Calculate the volume in Å³ of a face-centered cubic unit cell if it is composed of atoms with an atomic radius of 1.82 Å.

Name: Chemistry: The Central Science
Author: Brown
ISBN: 9780134414232

검증된 단계별 안내

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.

Step 2: Recognize that in an FCC unit cell, the face diagonal is equal to four times the atomic radius (4r), because the face diagonal passes through the centers of two corner atoms and one face-centered atom.

Step 3: Use the relationship between the face diagonal and the edge length (a) of the cube. The face diagonal can be expressed as \( \sqrt{2}a \). Therefore, set \( \sqrt{2}a = 4r \) and solve for the edge length \( a \).

Step 4: Substitute the given atomic radius (1.82 Å) into the equation from Step 3 to find the edge length \( a \) of the unit cell.

Step 5: Calculate the volume of the cubic unit cell using the formula \( V = a^3 \), where \( a \) is the edge length obtained in Step 4.

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Face-Centered Cubic (FCC) Structure

The face-centered cubic (FCC) structure is a type of crystal lattice where atoms are located at each corner and the centers of all the cube faces. This arrangement results in a high packing efficiency, with each unit cell containing four atoms. Understanding the geometry of the FCC unit cell is essential for calculating its volume.

Atomic Radius

The atomic radius is a measure of the size of an atom, typically defined as the distance from the nucleus to the outermost electron shell. In the context of FCC structures, the atomic radius is crucial for determining the dimensions of the unit cell, as it influences the length of the edges of the cube and the overall volume.

Volume of a Unit Cell

The volume of a unit cell is calculated using the formula V = a³, where 'a' is the length of one edge of the cubic cell. For FCC structures, the edge length can be derived from the atomic radius, specifically using the relationship a = 2√2r, where 'r' is the atomic radius. This relationship is key to finding the volume when given the atomic radius.

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