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.

Verified step by step guidance

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.

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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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Simple Cubic Unit Cell

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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.

a. Calculate the atomic radius of a calcium atom.

b. Calculate the density of Ca metal at this temperature.

Textbook Question

An element crystallizes in a face-centered cubic lattice. The edge of the unit cell is 4.078 Å, and the density of the crystal is 19.30 g>cm3. Calculate the atomic weight of the element and identify the element.

Textbook Question

Which of these statements about alloys and intermetallic compounds is false? (a) Bronze is an example of an alloy. (b) 'Alloy' is just another word for 'a chemical compound of fixed composition that is made of two or more metals.' (c) Intermetallics are compounds of two or more metals that have a definite composition and are not considered alloys. (d) If you mix two metals together and, at the atomic level, they separate into two or more different compositional phases, you have created a heterogeneous alloy. (e) Alloys can be formed even if the atoms that comprise them are rather different in size.