Question

You are given a small bar of an unknown metal X. You find the density of the metal to be 10.5 g/cm³. An X-ray diffraction experiment measures the edge of the face-centered cubic unit cell as 4.09 Å $\left(1\AA ={10}^{-10}m\right)$. Identify X.

Short answer

The metal X is silver.

Step-by-step solution

Density of a metal

The density of a metal is the ratio of the mass of its unit cell and its volume.

$\mathbf{Density}\mathbf{=}\frac{\mathbf{Mass}}{\mathbf{Volume}}$

Identification of X by using its molecular mass

In FCC crystal, there are four atoms per unit cell.

It is given that the density of metal X is $10.5\mathrm{gm}/{\mathrm{cm}}^3$.

The edge length a is $4.09\times {10}^{-10}\mathrm{m}$

$$\begin{gathered} \mathrm{Density}=\frac{\mathrm{Z}\times \mathrm{M}}{{\mathrm{N}}_{\mathrm{A}}\times {\mathrm{a}}^3} \\ 10.5\mathrm{gm}/{\mathrm{cm}}^3=\frac{4\times \mathrm{M}}{6.023\times {10}^{23}\mathrm{atoms}/\mathrm{mole}\times {\left(4.09\times {10}^{-8}\mathrm{cm}\right)}^3} \\ \mathrm{M}=108.1\mathrm{gm}/\mathrm{mole} \end{gathered}$$

Thus, metal X is Silver.