Lectures on Gas Theory


One of the great masterpieces of theoretical physics, this classic work contains a comprehensive exposition of the kinetic theory of gases that is still relevant today, nearly 100 years after its first publication. Although the modifications of quantum mechanics have rendered some parts of the work obsolete, many of the topics dealt with still yield to the classical-mechanics approach outlined by Boltzmann; moreover, a variety of problems in aerodynamics, nuclear reactors, and thermonuclear power generation are ...

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One of the great masterpieces of theoretical physics, this classic work contains a comprehensive exposition of the kinetic theory of gases that is still relevant today, nearly 100 years after its first publication. Although the modifications of quantum mechanics have rendered some parts of the work obsolete, many of the topics dealt with still yield to the classical-mechanics approach outlined by Boltzmann; moreover, a variety of problems in aerodynamics, nuclear reactors, and thermonuclear power generation are best solved by Boltzmann's famous transport equation.
The work is divided into two parts: Part I deals with the theory of gases with monatomic particles, whose dimensions are negligible compared to the mean free path. Topics include molecules as elastic spheres and as centers of force, external forces and visible motions of the gas and the repelling force between molecules. Part II covers van der Waals' theory, the principles of general mechanics needed for a gas theory, gases with compound molecules, derivation of van der Waals' equation by means of the virial concept, theory of dissociation and supplements to the laws of thermal equilibrium in gases with compound molecules.
Combining rigorous mathematical analysis with pragmatic treatment of physical and chemical applications, Lectures on Gas Theory was the standard work on kinetic theory in the first quarter of the 20th century. It remains "one of the greatest books in the history of exact sciences." ― Mark Kac.

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Editorial Reviews

Reprint of the U. of California Press original of 1964. Annotation c. Book News, Inc., Portland, OR (booknews.com)
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Product Details

  • ISBN-13: 9780486684550
  • Publisher: Dover Publications
  • Publication date: 2/17/2011
  • Series: Dover Books on Physics Series
  • Edition description: Reprint
  • Edition number: 1
  • Pages: 512
  • Sales rank: 941,493
  • Product dimensions: 5.38 (w) x 8.64 (h) x 0.99 (d)

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Lectures On Gas Theory

By Ludwig Boltzmann, Stephen G. Brush

Dover Publications, Inc.

Copyright © 1964 The Regents of the University of California
All rights reserved.
ISBN: 978-0-486-68455-0


The molecules are elastic spheres. External forces and visible mass motion are absent.

§3. Maxwell's proof of the velocity distribution law; frequency of collisions.

We shall suppose for a moment that in the container there is a single gas composed of completely identical molecules. The molecules will also from now on—unless we specify otherwise—be assumed to behave like completely elastic spheres when they collide with each other. Even if all the molecules initially had the same velocity, there would soon occur collisions in which the velocity of one colliding molecule is nearly in the direction of the line of centers, but that of the other is nearly perpendicular to it. The first molecule would thereby end up with nearly zero velocity, while the velocity of the second would become [square root of 2] times as large. In the course of further collisions it would soon happen, if the number of molecules were large enough, that all possible velocities would occur, from zero up to a velocity much larger than the original common velocity of all the molecules; it is then a question of calculating the law of distribution of velocities among the molecules in the final state thus reached, or, as one says more briefly, to find the velocity distribution law. In order to find it, we shall consider a more general case. We assume that we have two kinds of molecules in the container. Each molecule of the first kind has mass m, and each of the second has mass m1. The velocity distribution which prevails at any arbitrary time t will be represented by drawing as many straight lines (starting from the origin of coördinates) as there are m molecules in unit volume. Each line will be the same in length and direction as the velocity of the corresponding molecule. Its endpoint will be called the velocity point of the corresponding molecule. Now at time t let


be the number of m-molecules whose velocity components in the three coördinate directions lie between the limits


and for which the velocity points thus lie in the parallelepiped having at one corner the coördinates [xi], η, ζ and having edges of length d]xi], dη dζ parallel to the coördinate axes. We shall always call this the parallelepiped dω. We shall also use the abbreviations dω for the product d]xi] dη dζ, and f for f ([xi], η, ζ, t). If dω were a volume element of any other shape, though still infinitesimal, the number of m-molecules whose velocity points lie inside dω would still be equal to

(11) f([xi], η, ζ, t) dω,

as one can see by dividing the volume element dω into even smaller parallelepipeds. If the function f is known for one value of t, then the velocity distribution for the m-molecules is determined at time t. Similarly we can represent the velocity of each m1-molecule by a velocity point, and denote by


the number of molecules whose velocity components lie between any other limits


and for which therefore the velocity points lie in a similar parallelepiped dω1. Likewise we write dω1 for dω1d][xi1dη1 and F1 for F ([xi]1, η1, ζ1, t). Moreover, we shall completely exclude any external forces, and assume that the walls are completely smooth and elastic. Then the molecules reflected from the wall will move as if they came from a gas which is the mirror image of our gas, and which is thus completely equivalent to it; the container wall is thought of as a reflecting surface. According to these assumptions, the same conditions will prevail everywhere inside the container, and if the number of molecules in a volume element whose velocity components lie between the limits (10) is initially the same everywhere in the gas, then this will also be true for all subsequent times. If we assume this, then it follows that the number of m-molecules inside any volume Φ that satisfy the conditions (10) is proportional to the volume Φ and is therefore equal to

(14) Φfdω;

likewise the number of m1-molecules in the volume Φ that satisfy the conditions (13) is:

(14a) ΦF1dω1.

From these assumptions it follows that the molecules that leave any space as a result of their progressive motion will on the average be replaced by an equal number of molecules from the neighboring space or by reflection at the walls of the container, so that the velocity distribution is changed only by collisions, and not by the progressive motion of the molecules. We shall make ourselves independent of these restrictive conditions (made now to simplify the calculation) in §§15-18, where we shall take account of the effect of gravity and other external forces.

We next consider only the collisions of an m-molecule with an m1-molecule and indeed we shall single out, from all those collisions that can occur during the interval dt, only those for which the following three conditions are satisfied:

1. The velocity components of the m-molecule lie between the limits (10) before the collision, hence its velocity point lies in the parallelepiped dω.

2. The velocity components of the m1-molecule lie between the limits(13) before the collision, hence its velocity point lies in the parallelepipe dω1. All m-molecules for which the first condition is fulfilled will be called "m-molecules of the specified kind," and similarly we speak of "m1-molecules of the specified kind."

3. We construct a sphere of unit radius, whose center is at the origin of coördinates, and on it a surface element dλ. The line of centers of the colliding molecules drawn from m to m1 must, at the moment of collision, be parallel to a line drawn from the origin to some point of the surface element dλ. The aggregate of these lines constitutes the cone dλ.

(15) Direction mm1 in the cone dλ.

All collisions that take place in such a way that these three conditions are fulfilled will be called "collisions of the specified kind" and we have the problem of determining the number dv of collisions of the specified kind that take place during a time interval dt in unit volume. We shall represent these collisions in Figure 2. Let O be the origin of coördinates, C and C1 the velocity points of the two molecules before the collision, so that the lines OC and OC1 represent these velocities in magnitude and direction, before the collision. The point C must lie inside the parallelepiped dω and the point C1 inside the parallelepiped dω1. (The two parallelepipeds are not shown in the figure.) Let OK be a line of unit length which has the same direction as the line of centers of the two molecules at the instant of the collision, drawn from m to m1. The point K must therefore lie inside the surface element dλ, which is also not shown in the figure. The line C1C = g represents in magnitude and direction the relative velocity of the m-molecule with respect to the m1-molecule before the collision, since its projections on the coördinate axes are equal to [xi]- [xi]1, η- ηITL1, and ζ- ζ1, respectively. The frequency of collisions obviously depends only on the relative velocity. Hence if we wish to find the number of collisions of the specified kind, we can imagine that the specified m1-molecule is at rest, while the m molecule moves with velocity g. We imagine further that a sphere of radius σ (the sphere σ) is rigidly attached to each of the latter molecules, so that the center of the sphere always coincides with the center of the molecule, σ should be equal to the sum of the radii of the two molecules. Each time that the surface of such a sphere touches the center of an m1-molecule, a collision takes place. We now draw from the center of each sphere σ a cone, similar and similarly situated to the cone dλ. A surface element of area σ2dλ is thereby cut out from the surface of each of these spheres. Since all the spheres are rigidly attached to the corresponding molecules, all these surface elements move a distance gdt relative to the specified m1-molecule. A collision of the specified kind occurs whenever one of these surface elements touches the center of a specified m1-molecule, which is of course possible only if the angle [??] between the directions of the lines C1C and OK is acute. Each of these surface elements traverses by its relative motion toward the m1molecule an oblique cylinder of base σ2dλ and height g cos [??]dt. Since there are fdωm molecules of the specified kind in unit volume, all the oblique cylinders traversed in this manner by all the surface elements have total volume


All centers of m1-molecules of the specified kind lying inside the volume Φ will touch during the interval dt one surface element σ2dλ and hence the number dv of collisions of the specified kind which occur in the volume element during time dt is equal to the number ZΦ of centers of m1molecules of the specified kind that are in the volume Φ at the beginning of dt. But according to Equation (14a) this is

(17) ZΦ = ΦF1dω1.

In this formula there is contained a special assumption, as Burbury has clearly emphasized. From the standpoint of mechanics, any arrangement of molecules in the container is possible; in such an arrangement, the variables determining the motion of the molecules may have different average values in one part of the space filled by the gas than in another, where for example the density or mean velocity of a molecule may be larger in one half of the container than in the other, or more generally some finite part of the gas has different properties than another. Such a distribution will be called molar-ordered [molar-geordnete]. Equations (14) and (14a) pertain to the case of a molar-disordered distribution. If the arrangement of the molecules also exhibits no regularities that vary from one finite region to another—if it is thus molar-disordered—then nevertheless groups of two or a small number of molecules can exhibit definite regularities. A distribution that exhibits regularities of this kind will be called molecular-ordered. We have a molecular-ordered distribution if—to select only two examples from the infinite manifold of possible cases—each molecule is moving toward its nearest neighbor, or again if each molecule whose velocity lies between certain limits has ten much slower molecules as nearest neighbors. When these special groupings are not limited to particular places in the container but rather are found on the average equally often throughout the entire container, then the distribution would be called molar-disordered. Equations (14) and (14a) would then always be valid for individual molecules, but Equation (17) would not be valid, since the nearness of the m-molecule would be influenced by the probability that the m1-molecule lies in the space Φ. The presence of the m1-molecule in the space Φ cannot therefore be considered in the probability calculation as an event independent of the nearness of the m-molecule. The validity of Equation (17) and the two similar equations for collisions of m- or m1-molecules with each other can therefore be considered as defining the meaning of the expression: the distribution of states is molecular-disordered.

If the mean free path in a gas is large compared to the mean distance of two neighboring molecules, then in a short time, completely different molecules than before will be nearest neighbors to each other. A molecular-ordered but molar-disordered distribution will most probably be transformed into a molecular-disordered one in a short time. Each molecule flies from one collision to another one so far away that one can consider the occurrence of another molecule, at the place where it collides the second time, with a definite state of motion, as being an event completely independent (for statistical calculations) of the place from which the first molecule came (and similarly for the state of motion of the first molecule). However, if we choose the initial configuration on the basis of a previous calculation of the path of each molecule, so as to violate intentionally the laws of probability, then of course we can construct a persistent regularity or an almost molecular-disordered distribution which will become molecular-ordered at a particular time. Kirchhoff also makes the assumption that the state is molecular-disordered in his definition of the probability concept.

That it is necessary to the rigor of the proof to specify this assumption in advance was first noticed in the discussion of my so-called H-theorem or minimum theorem. However, it would be a great error to believe that this assumption is necessary only for the proof of this theorem. Because of the impossibility of calculating the positions of all the molecules at each time, as the astronomer calculates the positions of all the planets, it would be impossible without this assumption to prove the theorems of gas theory. The assumption is made in the calculation of the viscosity, heat conductivity, etc. Also, the proof that the Maxwell velocity distribution law is a possible one—i.e., that once established it persists for an infinite time—is not possible without this assumption. For one cannot prove that the distribution always remains molecular-disordered. In fact, when Maxwell's state has arisen from some other state, the exact recurrence of that other state will take place after a sufficiently long time (cf. the second half of §6). Thus one can have a state arbitrarily close to the Max-wellian state which finally is transformed into a completely different one. It is not a defect that the minimum theorem is tied to the assumption of disorder, rather it is a merit that this theorem has clarified our ideas so that one recognizes the necessity of this assumption.

We shall now explicitly make the assumption that the motion is molar- and molecular-disordered, and also remains so during all subsequent time. Equation (17) is then valid, and we obtain


This is the number of collisions of the specified kind in unit volume in time dt, which was to be calculated. We ignore the grazing collisions, whose number is in any case a higher-order infinitesimal, so that in each collision at least one velocity component of each molecule changes by a finite amount. In each collision of the specified kind, the number fdω of m-molecules whose velocity components lie between the limits (10), which we call m-molecules of the specified kind, and also the number F1dω1 of m1-molecules of the specified kind, both decrease by one. In order to find the total decrease ∫dv suffered by fdω during dt as a result of all collisions of m-molecules with m1-molecules (without restriction on the magnitude and direction of the line of centers), we must consider [xi], η, ζ, dω and dt as constant in Equation (18) and integrate dω1 and d]summation] over all possible values—i.e., we integrate d]ωITL1 over all space, and dλ over all surface elements for which the angle [??] is acute. We shall denote the result of this integration by ∫dv.


Excerpted from Lectures On Gas Theory by Ludwig Boltzmann, Stephen G. Brush. Copyright © 1964 The Regents of the University of California. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

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Table of Contents


Translator's Introduction,
PART I Theory of gases with monatomic molecules, whose dimensions are negligible compared to the mean free path.,
CHAPTER I The molecules are elastic spheres. Extebnal forces and visible mass motion are absent.,
CHAPTER II The molecules are centers of force. consideration of external forces and visible motions of the gas.,
CHAPTER III The molecules repel each other with a force inversely proportional to the fifth power of their distance.,
PART II Van Der Waals' Theory; Gases with compound molecules; Gas dissociation; concluding remarks.,
CHAPTER I Foundations of van der Waals' theory.,
CHAPTER II Physical discussion of the van der Waals' theory.,
CHAPTER III Principles of general mechanics needed for gas theory.,
CHAPTER IV Gases with compound molecules.,
CHAPTER V Derivation of van der Waals' equation by means of the virial concept.,
CHAPTER VI Theory of dissociation.,
CHAPTER VII Supplements to the laws of thermal equilibrium in gases with compound molecules.,

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