The kinetic theory of gases is a simple classical model of the thermodynamic behavior of gases. 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 the atoms or molecules of the gas. The kinetic theory of gases uses their collisions with each other and with the walls of their container to explain the relationship between 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 basic version of the model describes an ideal gas. It treats the collisions as perfectly elastic and as the only interaction between the particles, which are additionally assumed to be much smaller than their average distance apart. Due to the time reversibility of microscopic dynamics (microscopic reversibility), the kinetic theory is also connected to the principle of detailed balance, in terms of the fluctuation-dissipation theorem (for Brownian motion) and the Onsager reciprocal relations. The theory was historically significant as the first explicit exercise of the ideas of statistical mechanics and it forms the theoretical foundation for rarefied gas dynamics.
History
Kinetic theory of matter
Antiquity 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. This Epicurean atomistic point of view was rarely considered in the subsequent centuries, when Aristotelian ideas were dominant.
Modern era
"Heat is motion"
One of the first and boldest statements on the relationship between motion of particles and heat was by the English philosopher Francis 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." "not a ... motion of the whole, but of the small particles of the body." In 1623, in The 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.
In 1665, in Micrographia, the English polymath Robert Hooke repeated Bacon's assertion, and in 1675, his colleague, Anglo-Irish scientist Robert 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. 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. 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." In a manuscript published 1720, the English philosopher John Locke made a very similar statement: "What in our sensation is heat, in the object is nothing but motion." Locke too talked about the motion of the internal particles of the object, which he referred to as its "insensible parts". In his 1744 paper Meditations on the Cause of Heat and Cold, Russian polymath Mikhail Lomonosov made a relatable appeal to everyday experience to gain acceptance of the microscopic and kinetic nature of matter and heat: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 of dissolution, extraction and diffusion, providing as examples the dissolution and diffusion of salts by the action of water particles on the “molecules of salt”, the dissolution of metals in mercury, and the extraction of plant pigments by alcohol. Also the transfer of heat was explained by the motion of particles. Around 1760, Scottish physicist and chemist Joseph 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."
Kinetic theory of gases
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. Pioneers of the kinetic theory, whose work was also largely neglected by their contemporaries, were Mikhail Lomonosov (1747), Georges-Louis Le Sage (ca. 1780, published 1818), John Herapath (1816) and John James Waterston (1843), 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. 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. 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. This was the first-ever statistical law in physics. Maxwell also gave the first mechanical argument that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium. 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." 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, atoms were considered by many physicists to be purely hypothetical constructs, rather than real objects. An important turning point was Albert Einstein's (1905) and Marian Smoluchowski's (1906) papers on Brownian motion, which succeeded in making certain accurate quantitative predictions based on the kinetic theory. Following the development of the Boltzmann equation, a framework for its use in developing transport equations was developed independently by David Enskog and Sydney Chapman in 1917 and 1916. The framework provided a route to prediction of the transport properties of dilute gases, and became known as Chapman–Enskog theory. The framework was gradually expanded throughout the following century, eventually becoming a route to prediction of transport properties in real, dense gases.
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 during 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, and all these collisions are perfectly elastic. Interactions (i.e. collisions) between particles are strictly binary and uncorrelated, meaning that there are no three-body (or higher) interactions, and the particles have no memory. 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, so they are often neglected as a simplifying assumption in derivations (see below). More modern developments, such as the revised Enskog theory and the extended Bhatnagar–Gross–Krook model, relax one or more of the above assumptions. These can accurately describe the properties of dense gases, and gases with internal 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. 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.
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 individual gas atoms or molecules hitting and rebounding from the gas container's surface. Consider a gas particle traveling at velocity, v i {\textstyle v_{i}} , along the i ^ {\displaystyle {\hat {i}}} -direction in an enclosed volume with characteristic length, L i {\displaystyle L_{i}} , cross-sectional area, A i {\displaystyle A_{i}} , and volume, V = A i L i {\displaystyle V=A_{i}L_{i}} . The gas particle encounters a boundary after characteristic time
t = L i / v i . {\displaystyle t=L_{i}/v_{i}.}
The momentum of the gas particle can then be described as
p i = m v i = m L i / t . {\displaystyle p_{i}=mv_{i}=mL_{i}/t.}
We combine the above with Newton'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
F i = d p i d t = m L i t 2 = m v i 2 L i . {\displaystyle F_{i}={\frac {\mathrm {d} p_{i}}{\mathrm {d} t}}={\frac {mL_{i}}{t^{2}}}={\frac {mv_{i}^{2}}{L_{i}}}.}
Now consider a large number, N {\displaystyle N} , of gas particles with random orientation in a three-dimensional volume. Because the orientation is random, the average particle speed, v {\textstyle v} , in every direction is identical
v x 2 = v y 2 = v z 2 . {\displaystyle v_{x}^{2}=v_{y}^{2}=v_{z}^{2}.}
Further, assume that the volume is symmetrical about its three dimensions, i ^ , j ^ , k ^ {\displaystyle {\hat {i}},{\hat {j}},{\hat {k}}} , such that
V =
V i = V j = V k , F =
F i = F j = F k , A i = A j = A k . {\displaystyle {\begin{aligned}V={}&V_{i}=V_{j}=V_{k},\\F={}&F_{i}=F_{j}=F_{k},\\&A_{i}=A_{j}=A_{k}.\end{aligned}}}
The total surface area on which the gas particles act is therefore
A = 3 A i . {\displaystyle A=3A_{i}.}
The pressure exerted by the collisions of the N {\displaystyle N} 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,
P = N F ¯ A = N L F V {\displaystyle P={\frac {N{\overline {F}}}{A}}={\frac {NLF}{V}}}
⇒ P V = N L F = N 3 m v 2 . {\displaystyle \Rightarrow PV=NLF={\frac {N}{3}}mv^{2}.}
The total translational kinetic energy K t {\displaystyle K_{\text{t}}} of the gas is defined as
K t = N 2 m v 2 , {\displaystyle K_{\text{t}}={\frac {N}{2}}mv^{2},}
providing the result
P V = 2 3 K t . {\displaystyle PV={\frac {2}{3}}K_{\text{t}}.}
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. The mass density of a gas ρ {\displaystyle \rho } is expressed through the total mass of gas particles and through volume of this gas: ρ = N m V {\displaystyle \rho ={\frac {Nm}{V}}} . Taking this into account, the pressure is equal to
P = ρ v 2 3 . {\displaystyle P={\frac {\rho v^{2}}{3}}.}
Relativistic expression for this formula is
P = 2 ρ c 2 3 ( ( 1 − v 2 ¯ / c 2 ) − 1 / 2 − 1 ) , {\displaystyle P={\frac {2\rho c^{2}}{3}}\left({\left(1-{\overline {v^{2}}}/c^{2}\right)}^{-1/2}-1\right),}
where c {\displaystyle c} is speed of light. In the limit of small speeds, the expression becomes P ≈ ρ v 2 ¯ / 3 {\displaystyle P\approx \rho {\overline {v^{2}}}/3} .
Temperature and kinetic energy Rewriting the above result for the pressure as P V = 1 3 N m v 2 {\textstyle PV={\frac {1}{3}}Nmv^{2}} , we may combine it with the ideal gas law
where k B {\displaystyle k_{\mathrm {B} }} is the Boltzmann constant and T {\displaystyle T} is the absolute temperature defined by the ideal gas law, to obtain
k B T = 1 3 m v 2 , {\displaystyle k_{\mathrm {B} }T={\frac {1}{3}}mv^{2},}
which leads to a simplified expression of the average translational kinetic energy per molecule,
1 2 m v 2 = 3 2 k B T . {\displaystyle {\frac {1}{2}}mv^{2}={\frac {3}{2}}k_{\mathrm {B} }T.}
The translational kinetic energy of the system is N {\displaystyle N} times that of a molecule, namely K t = 1 2 N m v 2 {\textstyle K_{\text{t}}={\frac {1}{2}}Nmv^{2}} . The temperature, T {\displaystyle T} is related to the translational kinetic energy by the description above, resulting in
which becomes
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
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: The equipartition theorem requires that kinetic energy is partitioned equally between all kinetic degrees of freedom, D. A monatomic gas is axially symmetric about each spatial axis, so that D = 3 comprising translational motion along each axis. A diatomic gas is axially symmetric about only one axis, so that D = 5, comprising translational motion along three axes and rotational motion along two axes. A polyatomic gas, like water, is not radially symmetric about any axis, resulting in D = 6, comprising 3 translational and 3 rotational degrees of freedom. Because the equipartition theorem requires that kinetic energy is partitioned equally, the total kinetic energy is
K = D K t = D 2 N m v 2 . {\displaystyle K=DK_{\text{t}}={\frac {D}{2}}Nmv^{2}.}
Thus, the energy added to the system per gas particle kinetic degree of freedom is
K N D = 1 2 k B T . {\displaystyle {\frac {K}{ND}}={\frac {1}{2}}k_{\text{B}}T.}
Therefore, the kinetic energy per kelvin of one mole of monatomic ideal gas (D = 3) is
K = D 2 k B N A = 3 2 R , {\displaystyle K={\frac {D}{2}}k_{\text{B}}N_{\text{A}}={\frac {3}{2}}R,}
where N A {\displaystyle N_{\text{A}}} is the Avogadro constant, and R is the ideal gas constant. Thus, the ratio of the kinetic energy to the absolute temperature of an ideal monatomic gas 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. At higher temperatures (typically thousands of kelvins), vibrational modes become active to provide additional degrees of freedom, creating a temperature-dependence on D and the total molecular energy. Quantum statistical mechanics is needed to accurately compute these contributions.
Collisions with container wall 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 based on naive kinetic theory, and the results can be used for analyzing effusive flow rates, which is useful in applications such as the gaseous diffusion method for isotope separation. Assume that in the container, the number density (number per unit volume) is n = N / V {\displaystyle n=N/V} and that the particles obey Maxwell's velocity distribution:
f Maxwell ( v x , v y , v z ) d v x d v y d v z = ( m 2 π k B T ) 3 / 2 e − m v 2 2 k B T d v x d v y d v z {\displaystyle f_{\text{Maxwell}}(v_{x},v_{y},v_{z})\,dv_{x}\,dv_{y}\,dv_{z}=\left({\frac {m}{2\pi k_{\text{B}}T}}\right)^{3/2}e^{-{\frac {mv^{2}}{2k_{\text{B}}T}}}\,dv_{x}\,dv_{y}\,dv_{z}}
Then for a small area d A {\displaystyle dA} on the container wall, a particle with speed v {\displaystyle v} at angle θ {\displaystyle \theta } from the normal of the area d A {\displaystyle dA} , will collide with the area within time interval d t {\displaystyle dt} , if it is within the distance v d t {\displaystyle v\,dt} from the area d A {\displaystyle dA} . Therefore, all the particles with speed v {\displaystyle v} at angle θ {\displaystyle \theta } from the normal that can reach area d A {\displaystyle dA} within time interval d t {\displaystyle dt} are contained in the tilted pipe with a height of v cos ( θ ) d t {\displaystyle v\cos(\theta )dt} and a volume of v cos ( θ ) d A d t {\displaystyle v\cos(\theta )\,dA\,dt} . The total number of particles that reach area d A {\displaystyle dA} within time interval d t {\displaystyle dt} also depends on the velocity distribution; All in all, it calculates to be: n v cos ( θ ) d A d t × ( m 2 π k B T ) 3 / 2 e − m v 2 2 k B T ( v 2 sin ( θ ) d v d θ d ϕ ) . {\displaystyle nv\cos(\theta )\,dA\,dt\times \left({\frac {m}{2\pi k_{\text{B}}T}}\right)^{3/2}e^{-{\frac {mv^{2}}{2k_{\text{B}}T}}}\left(v^{2}\sin(\theta )\,dv\,d\theta \,d\phi \right).}
Integrating this over all appropriate velocities within the constraint v > 0 {\displaystyle v>0} , 0 < θ < π 2 {\textstyle 0<\theta <{\frac {\pi }{2}}} , 0 < ϕ < 2 π {\displaystyle 0<\phi <2\pi } yields the number of atomic or molecular collisions with a wall of a container per unit area per unit time:
J collision = ∫ 0 π / 2 cos ( θ ) sin ( θ ) d θ ∫ 0 π sin ( θ ) d θ × n v ¯ = 1 4 n v ¯ = n 4 8 k B T π m . {\displaystyle J_{\text{collision}}={\frac {\displaystyle \int _{0}^{\pi /2}\cos(\theta )\sin(\theta )\,d\theta }{\displaystyle \int _{0}^{\pi }\sin(\theta )\,d\theta }}\times n{\bar {v}}={\frac {1}{4}}n{\bar {v}}={\frac {n}{4}}{\sqrt {\frac {8k_{\mathrm {B} }T}{\pi m}}}.}
This quantity is also known as the "impingement rate" in vacuum physics. Note that to calculate the average speed v ¯ {\displaystyle {\bar {v}}} of the Maxwell's velocity distribution, one has to integrate over v > 0 {\displaystyle v>0} , 0 < θ < π {\displaystyle 0<\theta <\pi } , 0 < ϕ < 2 π {\displaystyle 0<\phi <2\pi } . The momentum transfer to the container wall from particles hitting the area d A {\displaystyle dA} with speed v {\displaystyle v} at angle θ {\displaystyle \theta } from the normal, in time interval d t {\displaystyle dt} is:
[ 2 m v cos ( θ ) ] × n v cos ( θ ) d A d t × ( m 2 π k B T ) 3 / 2 e − m v 2 2 k B T ( v 2 sin ( θ ) d v d θ d ϕ ) . {\displaystyle [2mv\cos(\theta )]\times nv\cos(\theta )\,dA\,dt\times \left({\frac {m}{2\pi k_{\text{B}}T}}\right)^{3/2}e^{-{\frac {mv^{2}}{2k_{\text{B}}T}}}\left(v^{2}\sin(\theta )\,dv\,d\theta \,d\phi \right).}
Integrating this over all appropriate velocities within the constraint v > 0 {\displaystyle v>0} , 0 < θ < π 2 {\textstyle 0<\theta <{\frac {\pi }{2}}} , 0 < ϕ < 2 π {\displaystyle 0<\phi <2\pi } yields the pressure (consistent with Ideal gas law):
P = 2 ∫ 0 π / 2 cos 2 ( θ ) sin ( θ ) d θ ∫ 0 π sin