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## Kinetic theory
## Additional recommended knowledge
## HistoryIn 1738, Dutch born Swiss physicist and mathematician Daniel Bernoulli published Other pioneers of the kinetic theory were Mikhail Lomonosov (1747), 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. For example, Herapath, considered how a system of colliding particles could give rise to 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. 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. ## PostulatesThe 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. 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. The kinetic theory has also been extended to include inelastic collisions in granular matter by Jenkins and others. ## PressurePressure 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 where x-component of the initial velocity of the particle.
The particle impacts the wall once every 2 l is the length of the container). Although the particle impacts a side wall once every 1l/v 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 2_{x}l/v time units.
_{x}The force due to this particle is: The total force acting on the wall is: where the summation is over all the gas molecules in the container. The magnitude of the velocity for each particle will follow: Now considering the total force acting on all six walls, adding the contributions from each direction we have: 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: The quantity can be written as , where the bar denotes an average, in this case an average over all particles. This quantity is also denoted by where Thus the force can be written as: Pressure, which is force per unit area, of the gas can then be written as: where Thus, as cross-sectional area multiplied by length is equal to volume, we have the following expression for the pressure where 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 (1/2 ^{2}), which is a microscopic property. Note that the product of pressure and volume is simply two thirds of the total kinetic energy.
## Temperature and kinetic energyFrom the ideal gas law, (1)
where is the Boltzmann constant, and the absolute temperature, it follows from the above result that the temperature takes the form (2)
and the kinetic energy of the system can now written as (3)
Eq.(3) From Eq.(1) and
Eq.(3) (4)
Thus, the product of pressure and volume per mole 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. Since there are degrees of freedom (dofs) in a monoatomic-gas system with particles, the kinetic energy per dof is (5)
In the kinetic energy per dof, the constant of proportionality of temperature is 1/2 times Boltzmann constant. This result is related to the equipartition theorem. 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. Thus the kinetic energy per kelvin (monatomic ideal gas) is: - per mole: 12.47 J
- per molecule: 20.7 yJ = 129 μeV
At standard temperature (273.15 K), we get: - per mole: 3406 J
- per molecule: 5.65 zJ = 35.2 meV
## Number of collisions with wallOne 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 ## RMS speeds of moleculesFrom the kinetic energy formula it can be shown that with ## See also## ReferencesThe 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 Categories: Gases | Thermodynamics |
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This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Kinetic_theory". A list of authors is available in Wikipedia. |