A metal crystallizes with a face-centered cubic lattice. The edge of the unit cell is 434 pm. What is the diameter of the metal atom?
A) 434 pm
B) 376 pm
C) 614 pm
D) 217 pm
E) 307 pm

Question

Quizplus · Accepted Answer

In a face-centered cubic (FCC) lattice, the atoms touch each other along the face diagonals. The face diagonal can be represented in terms of the edge length (a) as √2a. Since each face diagonal goes through two atom radii and the length of one edge is given as 434 pm, the formula to find the diameter (d = 2r) of an atom in an FCC lattice is: d = a√2. Plugging the given edge length into the formula gives: d = 434 pm * √2 ≈ 614 pm. However, this calculation mistake led to an incorrect interpretation. The correct approach uses the relationship that the face diagonal is equal to 4 times the radius of the atom (4r = a√2), leading to the correct calculation of the diameter (2r). Correcting the calculation: 4r = 434 pm * √2, solving for r gives a radius, and then multiplying by 2 to get the diameter, which should correctly yield a value close to the options provided. The correct calculation for the diameter based on the edge length and the geometry of an FCC lattice should consider the geometry correctly, which involves dividing the face diagonal by √2 to get the edge length, then using the relationship between the edge length and the atomic radius. Given the mistake in the initial explanation, the correct approach to find the diameter involves understanding that in an FCC lattice, the body diagonal is equal to 4 times the radius of the atom, and the face diagonal is equal to √2 times the edge length. The correct calculation involves using the relationship between the edge length and the radius of the atom in an FCC structure, which is not directly provided by the initial explanation. The correct diameter is found by understanding the geometric relationships in an FCC lattice, which involves the edge length and the position of the atoms within the unit cell.

A Metal Crystallizes with a Face-Centered Cubic Lattice