Pressure from Molecular Theory

Kinetic-theory derivation: P = (1/3)·n·m·⟨v²⟩, where n = number density, m = molecular mass, ⟨v²⟩ = mean-square speed.

Equivalently: P = (1/3)·ρ·v_rms², where ρ = mass density.

Derived from: molecules in random motion, elastic collisions with walls, momentum transfer 2m·v.

Connects MICROSCOPIC (molecular motion) to MACROSCOPIC (pressure).

Implies pressure depends ONLY on density and speed — independent of detailed molecular interactions (in ideal gas).

At fixed T: ⟨v²⟩ = 3k_BT/m ⇒ P = nk_BT (ideal gas law form).

Total KE per molecule: ⟨KE⟩ = (3/2)k_BT (in 3D, monatomic).

Temperature is a measure of MEAN MOLECULAR KE.