Aluminum metal has a density of \(2.70 \mathrm{~g} \cdot \mathrm{cm}\) and crystallizes in a lattice with a unit cell edge of \(404 \mathrm{pm}\). (a) What type of cubic unit cell is formed by aluminum? (b) What is the coordination number of aluminum?

When it comes to metals like aluminum, density is a fundamental property that reflects how tightly atoms are packed together in a solid form. The density of aluminum is famously low at 2.70 grams per cubic centimeter (g/cm³), which is why it is valued for its strength-to-weight ratio in industries such as aerospace and transportation.

The concept of density is crucial in understanding materials, and in the case of aluminum, it is directly related to its unit cell—the smallest repeating unit that shows the entire structure of the crystal. By examining the density in conjunction with the unit cell type and dimensions, we gain insight into the arrangement of atoms within the crystal lattice structure of aluminum.

The coordination number of an atom within a crystal lattice is the number of other atoms it directly bonds with or touches. This number can reveal a lot about the properties of the material. For instance, in metal crystals, a higher coordination number typically implies a denser atomic packing, which can affect the material's strength and melting point.

In the context of our problem, after determining the cubic unit cell that aluminum forms, we can then ascertain its coordination number. Since aluminum forms a face-centered cubic (FCC) lattice, as is commonly found for metals, each atom in an aluminum crystal touches 12 other atoms, yielding a coordination number of 12.

A crystal lattice represents the symmetrical three-dimensional arrangement of atoms in a crystal. In solid-state physics and materials science, the concept of a crystal lattice is paramount because it influences the material's properties, including strength, conductivity, and durability.

For aluminum, the lattice is face-centered cubic (FCC), one of the three types of cubic lattices. It consists of atoms at each corner of a cube and at the centers of each cube face, which distinguishes it from the other cubic lattice types, namely simple cubic and body-centered cubic. Each lattice type has a unique geometric arrangement that affects the material's physical attributes.

In a crystalline structure, the edge length of the unit cell is the distance between two adjacent corners along the edge of the cube. This edge length is vital as it is used to calculate the volume of the unit cell, which in turn is essential for finding the density of the material.

For aluminum with a face-centered cubic structure, the unit cell edge length given is 404 picometers (pm), or 404 x 10^-10 centimeters (cm). Understanding and measuring the edge length accurately is key to performing calculations related to the crystal structure, such as determining the theoretical density and the size of atoms or ions. This measurement is the foundation for understanding many of the material's macroscopic properties.

Theoretical density is calculated to predict the density of a crystalline material based on its atomic mass, the number of atoms per unit cell, and the volume of the unit cell. This value helps in verifying the type of cubic unit cell a material forms and in understanding its physical characteristics.

To calculate the theoretical density of aluminum, one would use the formula mentioned above. With the known atomic weight of aluminum, the number of atoms per FCC unit cell (four), Avogadro's number, and the volume derived from the unit cell edge length, you can determine the theoretical density of aluminum, which should align closely with its known, empirical density.

Avogadro's number, approximately 6.022 x 10²³, is the number of atoms, ions, or molecules in one mole of substance. This fundamental constant is pivotal in chemistry and physics as it serves as a bridge between the microscopic world of atoms and the macroscopic world we experience daily.

In the context of calculating the density of a crystalline structure, like that of aluminum, Avogadro's number is used to determine the mass of atoms within a unit cell. By understanding the number of atoms present and their individual masses, calculations can include the vast number of individual units involved, allowing the extrapolation of data from the atomic scale to real-world applications where materials are measured in grams and centimeters.