Calculate the atomic radius of each of the following elements (b) tantalum, bcc structure, density \(16.654 \mathrm{~g} \cdot \mathrm{cm}^{-3}\).

The body-centered cubic (BCC) structure is one of the fundamental crystalline arrangements found in metals and other solid materials. In a BCC structure, each unit cell—which is the smallest repeating unit of the crystal lattice—consists of one atom at each of the eight corners of a cube and one atom right at the center, hence the term ‘body-centered’. These atoms are not separate; they represent the positions that a repeating pattern of the same atom occupies within the crystal.

Understanding the BCC arrangement is crucial when studying material properties, as it directly affects the density and mechanical characteristics of the material. In the case of the BCC structure, atoms touch along the body diagonal of the cube, which helps in determining the relationship between the atoms’ radii and the cube's edge length. For students calculating atomic radius, comprehending this geometric relationship aids in visualizing how the atoms fit within the unit cell and interact with each other.

Volume plays a pivotal role in understanding the unit cell, as it denotes the space an individual cell occupies. It can reveal critical information about the structure when combined with density and mass. In a BCC crystal structure, the unit cell has a simple cubic geometry, making the calculation of its volume straightforward: it's the cube of the edge length, symbolically represented as V = a^3, where a is the edge length of the cube.

This volume, when multiplied by the density, yields the mass of the unit cell, a vital step in calculating the atomic radius from a known density. For educational purposes, it is crucial for students to internalize the relationship between edge length and volume—not only can it help figure out the number of atoms within a cell but also allows them to calculate other properties like atomic mass when combined with density and Avogadro's number.

Avogadro's number, N_A, is a fundamental constant in chemistry and physics, representing the number of atoms, ions, or molecules in one mole of a substance. The value is approximately 6.022 x 10^23 per mole, establishing a bridge between the microscopic world of atoms and the macroscopic world of grams and centimeters that we measure in the lab.

When solving problems in crystallography, Avogadro's number enables the conversion from molar mass (the mass of one mole of an element) to the mass of a single atom. This conversion is particularly useful when calculating the mass of a unit cell in a crystal lattice, as it requires knowing the mass of individual atoms to determine properties like atomic radius. The ability to use Avogadro’s number to switch between the macroscopic and microscopic is a cornerstone of chemistry that underpins many calculations and understanding it can help students with a vast range of topics, from gas laws to stoichiometry and beyond.