Worked example (conceptual + numerical): A solid has Debye temperature θD = 300 K. Estimate its molar heat capacity at T = 10 K using the Debye T³ law, and compare with the classical Dulong-Petit value. Step 1: The Debye T³ law says CV = (12π⁴/5)·R·(T/θD)³. Step 2: Compute the prefactor: 12π⁴/5 ≈ 233.8, and R = 8.314 J/(mol·K), so 233.8 × 8.314 ≈ 1944 J/(mol·K). Step 3: T/θD = 10/300 = 1/30, and (1/30)³ ≈ 3.70 × 10⁻⁵. Step 4: CV ≈ 1944 × 3.70 × 10⁻⁵ ≈ 0.072 J/(mol·K). Step 5: Compare with Dulong-Petit: 3R ≈ 24.9 J/(mol·K). At 10 K the measured heat capacity is hundreds of times smaller — exactly what the Debye model predicts, because at such low temperature only the lowest-frequency sound waves (long-wavelength phonons) can be thermally excited; the high-frequency vibrations are frozen out.