Thekinetic theory of gases is a simpleclassical model of thethermodynamic behavior ofgases. Its introduction allowed many principal concepts of thermodynamics to be established. It treats a gas as composed of numerous particles, too small to be seen with a microscope, in constant, random motion. These particles are now known to be theatoms ormolecules of the gas. The kinetic theory of gases uses their collisions with each other and with the walls of their container to explainthe relationship between themacroscopic properties of gases, such asvolume,pressure, andtemperature, as well astransport properties such asviscosity,thermal conductivity andmass diffusivity.
The basic version of the model describes anideal gas. It treats the collisions asperfectly elastic and as the only interaction between the particles, which are additionally assumed to be much smaller than their average distance apart.
Due to thetime reversibility of microscopic dynamics (microscopic reversibility), the kinetic theory is also connected to the principle ofdetailed balance, in terms of thefluctuation-dissipation theorem (forBrownian motion) and theOnsager reciprocal relations.
The theory was historically significant as the first explicit exercise of the ideas ofstatistical mechanics.
In about 50BCE, the Roman philosopherLucretius proposed that apparently static macroscopic bodies were composed on a small scale of rapidly moving atoms all bouncing off each other.[1] ThisEpicurean atomistic point of view was rarely considered in the subsequent centuries, whenAristotlean ideas were dominant.[citation needed]
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One of the first and boldest statements on the relationship between motion of particles andheat was by the English philosopherFrancis Bacon in 1620. "It must not be thought that heat generates motion, or motion heat (though in some respects this be true), but that the very essence of heat ... is motion and nothing else."[2] "not a ... motion of the whole, but of the small particles of the body."[3] In 1623, inThe Assayer,Galileo Galilei, in turn, argued that heat, pressure, smell and other phenomena perceived by our senses are apparent properties only, caused by the movement of particles, which is a real phenomenon.[4][5]
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In 1665, inMicrographia, the English polymathRobert Hooke repeated Bacon's assertion,[6][7] and in 1675, his colleague, Anglo-Irish scientistRobert Boyle noted that a hammer's "impulse" is transformed into the motion of a nail's constituent particles, and that this type of motion is what heat consists of.[8] Boyle also believed that all macroscopic properties, including color, taste and elasticity, are caused by and ultimately consist of nothing but the arrangement and motion of indivisible particles of matter.[9] In a lecture of 1681, Hooke asserted a direct relationship between the temperature of an object and the speed of its internal particles. "Heat ... is nothing but the internal Motion of the Particles of [a] Body; and the hotter a Body is, the more violently are the Particles moved."[10] In a manuscript published 1720, the English philosopherJohn Locke made a very similar statement: "What in our sensation isheat, in the object is nothing butmotion."[11][12] Locke too talked about the motion of the internal particles of the object, which he referred to as its "insensible parts".
In his 1744 paperMeditations on the Cause of Heat and Cold, Russian polymathMikhail Lomonosov made a relatable appeal to everyday experience to gain acceptance of the microscopic and kinetic nature of matter and heat:[13]
Movement should not be denied based on the fact it is not seen. Who would deny that the leaves of trees move when rustled by a wind, despite it being unobservable from large distances? Just as in this case motion remains hidden due to perspective, it remains hidden in warm bodies due to the extremely small sizes of the moving particles. In both cases, the viewing angle is so small that neither the object nor their movement can be seen.
Lomonosov also insisted that movement of particles is necessary for the processes ofdissolution,extraction anddiffusion, providing as examples the dissolution and diffusion of salts by the action of water particles on the of the “molecules of salt”, the dissolution of metals in mercury, and the extraction of plant pigments by alcohol.[14]
Also thetransfer of heat was explained by the motion of particles. Around 1760, Scottish physicist and chemistJoseph Black wrote: "Many have supposed that heat is a tremulous ... motion of the particles of matter, which ... motion they imagined to be communicated from one body to another."[15]
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In 1738Daniel Bernoulli publishedHydrodynamica, which laid the basis for thekinetic theory ofgases. 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 averagekinetic energy determines the temperature of the gas. The theory was not immediately accepted, in part becauseconservation of energy had not yet been established, and it was not obvious tophysicists how the collisions between molecules could be perfectly elastic.[16]: 36–37
Pioneers of the kinetic theory, whose work was also largely neglected by their contemporaries, were Mikhail Lomonosov (1747),[17]Georges-Louis Le Sage (ca. 1780, published 1818),[18]John Herapath (1816)[19] andJohn James Waterston (1843),[20] which connected their research with the development ofmechanical explanations of gravitation.
In 1856August Krönig created a simple gas-kinetic model, which only considered thetranslational motion of the particles.[21] In 1857Rudolf Clausius developed a similar, but more sophisticated version of the theory, which included translational and, contrary to Krönig, alsorotational and vibrational molecular motions. In this same work he introduced the concept ofmean free path of a particle.[22] In 1859, after reading a paper about thediffusion of molecules by Clausius, Scottish physicistJames Clerk Maxwell formulated theMaxwell distribution of molecular velocities, which gave the proportion of molecules having a certain velocity in a specific range.[23] This was the first-ever statistical law in physics.[24] Maxwell also gave the first mechanical argument that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium.[25] 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 calledpressure of air and other gases."[26]In 1871,Ludwig Boltzmann generalized Maxwell's achievement and formulated theMaxwell–Boltzmann distribution. Thelogarithmic connection betweenentropy andprobability was also first stated by Boltzmann.
At the beginning of the 20th century, atoms were considered by many physicists to be purely hypothetical constructs, rather than real objects. An important turning point wasAlbert Einstein's (1905)[27] andMarian Smoluchowski's (1906)[28] papers onBrownian motion, which succeeded in making certain accurate quantitative predictions based on the kinetic theory.
Following the development of theBoltzmann equation, a framework for its use in developing transport equations was developed independently byDavid Enskog andSydney Chapman in 1917 and 1916. The framework provided a route to prediction of the transport properties of dilute gases, and became known asChapman–Enskog theory. The framework was gradually expanded throughout the following century, eventually becoming a route to prediction of transport properties in real, dense gases.
The application of kinetic theory to ideal gases makes the following assumptions:
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 samemass 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 withDalton's law of partial pressures. Many of the model's predictions are the same whether or not collisions between particles are included, so they are often neglected as a simplifying assumption in derivations (see below).[29]
More modern developments, such as therevised Enskog theory and the extendedBhatnagar–Gross–Krook model,[30] relax one or more of the above assumptions. These can accurately describe the properties of dense gases, and gases withinternal degrees of freedom, 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.[31] While theories relaxing the assumptions that the gas particles occupy negligible volume and that collisions are strictly elastic have been successful, it has been shown that relaxing the requirement of interactions being binary and uncorrelated will eventually lead to divergent results.[32]
In the kinetic theory of gases, thepressure is assumed to be equal to the force (per unit area) exerted by the individual gas atoms or molecules hitting and rebounding from the gas container's surface.
Consider a gas particle traveling at velocity,, along the-direction in an enclosed volume withcharacteristic length,, cross-sectional area,, and volume,. The gas particle encounters a boundary after characteristic time
Themomentum of the gas particle can then be described as
We combine the above withNewton's second law, which states that the force experienced by a particle is related to the time rate of change of its momentum, such that
Now consider a large number,, of gas particles with random orientation in a three-dimensional volume. Because the orientation is random, the average particle speed,, in every direction is identical
Further, assume that the volume is symmetrical about its three dimensions,, such thatThe total surface area on which the gas particles act is therefore
The pressure exerted by the collisions of the gas particles with the surface can then be found by adding the force contribution of every particle and dividing by the interior surface area of the volume,
The total translationalkinetic energy of the gas is defined asproviding the result
This is an important, non-trivial result of the kinetic theory because it relates pressure, amacroscopic property, to the translational kinetic energy of the molecules, which is amicroscopic property.
The mass density of a gas is expressed through the total mass of gas particles and through volume of this gas:. Taking this into account, the pressure is equal to
Relativistic expression for this formula is[33]
where isspeed of light. In the limit of small speeds, the expression becomes.
Rewriting the above result for the pressure as, we may combine it with theideal gas law
| 1 |
where is theBoltzmann constant and is theabsolutetemperature defined by the ideal gas law, to obtainwhich leads to a simplified expression of the average translational kinetic energy per molecule,[34]The translational kinetic energy of the system is times that of a molecule, namely. The temperature, is related to the translational kinetic energy by the description above, resulting in
| 2 |
which becomes
| 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
| 4 |
Thus, the product of pressure and volume permole is proportional to the averagetranslational molecular kinetic energy.
Equations (1) and (4) are called the "classical results", which could also be derived fromstatistical mechanics;for more details, see:[35]
Theequipartition theorem requires that kinetic energy is partitioned equally between all kineticdegrees of freedom,D. A monatomic gas is axially symmetric about each spatial axis, so thatD = 3 comprising translational motion along each axis. A diatomic gas is axially symmetric about only one axis, so thatD = 5, comprising translational motion along three axes and rotational motion along two axes. A polyatomic gas, likewater, is not radially symmetric about any axis, resulting inD = 6, comprising 3 translational and 3 rotational degrees of freedom.
Because theequipartition theorem requires that kinetic energy is partitioned equally, the total kinetic energy is
Thus, the energy added to the system per gas particle kinetic degree of freedom is
Therefore, the kinetic energy per kelvin of one mole of monatomicideal gas (D = 3) iswhere is theAvogadro constant, andR is theideal gas constant.
Thus, the ratio of the kinetic energy to the absolute temperature of an ideal monatomic gas can be calculated easily:
Atstandard temperature (273.15 K), the kinetic energy can also be obtained:
At higher temperatures (typically thousands of kelvins), vibrational modes become active to provide additional degrees of freedom, creating a temperature-dependence onD and the total molecular energy. Quantumstatistical mechanics is needed to accurately compute these contributions.[36]
For an ideal gas in equilibrium, the rate of collisions with the container wall and velocity distribution of particles hitting the container wall can be calculated[37] based on naive kinetic theory, and the results can be used for analyzingeffusive flow rates, which is useful in applications such as thegaseous diffusion method forisotope separation.
Assume that in the container, the number density (number per unit volume) is and that the particles obeyMaxwell's velocity distribution:
Then for a small area on the container wall, a particle with speed at angle from the normal of the area, will collide with the area within time interval, if it is within the distance from the area. Therefore, all the particles with speed at angle from the normal that can reach area within time interval are contained in the tilted pipe with a height of and a volume of.
The total number of particles that reach area within time interval also depends on the velocity distribution; All in all, it calculates to be:
Integrating this over all appropriate velocities within the constraint,, 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. Note that to calculate the average speed of the Maxwell's velocity distribution, one has to integrate over,,.
The momentum transfer to the container wall from particles hitting the area with speed at angle from the normal, in time interval is:Integrating this over all appropriate velocities within the constraint,, yields thepressure (consistent withIdeal gas law):If this small area is punched to become a small hole, theeffusive flow rate will be:
Combined with theideal gas law, this yields
The above expression is consistent withGraham's law.
To calculate the velocity distribution of particles hitting this small area, we must take into account that all the particles with that hit the area within the time interval are contained in the tilted pipe with a height of and a volume of; Therefore, compared to the Maxwell distribution, the velocity distribution will have an extra factor of:with the constraint,,. The constant can be determined by the normalization condition to be, and overall:
From the kinetic energy formula it can be shown thatwherev is in m/s,T is in kelvin, andm is the mass of one molecule of gas in kg. The most probable (or mode) speed is 81.6% of the root-mean-square speed, and the mean (arithmetic mean, or average) speed is 92.1% of the rms speed (isotropicdistribution of speeds).
See:
In kinetic theory of gases, themean 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 collisioncross section of one molecule colliding with another. As in the previous section, the number density is defined as the number of molecules per (extensive) volume, or. 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 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 asviscosity,thermal conductivity,mass diffusivity andthermal diffusion.
In its most basic form, Kinetic gas theory is only applicable to dilute gases. The extension of Kinetic gas theory to dense gas mixtures,Revised Enskog Theory, was developed in 1983-1987 byE. G. D. Cohen,J. M. Kincaid andM. Lòpez de Haro,[38][39][40][41] building on work byH. van Beijeren andM. H. Ernst.[42]
In books on elementary kinetic theory[43] 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 aCouette 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 forceF. 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 aMaxwell-Boltzmann equilibrium distribution of molecular motions.
Inside a dilute gas in aCouette flow setup, let be the forward velocity of the gas at a horizontal flat layer (labeled as); is along the horizontal direction. The number of molecules arriving at the area on one side of the gas layer, with speed at angle from the normal, in time interval is
These molecules made their last collision at, where is themean free path. Each molecule will contribute a forward momentum ofwhere plus sign applies to molecules from above, and minus sign below. Note that the forward velocity gradient 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 asshear stress):
The net rate of momentum per unit area that is transported across the imaginary surface is thus
Combining the above kinetic equation withNewton's law of viscositygives 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
Maxwell-Boltzmann distribution gives the average (equilibrium) molecular speed aswhere is the most probable speed. We note that
and insert the velocity in the viscosity equation above. This gives the well known equation[44] (with subsequently estimated below) forshear viscosity for dilute gases:and is themolar mass. The equation above presupposes that the gas density is low (i.e. the pressure is low). This implies that the transport of momentum through the gas due to the translational motion of molecules is much larger than the transport due to momentum being transferred between molecules during collisions. The transfer of momentum between molecules is explicitly accounted for inRevised Enskog theory, which relaxes the requirement of a gas being dilute. 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 orkinetic diameter of a molecule in a monomolecular gas. There are no simple general relation between the collisioncross 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 theLennard-Jones potential orMorse 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 may then be used as a rough estimate for the kinetic radius. However, using this estimate will typically lead to an erroneous temperature dependency of the viscosity. For such interaction potentials, significantly more accurate results are obtained by numerical evaluation of the requiredcollision integrals.
The expression for viscosity obtained fromRevised Enskog Theory reduces to the above expression in the limit of infinite dilution, and can be written aswhere is a term that tends to zero in the limit of infinite dilution that accounts for excluded volume, and is a term accounting for the transfer of momentum over a non-zero distance between particles during a collision.
Following a similar logic as above, one can derive the kinetic model forthermal conductivity[43] 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 asthermal reservoirs. The upper plate has a higher temperature than the lower plate. The molecules in the gas layer have a molecular kinetic energy which increases uniformly with distance above the lower plate. The non-equilibrium energy flow is superimposed on aMaxwell-Boltzmann equilibrium distribution of molecular motions.
Let 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 on one side of the gas layer, with speed at angle from the normal, in time interval is
These molecules made their last collision at a distance above and below the gas layer, and each will contribute a molecular kinetic energy ofwhere is thespecific heat capacity. Again, plus sign applies to molecules from above, and minus sign below. Note that the temperature gradient 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 asheat flux):
Note that the energy transfer from above is in the 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 withFourier's lawgives the equation for thermal conductivity, which is usually denoted when it is a dilute gas:
Similarly to viscosity,Revised Enskog Theory yields an expression for thermal conductivity that reduces to the above expression in the limit of infinite dilution, and which can be written aswhere is a term that tends to unity in the limit of infinite dilution, accounting for excluded volume, and is a term accounting for the transfer of energy across a non-zero distance between particles during a collision.
Following a similar logic as above, one can derive the kinetic model formass diffusivity[43] of a dilute gas:
Consider asteady 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 uniformnumber 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 in the layer increases uniformly with distance above the lower plate. The non-equilibrium molecular flow is superimposed on aMaxwell–Boltzmann equilibrium distribution of molecular motions.
Let be the number density of the gas at an imaginary horizontal surface inside the layer. The number of molecules arriving at an area on one side of the gas layer, with speed at angle from the normal, in time interval is
These molecules made their last collision at a distance 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 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 asdiffusion flux):
Note that the molecular transfer from above is in the 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 withFick's first law of diffusiongives the equation for mass diffusivity, which is usually denoted when it is a dilute gas:
The corresponding expression obtained fromRevised Enskog Theory may be written aswhere is a factor that tends to unity in the limit of infinite dilution, which accounts for excluded volume and the variationchemical potentials with density.
The kinetic theory of gases entails that due to themicroscopic reversibility of the gas particles' detailed dynamics, the system must obey the principle ofdetailed balance. Specifically, thefluctuation-dissipation theorem applies to theBrownian motion (ordiffusion) and thedrag force, which leads to theEinstein–Smoluchowski equation:[45] where
Note that the mobilityμ =vd/F can be calculated based on the viscosity of the gas; Therefore, the Einstein–Smoluchowski equation also provides a relation between the mass diffusivity and the viscosity of the gas.
The mathematical similarities between the expressions for shear viscocity, thermal conductivity and diffusion coefficient of the ideal (dilute) gas is not a coincidence; It is a direct result of theOnsager reciprocal relations (i.e. the detailed balance of thereversible dynamics of the particles), when applied to theconvection (matter flow due to temperature gradient, and heat flow due to pressure gradient) andadvection (matter flow due to the velocity of particles, and momentum transfer due to pressure gradient) of the ideal (dilute) gas.
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