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Chapter 10: Problem 60

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

Expert verified
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

01

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 \(V = a^3\).
02

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.
03

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.
04

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.

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