In physics, the Wiedemann–Franz law states that the ratio of the electronic contribution of the thermal conductivity (κ) to the electrical conductivity (σ) of a metal is proportional to the temperature (T).
κ σ = L T {\displaystyle {\frac {\kappa }{\sigma }}=LT}
Theoretically, the proportionality constant L, known as the Lorenz number, is equal to
L = κ σ T = π 2 3 ( k B e ) 2 = 2.44 × 10 − 8 V 2 ⋅ K − 2 , {\displaystyle L={\frac {\kappa }{\sigma T}}={\frac {\pi ^{2}}{3}}\left({\frac {k_{\rm {B}}}{e}}\right)^{2}=2.44\times 10^{-8}\;\mathrm {V^{2}{\cdot }K} ^{-2},}
where kB is the Boltzmann constant and e is the elementary charge. This empirical law is named after Gustav Wiedemann and Rudolph Franz, who in 1853 reported that κ/σ has approximately the same value for different metals at the same temperature. The proportionality of κ/σ with temperature was discovered by Ludvig Lorenz in 1872.
Derivation
Qualitatively, this relationship is based upon the fact that the heat and electrical transport both involve the free electrons in the metal. The mathematical expression of the law can be derived as following. Electrical conduction of metals is a well-known phenomenon and is attributed to the free conduction electrons, which can be measured as sketched in the figure. The current density j is observed to be proportional to the applied electric field and follows Ohm's law where the prefactor is the specific electrical conductivity. Since the electric field and the current density are vectors Ohm's law is expressed here in bold face. The conductivity can in general be expressed as a tensor of the second rank (3×3 matrix). Here we restrict the discussion to isotropic, i.e. scalar conductivity. The specific resistivity is the inverse of the conductivity. Both parameters will be used in the following.
The thermal conductivity is given by κ = 1 3 c n ℓ ⟨ v ⟩ {\displaystyle \kappa ={\frac {1}{3}}cn\,\ell \,\langle v\rangle } where c {\displaystyle c} is the heat capacity per electron, n {\displaystyle n} is the number density of charge carriers, ℓ {\displaystyle \ell } is the mean free path of the electrons, and ⟨ v ⟩ {\displaystyle \langle v\rangle } is the mean speed of the carriers. The electrical conductivity is given by
σ = n e 2 τ m = n e 2 ℓ m ⟨ v ⟩ {\displaystyle \sigma ={\frac {ne^{2}\tau }{m}}={\frac {ne^{2}\ell }{m\langle v\rangle }}} . where τ {\displaystyle \tau } is the mean free time and m the mass of the charge carriers.
The ratio is given by κ σ = 1 3 c m ⟨ v ⟩ 2 e 2 . {\displaystyle {\frac {\kappa }{\sigma }}={\frac {1}{3}}{\frac {c\,m\langle v\rangle ^{2}}{e^{2}}}\,.}
Drude model derivation Paul Drude (c. 1900) realized that the phenomenological description of conductivity can be formulated quite generally (electron-, ion-, heat- etc. conductivity). Although the phenomenological description is incorrect for conduction electrons, it can serve as a preliminary treatment. The assumption is that the electrons move freely in the solid like in an ideal gas. The force applied to the electron by the electric field leads to an acceleration according to
F = − e E = m d v d t {\displaystyle \mathbf {F} =-e\mathbf {E} =m{\frac {\;d\mathbf {v} }{dt}}}
d v = − e E m d t {\displaystyle \;d\mathbf {v} =-{\frac {e\mathbf {E} }{m}}dt}
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