Real Gases & Van der Waals Equation Cheat Sheet

Key Facts

• The ideal gas law is $PV = nRT$, where $P$ is pressure, $V$ is volume, $n$ is moles, $R$ is the gas constant, and $T$ is temperature in kelvin.
• Real gases behave most ideally at high temperature and low pressure because particles are far apart and moving fast.
• The Van der Waals equation is $\left(P + a\frac{n^2}{V^2}\right)(V - nb) = nRT$.
• The pressure correction $a\frac{n^2}{V^2}$ accounts for attractive forces that make the measured pressure lower than ideal.
• The volume correction $nb$ accounts for the actual volume occupied by gas particles, so the free space is $V - nb$.
• A larger $a$ value means stronger intermolecular attractions and greater pressure deviation from ideal behavior.
• A larger $b$ value means larger particle volume and greater excluded-volume deviation from ideal behavior.
• The compressibility factor is $Z = \frac{PV}{nRT}$, with $Z = 1$ for an ideal gas, $Z < 1$ when attractions dominate, and $Z > 1$ when particle volume dominates.

Vocabulary

Real gas

A gas that deviates from ideal behavior because its particles have volume and experience intermolecular forces.

Ideal gas

A theoretical gas whose particles have no volume, no attractions, and obey $PV = nRT$ exactly.

Van der Waals equation

An equation of state, $\left(P + a\frac{n^2}{V^2}\right)(V - nb) = nRT$, that corrects the ideal gas law for real gas behavior.

Pressure correction

The term $a\frac{n^2}{V^2}$ added to pressure to account for attractions between gas particles.

Volume correction

The term $nb$ subtracted from volume to account for the space occupied by gas particles themselves.

Compressibility factor

The ratio $Z = \frac{PV}{nRT}$ that shows how much a gas deviates from ideal behavior.