The question that caused the greatest difficulty in the early development of the electron theory of metals concerns the heat capacity of the conduction electrons. Classical statistical mechanics predicts that a free particle should have a heat capacity of 3/2kB, where kB is the Boltzmann constant. If N atoms each give one valence electron to the electron gas, and the electrons are freely mobile, then the electronic contribution to the heat capacity should be 3/2NkB, just as for the atoms of a monatomic gas. But the observed electronic contribution at room temperature is usually less than 0.01 of this value.
This discrepancy was resolved only upon the discovery of the Pauli exclusion principle and the Fermi distribution function. When we heat the specimen from absolute zero not every electron gains an energy ~kBT as expected classically, but only those electrons, which have the energy within an energy range kBT of the Fermi level, can be excited thermally. These electrons gain an energy, which is itself of the order of kBT, as in Fig. 3. This gives a qualitative solution to the problem of the heat capacity of the conduction electron gas. If N is the total number of electrons, only a fraction of the order of kBT/EF can be excited thermally at temperature T, because only these lie within an energy range of the order of kBT of the top of the energy distribution.
Each of these NkBT/EF electrons has a thermal energy of the order of kBT. The total electronic thermal kinetic energy U is of the order of U≈(NkBT/EF)kBT. The electronic heat capacity is Cel=dU/dT≈NkB(kBT/EF) and is directly proportional to T, in agreement with the experimental results discussed in the following section. At room temperature C is smaller than the classical value ≈ NkB by a factor 0.01 or less.
Fig.3 Density of single-particle states as a function of energy, for a free electron gas in three dimensions. The dashed curve represents the density f(E,T)D(E) of filled orbitals at a finite temperature, but such that kT is small in comparison with EF. The shaded area represents the filled orbitals at absolute zero. The average energy is increased when the temperature is increased from 0 to T, for electrons are thermally excited from region 1 to region 2.