Thomson & Rudolf Clausius], has led to the most important developments in all departments of physical science. It has pointed out relations among the properties of bodies which could scarcely have been anticipated in any other way; it has laid the foundation for the Science of Chemical Physics; and, taken in connection with the kinetic theory of gases, as developed by Maxwell and Boltzmann, it has furnished a general view of the operations of the universe which is far in advance of any that could have been reached by purely dynamical reasoning.
  • The first edition of this book appeared in 1877, at the time of the most rapid and beautiful development of the kinetic theory of gases. About twenty years before, the founders... Kronig and Clausius, had explained the expansive tendency of gases, and had calculated their pressure on the assumption that the smallest particles of gases do not repel each other, but are in rapid motion. From the theory based on this supposition not only were the laws of gases... deduced... but also new laws, hitherto undreamt of, were discovered... [and] afterwards confirmed... by experiment. These results, which we owe to Maxwell and Clausius, quickly won to the theory many friends and adherents. ...I undertook ...to exhibit the ...theory ...such ...as to be more easily intelligible ...especially to chemists and other natural philosophers to whom mathematics are not congenial. ...I endeavoured ...not only to develop the theory by calculation, but ...to support it by observation and found it on experiment. I... collected... and summarised, the observations by which the admissibility of the theory might be tested and its correctness proved. ...The mathematical discussions form ...an Appendix which ...need not be studied by every reader ...
    • Oskar Emil Meyer, Die kinetische Theorie der Gase (1877) Tr. Robert E. Baynes as Kinetic Theory of Gases (1899) from the 2nd revised edition, Preface, pp. v-vi.
  • The researches of Galileo, followed up by Huygens and others, led to those modern conceptions of Force and Law, which have revolutionized the intellectual world. The great attention given to mechanics in the seventeenth century soon so emphasized these conceptions as to give rise to the Mechanical Philosophy, a doctrine that all the phenomena of the physical universe are to be explained upon mechanical principles. Newton's great discovery imparted a new impetus to this tendency. The old notion that heat consists in an agitation of corpuscles was now applied as an explanation to the chief properties of gases. The first suggestion in this direction was that the pressure of gases is explained by the battering of the particles against the walls of the containing vessel, which explained Boyle's law of the compressibility of air. Later, the expansion of gases, Avogadro's chemical law, the diffusion and viscosity of gases, and the action of Crooke's radiometer were shown to be consequences of the same kinetical theory; but other phenomena, such as the ratio of the specific heat at constant volume to that at constant pressure, require additional hypotheses, which we have little reason to suppose are simple, so that we find ourselves quite afloat. In like manner with regard to light...
  • The first constant... is connected with the definition of temperature. If temperature were defined as the mean kinetic energy of a molecule in a perfect gas, which is a minute energy indeed, this constant would have the value ⅔. But in the conventional scale of temperature the constant ...[instead] assumes an extremely small value... intimately connected with the energy of a single molecule... [I]ts accurate determination would lead to the calculation of the mass of a molecule and... associated magnitudes. This constant is frequently termed Boltzmann's constant, although to the best of my knowledge Boltzmann... never introduced it (...he, as appears from... his statements, never believed it would be possible to determine this constant accurately)...
  • If we study the history of science we see happen two inverse phenomena, so to speak. Sometimes simplicity hides under complex appearances; sometimes it is the simplicity which is apparent, and which disguises extremely complicated realities.
    ...What is more complicated than the confused movements of the planets? What simpler than Newton's law?
    ...In the kinetic theory of gases, one deals with molecules moving with great velocities, whose paths, altered by incessant collisions, have the most capricious forms... The observable result is Mariotte's simple law. ...The law of great numbers has reestablished simplicity in the average.
    ...No doubt, if our means of investigation should become more and more penetrating, we should discover the simple under the complex, then the complex under the simple, then again the simple under the complex, and so on, without our being able to foresee what will be the last term. We must stop somewhere, and that science may be possible, we must stop when we have found simplicity. This is the only ground on which we can rear the edifice of our generalizations.
  • The old mechanical and atomic hypotheses have, during recent years, become so plausible that they have ceased to seem like hypotheses; atoms are no longer just a convenient fiction. It seems almost as if we could see them, now that we know how to count them. ...The kinetic theory of gases has thus received unexpected corroboration. ...The remarkable counting of the number of atoms by Perrin completed the triumph of the atomic theory. ...In the processes used with the Brownian phenomenon, or in those used for the law of radiation, we do not deal directly with the number of atoms, but with their degrees of freedom of movement. In that process where we consider the blue of the sky, the mechanical properties of the atoms come into play; the atoms are looked upon as producing an optical discontinuity. ...The atom of the chemist is now a reality. But that does not mean that we have reached the ultimate limit of the divisibility of matter. When Democritus invented the atom he considered it as the absolutely indivisible element within which there would be nothing further to distinguish. That is what the word meant in Greek. ... the atom of the chemist would not have satisfied him since that is not indivisible; it is not a true element; it is not free from mystery, from secrets. The chemist's atom is a universe. Democritus would have considered, even after so much trouble in finding it, that we were still only at the beginning of our search—these philosophers are never satisfied. ...This atom disintegrates into yet smaller atoms. What we call radioactivity is the perpetual breaking up of atoms. ...Each atom is like a sort of solar system where the small negative electrons play the role of planets revolving around the great... sun. ...the atom of a radioactive body is a universe within itself and a world subject to chance.
  • The idea of a Kinetic Theory of Gases originated with J. Bernouilli about the middle of the last century, but the first establishment of the theory on a scientific basis is due to Professor Clausius.
    During the last few years the theory has been greatly developed by many physicists, especially by Professor Clerk Maxwell in England and Professor Clausius and Dr. Ludwig Boltzmann... and although still beset by formidable difficulties, it has succeeded in explaining most of the established laws of gases in so remarkable a manner as to render it well worthy of the attentive consideration of scientific men. ...For the most part I have followed the method of treatment adopted by Dr. Ludwig Boltzmann in some very interesting memoirs ...

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