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Wiedemann-Franz Law

June 17, 2024

Science is most useful when it makes predictions. Predictions are based on theory, and one problem with most theories in materials science is that they are based on data that might not exist for all of the intended class of materials. Early in my career I developed a theory of how the concentrations of elements in crystals grown from a molten solution of lead oxide would depend on their concentrations in that solution.[1] This theory was intended to assist in the epitaxial growth of magnetic garnets for magnetic bubble memories. One parameter I needed was the enthalpy of vaporization of some metal oxides, and many of these had not been measured.

Fortunately, Trouton's rule came to my rescue. This rule, developed by Irish physicist, Frederick Trouton (1863-1922), states that the molar entropy of vaporization ΔSv for most materials is quite close to 85 J/(mol-K) at their boiling points TB.[2] This gave me the ability to estimate the enthalpy of vaporization ΔHv from the known boiling points,
ΔHv = TB·ΔSv
While Trouton's rule is valid for many liquids, it fails when the liquid components interact peculiarly, as in hydrogen bonding. That's why the rule fails for water and other hydrogen bonded materials.

Frederick Trouton, Irish physicst, in 1926

Frederick Trouton, Irish physicist, in 1926.

There have been many important Irish physicists, a few of the most significant are J. D. Bernal (1901-1971), who applied X-ray crystallography to molecular biology; Robert Boyle (1627-1691), famous for Boyle's law; Nicholas Callan (1799-1864), who invented the Induction coil for conversion of low voltages to high-voltage; William Rowan Hamilton (1805-1865), a contributor to mechanics, especially for devising the Hamiltonian; John Joly (1857-1933), who perfected uranium-thorium dating.

Also, Thomas Preston (1860-1900), who discovered the anomalous Zeeman effect; Erwin Schrödinger (1887-1961), who created the eponymous Schrödinger equation; George Stokes (1819-1903), famous for the Navier-Stokes equations; George Stoney (1826-1911), most famous for naming the electron, but he was also the first to estimate the number of gas molecules in a volume; and, John Tyndall (1820-1893), whose research in infrared led to the Tyndall effect, proof of the heat trapping by CO2, and the greenhouse effect.

Wikimedia Commons image from the Proceedings of the Royal Society (London), A110, 4, 1926.

Another correlation between two material properties is so precise, in certain limits, that it's known as a law rather than a rule. This is the Wiedemann-Franz law relating electrical conductivity σ and thermal conductivity κ; viz.,
κ/σ = LT
in which T is the absolute temperature, and L is the proportionality constant, called the Lorenz number (2.44 x 10-8 V2-K-2). The Lorentz number can be given in fundamental units as
L = (π2/3)·(kB/e)2
in which kB is the Boltzmann constant (1.38065 x 10-23 J-K-1), and e is the elementary charge (1.60218 x 10-19 coulombs). This law is named after German physicists, Gustav Wiedemann (1826-1899) and Rudolph Franz (1826-1902), who discovered in 1853 that κ/σ was approximately constant for different metals at the same temperature.[3] Danish physicist, Ludvig Lorenz (1829-1891), deduced the proportionality constant in 1872.

The Wiedemann-Franz law gets its validity from the fact that in most cases both heat and electric current are transported by free electrons in metals. However, electron transport of heat is only dominated by electrons above a certain temperature, called the Debye temperature, developed by Peter Debye (1884-1966). Below the Debye temperature, the thermal conductivity of metals decreases more rapidly with increased temperature than the electrical conductivity. Debye also had a model for the heat capacity that accurately predicts its T3 behavior at low temperatures, and is known for his work on X-ray diffraction, for which he was awarded the 1936 Nobel Prize in Chemistry.

Debye Temperature for Selected Elements

Element T(K) Element T(K)
Aluminum 428      Beryllium 1440
Cadmium 209      Cesium 38
Diamond 2230      Chromium 630
Copper 343      Germanium 374
Gold 170      Iron 470
Lead 105      Manganese 410
Nickel 450      Platinum 240
Rubidium 56      Sapphire 1047
Selenium 90      Silicon 645
Silver 215      Tantalum 240
Tin (white) 200      Titanium 420
Tungsten 400      Zinc 327

Temperature dependence of the thermal conductivity of beryllium

Temperature dependence of the thermal conductivity of beryllium.

Beryllium has the very high Debye temperature of 1440 K (1167 °C), just below its melting point of 1,560 K (1277°C).

This is the usual trend of higher thermal conductivity at lower temperature, just as the trend for electrical conductivity.

(Plotted using Gnumeric from data in ref.4.[4] Click for larger image.)

Measurement of electrical conductivity of solids and liquids is somewhat easy, but thermal conductivity measurements are much more difficult. That's the reason that the Wiedemann-Franz law hasn't been tested over an extended range of temperature. A recent open access paper by scientists and engineers from the University of Virginia (Charlottesville, Virginia), the European Commission, Joint Research Centre (Karlsruhe, Germany), Laser Thermal Analysis, Inc. (Charlottesville, Virginia), and the University of Rhode Island (Kingston, Rhode Island) describes their differential radiometry technique used to validate the Wiedemann-Franz law for tungsten in both its solid and molten states to temperatures above 2000 K.[5-6] Their paper is published in Physical Review Letters.[5]

The research team has made the first direct thermal conductivity measurement for tungsten, the elemental metal with the highest melting point, 3,695 K (3,422°C).[6] To do this, they used a laser beam to gradually heat the center of a 2 millimeter thickness tungsten disk until it began to melt. The emitted radiation was monitored during heating to determine the temperature of the central hot spot. They monitored the required increase in the laser power required to heat the area from 2000 K to 4000 K.[6] Data from this technique allowed calculation of the thermal conductivity for tungsten over this range of temperature using Fourier's law of thermal conduction.[5-6]

Schematic of the thermal conductivity apparatus and comparison data.

Left, a schematic of the experimental apparatus for the thermal conductivity measurement in which the tungsten disk acts as its own crucible to contain the molten metal at its center. Right, measured thermal conductivity of solid and molten tungsten compared with literature values. (Fig. 1a and fig. 3 of ref. 5,[5] released under the Creative Commons Attribution 4.0 International license. Click for larger image.)

The technique is a direct measurement of thermal conductivity that avoids the requirement of knowing the material's heat capacity to calculate the thermal conductivity.[5] This single-sided method in which the heat is sourced and temperature is measured from the same side can be used for measurement of thermal conductivity of materials other than tungsten in their molten states.[5] The experiment validated the Wiedemann-Franz law in tungsten even into its molten state.[5]


  1. D.M. Gualtieri, Flux Growth of (Ca,Ge)-Substituted Rare- Earth Iron Garnets in the Regular Solution Approximation, J. Appl. Phys., vol. 50 (1979), pp. 2170-2172, https://doi.org/10.1063/1.327041. (Surprisingly, the publisher wants $40.00 to read this paper after 45 years!)
  2. Frederick Trouton, "On Molecular Latent Heat," Philosophical Magazine, vol. 18, no. 110 (1884), pp. 54-57, doi:10.1080/14786448408627563.
  3. G. Wiedemann and R. Franz, "Ueber die Wärme-Leitungsfähigkeit der Metalle," Annalen der Physik, vol. 165, no. 8 (1853), pp. 497-531, doi:10.1002/andp.18531650802. A free pdf file is available at the same URL.
  4. Beryllium, ARIES Archive (2013), Office of Fusion Energy Sciences, US Department of Energy.
  5. Milena Milich, Hunter B. Schonfeld, Konstantinos Boboridis, Davide Robba, Luka Vlahovic, Rudy J. M. Konings, Jeffrey L. Braun, John T. Gaskins, Niraj Bhatt, Ashutosh Giri, and Patrick E. Hopkins, "Validation of the Wiedemann-Franz Law in Solid and Molten Tungsten above 2000 K through Thermal Conductivity Measurements via Steady-State Temperature Differential Radiometry," Phys. Rev. Lett., vol. 132, no. 14 (April 5, 2024), Article no. 146303, DOI:https://doi.org/10.1103/PhysRevLett.132.146303. This is an open access article with a pdf file here.
  6. Rachel Berkowitz, "Thermal Conductivity Not Too Hot to Handle," Physics, vol. 17, no. 25 (April 5, 2024).

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