# 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 (also known as Molecular theory of gases or Kinetic molecular 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. The early versions of this theory make a few assumptions about the particles (see Assumptions section below), such that only the kinetic energy of particles are considered, hence the name "kinetic" theory.

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.[1]

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.[2] 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. Bernoulli also surmised that temperature was the effect of the kinetic energy of the molecules, and thus correlated with the ideal gas law.[3]

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.[4]:36–37 A competing theory favored by Newton was the repulsion theory, in which heat was a calorific fluid that repulsed molecules in proportion to its quantity (i.e. heat) and the inverse square of the distances between molecules.[3]

Other pioneers of the kinetic theory (which were neglected by their contemporaries) were Mikhail Lomonosov (1747),[5] Georges-Louis Le Sage (ca. 1780, published 1818),[6] John Herapath (1816)[7] and John James Waterston (1843),[8] 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.[9]

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. [10] 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.[11] This was the first-ever statistical law in physics.[12] Maxwell also gave the first mechanical argument that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium.[13] 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."[14] 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)[15] and Marian Smoluchowski's (1906)[16] 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. 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.
4. The collision between particles can be effectively ignored for most purposes, because the elastic collision between two particles of the same mass simply exchanges their momentum, and appears as if they never collided at all.
• 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.[1] An important approach to the subject is called Chapman–Enskog theory.[17] There have been many modern developments and there is an alternative approach developed by Grad based on moment expansions.[18] 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:

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

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

${\displaystyle \Delta t={\frac {2L}{v_{x}}},}$

where L is the distance between opposite walls.

The force due to this particle is

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

The total force on the wall is

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

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:

${\displaystyle {\overline {v_{x}^{2}}}={\overline {v_{y}^{2}}}={\overline {v_{z}^{2}}}.}$

By Pythagorean theorem in three dimensions the total squared speed v is given by

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

Therefore:

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

and the force can be written as:

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

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

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

where V = L3 is the volume of the box.

In terms of the kinetic energy of the gas K:

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

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

### Temperature and kinetic energy

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

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

(1)

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

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

which leads to simplified expression of the average kinetic energy per molecule,[19]

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

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

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

(2)

which becomes

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

(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

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

(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:[20]

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

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

(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 total number and velocity distribution of particles hitting the container wall can be calculated[21] based on kinetic theory, and the result can be used for analyzing effusion into vacuum:

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

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

The particles hitting a small area ${\displaystyle dA}$ on the container, with speed ${\displaystyle v}$ at angle ${\displaystyle \theta }$ from the normal, in time interval ${\displaystyle dt}$, is contained in a parallelepiped with base area ${\displaystyle dA}$ and height ${\displaystyle v\,dt{\times }\cos(\theta )}$, hence the total number of these particles is:

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

Note that only the particles within the following constraint are actually heading to hit the wall:

${\displaystyle v>0,\qquad 0<\theta <\pi /2,\qquad 0<\phi <2\pi .}$

Integrating over all appropriate velocities within the constraint yields the number of atomic or molecular collisions with a wall of a container:

{\displaystyle {\begin{aligned}\Delta N&=ndAdt\int _{0}^{+\infty }\left({\frac {m}{2\pi k_{B}T}}\right)^{3/2}v^{3}e^{-{\frac {mv^{2}}{2k_{B}T}}}dv\int _{0}^{\frac {\pi }{2}}\cos {\theta }\sin {\theta }d\theta \int _{0}^{2\pi }d\phi \\&=ndAdt\int _{0}^{+\infty }\left({\frac {m}{2\pi k_{B}T}}\right)^{3/2}v^{3}e^{-{\frac {mv^{2}}{2k_{B}T}}}dv\times {\frac {1}{2}}\times 2\pi \\&=ndAdt\int _{0}^{+\infty }{\frac {v}{4}}\left({\frac {m}{2\pi k_{B}T}}\right)^{3/2}e^{-{\frac {mv^{2}}{2k_{B}T}}}4\pi v^{2}dv\\&=ndAdt{\frac {v_{\text{avg}}}{4}}\end{aligned}}}

where ${\displaystyle v_{\text{avg}}}$ is the average speed from Maxwell's speed distribution.

Hence the particle flux hitting the container wall (also known as impingement rate in vacuum physics) is:

{\displaystyle {\begin{aligned}J_{\text{impingement}}&={\frac {\Delta N}{dAdt}}\\&={\frac {1}{4}}nv_{\text{avg}}={\frac {1}{4}}\left({\frac {N}{V}}\right){\sqrt {\frac {8k_{B}T}{\pi m}}}\\&={\frac {P}{\sqrt {2\pi mk_{B}T}}}={\sqrt {\frac {NP}{2\pi mV}}}\end{aligned}}}

The last line makes use of ideal gas law. It's inversely proportional to molecular mass (and therefore molar mass), which, in the context of Effusion, is known as Graham's law.

The velocity distribution of particles hitting this small area is:

{\displaystyle {\begin{aligned}f(v,\theta ,\phi )\,dv\,{d\theta }\,d\phi &\propto nv\cos {\theta }dAdt{\times }e^{-{\frac {mv^{2}}{2k_{B}T}}}{\times }(v^{2}\sin {\theta }\,dv\,{d\theta }\,d\phi )\\&\propto v\cos {\theta }{\times }e^{-{\frac {mv^{2}}{2k_{B}T}}}{\times }(v^{2}\sin {\theta }\,dv\,{d\theta }\,d\phi )\\&={\text{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 ${\displaystyle v>0,0<\theta <\pi /2,0<\phi <2\pi }$, and the constant can be determined by normalization condition.

In Cartesian coordinates, this is:

{\displaystyle {\begin{aligned}f(v,\theta ,\phi )\,dv\,{d\theta }\,d\phi &\propto v_{z}e^{-{\frac {mv^{2}}{2k_{B}T}}}{\times }(dv_{x}\,dv_{y}\,dv_{z})\\&={\text{const.}}{\times }(e^{-{\frac {mv_{x}^{2}}{2k_{B}T}}}dv_{x}){\times }(e^{-{\frac {mv_{y}^{2}}{2k_{B}T}}}dv_{y}){\times }(v_{z}e^{-{\frac {mv_{z}^{2}}{2k_{B}T}}}dv_{z})\end{aligned}}}

with the constraint ${\displaystyle v_{z}>0}$, and the constant can be determined by normalization condition.

From the above distribution, the average velocity of these impinging particles is:

{\displaystyle {\begin{aligned}{\overline {v_{x}}}&={\overline {v_{y}}}=0\\{\overline {v_{z}}}&={\frac {\int _{0}^{+\infty }v_{z}^{2}e^{-{\frac {mv_{z}^{2}}{2k_{B}T}}}dv_{z}}{\int _{0}^{+\infty }v_{z}e^{-{\frac {mv_{z}^{2}}{2k_{B}T}}}dv_{z}}}={\frac {_{1d}}{_{1d}}}={\sqrt {\frac {\pi k_{B}T}{2m}}}\end{aligned}}}

The last equal sign used the result from Maxwell's speed distribution in 1-Dimension.

When these particles bounce off the container wall, each of them transfers a momentum of ${\displaystyle 2m{\overline {v_{z}}}}$, hence the average force is:

${\displaystyle F=2m{\overline {v_{z}}}\times J_{\text{impingement}}\times A=PA}$

### Mean free path and collision frequency

The mean free path ${\displaystyle \ell }$ in an equilibrium is the average distance a particle travels between collisions with other moving particles. Assume that a particle with diameter ${\displaystyle d}$ is moving at speed ${\displaystyle v}$, and all neighboring identical particles are at rest but positioned randomly with a number density ${\displaystyle n}$. Any neighboring (identical) particle whose center gets closer than ${\displaystyle d}$ to the center of the moving particle would collide with it and hence, end its free path. Therefore, the mean free path (MFP) of the moving particle satisfies: ${\displaystyle n\pi d^{2}\ell =1\rightarrow \ell ={\frac {1}{n\pi d^{2}}}.}$

However, in equilibrium all particles are in random and uncorrelated motions, hence the relative velocity ${\displaystyle \mathbf {v} _{rel}}$ must be used instead. The ensemble average of the square of relative velocity is:

${\displaystyle {\overline {\mathbf {v} _{relative}^{2}}}={\overline {(\mathbf {v} _{1}-\mathbf {v} _{2})^{2}}}={\overline {\mathbf {v} _{1}^{2}+\mathbf {v} _{2}^{2}-2\mathbf {v} _{1}\cdot \mathbf {v} _{2}}}.}$

In equilibrium, ${\displaystyle \mathbf {v} _{1}}$ and ${\displaystyle \mathbf {v} _{2}}$ are random and uncorrelated, therefore ${\displaystyle {\overline {\mathbf {v} _{1}\cdot \mathbf {v} _{2}}}=0}$, and the relative speed is

${\displaystyle v_{rel}={\sqrt {\overline {\mathbf {v} _{relative}^{2}}}}={\sqrt {\overline {\mathbf {v} _{1}^{2}+\mathbf {v} _{2}^{2}}}}={\sqrt {2}}v.}$

This means, the number of collisions is ${\displaystyle {\sqrt {2}}}$ times the number with stationary targets. Therefore, the following relationship applies:[22]

${\displaystyle \ell ={\frac {1}{{\sqrt {2}}\,n\pi d^{2}}}={\frac {k_{\rm {B}}T}{{\sqrt {2}}\pi d^{2}P}},}$

where the last equal sign makes use of Ideal gas law.

For a more general situation, the cross section ${\displaystyle \sigma }$ of a particle may be different from ${\displaystyle \pi d^{2}}$, in which case ${\displaystyle \ell =({\sqrt {2}}\,n\sigma )^{-1}.}$ The mean free time ${\displaystyle \tau }$ is defined as ${\displaystyle \tau =\ell /{\overline {v}}}$, where

${\displaystyle {\bar {v}}={\frac {2}{\sqrt {\pi }}}v_{p}={\sqrt {{\frac {8}{\pi }}\cdot {\frac {k_{B}T}{m}}}}}$

is the average speed of particles.

The collision frequency ${\displaystyle Z}$, as defined in Collision theory, is the rate of collisions per unit volume in a certain equilibrium. Based on the calculation above, for a pure gas with identical particles in equilibrium:

{\displaystyle {\begin{aligned}Z&={\frac {n}{\tau }}=4n^{2}d^{2}{\sqrt {\frac {\pi k_{B}T}{m}}}\\&=4d^{2}{\sqrt {\frac {\pi Pn^{3}}{m}}}\\&=4P^{2}d^{2}{\sqrt {\frac {\pi }{m(k_{B}T)^{3}}}}\end{aligned}}}

### Speed of molecules

From the kinetic energy formula it can be shown that

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

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 ${\displaystyle v_{\text{p}}}$ is 81.6% of the rms speed ${\displaystyle v_{\text{rms}}}$, and the mean (arithmetic mean, or average) speed ${\displaystyle {\bar {v}}}$ 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 mass diffusivity, viscosity and thermal conductivity. For derivations of these properties, one important assumption is that even though gas molecules collide very often, an average molecule can travel a distance of mean free path ${\displaystyle \ell }$ in uniform linear motion without any interaction.

### Diffusion coefficient and diffusion flux

The diffusion coefficient ${\displaystyle D}$ is defined by Fick's first law of diffusion ${\displaystyle J=-D\nabla n}$, which describes a macroscopic motion of molecules from higher number density to lower number density, also known as mass transport. The diffusion coefficient ${\displaystyle D}$ of a gas can be calculated from kinetic theory as below.

Assume that, the number density of gas molecules ${\displaystyle n(\mathbf {r} )}$ is a function of position ${\displaystyle \mathbf {r} =(x,y,z)}$, and the molecules obey Maxwell's velocity distribution:

${\displaystyle f(\mathbf {v} )=f_{\text{Maxwell}}(v_{x},v_{y},v_{z})\,dv_{x}\,dv_{y}\,dv_{z}=\left({\frac {m}{2\pi k_{B}T}}\right)^{3/2}\,e^{-{\frac {m|v|^{2}}{2k_{B}T}}}\,dv_{x}\,dv_{y}\,dv_{z}}$

Consider an infinitesimal flat area ${\displaystyle dA}$ at level ${\displaystyle x=x_{0}}$. The molecules crossing this area from level ${\displaystyle x=x_{0}+\ell \cos \theta }$, with speed ${\displaystyle v}$ at angle ${\displaystyle 0<\theta <\pi /2}$ from the ${\displaystyle -x}$ direction (which is also the normal of area ${\displaystyle dA}$), in time interval ${\displaystyle dt}$, is contained in a parallelepiped with base area ${\displaystyle dA}$ and height ${\displaystyle v\,dt{\times }\cos \theta }$, hence the total number of these molecules is

${\displaystyle dN_{-}=n(x_{0}+\ell cos\theta )v\cos {\theta }dAdt{\times }\left({\frac {m}{2\pi k_{B}T}}\right)^{3/2}e^{-{\frac {mv^{2}}{2k_{B}T}}}(v^{2}\sin {\theta }dvd\theta d\phi ),}$

where ${\displaystyle \ell }$ is the mean free path.

In the meantime, the same analysis applies to the molecules crossing this area from level ${\displaystyle x=x_{0}-\ell \cos \theta }$, with speed ${\displaystyle v}$ at angle ${\displaystyle 0<\theta <\pi /2}$ from the ${\displaystyle +x}$ direction:

${\displaystyle dN_{+}=n(x_{0}-\ell cos\theta )v\cos {\theta }dAdt{\times }\left({\frac {m}{2\pi k_{B}T}}\right)^{3/2}e^{-{\frac {mv^{2}}{2k_{B}T}}}(v^{2}\sin {\theta }dvd\theta d\phi ).}$

Therefore, the net flux in ${\displaystyle x}$ direction crossing the area ${\displaystyle dA}$ is:

{\displaystyle {\begin{aligned}J_{x}&={\frac {dN_{+}-dN_{-}}{dAdt}}=\int _{0}^{+\infty }\left({\frac {m}{2\pi k_{B}T}}\right)^{3/2}v^{3}e^{-{\frac {mv^{2}}{2k_{B}T}}}dv\int _{0}^{\frac {\pi }{2}}[n(x_{0}-\ell \cos \theta )-n(x_{0}+\ell \cos \theta )]\cos {\theta }\sin {\theta }d\theta \int _{0}^{2\pi }d\phi \\&=\int _{0}^{+\infty }\left({\frac {m}{2\pi k_{B}T}}\right)^{3/2}v^{3}e^{-{\frac {mv^{2}}{2k_{B}T}}}dv\times \int _{0}^{\frac {\pi }{2}}{\frac {\partial n}{\partial x}}{\bigg |}_{x_{0}}(-2\ell \cos {\theta })\cos {\theta }\sin {\theta }d\theta \times 2\pi \\&=\int _{0}^{+\infty }\left({\frac {m}{2\pi k_{B}T}}\right)^{3/2}v^{3}e^{-{\frac {mv^{2}}{2k_{B}T}}}dv\times {\frac {\partial n}{\partial x}}{\bigg |}_{x_{0}}{\frac {-2\ell }{3}}\times 2\pi \\&=-{\frac {\ell }{3}}{\frac {\partial n}{\partial x}}{\bigg |}_{x_{0}}\int _{0}^{+\infty }v\times \left({\frac {m}{2\pi k_{B}T}}\right)^{3/2}4\pi v^{2}e^{-{\frac {mv^{2}}{2k_{B}T}}}dv=-{\frac {1}{3}}\ell \,{\overline {v}}{\frac {\partial n}{\partial x}}{\bigg |}_{x_{0}}\end{aligned}}}

Comparing to Fick's Law ${\displaystyle J=-D\nabla n}$, this gives kinetic theory prediction of diffusion coefficient:

${\displaystyle D={\frac {1}{3}}\,\ell \,{\overline {v}}={\frac {2}{3}}{\sqrt {\frac {k_{\rm {B}}^{3}}{\pi ^{3}m}}}{\frac {T^{3/2}}{Pd^{2}}}\,,}$

where ${\displaystyle m}$ is the molecular mass, ${\displaystyle d}$ is the effective diameter of a gas molecule, and ${\displaystyle {\overline {v}}}$ is the mean thermal speed (not ${\displaystyle v_{rms}}$).

A more general equation for predicting the diffusion coefficient is Einstein relation:

${\displaystyle D=\mu \,k_{\text{B}}T,}$

where

D is the diffusion constant;
μ is the "mobility", or the ratio of the particle's terminal drift velocity to an applied force, μ = vd/F;
kB is Boltzmann's constant;
T is the absolute temperature.

### Viscosity and kinetic momentum

Even for a highly dilute gas with perfectly elastic molecular collisions, viscosity still occurs as a result from the momentum transport during elastic molecular collisions. In books on elementary kinetic theory[23] one can find results for dilute gas modeling that have widespread use.

The 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 moves with constant velocity to the right due to a force ${\displaystyle F}$. 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 ${\displaystyle u}$ which increases uniformly with distance ${\displaystyle y}$ above the lower plate. The non-equilibrium flow is superimposed on a Maxwell–Boltzmann equilibrium distribution of molecular motions.

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

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

Notice that the unit of the collision cross section per volume ${\displaystyle C\sigma }$ 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 ${\displaystyle u_{0}}$ be the forward velocity of the gas at an imaginary horizontal surface inside the gas layer. On the average, a molecule that crosses the surface makes its last collision before crossing at a distance equal to two-thirds of the mean free path (i.e. ${\displaystyle 2l/3}$) away from the surface. At this distance above and below the surface, the forward momentum of the molecule is respectively

${\displaystyle \quad p_{x}^{\pm }=m\left(u_{0}\pm {\frac {2}{3}}l{du \over dy}\right)}$

where m is the molecular mass. The molecular flux ${\displaystyle \Phi }$ includes all molecules arriving at one side of an element of the surface within the gas layer. The incoming molecules are coming from all directions at the one side of the surface and with all speeds. This molecular flux (i.e. the number flux) is related to the average molecular speed ${\displaystyle {\bar {v}}}$ by

${\displaystyle \quad \Phi ={\frac {1}{4}}{\bar {v}}C}$

Notice that the forward velocity gradient ${\displaystyle du/dy}$ can be considered to be constant over a distance of mean free path. Next we multiply by the total flux to get the change of momentum per unit time and per unit area, that is carried by the molecules crossing from either above or below the surface area. This gives the equation

${\displaystyle \quad \Phi p_{x}^{\pm }={\frac {1}{4}}{\bar {v}}C\cdot m\left(u_{0}\pm {\frac {2}{3}}l{du \over dy}\right)}$

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

${\displaystyle \quad \tau =\Phi p_{x}^{+}-\Phi p_{x}^{-}={\frac {1}{3}}{\bar {v}}Cm\cdot l{du \over dy}}$

The defining equation for the (shear) viscosity ${\displaystyle \eta }$ of the gas is

${\displaystyle \quad {\frac {F}{A}}=\eta {du \over dy}}$

Combining the above kinetic equation with defining equation for (shear) viscosity by ${\displaystyle \tau =F/A}$ gives the equation for shear viscosity, which is usually denoted ${\displaystyle \eta _{0}}$ when it is a dilute gas:

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

Combining this equation with the equation for mean free path gives

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

From statistical thermodynamics for gases we have equations relating average molecular speed to most likely speed and further to temperature. These statistical results gives the average (equilibrium) molecular speed as

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

where ${\displaystyle v_{p}}$ is the most probable speed, ${\displaystyle k_{B}}$ is the Boltzmann constant. We note that

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

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

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

and ${\displaystyle M}$ 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

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

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

Local nomenclature list:

• ${\displaystyle A\ \ \ }$ : area of moving boundary in Couette flow experiment [m2]
• ${\displaystyle C\ \ \ }$ : number concentration or number density [1/m3]
• ${\displaystyle c\ \ \ \,}$ : molar concentration or molar density [mol/cm3]
• ${\displaystyle d\ \ \ \,}$ : kinetic diameter in collision cross section [m]
• ${\displaystyle F\ \ \ }$ : force that move a boundary in Couette flow experiment [N]
• ${\displaystyle k_{B}\ \,}$ : Boltzmann constant [JK−1]
• ${\displaystyle l\ \ \ \ \ }$ : mean free path [m]
• ${\displaystyle M\ \ }$ : molar mass [g/mol]
• ${\displaystyle m\ \ \ }$ : molecular mass [Da]
• ${\displaystyle N_{A}\,}$ : Avogadro constant [mol−1]
• ${\displaystyle P\ \ \ }$ : pressure [Pa or bar or atm]
• ${\displaystyle P_{c}\ \ }$ : critical pressure [Pa or bar or atm]
• ${\displaystyle p_{x}\ \ }$ : linear momentum in x-direction of a molecule [kgm/s]
• ${\displaystyle R\ \ \ }$ : gas constant [JK−1mol−1]
• ${\displaystyle r\ \ \ \,\,}$ : kinetic radius in collision cross section or hard core molecular radius [m]
• ${\displaystyle T\ \ \,\,}$ : temperature [K]
• ${\displaystyle T_{c}\ \,\,}$ : critical temperature [K]
• ${\displaystyle u\ \ \ \ }$ : macroscopic fluid velocity in x-direction [m/s]
• ${\displaystyle u_{0}\ \ }$ : macroscopic fluid velocity in x-direction on the imaginary surface [m/s]
• ${\displaystyle {\bar {v}}\ \ \ \ }$ : average molecular equilibrium speed [m/s]
• ${\displaystyle v_{p}\ \ \,}$ : most probable molecular equilibrium speed [m/s]
• ${\displaystyle V\ \ \ }$ : molar volume [cm3/mol]
• ${\displaystyle V_{c}\ \ }$ : critical molar volume [cm3/mol]
• ${\displaystyle V_{ext}}$ : extensive fluid volume [m3]
• ${\displaystyle y\ \ \ \ }$ : distance from non-moving boundary in direction normal to fluid flow [m]
• ${\displaystyle \eta \ \ \ \ }$ : viscosity [Pas or μP or cP]
• ${\displaystyle \eta _{0}\ \,\,}$ : viscosity of dilute gas [Pas or μP or cP]
• ${\displaystyle \Phi \ \ \,\,}$ : molecular flux across an imaginary surface [m−2s−1]
• ${\displaystyle \sigma \ \ \ \ }$ : collision cross section [m2]
• ${\displaystyle \tau \ \ \ \ }$ : shear stress [Nm−2]
• ${\displaystyle K_{r}\ }$ : dummy

### Thermal conductivity and heat flux

In reality, especially for gases, it's difficult to prevent conductive heat flow from be masked by heat convection. The gas layer must be very thin to have a negligible convection.

The thermal conductivity ${\displaystyle k}$ is defined by Fourier's law ${\displaystyle \mathbf {q} =-k\nabla T}$, which describes a macroscopic heat transfer from higher temperature to lower temperature, or equivalently, a microscopic kinetic energy transport. The thermal conductivity ${\displaystyle k}$ of a gas can be calculated from kinetic theory as below.

Assume that, the temperature of gas molecules ${\displaystyle T(\mathbf {r} )}$ is a function of position ${\displaystyle \mathbf {r} =(x,y,z)}$, and the molecules obey Maxwell's velocity distribution:

${\displaystyle f(\mathbf {v} )=f_{\text{Maxwell}}(v_{x},v_{y},v_{z})\,dv_{x}\,dv_{y}\,dv_{z}=\left({\frac {m}{2\pi k_{B}T}}\right)^{3/2}\,e^{-{\frac {m|v|^{2}}{2k_{B}T}}}\,dv_{x}\,dv_{y}\,dv_{z}.}$

For an ideal gas, the average molecular energy is given by ${\displaystyle E_{molecular}=c_{V}mT}$, where ${\displaystyle c_{V}}$ is the isochoric specific heat capacity and ${\displaystyle m}$ is the molecular mass.

Consider an infinitesimal flat area ${\displaystyle dA}$ at level ${\displaystyle x=x_{0}}$. The molecules crossing this area from level ${\displaystyle x=x_{0}+\ell \cos \theta }$, with speed ${\displaystyle v}$ at angle ${\displaystyle 0<\theta <\pi /2}$ from the ${\displaystyle -x}$ direction (which is also the normal of area ${\displaystyle dA}$), in time interval ${\displaystyle dt}$, is contained in a parallelepiped with base area ${\displaystyle dA}$ and height ${\displaystyle v\,dt{\times }\cos \theta }$, hence the total number of these molecules is

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

where ${\displaystyle \ell }$ is the mean free path.

In the meantime, the same analysis applies to the molecules crossing this area from level ${\displaystyle x=x_{0}-\ell \cos \theta }$, with speed ${\displaystyle v}$ at angle ${\displaystyle 0<\theta <\pi /2}$ from the ${\displaystyle +x}$ direction:

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

Therefore, the net heat flux in ${\displaystyle x}$ direction crossing the area ${\displaystyle dA}$ is:

{\displaystyle {\begin{aligned}q_{x}&={\frac {dE_{+}-dE_{-}}{dAdt}}=\int _{0}^{+\infty }n\left({\frac {m}{2\pi k_{B}T}}\right)^{3/2}v^{3}e^{-{\frac {mv^{2}}{2k_{B}T}}}dv\int _{0}^{\frac {\pi }{2}}[c_{V}mT(x_{0}-\ell \cos \theta )-c_{V}mT(x_{0}+\ell \cos \theta )]\cos {\theta }\sin {\theta }d\theta \int _{0}^{2\pi }d\phi \\&=\int _{0}^{+\infty }n\left({\frac {m}{2\pi k_{B}T}}\right)^{3/2}v^{3}e^{-{\frac {mv^{2}}{2k_{B}T}}}dv\times \int _{0}^{\frac {\pi }{2}}c_{V}m{\frac {\partial T}{\partial x}}{\bigg |}_{x_{0}}(-2\ell \cos {\theta })\cos {\theta }\sin {\theta }d\theta \times 2\pi \\&=\int _{0}^{+\infty }n\left({\frac {m}{2\pi k_{B}T}}\right)^{3/2}v^{3}e^{-{\frac {mv^{2}}{2k_{B}T}}}dv\times c_{V}m{\frac {\partial T}{\partial x}}{\bigg |}_{x_{0}}{\frac {-2\ell }{3}}\times 2\pi \\&=-c_{V}mn{\frac {\ell }{3}}{\frac {\partial T}{\partial x}}{\bigg |}_{x_{0}}\int _{0}^{+\infty }v\times \left({\frac {m}{2\pi k_{B}T}}\right)^{3/2}4\pi v^{2}e^{-{\frac {mv^{2}}{2k_{B}T}}}dv=-{\frac {1}{3}}c_{V}mn\ell \,{\overline {v}}{\frac {\partial T}{\partial x}}{\bigg |}_{x_{0}}\end{aligned}}}

Comparing to Fourier's law ${\displaystyle \mathbf {q} =-k\nabla T}$, this gives kinetic theory prediction of thermal conductivity:

${\displaystyle k={\frac {1}{3}}c_{V}mn\,\ell \,{\overline {v}}={\frac {2}{3}}c_{V}{\sqrt {\frac {mk_{\rm {B}}T}{\pi ^{3}}}}{\frac {1}{d^{2}}}\,,}$

where ${\displaystyle m}$ is the molecular mass, ${\displaystyle d}$ is the effective diameter of a gas molecule, and ${\displaystyle {\overline {v}}}$ is the mean thermal speed (not ${\displaystyle v_{rms}}$).

A more general prediction of the thermal conductivity of a gas is[24]:

${\displaystyle k=\beta \rho \ell c_{V}{\sqrt {\frac {2k_{B}T}{\pi m}}}={\frac {\beta }{2}}c_{V}mn\,\ell \,{\overline {v}}=\beta c_{V}{\sqrt {\frac {mk_{\rm {B}}T}{\pi ^{3}}}}{\frac {1}{d^{2}}},}$

where

• ${\displaystyle \beta }$ is a numerical constant of order ${\displaystyle 1}$
• ${\displaystyle \ell }$ is the mean free path
• ${\displaystyle \rho }$ is the mass Density
• ${\displaystyle c_{V}}$ is the specific heat
• ${\displaystyle m}$ is the molecular mass

## Notes

1. ^ a b Chapman, S., Cowling, T.G. (1939/1970).
2. ^ Maxwell, J. C. (1867). "On the Dynamical Theory of Gases". Philosophical Transactions of the Royal Society of London. 157: 49–88. doi:10.1098/rstl.1867.0004.
3. ^ a b
4. ^ L.I Ponomarev; I.V Kurchatov (1 January 1993). The Quantum Dice. CRC Press. ISBN 978-0-7503-0251-7.
5. ^ Lomonosov 1758
6. ^ Le Sage 1780/1818
7. ^ Herapath 1816, 1821
8. ^ Waterston 1843
9. ^ Krönig 1856
10. ^ Clausius 1857
11. ^ See:
12. ^ Mahon, Basil (2003). The Man Who Changed Everything – the Life of James Clerk Maxwell. Hoboken, NJ: Wiley. ISBN 978-0-470-86171-4. OCLC 52358254.
13. ^ 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. arXiv:1702.01411. Bibcode:2017SHPMP..57...53G. doi:10.1016/j.shpsb.2017.01.001.
14. ^ Maxwell 1875
15. ^ Einstein 1905
16. ^ Smoluchowski 1906
17. ^ Kauzmann, W. (1966). Kinetic Theory of Gasses, W.A. Benjamin, New York, pp. 232–235.
19. ^ The average kinetic energy of a fluid is proportional to the root mean-square velocity, which always exceeds the mean velocity - Kinetic Molecular Theory
20. ^
21. ^ "5.62 Physical Chemistry II" (PDF). MIT OpenCourseWare.
22. ^ S. Chapman and T. G. Cowling, The mathematical theory of non-uniform gases, 3rd. edition, Cambridge University Press, 1990, ISBN 0-521-40844-X, p. 88.
23. ^ Sears, F.W.; Salinger, G.L. (1975). "10". Thermodynamics, Kinetic Theory, and Statistical Thermodynamics (3 ed.). Reading, Massachusetts, USA: Addison-Wesley Publishing Company, Inc. pp. 286–291. ISBN 978-0201068948.
24. ^ Chapman, Sydney; Cowling, T.G. (1970), The Mathematical Theory of Non-Uniform Gases (3rd ed.), Cambridge University Press, pp. 100–101

## References

• 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
• 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 (ed.), 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 978-0-470-86171-4
• Waterston, John James (1843), Thoughts on the Mental Functions (reprinted in his Papers, 3, 167, 183.)

• Sydney Chapman and T. G. Cowling (1939/1970). The Mathematical Theory of Non-uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases, (first edition 1939, second edition 1952), third edition 1970 prepared in co-operation with D. Burnett, Cambridge University Press, London.
• J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird (1964). Molecular Theory of Gases and Liquids, second edition (Wiley).
• R. L. Liboff (2003). Kinetic Theory: Classical, Quantum, and Relativistic Descriptions, third edition (Springer).
• B. Rahimi and H. Struchtrup, Macroscopic and kinetic modelling of rarefied polyatomic gases, Journal of Fluid Mechanics, 806, 437–505, 2016. DOI: https://dx.doi.org/10.1017/jfm.2016.604