1. calcium crystallizes through cubic closest - packing. if the atomic radius of calcium is 197 pm, find the…

Explanation:

Step1: Calculate the edge length of the unit cell

In cubic closest - packing (face - centered cubic, FCC), the relationship between the edge length (a) and the atomic radius (r) is (a = 2\sqrt{2}r). Given (r=197\space pm), then (a = 2\sqrt{2}\times197\space pm\approx 2\sqrt{2}\times197\times10^{- 10}\space cm) [a\approx5.57\times10^{-8}\space cm]

Step2: Calculate the volume of the unit cell

The volume of a cube (V=a^{3}) [V=(5.57\times 10^{-8})^{3}\space cm^{3}\approx1.72\times10^{-22}\space cm^{3}]

Step3: Calculate the number of atoms per unit cell in FCC

In FCC, the number of atoms per unit cell (n = 4)

Step4: Calculate the molar mass of calcium

The molar mass of (Ca), (M = 40.08\space g/mol)

Step5: Use the density formula (\rho=\frac{nM}{V{N}_{A}})

where (N_{A}=6.022\times 10^{23}\space mol^{-1}) [ \begin{align*} \rho&=\frac{4\times40.08\space g/mol}{1.72\times10^{-22}\space cm^{3}\times6.022\times 10^{23}\space mol^{-1}}\ &=\frac{160.32\space g/mol}{1.036\times10^{2}\space cm^{3}/mol}\ &\approx1.55\space g/cm^{3} \end{align*} ]

Answer:

The density of solid calcium is approximately (1.55\space g/cm^{3})