Ideal gas law

The ideal gas law is useful for calculating temperatures, volumes, pressures or number of moles for many gases over a wide range of temperatures and pressures. Real gases differ from the ideal gas equation by extra term(s) which account for the attractive and repulsive factors that are especially significant in low temperatures or high pressures. When the ideal gas law is not accurate enough, one of the "real" gas equations, such as the van der Waals equation must be used. The ideal gas law is the combination of Boyle's law, Charles's law and Avogadro's law and is expressed mathematically as

$pV=nRT\,$

where p = pressure, V = volume, n = number of moles, and T = the absolute temperature, in degrees Kelvin. R is the molar gas constant, R = N_A k, where k is the Boltzmann constant and N_A is Avogadro's constant. R = 8.314472(15) J / mol K

Special cases of the ideal gas law

Amonton's law:   $\left({\frac {p}{T}}\right)=\mathrm {constant}$  (at a fixed volume and amount of gas)

Avogadro's law:   $V=\left(nV_{\mathrm {m} }\right)$  (at a fixed temperature and pressure) and $V_{m}$ is the molar volume of gas, about 22 liter/mole

Boyle's law:   $\left(pV\right)=\mathrm {constant}$  (at a fixed temperature and amount of gas)

Charles's law:   $\left({\frac {V}{T}}\right)=\mathrm {constant}$  (at a fixed pressure and amount of gas)

Boyle's + Charles's:   $\left({\frac {pV}{T}}\right)=\mathrm {constant}$  (at a fixed amount of gas)

An ideal gas

To be an ideal gas, several conditions must be met. First, the size of the gas molecules must be negligible compared to the average distance between them. This condition is not true at extremely high pressures or extremely cold temperatures. Second, the intermolecular forces of attraction or repulsion between molecules must be very weak or negligible except during collisions. And third, when the gas molecules do collide, thus must do so in an elastic manner. That is, they bounce right off of each other rather than sticking together.

When the ideal gas law fails

When the ideal gas law fails, a real gas law, such as the van der Waals equation must be used. However, this equation contains constants, $a$ and $b$, that are unique for each gas. This law also fails at extreme high pressures. When the coefficients $a$ and $b$ are set to zero, the van der Waals equation reduces to the ideal gas law.

van der Waals equation :$\left(p+{\frac {n^{2}a}{V^{2}}}\right)\left(V-nb\right)=nRT$

Background

The gas laws were developed in the 1660's, starting with Boyle's law, derived by Robert Boyle. Boyle's law states that "the volume of a sample of gas at a given temperature varies inversely with the applied pressure, or V = constant/p (at fixed temperature and amount of gas)". Jacques Alexandre Charles' experiments with hot-air balloons, and additional contributions by John Dalton (1801) and Joseph Louis Gay-Lussac (1802) showed that a sample of gas, at a fixed pressure, increases in volume linearly with the temperature, or V/T = a constant. This is known as Charles's law. Extrapolations of volume/temperature data for many gases, to a volume of zero, all cross at about -273 degrees Celsius, which is defined as absolute zero. Since real gases would liquefy before reaching this temperature, this temperature region remains a theoretical minimum.

In 1811 Amedeo Avogadro re-interpreted Gay-Lussac's law of combining volumes (1808) to state Avogadro's law : equal volumes of any two gases at the same temperature and pressure contain the same number of molecules.