The metal polonium (which was named by Maric Curie after her homeland, Poland) crystallizes in a primitive cubic strucrure, with an atom at each corner of a cubic unit cell. The atomic radius of polonium is \(167 \mathrm{pm}\). Sketch the unit cell and determine (a) the number of atoms per unit cell; (b) the coordination number of an atom of polonium; (c) the length of the side of the unit cell.

Polonium is a chemical element with the symbol Po and atomic number 84, discovered by Marie Curie and named in honor of her native country, Poland. It is a rare and highly radioactive metalloid with no stable isotopes. Polonium has unique properties, as it assumes a position in the periodic table where the metals transition to nonmetals, which contributes to its complex chemistry.

Interestingly, polonium is one of the few elements that crystallize in a primitive cubic structure, an attribute rare among metals. Despite its radioactivity, polonium's place in the study of crystal structures is valuable in understanding atomic behavior in a solid state. It is typically used in devices like anti-static brushes and has played a role in early nuclear research.

Crystallography is the scientific study of crystals and their structure. It involves looking at the arrangement of atoms within a crystalline solid. The arrangement of atoms is not random but follows a repetitive pattern that defines the particular crystal structure of the material.

In crystallography, the basic unit of a crystal's structure is called the unit cell, which is the smallest repeating pattern that shows the full symmetry of the crystalline substance. The primitive cubic structure, as seen in polonium, is one of the simplest crystal structures where the unit cell is shaped like a cube, with atoms located at each of the eight corners.

The atomic radius is a measure of the size of atoms. Usually, it's defined as half the distance between the nuclei of two adjacent or bonded atoms of the same element in the solid phase. For elements that do not occur as diatomic molecules in the solid phase, such as polonium in its primitive cubic arrangement, the atomic radius can also be determined by half the distance of the edge length of the unit cell when the atoms are in direct contact.

Knowing the atomic radius is essential in calculating other properties of the unit cell, including the cell's volume and the density of the element. In polonium's case, the atomic radius significantly influences the understanding of its crystal structure and how atoms interact within that structure.

The coordination number of an atom in a crystal structure is the number of nearest neighboring atoms to that atom. It's a key concept in understanding how atoms are packed together and how they interact with each other within a solid material.

In a primitive cubic lattice, each atom has a coordination number of 6 because it is immediately surrounded by 6 neighboring atoms, one on each face of the cube. This number does not change regardless of the location of the atom within the crystal lattice. It is a fundamental property that affects the physical and chemical properties of the element, such as its melting point, boiling point, and reactivity.