Final answer:
The average energy of the free electrons in copper at room temperature is close to 3/5 of the Fermi energy, while Maxwell-Boltzmann statistics would predict a much lower average energy of 0.0375 eV due to the classical equipartition theorem, which does not account for quantum mechanical principles.
Step-by-step explanation:
The student has asked to compare the average energy of the free electrons in copper at room temperature with their average energy if they followed Maxwell-Boltzmann statistics, given that the Fermi energy in copper is 7.04 eV and kT (thermal energy at room temperature) is 0.025 eV.
In real conditions, free electrons in a metal at zero temperature occupy energy states up to the Fermi energy. However, at finite temperatures like room temperature, the distribution of electrons among energy levels is described by the Fermi-Dirac distribution. This results in an average energy that is close to 3/5 of the Fermi energy, because the states are filled up to that level due to the Pauli exclusion principle.
If the electrons followed Maxwell-Boltzmann statistics, which is a classical approach that is not the actual case for electrons in a metal, the average energy would be derived from the equipartition theorem and would be (3/2)kT. For room temperature, where kT is 0.025 eV, this would imply an average energy of 0.0375 eV, much lower than the actual average energy of the electrons in the Fermi-Dirac distribution.
Therefore, the Maxwell-Boltzmann statistics significantly underestimate the average energy of free electrons in metals because it does not consider quantum mechanical principles such as the Pauli exclusion principle and the discrete energy levels within the atom.