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.

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Identify the type of lattice: The problem states that the element crystallizes in a face-centered cubic (FCC) lattice.

Determine the number of atoms per unit cell: In a face-centered cubic lattice, there are 4 atoms per unit cell.

Calculate the volume of the unit cell: Convert the edge length from Ångströms to centimeters (1 Å = 1 x 10^-8 cm) and use the formula for the volume of a cube, V = a^3, where a is the edge length.

Use the density formula to find the mass of the unit cell: Density (d) = mass/volume, rearrange to find mass = density x volume.

Calculate the atomic weight: Use the mass of the unit cell and the number of atoms per unit cell to find the atomic weight. Atomic weight = (mass of unit cell / number of atoms per unit cell) x Avogadro's number.

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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) Lattice

A face-centered cubic lattice is a type of crystal structure where atoms are located at each corner and the centers of all the cube faces. This arrangement results in a high packing efficiency of about 74%, meaning that a significant volume of the crystal is occupied by atoms. Understanding this structure is crucial for calculating the number of atoms per unit cell, which is essential for determining the atomic weight.

Density and Its Relation to Atomic Weight

Density is defined as mass per unit volume and is a critical property in material science. In the context of crystallography, the density of a crystal can be used to derive the atomic weight of the element by using the formula: density = (n * atomic weight) / (V * N_A), where n is the number of atoms per unit cell, V is the volume of the unit cell, and N_A is Avogadro's number. This relationship allows for the calculation of atomic weight from the known density and unit cell dimensions.

Unit Cell Volume Calculation

The volume of a unit cell is calculated using the formula V = a^3, where 'a' is the edge length of the cubic cell. For a face-centered cubic lattice, the volume is essential for determining how many atoms are present in the unit cell and subsequently calculating the atomic weight. In this case, with an edge length of 4.078 Å, the volume can be computed to facilitate the density and atomic weight calculations.

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Simple Cubic Unit Cell

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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

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: (b) Substitutional alloys have 'solute' atoms that replace 'solvent' atoms in a lattice, but interstitial alloys have 'solute' atoms that are in between the 'solvent' atoms in a lattice.

Textbook Question

For each of the following alloy compositions, indicate whether you would expect it to be a substitutional alloy, an interstitial alloy, or an intermetallic compound: (a) Fe_0.97Si_0.03 (b) Fe_0.60Ni_0.40 (c) SmCo₅.