Elemental magnesium crystallizes in a face-centered cubic lattice. The density of magnesium is 1.738 g/cm³. The unit cell length is 4.80 × 10² pm. What is the atomic radius of Mg? A) 340 pm B) 215 pm C) 242 pm D) 126 pm E) 170 pm

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Quizplus · Accepted Answer

In a face-centered cubic (FCC) lattice, each unit cell contains 4 atoms of the element. The formula to relate the density ($ \rho $), the molar mass (M), the Avogadro number ($ N_A $), and the volume of the unit cell (V) is $ \rho = \frac{M}{N_A \cdot V} $. The volume of the unit cell can be calculated using the length of the unit cell edge (a), $ V = a^3 $. For a FCC lattice, the atomic radius (r) is related to the unit cell length by the equation $ a = 2\sqrt{2}r $. Given the density of magnesium is 1.738 g/cm$^3$ and the unit cell length is 4.80 × 10$^{-8}$ cm (480 pm), we can find the atomic radius. Rearranging the equation for the atomic radius gives $ r = \frac{a}{2\sqrt{2}} $. Substituting the given unit cell length into this equation yields $ r = \frac{480 \, \text{pm}}{2\sqrt{2}} = 170 \, \text{pm} $.

Elemental Magnesium Crystallizes in a Face-Centered Cubic Lattice