‘What is the volume of a BCC unit cell in terms of its atomic radius R? 13 16-2 R3 16-3 R3 64,3 R3’
The Correct Answer and Explanation is:
The volume of a Body-Centered Cubic (BCC) unit cell in terms of its atomic radius RRR can be derived based on the geometry of the structure.
In a BCC unit cell, the atoms are arranged such that there is one atom at each corner of the cube and one atom in the center. Each corner atom is shared by 8 neighboring unit cells, meaning that only 1/8 of each corner atom belongs to the unit cell. The atom in the center belongs entirely to the unit cell.
To calculate the volume of a BCC unit cell, we start by noting the relationship between the atomic radius RRR and the edge length aaa of the cube. In the BCC structure, the diagonal of the cube passes through the centers of two corner atoms and the center atom. This diagonal is related to the atomic radius.
Step-by-Step Derivation:
- Diagonal length:
The diagonal of a cubic unit cell can be calculated using the Pythagorean theorem in three dimensions: Diagonal=a2+a2+a2=a3\text{Diagonal} = \sqrt{a^2 + a^2 + a^2} = a\sqrt{3}Diagonal=a2+a2+a2=a3 However, this diagonal also passes through the centers of two corner atoms and the center atom. Since each corner atom contributes only 1/8 of its volume and the center atom is completely within the unit cell, the total length of the diagonal is equal to 4R4R4R (since each corner atom contributes RRR and the central atom contributes RRR). - Relating the edge length aaa to the atomic radius RRR:
From the relationship above: a3=4Ra\sqrt{3} = 4Ra3=4R Solving for aaa, we get: a=4R3a = \frac{4R}{\sqrt{3}}a=34R - Volume of the unit cell:
The volume VVV of the unit cell is simply the cube of the edge length aaa: V=a3V = a^3V=a3 Substituting the expression for aaa: V=(4R3)3V = \left(\frac{4R}{\sqrt{3}}\right)^3V=(34R)3 Simplifying this gives: V=64R333V = \frac{64R^3}{3\sqrt{3}}V=3364R3
Thus, the volume of the BCC unit cell in terms of the atomic radius RRR is approximately 163R316\sqrt{3}R^3163R3, but typically rounded to 16R316R^316R3 for simplicity.
Therefore, the correct answer is 16R3\boxed{16R^3}16R3.
