Question

Titanium metal has a density of \(4.506 \mathrm{~g} / \mathrm{cm}^{3}\) and an atomic radius of \(144.8 \mathrm{pm}\). In what cubic unit cell does titanium crystallize?

Short answer

Titanium crystallizes in a hexagonal close-packed (HCP) structure.

Step-by-step solution

Understand the Problem

We need to determine in which cubic unit cell titanium crystallizes. We have the density of titanium and the atomic radius. Common types of cubic unit cells include simple cubic, body-centered cubic (BCC), and face-centered cubic (FCC).

Convert Units

Convert the atomic radius from picometers to centimeters, the standard unit in crystallography for comparing with density:\[144.8 \, \text{pm} = 144.8 \, \times 10^{-10} \, \text{cm}\]

Calculate the Atomic Volume

Calculate the volume of a single titanium atom assuming it as a sphere with radius \(144.8 \, \text{pm}\):\[V = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi (144.8 \times 10^{-10} \, \text{cm})^3\]

Identify the Number of Atoms per Unit Cell

Recognize that for a simple cubic cell, there is 1 atom per unit cell, for BCC there are 2 atoms per unit cell, and for FCC there are 4 atoms per unit cell. Compare the calculated volume to these structures.

Calculate the Mass of Titanium Atoms in Different Unit Cells

Find the mass of titanium atoms per unit cell:- For Simple cubic: \(1 \times 47.867 \, \text{g/mol}\)- For BCC: \(2 \times 47.867 \, \text{g/mol}\)- For FCC: \(4 \times 47.867 \, \text{g/mol}\)Convert to grams using Avogadro's number \(N_A = 6.022 \times 10^{23} \, \text{atoms/mol}\):\[\frac{47.867 \, \text{g/mol}}{N_A}\]

Calculate the Theoretical Density of Each Structure

Calculate the theoretical density for each cell type based on the masses and volumes:\[\text{Density} = \frac{\text{Total Mass (atoms per unit cell)}}{\text{Volume of Unit Cell}}\]

Match with Given Density

Compare the calculated densities with the given density of \(4.506 \, \text{g/cm}^3\). The unit cell that matches this density closely is the actual crystal structure of titanium.

Key concepts

Density

Density is a key property used to determine the type of cubic unit cell in which a metal crystallizes. It measures how much mass is contained in a given volume. For titanium, we know the density is given as \(4.506 \, \text{g/cm}^3\).
This value allows us to compare different potential structures and find which one matches closely.
- Density formula: \(\text{Density} = \frac{\text{Mass}}{\text{Volume}}\).- In crystallography, we calculate the density by considering how many atoms exist in one unit cell structure and their atomic mass.With the density of titanium and knowledge of different unit cell structures, you can determine the most likely structure that titanium forms.

Atomic Radius

The atomic radius is an important factor in crystallography as it helps determine what kind of unit cell a metal might form. The atomic radius of titanium is given as \(144.8 \, \text{pm}\).
Converting this to centimeters for crystallography use, we get \(144.8 \, \times 10^{-10} \, \text{cm}\).
Why is the atomic radius important?

• It helps to calculate the volume of an atom, assuming it is a perfect sphere.
• It's integral to calculating how tightly atoms pack together in a crystal structure, which consequently impacts density.

Taking the atomic radius through our calculations helps us see which structure fits with the known density. This provides insight into the physical building blocks of titanium metal.

Crystallography

Crystallography is the study of atomic and molecular structures and how they arrange themselves in space, especially in a crystal form.
It's an essential field for understanding materials on an atomic level.
- In the context of metals like titanium, it's about examining how atoms or ions are packed in a repeating pattern. - Each type of cubic unit cell, whether simple cubic, body-centered cubic, or face-centered cubic, represents different ways atoms can be stacked up to make a solid. In crystallography calculations, properties like density, atomic radius, and atom count in a unit cell are essential to predicting or confirming a material's crystal structure.

Body-Centered Cubic

The body-centered cubic (BCC) is one type of cubic unit cell found in metal crystallography. In a BCC structure, each unit cell contains two atoms.
One atom sits at the center of the cube, while eight atoms are located at the cube's corners, shared with adjoining cells.
BCC Characteristics:

• This layout creates a structure that balances effective packing and space available.
• Each atom in a BCC lattice coordinates with eight others, demonstrating a different packaging efficiency compared to FCC or simple cubic.

Calculating the density for a BCC structure involves multiplying the mass of two titanium atoms by the dimension of the unit cell. Comparing this density with the given value helps decide if the BCC structure matches the crystal structure of titanium.

Face-Centered Cubic

The face-centered cubic (FCC) is another common type of cubic unit cell. In this structure, each unit cell contains four atoms.
Atoms are located at each face center and at the eight corners of the cube.
FCC Characteristics:

• The atoms pack closely together, achieving efficient use of space.
• Each atom coordinates with 12 others, providing a strong, stable structure.

To check if titanium crystallizes in an FCC structure, we calculate the theoretical density by considering four titanium atoms per unit cell.
The close comparison of computed density with the known density of titanium assists in confirming if FCC is a feasible structure.