Question

Total volume of atoms present in a face-centred cubic unit cell of a metal is \((\mathrm{r}\) is atomic radius) (a) \(\frac{20}{3} \pi \mathrm{r}^{3}\) (b) \(\frac{24}{3} \pi \mathrm{r}^{3}\) (c) \(\frac{12}{3} \pi \mathrm{r}^{3}\) (d) \(\frac{16}{3} \pi \mathrm{r}^{3}\)

Short answer

The total volume of atoms in an FCC unit cell is \( \frac{16}{3} \pi r^3 \), so the answer is (d).

Step-by-step solution

Understanding the Structure

A face-centred cubic (FCC) unit cell consists of atoms positioned at each corner of the cube and one atom at the center of each face. To calculate the total volume of atoms in the FCC unit cell, we first need to know how many atoms are inside the cell.

Calculate Atoms in FCC

In an FCC unit cell, there are 8 corner atoms, each shared by 8 cells, and 6 face-centered atoms, each shared by 2 cells. Therefore, the total number of whole atoms per FCC unit cell is calculated as: \( \frac{8}{8} + \frac{6}{2} = 1 + 3 = 4 \). So, there are 4 atoms per FCC unit cell.

Volume of One Atom

Each atom can be approximated as a sphere with volume given by the formula \( V = \frac{4}{3} \pi r^3 \), where \( r \) is the radius of the atom.

Calculate Total Volume of Atoms in FCC Cell

Since there are 4 atoms in the cell, the total volume of all atoms in the FCC unit cell is: \( 4 \times \frac{4}{3} \pi r^3 = \frac{16}{3} \pi r^3 \).

Key concepts

Atomic Radius

Understanding atomic radius is essential when discussing the structure of atoms in different arrangements like FCC. The atomic radius defines the size of an atom from its nucleus to the outer boundary of electrons. In simpler terms, it's the edge of an atom where another atom can come into contact. This measurement helps to calculate other properties such as volume and packing efficiency in crystalline structures.

Atomic radius is particularly important in a face-centred cubic (FCC) structure because it affects how closely the atoms can pack together. In the context of FCC, knowing the atomic radius helps calculate the volume each atom occupies, which in turn is crucial for determining the overall packing density and other physical properties.

The radius is usually determined by using specific techniques and is generally denoted by the letter 'r' within mathematical formulas. This value is fundamental and used to find the volume of a sphere when considering the atom to be a sphere.

Volume of Atoms

Calculating the volume of an atom is a fundamental step when determining how atoms pack in a solid structure like an FCC unit cell. The volume of an atom can be often approximated by modeling it as a sphere, with the formula for the volume of a sphere being \[ V = \frac{4}{3} \pi r^3 \] where \(r\) is the atomic radius.

By using this equation, you can calculate the volume of a single atom, allowing you to understand how much space each atom occupies in the crystal lattice. In crystal structures like FCC, the volume that atoms occupy helps explain features like density, strength, and stability of the material.

Once you understand how to calculate the volume of one atom, extending this method to find the total volume occupied by atoms in a crystal becomes straightforward. Just multiply the volume of one atom by the total number of atoms in a single unit cell.

FCC Structure

The face-centred cubic (FCC) structure is a common and remarkably efficient way that atoms arrange themselves in a crystal lattice. In an FCC unit cell, you will find atoms located at each corner of the cube as well as one atom in the center of each face. This arrangement gives FCC structures high packing efficiency, meaning atoms are closely packed with minimal wasted space.

FCC structures are significant in materials science because this optimal arrangement leads to specific properties like high density and stability. Metals such as copper, aluminum, and nickel frequently display an FCC structure due to its efficient packing and low energy configuration.

By understanding the characteristics of an FCC unit cell, you can predict how materials behave under different conditions. This knowledge is also useful in determining how these materials interact and perform, impacting fields from industrial manufacturing to electronics.

Number of Atoms in Unit Cell

In a face-centred cubic (FCC) unit cell, it's essential to understand how atoms are counted and how they contribute to the overall structure. Each of the 8 corner atoms in an FCC structure is shared among 8 different unit cells, effectively contributing only \(\frac{1}{8}\) of an atom to a single cell. Moreover, each of the 6 face-centered atoms is shared between two unit cells, contributing \(\frac{1}{2}\) of an atom per cell.

Adding these contributions gives us the total number of atoms in one FCC unit cell as: \[ \frac{8}{8} + \frac{6}{2} = 1 + 3 = 4 \] Thus, each FCC unit cell contains a total of 4 atoms. This arrangement and distribution are crucial because they directly affect the calculation of properties like density and the volume of atoms within the cell. These properties help in determining how efficiently the atoms pack within the crystal, explaining why FCC structures possess high packing efficiency and stability.