The kinetic theory of gases describes a gas as a large number of submicroscopic particles (atoms or molecules), all of which are in constant, rapid, random motion. The randomness arises from the particles' many collisions with each other and with the walls of the container.

Kinetic theory of gases explains the macroscopic properties of gases, such as pressure, temperature, viscosity, thermal conductivity, and volume, by considering their molecular composition and motion. The theory posits that gas pressure results from particles' collisions with the walls of a container at different velocities.

Kinetic molecular theory defines temperature in its own way, in contrast with the thermodynamic definition.^[1]

Under an optical microscope, the molecules making up a liquid are too small to be visible. However, the jittery motion of pollen grains or dust particles in liquid are visible. Known as Brownian motion, the motion of the pollen or dust results from their collisions with the liquid's molecules.

In approximately 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.^[2] This Epicurean atomistic point of view was rarely considered in the subsequent centuries, when Aristotlean ideas were dominant.

In 1738 Daniel Bernoulli published Hydrodynamica, which laid the basis for the kinetic theory of gases. In this work, Bernoulli posited the argument, still used to this day, that gases consist of great numbers of molecules moving in all directions, that their impact on a surface causes the gas pressure that we feel, and that what we experience as heat is simply the kinetic energy of their motion. Bernoulli also surmised that temperature was the effect of the kinetic energy of the molecules, and thus correlated with the ideal gas law.^[3]

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.^{[4] []} A competing theory favored by Newton was the replussion theory, in which heat was a calorific fluid that repulsed molecules in proportion its quantity (i.e. heat) and the inverse square of the distances between molecules.^[3]

Other pioneers of the kinetic theory (which were neglected by their contemporaries) were Mikhail Lomonosov (1747),^[5] Georges-Louis Le Sage (ca. 1780, published 1818),^[6] John Herapath (1816)^[7] and John James Waterston (1843),^[8] which connected their research with the development of mechanical explanations of gravitation. In 1856 August Krönig (probably after reading a paper of Waterston) created a simple gas-kinetic model, which only considered the translational motion of the particles.^[9]

In 1857 Rudolf Clausius, according to his own words independently of Krönig, developed a similar, but much 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. ^[10] In 1859, after reading a paper on the diffusion of molecules by Rudolf 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.^[11] This was the first-ever statistical law in physics.^[12] Maxwell also gave the first mechanical argument that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium.^[13] 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."^[14] In 1871, Ludwig Boltzmann generalized Maxwell's achievement and formulated the Maxwell–Boltzmann distribution. Also the logarithmic connection between entropy and probability was first stated by him.

In the beginning of the twentieth century, however, atoms were considered by many physicists to be purely hypothetical constructs, rather than real objects. An important turning point was Albert Einstein's (1905)^[15] and Marian Smoluchowski's (1906)^[16] papers on Brownian motion, which succeeded in making certain accurate quantitative predictions based on the kinetic theory.

The theory for ideal gases makes the following assumptions:

More modern developments relax these assumptions and are based on the Boltzmann equation. These can accurately describe the properties of dense gases, because they include the volume of the molecules. The necessary assumptions are the absence of quantum effects, molecular chaos and small gradients in bulk properties. Expansions to higher orders in the density are known as virial expansions.

An important book on kinetic theory is that by Chapman and Cowling.^[1] An important approach to the subject is called Chapman–Enskog theory.^[17] There have been many modern developments and there is an alternative approach developed by Grad based on moment expansions.^[18] In the other limit, for extremely rarefied gases, the gradients in bulk properties are not small compared to the mean free paths. This is known as the Knudsen regime and expansions can be performed in the Knudsen number.

In kinetic model of gases, the pressure is equal to the force exerted by the atoms hitting and rebounding from a unit area of the gas container surface. Consider a gas of N molecules, each of mass m, enclosed in a cube of volume V = L3. When a gas molecule collides with the wall of the container perpendicular to the x axis and bounces off in the opposite direction with the same speed (an elastic collision), the change in momentum is given by:

where p is the momentum, i and f indicate initial and final momentum (before and after collision), x indicates that only the x direction is being considered, and v is the speed of the particle (which is the same before and after the collision).

The particle impacts one specific side wall once every

where L is the distance between opposite walls.

The force due to this particle is

The total force on the wall is

where the bar denotes an average over the N particles.

Since the motion of the particles is random and there is no bias applied in any direction, the average squared speed in each direction is identical:

By Pythagorean theorem in three dimensions the total squared speed v is given by

Therefore:

and the force can be written as:

This force is exerted on an area L2. Therefore, the pressure of the gas is

where V = L3 is the volume of the box.

In terms of the kinetic energy of the gas K:

which leads to simplified expression of the average kinetic energy per molecule,^[19]

which becomes

Eq.(3) is one important result of the kinetic theory: The average molecular kinetic energy is proportional to the ideal gas law's absolute temperature. From Eq.(1) and Eq.(3), we have

Thus, the product of pressure and volume per mole is proportional to the average (translational) molecular kinetic energy.

Eq.(1) and Eq.(4) are called the "classical results", which could also be derived from statistical mechanics; for more details, see:^[20]

In the kinetic energy per degree of freedom, the constant of proportionality of temperature is 1/2 times Boltzmann constant or R/2 per mole. In addition to this, the temperature will decrease when the pressure drops to a certain point. This result is related to the equipartition theorem.

As noted in the article on heat capacity, diatomic gases should have 7 degrees of freedom, but the lighter diatomic gases act as if they have only 5. Monatomic gases have 3 degrees of freedom.

Thus the kinetic energy per kelvin (monatomic ideal gas) is 3 [R/2] = 3R/2:

At standard temperature (273.15 K), we get:

The total number and velocity distribution of particles hitting the container wall can be calculated^[21] based on naive kinetic theory, and the result can be used for analyzing effusion into vacuum:

Note that only the particles within the following constraint are actually heading to hit the wall:

Integrating over all appropriate velocities within the constraint yields the number of atomic or molecular collisions with a wall of a container per unit area per unit time:

The last line makes use of ideal gas law. This quantity is also known as the impingement rate in vacuum physics.

The velocity distribution of particles hitting this small area is:

In Cartesian coordinates, this is:

From the above distribution, the average velocity of these impinging particles is:

From the kinetic energy formula it can be shown that

See:

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 mass diffusivity, viscosity and thermal conductivity.

The diffusion constant is related to viscosity by the Einstein relation (kinetic theory).

The net rate of momentum per unit area that is transported across the imaginary surface is thus

Combining this equation with the equation for mean free path gives

From statistical thermodynamics for gases we have equations relating average molecular speed to most likely speed and further to temperature. These statistical results gives the average (equilibrium) molecular speed as

and insert the velocity in the viscosity equation above. This gives the well known equation for shear viscosity for dilute gases:

Local nomenclature list: