The aim of this question is to calculate the **volume of a unit cell,** giving due attention to the **lettuce structure** of the given metal. The uniform **spatial arrangement scheme** of atoms, molecules, and/or ions is called **crystal structure**.

The overall crystal structure can be **divided** into smaller **basic elements** that can be **spatially repeated** to form the whole structure of the lettuce crystal. This basic unit has the **same properties** as the crystal. This basic unit structure is called the **unit cell**.

There are **many types** of unit cell structures depending upon the **number of bonds and type of atoms** such as **cubic, tetragonal, orthorhombic, rhombohedral, hexagonal, monoclinic, triclinic,** etc.

The metallic crystal structure is modeled by a **face-centered cubic (FCC) structure**. In such a structure, the metal atoms have such a spatial arrangement that **each corner and face contains an atom** at its center and all atoms are uniformly distributed over space.

The **volume of the unit cell with a face-centered cubic (FCC) structure** can be calculated using the following mathematical formula:

\[ V \ = \ 16 \ \sqrt{ 2 } \ r^{ 3 } \]

Where $ r $ is the **average radius** of the metal atom. If $ r $ is measured in meters, then the volume $ V $ will be in cubic meters.

## Expert Answer

Given:

\[ r \ = \ 0.175 \ nm \]

\[ \Rightarrow r \ = \ 1.75 \ \times \ 10^{ -10 } \ m \]

Since it has a **face-centered cubic crystal (FCC) structure**, the volume of the unit cell of lead can be calculated using the following formula:

\[ V \ = \ 16 \ \sqrt{ 2 } \ r^{ 3 } \]

Substituting the value of $ r $:

\[ V \ = \ 16 \ \sqrt{ 2 } \ ( 1.75 \ \times \ 10^{ -10 } \ m )^{ 3 } \]

\[ V \ = \ 1.21 \ \times \ 10^{ -28 } \ m^{ 3 } \]

Which is the required answer.

## Numerical Result

\[ V \ = \ 1.21 \ \times \ 10^{ -28 } \ m^{ 3 } \]

## Example

**Copper** has an atomic radius of 0.128 pm, if all metals have a face-centered cubic crystal (FCC) structure, then **find the volume of its unit cell cubic meters**.

Given:

\[ r \ = \ 128 \ pm \]

\[ \Rightarrow r \ = \ 1.28 \ \times \ 10^{ -10 } \ m \]

Since it has a **face centered cubic crystal (FCC) structure**, the volume of the unit cell of copper can be calculated using the following formula:

\[ V \ = \ 16 \ \sqrt{ 2 } \ r^{ 3 } \]

Substituting the value of $ r $:

\[ V \ = \ 16 \ \sqrt{ 2 } \ ( 1.28 \ \times \ 10^{ -10 } \ m )^{ 3 } \]

\[ V \ = \ 4.745 \ \times \ 10^{ -29 } \ m^{ 3 } \]

Which is the required answer.