Editing Kinetic theory of gases (section)

== Transport properties ==
{{See also|Transport phenomena}}

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]], [[mass diffusivity]] and [[Thermophoresis|thermal diffusion]].

In its most basic form, Kinetic gas theory is only applicable to dilute gases. The extension of Kinetic gas theory to dense gas mixtures, [[Chapman–Enskog theory#Revised Enskog Theory|Revised Enskog Theory]], was developed in 1983-1987 by [[E. G. D. Cohen]], [[J. M. Kincaid]] and [[M. Lòpez de Haro]],<ref>{{cite journal |last1=Lòpez de Haro |first1=M. |last2=Cohen |first2=E. G. D. |last3=Kincaid |first3=J. M. |title=The Enskog theory for multicomponent mixtures. I. Linear transport theory |journal=The Journal of Chemical Physics |date=1983 |volume=78 |issue=5 |pages=2746–2759 |doi=10.1063/1.444985 |bibcode=1983JChPh..78.2746L |url=https://doi.org/10.1063/1.444985|url-access=subscription }}</ref><ref>{{cite journal |last1=Kincaid |first1=J. M. |last2=Lòpez de Haro |first2=M. |last3=Cohen |first3=E. G. D. |title=The Enskog theory for multicomponent mixtures. II. Mutual diffusion |journal=The Journal of Chemical Physics |date=1983 |volume=79 |issue=9 |pages=4509–4521 |doi=10.1063/1.446388 |url=https://doi.org/10.1063/1.446388|url-access=subscription }}</ref><ref>{{cite journal |last1=Lòpez de Haro |first1=M. |last2=Cohen |first2=E. G. D. |title=The Enskog theory for multicomponent mixtures. III. Transport properties of dense binary mixtures with one tracer component |journal=The Journal of Chemical Physics |date=1984 |volume=80 |issue=1 |pages=408–415 |doi=10.1063/1.446463 |bibcode=1984JChPh..80..408L |url=https://doi.org/10.1063/1.446463|url-access=subscription }}</ref><ref>{{cite journal |last1=Kincaid |first1=J. M. |last2=Cohen |first2=E. G. D. |last3=Lòpez de Haro |first3=M. |title=The Enskog theory for multicomponent mixtures. IV. Thermal diffusion |journal=The Journal of Chemical Physics |date=1987 |volume=86 |issue=2 |pages=963–975 |doi=10.1063/1.452243 |bibcode=1987JChPh..86..963K |url=https://doi.org/10.1063/1.452243|url-access=subscription }}</ref> building on work by [[H. van Beijeren]] and [[M. H. Ernst]].<ref>{{cite journal |last1=van Beijeren |first1=H. |last2=Ernst |first2=M. H. |title=The non-linear Enskog-Boltzmann equation |journal=Physics Letters A |date=1973 |volume=43 |issue=4 |pages=367–368 |doi=10.1016/0375-9601(73)90346-0 |bibcode=1973PhLA...43..367V |hdl=1874/36979 |url=https://doi.org/10.1016/0375-9601(73)90346-0|hdl-access=free }}</ref>

=== Viscosity and kinetic momentum ===
{{See also|Viscosity#Momentum transport}}

In books on elementary kinetic theory<ref name="Sears1975">{{cite book|last1=Sears|first1=F.W.|last2=Salinger|first2=G.L.|year=1975|title=Thermodynamics, Kinetic Theory, and Statistical Thermodynamics|publisher=Addison-Wesley Publishing Company, Inc.|location= Reading, Massachusetts, USA |edition=3|chapter=10|pages=286–291|isbn=978-0201068948}}</ref> 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 <math>u</math> which increase uniformly with distance <math>y</math> above the lower plate. The non-equilibrium flow is superimposed on a [[Maxwell-Boltzmann distribution|Maxwell-Boltzmann equilibrium distribution]] of molecular motions.

Inside a dilute gas in a [[Couette flow]] setup, let <math> u_0 </math> be the forward velocity of the gas at a horizontal flat layer (labeled as <math>y=0</math>); <math> u_0 </math> is along the horizontal direction. The number of molecules arriving at the area <math>dA</math> on one side of the gas layer, with speed <math>v</math> at angle <math>\theta</math> from the normal, in time interval <math>dt</math> is
<math display="block">nv\cos({\theta})\, dA \, dt \times \left(\frac{m}{2 \pi k_\mathrm{B} T}\right)^{3/2} \, e^{- \frac{mv^2}{2 k_\mathrm{B} T}} (v^2\sin{\theta} \, dv \, d\theta \, d\phi)</math>

These molecules made their last collision at <math>y = \pm \ell\cos \theta</math>, where <math>\ell</math> is the [[Mean free path#Kinetic theory|mean free path]]. Each molecule will contribute a forward momentum of
<math display="block">p_x^{\pm} = m \left( u_0 \pm \ell \cos \theta \frac{du}{dy} \right), </math>
where plus sign applies to molecules from above, and minus sign below. Note that the forward velocity gradient <math>du/dy</math> can be considered to be constant over a distance of mean free path.

Integrating over all appropriate velocities within the constraint <math>v > 0</math>, <math display="inline">0 < \theta < \frac{\pi}{2}</math>, <math>0 < \phi < 2\pi</math> yields the forward momentum transfer per unit time per unit area (also known as [[shear stress]]):
<math display="block">\tau^{\pm} = \frac{1}{4} \bar{v} n \cdot m \left( u_0 \pm \frac{2}{3} \ell \frac{du}{dy} \right) </math>

The net rate of momentum per unit area that is transported across the imaginary surface is thus
<math display="block">\tau = \tau^{+} - \tau^{-} = \frac {1}{3} \bar v n m \cdot \ell \frac{du}{dy} </math>

Combining the above kinetic equation with [[Newton's law of viscosity]]
<math display="block">\tau = \eta \frac{du}{dy} </math>
gives the equation for shear viscosity, which is usually denoted <math> \eta_0 </math> when it is a dilute gas:
<math display="block">\eta_0 = \frac{1}{3} \bar{v} n m \ell </math>

Combining this equation with the equation for mean free path gives
<math display="block">\eta_0 = \frac {1} {3 \sqrt{2}} \frac{m \bar{v}}{\sigma}</math>

Maxwell-Boltzmann distribution gives the average (equilibrium) molecular speed as
<math display="block">\bar v = \frac{2}{\sqrt{\pi}} v_p = 2 \sqrt{\frac{2}{\pi} \frac {k_\mathrm{B}T}{m}} </math>
where <math>v_p</math> is the most probable speed. We note that
<math display="block">k_\text{B} N_\text{A} = R \quad \text{and} \quad M = m N_\text{A} </math>

and insert the velocity in the viscosity equation above. This gives the well known equation <ref name="Hildebrand1976">{{cite journal| journal = Proc Natl Acad Sci U S A |last1=Hildebrand |first1=J.H.|year=1976 | volume=76 |title=Viscosity of dilute gases and vapors|issue=12 | pages= 4302–4303 |doi=10.1073/pnas.73.12.4302 |pmid=16592372 |pmc=431439 |bibcode=1976PNAS...73.4302H |doi-access=free }}</ref> (with <math>\sigma</math> subsequently estimated below) for [[Viscosity models for mixtures#Dilute gas limit and scaled variables|shear viscosity for dilute gases]]:
<math display="block">\eta_0
= \frac {2} {3 \sqrt{\pi} } \cdot \frac {\sqrt{m k_\mathrm{B} T}} { \sigma }
= \frac {2} {3 \sqrt{\pi} } \cdot \frac {\sqrt{M R T}} { \sigma N_\text{A} } </math>
and <math> M </math> is the [[molar mass]]. The equation above presupposes that the gas density is low (i.e. the pressure is low). This implies that the transport of momentum through the gas due to the translational motion of molecules is much larger than the transport due to momentum being transferred between molecules during collisions. The transfer of momentum between molecules is explicitly accounted for in [[Revised Enskog theory]], which relaxes the requirement of a gas being dilute. The viscosity equation further presupposes that there is only one type of gas molecules, and that the gas molecules are perfect elastic and hard core particles of spherical shape. This assumption of elastic, hard core spherical molecules, like billiard balls, implies that the collision cross section of one molecule can be estimated by
<math display="block">\sigma = \pi \left( 2 r \right)^2 = \pi d^2 </math>

The radius <math>r</math> is called collision cross section radius or kinetic radius, and the diameter <math>d</math> 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 (physics)#Collision among particles|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. [[Mean free path#Mean free path in kinetic theory|The radius for zero Lennard-Jones potential]] may then be used as a rough estimate for the kinetic radius. However, using this estimate will typically lead to an erroneous temperature dependency of the viscosity. For such interaction potentials, significantly more accurate results are obtained by numerical evaluation of the required [[Cross section (physics)|collision integrals]].

The expression for viscosity obtained from [[Revised Enskog theory|Revised Enskog Theory]] reduces to the above expression in the limit of infinite dilution, and can be written as
<math display="block">\eta = (1 + \alpha_\eta)\eta_0 + \eta_c </math>
where <math>\alpha_\eta</math> is a term that tends to zero in the limit of infinite dilution that accounts for excluded volume, and <math>\eta_c</math> is a term accounting for the transfer of momentum over a non-zero distance between particles during a collision.

=== Thermal conductivity and heat flux ===
{{See also|Thermal conductivity}}
Following a similar logic as above, one can derive the kinetic model for [[thermal conductivity]]<ref name="Sears1975" /> 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 reservoir]]s. The upper plate has a higher temperature than the lower plate. The molecules in the gas layer have a molecular kinetic energy <math>\varepsilon</math> which increases uniformly with distance <math>y</math> above the lower plate. The non-equilibrium energy flow is superimposed on a [[Maxwell-Boltzmann distribution|Maxwell-Boltzmann equilibrium distribution]] of molecular motions.

Let <math> \varepsilon_0 </math> 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 <math>dA</math> on one side of the gas layer, with speed <math>v</math> at angle <math>\theta</math> from the normal, in time interval <math>dt</math> is
<math display="block"> nv \cos(\theta)\, dA \, dt \times \left(\frac{m}{2 \pi k_\mathrm{B}T}\right)^{3 / 2} e^{- \frac{mv^2}{2k_\text{B}T}} (v^2 \sin(\theta) \, dv \, d\theta \, d\phi)</math>

These molecules made their last collision at a distance <math>\ell\cos \theta</math> above and below the gas layer, and each will contribute a molecular kinetic energy of
<math display="block"> \varepsilon^{\pm} = \left( \varepsilon_0 \pm m c_v \ell \cos \theta \, \frac{dT}{dy} \right), </math>
where <math>c_v</math> is the [[specific heat capacity]]. Again, plus sign applies to molecules from above, and minus sign below. Note that the temperature gradient <math>dT/dy</math> can be considered to be constant over a distance of mean free path.

Integrating over all appropriate velocities within the constraint <math>v > 0 </math>, <math display="inline">0 < \theta < \frac{\pi}{2}</math>, <math>0 < \phi < 2\pi</math> yields the energy transfer per unit time per unit area (also known as [[heat flux]]):
<math display="block"> q_y^{\pm} = -\frac{1}{4} \bar v n \cdot \left( \varepsilon_0 \pm \frac {2}{3} m c_v \ell \frac{dT}{dy} \right) </math>

Note that the energy transfer from above is in the <math>-y</math> direction, and therefore the overall minus sign in the equation. The net heat flux across the imaginary surface is thus
<math display="block"> q = q_y^{+} - q_y^{-} = -\frac{1}{3} \bar{v} n m c_v \ell \,\frac{dT}{dy} </math>

Combining the above kinetic equation with [[Fourier's law]]
<math display="block"> q = -\kappa \, \frac{dT}{dy} </math>
gives the equation for thermal conductivity, which is usually denoted <math> \kappa_0 </math> when it is a dilute gas:
<math display="block"> \kappa_0 = \frac{1}{3} \bar{v} n m c_v \ell </math>

Similarly to viscosity, [[Revised Enskog theory]] yields an expression for thermal conductivity that reduces to the above expression in the limit of infinite dilution, and which can be written as
<math display="block"> \kappa = \alpha_\kappa \kappa_0 + \kappa_c </math>
where <math> \alpha_\kappa </math> is a term that tends to unity in the limit of infinite dilution, accounting for excluded volume, and <math> \kappa_c </math> is a term accounting for the transfer of energy across a non-zero distance between particles during a collision.

=== Diffusion coefficient and diffusion flux ===
{{See also|Fick's laws of diffusion}}Following a similar logic as above, one can derive the kinetic model for [[mass diffusivity]]<ref name="Sears1975" /> of a dilute gas:

Consider a [[Steady state|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 density|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 <math>n</math> in the layer increases uniformly with distance <math>y</math> above the lower plate. The non-equilibrium molecular flow is superimposed on a [[Maxwell–Boltzmann distribution|Maxwell–Boltzmann equilibrium distribution]] of molecular motions.

Let <math> n_0 </math> be the number density of the gas at an imaginary horizontal surface inside the layer. The number of molecules arriving at an area <math>dA</math> on one side of the gas layer, with speed <math>v</math> at angle <math>\theta</math> from the normal, in time interval <math>dt</math> is
<math display="block"> nv\cos(\theta) \, dA \, dt \times \left(\frac{m}{2 \pi k_\mathrm{B}T}\right)^{3 / 2} e^{- \frac{mv^2}{2k_\text{B}T}} (v^2\sin(\theta) \, dv\, d\theta \, d\phi)</math>

These molecules made their last collision at a distance <math>\ell\cos \theta</math> above and below the gas layer, where the local number density is
<math display="block"> n^{\pm} = \left( n_0 \pm \ell \cos \theta \, \frac{dn}{dy} \right) </math>

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

Integrating over all appropriate velocities within the constraint <math>v > 0</math>, <math display="inline">0 < \theta < \frac{\pi}{2} </math>, <math>0 < \phi < 2\pi</math> yields the molecular transfer per unit time per unit area (also known as [[diffusion flux]]):
<math display="block"> J_y^{\pm} = - \frac{1}{4} \bar v \cdot \left( n_0 \pm \frac{2}{3} \ell \, \frac{dn}{dy} \right) </math>

Note that the molecular transfer from above is in the <math>-y</math> direction, and therefore the overall minus sign in the equation. The net diffusion flux across the imaginary surface is thus
<math display="block"> J = J_y^{+} - J_y^{-} = -\frac {1}{3} \bar{v} \ell \frac{dn}{dy} </math>

Combining the above kinetic equation with [[Fick's laws of diffusion#Fick's first law|Fick's first law of diffusion]]
<math display="block"> J = - D \frac{dn}{dy} </math>
gives the equation for mass diffusivity, which is usually denoted <math> D_0 </math> when it is a dilute gas:
<math display="block"> D_0 = \frac{1}{3} \bar{v} \ell </math>

The corresponding expression obtained from [[Revised Enskog theory|Revised Enskog Theory]] may be written as
<math display="block"> D = \alpha_D D_0 </math>
where <math> \alpha_D </math> is a factor that tends to unity in the limit of infinite dilution, which accounts for excluded volume and the variation [[chemical potential]]s with density.