If \(Z\) is the number of atoms in the unit cell that represents the closest packing sequence ...A B C A B C..., the number of tetrahedral voids in the unit cell is equal to: (a) \(2 \mathrm{Z}\) (b) \(\mathrm{Z} / 2\) (c) \(\mathrm{Z}\) (d) \(\mathrm{Z} / 4\)

The packing sequence of atoms in crystals is fundamental to understanding their structures and properties. This sequence describes how atoms are arranged in the three-dimensional space of a crystal lattice. One of the most efficient ways atoms can pack together is in a sequence known commonly as closest packing.

There are two main types of closest packing sequences: hexagonal close-packed (hcp) and face-centered cubic (fcc), also known as cubic close-packed (ccp). In both types, each atom is surrounded by 12 others, but the stacking sequence differs. The fcc sequence follows an alternating order of three layers — typically labeled as A, B, and C — such that each layer is positioned over the voids of the preceding layer, creating an ABCABC... pattern.

This sequence results in high packing efficiency and thus, is favored by many metallic elements. Understanding the packing sequence is crucial for interpreting several properties of materials including their density, melting point, and how they interact with light and other forms of energy.

The face-centered cubic (fcc) structure is a type of crystal lattice that is particularly important in the context of metals and their alloys. In this arrangement, atoms are located at each of the eight corners of a cube, as well as at the centers of each of the cube's faces. This creates a unit cell with a total count of four whole atoms.

Despite the simple description, the fcc structure has a complex three-dimensional geometry. Each corner atom is shared among eight adjacent unit cells, while each face-centered atom is shared between just two unit cells. This leads to an intricate network of atomic positions that requires careful analysis to understand the unit cell's actual atomic count and the distribution of void spaces, which in turn influences material properties like ductility and tensile strength.

In materials science, the fcc structure is celebrated for its close-packed arrangement, permitting atoms to slide over each other under stress without fracturing, which accounts for the malleability of metals like gold and silver.

The concept of 'atoms in unit cell chemistry' is pivotal when studying crystalline materials. A unit cell is the smallest repeating unit of a crystal lattice, and its composition determines the entire structure of the crystal. When counting atoms in a unit cell, it is essential to remember that an atom at a corner is shared by eight neighboring unit cells and an atom on an edge is shared by four unit cells, while an atom on a face is shared by two.

The face-centered cubic (fcc) unit cell includes 8 corner atoms, each contributing an eighth of its volume to the unit cell, and 6 face atoms, each contributing half. When you do the math, this adds up to \( Z = 8 \times \frac{1}{8} + 6 \times \frac{1}{2} = 4 \) whole atoms per unit cell. This counting strategy is key in determining properties such as the cell's density and helps in calculating the number of void spaces, like tetrahedral and octahedral voids, which in turn can affect the behavior of the crystal under various conditions.