in , about 100 times smaller at room temperature and much smaller at lower . The good estimation of the Lorenz number in the Drude model was a result of the classical mean velocity of electron being about 100 larger than the quantum version, compensating the large value of the classical heat capacity. The free electron model calculation of the Lorenz factor is about twice the value of Drude's and its closer to the experimental value. With this heat capacity the free electron model is also able to predict the right order of magnitude and temperature dependence at low T for the Seebeck coefficient of the thermoelectric effect. Evidently, the electronic contribution alone does not predict the Dulong–Petit law, i.e. the observation that the heat capacity of a metal is constant at high temperatures. The free electron model can be improved in this sense by adding the lattice vibrations contribution. Two famous schemes to include the lattice into the problem are the Einstein solid model and Debye model. With the addition of the later, the volumetric heat capacity of a metal at low temperatures can be more precisely written in the form, , where and are constants related to the material. The linear term comes from the electronic contribution while the cubic term comes from Debye model. At high temperature this expression is no longer correct, the electronic heat capacity can be neglected, and the total heat capacity of the metal tends to a constant. Mean free path Notice that without the relaxation time approximation, there is no reason for the electrons to deflect their motion, as there are no interactions, thus the mean free path should be infinite. The Drude model considered the mean free path of electrons to be close to the distance between ions in the material, implying the earlier conclusion that the diffusive motion of the electrons was due to collisions with the ions. The mean free paths in the free electron model are instead given by (where is the Fermi speed) and are in the order of hundreds of ångströms, at least one order of magnitude larger than any possible classical calculation. The mean free path is then not a result of electron–ion collisions but instead is related to imperfections in the material, either due to defects and impurities in the metal, or due to thermal fluctuations. Inaccuracies and extensions The free electron model presents