Kinetic theory of gases

The kinetic theory of gases is a simple, historically significant 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 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 about 50 BCE, the Roman philosopher Lucretius proposed that apparently static macroscopic bodies were composed on a small scale of rapidly moving atoms all bouncing off each other. This Epicurean atomistic point of view was rarely considered in the subsequent centuries, when 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, 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.
Other pioneers of the kinetic theory, whose work was also largely neglected by their contemporaries, were Mikhail Lomonosov (1747), Georges-Louis Le Sage (ca. 1780, published 1818), John Herapath (1816) and John James Waterston (1843), which connected their research with the development of mechanical explanations of gravitation. In 1856 August Krönig created a simple gas-kinetic model, which only considered the translational motion of the particles.
In 1857 Rudolf Clausius developed a similar, but more sophisticated version of the theory, which included translational and, contrary to Krönig, also rotational and vibrational molecular motions. In this same work he introduced the concept of mean free path of a particle. In 1859, after reading a paper about the diffusion of molecules by Clausius, Scottish physicist James Clerk Maxwell formulated the Maxwell distribution of molecular velocities, which gave the proportion of molecules having a certain velocity in a specific range. This was the first-ever statistical law in physics. Maxwell also gave the first mechanical argument that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium. In his 1873 thirteen page article 'Molecules', Maxwell states: "we are told that an 'atom' is a material point, invested and surrounded by 'potential forces' and that when 'flying molecules' strike against a solid body in constant succession it causes what is called pressure of air and other gases."
In 1871, Ludwig Boltzmann generalized Maxwell's achievement and formulated the Maxwell–Boltzmann distribution. Also the logarithmic connection between entropy and probability was first stated by him.
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) and Marian Smoluchowski's (1906) 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:
Thus, the dynamics of particle motion can be treated classically, and the equations of motion are time-reversible.
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. 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.
Equilibrium properties.
Pressure and kinetic energy.
In the kinetic model of gases, the pressure is equal to the force (per unit area) exerted by the atoms hitting and rebounding from a unit area of the gas container surface. Consider a gas of a large number "N" of molecules, each of mass "m", enclosed in a cube of volume "V" = "L"3. 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:
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 during the time interval formula_2
where "L" is the distance between opposite walls.
The force of this particle's collision with the wall is
The total force on the wall is
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:
By the Pythagorean theorem, in three dimensions the total squared speed "v" is given by
Therefore
and the force can be written as
This force is exerted uniformly on an area "L"2. Therefore, the pressure of the gas is
where "V" = "L"3 is the volume of the box.
In terms of the kinetic energy of the gas "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 formula_13, which is a microscopic property.
Temperature and kinetic energy.
Rewriting the above result for the pressure as formula_14, 
we may combine it with the ideal gas law
where formula_15 is the Boltzmann constant and formula_16 the
absolute temperature defined by the ideal gas law, to obtain
which leads to simplified expression of the average kinetic energy per molecule,
The kinetic energy of the system is N times that of a molecule, namely formula_19.
Then the temperature formula_16 takes the form
which becomes
Eq.()
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.() and
Eq.(),
we have
Thus, the product of pressure and
volume per mole is proportional to the average
(translational) molecular kinetic energy.
Eq.() and Eq.()
are called the "classical results",
which could also be derived from statistical mechanics;
for more details, see:
Since there are
formula_21
degrees of freedom in a monatomic-gas system with
formula_22
particles,
the kinetic energy per degree of freedom per molecule is
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.
This result is related to the equipartition theorem.
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:
At standard temperature (273.15 K), we get:
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 formula_23 and particles obey Maxwell's velocity distribution:
Then the number of particles hitting the area formula_25 with speed formula_26 at angle formula_27 from the normal, in time interval formula_28 is: 
Integrating this over all appropriate velocities within the constraint formula_30 yields the number of atomic or molecular collisions with a wall of a container per unit area per unit time:
This quantity is also known as the "impingement rate" in vacuum physics. 
If this small area formula_32 is punched to become a small hole, the effusive flow rate will be:
Combined with ideal gas law, this yields:
The velocity distribution of particles hitting this small area is:
with the constraint formula_36, and can be determined by normalization condition to be formula_37.
Speed of molecules.
From the kinetic energy formula it can be shown that
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 formula_41 is 81.6% of the rms speed formula_42, and the mean (arithmetic mean, or average) speed formula_43 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 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 formula_44 which increase uniformly with distance formula_45 above the lower plate. The non-equilibrium flow is superimposed on a Maxwell-Boltzmann equilibrium distribution of molecular motions.
Let formula_46 be the collision cross section of one molecule colliding with another. The number density formula_47 is defined as the number of molecules per (extensive) volume formula_48. The collision cross section per volume or collision cross section density is formula_49, and it is related to the mean free path formula_50 by 
Notice that the unit of the collision cross section per volume formula_49 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 formula_53 be the forward velocity of the gas at an imaginary horizontal surface inside the gas layer. The number of molecules arriving at an area formula_25 on one side of the gas layer, with speed formula_26 at angle formula_27 from the normal, in time interval formula_28 is 
These molecules made their last collision at a distance formula_59 above and below the gas layer, and each will contribute a forward momentum of
where plus sign applies to molecules from above, and minus sign below. Note that the forward velocity gradient formula_61 can be considered to be constant over a distance of mean free path. 
Integrating over all appropriate velocities within the constraint 
yields the forward momentum transfer per unit time per unit area (also known as shear stress):
The net rate of momentum per unit area that is transported across the imaginary surface is thus
Combining the above kinetic equation with Newton's law of viscosity
gives the equation for shear viscosity, which is usually denoted formula_66 when it is a dilute gas:
Combining this equation with the equation for mean free path gives
Maxwell-Boltzmann distribution gives the average (equilibrium) molecular speed as
where formula_70 is the most probable speed. We note that
and insert the velocity in the viscosity equation above. This gives the well known equation for shear viscosity for dilute gases:
and formula_73 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
The radius formula_75 is called collision cross section radius or kinetic radius, and the diameter formula_76 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 formula_77 which increases uniformly with distance formula_45 above the lower plate. The non-equilibrium energy flow is superimposed on a Maxwell-Boltzmann equilibrium distribution of molecular motions. 
Let formula_79 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 formula_25 on one side of the gas layer, with speed formula_26 at angle formula_27 from the normal, in time interval formula_28 is 
formula_84
These molecules made their last collision at a distance formula_59 above and below the gas layer, and each will contribute a molecular kinetic energy of
formula_86
where formula_87 is the specific heat capacity. Again, plus sign applies to molecules from above, and minus sign below. Note that the temperature gradient formula_88 can be considered to be constant over a distance of mean free path. 
Integrating over all appropriate velocities within the constraint 
yields the energy transfer per unit time per unit area (also known as heat flux):
formula_90
Note that the energy transfer from above is in the formula_91 direction, and therefore the overall minus sign in the equation. The net heat flux across the imaginary surface is thus
formula_92
Combining the above kinetic equation with Fourier's law
formula_93
gives the equation for thermal conductivity, which is usually denoted formula_94 when it is a dilute gas:
formula_95
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 formula_23 in the layer increases uniformly with distance formula_45 above the lower plate. The non-equilibrium molecular flow is superimposed on a Maxwell-Boltzmann equilibrium distribution of molecular motions. 
Let formula_98 be the number density of the gas at an imaginary horizontal surface inside the layer. The number of molecules arriving at an area formula_25 on one side of the gas layer, with speed formula_26 at angle formula_27 from the normal, in time interval formula_28 is 
formula_84
These molecules made their last collision at a distance formula_59 above and below the gas layer, where the local number density is
formula_105
Again, plus sign applies to molecules from above, and minus sign below. Note that the number density gradient formula_106 can be considered to be constant over a distance of mean free path.
Integrating over all appropriate velocities within the constraint
yields the molecular transfer per unit time per unit area (also known as diffusion flux):
formula_108
Note that the molecular transfer from above is in the formula_91 direction, and therefore the overall minus sign in the equation. The net diffusion flux across the imaginary surface is thus
formula_110
Combining the above kinetic equation with Fick's first law of diffusion
formula_111
gives the equation for mass diffusivity, which is usually denoted formula_112 when it is a dilute gas:
formula_113