Total volume of atoms present in a face-centred cubic unit cell of a metal is ( \(\mathrm{r}\) is atomic radius): (a) \(\frac{20}{3} \pi r^{3}\) (b) \(\frac{24}{3} \pi r^{3}\) (c) \(\frac{12}{3} \pi r^{3}\) (d) \(\frac{16}{3} \pi r^{3}\)

Solid State Chemistry explores the synthesis, structure, and properties of solid materials. It's a vibrant area of chemistry that examines how the atoms in a solid are arranged and how this arrangement affects the material's properties. For example, solids can have an amorphous structure, where atoms are randomly distributed, or a crystalline structure, where atoms are arranged in an orderly pattern. The study of crystalline solids is particularly important because their well-ordered structures result in characteristic physical properties such as electrical conductivity and melting point. With the exercise in question, we dove into the realm of crystalline structures, focusing on the face-centered cubic unit cell, which is commonly found in metals like copper, aluminum, and nickel.

Crystal lattice structures are the repetitive geometric arrangement of atoms within a crystal. This three-dimensional structure extends in all directions and defines the material's properties. In solid state chemistry, there are many types of crystal lattices, the most common being simple cubic, body-centered cubic, and face-centered cubic (FCC). The FCC structure is particularly dense, making it an interesting subject of study for material science.

The FCC structure has atoms at each of the eight corners of a cube and at the center of each of the cube's faces. This configuration is crucial to understanding the properties and behaviors of metals. However, when calculating properties related to the FCC structure, one must take into account that the atoms at the faces and corners are shared with adjacent unit cells, as demonstrated in the textbook solution.

Unit cell calculations involve the mathematical quantification of the properties of the smallest unit of a crystal lattice that can replicate the entire structure when stacked in space. In FCC, the contributions from atoms on corners and faces of the unit cell differ due to sharing. For the corners, each atom is shared by eight unit cells, while each face-centered atom is shared by two.

Crucially, these calculations often require understanding the volume of individual atoms, treating them as spheres, and considering their packing in the unit cell. By combining these factors—effective atoms per cell and the volume of a single atom—you can calculate critical material properties, such as density. The solution in the exercise provided a clear instance of this by calculating the total volume of atoms within an FCC unit cell. Remember, these computations are fundamental in predicting how materials will behave under different conditions.