If \(\mathrm{R}\) is the radius of the sphere in the close packed arrangement and \(\mathrm{r}\) is the radius of the tetrahedral void, then: (a) \(\mathrm{R}=0.414 \mathrm{r}\) (b) \(\mathrm{r}=0.224 \mathrm{R}\) (c) \(\mathrm{R}=0.225 \mathrm{r}\) (d) \(\mathrm{r}=0.414 \mathrm{R}\)

Within the realm of crystallography and material science, a 'close packed arrangement' refers to a way of stacking identical spheres so that they occupy the maximum space possible. This configuration is highly efficient and can be found naturally in the structure of many metallic crystals.

There are two main types: hexagonal close packing (HCP) and face-centered cubic (FCC), each with a specific geometric order that allows for the minimal amount of space between spheres. In these arrangements, voids or empty spaces are formed, and the two most common types of voids are tetrahedral and octahedral voids.

In a close packed arrangement, each sphere touches several others, coordinating to create a stable structure. The tetrahedral void is formed between four spheres, resembling a tetrahedron – a pyramid with a triangular base – hence the name. Understanding the geometry of close packed arrangements is critical for various applications, such as calculating the density of materials or understanding the properties of alloys.

The 'radius of a sphere' is a fundamental concept in geometry that represents the distance from the center of the sphere to any point on its surface. When we talk about spheres in close packed arrangements, the radius denotes not only a measure of size but also a key factor in determining the arrangement's overall geometry.

For any given sphere, the radius (denoted as R) is half the diameter, which is crucial in calculating volume and surface area of the sphere itself. In chemistry and material science, particularly when examining crystalline structures, knowing the radius helps in determining interatomic distances and the size of voids within the lattice structure. The relationship between the radius of the spheres and the radius of the tetrahedral voids (r) is governed by geometric and algebraic equations derived from the arrangement's symmetry and dimensions.

Geometry plays an integral role in chemistry, especially in understanding molecular shapes, crystal structures, and the arrangement of atoms within compounds. The spatial considerations dictated by geometry are essential for predicting physical properties, behaviors, and reactivity.

In the context of crystal structures, geometry helps in visualizing the three-dimensional arrangement of atoms. For example, the close packed arrangement of atoms can be interpreted through geometric figures such as cubes and tetrahedra. The bonds angles, coordination numbers, and the distances between atoms in a molecule or crystal lattice are all geometric considerations that chemists must understand. Knowing these concepts allows chemists to design and synthesize new materials with desired properties, essentially 'engineering' substances at the molecular level.

The Pythagorean theorem, a fundamental principle in geometry, states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Although the theorem is primarily associated with mathematics, it finds a unique application in chemistry regarding the calculation of distances between atoms in lattice structures.

For instance, in the step by step solution for the tetrahedral void radius calculation, the Pythagorean theorem is used to relate the radii of the sphere and the tetrahedral void. It allows us to derive the exact mathematical relationship between the radius of the sphere (R) and the radius of the tetrahedral void (r) within the close packed arrangement. This application of the theorem illustrates how geometry and algebra are indispensable tools for solving complex spatial problems in chemistry.