(a) Calculate the number of free electrons per cubic meter for gold, assuming that there are \(1.5\) free electrons per gold atom. The electrical conductivity and density for Au are \(4.3 \times 10^{7}(\Omega \cdot \mathrm{m})^{-1}\) and \(19.32 \mathrm{~g} / \mathrm{cm}^{3}\), respectively. (b) Now compute the electron mobility for \(\mathrm{Au}\).

Electrical conductivity is a fundamental property that characterizes how well a material can conduct an electric current. In the context of metals, such as gold or silver, which the exercise explores, the property is highly significant. The ability of the metal to conduct electricity comes from the sea of free electrons that can move throughout the metal's crystal lattice. These free electrons are responsible for the transmission of electric current.

When an electric field is applied to a metal, these free electrons accelerate due to the electric force, leading to a flow of charge, which is the electric current. The measure of how easily these electrons can flow is called electrical conductivity, often denoted by the symbol \(\sigma\). It's directly proportional to both the number of free electrons available per unit volume (\

Electron mobility is a key concept when discussing the electrical properties of materials, especially conductors like metals. Electron mobility, denoted by \(\mu\), represents how quickly an electron can move through a material when subjected to an electric field. This is important because greater electron mobility means that the material can conduct electricity more efficiently.

The mathematical expression from the exercise relates electron mobility to electrical conductivity (\(\sigma\)) and the density of free electrons (n) in a given volume, using the elementary charge (e). This relationship shows how intrinsic electronic properties of a material influence its performance in electrical applications. In essence, higher mobility implies that less energy is required to move the electrons through the material, which translates to better electrical conductivity.

In summary, electron mobility can be indicative of a metal's effectiveness in applications such as electrical wiring or components where minimal energy loss is desired.

Avogadro's number is a constant that represents the number of constituent particles, usually atoms or molecules, contained in one mole of any substance. The value of Avogadro's number is approximately \(6.022 \times 10^{23}\) particles per mole. This number is foundational to chemistry and provides the link between the microscale world of atoms and the macroscale world we interact with every day.

In the context of the exercise, Avogadro's number is used to convert the number of moles of gold or silver, as calculated from a given mass or density, into the actual number of atoms. By knowing the number of atoms per cubic meter, and understanding that metals have free electrons associated with each atom, we can calculate the total number of free electrons in a given volume. This number is crucial for understanding and calculating a metal's electrical properties, as it directly affects the metal's electrical conductivity and, in turn, influences properties such as electron mobility.