At room temperature the electrical conductivity and the electron mobility for aluminum are \(3.8 \times 10^{7}(\Omega-\mathrm{m})^{-1}\) and \(0.0012 \mathrm{m}^{2} / \mathrm{V}-\mathrm{s}, \mathrm{re}\) spectively. (a) Compute the number of free electrons per cubic meter for aluminum at room temperature. (b) What is the number of free clectrons per aluminum atom? Assume a density of \(2.7 \mathrm{g} / \mathrm{cm}^{3}\)

When diving into the fundamentals of electrical conductivity, understanding charge carrier density is crucial. Charge carrier density, often symbolized as 'n', represents the number of charge carriers, such as electrons, within a certain volume of a material—specifically a cubic meter when dealing with the SI unit system.

In materials like metals, free electrons act as charge carriers, and the density of these carriers determines how easily an electric current can flow. A higher 'n' value signifies a larger population of charge carriers available to move and thus, typically correlates with improved conductivity.

To find the charge carrier density, as demonstrated in the exercise, the following formula is utilized: \[ n = \frac{\sigma}{e \mu} \]
In this equation, \(\sigma\) represents electrical conductivity, \(e\) stands for the elementary charge (the charge of a single electron, 1.6 x 10^-19 Coulombs), and \(\mu\) is electron mobility, which reflects how swiftly electrons can navigate through the material. Having high electron mobility enhances the charge carrier density's effect on the overall conductivity.

Electron mobility, denoted by \(\mu\), is a measure that portrays the velocity of an electron through a conductor when subjected to an electric field. The ease with which electrons can move through the material under the influence of this field directly affects the conductivity.

Factors such as the structure of the material, impurities, and temperature can influence electron mobility. For instance, higher temperatures tend to decrease \(\mu\) due to increased scattering events between electrons and lattice vibrations, called phonons.

In the context of the exercise, electron mobility is given and plays a pivotal role in determining the charge carrier density. The product of the charge carrier density 'n' and electron mobility \(\mu\), multiplied by the elementary charge 'e', yields the electrical conductivity of the material:\[ \sigma = n e \mu \]Thus, knowing the electron mobility allows us to understand and calculate how conductive a material is likely to be under certain conditions.

The concept of 'free electrons per atom' is an integral part of understanding metallic conductivity. In a conductor, not every electron contributes to the flow of electric current—only the outermost, or 'free', electrons do.

The sample problem calculates the free electrons per aluminum atom at room temperature. To grasp this concept, imagine that atoms in a metal are fixed in place and the free electrons move about, which is often referred to as the 'sea of electrons' model.

The number of free electrons available for conducting electricity can be understood in terms of the metal's atomic structure. Elements like aluminum contribute a certain number of free electrons (in aluminum's case, 3.29 free electrons per atom), which comes from the calculation provided in the exercise and is crucial for determining the material's overall conductivity:
Free electrons per aluminum atom \[ = \frac{\text{Charge carrier density (n)}}{\text{Number of aluminum atoms per cubic meter}} \]This attribute, alongside charge carrier density and electron mobility, completes the trio of factors that are paramount for analyzing a material's electrical conductivity.