Problem 62 - Chemistry

Question

The radius of tungsten is 137 \(\mathrm{pm}\) and the density is 19.3 \(\mathrm{g} / \mathrm{cm}^{3}\) . Does elemental tungsten have a face-centered cubic structure or a body- centered cubic structure?

Step-by-Step Solution

Answer

Elemental tungsten has a face-centered cubic (FCC) structure because the calculated density for FCC structure (19.31 g/cm³) is very close to the given density (19.3 g/cm³), while the calculated density for BCC structure (12.69 g/cm³) is far from the given density.

Step 1: Convert the radius of tungsten to cm

First, we must convert the given radius of tungsten from picometers (pm) to centimeters (cm) using the conversion factor: 1 pm = 1 × 10^-12 cm The radius of tungsten in cm is: r = 137 pm × (1 × 10^-12 cm / 1 pm) = 1.37 × 10^-8 cm

Step 2: Calculate the unit cell edge length (a) for both FCC and BCC structures

- For a face-centered cubic (FCC) structure: The diagonal of the half cube should be equal to 4 times the radius of the atom: \(\sqrt{2}a = 4r\) Hence, we can find the edge length (a) for FCC as follows: a_FCC = (4r) / √2 = (4 × 1.37 × 10^-8 cm) / √2 = 3.88 × 10^-8 cm - For a body-centered cubic (BCC) structure: The diagonal of the entire cube should be equal to 4 times the radius of the atom: \(\sqrt{3}a = 4r\) Hence, we can find the edge length (a) for BCC as follows: a_BCC = (4r) / √3 = (4 × 1.37 × 10^-8 cm) / √3 = 3.17 × 10^-8 cm

Step 3: Determine the number of atoms per unit cell for both FCC and BCC structures

- For a face-centered cubic (FCC) structure: There are 8 corner atoms (1/8 contribution each) and 6 face atoms (1/2 contribution each), resulting in: n_FCC = 8 × (1/8) + 6 × (1/2) = 4 atoms/unit cell - For a body-centered cubic (BCC) structure: There are 8 corner atoms (1/8 contribution each) and 1 body atom (full contribution), resulting in: n_BCC = 8 × (1/8) + 1 = 2 atoms/unit cell

Step 4: Calculate the density for both FCC and BCC structures

Given: Mass of one tungsten atom (m) = atomic mass of tungsten / Avogadro's number = 183.84 g/mol / 6.022 × 10^23 atoms/mol = 3.05 × 10^-22 g/atom - For a face-centered cubic (FCC) structure: Density_FCC = (n_FCC × m) / (a_FCC^3) Density_FCC = (4 × 3.05 × 10^-22 g) / ((3.88 × 10^-8 cm)^3) = 19.31 g/cm³ - For a body-centered cubic (BCC) structure: Density_BCC = (n_BCC × m) / (a_BCC^3) Density_BCC = (2 × 3.05 × 10^-22 g) / ((3.17 × 10^-8 cm)^3) = 12.69 g/cm³

Step 5: Compare the calculated densities with the given density to determine the crystal structure

The given density of tungsten is 19.3 g/cm³. - Calculated density for FCC structure: 19.31 g/cm³ (very close to the given density) - Calculated density for BCC structure: 12.69 g/cm³ (far from the given density) Therefore, the elemental tungsten has a face-centered cubic (FCC) structure.

Key Concepts

Face-Centered Cubic

The face-centered cubic (FCC) structure is a type of crystal arrangement where each cube face has an atom at its center. Additionally, there is one atom at each corner of the cube. This structure is characterized by having a high packing density, meaning atoms are densely packed together.

• In an FCC unit cell, each of the 8 corner atoms contributes 1/8th of its volume to the cell, while each of the 6 face-centered atoms contributes half of its volume.
• Adding these up gives a total of 4 atoms within the FCC unit cell.
• The packing efficiency, or the ratio of the occupied space to the available space in the FCC structure, is about 74%, which is quite high among crystal structures.

The geometric relationship in an FCC structure is that the diagonal across the face of the cube equals 4 times the atomic radius \( (\sqrt{2}a = 4r) \). This relationship helps to calculate the edge length \( a \) when the atomic radius \( r \) is known. The high packing efficiency makes FCC structures common in metals like aluminum, copper, and in our problem's case, tungsten.

Body-Centered Cubic

The body-centered cubic (BCC) structure also features atoms at the eight corners of a cube, with a distinct difference. It places one additional atom at the very center of the cube, giving the BCC its name.

• In this setup, each corner atom contributes 1/8th of its volume to the unit cell, adding up across the 8 corners.
• The central atom contributes its whole volume, resulting in a total of 2 atoms per BCC unit cell.
• The BCC structure is less densely packed compared to FCC, with a packing efficiency of about 68%.

The geometrical relationship is defined by the cube's body diagonal, which equals 4 times the atomic radius \( (\sqrt{3}a = 4r) \). This allows us to calculate the cube's edge length \( a \) from the atomic radius \( r \). This structure is prevalent in metals such as iron and tungsten, although for tungsten in its stable form, FCC is the correct configuration as revealed through density comparison with BCC.

Density Calculation

Determining a crystal's structure involves calculating its density, a measurement of how much mass exists per volume unit. Density calculations help to ascertain the likely form that an element takes in its solid phase.

• First, we compute the mass of one atom by dividing its atomic mass (g/mol) by Avogadro's number (atoms/mol). This conversion gives us the mass per atom in grams.
• Using this mass, multiply it by the number of atoms per unit cell for FCC or BCC to find the total mass of the unit cell.
• Next, calculate the volume of the unit cell from the cube's edge length \( a \), which has been derived from the atomic structure equations. The volume is \( a^3 \).
• Lastly, divide the total mass of the unit cell by its volume to find the density.

By comparing these calculated densities to the known density, one can infer the crystal structure. For tungsten, the calculated density for FCC matches the given density closely, indicating it adopts this structure rather than BCC.

Unit Cell

A unit cell refers to the smallest, repeating structure in a crystal that shows the crystal's whole symmetry. Understanding unit cells is fundamental to grasping how atoms pack together to form crystals. This small building block determines the arrangement of atoms within the larger crystalline material.

• By repeating unit cells in three dimensions, we can create the overall crystal lattice.
• In both FCC and BCC structures, the unit cell is cubic, but the positioning and number of atoms within differ between the two.
• The unit cell's geometry, specifically dimensions like edge length \( a \), directly relates to the atomic radius and the crystal's packing efficiency.

These parameters help identify which type of unit cell a material presents, making unit cells instrumental in studying metallic and other crystalline structures, such as in determining the structure of tungsten.

Atomic Radius

The atomic radius is the distance from the center of an atom to the outer boundary of the electron cloud. This measurement is crucial when considering crystal structures as it helps define the parameters of unit cells. In essence, the atomic radius influences the size of the cube in a crystal lattice and is pivotal in crystal density calculations.

• In crystal structures, the atomic radius affects how closely a unit cell's atoms can pack together.
• For FCC crystals, knowing the radius lets us determine the relationship for edge length \( a \): \( \sqrt{2}a = 4r \).
• For BCC, this becomes \( \sqrt{3}a = 4r \), linking the radius with the internal diagonal of the cube.

The atomic radius is generally measured in picometers (pm) and is converted when necessary for calculations. For tungsten, using an atomic radius of 137 pm was key to the exercise in identifying its face-centered cubic structure based on its density.