Indium (In) has a tetragonal unit cell for which the \(a\) and \(c\) lattice parameters are \(0.459\) and \(0.495\) \(\mathrm{nm}\), respectively. (a) If the atomic packing factor and atomic radius are \(0.693\) and \(0.1625 \mathrm{~nm}\), respectively, determine the number of atoms in each unit cell. (b) The atomic weight of In is \(114.82 \mathrm{~g} / \mathrm{mol} ;\) compute its theoretical density.

The atomic packing factor (APF) provides vital insight into how tightly atoms are packed in a solid material. In essence, it's a ratio representing the fraction of volume in a unit cell occupied by atoms. To calculate it, we must first understand that the unit cell represents the smallest repeating structure of the crystal lattice, giving us a three-dimensional snapshot of the material's overarching structure.

When students use the APF and the volume of the unit cell, they can deduce the would-be volume of atoms, assuming the space is entirely filled by them. Hence, for the tetragonal unit cell of Indium, the APF is given as 0.693, suggesting that 69.3% of the unit cell's volume is occupied by Indium atoms. Calculations involving APF are foundational for predicting a variety of material characteristics, including density.

Lattice parameters are constants that define the dimensions of a unit cell in a crystal structure. Every crystal system, from cubic to tetragonal, has specific parameters that reveal the size and shape of the unit cell. For a tetragonal unit cell like that of Indium, we need to know two parameters – denoted as 'a' for the base and 'c' for the height.

These parameters not only impact the overall properties of a material but also play a crucial role in calculating vital statistics such as the unit cell's volume. For our tetragonal unit cell of Indium, with lattice parameters of 0.459 nm and 0.495 nm respectively, we can already predict certain physical properties of the material, including how these dimensions can affect its strength and reactivity.

Tetragonal unit cells are a distinct type of crystal lattice structure characterized by three axes, where two are of equal length (a and b) and the third (c) is of a different length. This difference sets it apart from cubic lattices, where all three axes are equal. Knowing the type of crystal structure is important when trying to determine many physical properties of a material.

The volume of such a unit cell is the product of its lattice parameters, which is particularly relevant for calculations of density and the number of atoms per unit cell. In the case of Indium, its structure determines its electrical and thermal properties, affecting its applications in various technologies.

The atomic radius refers to the size of an atom, typically measured from the center of the nucleus to the outermost shell of electrons. This measurement, although seemingly simple, affects how atoms interact with one another and how they pack together in a solid. The atomic radius for Indium is given as 0.1625 nm. This dimension is not arbitrary; it helps us understand how Indium atoms fit within the unit cell and interact with each other, influencing not only the material's APF but also its overall physical characteristics, like density and atomic arrangement in the crystal lattice.

Avogadro's number, denoted as \(N_A\), is the number of entities (could be atoms, molecules, ions, etc.) in one mole of a substance. This constant is fundamental in converting between atomic or molecular scale and the macroscopic scale. It allows us to relate the amount of substance in moles to the actual number of atoms or molecules present.

For our theoretical density calculation of Indium, Avogadro's number is used to scale the number of unit cells to a macroscopic level. This is because density is mass per unit volume, and in a substance, we account for the mass of moles of the material in a given volume. By multiplying the number of atoms by the atomic weight and dividing by the volume of the unit cell then by Avogadro's number, we end up with a property – theoretical density – that's enormously useful in material science for predicting how a substance will behave in the real world.