Textbook Question
Consider the unit cells shown here for three different structures that are commonly observed for metallic elements. (b) Which structure(s) corresponds to the least dense packing of atoms?
Ch.12 - Solids and Modern Materials
Chapter 12, Problem 36
Calcium crystallizes in a body-centered cubic structure at 467°C. (b) How many nearest neighbors does each Ca atom possess? (c) Estimate the length of the unit cell edge, a, from the atomic radius of calcium (1.97 Å). (d) Estimate the density of Ca metal at this temperature.
Verified step by step guidance1
To determine the number of nearest neighbors in a body-centered cubic (BCC) structure, recall that each atom in a BCC lattice has 8 nearest neighbors. This is because the atom at the center of the cube is surrounded by the 8 corner atoms.
To estimate the length of the unit cell edge, a, in a BCC structure, use the relationship between the atomic radius (r) and the edge length (a). In a BCC lattice, the diagonal of the cube is equal to 4 times the atomic radius. The diagonal can also be expressed in terms of the edge length as \( \sqrt{3}a \). Set these equal: \( \sqrt{3}a = 4r \). Solve for \( a \) using the atomic radius of calcium, \( r = 1.97 \text{ Å} \).
To estimate the density of calcium metal, use the formula for density: \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \). First, calculate the mass of one unit cell. Since the BCC structure has 2 atoms per unit cell, find the mass of 2 calcium atoms using the atomic mass of calcium (40.08 g/mol) and Avogadro's number (\( 6.022 \times 10^{23} \text{ atoms/mol} \)).
Next, calculate the volume of the unit cell using the edge length \( a \) obtained in step 2. The volume of the cubic unit cell is \( a^3 \).
Finally, substitute the mass and volume into the density formula to estimate the density of calcium metal at 467°C.
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Body-Centered Cubic Structure
A body-centered cubic (BCC) structure is a type of crystal lattice where atoms are located at each corner of a cube and a single atom is positioned at the center of the cube. In a BCC lattice, each corner atom is shared among eight adjacent cubes, leading to a total of two atoms per unit cell. This arrangement influences the properties of the material, including its density and atomic packing efficiency.
Recommended video:
Guided course
00:40
Body Centered Cubic Example
Unit Cell and Atomic Radius
The unit cell is the smallest repeating unit in a crystal lattice that reflects the symmetry and structure of the entire crystal. For BCC structures, the relationship between the atomic radius and the unit cell edge length, 'a', can be derived from geometric considerations. Specifically, in a BCC lattice, the diagonal of the cube can be expressed in terms of the atomic radius, allowing for the calculation of 'a' using the formula: a = 4r/√3, where 'r' is the atomic radius.
Recommended video:
Guided course
02:02Atomic Radius
Density Calculation
Density is defined as mass per unit volume and is a critical property of materials. To estimate the density of calcium metal, one must first determine the mass of the atoms in the unit cell and the volume of the unit cell. The density can be calculated using the formula: density = (mass of atoms in unit cell) / (volume of unit cell). For BCC, the mass can be calculated by multiplying the number of atoms per unit cell by the molar mass of calcium and converting it to grams.
Recommended video:
Guided course
01:56Density Concepts
Related Practice
Textbook Question
Sodium metal (atomic weight 22.99 g/mol) adopts a body-centered cubic structure with a density of 0.97 g/cm3. (a) Use this information and Avogadro’s number (NA = 6.022 × 1023/mol) to estimate the atomic radius of sodium. (b) If sodium didn't react so vigorously, it could float on water. Use the answer from part (a) to estimate the density of Na if its structure were that of a cubic close-packed metal. Would it still float on water?
Textbook Question
Calcium crystallizes in a body-centered cubic structure at 467°C. (a) How many Ca atoms are contained in each unit cell?
Textbook Question
Calcium crystallizes in a face-centered cubic unit cell at room temperature that has an edge length of 5.588 Å. (b) Calculate the density of Ca metal at this temperature.