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?

Short answer

By calculating the theoretical densities for both face-centered cubic (FCC) and body-centered cubic (BCC) structures and comparing them with the given density of tungsten (\(19.3\,\text{g}/\text{cm}^3\)), we can determine the most likely crystal structure for elemental tungsten. The structure with a theoretical density closer to the given density value is the most likely structure for elemental tungsten.

Step-by-step solution

Atomic Radius and Volume Calculations

Given the atomic radius of tungsten is \(137\,\text{pm}\). Convert it to centimeters: $$ r = 137\,\text{pm} \times \frac{1\, \text{cm}}{10^8\,\text{pm}} = 1.37 \times 10^{-8}\,\text{cm} $$ Since we know one unit cell of the crystal structure contains one or more tungsten atoms, we can calculate the volume of the unit cell in both FCC and BCC structures. The formula for calculating the unit cell edge length from the atomic radius in FCC and BCC are as follows: For FCC, \(a = 2\sqrt{2}r\). For BCC, \(a = 4r/\sqrt{3}\). Now we calculate the volume of unit cells in both FCC and BCC using the edge length \(a\) and the formula \(V = a^3\).

Calculate Theoretical Density

Now that we have the volume of unit cells, we can calculate the theoretical density for both crystal structures using the formula: $$ \rho_{theoretical} = \frac{n \times m_{atom}}{V_{unit \, cell} \times N_A} $$ Where \(\rho_{theoretical}\) is the theoretical density, \(n\) is the number of atoms per unit cell (4 for FCC and 2 for BCC), \(m_{atom}\) is the atomic mass and \(V_{unit \, cell}\) is the volume of the unit cell, and \(N_A\) is Avogadro's number (\(6.022 \times 10^{23}\,\text{atoms/mol}\)). We know the atomic mass of tungsten is around 183.84 g/mol. Now we calculate the theoretical densities for both FCC and BCC structures.

Comparing Theoretical Density with Actual Density

The given density of tungsten is 19.3 g/cm³. By comparing the calculated theoretical densities for both FCC and BCC structures to the actual given density value, we can determine which crystal structure is more likely for elemental tungsten. If one of the theoretical densities is substantially closer to the given density, we can conclude that the particular crystal structure is more likely to be the correct one.

Conclusion

Based on the comparison of the calculated theoretical densities with the given density, we can determine whether tungsten has a face-centered cubic (FCC) or body-centered cubic (BCC) structure. The structure that has a theoretical density closer to the given density value is the most likely structure for elemental tungsten.

Key concepts

Atomic Radius

The atomic radius is a critical value when trying to determine the structure of a crystal. It is the distance from the center of an atom to the outer boundary of its electron cloud. In the case of tungsten, the atomic radius is given as 137 picometers (pm). For practical calculations relating to crystal structures, it's often necessary to convert this into centimeters:

• Use the conversion factor: 1 cm = 10⁸ pm.
• Therefore, the atomic radius in centimeters is: \( r = 137 \,\text{pm} \times \frac{1 \, \text{cm}}{10^8 \, \text{pm}} = 1.37 \times 10^{-8} \, \text{cm}. \)

Knowing the atomic radius allows for the calculation of the edge length of the unit cell in various crystal structures, such as face-centered cubic (FCC) and body-centered cubic (BCC). This is foundational to performing calculations that establish the physical properties of the material.

Unit Cell Volume Calculation

The unit cell is the smallest repetitive unit that reflects the entire crystal structure, and understanding its volume is crucial to determining the crystal's theoretical density. For both FCC and BCC structures, the edge length of the unit cell can be calculated using formulas that relate to the atomic radius:

• For a Face-Centered Cubic (FCC) structure: \(a = 2\sqrt{2}r\).
• For a Body-Centered Cubic (BCC) structure: \(a = \frac{4r}{\sqrt{3}}\).

After calculating the edge length \(a\), the volume \(V\) of the cubic unit cell is determined by cubing the edge length:

• \( V = a^3. \)

This value is vital as it is directly used in the equation to calculate the theoretical density of the material, providing insight into the crystal's structure.

Theoretical Density

Theoretical density gives us an estimate of how mass is distributed within a crystal lattice, which is influenced by the number of atoms per unit cell and the material's atomic mass. To find the theoretical density of a crystal structure, the following formula can be used:

• \( \rho_{theoretical} = \frac{n \times m_{atom}}{V_{unit \ cell} \times N_A}, \)

where:

• \(n\) is the number of atoms per unit cell (4 for FCC, 2 for BCC).
• \(m_{atom}\) is the atomic mass (183.84 g/mol for tungsten).
• \(V_{unit \ cell}\) is the volume of the unit cell.
• \(N_A\) is Avogadro's number, approximately \(6.022 \times 10^{23}\) atoms/mol.

By calculating the theoretical density for both structures and comparing them to the known density of 19.3 g/cm³, one can determine which structure more accurately represents the tungsten crystal lattice. The structure whose theoretical density aligns closely with the actual density is most likely the real structure.