Kinetic Theory of Gases

kinetic theory of gases

Theory based on a simple description of a gas, from which many properties of gases can be derived. Established primarily by James Clerk Maxwell and Ludwig Boltzmann, the theory is one of the most important concepts in modern science. The simplest kinetic model is based on the assumptions that (1) a gas is composed of a large number of identical molecules moving in random directions, separated by distances that are large compared to their size; (2) the molecules undergo perfectly elastic (no energy loss) collisions with each other and with the walls of the container; and (3) the transfer of kinetic energy between molecules is heat. This model describes a perfect gas but is a reasonable approximation to a real gas. Using the kinetic theory, scientists can relate the independent motion of molecules of gases to their pressure, volume, temperature, viscosity, and heat conductivity.

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Kinetic Theory of Gases 

a branch of theoretical physics concerned with the statistical study of the properties of gases on the basis of concepts of the molecular structure of the gas and of a definite law of interaction between its molecules. The term “kinetic theory of gases” usually implies the theory of nonequilibrium processes in gases, whereas the theory of equilibrium states is part of the equilibrium statistical mechanics. The kinetic theory of gases is valid for gases themselves, gaseous mixtures, and plasma. The foundations of the kinetic theory of gases were formulated by L. Boltzmann in the second half of the 19th century.

A gas is a simpler system than either a liquid or a solid. The mean distance between the molecules of a gas is many times greater than their dimensions. Since the forces of interaction between electrically neutral atoms or molecules are of a short-range nature (that is, they decrease very rapidly with increasing distance between the particles and have virtually no effect at distances of several molecular diameters), interactions between molecules occur only when they are brought closer together— during collisions. The time of a collision is much shorter than the time required for traversing the free path, which is the time between two subsequent collisions. For this reason, the molecules are in free motion most of the time.

In the kinetic theory of gases the observed macroscopic effects (pressure, diffusion, heat conduction, and so on) are regarded as the mean result of the action of all of the molecules in the gas being studied. For calculating these mean quantities, Boltzmann introduced the distribution function f(v, r, t), which depends on the velocities ν and coordinates r of the molecules and on the time t. The product f(v, r, t )ΔvΔr gives the average number of molecules with velocities over the range from v to v + Δv and coordinates over the range from r to r + Δr. The distribution function f obeys the Boltzmann kinetic equation, in which the time variation of f is regarded as a result of the motion of particles, the action of external forces on the particles, and collisions involving pairs of particles.

The Boltzmann equation is applicable only to sufficiently rarefied gases. In a state of static equilibrium and in the absence of external forces, the distribution function depends only on the velocities of the molecules and is called the Maxwellian distribution.

The main task of the kinetic theory of gases is the determination (from the Boltzmann equation) of the form of the distribution function f, since knowledge of f(v, r, t) makes possible calculation of the mean quantities that characterize the state of the gas and the processes occurring in it—the mean particle velocity, the coefficient of diffusion, viscosity, thermal conductivity, and so on. Methods for solving the Boltzmann equation have been developed by the British scientists S. Chapman and D. Enskog. The Boltzmann equation describes the evolution of the system to the equilibrium state in the special case when external forces are absent.

In ionized gases (plasmas), particles interact with each other by means of Coulomb forces, which slowly decrease with increasing distance. In the case of such forces, it is impossible to speak of pair collisions, since a large number of particles interact simultaneously. But it is possible also in this case to derive a kinetic equation (called the Landau equation), if the fact that, in the overwhelming majority of cases, momentum transport is small during collision is taken into account. If collisions may be neglected altogether, the principal role will be played by Coulomb forces exerted on a given particle by all other particles of the system (self-consistent field approximation). In this case, the Vlasov kinetic equation applies to the plasma. N. N. Bogoliubov developed the most consistent and effective methods for the derivation of kinetic equations based on the dynamics of systems consisting of a large number of particles.

REFERENCES

Boltzmann, L. Lektsii po teorii gazov. Moscow, 1953. (Translated from German.)
Chapman, S., and T. Cowling. Matematicheskaia teoriia neodnorodnykh gazov. Moscow, 1960. (Translated from English.)
Bogoliubov, N. N. Problemy dinamicheskoi teorii ν statisticheskoi fizike. Moscow-Leningrad, 1946.
Silin, V. P. Vvedenie ν kineticheskuiu teoriiu gazov. Moscow, 1971.
Kogan, M. N. Dinamika razrezhennogo gaza. Moscow, 1967.
Nekotorye voprosy kineticheskoi teorii gazov. Moscow, 1965. (Translated from English.)
Klimontovich, Iu. L. Statisticheskaia teoriia neravnovesnykh protsessov ν plazme. Moscow, 1964.
Sommerfeld, A. Termodinamika i statisticheskaia fizika. Moscow, 1955. (Translated from German.)
Kikoin, I. K., and A. K. Kikoin. Molekuliarnaia fizika. Moscow, 1963. Chapters 1 and 2.

G. IA. MIAKISHEV