Temperature
Kinetic theory of gases

Maxwell and Boltzmann developed a kinetic theory that yields a fundamental understanding of temperature in gases. This theory also explains the ideal gas law and the observed heat capacity of monatomic (or 'noble') gases.

The ideal gas law is based on observed empirical relationships between pressure (p), volume (V), and temperature (T), and was recognized long before the kinetic theory of gases was developed (see Boyle's and Charles's laws). The ideal gas law states:

where n is the number of moles of gas and R = 8.314 462 618... J⋅mol ⋅K is the gas constant.

This relationship gives us our first hint that there is an absolute zero on the temperature scale, because it only holds if the temperature is measured on an absolute scale such as Kelvin's. The ideal gas law allows one to measure temperature on this absolute scale using the gas thermometer. The temperature in kelvins can be defined as the pressure in pascals of one mole of gas in a container of one cubic meter, divided by the gas constant.

Although it is not a particularly convenient device, the gas thermometer provides an essential theoretical basis by which all thermometers can be calibrated. As a practical matter, it is not possible to use a gas thermometer to measure absolute zero temperature since the gases condense into a liquid long before the temperature reaches zero. It is possible, however, to extrapolate to absolute zero by using the ideal gas law, as shown in the figure.

The kinetic theory assumes that pressure is caused by the force associated with individual atoms striking the walls, and that all energy is translational kinetic energy. Using a sophisticated symmetry argument, Boltzmann deduced what is now called the Maxwell–Boltzmann probability distribution function for the velocity of particles in an ideal gas. From that probability distribution function, the average kinetic energy (per particle) of a monatomic ideal gas is

where the Boltzmann constant k is the ideal gas constant divided by the Avogadro number, and v rms = ⟨ v 2 ⟩ = ⟨ v ⋅ v ⟩ {\textstyle v_{\text{rms}}={\sqrt {\langle v^{2}\rangle }}={\sqrt {\langle \mathbf {v\cdot v} \rangle }}} is the root-mean-square speed. This direct proportionality between temperature and mean molecular kinetic energy is a special case of the equipartition theorem, and holds only in the classical limit of a perfect gas. It does not hold exactly for most substances.
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