# Selected Works of Yakov Borisovich Zeldovich, Volume I: Chemical Physics and Hydrodynanics

YAKOV BORISOVICH ZELDOVICH
G. I. BARENBLATT
RASHID ALIEVICH SUNYAEV
Editor of English Edition J. P. Ostriker
G. I. Barenblatt
R. A. Sunyaev
Technical Supervisor of English Edition E. Jackson
A. Granik
E. Jackson
Pages: 489
https://www.jstor.org/stable/j.ctt7zvs4j

1. Front Matter
(pp. i-iv)
(pp. v-vi)
3. Preface to the English Edition of the Selected Works of Ya. B. Zeldovich
(pp. vii-2)
J. P. Ostriker
4. The Scientific and Creative Career of Yakov Borisovich Zeldovich (1984)
(pp. 3-56)

The Editors feel that this volume of selected works by Ya.B. Zeldovich,Chemical Physics and Hydrodynamics,as well as a second volume,Particles, Nuclei, and the Universe,will be of great scientific interest to the reader, be he a chemist, physicist, or astronomer.

The present edition, undertaken in connection with the seventieth birthday of Academician Yakov Borisovich Zeldovich, is an original exposition of the now-classical results of a scientist, results which remain actual even today. Some were obtained by Ya.B.¹ as early as the thirties, before World War II, while many papers were published between 1947–1986. Some have already...

• ### I. Adsorption and Catalysis

• 1 On the Theory of the Freundlich Adsorption Isotherm
(pp. 58-67)

§1. By far the majority of available experimental data on the adsorption of gases on the surface of a solid substance totally fails to agree with the simple laws derived by Langmuir. Moreover, even the linearity of the adsorption isotherm at very low surface coverage, which follows for adsorption on a uniform surface from the general laws of statistical mechanics, independently of any assumptions regarding the interaction between adsorbed particles, the dependence of the potential energy on distance, etc.¹—even this linearity of the initial portion of the isotherm represents not so much an unambiguous result of experiments as the...

• 2 Adsorption on a Uniform Surface
(pp. 68-70)

The concept of a uniform surface is widely used in theoretical work on adsorption. A closer examination of this concept, and of the unexpected conclusions to which it may lead upon suitable selection of the various possibilities, may therefore be of some interest.

Let us start with an analogy. An ideal crystal, in which all the atoms are exactly located at the nodes of a geometrically perfect space lattice, can be conceived only on classical grounds and at absolute zero. However, it is impossible to accept this somewhat naive concept because of the uncertainty principle and thermal agitation atT...

• 3 On the Theory of Reactions on Powders and Porous Substances
(pp. 71-77)

A chemical reaction frequently turns out to be related to the process of transport of the reacting substances (for instance, in the case of a reaction between a gas and a condensed phase or in heterogeneous catalysis).

Two limiting cases may be considered: 1) for a reaction rate which is considerably larger than the rate of transport, the observed macroscopic reaction kinetics is completely determined by the transport conditions and does not reflect in any way the actual reaction rate on the surface or its dependence on temperature, concentration of the reacting substances, surface activity, etc. (“diffusion region”); 2) for...

• ### II. Hydrodynamics. Magnetohydrodynamics. Heat Transfer. Self-Similarity

• 4 The Asymptotic Law of Heat Transfer at Small Velocities in the Finite Domain Problem
(pp. 78-81)

As was shown by G. Helmholtz [1], in a purely viscous regime the energy dissipation, and hence the drag as well, are minimal. Similarly, it may be shown that heat transfer is minimal in a purely conductive regime in a fluid at rest.

Let us consider heat transfer in a closed vessel filled with fluid. All of the vessel’s walls are impermeable to the fluid. Some of the walls are thermally insulated, the rest are maintained at different fixed temperatures. It is the heat transfer between these regions heated to different temperatures which interests us. We will assume that the...

• 5 The Asymptotic Laws of Freely-Ascending Convective Flows
(pp. 82-85)

In studying the motion of submerged jets it was natural to assume that at large distances from the nozzle the phenomenon would become self-similar, and that the nozzle’s diameter would disappear as a governing parameter. Combining this self-similarity with the assumption of proportionality of the mixing path lengthl(Prandtl) to the submerged jet widthb,Tollmien [1] easily derived the asymptotic laws of turbulent propagation of the jet:

$u = {\left( {\frac{{{p_1}}} {{\rho x}}} \right)^{1/2}}{f_1}\left( {\frac{y} {x}} \right)$for a slit (1)

and

$u = \frac{{{{({p_2}/\rho )}^{1/2}}}} {x}{f_2}\left( {\frac{y} {x}} \right)$for a round nozzle (1a)

The jet is directed along thex-axis,uis velocity,P1is the momentum of the round jet. The...

• 6 Exact Solution of the Diffusion Problem in a Periodic Velocity Field and Turbulent Diffusion
(pp. 86-92)

We write the exact solution of the diffusion in a velocity field with a single Fourier-component. The expression for the effective diffusion contains the molecular diffusion coefficient as a factor; this ensures correct behavior of the result with respect to time reversal.

The diffusion in a velocity field with a wide velocity spectrum supposedly describing turbulence is considered in the spirit of cascade-renormalization ideas. For the latter case of isotropic turbulence, we construct an ordinary differential equation for the turbulent diffusion coefficient.

With certain restrictions on the parameters, we obtain an answer which agrees with elementary conceptions of the characteristic...

• 7 A Magnetic Field in the Two-Dimensional Motion of a Conducting Turbulent Fluid
(pp. 93-96)

The problem of magnetic fields arising spontaneously in the motion of a fluid was considered by Batchelor [1], He came to the conclusion that the magnetic field increases without limit for sufficient conductivity in a given velocity field. His conclusion was based on nonrigorous considerations of the analogy between the magnetic field and a velocity vortex.

In the present work, the special case of two-dimensional motion is considered:${u_z} = 0,$and${u_x}$depend only onxandy; the fluid is incompressible, div v = 0. In this case, we have succeeded in treating the problem completely rigorously. The results differ...

• 8 The Magnetic Field in a Conducting Fluid Moving in Two Dimensions
(pp. 97-105)
A. A. Ruzmaikin

The classical problem of the magnetic dynamo, whether amplification or maintenance of a magnetic field is possible in a moving conducting fluid, has received, as we know, an affirmative solution. Many concrete examples of magnetic dynamos have been constructed, and this has stimulated applications of the theory to the explanation of the origin and maintenance of magnetic fields of planets, stars, and galaxies. On the other hand, even in its simplest kinematic form (i.e., when the velocity field is supposed known), the theory of the magnetic dynamo remains incomplete since necessary and sufficient conditions for the action of a dynamo...

• 9 Gas Motion Under the Action of Short-Duration Pressure (Impulse)
(pp. 106-119)

Let us consider a half-space (the regionx> 0) occupied by an ideal monatomic gas. At an initial moment t = 0 the gas is at rest, its density is the same everywhere$(\rho = {\rho _0}$forx> 0;ρ= 0,x< 0), and the temperature and pressure are equal to zero. The outer surface of this gas is acted upon by a short pressure impulse,π(t), where the functionπ(t)is such thatπ(t)= 0 fort< 0, then it attains a maximum valueP,after which π(t) rapidly decreases. The time of the decrease...

• ### III. Phase Transitions. Molecular Physics

• 10 On the Theory of New Phase Formation. Cavitation
(pp. 120-137)

The critical nucleus of a new phase (Gibbs) is an activated complex (a transitory state) of a system. The motion of the system across the transitory state is the result of fluctuations and has the character of Brownian motion, in accordance with Kramers’ theory, and in contrast to the inertial motion in Eyring’s theory of chemical reactions. The relationship between the rate (probability) of the direct and reverse processes—the growth and the decrease of the nucleus—is determined from the condition of steadiness of the equilibrium distribution, which leads to an equation of the Fourier-Fick type (heat conduction or...

• 11 Theory of Interaction Between an Atom and a Metal
(pp. 138-143)

The problem of the interaction of an atomic system with the surface of a metal at large distances is of significant interest for the theory of gas and vapor adsorption on solids surfaces. Just as in the interaction of two atomic systems, the universal attraction at large distances for neutral and non-polar systems may be obtained only in the second approximation of perturbation theory [1], Hitherto, only one attempt has been made in this direction, but the untenability of the assumptions, methods, and results of this attempt were obvious [2].

The perturbation energy may be written in the form

$V = \mu e\sum\limits_i {\frac{1} {{r_i^2}}}$...

• 12 Proof of the Uniqueness of the Solution of the Equations of the Law of Mass Action
(pp. 144-147)

Intuitively, the uniqueness of the chemical equilibrium state of a mixture of reacting gases is more or less obvious. However, it may be of some interest to rigorously prove that the system of equations of the law of mass action (LMA), together with the imposed conditions of conservation of matter for givenTandνorTandp, has one and only one real-valued and positive solution.

To carry out the proof, we note that the LMA equations are equivalent to imposing an extremum on some function of the concentrations—the free energyFforv= const or...

• 13 On the Relation Between Liquid and Gaseous States of Metals
(pp. 148-151)
L. D. Landau

General considerations regarding the character of the transition of a substance from a metal to a dielectric state lead to the conclusion that such a transition occurs as a normal phase transition even up to high temperatures. For mercury and other low-boiling metals the critical point of transition from a liquid to a gaseous state probably corresponds to a lower temperature. One should expect the existence in some region of two separate (at different pressures and temperatures) transitions, from a metallic to a nonmetallic state, and from a liquid to a gaseous state, i.e., the existence of a liquid nonmetallic...

• ### IV. Theory of Shock Waves

• 14 On the Possibility of Rarefaction Shock Waves
(pp. 152-154)

A shock wave is the surface of a sudden very sharp variation in the motion and state (pressure, density, etc.) of a material, which moves with respect to this material.

The relations between the state of the material before the wave passes (I), the properties of the wave, and the state after the wave passes (II) are easily obtained from the laws of conservation of mass, momentum, and energy. These relations are symmetrical with respect to the quantities describing I and II, and are equally suited to description of the transition from I to II and the reverse transition from...

• 15 On the Propagation of Shock Waves in a Gas with Reversible Chemical Reactions
(pp. 155-160)

The acoustics of a gas in which occur reversible chemical reactions whose equilibrium shifts with pressure and temperature variations in the acoustic wave was studied by Albert Einstein [1]. His results were later applied to the study of the very fast processes encountered at the boundary between physics and chemistry—the transition of molecules from one vibrational state to another [2].

The propagation of shock waves accompanied by an irreversible chemical reaction with substantial release of heat is the subject of the theory of detonation [3-6].

Below we consider the question of shock wave propagation in a gas with a...

6. ### Part Two Theory of Combustion and Detonation

• 16 Theory of Combustion and Detonation of Gases
(pp. 162-232)

At every stage in the development of science the study of combustion has been very closely related to general and physical chemistry.

The first period in the development of combustion science was a period of determination of the basic chemical facts; to this period belong the refutation of the phlogiston theory and the discovery of oxygen, the discovery and study of the properties of carbon monoxide and carbon dioxide, and the so-called “pneumatic chemistry”—the investigation of various gases and determination of the stoichiometric laws (1650–1820).

Later, the energy aspects of combustion were studied, beginning with determination of the...

• ### I. Ignition and Thermal Explosion

• 17 On the Theory of Thermal Intensity. Exothermic Reaction in a Jet I
(pp. 233-242)

In 1908 an elementary theory of chemical reactions in a jet was given in a paper by Bodenstein and Wohlgast [1],

Only in the case when the reaction vessel is a long tube do the laws of chemical reaction in a jet approach the laws of chemical kinetics in a closed vessel uncomplicated by diffusion exchange, i.e., the laws which are obtained by integration of chemical kinetics equations of the form

$W = \frac{{dc}} {{dt}} = f(c).$(1)

For the simplest cases, for example, a monomolecular reaction,

$f(c) = kc,$, (2)

$c = {c_0}{e^{ - kt}},$(3)

they were found in integral form [e.g., (3)] as long ago as 1850 by...

• 17a On the Theory of Thermal Intensity. Exothermic Reaction in a Jet II. Consideration of Heat Transfer in the Reaction
(pp. 243-254)
Yu. A. Zysin

The theory of adiabatic reaction developed in the previous article is here generalized to the case when heat transfer is present. Consideration of the heat transfer leads to the appearance of new features in the “consumption-time” kinetic curves, specifically, the possibility of extinction as the residence time is increased and of self-ignition when the reaction time is decreased (in the previous article, in the adiabatic case, extinction occurred only for a decrease in the reaction time, and self-ignition only for an increase).

In a number of regions of the values of the initial concentration and of the constant which characterizes...

• 18 The Theory of Ignition by a Heated Surface
(pp. 255-261)

The problem of self-ignition of a gas confined in a vessel with two plane-parallel walls of identical temperature was solved by Frank-Kamenetskiĭ by investigating a steady regime which preserves equality between the released and evacuated heat at all points of the gas mixture, and by considering the limit of existence of this steady regime [1].

Generalizing the problem to the case of walls of different temperatures, we come to the question of ignition of a gas by a hot wall for given conditions of heat evacuation which are determined by the presence of a cold wall at a certain distance....

• ### II. Flame Propagation

• 19 A Theory of Thermal Flame Propagation
(pp. 262-270)
D. A. Frank-Kamenetskiĭ

The conception of the thermal propagation of a flame as the most common mechanism of combustion, related to successive ignition of the explosive mixture by heat generated in the reaction, is very far from new [1].

Existing theories [2–4], however, are unsatisfactory [5] since they make use of the concept of the “ignition temperature” of the mixture. Meanwhile, as we know [6,7], the phenomenon of self-ignition is naturally explained by the assumption of smooth growth of the rate of exothermic reaction with increasing temperature if the heat balance of the system is considered. The ignition temperature itself proves here...

• 20 The Theory of the Limit of Propagation of a Slow Flame
(pp. 271-287)

This paper is a continuation of a series of theoretical studies carried out at the Institute of Chemical Physics which seek to give a description of various phenomena of combustion and explosion under the simplest realistic assumptions about the kinetics of the chemical reaction. A characteristic feature of the specific rate (rate constant) of chemical combustion reactions is its strong Arrhenius-Iike dependence on the temperature with a large value of the activation heat, related to the large thermal effect of the combustion reaction.

Works published on this subject include Semenov’s fundamental theory of thermal explosion [1], Todes’ analysis of the...

• 21 Diffusion Phenomena at the Limits of Flame Propagation. An Experimental Study of Flegmatization of Explosive Mixtures of Carbon Monoxide
(pp. 288-303)
N. P. Drozdov

In a hydrogen-lean explosive mixture one observes a significant difference between the concentration at which downward flame propagation ceases (for ignition in the upper part of the vessel), and the concentration at which upward flame propagation ceases (for ignition in the lower part of the vessel). In the concentration interval in which combustion is impossible when ignited from below, whereas when ignited from above the mixture does burn, the combustion process is characterized by a number of peculiarities. Observation shows that the flame propagates in the form of separate caps or pellets without covering the full cross-section of the vessel....

• 22 On the Theory of Combustion of Non-Premixed Gases
(pp. 304-319)

Let us consider a chemical reaction between two substances (a fuel and oxygen) accompanied by the formation of new substances—the combustion products—and release of heat.

We will consider the stationary process with continuous supply of the original substances and output of the products. A distinctive feature of this case is that the fuel and oxygen (or air) are supplied separately, i.e., are not premixed. Therefore, even in the case when the reaction rate constant for oxygen with the fuel is large, the intensity of combustion does not exceed some limit which depends on the rate of mixing of...

• 23 Numerical Study of Flame Propagation in a Mixture Reacting at the Initial Temperature
(pp. 320-329)
A. P. Aldushin and S. I. Khudyaev

A reaction at the initial temperature changes the characteristics of an explosive mixture before the flame front and introduces an element of non-steadiness into the process of propagation of the combustion wave. The method proposed in [1] to describe this effect consists in replacing the original non-steady problem by a quasi-steady one with adiabatically increasing initial temperature${T_a}(t)$and an effective source of heat release which takes this increase into account. We test this method below by comparing it directly with the results of a numerical solution of the original non-steady problem.

For simplicity we consider the case of similarity...

• ### III. Combustion of Powders. Oxidation of Nitrogen

• 24 On the Theory of Combustion of Powders and Explosives
(pp. 330-363)

The very interesting experimental work carried out in recent years by Belyaev at the explosives laboratory of the Institute of Chemical Physics of the USSR Academy of Sciences provides starting points for theoretical investigation of a number of important problems, namely:

1. the spatial temperature distribution and state of matter in the combustion zone,

2. the combustion rate as a function of the conditions,

3. the conditions of the transition from combustion to detonation,

4. the ignition of an explosive material or powder and the conditions required for its combustion.

As is known, Belyaev [1] used convincing arguments to prove...

• 25 The Oxidation of Nitrogen in Combustion and Explosions
(pp. 364-403)

§1.Throughout the XIX century attention was repeatedly drawn to the fact that nitric oxide is formed during combustion and explosions. A theoretical approach to this problem, however, became possible only in the beginning of the XX century when the problem of the fixation of atmospheric nitrogen and the development of chemical thermodynamics led to a study of the equilibrium oxygen-nitrogen-nitric oxide [1]. Fink [2] investigated explosions of mixtures of 2H2+ O2with oxygen and nitrogen. Nernst [1] was of the opinion that a reversible bimolecular reaction takes place at the highest temperature of the explosion, the composition approaching...

• 26 Oxidation of Nitrogen in Combustion and Explosions
(pp. 404-410)

The formation of nitrogen oxides in combustion and explosions has been studied by many scientists, but to date there is no uniform opinion on the question of the nature and mechanism of this process. Nernst [1] believed that in an explosion, as a result of the high temperature, a direct bimolecular reaction of molecular oxygen with molecular nitrogen occurs. Haber [2] ascribed a catalytic role to the charged particles in the flame. Bone [3] considered that activation of nitrogen occurs in the flame. Polyakov [4, 5] considered nitrogen oxide to be a product of chain breaking in a chain combustion...

• ### IV. Detonation

• 27 On the Theory of Detonation Propagation in Gaseous Systems
(pp. 411-451)

In this article existing theories of detonation are critically examined. It is shown that the considerations which are used to select the steady value of the detonation velocity are unconvincing. In connection with the problem of a chemical reaction in a detonation wave, we refute objections against the idea of gas ignition by a shock wave which were expressed in the nineteenth century by Le Chatelier and Vieille. On the basis of this idea, we are able to rigorously justify the existing method of calculating the detonation velocity. We analyze the distribution of temperature, pressure and mass velocity at the...

• 28 On Detonation of Gas Mixtures
(pp. 452-458)
S. M. Kogarko

Rivin and Sokolik [1] were the first to notice that in a detonation wave the chemical reaction zone, in which the transformation of the original mixture into the reaction products occurs, must have a finite width which depends on the reaction rate.

Theoretical analysis of gas detonation leads to the conclusion that a shock wave propagates at the detonation front, compressing and heating the gas mixture. The chemical reaction runs in the already compressed gas, and it is only after completion of the reaction that the state of the explosion products calculated in the classical theory is attained (pressurepc,...

• 29 Flame Propagation in Tubes: Hydrodynamics and Stability
(pp. 459-479)
A. G. Istratov, N. I. Kidin and V. B. Librovich

The propagation of flames in channels has long attracted the attention of many investigators (Zeldovich, 1944; Tsien, 1951; Chernyĭ, 1954; Uberoi, 1959; Maxworthy, 1962; Borisov, 1978). The photographs show a curved shape of the flame front in channels up to considerable values of the Reynolds number, Re (Re is calculated from the flame propagation rate along the tube, the tube diameter, and the cold gas viscosity). The flame front has a stationary curved surface at Reynolds numbers equal to a few hundreds, although the hydrodynamic flame instability theory developed by Landau (1944), Darrieus (1944) and Markstein (1951) predicts stable propagation...