ideal gas(redirected from Boltzmann gas)
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ideal gas:see gas lawsgas laws,
physical laws describing the behavior of a gas under various conditions of pressure, volume, and temperature. Experimental results indicate that all real gases behave in approximately the same manner, having their volume reduced by about the same proportion of the
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physical theory that explains the behavior of gases on the basis of the following assumptions: (1) Any gas is composed of a very large number of very tiny particles called molecules; (2) The molecules are very far apart compared to their sizes,
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a theoretical model of a gas in which the interaction of the gas particles is disregarded (the mean kinetic energy of the particles is much greater than the energy of their interaction).
A distinction is made between the classical ideal gas, whose properties are described by the laws of classical physics, and a quantum ideal gas, which conforms to the laws of quantum mechanics.
The particles of a classical ideal gas move independently of each other, so that the pressure of the ideal gas on a wall is equal to the sum of the momenta transmitted per unit time by the individual particles upon colliding with the wall and the energy is equal to the sum of the energies of the individual particles. The classical ideal gas conforms to the Clapeyron equation of state p = nkT, where p is the pressure, n is the number of particles per unit volume, k is the Boltzmann constant, and T is the absolute temperature. The Boyle-Mariotte and Gay-Lussac laws are particular cases of this equation. The particles of the classical ideal gas are distributed by energy according to the Boltzmann distribution. Real gases are described well by the model of the classical ideal gas if they are sufficiently rarefied.
When the temperature T of the gas is lowered or its density n is increased to a certain value, the wave (quantum) properties of the particles of an ideal gas become significant. The transition from the classical ideal gas to the quantum ideal gas takes place for those values of T and n at which the de Broglie wavelengths of particles moving with velocities of the order of thermal velocities are comparable to the distance between particles.
In the quantum case a distinction is made between two types of ideal gas; the particles of one type of gas have integral spin, and Bose-Einstein statistics is applicable to them, whereas Fer-mi-Dirac statistics is applicable to particles of the other type (those having half-integral spin).
The Fermi-Dirac ideal gas differs from the classical ideal gas in that even at absolute zero its pressure and energy density are not zero and increase as the density increases. At absolute zero there exists a maximum (boundary) energy, which particles of the Fermi-Dirac ideal gas may have (the so-called Fermi energy). If the energy of thermal motion of the particles of a Fermi-Dirac ideal gas is much smaller than the Fermi energy, the gas is called a degenerate gas. According to the theory of stellar structure, a degenerate Fermi-Dirac ideal electron gas exists in stars whose density exceeds 1–10 kg/cm3, and in stars with a density exceeding 109 kg/cm3 matter is converted into a Fermi-Dirac ideal neutron gas.
The application of the theory of Fermi-Dirac ideal gases to electrons in metals makes possible the explanation of many properties of the metallic state. The more dense a real degenerate Fermi-Dirac gas, the closer it is to ideal.
At absolute zero the particles of a Bose-Einstein ideal gas occupy the lowest energy level and have a momentum of zero (an ideal gas in a condensate state). As T increases, the number of particles in the condensate gradually decreases, and at some temperature To (the phase transition temperature) the condensate disappears (all particles of the condensate acquire momentum). When T < T0 the pressure of a Bose-Einstein ideal gas depends only on the temperature. Helium has the properties of such an ideal gas at temperatures close to absolute zero. Electromagnetic radiation (an ideal photon gas) that is in thermal equilibrium with the radiating body is another example of a Bose-Einstein ideal gas. An ideal photon gas is also an example of an ultrarelativistic ideal gas—that is, a set of particles moving with velocities equal or close to the speed of light. The equation of state of such a gas is p = ε/3, where « is the energy density of the gas. At sufficiently low temperatures collective motions of various types in liquids and solids (for example, the oscillations of the atoms of a crystal lattice) can be represented as an ideal gas of weak perturbations (quasi-particles), whose energy contributes to the energy of the body.
V. L. POKROVSKII