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.
Ch.12 - Solids and Modern Materials
Chapter 12, Problem 39
Aluminum metal crystallizes in a face-centered cubic unit cell. (a) How many aluminum atoms are in a unit cell? (b) Estimate the length of the unit cell edge, a, from the atomic radius of aluminum (1.43 Å). (c) Calculate the density of aluminum metal.
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 of the corners and the centers of all the faces of the cube. Calculate the total number of atoms per unit cell by considering the contribution of each atom at the corners and faces.
Step 2: Calculate the number of atoms in the FCC unit cell. Each corner atom is shared by eight adjacent unit cells, and each face-centered atom is shared by two unit cells. Therefore, the total number of atoms in one FCC unit cell is calculated as follows: 8 corner atoms * (1/8) + 6 face atoms * (1/2).
Step 3: Use the atomic radius to estimate the length of the unit cell edge, a. In an FCC structure, the face diagonal is equal to four times the atomic radius. Use the relationship between the face diagonal and the edge length: face diagonal = a√2 = 4 * atomic radius.
Step 4: Rearrange the equation from Step 3 to solve for the edge length, a. Substitute the given atomic radius of aluminum (1.43 Å) into the equation to find the edge length.
Step 5: Calculate the density of aluminum metal. Use the formula for density: density = mass/volume. First, calculate the mass of the aluminum atoms in the unit cell using the molar mass of aluminum and Avogadro's number. Then, calculate the volume of the unit cell using the edge length from Step 4. Finally, divide the mass by the volume to find the density.
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 results in a higher packing efficiency, with each unit cell containing four atoms. Understanding this structure is crucial for determining the number of atoms in a unit cell and for calculating properties like density.
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00:51
Face Centered Cubic Example
Atomic Radius and Unit Cell Edge Length
The atomic radius is the distance from the nucleus of an atom to the outermost shell of electrons. In an FCC structure, the relationship between the atomic radius and the unit cell edge length (a) can be expressed as a = 2√2r, where r is the atomic radius. This relationship allows us to estimate the edge length of the unit cell based on the known atomic radius of aluminum.
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01:27Simple Cubic Unit Cell
Density Calculation
Density is defined as mass per unit volume. For a crystalline solid like aluminum, density can be calculated using the formula: density = (mass of atoms in unit cell) / (volume of unit cell). The mass can be determined from the number of atoms in the unit cell and the molar mass of aluminum, while the volume is derived from the cube of the edge length (a^3).
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Related Practice
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.
Textbook Question
Determine if each statement is true or false: (a) Substitutional alloys are solid solutions, but interstitial alloys are heterogenous alloys. (c) The atomic radii of the atoms in a substitutional alloy are similar to each other, but in an interstitial alloy, the interstitial atoms are a lot smaller than the host lattice atoms.