Kinetic theory of gases

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 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 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.

Assumptions

The theory for ideal gases makes the following assumptions:

• The gas consists of very small particles known as molecules. This smallness of their size is such that the total volume of the individual gas molecules added up is negligible compared to the volume of the smallest open ball containing all the molecules. This is equivalent to stating that the average distance separating the gas particles is large compared to their size.
• These particles have the same mass.
• The number of molecules is so large that statistical treatment can be applied.
• The rapidly moving particles constantly collide among themselves and with the walls of the container. All these collisions are perfectly elastic. This means the molecules are considered to be perfectly spherical in shape and elastic in nature.
• Except during collisions, the interactions among molecules are negligible. (That is, they exert no forces on one another.)
This implies:
1. Relativistic effects are negligible.
2. Quantum-mechanical effects are negligible. This means that the inter-particle distance is much larger than the thermal de Broglie wavelength and the molecules are treated as classical objects.
3. Because of the above two, their dynamics can be treated classically. This means that the equations of motion of the molecules are time-reversible.
• The average kinetic energy of the gas particles depends only on the absolute temperature of the system. The kinetic theory has its own definition of temperature, not identical with the thermodynamic definition.
• The elapsed time of a collision between a molecule and the container's wall is negligible when compared to the time between successive collisions.
• There are negligible gravitational force on molecules.

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.[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

Pressure and kinetic energy

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:

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

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

(1)

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

(2)

which becomes

(3)

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

(4)

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

(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. 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:

• 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

One can calculate the number of atomic or molecular collisions with a wall of a container per unit area per unit time.

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

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

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 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).

See:

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 considering what are known as "transport properties", such a viscosity and thermal conductivity.

Viscosity and kinetic momentum

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

Notice that the forward velocity gradient can be considered to be constant over a distance of mean free path. Next we multiply by the total flux to get the change of momentum per unit time and per unit area, that is carried by the molecules crossing from either above or below the surface area. This gives the equation

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

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

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

where is the most probable speed, is the Boltzmann constant. 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 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 is called collision cross section radius or kinetic radius, and the diameter 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.

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]
•  : 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]
•  : macroscopic fluid velocity in x-direction on the imaginary surface [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

History

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.

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. 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.[9]:36–37

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.

Notes

1. Chapman, S., Cowling, T.G. (1939/1970).
2. Kauzmann, W. (1966). Kinetic Theory of Gasses, W.A. Benjamin, New York, pp. 232–235.
4. The average kinetic energy of a fluid is proportional to the root mean-square velocity, which always exceeds the mean velocity - Kinetic Molecular Theory
5. Configuration integral (statistical mechanics) Archived 2012-04-28 at the Wayback Machine.
6. 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.
7. Maxwell, J. C. (1867). "On the Dynamical Theory of Gases". Philosophical Transactions of the Royal Society of London. 157: 49. doi:10.1098/rstl.1867.0004.
8. L.I Ponomarev; I.V Kurchatov (1 January 1993). The Quantum Dice. CRC Press. ISBN 978-0-7503-0251-7.
9. Lomonosov 1758
10. Le Sage 1780/1818
11. Herapath 1816, 1821
12. Waterston 1843
13. Krönig 1856
14. Clausius 1857
15. See:
16. Mahon, Basil (2003). The Man Who Changed Everything – the Life of James Clerk Maxwell. Hoboken, NJ: Wiley. ISBN 0-470-86171-1. OCLC 52358254.
17. 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:. Bibcode:2017SHPMP..57...53G. doi:10.1016/j.shpsb.2017.01.001.
18. Maxwell 1875
19. Einstein 1905
20. Smoluchowski 1906

References

• 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, 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.
• Sydney Chapman and T. G. 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.
• J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird (1964). Molecular Theory of Gases and Liquids, second edition (Wiley).
• R. L. Liboff (2003). Kinetic Theory: Classical, Quantum, and Relativistic Descriptions, third edition (Springer).
• B. Rahimi and H. Struchtrup, Macroscopic and kinetic modelling of rarefied polyatomic gases, Journal of Fluid Mechanics, 806, 437–505, 2016. DOI: https://dx.doi.org/10.1017/jfm.2016.604