Question

Network Covalent Atomic Solids When spheres of radius \(r\) are packed in a body-centered cubic arrangement, they occupy 68.0\(\%\) of the available volume. Use the fraction of occupied volume to calculate the value of \(a\) , the length of the edge of the cube, in terms of \(r .\)

Short answer

\(a = \frac{4r}{\sqrt{3}}\)

Step-by-step solution

Understand the Geometry of Body-Centered Cubic Structure

A body-centered cubic (bcc) unit cell has spheres at each corner of the cube and one sphere in the center. The spheres in the corners each contribute an eighth of their volume to the unit cell, and the central sphere contributes its entire volume. The diagonal of the cube passes through the radius of three spheres.

Express Diagonal in Terms of Sphere Radius

The body diagonal can be expressed in terms of the radius by adding the diameter of one sphere and twice the radius (the diameter of the sphere at the center), which results in four times the sphere's radius. So the body diagonal is given by: body diagonal = 4r.

Relate Diagonal to the Edge of the Cube

Using Pythagoras' theorem in three dimensions, the body diagonal in terms of the edge length of the cube (a) is given by the formula: body diagonal = \(\sqrt{3}a\).

Combine Equations to Solve for the Edge Length

Set the two expressions for the body diagonal equal to each other to solve for a, and get: \(4r = \sqrt{3}a\).

Calculate the Length of the Cube Edge in Terms of Sphere Radius

Isolate a and solve for it in terms of r: \(a = \frac{4r}{\sqrt{3}}\). This expresses the edge length of the cube in terms of the sphere's radius.

Key concepts

Understanding Geometric Crystal Lattices

Geometric crystal lattices represent the ordering of particles in a crystalline solid, revealing a periodic arrangement that extends in all three spatial dimensions. In the context of body-centered cubic (bcc) structures, this lattice pattern features spheres (or atoms) situated at each corner of a cube with an additional sphere occupying the very center, akin to a nucleus of a cell. The meticulous placement of these spheres within the lattice is not arbitrary; it ensures maximum space utilization while maintaining the stability of the crystal structure.

It is pivotal to envision how each sphere at the corners of the cube only contributes a fraction of its volume (an eighth) to the unit cell, since these corners are shared by eight adjacent unit cells. In contrast, the centrally-located sphere is entirely enclosed within the unit cell and accordingly donates its full volume. This specific arrangement leads to a distinct packing efficiency—a measure of how well the spheres fill the space—which is 68.0% for bcc crystals. In essence, understanding the geometric crystal lattice is instrumental in comprehending the subsequent calculations involving unit cell dimensions and their relationships to the constituent particles.

Unit Cell Volume Calculation Explained

To fathom the nature of unit cell volume calculations, we begin by recognizing that the unit cell is the fundamental repeating unit of the crystal lattice, epitomizing the smallest portion of the lattice that still retains the overall symmetry and properties of the entire crystal.

The volume of the unit cell in a body-centered cubic structure can be calculated by cubing the edge length, denoted as 'a'. Because the bcc lattice is constituted of spheres that snugly fill up 68.0% of the volume, translating this percentage into a workable volume requires a relation between the sphere radius, 'r', and the cube's edge length. By leveraging this relationship and the known packing efficiency, we can extract formulaic expressions to calculate the unit cell's volume, which is crucial in the study of material properties such as density and is instrumental for applications in fields like metallurgy and pharmacology.

The Pythagorean Theorem in Three Dimensions

Known to many from geometry class, the Pythagorean theorem is a bedrock principle that relates the lengths of sides in a right-angled triangle. When extrapolated into three dimensions, as in the case of a body-centered cubic crystal, this theorem becomes an indispensable tool for relating the body diagonal to the edge of the cube.

In a three-dimensional context, the theorem posits that the square of the body diagonal (\body diagonal^2) is equivalent to the sum of the squares of the cube edges (\body diagonal^2 = a^2 + a^2 + a^2). This leads to \body diagonal = \(\sqrt{3}a\), establishing a direct link between the sphere radius and the cube edge length, ultimately guiding us to a mathematical expression for 'a' in terms of 'r', specifically \a = \(\frac{4r}{\sqrt{3}}\). This geometric relationship is a cornerstone in crystallography, enabling scientists and engineers to explore and manipulate the underlying framework of materials at the atomic level.