FormulaProved

The ideal gas law

Statement

$pV = nRT$ — with $p$ pressure, $V$ volume, $n$ number of moles, $R \approx 8.314\ \text{J/(mol·K)}$ , $T$ absolute temperature.

Why is it true?

Combines three separate empirical laws (Boyle–Mariotte, Charles, Gay-Lussac) into a single equation, approximately valid for dilute gases at not-too-high pressure.

Gas in a piston cylinder: change T, V, n and watch pressure follow p = nRT/V; isotherms on the p–V diagram.

Proof sketch

From kinetic theory: gas pressure arises from molecular collisions with the container walls. Computing the momentum delivered to the walls per second gives $pV = \tfrac13 Nm\overline{v^2}$ . Comparing with the definition of temperature via mean kinetic energy $\tfrac12 m\overline{v^2} = \tfrac32 k_BT$ gives $pV = Nk_BT = nRT$ .

Topics that use this theorem

Ideal gas equation of state

Step-by-step proofs

Deriving pV = nRT from molecular collisions

PhysicLabs (theo lập luận cổ điển của Bernoulli, Clausius, Maxwell), 2026

References

Daniel V. Schroeder (2000). An Introduction to Thermal Physics