Ideal Gas

An ideal gas is a theoretical gas that obeys the equation of state $pV=nRT$, relating pressure, volume, amount of substance, and absolute temperature. Its classical microscopic model neglects the volume occupied by particles and their intermolecular interactions, except for idealized collisions. It connects thermodynamics with molecular motion and provides an approximation for many real gases at sufficiently low density, away from condensation. (goldbook.iupac.org)

Equation of state

The ideal gas law has two equivalent forms:

Here $p$ is absolute pressure, $V$ is volume, $n$ is the amount of substance in moles, and $N$ is the number of particles. The gas constant $R$ and Boltzmann constant $k_{\mathrm B}$ satisfy $R=N_{\mathrm A}k_{\mathrm B}$, where $N_{\mathrm A}$ is the Avogadro constant. Temperature is expressed in kelvin, and pressure is measured relative to vacuum rather than atmospheric pressure. (openstax.org)

For a fixed amount of gas, the equation implies $pV/T=\text{constant}$. At constant temperature, pressure varies inversely with volume; at constant pressure, volume varies directly with temperature; and at constant volume, pressure varies directly with temperature. These relationships are limiting descriptions of gas behavior, not universal laws for gases under all conditions. (openstax.org)

For an ideal mixture, each component has partial pressure $p_i=n_iRT/V=x_i p$, where $x_i$ is its mole fraction. Total pressure is the sum of these partial pressures, a relationship known as Dalton’s law. Thus each component contributes the pressure it would exert alone at the same temperature and volume. (goldbook.iupac.org)

Microscopic interpretation

The kinetic theory of gases represents a gas as many atoms or molecules moving randomly. In the classical model, they follow classical mechanics, and collisions are elastic. Their size is negligible compared with typical separations, while forces between particles are neglected between collisions. Pressure arises from the transfer of momentum when particles strike container walls. (openstax.org)

For identical particles of mass $m$, an isotropic velocity distribution gives

Combining this with the ideal gas law yields the mean translational kinetic energy per particle and root-mean-square speed:

At the same temperature, different species have the same mean translational kinetic energy, but lighter particles have higher root-mean-square speeds. This speed is distinct from both the mean speed and the most probable speed. (openstax.org)

Internal energy and heat capacities

For a fixed amount and composition of classical ideal gas, internal energy depends only on temperature, not independently on volume. For a monatomic gas with negligible electronic excitation,

up to an arbitrary additive reference constant. Consequently, its molar heat capacities are

More generally, ideal gases satisfy $C_p-C_V=R$. At constant pressure, some supplied heat supports expansion work, explaining why $C_p$ exceeds $C_V$. (openstax.org)

Molecular gases can also store energy in rotation and vibration. The equipartition theorem assigns $k_{\mathrm B}T/2$ to each active quadratic energy term. A diatomic gas with active translation and rotation but negligible vibration therefore has $C_V\approx5R/2$. However, quantum mechanics restricts excitation of molecular energy levels: rotational or vibrational contributions may be suppressed at low temperatures and become significant as temperature rises. Ideal behavior therefore does not require temperature-independent heat capacities. (openstax.org)

Expansion and compression

For a quasistatic volume change, work done by the gas is

In an isothermal process, temperature remains constant. For an ideal gas,

Internal energy is unchanged, so heat absorbed equals work done by the gas. Work depends on the process followed, rather than solely on its endpoints. (openstax.org)

An adiabatic process involves no heat transfer. For a reversible adiabatic change with constant heat capacities,

where $\gamma=C_p/C_V$. Expansion lowers temperature, while compression raises it. These equations do not describe every adiabatic process; reversibility and constant heat capacities are essential qualifications. (openstax.org)

Real gases and limits of the model

Real gases depart from ideal behavior when intermolecular attractions or excluded volume become important. Such departures commonly increase at high density or near liquefaction. The ideal gas law itself cannot describe a liquid–gas phase transition or a liquid–gas critical point. (openstax.org)

The van der Waals equation introduces approximate corrections:

The parameter $a$ represents attractive interactions, while $b$ accounts for excluded volume. When both corrections are negligible, the equation approaches the ideal gas law. Its modified isotherms can describe qualitative features of condensation absent from the ideal model. (openstax.org)

In statistical mechanics, “ideal” can also mean noninteracting without implying classical behavior. Quantum gases of bosons obey Bose–Einstein statistics, whereas gases of fermions obey Fermi–Dirac statistics. When quantum occupation effects become important, their thermodynamic properties differ from those of a classical ideal gas even without interparticle forces. The familiar $pV=Nk_{\mathrm B}T$ law belongs to their dilute classical limit. (theory.physics.manchester.ac.uk)