Ideal gas law

Values of R	Units
8.314472	J·K-1·mol-1
0.082057	L·atm·K-1·mol-1
8.205745 × 10-5	m3·atm·K-1·mol-1
8.314472	L·kPa·K-1·mol-1
8.314472	m3·Pa·K-1·mol-1
62.36367	L·mmHg·K-1·mol-1
62.36367	L·Torr·K-1·mol-1
83.14472	L·mbar·K-1·mol-1
10.7316	ft3·psi· °R-1·lb-mol-1
0.73024	ft3·atm·°R-1·lb-mol-1

The ideal gas law is an equation applicable to hypothetical gases that consist of atoms or molecules with no intermolecular forces, that undergo perfectly inelastic collisions, and that are in constant random motion. It is a useful approximation for calculating temperatures, volumes, pressures or number of moles for many gases over a wide range of temperatures and pressures. 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 is the absolute pressure, V is the volume, n is the number of moles, T is the absolute temperature, and R is the molar gas constant defined as N_A k, where k is the Boltzmann constant and N_A is Avogadro's constant. Currently, the most accurate value of R is: 8.314472 ± 0.000015 J·K^-1·mol^-1.

Real gases deviate from ideal gas behavior because of the intermolecular attractive and repulsive forces. The deviation is especially significant at low temperatures or high pressures. There are many equations of state (EOS) available for use with real gases, the simplest of which is the van der Waals equation.

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, an equation of state such as the van der Waals equation may be used. However, this equation contains constants, $a$ and $b$, that are unique for each gas. The van der Waals equation 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 a 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.

Special cases of the ideal gas law

Because the ideal gas law reflects the combined contributions of several scientists, a number of gas laws are special cases of the ideal gas law. Amonton's law states that at a fixed volume and moles of gas, that the absolute pressure and temperature are inversely related ($p\propto 1/T$), while Boyle's law states that at fixed temperature and moles of gas, the pressure and volume are inversely related ($p\propto 1/V$). Charles' law states that the volume and temperature are directly related ($V\propto T$) at fixed absolute pressure and moles of gas. Avogradro's law simply states that at fixed temperature and pressure, the volume of gas is related to a molar gas volume.