The Ideal Gas Law is one of the Equations of State. Although the law describes the behavior of an ideal gas, the equation is applicable to real gases under many conditions, so it is a useful equation to learn to use. The Ideal Gas Law may be expressed as:

PV = NkT

where:

P = absolute pressure in atmospheres

V = volume (usually in liters)

n = number of particles of gas

k = Boltzmann's constant (1.38·10^{−23} J·K^{−1})

T = temperature in Kelvin

The Ideal Gas Law may be expressed in SI units where pressure is in pascals, volume is in cubic meters, N becomes n and is expressed as moles, and k is replaced by R, the Gas Constant (8.314 J·K^{−1}·mol^{−1}):

PV = nRT

## Ideal Gases Versus Real Gases

The Ideal Gas Law applies to ideal gases. An ideal gas contains molecules of a negligible size that have an average molar kinetic energy that depends only on temperature. Intermolecular forces and molecular size are not considered by the Ideal Gas Law. The Ideal Gas Law applies best to monoatomic gases at low pressure and high temperature. Lower pressure is best because then the average distance between molecules is much greater than the molecular size. Increasing the temperature helps because of the kinetic energy of the molecules increases, making the effect of intermolecular attraction less significant.

## Derivation of the Ideal Gas Law

There are a couple of different ways to derive the Ideal as Law. A simple way to understand the law is to view it as a combination of Avogadro's Law and the Combined Gas Law. The Combined Gas Law may be expressed as:

PV / T = C

where C is a constant that is directly proportional to the quantity of the gas or number of moles of gas, n. This is Avogadro's Law:

C = nR

where R is the universal gas constant or proportionality factor. Combining the laws:

PV / T = nR

Multiplying both sides by T yields:

PV = nRT

## Ideal Gas Law - Worked Example Problems

Ideal vs Non-Ideal Gas Problems

Ideal Gas Law - Constant Volume

Ideal Gas Law - Partial Pressure

Ideal Gas Law - Calculating Moles

Ideal Gas Law - Solving for Pressure

Ideal Gas Law - Solving for Temperature

**Ideal Gas Equation for Thermodynamic Processes**

Process(Constant) |
KnownRatio |
P_{2} |
V_{2} |
T_{2} |

Isobaric
(P) |
V
_{2}/V
_{1}
T _{2}/T
_{1} |
P
_{2}=P
_{1}
P _{2}=P
_{1} |
V
_{2}=V
_{1}(V
_{2}/V
_{1})
V _{2}=V
_{1}(T
_{2}/T
_{1}) |
T
_{2}=T
_{1}(V
_{2}/V
_{1})
T _{2}=T
_{1}(T
_{2}/T
_{1}) |

Isochoric
(V) |
P
_{2}/P
_{1}
T _{2}/T
_{1} |
P
_{2}=P
_{1}(P
_{2}/P
_{1})
P _{2}=P
_{1}(T
_{2}/T
_{1}) |
V
_{2}=V
_{1}
V _{2}=V
_{1} |
T
_{2}=T
_{1}(P
_{2}/P
_{1})
T _{2}=T
_{1}(T
_{2}/T
_{1}) |

Isothermal
(T) |
P
_{2}/P
_{1}
V _{2}/V
_{1} |
P
_{2}=P
_{1}(P
_{2}/P
_{1})
P _{2}=P
_{1}/(V
_{2}/V
_{1}) |
V
_{2}=V
_{1}/(P
_{2}/P
_{1})
V _{2}=V
_{1}(V
_{2}/V
_{1}) |
T
_{2}=T
_{1}
T _{2}=T
_{1} |

isoentropic
reversible adiabatic (entropy) |
P
_{2}/P
_{1}
V _{2}/V
_{1}
T _{2}/T
_{1} |
P
_{2}=P
_{1}(P
_{2}/P
_{1})
P _{2}=P
_{1}(V
_{2}/V
_{1})
^{−γ}
P _{2}=P
_{1}(T
_{2}/T
_{1})
^{γ/(γ − 1)} |
V
_{2}=V
_{1}(P
_{2}/P
_{1})
^{(−1/γ)}
V _{2}=V
_{1}(V
_{2}/V
_{1})
V _{2}=V
_{1}(T
_{2}/T
_{1})
^{1/(1 − γ)} |
T
_{2}=T
_{1}(P
_{2}/P
_{1})
^{(1 − 1/γ)}
T _{2}=T
_{1}(V
_{2}/V
_{1})
^{(1 − γ)}
T _{2}=T
_{1}(T
_{2}/T
_{1}) |

polytropic
(PV ^{n}) |
P
_{2}/P
_{1}
V _{2}/V
_{1}
T _{2}/T
_{1} |
P
_{2}=P
_{1}(P
_{2}/P
_{1})
P _{2}=P
_{1}(V
_{2}/V
_{1})
^{−n}
P _{2}=P
_{1}(T
_{2}/T
_{1})
^{n/(n − 1)} |
V
_{2}=V
_{1}(P
_{2}/P
_{1})
^{(-1/n)}
V _{2}=V
_{1}(V
_{2}/V
_{1})
V _{2}=V
_{1}(T
_{2}/T
_{1})
^{1/(1 − n)} |
T
_{2}=T
_{1}(P
_{2}/P
_{1})
^{(1 - 1/n)}
T _{2}=T
_{1}(V
_{2}/V
_{1})
^{(1−n)}
T _{2}=T
_{1}(T
_{2}/T
_{1}) |