Question

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

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 \(

ho=\frac{nM}{V{N}_{A}}\)
where \(N_{A}=6.022\times 10^{23}\space mol^{-1}\)

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Answer:

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