Discussion
Consider one mole of a monovalent metal at absolute zero temperature, obeying the free electron model. Its Fermi energy is \(E_{F}\). The energy corresponding to the filling of \(\frac{N_{A}}{2}\) electrons, where \(N_{A}\) is the Avogadro number, is \(2^{n}E_{F}\). The value of \(n\) is
Answer: A
The value is \(n = -\frac{2}{3}\).
In the free electron model the Fermi energy depends on the electron density as \(E_{F} \propto N^{2/3}\) (for a fixed volume), since \(E_{F} = \frac{\hbar^{2}}{2m}(3\pi^{2}N/V)^{2/3}\). Filling only half the electrons gives
\[ E = E_{F}\left(\frac{1}{2}\right)^{2/3} = 2^{-2/3}E_{F} \]
so \(n = -\frac{2}{3}\). The energy must be less than \(E_{F}\), which rules out \(+\frac{2}{3}\); \(-\frac{1}{3}\) and \(-1\) use the wrong power of the density.
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