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Chapter 18: Problem 29

What is the length of an edge of the unit cell when barium (atomic radius \(222 \mathrm{pm}\) ) crystallizes in a crystal lattice of bcc unit cells?

Short Answer

Expert verified
Answer: The length of an edge of the barium unit cell in a body-centered cubic lattice is approximately 509.4 pm.

Step by step solution

01

Recall the body-centered cubic (bcc) structure

In a bcc unit cell, there is an atom at each corner of a cube and one atom in the center. So, there are 8 corner atoms and 1 center atom. However, each corner atom is shared by 8 neighboring cells, so in total, there are 2 atoms per unit cell (1/8 from each corner atom and 1 from the center atom).
02

Use the relationship between the atomic radius (r) and the edge length (a) in a bcc lattice

In a bcc structure, the relationship between the atomic radius (r) and the edge length (a) is given by the formula: \[a = 4r \sqrt{3}/3\] Here, we are given the atomic radius of barium as 222 pm. We will plug this value into the formula to find the edge length.
03

Calculate the edge length (a)

Substitute the atomic radius of barium (r = 222 pm) into the formula: \[a = 4(222\,\mathrm{pm}) \frac{\sqrt{3}}{3}\] Now, we can calculate the edge length (a): \[a \approx 509.4\,\mathrm{pm}\] Hence, the length of an edge of the barium unit cell when it crystallizes in a bcc lattice is approximately 509.4 pm.

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