# Kinetic theory of gases

The kinetic theory of gases is a simple, historically significant classical 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.

The temperature of the ideal gas is proportional to the average kinetic energy of its particles. The size of helium atoms relative to their spacing is shown to scale under 1950 atmospheres of pressure. The atoms have an average speed relative to their size slowed down here two trillion fold from that at room temperature.

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. 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.

Historically, the kinetic theory of gases was the first explicit exercise of the ideas of statistical mechanics.

## History

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.[1] 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, 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.[2]: 36–37

Other pioneers of the kinetic theory, whose work was also largely neglected by their contemporaries, were Mikhail Lomonosov (1747),[3] Georges-Louis Le Sage (ca. 1780, published 1818),[4] John Herapath (1816)[5] and John James Waterston (1843),[6] 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.[7]

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.[8] 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.[9] This was the first-ever statistical law in physics.[10] Maxwell also gave the first mechanical argument that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium.[11] 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."[12] 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, 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)[13] and Marian Smoluchowski's (1906)[14] papers on Brownian motion, which succeeded in making certain accurate quantitative predictions based on the kinetic theory.

## 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 of 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. All these collisions are perfectly elastic, which means the molecules are perfect hard spheres.
• 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).[15]

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 particles as well as contributions from intermolecular and intramolecular forces as well as quantized molecular rotations, quantum rotational-vibrational symmetry effects, and electronic excitation.[16]

## 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 atoms hitting and rebounding from the gas container's surface. Consider a gas of a large number N of 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 ${\displaystyle v_{x}}$  is the speed of the particle in the x direction (which is the same before and after the collision).

The particle impacts one specific side wall once during the time interval ${\displaystyle \Delta t}$

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

where L is the distance between opposite walls.

The force of this particle's collision with the wall is

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

The total force on the wall due to collisions by molecules impacting the walls with a range of possible values of ${\displaystyle v_{x}}$  is

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

where the bar denotes an average over the possible velocities of 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 the Pythagorean theorem, in three dimensions the average squared speed ${\displaystyle v^{2}}$  is given by

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

Therefore

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

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

and so the force can be written as

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

This force is exerted uniformly 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 translational kinetic energy K of the gas, since

${\displaystyle K=N{\frac {1}{2}}m{\overline {v^{2}}}}$

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

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.

### Temperature and kinetic energy

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

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

(1)

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

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

which leads to a simplified expression of the average kinetic energy per molecule,[17]
${\displaystyle {\frac {1}{2}}m{\overline {v^{2}}}={\frac {3}{2}}k_{\mathrm {B} }T.}$

The kinetic energy of the system is ${\displaystyle N}$  times that of a molecule, namely ${\textstyle K={\frac {1}{2}}Nm{\overline {v^{2}}}}$ . Then the temperature ${\displaystyle T}$  takes the form
${\displaystyle T={m{\overline {v^{2}}} \over 3k_{\mathrm {B} }}}$

(2)

which becomes

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

(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

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

(4)

Thus, the product of pressure and volume per mole is proportional to the average (translational) molecular kinetic energy.

Equations (1) and (4) are called the "classical results", which could also be derived from statistical mechanics; for more details, see:[18]

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

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

(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. This result is related to the equipartition theorem.

Thus the kinetic energy per Kelvin of one mole of (monatomic ideal gas) is 3 [R/2] = 3R/2. Thus the kinetic energy per Kelvin 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.

Although monatomic gases have 3 (translational) degrees of freedom per atom, diatomic gases should have 6 degrees of freedom per molecule (3 translations, two rotations, and one vibration). However, the lighter diatomic gases (such as diatomic oxygen) may act as if they have only 5 due to the strongly quantum-mechanical nature of their vibrations and the large gaps between successive vibrational energy levels. Quantum statistical mechanics is needed to accurately compute these contributions. [19]

### 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[20] 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 ${\displaystyle n=N/V}$  and that the particles obey Maxwell's velocity distribution:

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

Then for a small area ${\displaystyle dA}$  on the container wall, a particle with speed ${\displaystyle v}$  at angle ${\displaystyle \theta }$  from the normal of the area ${\displaystyle dA}$ , will collide with the area within time interval ${\displaystyle dt}$ , if it is within the distance ${\displaystyle vdt}$  from the area ${\displaystyle dA}$ . Therefore, all the particles with speed ${\displaystyle v}$  at angle ${\displaystyle \theta }$  from the normal that can reach area ${\displaystyle dA}$  within time interval ${\displaystyle dt}$  are contained in the tilted pipe with a height of ${\displaystyle v\cos(\theta )dt}$  and a volume of ${\displaystyle v\cos(\theta )dAdt}$ .

The total number of particles that reach area ${\displaystyle dA}$  within time interval ${\displaystyle dt}$  also depends on the velocity distribution; All in all, it calculates to be:

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

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

${\displaystyle J_{\text{collision}}={\frac {\int _{0}^{\pi /2}\cos \theta \sin \theta d\theta }{\int _{0}^{\pi }\sin \theta d\theta }}\times n{\bar {v}}={\frac {1}{4}}n{\bar {v}}={\frac {n}{4}}{\sqrt {\frac {8k_{\mathrm {B} }T}{\pi m}}}.}$

This quantity is also known as the "impingement rate" in vacuum physics. Note that to calculate the average speed ${\displaystyle {\bar {v}}}$  of the Maxwell's velocity distribution, one has to integrate over${\displaystyle v>0,0<\theta <\pi ,0<\phi <2\pi }$ .

The momentum transfer to the container wall from particles hitting the area ${\displaystyle dA}$  with speed ${\displaystyle v}$  at angle ${\displaystyle \theta }$  from the normal, in time interval ${\displaystyle dt}$  is:

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

Integrating this over all appropriate velocities within the constraint ${\displaystyle v>0,0<\theta <\pi /2,0<\phi <2\pi }$  yields the pressure (consistent with Ideal gas law):
${\displaystyle P={\frac {2\int _{0}^{\pi /2}\cos ^{2}\theta \sin \theta d\theta }{\int _{0}^{\pi }\sin \theta d\theta }}\times nmv_{\text{rms}}^{2}={\frac {1}{3}}nmv_{\text{rms}}^{2}={\frac {2}{3}}n\langle E_{kin}\rangle =nk_{\mathrm {B} }T}$

If this small area ${\displaystyle A}$  is punched to become a small hole, the effusive flow rate will be:
${\displaystyle \Phi _{\text{effusion}}=J_{\text{collision}}A=nA{\sqrt {\frac {k_{\mathrm {B} }T}{2\pi m}}}.}$

Combined with the ideal gas law, this yields

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

The above expression is consistent with Graham's law.

To calculate the velocity distribution of particles hitting this small area, we must take into account that all the particles with ${\displaystyle (v,\theta ,\phi )}$  that hit the area ${\displaystyle dA}$  within the time interval ${\displaystyle dt}$  are contained in the tilted pipe with a height of ${\displaystyle v\cos(\theta )dt}$  and a volume of ${\displaystyle v\cos(\theta )dAdt}$ ; Therefore, compared to the Maxwell distribution, the velocity distribution will have an extra factor of ${\displaystyle v\cos \theta }$ :

{\displaystyle {\begin{aligned}f(v,\theta ,\phi )\,dv\,d\theta \,d\phi &=\lambda v\cos {\theta }{\times }\left({\frac {m}{2\pi kT}}\right)^{3/2}e^{-{\frac {mv^{2}}{2k_{\mathrm {B} }T}}}(v^{2}\sin {\theta }\,dv\,d\theta \,d\phi )\\\end{aligned}}}

with the constraint ${\textstyle v>0,\,0<\theta <{\frac {\pi }{2}},\,0<\phi <2\pi }$ . The constant ${\displaystyle \lambda }$  can be determined by the normalization condition ${\displaystyle \int f(v,\theta ,\phi )\,dv\,d\theta \,d\phi =1}$  to be ${\textstyle {4}/{\bar {v}}}$ , and overall:
{\displaystyle {\begin{aligned}f(v,\theta ,\phi )\,dv\,d\theta \,d\phi &={\frac {1}{2\pi }}\left({\frac {m}{k_{\mathrm {B} }T}}\right)^{2}e^{-{\frac {mv^{2}}{2k_{\mathrm {B} }T}}}(v^{3}\sin {\theta }\cos {\theta }\,dv\,d\theta \,d\phi )\\\end{aligned}};\quad v>0,\,0<\theta <{\frac {\pi }{2}},\,0<\phi <2\pi }

### Speed of molecules

From the kinetic energy formula it can be shown that

${\displaystyle v_{\text{p}}={\sqrt {2\cdot {\frac {k_{\mathrm {B} }T}{m}}}},}$

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

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

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 root-mean-square 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:

### Mean free path

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

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

Notice that the unit of the collision cross section per volume ${\displaystyle n\sigma }$  is reciprocal of length.

## 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[21] 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 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 ${\displaystyle u}$  which increase uniformly with distance ${\displaystyle y}$  above the lower plate. The non-equilibrium flow is superimposed on a Maxwell-Boltzmann equilibrium distribution of molecular motions.

Inside a dilute gas in a Couette flow setup, let ${\displaystyle u_{0}}$  be the forward velocity of the gas at a horizontal flat layer (labeled as ${\displaystyle y=0}$ ); ${\displaystyle u_{0}}$  is along the horizontal direction. The number of molecules arriving at the area ${\displaystyle dA}$  on one side of the gas layer, with speed ${\displaystyle v}$  at angle ${\displaystyle \theta }$  from the normal, in time interval ${\displaystyle dt}$  is

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

These molecules made their last collision at ${\displaystyle y=\pm l\cos \theta }$ , where ${\displaystyle l}$  is the mean free path. Each molecule will contribute a forward momentum of

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

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

Integrating over all appropriate velocities within the constraint

${\displaystyle {\begin{cases}v>0\\0<\theta <\pi /2\\0<\phi <2\pi \end{cases}}}$

yields the forward momentum transfer per unit time per unit area (also known as shear stress):
${\displaystyle \tau ^{\pm }={\frac {1}{4}}{\bar {v}}n\cdot m\left(u_{0}\pm {\frac {2}{3}}l\,{du \over dy}\right)}$

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

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

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

${\displaystyle \tau =\eta \,{du \over dy}}$

gives the equation for shear viscosity, which is usually denoted ${\displaystyle \eta _{0}}$  when it is a dilute gas:
${\displaystyle \eta _{0}={\frac {1}{3}}{\bar {v}}nml}$

Combining this equation with the equation for mean free path gives

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

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

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

where ${\displaystyle v_{p}}$  is the most probable speed. We note that
${\displaystyle k_{B}\cdot N_{A}=R\quad {\text{and}}\quad M=m\cdot N_{A}}$

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

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

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

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

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 an 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[21] 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 ${\displaystyle \varepsilon }$  which increases uniformly with distance ${\displaystyle y}$  above the lower plate. The non-equilibrium energy flow is superimposed on a Maxwell-Boltzmann equilibrium distribution of molecular motions.

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

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

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

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

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

Integrating over all appropriate velocities within the constraint

${\displaystyle {\begin{cases}v>0\\0<\theta <\pi /2\\0<\phi <2\pi \end{cases}}}$

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

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

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

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

Combining the above kinetic equation with Fourier's law

${\displaystyle q=-\kappa \,{dT \over dy}}$

gives the equation for thermal conductivity, which is usually denoted ${\displaystyle \kappa _{0}}$  when it is a dilute gas:
${\displaystyle \kappa _{0}={\frac {1}{3}}{\bar {v}}nmc_{v}l}$

### Diffusion Coefficient and diffusion flux

Following a similar logic as above, one can derive the kinetic model for mass diffusivity[21] 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 ${\displaystyle n}$  in the layer increases uniformly with distance ${\displaystyle y}$  above the lower plate. The non-equilibrium molecular flow is superimposed on a Maxwell-Boltzmann equilibrium distribution of molecular motions.

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

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

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

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

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

Integrating over all appropriate velocities within the constraint

${\displaystyle {\begin{cases}v>0\\0<\theta <\pi /2\\0<\phi <2\pi \end{cases}}}$

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

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

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

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

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

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

gives the equation for mass diffusivity, which is usually denoted ${\displaystyle D_{0}}$  when it is a dilute gas:
${\displaystyle D_{0}={\frac {1}{3}}{\bar {v}}l}$

## Detailed balance

### Fluctuation and dissipation

The kinetic theory of gases entails that due to the microscopic reversibility of the gas particles' detailed dynamics, the system must obey the principle of detailed balance. Specifically, the fluctuation-dissipation theorem applies, which leads to the Einstein–Smoluchowski equation:[22]

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

where

### Onsager reciprocal relations

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 the Onsager reciprocal relations, which is connected to the detailed balance of the system.

## Notes

1. ^ 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. S2CID 96568430.
2. ^ L.I Ponomarev; I.V Kurchatov (1 January 1993). The Quantum Dice. CRC Press. ISBN 978-0-7503-0251-7.
3. ^ Lomonosov 1758
4. ^ Le Sage 1780/1818
5. ^ Herapath 1816, 1821
6. ^ Waterston 1843
7. ^ Krönig 1856
8. ^ Clausius 1857
9. ^ See:
10. ^ Mahon, Basil (2003). The Man Who Changed Everything – the Life of James Clerk Maxwell. Hoboken, NJ: Wiley. ISBN 0-470-86171-1. OCLC 52358254.
11. ^ 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. S2CID 38272381.
12. ^ Maxwell 1873
13. ^ Einstein 1905
14. ^ Smoluchowski 1906
15. ^ Chang, Raymond; Thoman, John W. Jr. (2014). Physical Chemistry for the Chemical Sciences. New York, NY: University Science Books. p. 37.
16. ^ McQuarrie, Donald A. (1976). Statistical Mechanics. New York, NY: University Science Press.
17. ^ The average kinetic energy of a fluid is proportional to the root mean-square velocity, which always exceeds the mean velocity - Kinetic Molecular Theory
18. ^
19. ^ Chang, Raymond; Thoman, John W. Jr. (2014). Physical Chemistry for the Chemical Sciences. New York: University Science Books. pp. 56–61.
20. ^ "5.62 Physical Chemistry II" (PDF). MIT OpenCourseWare.
21. ^ a b c 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.
22. ^ Dill, Ken A.; Bromberg, Sarina (2003). Molecular Driving Forces: Statistical Thermodynamics in Chemistry and Biology. Garland Science. p. 327. ISBN 9780815320517.

## 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
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