CsBr has b.c.c. structure with edge length 4.3 \(\AA\). The shortest inter ionic distance in between \(\mathrm{Cs}^{+}\)and \(\mathrm{Br}^{-}\)is : (1) \(3.72\) (2) \(1.86\) (3) \(7.44\) (4) \(4.3\)

The body-centered cubic (b.c.c.) structure is one of the simplest and most common types of crystal lattice. In this structure, ions or atoms are placed at each of the eight corners of a cube and one ion or atom is positioned at the center. This central position plays a key role in the stability and properties of the material.
In a b.c.c. lattice:

• Each corner atom is shared among eight adjacent cubes, effectively contributing one-eighth of its volume to each cube.
• The central atom is entirely within the cube, thus not shared with neighboring cells.

It's important to note that the positioning of atoms affects the overall interatomic distances and physical properties, such as density and coordination number. In b.c.c., each atom touches four atoms in its layer, four above, and four below, resulting in a coordination number of 8.

In solid-state chemistry, the concept of inter-ionic distance is crucial as it influences the nature of bonding and material properties. Inter-ionic distance refers to the distance between the centers of adjacent ions in the lattice structure.
For example, in a b.c.c. lattice like that of Cesium Bromide (CsBr), the inter-ionic distance is determined by the geometry of the cubic structure. The shortest inter-ionic distance in a b.c.c. arrangement is found along the body diagonal. To calculate this distance:

• Use the edge length of the cube.
• Multiply the edge length by \(\root{3}\) (since the body diagonal spans across three sides of the cube).

In our given problem with CsBr, the edge length is 4.3 \(\text{\r{A}}\). So, the body diagonal is calculated as 4.3 \(\times\) \(\root{3} \) = 7.44 \(\text{\r{A}}\).
An important detail is that the shortest inter-ionic distance is half the length of the body diagonal, i.e., 7.44 \(\text{\r{A}}\) ÷ 2 = 3.72 \(\text{\r{A}}\). This calculated distance signifies the closest approach between a Cesium ion and a Bromide ion within the structure.

Understanding crystal lattice calculations is essential for accurately determining structural properties of a material. These calculations often involve distances within the unit cell, as in the case of inter-ionic distances, which directly affect the material’s characteristics.
In a body-centered cubic structure like that of CsBr, key calculations include:

• Calculating the body diagonal, which involves understanding the relationship within the cubic structure.
• Applying geometric principles, such as multiplying the edge length by \(\root{3}\).
• Dividing the body diagonal by 2 to find the shortest inter-ionic distance.

For example, let’s review the steps again for CsBr with an edge length of 4.3 \(\text{\r{A}}\). First, calculate the body diagonal: 4.3 \(\times\) \(\root{3}\) = 7.44 \(\text{\r{A}}\).
Next, determine the shortest inter-ionic distance: 7.44 \(\text{\r{A}}\) ÷ 2 = 3.72 \(\text{\r{A}}\).
These calculations are foundational in solid-state chemistry, allowing scientists and engineers to predict material behaviors and properties accurately.