Question

The effective number of atoms per unit cell in a simple cube, face centred cube and body centred cube are respectively: (a) \(1,4,2\) (b) \(1,2,4\) (c) \(8,14,9\) (d) \(8,4,2\)

Short answer

The effective number of atoms per unit cell in a simple cube (sc), face-centered cube (fcc), and body-centered cube (bcc) are 1, 4, and 2, respectively, so the answer is (a) 1,4,2.

Step-by-step solution

Understanding the Crystal Structures

A simple cubic (sc) lattice has atoms at the corners of the cube, face-centered cubic (fcc) has atoms at the corners and at the center of each face, and the body-centered cubic (bcc) has atoms at the corners and one atom at the center of the cube.

Calculating Atoms in a Simple Cubic Unit Cell

In a simple cubic structure, there is 1/8 of an atom at each corner of the cube and there are 8 corners. Therefore, the number of atoms per unit cell in a simple cubic lattice is 8 * (1/8) = 1.

Calculating Atoms in a Face-Centered Cubic Unit Cell

In a face-centered cubic structure, besides the 8 corners, there is 1/2 of an atom on each face, and there are 6 faces. Therefore, the number of atoms per unit cell in an fcc lattice is 8 * (1/8) + 6 * (1/2) = 1 + 3 = 4.

Calculating Atoms in a Body-Centered Cubic Unit Cell

In a body-centered cubic structure, there is one full atom at the body center and the corner atoms. Thus, the number of atoms per unit cell in a bcc lattice is 8 * (1/8) + 1 = 1 + 1 = 2.

Key concepts

Understanding Simple Cubic Unit Cells

A simple cubic unit cell represents one of the most fundamental crystal structures in physical chemistry. As the name suggests, this structure features a cube with atoms situated only at its eight corners. Each corner atom is shared among eight adjacent unit cells, meaning only a fraction of each atom actually belongs to one particular cell.

Now, picture a single atom as a pie. When it comes to spreading out that pie among different unit cells, in the simple cubic structure, each unit cell gets one eighth (1/8) of that pie, as it is shared by eight cells. With an atom at each corner, the simple cubic cell has eight 'slices' of the atomic 'pie.' That's how we derive the effective number of atoms per unit cell in a simple cubic lattice, which is simply the sum of these slices: 8 * (1/8) = 1 atom per unit cell.

Discovering Face-Centered Cubic Unit Cells

Moving on to the face-centered cubic (fcc) unit cell, we step into a more densely packed arrangement. Here, in addition to the corner atoms seen in our simple cubic, we have atoms centrally located on each of the cube's six faces.

In the face-centered cubic cell, we still count the corner atoms as one eighth of an atom per unit cell, but the face atoms are shared between only two unit cells. This means for each face atom, half (1/2) belongs to one unit cell. There are six such faces, so we have: 8 corners * (1/8 of an atom) + 6 faces * (1/2 of an atom) = 1 + 3 = 4 atoms effectively in one fcc unit cell.

Exploring Body-Centered Cubic Unit Cells

The body-centered cubic (bcc) unit cell adds an interesting twist to our understanding of crystal structures. Still maintaining atoms at its corners, the bcc cell also houses one additional atom smack in the center of the unit. This central atom is not shared with any other unit cells and therefore, counts as one whole atom.

As with the other structures, the corner atoms contribute one eighth of an atom for the unit cell. In the body-centered cubic structure, combining the contributions from the corners and the center, we find our calculation is 8 * (1/8) + 1 (central atom) = 1 + 1 = 2 effective atoms per unit cell.

Calculation of Atoms in a Unit Cell

Calculating the effective number of atoms in a unit cell is critical for understanding the properties of crystal lattices. This entails calculating not just the atoms located at the lattice points, but how those atoms are shared with adjacent cells.

The process involves accounting for each atom's position—be it corner, face, or body—and determining its contribution to a single unit cell. By summing these fractional contributions, we arrive at the effective number of atoms. This vital piece of knowledge helps in determining the density, stability, and other characteristics of the material. Furthermore, grasping the calculation method can reinforce one's understanding of the three-dimensional structure in crystal lattices.