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Question: Calculate the total volume occupied by all atoms in simple cubic unit cell if radius of atom is 3 x ...

Calculate the total volume occupied by all atoms in simple cubic unit cell if radius of atom is 3 x 10⁻⁸cm.

A

1.13 x 10⁻²²cm³

B

2.25 x 10⁻²²cm³

C

3.15 x 10⁻²²cm³

D

4.37 x 10⁻²²cm³

Answer

1.13 x 10⁻²²cm³

Explanation

Solution

In a simple cubic cell, only 18\frac{1}{8} of each of the 8 corner atoms is inside the cell. Thus, the total number of atoms per cell is 1.

The volume of one atom (a sphere) with radius r=3×10−8r = 3 \times 10^{-8} cm is:

Vatom=43πr3=43π(3×10−8)3V_{\text{atom}} = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi (3 \times 10^{-8})^3 (3×10−8)3=27×10−24=2.7×10−23 cm3(3 \times 10^{-8})^3 = 27 \times 10^{-24} = 2.7 \times 10^{-23} \text{ cm}^3 Vatom=43π(2.7×10−23)=3.6π×10−23 cm3V_{\text{atom}} = \frac{4}{3} \pi (2.7 \times 10^{-23}) = 3.6 \pi \times 10^{-23} \text{ cm}^3 ≈3.6×3.14×10−23 cm3≈11.3×10−23 cm3=1.13×10−22 cm3\approx 3.6 \times 3.14 \times 10^{-23} \text{ cm}^3 \approx 11.3 \times 10^{-23} \text{ cm}^3 = 1.13 \times 10^{-22} \text{ cm}^3

Since there is 1 atom per unit cell, the total volume occupied by all atoms in the cell is 1.13×10−221.13 \times 10^{-22} cm³.