Question

The radius of gold is 144 pm and the density is $\mathbf{19}\mathbf{.}\mathbf{32}\mathbf{}\mathbf{gm}\mathbf{/}{\mathbf{cm}}^{\mathbf{3}}$. Does elemental gold have a face-centered cubic structure or a body-centered cubic structure?

Short answer

Elemental gold has a face-centered cubic structure.

Step-by-step solution

If it has a face-centered cubic structure

In a face-centered cubic unit cell, atoms touch along the face diagonal

4r is diagonal and L is length.

$$\begin{gathered} 2{\mathrm{L}}^2=16{\mathrm{r}}^2 \\ \mathrm{L}=\mathrm{r}\sqrt{8} \\ \mathrm{L}=\left(144\mathrm{pm}\right)\sqrt{8} \\ \mathrm{L}=407\mathrm{pm}\left(\frac{{10}^{-10}\mathrm{cm}}{1\mathrm{pm}}\right) \\ \mathrm{L}=4.07\times {10}^{-8}\mathrm{cm} \end{gathered}$$

role="math" localid="1664128846164" $$\begin{gathered} \mathrm{Density}=\frac{4\times 197.0\mathrm{gm}/\mathrm{mole}}{6.023\times {10}^{23}\mathrm{atoms}/\mathrm{mole}\times {\left(4.07\times {10}^{-8}\mathrm{cm}\right)}^3} \\ \mathrm{Density}=19.4\mathrm{gm}/{\mathrm{cm}}^3 \end{gathered}$$

If it has a body-centered cubic structure

If gold has a body-centered cubic structure then there are two atoms per unit cell.

$$\begin{gathered} \mathrm{L}=4\mathrm{r}\sqrt{3} \\ \mathrm{L}=333\mathrm{pm}\left(\frac{{10}^{-10}\mathrm{cm}}{1\mathrm{pm}}\right) \\ \mathrm{L}=3.33\times {10}^{-8}\mathrm{cm} \end{gathered}$$

$$\begin{gathered} \mathrm{Density}=\frac{4\times 197.0\mathrm{gm}/\mathrm{mole}}{6.023\times {10}^{23}\mathrm{atoms}/\mathrm{mole}\times {(3.33\times {10}^{-8}\mathrm{cm})}^3} \\ \mathrm{Density}=17.7\mathrm{gm}/{\mathrm{cm}}^3 \end{gathered}$$

The density of gold is consistent with a face-centered cubic unit cell.