Engineering & Science
Wiedemann-Franz Law & Sommerfeld Lorenz Number (Metal Thermal Conductivity)
Calculate the electronic thermal conductivity kappa_e = L0 * sigma * T in W/(m*K) from electrical conductivity sigma using the Sommerfeld Lorenz number L0 = 2.44e-8 W*Ohm/K^2, and extract the empirical Lorenz ratio L = kappa / (sigma * T).
Why great electrical conductors like silver and copper are also great heat conductors!
In pure metals, the exact same Fermi-surface conduction electrons carry BOTH electric charge and thermal energy! Gustav Wiedemann and Rudolph Franz (1853) noticed that at a given absolute temperature T, the ratio of thermal conductivity kappa to electrical conductivity sigma is nearly identical across all metals, and Arnold Sommerfeld (1928) applied Fermi-Dirac quantum statistics to derive the universal Sommerfeld Lorenz number: L0 = (PI()^2 / 3) * (kB / e)^2 = 2.44e-8 W*Ohm/K^2!
In cryogenic spacecraft straps, superconducting magnet busbars, and thin metal interconnects—where direct thermal conductivity measurement is notoriously hard—engineers measure four-point electrical conductivity sigma (S/m = 1 / rho_elec) and compute electronic thermal conductivity immediately via kappa_e = L0 * sigma * T! For annealed copper at T = 300 K (sigma = 5.96e7 S/m), Wiedemann-Franz predicts kappa_e = 436.27 W/(m*K) (close to measured total 401 W/(m*K), where slight inelastic electron-phonon scattering lowers empirical L to 2.24e-8 W*Ohm/K^2)!
| Metal / Cryogenic Regime | Conductivity sigma (S/m) & Temp T | Predicted kappa_e (W/m*K) | Empirical Lorenz Check |
|---|---|---|---|
| Copper Busbar at Room Temp | 5.96e7 S/m, T = 300 K | 436.27 W/(m*K) | Measured 401 -> L = 2.243e-8 |
| Same Conductivity at LN2 (77 K) | 5.96e7 S/m, T = 77 K | 111.98 W/(m*K) | Linear T factor at fixed sigma |
| Invalid Absolute Temp T <= 0 | Non-Positive Kelvin Guard | Guarded Input Error | "Invalid Metal Inputs" |
Sommerfeld Electronic Thermal Conductivity kappa_e = L0 * sigma * T (W/(m*K))
Multiplies electrical conductivity B2 (S/m), absolute temperature B3 (K), and L0 = 2.44e-8 W*Ohm/K^2.
=IF(OR(B2 <= 0, B3 <= 0), "Invalid Metal Inputs", ROUND(0.0000000244 * B2 * B3, 2))Empirical Lorenz Ratio L_obs / L0 Relative to Sommerfeld Value
Compares measured total thermal conductivity B4 against L0 * sigma * T (values > 1 indicate strong phonon lattice contribution; values < 1 indicate inelastic scattering).
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0), "Invalid Metal Inputs", ROUND(B4 / (0.0000000244 * B2 * B3), 4))Try the example
Use the original values or change them to test your understanding.