C_V from quantized lattice vibrations: T³ law to Dulong-Petit limit
The Debye model treats a crystal as a gas of phonons with linear dispersion up to a cutoff ω_D. Heat capacity C_V = 9Nk_B(T/θ_D)³ ∫₀^(θ_D/T) x⁴eˣ/(eˣ−1)² dx. At low T: C_V ∝ T³ (Debye law); at high T: C_V → 3Nk_B = 24.9 J/mol·K (Dulong-Petit). The lattice animation shows increasing vibrational amplitude with temperature.