Engineering & Science
Einstein Quantum Solid Molar Heat Capacity (CV) vs. Dulong-Petit 3R Limit
Calculate quantized lattice molar heat capacity C_V = 3 * R * (theta_E / T)^2 * EXP(theta_E / T) / (EXP(theta_E / T) - 1)^2 (J/(mol*K)) and % of the classical Dulong-Petit 3R = 24.943 J/(mol*K) limit.
Why room-temperature diamond has less than 20% of the classical Dulong-Petit heat capacity!
Classical 19th-century equipartition (Dulong and Petit, 1819) predicted that every monatomic crystal must have the exact same constant-volume molar heat capacity C_V = 3 * R = 3 * 8.31446 = 24.943 J/(mol*K)—yet Heinrich Weber observed that diamond at room temperature (T = 300 K) holds barely ~6 J/(mol*K) and plunges toward zero at cryogenic temperatures!
Albert Einstein (1907) launched quantum solid-state physics by modeling 3N lattice atoms as quantized harmonic oscillators with characteristic Einstein temperature theta_E = hbar * omega_E / k_B: C_V = 3 * R * (theta_E / T)^2 * EXP(theta_E / T) / (EXP(theta_E / T) - 1)^2! Because stiff carbon-carbon bonds give diamond a huge theta_E ~ 1,450 K, at T = 300 K its quantum oscillators are mostly "frozen out" at C_V = 4.713 J/(mol*K) (18.89% of 3R), whereas softer Copper/Silver lattices (theta_E = 240 K) reach C_V = 23.655 J/(mol*K) (94.83% of Dulong-Petit 3R)!
| Crystal Lattice at T = 300 K | Einstein Temp thetaE (K) | Quantum CV (J/(mol*K)) | % of Classical 3R (24.943) |
|---|---|---|---|
| Diamond (Stiff Covalent Bonds) | 1,450 K | 4.713 J/(mol*K) | 18.89% (Quantum Frozen-Out) |
| Copper / Silver Lattice | 240 K | 23.655 J/(mol*K) | 94.83% (Near Dulong-Petit) |
| Absolute Zero / Invalid T <= 0 | Guarded Singularity | "Invalid Thermodynamic Temp" | Requires thetaE > 0 and T > 0 |
Einstein Quantum Solid Molar Heat Capacity C_V (J/(mol*K))
Evaluates 3 * 8.31446 * (theta_E / T)^2 * EXP(theta_E / T) / (EXP(theta_E / T) - 1)^2 for theta_E (B2) and T (B3).
=IF(OR(B2 <= 0, B3 <= 0), "Invalid Thermodynamic Temp", ROUND(3 * 8.31446 * ((B2 / B3) ^ 2) * EXP(B2 / B3) / ((EXP(B2 / B3) - 1) ^ 2), 3))Fraction of Classical Dulong-Petit 3R Limit (%)
Expresses quantum C_V as a percentage of classical 3R = 24.94338 J/(mol*K).
=IF(OR(B2 <= 0, B3 <= 0), "Invalid Thermodynamic Temp", ROUND(100 * (((B2 / B3) ^ 2) * EXP(B2 / B3) / ((EXP(B2 / B3) - 1) ^ 2)), 2))Try the example
Use the original values or change them to test your understanding.