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If the atomic radius of lead is 0.175 nm, calculate the volume of its unit cell in cubic meters.

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.

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