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Encyclopedia > Kinetic theory

Kinetic theory or kinetic theory of gases attempts to explain macroscopic 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's conjecture, but due to collisions between molecules moving at different velocities. Kinetic theory is also known as kinetic-molecular theory or collision theory. Macroscopic is commonly used to describe physical objects that are measurable and observable by the naked eye. ... For other uses, see Gas (disambiguation). ... 3D (left and center) and 2D (right) representations of the terpenoid molecule atisane. ... This article or section is in need of attention from an expert on the subject. ... Sir Isaac Newton FRS (4 January 1643 – 31 March 1727) [ OS: 25 December 1642 – 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ...

Contents

History

In 1738, Dutch born Swiss physicist and mathematician Daniel Bernoulli published Hydrodynamica, which laid the basis for the kinetic theory of gases. In this work, Bernoulli positioned the argument, still used to this day, that gases consist of great numbers of molecules moving in all directions, that their impact on a surface causes the gas pressure that we feel, and that what we experience as heat is simply the kinetic energy of their motion. 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. Daniel Bernoulli Daniel Bernoulli (February 8, 1700 – March 17, 1782) was a Dutch-born mathematician who spent much of his life in Basel, Switzerland where he died. ... For other uses, see Heat (disambiguation) In physics, heat, symbolized by Q, is energy transferred from one body or system to another due to a difference in temperature. ... Look up conservation of energy in Wiktionary, the free dictionary. ...


Other pioneers of the kinetic theory were Mikhail Lomonosov (1745), Georges-Louis Le Sage (1818), John Herapath (1820) and John James Waterston (1843), which connected their research with the development of mechanical explanations of gravitation. However, those scientists were neglected by their contemporaries. Mikhail Vasilyevich Lomonosov Mikhail Vasilyevich Lomonosov (Михаи́л Васи́льевич Ломоно́сов) (November 19 (November 8, Old Style), 1711 – April 15 (April 4, Old Style), 1765) was a Russian writer and polymath who made important contributions to literature, education, and science. ... Georges-Louis Lesage (1724 - 1803) was a Swiss physicist. ... John Herapath (May 30, 1790 - February 24, 1868) 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. ... John James Waterston (1811 - June 18, 1883) was a Scottish physicist, a neglected pioneer of the kinetic theory of gases. ... The mechanical theories or explanations of gravitation are attempts to explain the law of gravity by aid of basic mechanical processes, such as pushes, and without the use of any action at a distance. ...


For example, Herapath, considered how a system of colliding particles could give rise to action at a distance. In this direction, when thinking about the effect of the high temperatures near the Sun on his gravific particles he was led to a relationship between temperature and particle velocity. Herapath postulated that the momentum of a particle in a gas is a measure of the absolute temperature of the gas. He used momentum, rather than the kinetic energy on which the later established theory is based, as it seemed to him to avoid some difficulties around whether elastic collisions were possible between indivisible atoms. Apparently ignorant of Daniel Bernoulli's work, he was led to the incorrect, but suggestive, relationship that expresses the product of pressure P and volume V as proportional to the square of his true temperature. The correct relationship is proportional to the absolute temperature, not its square, the error arising from his identification of momentum, rather than energy, with temperature. In physics, action at a distance is the interaction of two objects which are separated in space with no known mediator of the interaction. ... For other uses, see Temperature (disambiguation). ... Sol redirects here. ... This article is about velocity in physics. ... This article is about momentum in physics. ... For other uses, see Gas (disambiguation). ... Absolute zero is the lowest temperature that can be obtained in any macroscopic system. ... The kinetic energy of an object is the extra energy which it possesses due to its motion. ... An elastic collision is a collision in which the total kinetic energy of the colliding bodies after collision is equal to their total kinetic energy before collision. ... For other uses, see Atom (disambiguation). ... Daniel Bernoulli Daniel Bernoulli (February 8, 1700 – March 17, 1782) was a Dutch-born mathematician who spent much of his life in Basel, Switzerland where he died. ... This article is about pressure in the physical sciences. ... For other uses, see Volume (disambiguation). ...


In 1859, after reading a paper on the diffusion of molecules by Rudolf Clausius, Scottish mathematical 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.[1] In his 1873 thirteen page article 'Molecules', published in the September issue of Nature, 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.”[2] Rudolf Clausius - physicist and mathematician Rudolf Julius Emanuel Clausius (January 2, 1822 – August 24, 1888), was a German physicist and mathematician. ... James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and theoretical physicist from Edinburgh, Scotland, UK. His most significant achievement was aggregating a set of equations in electricity, magnetism and inductance — eponymously named Maxwells equations — including an important modification (extension) of the Ampères... The Maxwell-Boltzmann distribution is a probability distribution with applications in physics and chemistry. ... This article is about pressure in the physical sciences. ...


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's 1905 paper on Brownian motion, which succeeded in making certain accurate quantitative predictions based on the kinetic theory. “Einstein” redirects here. ... Three different views of Brownian motion, with 32 steps, 256 steps, and 2048 steps denoted by progressively lighter colors. ...


Postulates

The theory for ideal gases makes the following assumptions:

  • The gas consists of very small particles, each of which has a mass.
  • The number of molecules is large such that statistical treatment can be applied.
  • These molecules are in constant, random motion. The rapidly moving particles constantly collide with each other and with the walls of the container.
  • The collisions of gas particles with the walls of the container holding them are perfectly elastic.
  • The interactions between molecules are negligible. They exert no forces on one another except during collisions.
  • The total volume 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 relatively large compared to their size.
  • The molecules are perfectly spherical in shape, and elastic in nature .
  • The average kinetic energy of the gas particles depends only on the temperature of the system.
  • Relativistic effects are negligible.
  • Quantum-mechanical effects are negligible. This means that the inter-particle distance is much larger than the thermal de Broglie wavelength and the molecules can be treated as classical objects.
  • The time during collision of molecule with the container's wall is negligible as comparable to the time between successive collisions.
  • The equations of motion of the molecules are time-reversible.

In addition, if the gas is in a container, the collisions with the walls are assumed to be instantaneous and elastic. This article or section is in need of attention from an expert on the subject. ... “Random” redirects here. ... Interaction is a kind of action that occurs as two or more objects have an effect upon one another. ... Although related to the more mathematical concepts of infinitesimal , the idea of something being negligible is particularly useful in practical disciplines like physics, chemistry, mechanical and electronic engineering, computer programming and in everyday decision-making. ... its made by jaypeeng magandang google wikepedia For other uses, see Force (disambiguation). ... For other uses, see Volume (disambiguation). ... Although related to the more mathematical concepts of infinitesimal , the idea of something being negligible is particularly useful in practical disciplines like physics, chemistry, mechanical and electronic engineering, computer programming and in everyday decision-making. ... In mathematics, an average or central tendency of a set (list) of data refers to a measure of the middle of the data set. ... Distance is a numerical description of how far apart objects are at any given moment in time. ... 2-dimensional renderings (ie. ... The kinetic energy of an object is the extra energy which it possesses due to its motion. ... Thermodynamic temperature is the absolute measure of temperature and is one of the principal parameters of thermodynamics. ... For other uses, see System (disambiguation). ... For a less technical and generally accessible introduction to the topic, see Introduction to special relativity. ... For a less technical and generally accessible introduction to the topic, see Introduction to quantum mechanics. ... In physics, the Thermal de Broglie wavelength is defined for a free ideal gas of massive particles in equilibrium as: where h is Plancks constant m is the mass of a gas particle k is Boltzmanns constant T is the Temperature of the gas The thermal de Broglie... Classical mechanics (commonly confused with Newtonian mechanics, which is a subfield thereof) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. ... A physical body is an object which can be described by the theories of classical mechanics, or quantum mechanics, and experimented upon by physical instruments. ...


More modern developments relax these assumptions and are based on the Boltzmann equation. These can accurately describe the properties of dense gases, because they include the volume of the molecules. The necessary assumptions are the absence of quantum effects, molecular chaos and small gradients in bulk properties. Expansions to higher orders in the density are known as virial expansions. 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. The Boltzmann equation describes the statistical distribution of particles in a fluid. ... In kinetic theory in physics, molecular chaos is the assumption that the velocities of colliding particles are uncorrelated, and independent of position. ... The Knudsen number (Kn) is the ratio of the molecular mean free path length to a representative physical length scale. ...


The kinetic theory has also been extended to include inelastic collisions in granular matter by Jenkins and others. A granular material is a conglomeration of discrete solid, macroscopic particles characterized by a loss of energy whenever the particles interact (the most common example would be friction when grains collide). ...


Pressure

Pressure is explained by kinetic theory as arising from the force exerted by gas molecules impacting on the walls of the container. Consider a gas of N molecules, each of mass m, enclosed in a cuboidal container of volume V. 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), then the momentum lost by the particle and gained by the wall is: This article is about pressure in the physical sciences. ... An elastic collision is a collision in which the total kinetic energy of the colliding bodies after collision is equal to their total kinetic energy before collision. ... This article is about momentum in physics. ...

Delta p_x = p_i - p_f = 2 m v_x,

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


The particle impacts the wall once every 2l/vx time units (where l is the length of the container). Although the particle impacts a side wall once every 1l/vx time units, only the momentum change on one wall is considered so that the particle produces a momentum change on a particular wall once every 2l/vx time units.

Delta t = frac{2l}{v_x}

The force due to this particle is: its made by jaypeeng magandang google wikepedia For other uses, see Force (disambiguation). ...

F = frac{Delta p}{Delta t} = frac{2 m v_x}{frac{2l}{v_x}} = frac{m v_x^2}{l}

The total force acting on the wall is:

F = frac{msum_j v_{jx}^2}{l}

where the summation is over all the gas molecules in the container.


The magnitude of the velocity for each particle will follow:

v^2 = v_x^2 + v_y^2 + v_z^2

Now considering the total force acting on all six walls, adding the contributions from each direction we have:

mbox{Total Force} = 2 cdot frac{m}{l}(sum_j v_{jx}^2 + sum_j v_{jy}^2 + sum_j v_{jz}^2) = 2 cdot frac{m}{l} sum_j (v_{jx}^2 + v_{jy}^2 + v_{jz}^2) = 2 cdot frac{m sum_j v_{j}^2}{l}

where the factor of two arises from now considering both walls in a given direction.


Assuming there are a large number of particles moving sufficiently randomly, the force on each of the walls will be approximately the same and now considering the force on only one wall we have:

F = frac{1}{6} left(2 cdot frac{m sum_j v_{j}^2}{l}right) = frac{m sum_j v_{j}^2}{3l}

The quantity sum_j v_{j}^2 can be written as {N} overline{v^2}, where the bar denotes an average, in this case an average over all particles. This quantity is also denoted by v_{rms}^2 where vrms is the root-mean-square velocity of the collection of particles. In mathematics, the root mean square or rms is a statistical measure of the magnitude of a varying quantity. ...


Thus the force can be written as:

F = frac{Nmv_{rms}^2}{3l}

Pressure, which is force per unit area, of the gas can then be written as:

P = frac{F}{A} = frac{Nmv_{rms}^2}{3Al}

where A is the area of the wall of which the force exerted on is considered.


Thus, as cross-sectional area multiplied by length is equal to volume, we have the following expression for the pressure

P = {Nmv_{rms}^2 over 3V}

where V is the volume. Also, as Nm is the total mass of the gas, and mass divided by volume is density

P = {1 over 3} rho v_{rms}^2

where ρ is the density of the gas.


This result is interesting and significant, because it relates pressure, a macroscopic property, to the average (translational) kinetic energy per molecule (1/2mvrms2), which is a microscopic property. Note that the product of pressure and volume is simply two thirds of the total kinetic energy. Macroscopic is commonly used to describe physical objects that are measurable and observable by the naked eye. ... The kinetic energy of an object is the extra energy which it possesses due to its motion. ... A microscope (Greek: micron = small and scopos = aim) is an instrument for viewing objects that are too small to be seen by the naked or unaided eye. ...


Number of collisions with wall

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 of this[1] results in an equation for total number of collisions per unit time per area:

A = frac{1}{4}frac{N}{V} v_{ave} = frac{rho}{4} sqrt{frac{8 k T}{pi m}} frac{1}{m} ,

Temperature

The above equation tells us that the product of pressure and volume per mole is proportional to the average (translational) molecular kinetic energy. Further, the ideal gas equation tells us that this product is proportional to the absolute temperature. Putting the two together, we arrive at one important result of the kinetic theory: average molecular kinetic energy is proportional to the absolute temperature. The constant of proportionality per degree of freedom is 1/2 times Boltzmann's constant. This result is related to the equipartition theorem. Monatomic gases have 3 degrees of freedom. As noted in the article on heat capacity, diatomic gases should have 7 degrees of freedom, but the lighter gases act as if they have only 5. The mole (symbol: mol) is the SI base unit that measures an amount of substance. ... The ideal gas law, or universal gas equation, is an equation of state of an ideal gas. ... Thermodynamic temperature is the absolute measure of temperature and is one of the principal parameters of thermodynamics. ... For other uses, see Temperature (disambiguation). ... Degrees of freedom is a general term used in explaining dependence on parameters, and implying the possibility of counting the number of those parameters. ... The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ... Figure 1. ... To meet Wikipedias quality standards, this article or section may require cleanup. ...


Thus the kinetic energy per kelvin (monatomic ideal gas) is: An ideal gas or perfect gas is a hypothetical gas consisting of identical particles of zero volume, with no intermolecular forces. ...

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

At standard temperature (273.15 K), we get: In chemistry and other sciences, STP or standard temperature and pressure is a standard set of conditions for experimental measurements, to enable comparisons to be made between sets of data. ...

  • per mole: 3406 J
  • per molecule: 5.65 zJ = 35.2 meV

RMS speeds of molecules

From the kinetic energy formula it can be shown that

v_{rms}^2 = frac{3RT}{mbox{molar mass}}

with v in m/s, T in kelvins, and R is the gas constant. 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). The gas constant (also known as the universal or ideal gas constant, usually denoted by symbol R) is a physical constant used in equations of state to relate various groups of state functions to one another. ... The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ...


See also

The gas laws are a set of laws that describe the relationship between thermodynamic temperature (T), pressure (P) and volume (V) of gases. ... For other uses, see Heat (disambiguation) In physics, heat, symbolized by Q, is energy transferred from one body or system to another due to a difference in temperature. ... The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ... Thermodynamics (from the Greek θερμη, therme, meaning heat and δυναμις, dynamis, meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ... Collision theory is a theory, proposed by Max Trautz and William Lewis in 1916 that qualitatively explains how chemical reactions occur and why reaction rates differ for different reactions. ... The critical temperature, Tc, of a material is the temperature above which distinct liquid and gas phases do not exist. ...

References

  1. ^ Mahon, Basil (2003). The Man Who Changed Everything – the Life of James Clerk Maxwell. Hoboken, NJ: Wiley. ISBN 0-470-86171-1. 
  2. ^ Maxwell, James Clerk, "Molecules". Nature, September, 1873.

The Mathematical Theory of Non-uniform Gases : An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases Sydney Chapman, T. G. Cowling


External links


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