Problem 19 - Chemistry The Science in Context

Question

Which unit cell has the greater packing efficiency, simple cubic or body- centered cubic?

Step-by-Step Solution

Answer

Answer: The body-centered cubic unit cell has a greater packing efficiency (68.0%) compared to the simple cubic unit cell (52.4%).

Step 1: Define the packing efficiency formula

The packing efficiency (PE) of a unit cell can be calculated using the following formula: PE = (Volume of spheres in unit cell) / (Total volume of unit cell) * 100

Step 2: Calculate the volume of spheres in a simple cubic unit cell

In a simple cubic unit cell, there is one sphere at each corner, and there are eight corners. However, each corner sphere belongs to eight adjacent unit cells, so only 1/8 of the sphere's volume is inside the unit cell. Thus, the total number of spheres in a simple cubic unit cell is: 1 sphere = 8 * (1/8) The volume of a sphere is given by the formula (4/3)πr^3, where r is the radius. Therefore, the volume of spheres in a simple cubic unit cell is: Volume of spheres (SC) = 1 * (4/3)πr^3

Step 3: Calculate the total volume of a simple cubic unit cell

A simple cubic unit cell has an edge length equal to twice the sphere's radius. Therefore, the total volume of a simple cubic unit cell is: Total Volume (SC) = (2r)^3 = 8r^3

Step 4: Calculate the packing efficiency of a simple cubic unit cell

Now, we plug the obtained values back into the packing efficiency formula for a simple cubic unit cell: PE(SC) = (Volume of spheres(SC))/ (Total volume(SC)) * 100 PE(SC) = ((4/3)πr^3) / (8r^3) * 100 = 52.4%

Step 5: Calculate the volume of spheres in a body-centered cubic unit cell

In a body-centered cubic unit cell, in addition to the corner spheres, there is also a sphere located at the center. So, the total number of spheres in a body-centered cubic unit cell is: 2 spheres = 1 (8 * (1/8)) + 1 The volume of spheres in a body-centered cubic unit cell is: Volume of spheres (BCC) = 2 * (4/3)πr^3

Step 6: Calculate the total volume of a body-centered cubic unit cell

The total volume of a body-centered cubic unit cell is calculated by obtaining the lattice edge length using the relationship between the diagonal (d) and the sphere's radius (r) in a BCC unit cell: d = 4r (the diagonal passes through the center sphere, so there are four radii in the diagonal) Using Pythagorean theorem with the edge length (a) for a BCC unit cell: a^2 + a^2 + a^2 = d^2, so: a = (4/√3)r The total volume of a body-centered cubic unit cell is: Total Volume (BCC) = a^3 = ((4/√3)r)^3 = (64/√3)r^3

Step 7: Calculate the packing efficiency of a body-centered cubic unit cell

Plug the obtained values back into the packing efficiency formula for a body-centered cubic unit cell: PE(BCC) = (Volume of spheres(BCC))/ (Total volume(BCC)) * 100 PE(BCC) = (8/3)πr^3 / (64/√3)r^3 * 100 = 68.0%

Step 8: Compare the packing efficiencies of simple cubic and body-centered cubic unit cells

Now that we have calculated the packing efficiencies of both unit cell types, we can compare them: PE(SC) = 52.4% PE(BCC) = 68.0% Since the packing efficiency of the body-centered cubic (BCC) unit cell (68.0%) is greater than that of the simple cubic (SC) unit cell (52.4%), we can conclude that the body-centered cubic unit cell has a greater packing efficiency than the simple cubic unit cell.

Key Concepts

Simple Cubic Unit Cell

The simple cubic unit cell is the most basic type of crystal lattice structure. In this arrangement, spheres (representing atoms or ions) are positioned at each corner of a cube. As such, each unit cell shares these corner spheres with seven other unit cells, allowing only 1/8 of each sphere's volume to occupy any single unit cell. As a result, a simple cubic unit cell technically contains just one sphere worth of material within its boundaries.

The geometry of the simple cubic cell is characterized by its edge length, which is precisely twice the radius of the spheres:

• Edge length = 2r
• Total volume of unit cell = \[ (2r)^3 = 8r^3 \]
• Volume of spheres in unit cell = \[ \frac{4}{3} \pi r^3 \]

Given this setup, the packing efficiency of a simple cubic unit cell is calculated by dividing the volume occupied by the spheres by the total volume of the unit cell and expressing it as a percentage. With a packing efficiency of approximately 52.4%, the simple cubic unit cell is not very efficient compared to other types of unit cells.

Body-Centered Cubic Unit Cell

The body-centered cubic (BCC) unit cell offers a more efficient packing of spheres compared to the simple cubic arrangement. In a BCC cell, along with spheres at each corner, there is an additional sphere located at the center of the cube. This arrangement leads to two complete spheres being considered within each unit cell.

The arithmetic for the BCC is slightly more intricate than for the simple cubic:

• Besides sharing corner spheres, the sphere at the center fully belongs to the unit cell.
• The body diagonal of the BCC unit cell is equal to four times the radius and can be calculated using the relation \[ d = 4r \].
• To determine the edge length, the equation \[ a = \frac{4}{\sqrt{3}}r \] is used.
• Total volume of unit cell = \[ \left( \frac{4}{\sqrt{3}}r \right)^3 = \frac{64}{3\sqrt{3}}r^3 \]
• Volume of spheres in unit cell = \[ \frac{8}{3} \pi r^3 \]

The BCC's packing efficiency, calculated by comparing the volume of filled space with total unit cell volume, stands at about 68.0%. This higher percentage means that BCC structures are more tightly packed than simple cubic ones, allowing for better occupancy of space.

Crystal Lattice Structure

Crystal lattice structures describe the orderly and repeated arrangement of atoms in a crystalline solid. A crystal lattice is essentially a framework or scaffold upon which these atoms sit. The unit cells, such as simple cubic and body-centered cubic, act as the basic building blocks for these lattices.

Key characteristics of crystal lattice structures include:

• Repetition: Each pattern or arrangement repeats itself in the three-dimensional structure.
• Symmetry: Lattices strive to be as symmetrical as possible, often leading to distinct geometric patterns.
• Determinance of Properties: Structural properties, like melting point, density, and strength, are directly influenced by the lattice arrangement.

The significance of understanding crystal lattice structures lies in their application to material science and engineering. It helps predict how materials might behave, interact with other substances, and provide insights into developing new materials with desired traits. Thus, comprehending concepts like simple cubic and body-centered cubic structures is fundamental to employing crystals effectively in technology and research.