Kinetic theory of gases

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Kinetic 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. As analyzed by Albert Einstein in 1905, this experimental evidence for the kinetic theory confirmed the existence of atoms and molecules.


The theory for ideal gases makes the following assumptions:

  • These particles have the same mass.
This implies:
1. Relativistic effects are negligible.
  • There are negligible gravitational force on molecules.


An important book on kinetic theory is that by Chapman and Cowling.[1] An important approach to the subject is called Chapman–Enskog theory.[2] There have been many modern developments and there is an alternative approach developed by Grad based on moment expansions.[3] 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.

Equilibrium properties[edit]

Pressure and kinetic energy[edit]



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.




and the force can be written as:


where V = L3 is the volume of the box.

In terms of the kinetic energy of the gas K:

This is a first 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.

Temperature and kinetic energy[edit]

Rewriting the above result for the pressure as , we may combine it with the ideal gas law






where is the Boltzmann constant and the absolute temperature defined by the ideal gas law, to obtain


which leads to simplified expression of the average kinetic energy per molecule,[4]


The kinetic energy of the system is N times that of a molecule, namely . Then the temperature takes the form






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:[5]

Since there are degrees of freedom in a monatomic-gas system with particles, the kinetic energy per degree of freedom per molecule is






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.[why?] 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.


  • per mole: 12.47 J
  • per molecule: 20.7 yJ = 129 μeV.

At standard temperature (273.15 K), we get:

  • per mole: 3406 J
  • per molecule: 5.65 zJ = 35.2 meV.

Collisions with container[edit]


Assuming an ideal gas, a derivation[6] results in an equation for total number of collisions per unit time per area:


Speed of molecules[edit]

From the kinetic energy formula it can be shown that

where v is in m/s, T is in kelvins, and m is the mass of one molecule of gas. The most probable (or mode) speed is 81.6% of the rms speed , and the mean (arithmetic mean, or average) speed is 92.1% of the rms speed (isotropic distribution of speeds).


Transport properties[edit]


Viscosity and kinetic momentum[edit]

In books on elementary kinetic theory[7] one can find results for dilute gas modeling that has widespread use. 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. The lower plate is stationary, and an equal and opposite force must therefore be acting on it to keep it at rest. The molecules in the gas layer have a forward velocity component which increase uniformly with distance above the lower plate. The non-equilibrium flow is superimposed on a Maxwell-Boltzmann equilibrium distribution of molecular motions.

Let be the collision cross section of one molecule colliding with another. The number density is defined as the number of molecules per (extensive) volume . The collision cross section per volume or collision cross section density is , and it is related to the mean free path by

Notice that the unit of the collision cross section per volume is reciprocal of length. The mean free path is the average distance traveled by a molecule, or a number of molecules per volume, before they make their first collision.

Let be the forward velocity of the gas at an imaginary horizontal surface inside the gas layer. On the average, a molecule that crosses the surface makes its last collision before crossing at a distance equal to two-thirds of the mean free path (i.e. ) away from the surface. At this distance above and below the surface, the forward momentum of the molecule is respectively

where m is the molecular mass. The molecular flux includes all molecules arriving at one side of an element of the surface within the gas layer. The incoming molecules are coming from all directions at the one side of the surface and with all speeds. This molecular flux (i.e. the number flux) is related to the average molecular speed by



The defining equation for the (shear) viscosity of the gas is

Combining the above kinetic equation with defining equation for (shear) viscosity by gives the equation for shear viscosity, which is usually denoted when it is a dilute gas:

Combining this equation with the equation for mean free path gives


where is the most probable speed, is the Boltzmann constant. We note that




Local nomenclature list:

  •  : area of moving boundary in Couette flow experiment [m2]
  •  : number concentration or number density [1/m3]
  •  : molar concentration or molar density [mol/cm3]
  •  : kinetic diameter in collision cross section [m]
  •  : force that move a boundary in Couette flow experiment [N]
  •  : Boltzmann constant [JK−1]
  •  : mean free path [m]
  •  : molar mass [g/mol]
  •  : molecular mass [Da]
  •  : Avogadro constant [mol−1]
  •  : pressure [Pa or bar or atm]
  •  : critical pressure [Pa or bar or atm]
  •  : linear momentum in x-direction of a molecule [kgm/s]
  •  : gas constant [JK−1mol−1]
  •  : kinetic radius in collision cross section or hard core molecular radius [m]
  •  : temperature [K]
  •  : critical temperature [K]
  •  : macroscopic fluid velocity in x-direction [m/s]
  •  : average molecular equilibrium speed [m/s]
  •  : most probable molecular equilibrium speed [m/s]
  •  : molar volume [cm3/mol]
  •  : critical molar volume [cm3/mol]
  •  : extensive fluid volume [m3]
  •  : distance from non-moving boundary in direction normal to fluid flow [m]
  •  : viscosity [Pas or μP or cP]
  •  : viscosity of dilute gas [Pas or μP or cP]
  •  : molecular flux across an imaginary surface [m−2s−1]
  •  : collision cross section [m2]
  •  : shear stress [Nm−2]
  •  : dummy


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

Hydrodynamica front cover


Other pioneers of the kinetic theory (which were neglected by their contemporaries) were Mikhail Lomonosov (1747),[10] Georges-Louis Le Sage (ca. 1780, published 1818),[11] John Herapath (1816)[12] and John James Waterston (1843),[13] 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.[14]

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. [15] 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.[16] This was the first-ever statistical law in physics.[17] Maxwell also gave the first mechanical argument that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium.[18] 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."[19] 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)[20] and Marian Smoluchowski's (1906)[21] papers on Brownian motion, which succeeded in making certain accurate quantitative predictions based on the kinetic theory.

See also[edit]


  1. ^ a b Chapman, S., Cowling, T.G. (1939/1970).
  2. ^ Kauzmann, W. (1966). Kinetic Theory of Gasses, W.A. Benjamin, New York, pp. 232–235.
  3. ^ Grad 1949
  5. ^ Configuration integral (statistical mechanics) Archived 2012-04-28 at the Wayback Machine.
  6. ^ Collisions With a Surface.
  7. ^ Sears, F.W.; Salinger, G.L. (1975). "10". Thermodynamics, Kinetic Theory, and Statistical Thermodynamics (3 ed.). Reading, Massachusetts, USA: Addison-Wesley Publishing Company, Inc. pp. 286–291. ISBN 978-0201068948. 
  9. ^ L.I Ponomarev; I.V Kurchatov (1 January 1993). The Quantum Dice. CRC Press. ISBN 978-0-7503-0251-7. 
  10. ^ Lomonosov 1758
  11. ^ Le Sage 1780/1818
  12. ^ Herapath 1816, 1821
  13. ^ Waterston 1843
  14. ^ Krönig 1856
  15. ^ Clausius 1857
  16. ^ See:
  18. ^ Gyenis, Balazs (2017). "Maxwell and the normal distribution: A colored story of probability, independence, and tendency towards equilibrium". Studies in History and Philosophy of Modern Physics. 57: 53–65. arXiv:1702.01411Freely accessible. Bibcode:2017SHPMP..57...53G. doi:10.1016/j.shpsb.2017.01.001. 
  19. ^ Maxwell 1875
  20. ^ Einstein 1905
  21. ^ Smoluchowski 1906


  • Lomonosov, M. (1970) [1758], "On the Relation of the Amount of Material and Weight", in Henry M. Leicester, Mikhail Vasil'evich Lomonosov on the Corpuscular Theory, Cambridge: Harvard University Press, pp. 224–233 
  • Mahon, Basil (2003), The Man Who Changed Everything – the Life of James Clerk Maxwell, Hoboken, New Jersey: Wiley, ISBN 0-470-86171-1 
  • Waterston, John James (1843), Thoughts on the Mental Functions  (reprinted in his Papers, 3, 167, 183.)
  • Williams, M. M. R. (1971). Mathematical Methods in Particle Transport Theory. Butterworths, London. ISBN 9780408700696. 

Further reading[edit]

  • B. Rahimi and H. Struchtrup, Macroscopic and kinetic modelling of rarefied polyatomic gases, Journal of Fluid Mechanics, 806, 437–505, 2016. DOI:

External links[edit]