Engineering & Science
Debye T³ Law: Low-Temperature Phonon Molar Heat Capacity Formula
Calculate cryogenic lattice molar heat capacity C_V = (12/5) * π^4 * R * (T / Θ_D)^3 (J/(mol·K)) from crystal Debye temperature Θ_D vs. the classical 3R Dulong-Petit ceiling.
Why Einstein’s optical-oscillator model froze out too fast—and Peter Debye’s acoustic phonons hit T³!
Albert Einstein’s single-frequency oscillator model predicted exponential heat-capacity freeze-out EXP(-Θ_E / T) near absolute zero, contradicting cryogenic calorimetry! Peter Debye (Annalen der Physik, 1912) integrated three acoustic phonon branches with linear dispersion ω = v_s * k up to a spherical Brillouin-zone cutoff Θ_D, proving that for T << Θ_D (typically T < Θ_D / 10) lattice molar heat capacity follows the universal cubic law: C_V = (12 / 5) * π^4 * R * (T / Θ_D)^3 ≈ 1,943.78 * (T / Θ_D)^3 J/(mol·K)!
Size a liquid-hydrogen / helium cryostat stage made of oxygen-free copper (Θ_D = 343 K): at T = 20 K, C_V = 0.3853 J/(mol·K) (just 1.54% of classical Dulong-Petit 3R = 24.94 J/(mol·K)), and halving temperature to 10 K cuts C_V by 2^3 = 8× down to 0.0482 J/(mol·K)!
| Cryogenic Temperature T (Copper Θ_D = 343 K) | Reduced Ratio T / Θ_D | Debye Phonon C_V (J/(mol·K)) | Share of Classical 3R (24.94) |
|---|---|---|---|
| T = 20 K (Liquid H2 Boil-Off) | 0.0583 | 0.3853 J/(mol·K) | 1.54% of Dulong-Petit |
| T = 10 K (8× Cubic Drop!) | 0.0292 | 0.0482 J/(mol·K) | 0.19% of Dulong-Petit |
| Invalid T <= 0 or Θ_D <= 0 | Undefined | "Invalid Temperature Inputs" | Cryogenic Guard |
Debye T³ Cryogenic Lattice Molar Heat Capacity C_V (J/(mol·K))
Evaluates (12 / 5) * PI()^4 * 8.31446 * (T / Θ_D)^3 across operating temperature B2 (K) and Debye temperature B3 (K).
=IF(OR(B2 <= 0, B3 <= 0), "Invalid Temperature Inputs", ROUND((12 / 5) * (PI() ^ 4) * 8.31446 * ((B2 / B3) ^ 3), 4))Classical High-Temperature Dulong-Petit Limit 3R (J/(mol·K))
Returns the high-temperature equipartition asymptote 3 * R = 24.94 J/(mol·K) per mole of atoms.
=ROUND(3 * 8.31446, 2)Try the example
Use the original values or change them to test your understanding.