Textbook Question
Calcium crystallizes in a body-centered cubic structure at 467°C. (a) How many Ca atoms are contained in each unit cell?
Ch.12 - Solids and Modern Materials
Chapter 12, Problem 37
Calcium crystallizes in a face-centered cubic unit cell at room temperature that has an edge length of 5.588 Å.
a. Calculate the atomic radius of a calcium atom.
b. Calculate the density of Ca metal at this temperature.
Verified step by step guidance1
Step 1: Understand the structure of a face-centered cubic (FCC) unit cell. In an FCC unit cell, atoms are located at each corner and the centers of all the faces of the cube. Each face-centered atom is shared between two adjacent unit cells, and each corner atom is shared among eight unit cells.
Step 2: Calculate the atomic radius of a calcium atom. In an FCC unit cell, the face diagonal is equal to four times the atomic radius (4r). The face diagonal can also be expressed in terms of the edge length (a) as \( \sqrt{2}a \). Set these equal to each other: \( 4r = \sqrt{2}a \). Solve for the atomic radius \( r \).
Step 3: Calculate the volume of the unit cell. The volume \( V \) of a cube is given by \( a^3 \), where \( a \) is the edge length of the cube. Use the given edge length to find the volume.
Step 4: Determine the number of atoms per unit cell in an FCC structure. An FCC unit cell contains 4 atoms per unit cell (1/8 of each of the 8 corner atoms and 1/2 of each of the 6 face-centered atoms).
Step 5: Calculate the density of calcium metal. Use the formula for density \( \rho = \frac{\text{mass of atoms in unit cell}}{\text{volume of unit cell}} \). Find the mass of the atoms in the unit cell by multiplying the number of atoms per unit cell by the atomic mass of calcium (in grams per mole) and dividing by Avogadro's number to convert to grams. Then, divide by the volume of the unit cell (converted to cm³) to find the density.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Face-Centered Cubic (FCC) Structure
In a face-centered cubic (FCC) structure, atoms are located at each corner of the cube and at the center of each face. This arrangement allows for a high packing efficiency, with each unit cell containing four atoms. Understanding this structure is crucial for calculating properties like atomic radius and density, as it defines how atoms are arranged in the crystal lattice.
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Atomic Radius Calculation
The atomic radius in an FCC lattice can be determined using the relationship between the edge length of the unit cell and the atomic radius. Specifically, the atomic radius (r) is related to the edge length (a) by the formula: a = 2√2r. This relationship arises because the face diagonal of the cube contains four atomic radii, allowing for the calculation of the radius from the known edge length.
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Density Calculation
Density is defined as mass per unit volume and can be calculated for a crystalline solid using the formula: density = (mass of atoms in unit cell) / (volume of unit cell). For FCC structures, the mass can be determined by multiplying the number of atoms per unit cell by the molar mass and dividing by Avogadro's number. The volume is simply the cube of the edge length, making it essential to understand both mass and volume for density calculations.
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