The classical free-electron gas

Band theory started with a gross approximation the idea that a metal contained a gas of completely free electrons, uninfluenced by the other electrons or anything else. The model, an adaptation of the kinetic theory of gases, was very successful. The electrons were supposed to be flying about in the 

However, it could not explain some important properties. In particular, the theory was unable to account for the fact that metals had a similar specific heat to insulators. If an electron gas occurs in metals, but not in insulators, it could be proved that metals should have a molar specific heat of about 4.5R whereas insulators would have a molar specific heat of only 3R, where R is the gas constant. 

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The sp-valent metals such as sodium, magnesium and aluminium constitute the simplest form of condensed matter. They are archetypal of the textbook metallic bond in which the outer shell of electrons form a gas of free particles that are only very weakly perturbed by the underlying ionic lattice. The classical free-electron gas model of Drude accounted very well for the electrical and thermal conductivities of metals, linking their ratio in the very simple form of the Wiedemann-Franz law. However, we shall now see that a proper quantum mechanical treatment is required in order to explain not only the binding properties of a free-electron gas at zero temperature but also the observed linear temperature dependence of its heat capacity. According to classical mechanics the heat capacity should be temperature-independent, taking the constant value of kB per free particle. 

Problems with the Classical Free Electron Gas Theory 18.3.1 Temperature Dependence... 

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