Kinetic theory

Kinetic theory

Overview

This article applies to gases; see also Kinetic theory of solids


The kinetic theory of gases describes a gas as a large number of small particles (atom
Atom
The atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...

s or molecule
Molecule
A molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...

s), all of which are in constant, random
Randomness
Randomness has somewhat differing meanings as used in various fields. It also has common meanings which are connected to the notion of predictability of events....

 motion
Motion (physics)
In physics, motion is a change in position of an object with respect to time. Change in action is the result of an unbalanced force. Motion is typically described in terms of velocity, acceleration, displacement and time . An object's velocity cannot change unless it is acted upon by a force, as...

. The rapidly moving particles constantly collide with each other and with the walls of the container. Kinetic theory explains macroscopic
Macroscopic
The macroscopic scale is the length scale on which objects or processes are of a size which is measurable and observable by the naked eye.When applied to phenomena and abstract objects, the macroscopic scale describes existence in the world as we perceive it, often in contrast to experiences or...

 properties of gases, such as pressure, temperature, or volume, by considering their molecular composition and motion.
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Encyclopedia

This article applies to gases; see also Kinetic theory of solids


The kinetic theory of gases describes a gas as a large number of small particles (atom
Atom
The atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...

s or molecule
Molecule
A molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...

s), all of which are in constant, random
Randomness
Randomness has somewhat differing meanings as used in various fields. It also has common meanings which are connected to the notion of predictability of events....

 motion
Motion (physics)
In physics, motion is a change in position of an object with respect to time. Change in action is the result of an unbalanced force. Motion is typically described in terms of velocity, acceleration, displacement and time . An object's velocity cannot change unless it is acted upon by a force, as...

. The rapidly moving particles constantly collide with each other and with the walls of the container. Kinetic theory explains macroscopic
Macroscopic
The macroscopic scale is the length scale on which objects or processes are of a size which is measurable and observable by the naked eye.When applied to phenomena and abstract objects, the macroscopic scale describes existence in the world as we perceive it, often in contrast to experiences or...

 properties of gases, such as pressure, temperature, or volume, by considering their molecular composition and motion. Essentially, the theory posits that pressure is due not to static repulsion between molecules, as was Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

's conjecture, but due to collision
Collision
A collision is an isolated event which two or more moving bodies exert forces on each other for a relatively short time.Although the most common colloquial use of the word "collision" refers to accidents in which two or more objects collide, the scientific use of the word "collision" implies...

s between molecules moving at different velocities. Referring to this theory, all it is trying to say is that the collision between gas molecules and their container affects the pressure of the gas. As well as the temperature changes affect the volume and pressure of such gases.

While the particles making up a gas are too small to be visible, the jiterring motion of pollen grains or dust particles which can be seen under a microscope, known as Brownian motion
Brownian motion
Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...

, results directly from collisions between the particle and gas molecules. As pointed out by Albert Einstein
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

 in 1905, this experimental evidence for kinetic theory is generally seen as having confirmed the existence of atoms and molecules.

Postulates


The theory for ideal gases makes the following assumptions:
  • The gas consists of very small particles. This smallness of their size is such that the total volume
    Volume
    Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....

     of the individual gas molecules added up is negligible compared to the volume of the container. This is equivalent to stating that the average distance separating the gas particles is large compared to their size
    Dimension
    In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...

    .

  • These particles have the same mass
    Mass
    Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...

    .
  • The number of molecules is so large that statistical treatment can be applied.
  • These molecules are in constant, random
    Randomness
    Randomness has somewhat differing meanings as used in various fields. It also has common meanings which are connected to the notion of predictability of events....

     and rapid motion.
  • The rapidly moving particles constantly collide among themselves and with the walls of the container. All these collisions are perfectly elastic. This means, the molecules are considered to be perfectly spherical in shape, and elastic in nature.
  • Except during collisions, the interaction
    Interaction
    Interaction is a kind of action that occurs as two or more objects have an effect upon one another. The idea of a two-way effect is essential in the concept of interaction, as opposed to a one-way causal effect...

    s among molecules are negligible
    Negligible
    Negligible refers to the quantities so small that they can be ignored when studying the larger effect. Although related to the more mathematical concepts of infinitesimal, the idea of negligibility is particularly useful in practical disciplines like physics, chemistry, mechanical and electronic...

    . (That is, they exert no force
    Force
    In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...

    s on one another.)
This implies:
1. Relativistic
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

 effects are negligible.
2. Quantum-mechanical
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

 effects are negligible. This means that the inter-particle distance
Mean inter-particle distance
Mean inter-particle distance is the mean distance between microscopic particles in a macroscopic body.-Ambiguity:...

 is much larger than the thermal de Broglie wavelength and the molecules are treated as classical
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...

 objects
Physical body
In physics, a physical body or physical object is a collection of masses, taken to be one...

.
3. Because of the above two, their dynamics can be treated classically. This means, the equations of motion of the molecules are time-reversible.

  • The average kinetic energy
    Kinetic energy
    The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...

     of the gas particles depends only on the temperature
    Thermodynamic temperature
    Thermodynamic temperature is the absolute measure of temperature and is one of the principal parameters of thermodynamics. Thermodynamic temperature is an "absolute" scale because it is the measure of the fundamental property underlying temperature: its null or zero point, absolute zero, is the...

     of the system
    System
    System is a set of interacting or interdependent components forming an integrated whole....

    .
  • The time during collision of molecule with the container's wall is negligible as compared to the time between successive collisions.


More modern developments relax these assumptions and are based on the Boltzmann equation
Boltzmann equation
The Boltzmann equation, also often known as the Boltzmann transport equation, devised by Ludwig Boltzmann, describes the statistical distribution of one particle in rarefied gas...

. 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
Molecular chaos
In kinetic theory in physics, molecular chaos is the assumption that the velocities of colliding particles are uncorrelated, and independent of position...

 and small gradients in bulk properties. Expansions to higher orders in the density are known as virial expansion
Virial expansion
The classical virial expansion expresses the pressure of a many-particle system in equilibrium as a power series in the density.The virial expansion was introduced in 1901 by Heike Kamerlingh Onnesas a generalization of the ideal gas law...

s. The definitive work is the book by Chapman and Enskog but there have been many modern developments and there is an alternative approach developed by Grad based on moment expansions.
In the other limit, for extremely rarefied gases, the gradients in bulk properties are not small compared to the mean free paths. This is known as the Knudsen regime and expansions can be performed in the Knudsen number
Knudsen number
The Knudsen number is a dimensionless number defined as the ratio of the molecular mean free path length to a representative physical length scale. This length scale could be, for example, the radius of a body in a fluid...

.

Pressure and kinetic energy


Pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

 is explained by kinetic theory as arising from the force exerted by molecules or atoms impacting on the walls of a container. Consider a gas of N molecules, each of mass m, enclosed in a cuboidal container of volume V=L3. When a gas molecule collides with the wall of the container perpendicular to the x coordinate axis and bounces off in the opposite direction with the same speed (an elastic collision
Elastic collision
An elastic collision is an encounter between two bodies in which the total kinetic energy of the two bodies after the encounter is equal to their total kinetic energy before the encounter...

), then the momentum
Momentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...

 lost by the particle and gained by the wall is:


where vx is the x-component of the initial velocity of the particle.

The particle impacts one specific side wall once every


(where L is the distance between opposite walls).

The force
Force
In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...

 due to this particle is:


The total force on the wall is


where the bar denotes an average over the N particles.
Since the assumption is that the particles move in random directions, we will have to conclude that if we divide the velocity vectors of all particles in three mutually perpendicular directions, the average value along each direction must be same. (This does not mean that each particle always travel in 45 degrees to the coordinate axes.)

.

We can rewrite the force as


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


where V=L3 is the volume of the box.
The fraction n=N/V is the number density
Number density
In physics, astronomy, and chemistry, number density is an intensive quantity used to describe the degree of concentration of countable objects in the three-dimensional physical space...

 of the gas (the mass density ρ=nm is less convenient for theoretical derivations on atomic level). Using n, we can rewrite the pressure as


This is a first non-trivial result of the kinetic theory because it relates pressure, a macroscopic
Macroscopic
The macroscopic scale is the length scale on which objects or processes are of a size which is measurable and observable by the naked eye.When applied to phenomena and abstract objects, the macroscopic scale describes existence in the world as we perceive it, often in contrast to experiences or...

 property, to the average (translational) kinetic energy
Kinetic energy
The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...

 per molecule which is a microscopic
Microscopic
The microscopic scale is the scale of size or length used to describe objects smaller than those that can easily be seen by the naked eye and which require a lens or microscope to see them clearly.-History:...

 property.

Temperature and kinetic energy


From the ideal gas law
Ideal gas law
The ideal gas law is the equation of state of a hypothetical ideal gas. It is a good approximation to the behavior of many gases under many conditions, although it has several limitations. It was first stated by Émile Clapeyron in 1834 as a combination of Boyle's law and Charles's law...



(1)

where is the Boltzmann constant, and the
absolute
Thermodynamic temperature
Thermodynamic temperature is the absolute measure of temperature and is one of the principal parameters of thermodynamics. Thermodynamic temperature is an "absolute" scale because it is the measure of the fundamental property underlying temperature: its null or zero point, absolute zero, is the...

 temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...

,

and from the above result we have

we have

then the temperature takes the form

(2)

which leads to the expression of the kinetic energy of a molecule

The kinetic energy of the system is N times that of a molecule

The temperature becomes

(3)

Eq.(3)1
is one important result of the kinetic
theory:
The average molecular kinetic energy is proportional to
the absolute temperature
.
From Eq.(1) and
Eq.(3)1,
we have

(4)

Thus, the product of pressure and
volume per mole
Mole (unit)
The mole is a unit of measurement used in chemistry to express amounts of a chemical substance, defined as an amount of a substance that contains as many elementary entities as there are atoms in 12 grams of pure carbon-12 , the isotope of carbon with atomic weight 12. This corresponds to a value...

 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
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...

;
for more details, see
.

Since there are

degrees of freedom
Degrees of freedom (physics and chemistry)
A degree of freedom is an independent physical parameter, often called a dimension, in the formal description of the state of a physical system...

 in a monoatomic-gas system with

particles,
the kinetic energy per degree of freedom per molecule is

(5)

In the kinetic energy per degree of freedom,
the constant of proportionality of temperature
is 1/2 times
Boltzmann constant. In addition to this, the temperature will decrease when the pressure drops to a certain point.
This result is related
to the equipartition theorem
Equipartition theorem
In classical statistical mechanics, the equipartition theorem is a general formula that relates the temperature of a system with its average energies. The equipartition theorem is also known as the law of equipartition, equipartition of energy, or simply equipartition...

.

As noted in the article on heat capacity
Heat capacity
Heat capacity , or thermal capacity, is the measurable physical quantity that characterizes the amount of heat required to change a substance's temperature by a given amount...

, diatomic
gases should have 7 degrees of freedom, but the lighter gases act
as if they have only 5.

Thus the kinetic energy per kelvin (monatomic ideal gas
Ideal gas
An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At normal conditions such as...

) is:
  • per mole: 12.47 J
  • per molecule: 20.7 yJ = 129 μeV.


At standard temperature
Standard conditions for temperature and pressure
Standard condition for temperature and pressure are standard sets of conditions for experimental measurements established to allow comparisons to be made between different sets of data...

 (273.15 K), we get:
  • per mole: 3406 J
  • per molecule: 5.65 zJ = 35.2 meV.

Collisions with container


One can calculate the number of atomic or molecular collisions with a wall of a container per unit area per unit time.

Assuming an ideal gas, a derivation results in an equation for total number of collisions per unit time per area:

Speed of molecules


From the kinetic energy formula it can be shown that


with v in m/s, T in kelvins, and R is the gas constant
Gas constant
The gas constant is a physical constant which is featured in many fundamental equations in the physical sciences, such as the ideal gas law and the Nernst equation. It is equivalent to the Boltzmann constant, but expressed in units of energy The gas constant (also known as the molar, universal,...

. The molar mass is given as kg/mol. The most probable speed is 81.6% of the rms speed, and the mean speeds 92.1% (distribution of speeds).

History



In 1738 Daniel Bernoulli
Daniel Bernoulli
Daniel Bernoulli was a Dutch-Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics...

 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
Heat
In physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...

 is simply the kinetic energy
Kinetic energy
The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...

 of their motion. The theory was not immediately accepted, in part because conservation of energy
Conservation of energy
The nineteenth century law of conservation of energy is a law of physics. It states that the total amount of energy in an isolated system remains constant over time. The total energy is said to be conserved over time...

 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 (which were neglected by their contemporaries) were Mikhail Lomonosov
Mikhail Lomonosov
Mikhail Vasilyevich Lomonosov was a Russian polymath, scientist and writer, who made important contributions to literature, education, and science. Among his discoveries was the atmosphere of Venus. His spheres of science were natural science, chemistry, physics, mineralogy, history, art,...

 (1747), Georges-Louis Le Sage
Georges-Louis Le Sage
Georges-Louis Le Sage was a physicist and is most known for his theory of gravitation, for his invention of an electric telegraph and his anticipation of the kinetic theory of gases....

 (ca. 1780, published 1818), John Herapath
John Herapath
John Herapath was an English physicist who gave a partial account of the kinetic theory of gases in 1820 though it was neglected by the scientific community at the time....

 (1816) and John James Waterston
John James Waterston
John James Waterston was a Scottish physicist, a neglected pioneer of the kinetic theory of gases.-Early life:Waterston's father, George, was an Edinburgh sealing wax manufacturer and stationer, a relative of the Sandeman family Robert and his brother, George...

 (1843), which connected their research with the development of mechanical explanations of gravitation
Mechanical explanations of gravitation
Mechanical explanations of gravitation are attempts to explain the action of gravity by aid of basic mechanical processes, such as pressure forces caused by pushes, and without the use of any action at a distance. These theories were developed from the 16th until the 19th century in connection...

. In 1856 August Krönig
August Krönig
August Karl Krönig was a German chemist and physicist who published an account of the kinetic theory of gases in 1856, probably after reading a paper by John James Waterston....

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

In 1857 Rudolf Clausius
Rudolf Clausius
Rudolf Julius Emanuel Clausius , was a German physicist and mathematician and is considered one of the central founders of the science of thermodynamics. By his restatement of Sadi Carnot's principle known as the Carnot cycle, he put the theory of heat on a truer and sounder basis...

, 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
Mean free path
In physics, the mean free path is the average distance covered by a moving particle between successive impacts which modify its direction or energy or other particle properties.-Derivation:...

 of a particle.

In 1859, after reading a paper by Clausius, James Clerk Maxwell
James Clerk Maxwell
James Clerk Maxwell of Glenlair was a Scottish physicist and mathematician. His most prominent achievement was formulating classical electromagnetic theory. This united all previously unrelated observations, experiments and equations of electricity, magnetism and optics into a consistent theory...

 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. 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
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

 of air and other gases.”
In 1871, Ludwig Boltzmann
Ludwig Boltzmann
Ludwig Eduard Boltzmann was an Austrian physicist famous for his founding contributions in the fields of statistical mechanics and statistical thermodynamics...

 generalized Maxwell's achievement and formulated the Maxwell–Boltzmann distribution. Also the logarithm
Logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...

ic connection between entropy
Entropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...

 and probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

 was first stated by him.

In the beginning of 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
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

's (1905) and Marian Smoluchowski
Marian Smoluchowski
Marian Smoluchowski was an ethnic Polish scientist in the Austro-Hungarian Empire. He was a pioneer of statistical physics and an avid mountaineer.-Life:...

's (1906)
papers on Brownian motion
Brownian motion
Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...

, which succeeded in making certain accurate quantitative predictions based on the kinetic theory.

See also


  • Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy of equations
    BBGKY hierarchy
    In statistical physics, the BBGKY hierarchy is a set of equations describing the dynamics of a system of a large number of interacting particles...

  • Boltzmann equation
    Boltzmann equation
    The Boltzmann equation, also often known as the Boltzmann transport equation, devised by Ludwig Boltzmann, describes the statistical distribution of one particle in rarefied gas...

  • Collision theory
    Collision theory
    Collision theory is a theory proposed by Max Trautz and William Lewis in 1916 and 1918, that qualitatively explains how chemical reactions occur and why reaction rates differ for different reactions. For a reaction to occur the reactant particles must collide. Only a certain fraction of the total...

  • Critical temperature
  • Gas laws
    Gas laws
    The early gas laws were developed at the end of the 18th century, when scientists began to realize that relationships between the pressure, volume and temperature of a sample of gas could be obtained which would hold for all gases...

  • Heat
    Heat
    In physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...

  • Maxwell-Boltzmann distribution
  • Mixmaster dynamics
  • Thermodynamics
    Thermodynamics
    Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...

  • Vlasov equation
    Vlasov equation
    The Vlasov equation is a differential equation describing time evolution of the distribution function of plasma consisting of charged particles with long-range interaction...


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