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Kinetic theory of gases

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Kinetic theory of gases
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upright=1.4|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.
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.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.

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.JOURNAL, Maxwell, J. C., 1867, On the Dynamical Theory of Gases, Philosophical Transactions of the Royal Society of London, 157, 49, 10.1098/rstl.1867.0004, This Epicurean atomistic point of view was rarely considered in the subsequent centuries, when Aristotlean ideas were dominant.
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- HYDRODYNAMICA, Danielis Bernoulli.png -
upHydrodynamica 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.BOOK, The Quantum Dice, L.I Ponomarev, I.V Kurchatov, 1 January 1993, CRC Press, 978-0-7503-0251-7, {{rp|36–37}}Other pioneers of the kinetic theory (which were neglected by their contemporaries) were Mikhail Lomonosov (1747),Lomonosov 1758 Georges-Louis Le Sage (ca. 1780, published 1818),Le Sage 1780/1818 John Herapath (1816)Herapath 1816, 1821 and John James Waterston (1843),Waterston 1843 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.Krönig 1856In 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.Clausius 1857 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.See:
  • Maxwell, J.C. (1860) "Illustrations of the dynamical theory of gases. Part I. On the motions and collisions of perfectly elastic spheres," Philosophical Magazine, 4th series, 19 : 19–32.
  • Maxwell, J.C. (1860) "Illustrations of the dynamical theory of gases. Part II. On the process of diffusion of two or more kinds of moving particles among one another," Philosophical Magazine, 4th series, 20 : 21–37. This was the first-ever statistical law in physics.BOOK, The Man Who Changed Everything – the Life of James Clerk Maxwell, Mahon, Basil, Wiley, 2003, 0-470-86171-1, Hoboken, NJ, 52358254, Maxwell also gave the first mechanical argument that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium.JOURNAL, 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, 1702.01411, 2017SHPMP..57...53G, 10.1016/j.shpsb.2017.01.001, 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."Maxwell 1875
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)Einstein 1905 and Marian Smoluchowski's (1906)Smoluchowski 1906papers on Brownian motion, which succeeded in making certain accurate quantitative predictions based on the kinetic theory.

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.Chapman, S., Cowling, T.G. (1939/1970). An important approach to the subject is called Chapman–Enskog theory.Kauzmann, W. (1966). Kinetic Theory of Gasses, W.A. Benjamin, New York, pp. 232–235. There have been many modern developments and there is an alternative approach developed by Grad based on moment expansions.Grad 1949In 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

{{anchor|Pressure and Kinetic Energy}}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:
Delta p = p_{i,x} - p_{f,x} = p_{i,x} - (-p_{i,x}) = 2 p_{i,x} = 2 mv_x,
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
Delta t = frac{2L}{v_x},
where L is the distance between opposite walls.The force due to this particle is
F = frac{Delta p}{Delta t} = frac{m v_x^2}{L}.
The total force on the wall is
F = frac{Nmoverline{v_x^2}}{L},
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:
overline{v_x^2} = overline{v_y^2} = overline{v_z^2}.
By Pythagorean theorem in three dimensions the total squared speed v is given by
overline{v^2} = overline{v_x^2} + overline{v_y^2} + overline{v_z^2},
overline{v^2} = 3overline{v_x^2}.
Therefore:
overline{v_x^2} = frac{overline{v^2}}{3},
and the force can be written as:
F = frac{Nmoverline{v^2}}{3L}.
This force is exerted on an area L2. Therefore, the pressure of the gas is
P = frac{F}{L^2} = frac{Nmoverline{v^2}}{3V},
where V = L3 is the volume of the box.In terms of the kinetic energy of the gas K:
PV = frac{2}{3} times{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 Nfrac{1}{2} moverline{v^2}, which is a microscopic property.

Temperature and kinetic energy

Rewriting the above result for the pressure as PV = {Nmoverline{v^2}over 3} , we may combine it with the ideal gas law{{NumBlk|:|displaystyle PV = N k_B T ,|{{EquationRef|1}}}}where displaystyle k_B is the Boltzmann constant and displaystyle T theabsolute temperature defined by the ideal gas law, to obtain
k_B T = {moverline{v^2}over 3} ,
which leads to simplified expression of the average kinetic energy per molecule,The average kinetic energy of a fluid is proportional to the root mean-square velocity, which always exceeds the mean velocity - Kinetic Molecular Theory
displaystyle frac {1} {2} moverline{v^2} = frac {3} {2} k_B T.
The kinetic energy of the system is N times that of a molecule, namely K= frac {1} {2} N m overline{v^2} .Then the temperature displaystyle T takes the form{{NumBlk|:| displaystyle T = {moverline{v^2}over 3 k_B}|{{EquationRef|2}}}}which becomes{{NumBlk|:| displaystyle T = frac {2} {3} frac {K} {N k_B}.|{{EquationRef|3}}}}Eq.({{EquationNote|3}})is one important result of the kinetictheory:The average molecular kinetic energy is proportional to the ideal gas law's absolute temperature.From Eq.({{EquationNote|1}}) andEq.({{EquationNote|3}}),we have{{NumBlk|:|
displaystyle
PV
=
frac
{2}
{3}
K.
4}}}}Thus, the product of pressure andvolume per mole is proportional to the average(translational) molecular kinetic energy.Eq.({{EquationNote|1}}) and Eq.({{EquationNote|4}})are called the "classical results",which could also be derived from statistical mechanics;for more details, see:Configuration integral (statistical mechanics) {{webarchive|url=https://web.archive.org/web/20120428193950weblink |date=2012-04-28 }}Since there aredisplaystyle 3Ndegrees of freedom in a monatomic-gas system withdisplaystyle Nparticles,the kinetic energy per degree of freedom per molecule is{hide}NumBlk|:|
displaystyle
frac
{K}
{3 N}
=
frac
{k_B T}
{2}
5{edih}}}In the kinetic energy per degree of freedom,the constant of proportionality of temperatureis 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?|date=June 2014}}This result is relatedto the equipartition theorem.As noted in the article on heat capacity, diatomicgases should have 7 degrees of freedom, but the lighter diatomic gases actas 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

The total number and velocity distribution of particles hitting the container wall can be calculatedWEB, 5.62 Physical Chemistry II,weblink MIT OpenCourseWare, based on naive kinetic theory, and the result can be used for analyzing effusion into vacuum:Assume that, in the container, the number density is rho and particles obey Maxwell's velocity distribution:f_text{Maxwell}(v_x,v_y,v_z)dv_xdv_ydv_z=left(frac{m}{2 pi k_BT}right)^{3/2}, e^{- frac{m|v|^2}{2k_BT}} dv_xdv_ydv_zThe particles hitting a small area dA on the container, with speed v at angle theta from the normal, in time interval dt is contained in a parallelepiped with base area dAand height vdt{times}cos(theta), hence the total number of these particle is:{rho}vcos{theta}dAdt{times}left(frac{m}{2 pi k_BT}right)^{3/2}, e^{- frac{mv^2}{2k_BT}} (v^2sin{theta}dv{dtheta}dphi) Note that only the particles within the following constraint are actually heading to hit the wall:v>0,0


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