Aluminium crystallizes in a cubic close-packed structure. Its metallic radius is \(125 \mathrm{pm}\). (a) What is the length of the side of the unit cell? (b) How many unit cells are there in \(1.00 \mathrm{~cm}^{3}\) of aluminium?

The concept of metallic radius is crucial for understanding the arrangement of atoms in a metal. It refers to half the distance between the nuclei of two adjacent atoms in a crystal lattice. In the case of aluminium, which crystallizes in a cubic close-packed (ccp) structure, its metallic radius is given as 125 pm (picometers). This measurement is significant as it determines the size of the unit cell and the arrangement of atoms within the cell.

For metals with a ccp structure, the metallic radius aligns closely with how atoms are tightly packed in the lattice. Since each atom is surrounded by an array of other atoms in this formation, learning about metallic radius gives insights into the density and compactness of the metal. This basic understanding is a stepping stone for detailed calculations related to the unit cell and how it builds up the entire crystal structure of the metal.

The unit cell is the smallest repeating unit that composes the crystal structure of a metal and fully defines its crystallography. When we refer to unit cell calculation, we are looking to determine key parameters like the cell's edge length, volume, and the number of atoms per cell. For aluminium, with its given metallic radius, we can compute the length of the edge of the unit cell by employing a geometric relationship involving the face diagonal.

In ccp or fcc structures, the atoms touch along the face diagonals. By knowing that the face diagonal is four times the metallic radius, we can use the Pythagorean theorem to relate the face diagonal to the side length of the cell, which in turn provides the basis to calculate the volume of the unit cell. These computations offer valuable quantitative details about the metal's structure, influencing its properties and practical applications.

The face diagonal is an integral line segment within the geometry of the cubic crystal system. It stretches across the face of the cube, connecting opposite corners. In a ccp structure, it is inherently linked to the metallic radius as it is composed of four atomic radii lined up in a straight path across the center of a face.

To calculate the length of the face diagonal, we invoke the metallic radius and the geometry of a right-angled triangle. By understanding this connection, and knowing that the cube's face diagonal is the hypotenuse of a right triangle whose legs are the cube's edges, we can tease out the length of those edges, which define the parameters of the entire crystal lattice. This characteristic face diagonal is instrumental in solving many problems related to the crystalline structure of metals and helps elucidate the relationship between atomic spacing and the properties of the metal.