12.99P - Chemistry: Molecular Nature Of Matter And Change

Question

The coinage metals—copper, silver, and gold—crystallize in a cubic closest packed structure. Use the density of copper (8.95 g/cm3 ) and its molar mass (63.55 g/mol) to calculate an approximate atomic radius for copper.

Step-by-Step Solution

Answer

The atomic radius of copper is 1.41 Ǻ.

Step 1: Copper unit cell

Copper (Cu) has a simple cubic unit cell with Cu atoms in the corner of the cubic unit cell. The simple cubic unit cell consists of 8 atoms in the corner, thus one atom contributing 1/8 to the unit cell.

Thus, each unit cell consists of $8\times \left(\frac{1}{8}\right)=1$, i.e., one atom in the unit cell.

Step 2: Calculation of atomic radius for copper

As in SCC, in total, one atom contributes to the unit cell. The mass of the unit cell would be the same as the mass of a single atom of Cu.

Now, calculating the volume of the atom and thus the radius of the atom,

$$\begin{gathered} \mathrm{Volume}=\frac{\mathrm{Mass}}{\mathrm{Density}} \\ \frac{4}{3}{\mathrm{\pi r}}^3=\frac{63.55}{\left(8.95\times 6.023\times {10}^{23}\right)} \\ {\mathrm{r}}^3=0.28\times {10}^{-23} {\mathrm{cm}}^3 \\ \mathrm{r}=1.41\times {10}^{-10} \mathrm{m} \end{gathered}$$