The kinetic theory of gases is a simple classical model of the thermodynamic behavior of gases. Its introduction allowed many principal concepts of thermodynamics to be established. It treats a gas as composed of numerous particles, too small to be seen with a microscope, in constant, random motion. These particles are now known to be the atoms or molecules of the gas. The kinetic theory of gases uses their collisions with each other and with the walls of their container to explain the relationship between the macroscopic properties of gases, such as volume, pressure, and temperature, as well as transport properties such as viscosity, thermal conductivity and mass diffusivity.

The basic version of the model describes an ideal gas. It treats the collisions as perfectly elastic and as the only interaction between the particles, which are additionally assumed to be much smaller than their average distance apart.

Due to the time reversibility of microscopic dynamics (microscopic reversibility), the kinetic theory is also connected to the principle of detailed balance, in terms of the fluctuation-dissipation theorem (for Brownian motion) and the Onsager reciprocal relations.

The theory was historically significant as the first explicit exercise of the ideas of statistical mechanics and it forms the theoretical foundation for rarefied gas dynamics.

History

Kinetic theory of matter

Antiquity

In about 50 BCE, the Roman philosopher Lucretius proposed that apparently static macroscopic bodies were composed on a small scale of rapidly moving atoms all bouncing off each other. This Epicurean atomistic point of view was rarely considered in the subsequent centuries, when Aristotelian ideas were dominant.

Modern era

"Heat is motion"

One of the first and boldest statements on the relationship between motion of particles and heat was by the English philosopher Francis Bacon in 1620. "It must not be thought that heat generates motion, or motion heat (though in some respects this be true), but that the very essence of heat ... is motion and nothing else." "not a ... motion of the whole, but of the small particles of the body." In 1623, in The Assayer, Galileo Galilei, in turn, argued that heat, pressure, smell and other phenomena perceived by our senses are apparent properties only, caused by the movement of particles, which is a real phenomenon. In 1665, in Micrographia, the English polymath Robert Hooke repeated Bacon's assertion, and in 1675, his colleague, Anglo-Irish scientist Robert Boyle noted that a hammer's "impulse" is transformed into the motion of a nail's constituent particles, and that this type of motion is what heat consists of. Boyle also believed that all macroscopic properties, including color, taste and elasticity, are caused by and ultimately consist of nothing but the arrangement and motion of indivisible particles of matter. In a lecture of 1681, Hooke asserted a direct relationship between the temperature of an object and the speed of its internal particles. "Heat ... is nothing but the internal Motion of the Particles of [a] Body; and the hotter a Body is, the more violently are the Particles moved." In a manuscript published 1720, the English philosopher John Locke made a very similar statement: "What in our sensation is heat, in the object is nothing but motion." Locke too talked about the motion of the internal particles of the object, which he referred to as its "insensible parts". In his 1744 paper Meditations on the Cause of Heat and Cold, Russian polymath Mikhail Lomonosov made a relatable appeal to everyday experience to gain acceptance of the microscopic and kinetic nature of matter and heat:Lomonosov also insisted that movement of particles is necessary for the processes of dissolution, extraction and diffusion, providing as examples the dissolution and diffusion of salts by the action of water particles on the “molecules of salt”, the dissolution of metals in mercury, and the extraction of plant pigments by alcohol.

Also the transfer of heat was explained by the motion of particles. Around 1760, Scottish physicist and chemist Joseph Black wrote: "Many have supposed that heat is a tremulous ... motion of the particles of matter, which ... motion they imagined to be communicated from one body to another."

Kinetic theory of gases

In 1738 Daniel Bernoulli published Hydrodynamica, which laid the basis for the kinetic theory of gases. In this work, Bernoulli posited the argument, that gases consist of great numbers of molecules moving in all directions, that their impact on a surface causes the pressure of the gas, and that their average kinetic energy determines the temperature of the gas. The theory was not immediately accepted, in part because conservation of energy had not yet been established, and it was not obvious to physicists how the collisions between molecules could be perfectly elastic.

Pioneers of the kinetic theory, whose work was also largely neglected by their contemporaries, were Mikhail Lomonosov (1747), Georges-Louis Le Sage (ca. 1780, published 1818), John Herapath (1816) and John James Waterston (1843), which connected their research with the development of mechanical explanations of gravitation.

In 1856 August Krönig created a simple gas-kinetic model, which only considered the translational motion of the particles. In 1857 Rudolf Clausius developed a similar, but more sophisticated version of the theory, which included translational and, contrary to Krönig, also rotational and vibrational molecular motions. In this same work he introduced the concept of mean free path of a particle. In 1859, after reading a paper about the diffusion of molecules by Clausius, Scottish physicist James Clerk Maxwell formulated the Maxwell distribution of molecular velocities, which gave the proportion of molecules having a certain velocity in a specific range. This was the first-ever statistical law in physics. Maxwell also gave the first mechanical argument that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium. In his 1873 thirteen page article 'Molecules', Maxwell states: "we are told that an 'atom' is a material point, invested and surrounded by 'potential forces' and that when 'flying molecules' strike against a solid body in constant succession it causes what is called pressure of air and other gases." In 1871, Ludwig Boltzmann generalized Maxwell's achievement and formulated the Maxwell–Boltzmann distribution. The logarithmic connection between entropy and probability was also first stated by Boltzmann.

At the beginning of the 20th century, atoms were considered by many physicists to be purely hypothetical constructs, rather than real objects. An important turning point was Albert Einstein's (1905) and Marian Smoluchowski's (1906) papers on Brownian motion, which succeeded in making certain accurate quantitative predictions based on the kinetic theory.

Following the development of the Boltzmann equation, a framework for its use in developing transport equations was developed independently by David Enskog and Sydney Chapman in 1917 and 1916. The framework provided a route to prediction of the transport properties of dilute gases, and became known as Chapman–Enskog theory. The framework was gradually expanded throughout the following century, eventually becoming a route to prediction of transport properties in real, dense gases.

Assumptions

The application of kinetic theory to ideal gases makes the following assumptions:

• The gas consists of very small particles. This smallness of their size is such that the sum of the volume of the individual gas molecules is negligible compared to the volume of the container of the gas. This is equivalent to stating that the average distance separating the gas particles is large compared to their size, and that the elapsed time during a collision between particles and the container's wall is negligible when compared to the time between successive collisions.

• The number of particles is so large that a statistical treatment of the problem is well justified. This assumption is sometimes referred to as the thermodynamic limit.

• The rapidly moving particles constantly collide among themselves and with the walls of the container, and all these collisions are perfectly elastic.

• Interactions (i.e. collisions) between particles are strictly binary and uncorrelated, meaning that there are no three-body (or higher) interactions, and the particles have no memory.

• Except during collisions, the interactions among molecules are negligible. They exert no other forces on one another.

Thus, the dynamics of particle motion can be treated classically, and the equations of motion are time-reversible.

As a simplifying assumption, the particles are usually assumed to have the same mass as one another; however, the theory can be generalized to a mass distribution, with each mass type contributing to the gas properties independently of one another in agreement with Dalton's law of partial pressures. Many of the model's predictions are the same whether or not collisions between particles are included, so they are often neglected as a simplifying assumption in derivations (see below).

More modern developments, such as the revised Enskog theory and the extended Bhatnagar–Gross–Krook model, relax one or more of the above assumptions. These can accurately describe the properties of dense gases, and gases with internal degrees of freedom, because they include the volume of the particles as well as contributions from intermolecular and intramolecular forces as well as quantized molecular rotations, quantum rotational-vibrational symmetry effects, and electronic excitation. While theories relaxing the assumptions that the gas particles occupy negligible volume and that collisions are strictly elastic have been successful, it has been shown that relaxing the requirement of interactions being binary and uncorrelated will eventually lead to divergent results.

Equilibrium properties

Pressure and kinetic energy

In the kinetic theory of gases, the pressure is assumed to be equal to the force (per unit area) exerted by the individual gas atoms or molecules hitting and rebounding from the gas container's surface.

-direction in an enclosed volume with characteristic length

The momentum

We combine the above with Newton's second law

The total translational kinetic energy

This is an important, non-trivial result of the kinetic theory because it relates pressure, a macroscopic property, to the translational kinetic energy of the molecules, which is a microscopic property.

Relativistic expression for this formula is

is speed of light.

Temperature and kinetic energy

, we may combine it with the ideal gas law is the Boltzmann constant is the absolute temperature which leads to a simplified expression of the average translational kinetic energy per molecule, is related to the translational kinetic energy by the description above, resulting in which becomes Equation () is one important result of the kinetic theory: The average molecular kinetic energy is proportional to the ideal gas law's absolute temperature. From equations () and (), we have Thus, the product of pressure and volume per mole is proportional to the average translational molecular kinetic energy.

Equations () and () are called the "classical results", which could also be derived from statistical mechanics; for more details, see:

The equipartition theorem requires that kinetic energy is partitioned equally between all kinetic degrees of freedom, D. A monatomic gas is axially symmetric about each spatial axis, so that D = 3 comprising translational motion along each axis. A diatomic gas is axially symmetric about only one axis, so that D = 5, comprising translational motion along three axes and rotational motion along two axes. A polyatomic gas, like water, is not radially symmetric about any axis, resulting in D = 6, comprising 3 translational and 3 rotational degrees of freedom.

Because the equipartition theorem

Therefore, the kinetic energy per kelvin of one mole of monatomic ideal gas (D is the Avogadro constant, and R is the ideal gas constant.

Thus, the ratio of the kinetic energy to the absolute temperature of an ideal monatomic gas can be calculated easily:

• per mole: 12.47J/K

• per molecule: 20.7yJ/K = 129μeV/K

At standard temperature (273.15K), the kinetic energy can also be obtained:

• per mole: 3406J

• per molecule: 5.65zJ = 35.2meV.

At higher temperatures (typically thousands of kelvins), vibrational modes become active to provide additional degrees of freedom, creating a temperature-dependence on D and the total molecular energy. Quantum statistical mechanics is needed to accurately compute these contributions.

Collisions with container wall

For an ideal gas in equilibrium, the rate of collisions with the container wall and velocity distribution of particles hitting the container wall can be calculated based on naive kinetic theory, and the results can be used for analyzing effusive flow ratess, which is useful in applications such as the gaseous diffusion method for isotope separation.

and that the particles obey Maxwell's velocity distribution

.

.

yields the pressure (consistent with Ideal gas law is punched to become a small hole, the effusive flow rate

Combined with the ideal gas law

The above expression is consistent with Graham's law.

Speed of molecules

where v is in m/s, T is in kelvin, and m is 92.1% of the rms speed (isotropic distribution of speeds).

See:

• Average,

• Root-mean-square speed

• Arithmetic mean

• Mean

• Mode (statistics)

Mean free path

In kinetic theory of gases, the mean free path be the collision cross section

is reciprocal of length.

Transport properties

The kinetic theory of gases deals not only with gases in thermodynamic equilibrium, but also very importantly with gases not in thermodynamic equilibrium. This means using Kinetic Theory to consider what are known as "transport properties", such as viscosity, thermal conductivity, mass diffusivity and thermal diffusion.

In its most basic form, kinetic gas theory is only applicable to dilute gases. The extension of kinetic gas theory to dense gas mixtures, Revised Enskog Theory, was developed in 1983–1987 by E. G. D. Cohen, J. M. Kincaid and M. Lòpez de Haro, building on work by H. van Beijeren and M. H. Ernst.

Viscosity and kinetic momentum

In books on elementary kinetic theory one can find results for dilute gas modeling that are used in many fields. Derivation of the kinetic model for shear viscosity usually starts by considering a Couette flow where two parallel plates are separated by a gas layer. The upper plate is moving at a constant velocity to the right due to a force F above the lower plate. The non-equilibrium flow is superimposed on a Maxwell-Boltzmann equilibrium distribution of molecular motions.

Inside a dilute gas in a Couette flow

is the mean free path can be considered to be constant over a distance of mean free path.

yields the forward momentum transfer per unit time per unit area (also known as shear stress

Combining the above kinetic equation with Newton's law of viscosity

and insert the velocity in the viscosity equation above. This gives the well known equation subsequently estimated below) for shear viscosity for dilute gases is the molar mass. The equation above presupposes that the gas density is low (i.e. the pressure is low). This implies that the transport of momentum through the gas due to the translational motion of molecules is much larger than the transport due to momentum being transferred between molecules during collisions. The transfer of momentum between molecules is explicitly accounted for in Revised Enskog theory

is called collision cross section diameter or kinetic diameter of a molecule in a monomolecular gas. There are no simple general relation between the collision cross section and the hard core size of the (fairly spherical) molecule. The relation depends on shape of the potential energy of the molecule. For a real spherical molecule (i.e. a noble gas atom or a reasonably spherical molecule) the interaction potential is more like the Lennard-Jones potential or Morse potential which have a negative part that attracts the other molecule from distances longer than the hard core radius. The radius for zero Lennard-Jones potential may then be used as a rough estimate for the kinetic radius. However, using this estimate will typically lead to an erroneous temperature dependency of the viscosity. For such interaction potentials, significantly more accurate results are obtained by numerical evaluation of the required collision integrals.

The expression for viscosity obtained from Revised Enskog Theory is a term accounting for the transfer of momentum over a non-zero distance between particles during a collision.

Thermal conductivity and heat flux

Following a similar logic as above, one can derive the kinetic model for thermal conductivity of a dilute gas:

Consider two parallel plates separated by a gas layer. Both plates have uniform temperatures, and are so massive compared to the gas layer that they can be treated as thermal reservoirs above the lower plate. The non-equilibrium energy flow is superimposed on a Maxwell-Boltzmann equilibrium distribution of molecular motions.

is the specific heat capacity can be considered to be constant over a distance of mean free path.

yields the energy transfer per unit time per unit area (also known as heat flux

Combining the above kinetic equation with Fourier's law

Similarly to viscosity, Revised Enskog theory is a term accounting for the transfer of energy across a non-zero distance between particles during a collision.

Diffusion coefficient and diffusion flux

Following a similar logic as above, one can derive the kinetic model for mass diffusivity of a dilute gas:

Consider a steady diffusion between two regions of the same gas with perfectly flat and parallel boundaries separated by a layer of the same gas. Both regions have uniform number densities above the lower plate. The non-equilibrium molecular flow is superimposed on a Maxwell–Boltzmann equilibrium distribution of molecular motions.

can be considered to be constant over a distance of mean free path.

yields the molecular transfer per unit time per unit area (also known as diffusion flux

Combining the above kinetic equation with Fick's first law of diffusion

The corresponding expression obtained from Revised Enskog Theory is a factor that tends to unity in the limit of infinite dilution, which accounts for excluded volume and the variation chemical potentials with density.

Detailed balance

Fluctuation and dissipation

The kinetic theory of gases entails that due to the microscopic reversibility of the gas particles' detailed dynamics, the system must obey the principle of detailed balance. Specifically, the fluctuation-dissipation theorem applies to the Brownian motion (or diffusion) and the drag force, which leads to the Einstein–Smoluchowski equation: where

• is the mass diffusivity;

• is the "mobility", or the ratio of the particle's terminal drift velocity to an applied force, ;

• is the Boltzmann constant;

• is the absolute temperature.

Note that the mobility can be calculated based on the viscosity of the gas; Therefore, the Einstein–Smoluchowski equation also provides a relation between the mass diffusivity and the viscosity of the gas.

Onsager reciprocal relations

The mathematical similarities between the expressions for shear viscosity, thermal conductivity and diffusion coefficient of the ideal (dilute) gas is not a coincidence; It is a direct result of the Onsager reciprocal relations (i.e. the detailed balance of the reversible dynamics of the particles), when applied to the convection (matter flow due to temperature gradient, and heat flow due to pressure gradient) and advection (matter flow due to the velocity of particles, and momentum transfer due to pressure gradient) of the ideal (dilute) gas.

See also

• Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy of equations

• Boltzmann equation

• Chapman–Enskog theory

• Collision theory

• Critical temperature

• Gas laws

• Heat

• Interatomic potential

• Magnetohydrodynamics

• Maxwell–Boltzmann distribution

• Mixmaster universe

• Thermodynamics

• Vicsek model

• Vlasov equation

References

Citations

Sources cited

• de Groot, S. R., W. A. van Leeuwen and Ch. G. van Weert (1980), Relativistic Kinetic Theory, North-Holland, Amsterdam.

• Liboff, R. L. (1990), Kinetic Theory, Prentice-Hall, Englewood Cliffs, N. J.

• (reprinted in his Papers, 3, 167, 183.)

Further reading

• Sydney Chapman and Thomas George Cowling (1939/1970), The Mathematical Theory of Non-uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases, (first edition 1939, second edition 1952), third edition 1970 prepared in co-operation with D. Burnett, Cambridge University Press, London

• Joseph Oakland Hirschfelder, Charles Francis Curtiss, and Robert Byron Bird (1964), Molecular Theory of Gases and Liquids, revised edition (Wiley-Interscience),

• Richard Lawrence Liboff (2003), Kinetic Theory: Classical, Quantum, and Relativistic Descriptions, third edition (Springer),

• Behnam Rahimi and Henning Struchtrup (2016), "Macroscopic and kinetic modelling of rarefied polyatomic gases", Journal of Fluid Mechanics, 806, 437–505,

External links

• Early Theories of Gases

• Thermodynamics - a chapter from an online textbook

• on Project PHYSNET.

• Introduction to the kinetic molecular theory of gases, from The Upper Canada District School Board

• Java animation illustrating the kinetic theory from University of Arkansas

• Flowchart linking together kinetic theory concepts, from HyperPhysics

• Interactive Java Applets allowing high school students to experiment and discover how various factors affect rates of chemical reactions.

• https://www.youtube.com/watch?v=47bF13o8pb8&list=UUXrJjdDeqLgGjJbP1sMnH8A A demonstration apparatus for the thermal agitation in gases.