Indium 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 indium is 114.82 \(\mathrm{g} / \mathrm{mol} ;\) compute its theoretical density.

Understanding how atoms are arranged within a crystal structure is pivotal for interpreting a number of physical properties of materials. One key concept in this realm is the atomic packing factor (APF), which reflects how closely packed the atoms are within a unit cell. The APF is calculated by taking the ratio of the total volume of atoms in a unit cell to the volume of the unit cell itself. For a tetragonal unit cell, like the one we see with indium, the calculation becomes slightly more nuanced due to the distinct dimensions along different axes.

To put it simply, if you were packing spherical objects (atoms) into a box (unit cell), the APF tells you what fraction of the box is filled by the spheres. This has significant implications on the material's density and structural stability. In the case of indium, the APF is given as 0.693, which implies a relatively high density of packing. However, since the APF cannot exceed 1, we know that there must be some empty space within the unit cell, which is characteristic of atomic arrangement in crystals.

When diving into crystallography, you'll find that the tetragonal unit cell is one of the seven basic unit cell shapes. What sets the tetragonal unit cell apart is its rectangular base, characterized by two equal axes (often denoted as 'a') with a height (denoted as 'c') that is different. This gives the cell its distinct prismatic shape.

The tetragonal crystal system is more complex than the simple cubic system, yet not as complex as the orthorhombic or monoclinic systems. When calculating properties such as volume, which requires the formula \(V = a^2c\), it's important to note the difference in length between the 'a' and 'c' axes. These distinctions play into how atoms are arranged within the cell and ultimately affect the material's properties. In practice, when solving for specifics like the number of atoms per unit cell or the theoretical density, being mindful of the unit cell geometry is indispensable.

A cornerstone of chemical and physical science is Avogadro's number, an invaluable constant that helps bridge the gap between the atomic scale and the macroscopic world we experience. Representing the number of constituent particles, usually atoms or molecules, in one mole of a substance, Avogadro's number is precisely defined as \(6.022 \times 10^{23}\) entities per mole.

This constant allows us to calculate the mass of a single atom by relating it to the molar mass, a value easily determined from the periodic table. When calculating the theoretical density of a substance, Avogadro's number enables us to move from the atomic or molecular scale, where the mass of an atom is miniscule, to a scale we can work with and measure in the laboratory. Therefore, it is a bridge from the abstract concept of atoms to tangible measurements, and in the context of the given exercise, it brings us the theoretical density of indium when combined with the substance's molar mass and the volume of its unit cell.