Question

AB Crystallises in a B.C.C lattice with edge length. a as \(387 \mathrm{Pm}\). The distance between two oppositely charged ions in the lattice is (a) \(300 \mathrm{Pm}\) (b) \(335 \mathrm{Pm}\) (c) \(250 \mathrm{Pm}\) (d) \(200 \mathrm{Pm}\)

Short answer

The distance is approximately \(335 \mathrm{Pm}\), which is option (b).

Step-by-step solution

Understanding B.C.C Lattice

A B.C.C (Body-Centered Cubic) lattice structure consists of atoms at each corner of a cube and one atom at the center. The ions in question are located at the center of the cube and at the corner of the cube.

Formula for Distance between Nearest Neighbors in B.C.C

In a B.C.C lattice, the nearest neighbors are along the body diagonal of the cube. The body diagonal of a cube with edge length \(a\) is \(\sqrt{3}a\). However, the distance between the center atom and a corner atom (oppositely charged ions) is half of the body diagonal, or \(\frac{\sqrt{3}a}{2}\).

Substitute Edge Length in Formula

Given the edge length \(a\) is \(387 \mathrm{Pm}\), substitute this value into the formula to find the distance: \[\text{Distance} = \frac{\sqrt{3} \times 387\,\mathrm{Pm}}{2}\]

Calculation

Calculate the exact distance: \[\text{Distance} = \frac{1.732 \times 387}{2} \approx \frac{670.524}{2} \approx 335.262 \mathrm{Pm}\]

Conclusion

Compare the calculated distance to the given options. The closest value is \(335 \mathrm{Pm}\), which corresponds to option (b).

Key concepts

Body-Centered Cubic Structure

In a body-centered cubic (B.C.C) structure, atoms are arranged with unique precision. Imagine a cube where each of the eight corners hosts an atom. Besides the atoms at the corners, there's a solitary atom residing at the very center of this cube. This setup forms the foundation of the B.C.C lattice.

The B.C.C structure is prevalent in various metals and alloys, thanks to its effective space utilization. The central atom ensures that every part of the cube's interior is filled, contributing to stable and sturdy material properties.

• Eight corner atoms appear in each cube.
• One atom sits in the nucleus (center) of the cube.
• This central atom is equidistant from all corner atoms.

Nearest Neighbors

The concept of nearest neighbors is crucial in understanding interactions in a B.C.C lattice. In this crystalline structure, the nearest neighbors are not just limited to touching atoms.They extend along specific directions, which helps determine physical properties like conductivity and strength.

The nearest neighbor for any atom in a B.C.C structure is found along the body diagonal of the cube. This means that the closest atoms you can get to the central atom are along the line going from corner to corner through the cube's center.

• The nearest neighbors for the center atom would be the atoms at the corners in the line of the body diagonal.
• The line of the body diagonal is longer than the cube's edge, precisely \(\sqrt{3}a\), making neighbor distances significant.

Edge Length Calculation

Calculating the distance between certain atoms in a B.C.C structure hinges on knowing the edge length of the cube. For the B.C.C lattice, the edge length is critical to finding the accurate distance between neighbors, particularly between a central and a corner atom.

Let's break it down: in calculations, we rely on the body diagonal to determine distances. Given an edge length \(a\), the complete length of the body diagonal is \(\sqrt{3}a\), but the interest often is in the distance between the center and a corner atom, which is merely half of this diagonal.
For instance, substituting the edge length \(387 \, \mathrm{Pm}\) gives the distance calculated as follows: \[ \text{Distance} = \frac{\sqrt{3} \times 387}{2} \approx 335.262 \, \mathrm{Pm} \]This numeric result harmonizes with practical options, helping decide appropriate scenarios in theoretical and practical frameworks.