Physics · Ch 9 — Kinetic Theory of Gases

Summary

9.7

Summary

Kinetic theory explains the microscopic origin of macroscopic parameters like temperature and pressure, starting from the idea that a gas is an enormous collection of tiny, randomly moving, elastically colliding molecules.
The pressure exerted on the walls of a gas container arises from the momentum imparted by colliding gas molecules: $P=\tfrac13 nm\overline{v^2}$ , directly proportional to the number density, the mass of each molecule, and the mean square speed.
The temperature of a gas is a measure of the average translational kinetic energy per molecule: $\overline{KE}=\tfrac32 kT$ , directly proportional to absolute temperature and independent of the nature (mass) of the molecule.
Pressure also equals two-thirds of the internal energy per unit volume, $P=\tfrac23 u$ .
The three characteristic speeds of a gas, all sharing the same $\sqrt{kT/m}$ dependence: $v_{rms}=\sqrt{3kT/m}=1.73\sqrt{kT/m}$ , $\bar v=\sqrt{8kT/\pi m}=1.60\sqrt{kT/m}$ , and $v_{mp}=\sqrt{2kT/m}=1.41\sqrt{kT/m}$ , always ordered $v_{rms}>\bar v>v_{mp}$ .
The Maxwell-Boltzmann speed distribution function gives the number of molecules with speeds between $v$ and $v+dv$ : $dN_v=4\pi N(m/2\pi kT)^{3/2}v^2e^{-mv^2/2kT}\,dv$ .
Degrees of freedom: the minimum number of independent coordinates needed to fix a system's position and configuration; $f=3N$ for $N$ free molecules, or $f=3N-q$ with $q$ constraints. Monatomic: $f=3$ . Diatomic (normal T): $f=5$ . Diatomic (high T): $f=7$ . Linear triatomic: $f=7$ . Non-linear triatomic: $f=6$ .
The law of equipartition of energy: the average kinetic energy of a system in thermal equilibrium is shared equally over every degree of freedom, each getting exactly $\tfrac12 kT$ .
The ratio of specific heats $\gamma=C_P/C_V$ : monatomic 1.67, diatomic (normal T) 1.40, diatomic (high T) 1.28, linear triatomic 1.28, non-linear triatomic 1.33. …

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