Calculate Packing Fraction of Face Centered Unit Cell Salt Structure

To calculate the packing fraction of the given salt structure, we analyze the arrangement of ions $X$ and $Y$ in the cubic unit cell based on the provided figure.

1. Structure Analysis

From the diagram:

• The anion $Y$ spheres are positioned at the 8 corners of the cubic unit cell and touch each other along the edges.
• The cation $X$ spheres are positioned at the 6 face-centers of the cubic unit cell and touch the surrounding $Y$ spheres along the face diagonals.

2. Determination of Ionic Radii

Let $a$ be the edge length of the cubic unit cell, $r_Y$ be the radius of anion $Y$, and $r_X$ be the radius of cation $X$.

• Along the edge of the unit cell: Since adjacent $Y$ ions touch each other along the edge: $a = 2 r_Y \implies r_Y = \frac{a}{2}$

• Along the face diagonal of the unit cell: The face diagonal has a length of $\sqrt{2}a$. Along the face diagonal, the central $X$ ion touches two corner $Y$ ions: $2 r_Y + 2 r_X = \sqrt{2} a$

  Substituting $a = 2 r_Y$: $2 r_Y + 2 r_X = 2\sqrt{2} r_Y$ $r_X = r_Y (\sqrt{2} - 1)$

3. Number of Ions per Unit Cell

• Number of $Y$ ions ($Z_Y$): $Z_Y = 8 \text{ corners} \times \frac{1}{8} = 1$

• Number of $X$ ions ($Z_X$): $Z_X = 6 \text{ face centers} \times \frac{1}{2} = 3$

4. Calculation of Packing Fraction

The total volume occupied by the ions in the unit cell is: $V_{\text{occupied}} = Z_Y \left(\frac{4}{3} \pi r_Y^3\right) + Z_X \left(\frac{4}{3} \pi r_X^3\right)$ $V_{\text{occupied}} = \frac{4}{3} \pi \left[ r_Y^3 + 3 r_X^3 \right]$

Substituting $r_X = r_Y(\sqrt{2} - 1)$: $V_{\text{occupied}} = \frac{4}{3} \pi r_Y^3 \left[ 1 + 3(\sqrt{2} - 1)^3 \right]$

The total volume of the cubic unit cell is: $V_{\text{cell}} = a^3 = (2 r_Y)^3 = 8 r_Y^3$

Thus, the packing fraction is given by: $\text{Packing Fraction} = \frac{V_{\text{occupied}}}{V_{\text{cell}}} = \frac{\frac{4}{3} \pi r_Y^3 \left[ 1 + 3(\sqrt{2} - 1)^3 \right]}{8 r_Y^3} = \frac{\pi}{6} \left[ 1 + 3(\sqrt{2} - 1)^3 \right]$

5. Numerical Computation

Substitute $\sqrt{2} \approx 1.414$: $\sqrt{2} - 1 \approx 0.414$ $(\sqrt{2} - 1)^3 \approx (0.414)^3 \approx 0.071$ $1 + 3(0.071) = 1 + 0.213 = 1.213$

Now, calculating the packing fraction: $\text{Packing Fraction} \approx \frac{3.1416}{6} \times 1.213 \approx 0.5236 \times 1.213 \approx 0.635 \approx 0.63$

Correct Answer: (B) 0.63