Ideal gas law

The macroscopic gas law

The ideal gas law is

$PV=nRT$

where $P$ is pressure, $V$ is volume, $n$ is number of moles, $R$ is the gas constant, and $T$ is absolute temperature in kelvins.

An equivalent microscopic form is

$PV=Nk_BT$

where $N$ is the number of molecules and $k_B$ is Boltzmann's constant. The constants are related by

$R=N_Ak_B$

where $N_A$ is Avogadro's number.

Meaning of pressure

Pressure is force per unit area:

P=rac{F}{A}

In a gas, pressure comes from countless molecular collisions with container walls. Each molecule transfers momentum when it bounces off a wall. The total effect of many collisions produces a steady macroscopic pressure.

Ideal gas assumptions

An ideal gas is a simplified model with several assumptions. Molecules are treated as point particles with negligible volume compared with the container. They interact only during brief elastic collisions. Collisions conserve kinetic energy and momentum. The gas is dilute enough that long-range intermolecular forces are negligible.

These assumptions are most accurate at low density and high temperature, away from condensation.

Empirical gas laws

The ideal gas law combines earlier empirical relationships. Boyle's law says that at constant temperature and amount,

$PV=constant$

Charles's law says that at constant pressure,

$Vpropto T$

Avogadro's law says that at fixed pressure and temperature,

$Vpropto n$

Together, they lead to $PV=nRT$.

Absolute temperature is essential

Temperature must be in kelvins. If Celsius were used, proportionalities such as $Vpropto T$ would fail because $0^circ C$ is not zero thermal energy. Absolute temperature connects directly to microscopic kinetic energy.

For an ideal monatomic gas,

U=rac{3}{2}nRT

This internal energy depends only on temperature.

Density form

Using mass density, the ideal gas law can be written in useful alternate forms. If $m_{mol}$ is molecular mass and $ho$ is mass density, then

ho}{m_{mol}}k_BT$$ For molar mass $M$, another form is $$P=rac{ ho RT}{M}$$ These forms are useful in fluids, atmospheres, and astrophysics. ## Limits of the ideal gas law Real gases deviate from ideal behavior at high pressure or low temperature. At high pressure, molecular volume is no longer negligible. At low temperature, attractive forces become important and gases may liquefy. More advanced equations, such as the van der Waals equation, $$left(P+arac{n^2}{V^2} ight)(V-nb)=nRT$$ include corrections for attractions and molecular volume. ## The big idea The ideal gas law is a bridge between macroscopic variables and microscopic particle motion. It works well for dilute gases and provides the foundation for kinetic theory, thermodynamic processes, engines, atmospheres, and statistical mechanics.