Problem 21 - Chemistry: An Atoms-Focused Approach

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 (BCC) unit cell has a greater packing efficiency than the simple cubic (SC) unit cell.

Step 1: Determine the number of spheres in the unit cells

For a simple cubic unit cell, there is only one sphere located at each corner of the cube. Since there are 8 corners, and each corner is shared by 8 adjacent cells, there is a total of (8 corners × 1/8) = 1 sphere in the unit cell. For a body-centered cubic unit cell, there is one sphere at each corner of the cube (8 corners × 1/8) and one sphere at the center of the cube, giving a total of (8 corners × 1/8) + 1 = 2 spheres in the unit cell.

Step 2: Calculate the volume of a sphere

The volume of a sphere can be calculated using the formula: Volume = (4/3) × π × r³, where r is the radius of a sphere (or atom). Since both the SC and BCC unit cells have spheres (atoms) of equal size, we can simplify our calculations by comparing their packing efficiency with respect to their respective volume ratios.

Step 3: Determine the edge length and total volume of the unit cells

For a simple cubic unit cell, the edge length (a) is equal to 2r, where r is the radius of the sphere (or atom). The total volume of the unit cell can be calculated as: Total volume (SC) = a³ = (2r)³ = 8r³ For a body-centered cubic unit cell, the edge length (a) is related to the radius by the equation: a = 4r/√3 The total volume of the unit cell can be calculated as: Total volume (BCC) = a³ = (4r/√3)³ = 64r³/3√3

Step 4: Calculate packing efficiency

Now we can calculate the packing efficiency for both unit cells: Packing Efficiency (SC) = (Volume of spheres in SC / Total volume of SC) × 100 = ((4/3)πr³ / 8r³) × 100 ≈ 52.4% Packing Efficiency (BCC) = (Volume of spheres in BCC / Total volume of BCC) × 100 = (2 × (4/3)πr³ / (64r³/3√3)) × 100 ≈ 68.0%

Step 5: Compare packing efficiencies

Comparing the packing efficiencies of the simple cubic (52.4%) and body-centered cubic (68.0%) unit cells, we can conclude that the body-centered cubic unit cell has a greater packing efficiency than the simple cubic unit cell.