Aluminum has a density of 2.699 g>cm3 and crystallizes with a face-centered cubic unit cell. What is the edge length of a unit cell in picometers?

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Identify the type of crystal structure: Aluminum crystallizes in a face-centered cubic (FCC) unit cell.

Recall that in an FCC unit cell, there are 4 atoms per unit cell.

Use the formula for density: \( \text{Density} = \frac{\text{Mass of unit cell}}{\text{Volume of unit cell}} \).

Calculate the mass of the unit cell: Multiply the number of atoms per unit cell by the atomic mass of aluminum (26.98 g/mol) and divide by Avogadro's number (6.022 \(\times\) 10^{23} \(\text{ atoms/mol}\)).

Calculate the volume of the unit cell using the density and mass, then find the edge length by taking the cube root of the volume. Convert the edge length from cm to pm (1 cm = 10^{10} pm).

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주요 개념

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Density

Density is defined as mass per unit volume, typically expressed in grams per cubic centimeter (g/cm³) for solids. In this context, the density of aluminum (2.699 g/cm³) is crucial for determining the mass of the aluminum atoms in the unit cell, which will help in calculating the edge length of the cubic structure.

Face-Centered Cubic (FCC) Structure

A face-centered cubic (FCC) structure is a type of crystal lattice where atoms are located at each of the corners and the centers of all the faces of the cube. This arrangement is known for its high packing efficiency, with each unit cell containing four atoms. Understanding this structure is essential for calculating the edge length based on the number of atoms and their arrangement.

Unit Cell Volume

The unit cell volume is the volume occupied by one repeating unit of a crystal lattice. For a cubic unit cell, the volume can be calculated using the formula V = a³, where 'a' is the edge length. By relating the unit cell volume to the density and the molar mass of aluminum, one can derive the edge length in picometers, which is necessary for solving the given problem.