Copper has one conduction electron per atom, a density of 9...

Copper has one conduction electron per atom, a density of 9 and an atomic weight of 64 . Show that $n_{0}$, the number of free electrons per unit volume is $\approx 8 \times 10^{28} \mathrm{~m}^{-3}$ and that the value of its Fermi energy level is about $7 \mathrm{eV}\left(1 \mathrm{eV}=1.6 \times 10^{-19} \mathrm{~J}\right)$.

Key Concepts

Fermi Energy

Fermi energy is the highest occupied energy level of electrons in a material at absolute zero temperature. It is a crucial concept in understanding the electronic properties of metals because it determines many physical characteristics such as electrical conductivity and heat capacity. The Fermi energy is calculated using the free electron density and reflects the energy distribution of electrons in the conduction band.

Free Electron Density

This concept involves determining the number of conduction electrons per unit volume in a metal. It combines the idea that each atom contributes a conduction electron with the use of material properties (density and atomic weight) and Avogadro’s number to convert mass density into atomic density, which in turn gives the number density of free electrons.

Conversion of Mass Density to Atomic Density

This key concept explains how to convert the large-scale, measurable properties of a material (its density and atomic weight) into microscopic quantities like the number of atoms per unit volume. By using Avogadro’s number, one can relate the mass of a substance to the number of constituent particles, allowing for the calculation of free electron density in a metal.

Free Electron Model (Fermi Gas Model)

This model treats the conduction electrons in a metal as a gas of non-interacting particles obeying Fermi-Dirac statistics. It provides a framework for calculating the electronic properties, including the Fermi energy, by considering electrons in a 'box' with quantized energy levels. The model simplifies the complex interactions in a solid and is fundamental in describing the behavior of conduction electrons in metals.

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