Question

CsBr has been structure with edge length 4.3. The shortest inter ionic distance in between \(\mathrm{Cs}^{+}\) and \(\mathrm{Br}^{-}\) is (a) \(3.72\) (b) \(1.86\) (c) \(7.44\) (d) \(4.3\)

Short answer

3.72 Å

Step-by-step solution

Identify the crystal structure

CsBr crystallizes in a body-centered cubic (bcc) structure. In a bcc structure, the unit cell consists of one atom at each corner of the cube and one atom at the center of the cube.

Calculate the shortest interionic distance

The shortest distance between the central ion and one of the corner ions in a bcc structure is along the body diagonal of the cube. The body diagonal can be calculated using the cube edge length (a) and the formula for the body diagonal of the cube which is \(\sqrt{3} \times a\).

Find the interionic distance for CsBr

For CsBr with an edge length of 4.3 Å, the body diagonal is \(\sqrt{3} \times 4.3\) Å. This diagonal is shared by two interionic distances, so the shortest interionic distance is half of the body diagonal. Therefore, the shortest interionic distance is \(\frac{\sqrt{3} \times 4.3}{2}\) Å.

Perform the calculation

Now calculate the shortest interionic distance using the given edge length: \(\frac{\sqrt{3} \times 4.3}{2} =\frac{\sqrt{3} \times 4.3}{2}\approx \frac{1.732 \times 4.3}{2} \approx \frac{7.448}{2} \approx 3.724\) Å. Therefore, the correct answer is approximately 3.72 Å.

Key concepts

Crystal Structure

Understanding crystal structure is crucial for studying materials at the microscopic level, and this knowledge extends to many practical applications in chemistry, physics, and materials science. In simple terms, the crystal structure describes the orderly and repeating arrangement of atoms, molecules, or ions within a crystalline material. This pattern extends in all three spatial dimensions and defines many of the material's properties, such as electrical conduction, strength, and melting point. Different materials can crystallize in various structures based on factors like the size and valence of their constituent particles and the type of chemical bonding that holds them together. Learning to recognize and describe these structures helps in understanding how a material will behave in different conditions.

Body-Centered Cubic (BCC)

One of the fundamental crystal structures in materials science is the body-centered cubic (bcc) structure. In a bcc crystal, atoms are arranged at the corners of a cube and there is one additional atom at the very center of the cube, making an arrangement that visually resembles a box with an object at its heart. This specific arrangement affects the material properties, as it has a different packing density and coordination number compared to other arrangements like face-centered cubic (fcc) or hexagonal close-packed (hcp). The bcc structure is common among metals such as iron at certain temperatures, which contributes to its magnetic and structural attributes. The distinctive placement of atoms in bcc crystals plays a significant role in determining the shortest interionic distances within them.

Unit Cell

The unit cell is the smallest portion of a crystal lattice that, when repeated in all directions, forms the entire crystal. Think of it as a kind of 'building block' for crystals. This three-dimensional geometric figure captures the essence of the crystal's symmetry and structure. In a bcc crystal, the unit cell is a cube with atoms at each of its eight corners and one atom in the center. Identifying the unit cell helps to calculate various other properties of the crystal, including its volume, density, and - as in the case of this exercise - the shortest interionic distance. This makes the understanding of unit cells imperative for students diving into the field of crystallography and material science.

Edge Length Calculation

Calculating the edge length of a unit cell is a vital step in order to determine many other measurements related to crystal structures, including density and interatomic distances. The edge length refers to the length of one side of the unit cell and is commonly denoted by the letter 'a' in crystallography. In a bcc crystal, understanding the edge length is crucial for determining the shortest distance between ions, which is generally along the body diagonal, that is, a line connecting one corner of the cube to the opposite corner. The body diagonal's length is related to the cube's edge length 'a' through the geometric relationship involving the square root of three, as illustrated in the solution to the problem provided. In essence, the edge length calculation is a bridge to unraveling crucial information about a crystal's internal structure and the interactions between its components.