Derivation of the ideal gas law

Kinetic theory allows us to explicitly calculate the pressure exerted by $N$ particles in a volume $V$, as a function of their total kinetic energy $E_\text{cin}$. The result is surprisingly simple:

$P = \frac{2}{3}\,\frac{E_{\text{cin}}}{V}$

At this point it is enough to use the link between kinetic energy and temperature. For an ideal gas the total translational kinetic energy is $E_{\text{cin}} = \frac{3}{2}Nk_BT$; substituting:

$PV = \frac{2}{3}\cdot\frac{3}{2}Nk_BT = Nk_BT = nRT$

where in the last step $N = n\,N_A$ and $R = N_A k_B$ were used to go from the number of molecules $N$ to the number of moles $n$.

Law — Ideal gas equation of state

$\ev{PV = nRT}$

The significance is profound: an equation between macroscopic, measurable quantities (pressure, volume, temperature) emerges entirely from a microscopic model of bouncing balls. The microscopic derivation links the invisible world of atoms to laboratory instruments such as a pressure gauge and a thermometer.

Historical context

This equation was discovered experimentally by Boyle (1662), Charles (1787) and Gay-Lussac (1802), assembling the empirical gas laws, and only later derived from kinetic theory by Clausius in 1857. It is a classic example in which the theoretical explanation arrives afterwards, unifying a series of already known experimental regularities.

Collegamenti

Argomenti: Teoria cinetica dei gas Concetti: Legge dei gas perfetti · Pressione Competenze: Interpretazione micro-macro

Esercizi collegati: Problem — Temperature that doubles the pressure · Problem — Ranking of pressures by temperature · Problem — Pressure of an ideal gas