気体分子運動論

WordNet

attack with gas; subject to gas fumes; "The despot gassed the rebellious tribes"
the state of matter distinguished from the solid and liquid states by: relatively low density and viscosity; relatively great expansion and contraction with changes in pressure and temperature; the ability to diffuse readily; and the spontaneous tendency to become distributed uniformly throughout any container (同)gaseous state
a fluid in the gaseous state having neither independent shape nor volume and being able to expand indefinitely
a belief that can guide behavior; "the architect has a theory that more is less"; "they killed him on the theory that dead men tell no tales"
a well-substantiated explanation of some aspect of the natural world; an organized system of accepted knowledge that applies in a variety of circumstances to explain a specific set of phenomena; "theories can incorporate facts and laws and tested hypothes
relating to the motion of material bodies and the forces associated therewith; "kinetic energy"
characterized by motion; "modern dance has been called kinetic pantomime"

PrepTutorEJDIC

〈U〉〈C〉『気体』,『ガス』 / 〈U〉(灯用・燃料用)ガス / 〈U〉《米話》=gasoline / 〈U〉(麻酔用の)亜酸化窒素,笑気(laughing gas);毒ガス(mustard gas, tear gasなど) / 〈U〉《話》むだ話,ばか話 / 〈C〉《単数形で》とても楽しいこと / 〈敵など〉‘を'毒ガスで攻撃する / 《米話》〈車〉‘に'ガソリンを入れる《+『up』+『名,』+『名』+『up』》 / 長々とむだ話をする
《所有・所属》…『の』,…のものである,…に属する・《材料・要素》…『でできた』,から成る・《部分》…『の』[『中の』] ・《数量・単位・種類を表す名詞に付いて》…の・《原因・動機》…『で』,のために(because of) ・《主格関係》…『の』,による,によって・《目的格関係》…『を』,の・《同格関係》…『という』・《関係・関連》…『についての』[『の』],の点で・《抽象名詞などと共に》…の[性質をもつ] ・《『It is』+『形』+『of』+『名』+『to』 doの形で,ofの後の名詞を意味上の主語として》・《分離》…『から』・《起原・出所》…『から』[『の』](out of) ・《『名』+『of』+『a』(『an』)+『名』の形で》…のような・《『名』+『of』+『mine』(『yours, his』など独立所有格)の形で》…の…・《時》(1)《副詞句を作って》…に《形容詞句を作って》…の・《時刻》《米》…前(to,《米》before)
〈C〉(…の)(学問的な)『理論』,学説《+of+名》 / 〈C〉〈U〉(…の実地に対する)『理論』《+of+名》 / 〈C〉(…という)『推測』,憶測;見解,孝え《+that節》
運動の / 動きによる,動的な
動力学

Wikipedia preview

出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2015/07/27 21:10:58」(JST)

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The temperature of an ideal monatomic gas is proportional to the average kinetic energy of its atoms. The size of helium atoms relative to their spacing is shown to scale under 1950 atmospheres of pressure. The atoms have a certain, average speed, slowed down here two trillion fold from room temperature.

This article applies to gases; see also Kinetic theory of solids

The kinetic theory of gases describes a gas as a large number of small particles (atoms or molecules), all of which are in constant, random motion. The rapidly moving particles constantly collide with each other and with the walls of the container. Kinetic theory explains 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 is due to the impacts, on the walls of a container, of molecules or atoms moving at different velocities.

Kinetic theory defines temperature in its own way, not identical with the thermodynamic definition.^[1]

While the particles making up a gas are too small to be visible, the jittering motion of pollen grains or dust particles which can be seen under a microscope, known as Brownian motion, results directly from collisions between the particles and gas molecules. As analyzed by Albert Einstein in 1905, this experimental evidence for kinetic theory is generally seen as having 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.
These molecules are in constant, random, and rapid motion.
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, 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 time during collision of molecule with the container's wall is negligible as compared to the time between successive collisions.
Because they have mass, the gas molecules will be affected by gravity.

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.

Properties

Pressure and kinetic energy

Pressure is explained by kinetic theory as arising from the force exerted by molecules or atoms impacting on the walls of a container. Consider a gas of N molecules, each of mass m, enclosed in a cuboidal container of volume V=L³. When a gas molecule collides with the wall of the container perpendicular to the x coordinate axis and bounces off in the opposite direction with the same speed (an elastic collision), then the momentum lost by the particle and gained by the wall is:

where v_x is the x-component of the initial velocity of the particle.

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 assumption is that the particles move in random directions, we will have to conclude that if we divide the velocity vectors of all particles in three mutually perpendicular directions, the average value along each direction must be equal (though their proportions are arbitrary for individual particles). That is,

.

We can rewrite the force 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. The ratio n=N/V is the number density of the gas (the mass density ρ=nm is less convenient for theoretical derivations on atomic level). Using n, we can rewrite the pressure as

This is a first non-trivial result of the kinetic theory because it relates pressure, a macroscopic property, to the average (translational) kinetic energy per molecule 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 the expression of the average kinetic energy of a molecule,

.

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

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

(4)

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

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

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

$\displaystyle \frac {K} {3 N} = \frac {k_B T} {2}$

(5)

In the kinetic energy per degree of freedom, the constant of proportionality of temperature is 1/2 times Boltzmann constant. 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 gases act as if they have only 5.

Thus the kinetic energy per kelvin (monatomic ideal gas) is:

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^[5] 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

with v in m/s, T in kelvins, and m is the molecular mass (kg). The most probable speed is 81.6% of the rms speed, and the mean speeds 92.1% (isotropic distribution of speeds).

See: Root-mean-square speed

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.

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.^[6] This Epicurean atomistic point of view was rarely considered in the subsequent centuries, when Aristotlean ideas were dominant.

Hydrodynamica front cover

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.^[7]^:36–37

Other pioneers of the kinetic theory (which were neglected by their contemporaries) were Mikhail Lomonosov (1747),^[8] Georges-Louis Le Sage (ca. 1780, published 1818),^[9] John Herapath (1816)^[10] and John James Waterston (1843),^[11] 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.^[12]

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. ^[13] In 1859, after reading a paper by Clausius, 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.^[14] 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."^[15] 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)^[16] and Marian Smoluchowski's (1906)^[17] papers on Brownian motion, which succeeded in making certain accurate quantitative predictions based on the kinetic theory.

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, 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, 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" (– ^{Scholar search}), Nature 417 (6892): 903, Bibcode:2002Natur.417..903M, doi:10.1038/417903a, PMID 12087385 ^{[dead link]}

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.

Endnotes

^ ^a ^b Chapman, S., Cowling, T.G. (1939/1970).
^ Kauzmann, W. (1966). Kinetic Theory of Gases, W.A. Benjamin, New York, pp. 232–235.
^ Grad 1949
^ Configuration integral (statistical mechanics)
^ Collisions With a Surface
^ 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. edit
^ L.I Ponomarev; I.V Kurchatov (1 January 1993). The Quantum Dice. CRC Press. ISBN 978-0-7503-0251-7.
^ Lomonosov 1758
^ Le Sage 1780/1818
^ Herapath 1816, 1821
^ Waterston 1843
^ Krönig 1856
^ Clausius 1857
^ Mahon 2003
^ Maxwell 1875
^ Einstein 1905
^ Smoluchowski 1906

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.

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Thermochemical Properties and Bond Dissociation Enthalpies of 3 to 5 Member Ring Cyclic Ether Hydroperoxides, Alcohols and Peroxy Radicals: Cyclic Ether Radical + 3O2 Reaction Thermochemistry.

Auzmendi-Murua I, Bozzelli JW.AbstractThe formation of cyclic ethers is a major product in the oxidation of hydrocarbons, and the oxidation of biomass derived alcohols. Cyclic ethers are formed in the initial reactions of alkyl radicals with dioxygen in combustion and pre-combustion processes that occur at moderate temperatures. They represent a significant part of the oxygenated pollutants found in the exhaust gases of engines. Cyclic ethers can also be formed from atmospheric reactions of olefins. Additionally, cyclic ethers have been linked to the formation of the secondary organic aerosol (SOA) in the atmosphere. In combustion and thermal oxidation processes these cyclic ethers will form radicals that react with 3O2 to form peroxy radicals. Density Functional theory and higher level ab initio calculations are used to calculate thermochemical properties and bond dissociation enthalpies of 3 to 5 member ring cyclic ethers (oxirane, yC2O, oxetane, yC3O, and oxolane, yC4O), corresponding hydroperoxides, alcohols, hydroperoxy alkyl and alkyl radicals which are formed in these oxidation reaction systems. Trends in carbon - hydrogen bond energies for the ring and hydroperoxide groups relative to ring size and to distance from the ether group are determined. Bond energies are calculated for use in understanding effects of the ether oxygen in the cyclic ethers, their stability and kinetic properties. Geometries, vibration frequencies and enthalpies of formation, ΔHfº298, are calculated at the B3LYP/6-31G(d,p), B3LYP/6-31G(2d,2p), the composite CBS-QB3, and G3MP2B3 methods. Entropy and heat capacities, S (T) and Cp(T) (5K ≤ T ≤ 5000), are determined using geometric parameters and frequencies from the B3LYP/6-31G(d,p) calculations. The strong effects of ring strain on the bond energies in these peroxy systems are also of fundamental interest. Oxetane and oxolane exhibit a significant stabilization, 10 kcal mol-1, lower ∆fHo(298) when an oxygen group is on the ether carbon relative to the isomer with the oxygen group on a secondary carbon. Relative to alkane systems the ether oxygen decreases bond dissociation energies (BDE) on carbon sites adjacent to the ether by ~5 kcal mol-1, and increases BDEs on non-ether carbons ~1 kcal mol-1. The cyclic structures have significant effects on the C-H, CO-OH, COO-H and CO-H bond dissociation enthalpies. These values can be used to help calibrate calculations of larger more complex bicyclic and tri-cyclic hydrocarbon and ether species. Beta scission reactions from the three and four member ring radicals to form ring opened carbonyl and olefin radicals are all exothermic and endothermicity of the five member rings to ring opening ranges from 1.5 to 21.5 kcal mol-1.
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Tunable Hydrogen Separation in Porous Graphene Membrane: First-Principle and Molecular Dynamic Simulation.

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Japanese Journal

Parabolic temperature profile and second-order temperature jump of a slightly rarefied gas in an unsteady two-surface problem

Takata Shigeru,Aoki Kazuo,Hattori Masanari,Hadjiconstantinou Nicolas G.
PHYSICS OF
… The behavior of a slightly rarefied monatomic gas between two parallel plates whose temperature grows slowly and linearly in time is investigated on the basis of the kinetic theory of gases. … This problem is shown to be equivalent to a boundary-value problem of the steady linearized Boltzmann equation describing a rarefied gas subject to constant volumetric heating. …
NAID 120004920388

Numerical demonstration of the reciprocity among elemental relaxation and driven-flow problems for a rarefied gas in a channel

Takata Shigeru,Oishi Masashi
PHYSICS OF
… Relaxations from a uniform mass/heat flow and flows driven by an external force/temperature-gradient for a rarefied gas between two parallel plates are studied on the basis of the kinetic theory of gases. …
NAID 120004920390

カルーセルスパッタ法で作製したＣｏＦｅ系ＧＨｚ帯磁性薄膜の異方性入射効果と原子配列

今泉良一,宗像誠,大越正敏 [他],槙孝一郎
電気学会論文誌 A 基礎・材料・共通部門誌 131(7), 499-504, 2011
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NAID 130000754192

「気体分子運動論」

　　[★]

英: kinetic theory of gases

「kinetics」

　　[★]

動態学、動力学、速度論、運動論、速度式、動態、キネティクス、カイネティクス

関: dynamics、kinetic

同: enzyme
関: See also Catalysis／catalytic reactions (enzymatic)

「kinetic」

　　[★]

動力学的な、動力学の、動態学的な

関: dynamic、dynamical、dynamically、dynamics、kinetically、kinetics

「gas」

　　[★]

ガス、気体

関: air

[C_.26_C-1] Chapman, S., Cowling, T.G. (1939/1970).

[2] Kauzmann, W. (1966). Kinetic Theory of Gases, W.A. Benjamin, New York, pp. 232–235.

[3] Grad 1949

[4] Configuration integral (statistical mechanics)

[5] Collisions With a Surface

[6] 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. edit

[PonomarevKurchatov1993-7] L.I Ponomarev; I.V Kurchatov (1 January 1993). The Quantum Dice. CRC Press. ISBN 978-0-7503-0251-7.

[8] Lomonosov 1758

[9] Le Sage 1780/1818

[10] Herapath 1816, 1821

[11] Waterston 1843

[12] Krönig 1856

[13] Clausius 1857

[14] Mahon 2003

[15] Maxwell 1875

[16] Einstein 1905

[17] Smoluchowski 1906

リンク元	「気体分子運動論」
関連記事	「kinetics」「kinetic」「gas」

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案内

kinetic theory of gases