Question

Lead (atomic radius \(=0.181 \mathrm{nm})\) crystallizes with a face-centered cubic unit cell. What is the length of a side of the cell?

Short answer

Answer: The length of a side of the face-centered cubic unit cell of Lead is approximately 0.32357 nm.

Step-by-step solution

Understand the face-centered cubic structure

In a face-centered cubic (fcc) structure, atoms are located at the corner and center of each face of the cube. An important feature of this structure is that atoms at the corners and face centers come into contact with each other along the face diagonal.

Determine the length of the face diagonal

In the fcc structure, since the atoms at the corners and face centers are touching along the face diagonal, we can determine the length of the face diagonal using given atomic radius. There are a total of two corner atoms (each contributes 1/2 of its diameter) and one face-centered atom (contributing its diameter) along the face diagonal: Length of face diagonal = 1 diameter (corner atoms) + 1 diameter (face-centered atom) = 2(radius + radius) + 2(radius + radius) = 4 * atomic radius We are given that the atomic radius of Lead is 0.181 nm. So, the length of the face diagonal is: Face diagonal = 4 * atomic radius = 4 * 0.181 = 0.724 nm

Apply Pythagorean theorem to find the side length

To find the length of a side of the cube, we can apply the Pythagorean theorem in a right-angled triangle formed by half of the face diagonal, half of the cube edge, and the cube edge. Which means: (Half of face diagonal)^2 = (Half of cube side)^2 + (Cube side)^2 Substitute the expression for half of the face diagonal (2 * atomic radius) and solve for the cube side (let it be x): (2 * atomic radius)^2 = (x/2)^2 + x^2 Plug in the value of atomic radius, and solve for x: (2 * 0.181)^2 = (x/2)^2 + x^2

Solve for the cube side length

From the equation in Step 3, we can now solve for x: (2 * 0.181)^2 = (x/2)^2 + x^2 0.130944 = (x^2)/4 + x^2 0.130944 = (5x^2)/4 x^2 = (0.130944 * 4) / 5 x^2 = 0.1047552 x = √0.1047552 = 0.323573 nm The length of a side of the face-centered cubic unit cell of Lead is approximately 0.32357 nm.