and is the average speed of electrons; From Drude model, . Therefore, , which is the Wiedemann–Franz law with an erroneous proportionality constant ; Free electron model After taking into account the quantum effects, as in the free electron model, the heat capacity, mean free path and average speed of electrons are modified and the proportionality constant is then corrected to , which agrees with experimental values. Temperature dependence The value L0 = 2.44×10−8 V2K−2 results from the fact that at low temperatures ( K) the heat and charge currents are carried by the same quasi-particles: electrons or holes. At finite temperatures two mechanisms produce a deviation of the ratio from the theoretical Lorenz value L0: (i) other thermal carriers such as phonon or magnons, (ii) Inelastic scattering. As the temperature tends to 0K, inelastic scattering becomes weak and promotes large q scattering values (trajectory a in the figure). For each electron transported a thermal excitation is also carried and the Lorenz number is reached L = L0. Note that in a perfect metal, inelastic scattering would be completely absent in the limit K and the thermal conductivity would vanish . At finite temperature small q scattering values are possible (trajectory b in the figure) and electron can be transported without the transport of a thermal excitation L(T) < L0. At higher temperatures, the contribution of phonon to thermal transport in a system becomes important. This can lead to L(T) > L0. Above the Debye temperature the phonon contribution to thermal transport is constant and the ratio L(T) is again found constant. Limitations of the theory Experiments have shown that the value of L, while roughly constant, is not exactly the same for all materials. Kittel gives some values of L ranging from L = 2.23×10−8V2K−2 for copper at 0 °C to L = 3.2×10−8V2K−2 for tungsten at 100 °C. Rosenberg notes that the Wiedemann–Franz law is generally valid for high temperatures and for low (i.e., a few Kelvins) temperatures, but may not hold at intermediate temperatures. In many high purity metals both the electrical and thermal conductivities rise as temperature is decreased. In certain materials (such as silver or aluminum) however, the value of L also may decrease with temperature. In the purest samples of silver and at very low temperatures, L can drop by as much as a factor of 10. In degenerate semiconductors, the Lorenz number L