In cubic closest-packed solids, what percentage of space is occupied by the spheres?
A) 68.0%
B) 74.0%
C) 52.4%
D) 43.8%
E) none of these

Question

Quizplus · Accepted Answer

In cubic closest-packed solids, each sphere is surrounded by 12 nearest neighboring spheres. The spheres touch each other along the face diagonal of the cube, and the face diagonal is the twice the sphere radius. For a single sphere, the volume of the cube it is in is $d^3$, where $d$ is the diameter of the sphere. Therefore, the volume of a single sphere is $\frac{\pi}{6}d^3$. The volume of 12 nearest neighboring spheres is 12 times the volume of a single sphere, and this is how much space a sphere takes up in the solid. The volume of the cube that encloses these 12 spheres is $4d^3$. Therefore, the percentage of space occupied by the spheres is $\frac{3\pi \sqrt{2}}{4} \approx 0.74$ or 74.0%. So the answer is B.

In Cubic Closest-Packed Solids, What Percentage of Space Is Occupied