Question

The radius of tungsten is 137 pm and the density is 19.3 g/cm³. Does elemental tungsten have a face-centered cubic structure or a body-centered cubic structure?

Short answer

Elemental tungsten has a body-centered cubic structure.

Step-by-step solution

Given Information

The radius of tungsten is r = 137 pm and the density is d = 19.3 g/cm³.

If it is a face-centered cubic

$$\begin{gathered} \mathrm{L}=\mathrm{r}\sqrt{8} \\ \mathrm{L}=\left(137\mathrm{pm}\right)\sqrt{8} \\ \mathrm{L}=387\mathrm{pm}\left(\frac{{10}^{-10}\mathrm{cm}}{1\mathrm{pm}}\right) \\ \mathrm{L}=3.87\times {10}^{-10}\mathrm{cm} \end{gathered}$$

$$\begin{gathered} \mathrm{Density}=\frac{4\times 183.9\mathrm{gm}/\mathrm{mole}}{6.023\times {10}^{23}\mathrm{atoms}/\mathrm{mole}\times {\left(3.87\times {10}^{-8}\mathrm{cm}\right)}^3} \\ \mathrm{Density}=21.1\mathrm{gm}/{\mathrm{cm}}^3 \end{gathered}$$

If it is a body-centered cubic

$$\begin{gathered} \mathrm{L}=\frac{4\mathrm{r}}{\sqrt{3}} \\ \mathrm{L}=\frac{137\mathrm{pm}\times 4}{\sqrt{3}} \\ \mathrm{L}=316\mathrm{pm}\left(\frac{{10}^{-10}\mathrm{cm}}{1\mathrm{pm}}\right) \\ \mathrm{L}=3.16\times {10}^{-8}\mathrm{cm} \end{gathered}$$

$$\begin{gathered} \mathrm{Density}=\frac{4\times 183.9\mathrm{gm}/\mathrm{mole}}{6.023\times {10}^{23}\mathrm{atoms}/\mathrm{mole}\times {\left(3.16\times {10}^{-8}\mathrm{cm}\right)}^3} \\ \mathrm{Density}=19.4\mathrm{gm}/{\mathrm{cm}}^3 \end{gathered}$$

The density of tungsten is consistent with a body-centered unit cell.