Wiedemann-Franz Law Calculator
Enter the electrical conductivity and temperature to compute the metal thermal conductivity κ=L·σ·T. Copper σ=5.96e7, T=300K → κ≈436 W/(m·K).
Input Data
Results
At a glance:The Wiedemann-Franz law (1853) states that for metals the ratio of thermal conductivity κ to electrical conductivity σ is proportional to absolute temperature T: κ/σ=L·T, i.e. κ=L·σ·T. L=2.44×10⁻⁸ W·Ω/K² is the Lorenz number, derived by Sommerfeld (1928) from the free-electron theory as L=(π²/3)(k_B/e)². Physical meaning: the same electrons carry both charge and heat, so the two conduction abilities are proportional. Classic example: copper σ=5.96e7 S/m, T=300K → κ=2.44e-8×5.96e7×300≈436 W/(m·K) (measured ~401, ~9% off). History: Wiedemann and Franz (1853) found κ/σ∝T; Lorenz (1872) fixed the constant; Sommerfeld (1928) derived L from quantum free-electron theory. Applications: (1) metal thermal management — heat sinks, thermal-interface materials; (2) electronics cooling — CPU heat spreaders, heat pipes; (3) low-temperature physics — the breakdown of the WF law signals a superconducting transition; (4) materials science — infer κ from σ or vice versa; (5) aerospace — satellite thermal-control metal choice.
Formula
Thermal conductivity: κ = L·σ·T (W/(m·K))
Lorenz number: L = (π²/3)(k_B/e)² ≈ 2.44e-8 W·Ω/K²
Ratio: κ/σ = L·T
$$\kappa = L\sigma T, \quad L = \frac{\pi^2}{3}\left(\frac{k_B}{e}\right)^2 \approx 2.44\times 10^{-8}\,\mathrm{W\Omega/K^2}$$How to Use
- Enter the electrical conductivity σ (S/m) and temperature T (K).
- The tool computes κ=L·σ·T with L=2.44e-8.
- Typical: copper σ=5.96e7, T=300 → κ=436; silver σ=6.3e7 → 461; aluminum σ=3.5e7 → 256.
Case Studies
CPU heat-spreader material choice
Copper σ=5.96e7 S/m, T=350K (operating).
κ=2.44e-8×5.96e7×350=508 W/(m·K), above the room-temperature 436.
Aluminum σ=3.5e7 → κ=298; copper conducts 1.7× better but is 3.3× denser, so high-end CPUs use copper, thin devices use aluminum.
Superconducting-transition test
Below Tc the WF law breaks down because the electron gas no longer carries heat normally.
Measuring κ/σ vs T reveals the superconducting transition.
This deviation is a classic signature used to confirm superconductivity.
FAQ
What does the Wiedemann-Franz law say?
For metals κ/σ=L·T, so κ=LσT, where L≈2.44e-8 W·Ω/K² is the Lorenz number. It means thermal and electrical conductivities are proportional.
Why do good conductors also conduct heat well?
The same free electrons transport both charge and heat, so a high electrical conductivity (copper, silver) goes with high thermal conductivity.
What is the Lorenz number?
L=(π²/3)(k_B/e)²≈2.44e-8 W·Ω/K², derived by Sommerfeld from the free-electron model; it is roughly constant for metals.
When does it fail?
In superconducting and some strongly correlated or very low-temperature regimes the simple proportionality breaks down — a useful diagnostic of novel electronic behavior.
Related Tools
References
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.