Kinetic theory of gases

The kinetic theory of gases is a simple, historically significant classical model of the thermodynamic behavior of gases, with which many principal concepts of thermodynamics were established. The model describes a gas as a large number of identical submicroscopic particles (atoms or molecules), all of which are in constant, rapid, random motion. Their size is assumed to be much smaller than the average distance between the particles. The particles undergo random elastic collisions between themselves and with the enclosing walls of the container. The basic version of the model describes the ideal gas, and considers no other interactions between the particles.

The kinetic theory of gases explains 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 model also accounts for related phenomena, such as Brownian motion.

History[]

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.^[1] 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, 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.^[2]:36–37

Other pioneers of the kinetic theory, whose work was also largely neglected by their contemporaries, were Mikhail Lomonosov (1747),^[3] Georges-Louis Le Sage (ca. 1780, published 1818),^[4] John Herapath (1816)^[5] and John James Waterston (1843),^[6] 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.^[7]

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.^[8] 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.^[9] This was the first-ever statistical law in physics.^[10] Maxwell also gave the first mechanical argument that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium.^[11] 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."^[12] 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, 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)^[13] and Marian Smoluchowski's (1906)^[14] papers on Brownian motion, which succeeded in making certain accurate quantitative predictions based on the kinetic theory.

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 of 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. All these collisions are perfectly elastic, which means the molecules are perfect hard spheres.
• 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 and so often they are neglected as a simplifying assumption in derivations (see below).^[15]

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 particles as well as contributions from intermolecular and intramolecular forces as well as quantized molecular rotations, quantum rotational-vibrational symmetry effects, and electronic excitation.^[16]

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 atoms hitting and rebounding from the gas container's surface. Consider a gas of a large number N of 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 ${\displaystyle v_{x}}$ is the speed of the particle in the x direction (which is the same before and after the collision).

The particle impacts one specific side wall once during the time interval ${\displaystyle \Delta t}$

where L is the distance between opposite walls.

The force of this particle's collision with the wall is

The total force on the wall due to collisions by molecules impacting the walls with a range of possible values of ${\displaystyle v_{x}}$ is

where the bar denotes an average over the possible velocities of 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 the Pythagorean theorem, in three dimensions the average squared speed ${\displaystyle v^{2}}$ is given by

Therefore

and

and so the force can be written as

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

where V = L³ is the volume of the box.

In terms of the translational kinetic energy K of the gas, since

we have

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.

Temperature and kinetic energy[]

Rewriting the above result for the pressure as ${\textstyle PV={\frac {Nm{\overline {v^{2}}}}{3}}}$, we may combine it with the ideal gas law

${\displaystyle PV=Nk_{\mathrm {B} }T,}$

(1)

where ${\displaystyle k_{\mathrm {B} }}$ is the Boltzmann constant and ${\displaystyle T}$ the absolute temperature defined by the ideal gas law, to obtain

which leads to a simplified expression of the average kinetic energy per molecule,^[17] The kinetic energy of the system is ${\displaystyle N}$ times that of a molecule, namely ${\textstyle K={\frac {1}{2}}Nm{\overline {v^{2}}}}$. Then the temperature ${\displaystyle T}$ takes the form

${\displaystyle T={m{\overline {v^{2}}} \over 3k_{\mathrm {B} }}}$

(2)

which becomes

${\displaystyle T={\frac {2}{3}}{\frac {K}{Nk_{\mathrm {B} }}}.}$

(3)

Equation (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 equations (1) and (3), we have

${\displaystyle PV={\frac {2}{3}}K.}$

(4)

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

Equations (1) and (4) are called the "classical results", which could also be derived from statistical mechanics; for more details, see:^[18]

Since there are ${\displaystyle 3N}$ degrees of freedom in a monatomic-gas system with ${\displaystyle N}$ particles, the kinetic energy per degree of freedom per molecule is

${\displaystyle {\frac {K}{3N}}={\frac {k_{\mathrm {B} }T}{2}}}$

(5)

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. This result is related to the equipartition theorem.

Thus the kinetic energy per Kelvin of one mole of (monatomic ideal gas) is 3 [R/2] = 3R/2. Thus the kinetic energy per Kelvin can be calculated easily:

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

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

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

Although monatomic gases have 3 (translational) degrees of freedom per atom, diatomic gases should have 6 degrees of freedom per molecule (3 translations, two rotations, and one vibration). However, the lighter diatomic gases (such as diatomic oxygen) may act as if they have only 5 due to the strongly quantum-mechanical nature of their vibrations and the large gaps between successive vibrational energy levels. Quantum statistical mechanics is needed to accurately compute these contributions. ^[19]

Collisions with container[]

The velocity distribution of particles hitting the container wall can be calculated^[20] based on naive kinetic theory, and the result can be used for analyzing effusive flow rates.

Assume that in the container, the number density (number per unit volume) is ${\displaystyle n}$ and that the particles obey Maxwell's velocity distribution:

Then the number of particles hitting the area ${\displaystyle dA}$ with speed ${\displaystyle v}$ at angle ${\displaystyle \theta }$ from the normal, in time interval ${\displaystyle dt}$ is:

Integrating this over all appropriate velocities within the constraint ${\displaystyle v>0,0<\theta <\pi /2,0<\phi <2\pi }$ yields the number of atomic or molecular collisions with a wall of a container per unit area per unit time:

This quantity is also known as the "impingement rate" in vacuum physics.

If this small area ${\displaystyle A}$ is punched to become a small hole, the effusive flow rate will be:

Combined with the ideal gas law, this yields

The velocity distribution of particles hitting this small area is

with the constraint ${\textstyle v>0,\,0<\theta <{\frac {\pi }{2}},\,0<\phi <2\pi }$. The constant (const.) can be determined by the normalization condition to be ${\textstyle {\frac {1}{2\pi }}\left({\frac {m}{k_{\mathrm {B} }T}}\right)^{2}}$.

Speed of molecules[]

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 ${\displaystyle v_{\text{p}}}$ is 81.6% of the root-mean-square speed ${\displaystyle v_{\text{rms}}}$, and the mean (arithmetic mean, or average) speed ${\displaystyle {\bar {v}}}$ is 92.1% of the rms speed (isotropic distribution of speeds).

See:

• Average,
• Root-mean-square speed
• Arithmetic mean
• Mean
• Mode (statistics)

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 and mass diffusivity.

Viscosity and kinetic momentum[]

In books on elementary kinetic theory^[21] 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. 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 ${\displaystyle u}$ which increase uniformly with distance ${\displaystyle y}$ above the lower plate. The non-equilibrium flow is superimposed on a Maxwell-Boltzmann equilibrium distribution of molecular motions.

Let ${\displaystyle \sigma }$ be the collision cross section of one molecule colliding with another. The number density ${\displaystyle n}$ is defined as the number of molecules per (extensive) volume ${\displaystyle n=N/V}$. The collision cross section per volume or collision cross section density is ${\displaystyle n\sigma }$, and it is related to the mean free path ${\displaystyle l}$ by

Notice that the unit of the collision cross section per volume ${\displaystyle n\sigma }$ 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 ${\displaystyle u_{0}}$ be the forward velocity of the gas at an imaginary horizontal surface inside the gas layer. The number of molecules arriving at an area ${\displaystyle dA}$ on one side of the gas layer, with speed ${\displaystyle v}$ at angle ${\displaystyle \theta }$ from the normal, in time interval ${\displaystyle dt}$ is

These molecules made their last collision at a distance ${\displaystyle l\cos \theta }$ above and below the gas layer, and each will contribute a forward momentum of

where plus sign applies to molecules from above, and minus sign below. Note that the forward velocity gradient ${\displaystyle du/dy}$ can be considered to be constant over a distance of mean free path.

Integrating over all appropriate velocities within the constraint

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

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

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

gives the equation for shear viscosity, which is usually denoted ${\displaystyle \eta _{0}}$ when it is a dilute gas:

Combining this equation with the equation for mean free path gives

Maxwell-Boltzmann distribution gives the average (equilibrium) molecular speed as

where ${\displaystyle v_{p}}$ is the most probable speed. We note that

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

and ${\displaystyle M}$ is the molar mass. The equation above presupposes that the gas density is low (i.e. the pressure is low). This implies that the kinetic translational energy dominates over rotational and vibrational molecule energies. The viscosity equation further presupposes that there is only one type of gas molecules, and that the gas molecules are perfect elastic and hard core particles of spherical shape. This assumption of elastic, hard core spherical molecules, like billiard balls, implies that the collision cross section of one molecule can be estimated by

The radius ${\displaystyle r}$ is called collision cross section radius or kinetic radius, and the diameter ${\displaystyle d}$ 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 is then appropriate to use as estimate for the kinetic radius.

Thermal conductivity and heat flux[]

Following a similar logic as above, one can derive the kinetic model for thermal conductivity^[21] 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. The upper plate has a higher temperature than the lower plate. The molecules in the gas layer have a molecular kinetic energy ${\displaystyle \varepsilon }$ which increases uniformly with distance ${\displaystyle y}$ above the lower plate. The non-equilibrium energy flow is superimposed on a Maxwell-Boltzmann equilibrium distribution of molecular motions.

Let ${\displaystyle \varepsilon _{0}}$ be the molecular kinetic energy of the gas at an imaginary horizontal surface inside the gas layer. The number of molecules arriving at an area ${\displaystyle dA}$ on one side of the gas layer, with speed ${\displaystyle v}$ at angle ${\displaystyle \theta }$ from the normal, in time interval ${\displaystyle dt}$ is

These molecules made their last collision at a distance ${\displaystyle l\cos \theta }$ above and below the gas layer, and each will contribute a molecular kinetic energy of

where ${\displaystyle c_{v}}$ is the specific heat capacity. Again, plus sign applies to molecules from above, and minus sign below. Note that the temperature gradient ${\displaystyle dT/dy}$ can be considered to be constant over a distance of mean free path.

Integrating over all appropriate velocities within the constraint

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

Note that the energy transfer from above is in the ${\displaystyle -y}$ direction, and therefore the overall minus sign in the equation. The net heat flux across the imaginary surface is thus

Combining the above kinetic equation with Fourier's law

gives the equation for thermal conductivity, which is usually denoted ${\displaystyle \kappa _{0}}$ when it is a dilute gas:

Diffusion Coefficient and diffusion flux[]

Following a similar logic as above, one can derive the kinetic model for mass diffusivity^[21] 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, but the upper region has a higher number density than the lower region. In the steady state, the number density at any point is constant (that is, independent of time). However, the number density ${\displaystyle n}$ in the layer increases uniformly with distance ${\displaystyle y}$ above the lower plate. The non-equilibrium molecular flow is superimposed on a Maxwell-Boltzmann equilibrium distribution of molecular motions.

Let ${\displaystyle n_{0}}$ be the number density of the gas at an imaginary horizontal surface inside the layer. The number of molecules arriving at an area ${\displaystyle dA}$ on one side of the gas layer, with speed ${\displaystyle v}$ at angle ${\displaystyle \theta }$ from the normal, in time interval ${\displaystyle dt}$ is

These molecules made their last collision at a distance ${\displaystyle l\cos \theta }$ above and below the gas layer, where the local number density is

Again, plus sign applies to molecules from above, and minus sign below. Note that the number density gradient ${\displaystyle dn/dy}$ can be considered to be constant over a distance of mean free path.

Integrating over all appropriate velocities within the constraint

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

Note that the molecular transfer from above is in the ${\displaystyle -y}$ direction, and therefore the overall minus sign in the equation. The net diffusion flux across the imaginary surface is thus

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

gives the equation for mass diffusivity, which is usually denoted ${\displaystyle D_{0}}$ when it is a dilute gas:

See also[]

Statistical mechanics
Thermodynamics Kinetic theory
show Particle statistics Spin–statistics theorem Identical particles Maxwell–Boltzmann Bose–Einstein Fermi–Dirac Parastatistics Anyonic statistics Braid statistics
show Thermodynamic ensembles NVE Microcanonical NVT Canonical µVT Grand canonical NPH Isoenthalpic–isobaric NPT Isothermal–isobaric
show Models Debye Einstein Ising Potts
show Potentials Internal energy Enthalpy Helmholtz free energy Gibbs free energy Grand potential / Landau free energy
show Scientists Maxwell Boltzmann Gibbs Einstein Ehrenfest von Neumann Tolman Debye Fermi Bose
v t

• Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy of equations
• Boltzmann equation
• Collision theory
• Critical temperature
• Gas laws
• Heat
• Interatomic potential
• Magnetohydrodynamics
• Maxwell–Boltzmann distribution
• Mixmaster dynamics
• Thermodynamics
• Vicsek model
• Vlasov equation

Notes[]

References[]

• Clausius, R. (1857), "Ueber die Art der Bewegung, welche wir Wärme nennen", Annalen der Physik, 176 (3): 353–379, Bibcode:1857AnP...176..353C, doi:10.1002/andp.18571760302

• de Groot, S. R., W. A. van Leeuwen and Ch. G. van Weert (1980), Relativistic Kinetic Theory, North-Holland, Amsterdam.
• Einstein, A. (1905), "Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen" (PDF), Annalen der Physik, 17 (8): 549–560, Bibcode:1905AnP...322..549E, doi:10.1002/andp.19053220806

• Grad, Harold (1949), "On the Kinetic Theory of Rarefied Gases.", Communications on Pure and Applied Mathematics, 2 (4): 331–407, doi:10.1002/cpa.3160020403

• Herapath, J. (1816), "On the physical properties of gases", Annals of Philosophy, Robert Baldwin: 56–60

• Herapath, J. (1821), "On the Causes, Laws and Phenomena of Heat, Gases, Gravitation", Annals of Philosophy, Baldwin, Cradock, and Joy, 9: 273–293

• Krönig, A. (1856), "Grundzüge einer Theorie der Gase", Annalen der Physik, 99 (10): 315–322, Bibcode:1856AnP...175..315K, doi:10.1002/andp.18561751008

• Le Sage, G.-L. (1818), "Physique Mécanique des Georges-Louis Le Sage", in Prévost, Pierre (ed.), Deux Traites de Physique Mécanique, Geneva & Paris: J.J. Paschoud, pp. 1–186

• Liboff, R. L. (1990), Kinetic Theory, Prentice-Hall, Englewood Cliffs, N. J.
• Lomonosov, M. (1970) [1758], "On the Relation of the Amount of Material and Weight", in Henry M. Leicester (ed.), 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

• Maxwell, James Clerk (1873), "Molecules", Nature, 417 (6892): 903, Bibcode:2002Natur.417..903M, doi:10.1038/417903a, PMID 12087385, S2CID 4417753, archived from the original (– ^{Scholar search}) on February 9, 2007
• Smoluchowski, M. (1906), "Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen", Annalen der Physik, 21 (14): 756–780, Bibcode:1906AnP...326..756V, doi:10.1002/andp.19063261405

• 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[]

• 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), ISBN 978-0471400653
• Richard Lawrence Liboff (2003), Kinetic Theory: Classical, Quantum, and Relativistic Descriptions, third edition (Springer), ISBN 978-0-387-21775-8
• and Henning Struchtrup (2016), "Macroscopic and kinetic modelling of rarefied polyatomic gases", Journal of Fluid Mechanics, 806, 437–505, DOI 10.1017/jfm.2016.604

External links[]

• Early Theories of Gases
• Thermodynamics - a chapter from an online textbook
• Temperature and Pressure of an Ideal Gas: The Equation of State 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.