Chapter 39: Kinetic Theory of Gases

Kinetic Theory of Gases on the properties of matter begins by establishing the physical perspective that all substances consist of countless atoms and elementary particles governed by underlying mechanical laws. While acknowledging the fundamental accuracy of quantum mechanics, the chapter uses classical concepts to study the behavior of large systems, noting that results for macroscopic aggregates are often approximations. A crucial goal is to connect the mechanical laws governing atomic motion with the principles of thermodynamics. The discussion develops a microscopic understanding of gas pressure, modeling it as the result of continuous momentum transfer from vast numbers of gas molecules bombarding a surface. This analysis requires calculating the average squared velocity of the molecules, demonstrating that the pressure exerted by a gas is directly proportional to the total kinetic energy derived from the center-of-mass motion of its molecules. For a monatomic gas, where internal motions are disregarded, the internal energy is equal to this total kinetic energy. The theory is then applied to adiabatic compression, where no heat is exchanged, establishing a key thermodynamic relationship where the product of pressure and volume raised to a specific constant exponent remains fixed; this exponent is five-thirds for monatomic gases. The principles of kinetic theory are generalized to explain the pressure exerted by radiation (photons), where the pressure is found to be proportional to one-third of the total energy density. Furthermore, the compressibility condition for radiation follows a similar volume-pressure relationship, but with a different exponent: four-thirds. Crucially, the chapter defines temperature in terms of thermal equilibrium, proving that when different types of gases reach the same temperature, the average kinetic energy of the center-of-mass motion for all molecule types must be equal. This average kinetic energy of the center-of-mass motion is shown to be a fixed multiple of the absolute temperature. Finally, the concept of degrees of freedom—the independent directions of motion for a system—is introduced, stating that the average kinetic energy associated with each degree of freedom is equal to a constant times one-half of the temperature. The total energy of a molecule is comprised of its center-of-mass kinetic energy plus rotational and vibrational internal kinetic energies.