The densities of the elements K, Ca, Sc, and Ti are 0.86, 1.5, 3.2, and 4.5 g/cm³, respectively. One of these elements crystallizes in a body-centered cubic structure; the other three crystallize in a face-centered cubic structure. Which one crystallizes in the body-centered cubic structure? Justify your answer.

Verified step by step guidance

1. Step 1: Understand the difference between body-centered cubic (BCC) and face-centered cubic (FCC) structures. BCC has atoms at each corner of a cube and one atom in the center, while FCC has atoms at each corner and one atom at the center of each face.
2. Step 2: Recall that the density of a crystalline structure is related to the number of atoms per unit cell, the atomic mass, and the volume of the unit cell. BCC structures typically have lower packing efficiency compared to FCC structures.
3. Step 3: Calculate the packing efficiency for both BCC and FCC structures. BCC has a packing efficiency of about 68%, while FCC has a packing efficiency of about 74%.
4. Step 4: Compare the given densities of the elements (K: 0.86 g/cm³, Ca: 1.5 g/cm³, Sc: 3.2 g/cm³, Ti: 4.5 g/cm³) with typical densities for BCC and FCC structures. BCC structures generally have lower densities due to their lower packing efficiency.
5. Step 5: Identify the element with the lowest density, which is potassium (K) with a density of 0.86 g/cm³. This suggests that potassium is the element that crystallizes in a body-centered cubic structure due to its lower density compared to the others.