Temperature ↔ Kinetic Energy

Temperature is a measure of AVERAGE KINETIC ENERGY of molecules.

For an ideal gas (monatomic): ⟨KE⟩ = (3/2)k_BT per molecule.

Per mole: ⟨KE⟩ × N_A = (3/2)RT (using k_B·N_A = R).

Linear relation: T ∝ ⟨KE⟩. Absolute T = 0 ⇒ molecules at rest (classical limit).

For diatomic gas (5 DoF): U = (5/2)nRT — translation + rotation.

Polyatomic: more DoF, more energy at same T.

Equipartition theorem: each quadratic degree of freedom gets ½k_BT per molecule.

Temperature is a STATISTICAL concept — needs many molecules to be meaningful.