# Kinetic theory of gases 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 is a historically significant, but simple, model of the thermodynamic behavior of gases, with which many principal concepts of thermodynamics were established. The model describes a gas as a large number of identical submicroscopic particles (atoms or molecules), all of which are in constant, rapid, random motion. Their size is assumed to be much smaller than the average distance between the particles. The particles undergo random elastic collisions between themselves and with the enclosing walls of the container. The basic version of the model describes the ideal gas, and considers no other interactions between the particles and, thus, the nature of kinetic energy transfers during collisions is strictly thermal.

The kinetic theory of gases explains 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 model also accounts for related phenomena, such as Brownian motion.

## 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. 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.:36–37

Other pioneers of the kinetic theory (whose work was 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 (probably after reading a paper of Waterston) created a simple gas-kinetic model, which only considered the translational motion of the particles.

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.  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. 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. 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) and Marian Smoluchowski's (1906) papers 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.
• Because they have mass, gravity will accelerate molecules. (If this were not the case then there would be no density gradient in a planet's troposphere and it would collapse to the surface.)

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. An important approach to the subject is called Chapman–Enskog theory. There have been many modern developments and there is an alternative approach developed by Grad based on moment expansions. 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:

$\Delta p=p_{i,x}-p_{f,x}=p_{i,x}-(-p_{i,x})=2p_{i,x}=2mv_{x},$

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

$F={\frac {\Delta p}{\Delta t}}={\frac {mv_{x}^{2}}{L}}.$

The total force on the wall is

$F={\frac {Nm{\overline {v_{x}^{2}}}}{L}},$

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

${\overline {v^{2}}}={\overline {v_{x}^{2}}}+{\overline {v_{y}^{2}}}+{\overline {v_{z}^{2}}},$
${\overline {v^{2}}}=3{\overline {v_{x}^{2}}}.$

Therefore:

${\overline {v_{x}^{2}}}={\frac {\overline {v^{2}}}{3}},$

and the force can be written as:

$F={\frac {Nm{\overline {v^{2}}}}{3L}}.$

This force is exerted on an area L2. Therefore, the pressure of the gas is

$P={\frac {F}{L^{2}}}={\frac {Nm{\overline {v^{2}}}}{3V}},$

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 $N{\frac {1}{2}}m{\overline {v^{2}}}$ , which is a microscopic property.

### Temperature and kinetic energy

Rewriting the above result for the pressure as $PV={Nm{\overline {v^{2}}} \over 3}$ , we may combine it with the ideal gas law

$\displaystyle PV=Nk_{B}T,$

(1)

where $\displaystyle k_{B}$  is the Boltzmann constant and $\displaystyle T$  the absolute temperature defined by the ideal gas law, to obtain

$k_{B}T={m{\overline {v^{2}}} \over 3}$ ,

which leads to simplified expression of the average kinetic energy per molecule,

$\displaystyle {\frac {1}{2}}m{\overline {v^{2}}}={\frac {3}{2}}k_{B}T$ .

The kinetic energy of the system is N times that of a molecule, namely $K={\frac {1}{2}}Nm{\overline {v^{2}}}$ . Then the temperature $\displaystyle T$  takes the form

$\displaystyle T={m{\overline {v^{2}}} \over 3k_{B}}$

(2)

which becomes

$\displaystyle T={\frac {2}{3}}{\frac {K}{Nk_{B}}}.$

(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:

Since there are $\displaystyle 3N$  degrees of freedom in a monatomic-gas system with $\displaystyle N$  particles, the kinetic energy per degree of freedom per molecule is

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

(5)

In the kinetic energy per degree of freedom, the constant of proportionality of temperature is 1/2 times Boltzmann constant or R/2 per mole. 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 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

The velocity distribution of particles hitting the container wall can be calculated based on naive kinetic theory, and the result can be used for analyzing effusive flow rate:

Assume that, in the container, the number density is $n$  and particles obey Maxwell's velocity distribution:

$f_{Maxwell}(v_{x},v_{y},v_{z})dv_{x}dv_{y}dv_{z}=\left({\frac {m}{2\pi kT}}\right)^{3/2}\,e^{-{\frac {mv^{2}}{2k_{B}T}}}dv_{x}dv_{y}dv_{z}$

Then the number of particles hitting the area $dA$  with speed $v$  at angle $\theta$  from the normal, in time interval $dt$  is:

$nv\cos {\theta }dAdt{\times }\left({\frac {m}{2\pi k_{B}T}}\right)^{3/2}\,e^{-{\frac {mv^{2}}{2k_{B}T}}}(v^{2}\sin {\theta }dv{d\theta }d\phi )$ .

Integrating this over all appropriate velocities within the constraint $v>0,0<\theta <\pi /2,0<\phi <2\pi$  yields the number of atomic or molecular collisions with a wall of a container per unit area per unit time:

$J_{collision}={\frac {1}{4}}n{\bar {v}}={\frac {n}{4}}{\sqrt {\frac {8k_{B}T}{\pi m}}}.$

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

If this small area $A$  is punched to become a small hole, the effusive flow rate will be:

$\Phi _{effusion}=J_{collision}A=nA{\sqrt {\frac {k_{B}T}{2\pi m}}}.$

Combined with ideal gas law, this yields:

$\Phi _{effusion}={\frac {PA}{\sqrt {2\pi mk_{B}T}}}.$

The velocity distribution of particles hitting this small area is:

{\begin{aligned}f(v,\theta ,\phi )dv{d\theta }d\phi &=const.{\times }(v\cos {\theta }){\times }e^{-{\frac {mv^{2}}{2k_{B}T}}}{\times }(v^{2}\sin {\theta }dv{d\theta }d\phi )\\&=const.{\times }(v^{3}e^{-{\frac {mv^{2}}{2k_{B}T}}}dv){\times }(\cos {\theta }\sin {\theta }{d\theta }){\times }d\phi \end{aligned}}

with the constraint $v>0,0<\theta <\pi /2,0<\phi <2\pi$ , and $const.$  can be determined by normalization condition to be ${\frac {1}{2\pi }}\left({\frac {m}{k_{B}T}}\right)^{2}$ .

### Speed of molecules

From the kinetic energy formula it can be shown that

$v_{\text{p}}={\sqrt {2\cdot {\frac {k_{B}T}{m}}}},$
${\bar {v}}={\frac {2}{\sqrt {\pi }}}v_{p}={\sqrt {{\frac {8}{\pi }}\cdot {\frac {k_{B}T}{m}}}},$
$v_{\text{rms}}={\sqrt {\frac {3}{2}}}v_{p}={\sqrt {{3}\cdot {\frac {k_{B}T}{m}}}},$

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 $v_{\text{p}}$  is 81.6% of the rms speed $v_{\text{rms}}$ , and the mean (arithmetic mean, or average) speed ${\bar {v}}$  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 using Kinetic Theory to consider what are known as "transport properties", such as viscosity, thermal conductivity and mass diffusivity.

### Viscosity and kinetic momentum

In books on elementary kinetic theory 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 $u$  which increase uniformly with distance $y$  above the lower plate. The non-equilibrium flow is superimposed on a Maxwell-Boltzmann equilibrium distribution of molecular motions.

Let $\sigma$  be the collision cross section of one molecule colliding with another. The number density $n$  is defined as the number of molecules per (extensive) volume $n=N/V$ . The collision cross section per volume or collision cross section density is $n\sigma$ , and it is related to the mean free path $l$  by

$\quad l={\frac {1}{{\sqrt {2}}n\sigma }}$

Notice that the unit of the collision cross section per volume $n\sigma$  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 $u_{0}$  be the forward velocity of the gas at an imaginary horizontal surface inside the gas layer. The number of molecules arriving at an area $dA$  on one side of the gas layer, with speed $v$  at angle $\theta$  from the normal, in time interval $dt$  is

$\quad nv\cos {\theta }dAdt{\times }\left({\frac {m}{2\pi k_{B}T}}\right)^{3/2}\,e^{-{\frac {mv^{2}}{2k_{B}T}}}(v^{2}\sin {\theta }dv{d\theta }d\phi )$

These molecules made their last collision at a distance $l\cos \theta$  above and below the gas layer, and each will contribute a forward momentum of

$\quad p_{x}^{\pm }=m\left(u_{0}\pm l\cos \theta {du \over dy}\right),$

where plus sign applies to molecules from above, and minus sign below. Note that the forward velocity gradient $du/dy$  can be considered to be constant over a distance of mean free path.

Integrating over all appropriate velocities within the constraint

$\quad v>0,0<\theta <\pi /2,0<\phi <2\pi$

yields the forward momentum transfer per unit time per unit area (also known as shear stress):

$\quad \tau ^{\pm }={\frac {1}{4}}{\bar {v}}n\cdot m\left(u_{0}\pm {\frac {2}{3}}l{du \over dy}\right)$

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

$\quad \tau =\tau ^{+}-\tau ^{-}={\frac {1}{3}}{\bar {v}}nm\cdot l{du \over dy}$

Combining the above kinetic equation with Newton's law of viscosity

$\quad \tau =\eta {du \over dy}$

gives the equation for shear viscosity, which is usually denoted $\eta _{0}$  when it is a dilute gas:

$\quad \eta _{0}={\frac {1}{3}}{\bar {v}}nml$

Combining this equation with the equation for mean free path gives

$\quad \eta _{0}={\frac {1}{3{\sqrt {2}}}}{\frac {m\cdot {\bar {v}}}{\sigma }}$

Maxwell-Boltzmann distribution gives the average (equilibrium) molecular speed as

$\quad {\bar {v}}={\frac {2}{\sqrt {\pi }}}v_{p}=2{\sqrt {{\frac {2}{\pi }}\cdot {\frac {k_{B}T}{m_{}}}}}$

where $v_{p}$  is the most probable speed. We note that

$\quad k_{B}\cdot N_{A}=R\quad {\text{and}}\quad M=m\cdot N_{A}$

and insert the velocity in the viscosity equation above. This gives the well known equation for shear viscosity for dilute gases:

$\quad \eta _{0}={\frac {2}{3{\sqrt {\pi }}}}\cdot {\frac {\sqrt {mk_{B}T}}{\sigma }}={\frac {2}{3{\sqrt {\pi }}}}\cdot {\frac {\sqrt {MRT}}{\sigma \cdot N_{A}}}$

and $M$  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

$\quad \sigma =\pi \left(2r\right)^{2}=\pi d^{2}$

The radius $r$  is called collision cross section radius or kinetic radius, and the diameter $d$  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.

### Thermal conductivity and heat flux

Following a similar logic as above, one can derive the kinetic model for thermal conductivity 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 as thermal reservoirs. The upper plate has a higher temperature than the lower plate. The molecules in the gas layer have a molecular kinetic energy $\varepsilon$  which increases uniformly with distance $y$  above the lower plate. The non-equilibrium energy flow is superimposed on a Maxwell-Boltzmann equilibrium distribution of molecular motions.

Let $\varepsilon _{0}$  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 $dA$  on one side of the gas layer, with speed $v$  at angle $\theta$  from the normal, in time interval $dt$  is

$\quad nv\cos {\theta }dAdt{\times }\left({\frac {m}{2\pi k_{B}T}}\right)^{3/2}\,e^{-{\frac {mv^{2}}{2k_{B}T}}}(v^{2}\sin {\theta }dv{d\theta }d\phi )$

These molecules made their last collision at a distance $l\cos \theta$  above and below the gas layer, and each will contribute a molecular kinetic energy of

$\quad \varepsilon ^{\pm }=\left(\varepsilon _{0}\pm mc_{v}l\cos \theta {dT \over dy}\right),$

where $c_{v}$  is the specific heat capacity. Again, plus sign applies to molecules from above, and minus sign below. Note that the temperature gradient $dT/dy$  can be considered to be constant over a distance of mean free path.

Integrating over all appropriate velocities within the constraint

$\quad v>0,0<\theta <\pi /2,0<\phi <2\pi$

yields the energy transfer per unit time per unit area (also known as heat flux):

$\quad q_{y}^{\pm }=-{\frac {1}{4}}{\bar {v}}n\cdot \left(\varepsilon _{0}\pm {\frac {2}{3}}mc_{v}l{dT \over dy}\right)$

Note that the energy transfer from above is in the $-y$  direction, and therefore the overall minus sign in the equation. The net heat flux across the imaginary surface is thus

$\quad q=q_{y}^{+}-q_{y}^{-}=-{\frac {1}{3}}{\bar {v}}nmc_{v}l{dT \over dy}$

Combining the above kinetic equation with Fourier's law

$\quad q=-\kappa {dT \over dy}$

gives the equation for thermal conductivity, which is usually denoted $\kappa _{0}$  when it is a dilute gas:

$\quad \kappa _{0}={\frac {1}{3}}{\bar {v}}nmc_{v}l$

### Diffusion Coefficient and diffusion flux

Following a similar logic as above, one can derive the kinetic model for mass diffusivity of a dilute gas:

Consider a steady 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 uniform number 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 $n$  in the layer increases uniformly with distance $y$  above the lower plate. The non-equilibrium molecular flow is superimposed on a Maxwell-Boltzmann equilibrium distribution of molecular motions.

Let $n_{0}$  be the number density of the gas at an imaginary horizontal surface inside the layer. The number of molecules arriving at an area $dA$  on one side of the gas layer, with speed $v$  at angle $\theta$  from the normal, in time interval $dt$  is

$\quad nv\cos {\theta }dAdt{\times }\left({\frac {m}{2\pi k_{B}T}}\right)^{3/2}\,e^{-{\frac {mv^{2}}{2k_{B}T}}}(v^{2}\sin {\theta }dv{d\theta }d\phi )$

These molecules made their last collision at a distance $l\cos \theta$  above and below the gas layer, where the local number density is

$\quad n^{\pm }=\left(n_{0}\pm l\cos \theta {dn \over dy}\right)$

Again, plus sign applies to molecules from above, and minus sign below. Note that the number density gradient $dn/dy$  can be considered to be constant over a distance of mean free path.

Integrating over all appropriate velocities within the constraint

$\quad v>0,0<\theta <\pi /2,0<\phi <2\pi$

yields the molecular transfer per unit time per unit area (also known as diffusion flux):

$\quad J_{y}^{\pm }=-{\frac {1}{4}}{\bar {v}}\cdot \left(n_{0}\pm {\frac {2}{3}}l{dn \over dy}\right)$

Note that the molecular transfer from above is in the $-y$  direction, and therefore the overall minus sign in the equation. The net diffusion flux across the imaginary surface is thus

$\quad J=J_{y}^{+}-J_{y}^{-}=-{\frac {1}{3}}{\bar {v}}l{dn \over dy}$

Combining the above kinetic equation with Fick's first law of diffusion

$\quad J=-D{dn \over dy}$

gives the equation for mass diffusivity, which is usually denoted $D_{0}$  when it is a dilute gas:

$\quad D_{0}={\frac {1}{3}}{\bar {v}}l$