When spheres of radius r are packed in a body-centered cubic arrangement, they occupy 68.0% of the available volume. Use the fraction of occupied volume to calculate the value of a, the length of the edge of the cube, in terms of r.

Verified step by step guidance

1. Understand that in a body-centered cubic (BCC) arrangement, there is one atom at each corner of the cube and one atom in the center.
2. Recognize that the volume occupied by the spheres is 68.0% of the total volume of the cube.
3. The volume of a single sphere is given by \( \frac{4}{3} \pi r^3 \). In a BCC unit cell, there are effectively 2 spheres (1/8 of a sphere at each of the 8 corners and 1 whole sphere in the center).
4. The total volume of the cube is \( a^3 \), where \( a \) is the edge length of the cube.
5. Set up the equation for the fraction of occupied volume: \( \frac{2 \times \frac{4}{3} \pi r^3}{a^3} = 0.68 \) and solve for \( a \) in terms of \( r \).