Chapter 9, Problem 62QAP

(a)

Interpretation Introduction

Interpretation:

The packing efficiency of the simple cubic cell (1atom/cell) is to be determined.

Concept Introduction:

Packing efficiency of the cubic cell is calculated by the following formula:

  $\text{Packing efficiency=}\frac{\text{volume of atom in cell}}{\text{volume of cell}}\text{×100}$

Answer to Problem 62QAP

The packing efficiency of the simple cubic cell (1atom/cell) is 52 %.

Explanation of Solution

Given:

The volume of one atom is $\frac{4}{3}\pi r^3$ and the volume of the cell is s³.

The percentage of the total of solid occupied by the spherical atoms in the cell is said to be packing efficiency.

The mathematical formula for the packing efficiency is:

  $\text{Packing efficiency=}\frac{\text{volume of atom in cell}}{\text{volume of cell}}\text{×100}$ .......(1)

In cubic cell, edge length is denoted as ‘s’ and the radius as ‘r’.

The relation between edge length and radius of the simple cubic cell is:

  $2r=s$

Put the above values in (1):

  $\text{Packing efficiency = }\frac{\frac{\text{4}}{\text{3}}{\text{πr}}^{\text{3}}}{{\text{s}}^{\text{3}}}\text{×100}$

And the relation between r and s:

2r = s

There is only one atom present in the simple cubic cell.

Therefore,

  $\text{Packing efficiency = }\frac{\frac{\text{4}}{\text{3}}{\text{πr}}^{\text{3}}}{{\left(\text{2r}\right)}^{\text{3}}}\text{×100}$

Packing efficiency = 52 %.

(b)

Interpretation Introduction

Interpretation:

The packing efficiency of face-centered cubic cell (4 atoms/cell) is to be determined.

Concept Introduction:

Packing efficiency of the cubic cell is calculated by the following formula:

  $\text{Packing efficiency=}\frac{\text{volume of atom in cell}}{\text{volume of cell}}\text{×100}$

Answer to Problem 62QAP

The packing efficiency of the face-centered cubic cell (4atom/cell) is 74%.

Explanation of Solution

Given:

The volume of one atom is $\frac{4}{3}\pi r^3$ and the volume of the cell is s³.

The percentage of the total of solid occupied by the spherical atoms in the cell is said to be packing efficiency.

The mathematical formula for the packing efficiency is:

  $\text{Packing efficiency=}\frac{\text{volume of atom in cell}}{\text{volume of cell}}\text{×100}$ .......(1)

In cubic cell, edge length is denoted as ‘s’ and the radius as ‘r’.

The relation between edge length and radius of the face-centered cubic cell is:

  $4r=\sqrt{2}s$

  $s=\frac{4r}{\sqrt{2}}$

Put the above values in (1):

  $\text{Packing efficiency=}\frac{\frac{\text{4}}{\text{3}}{\text{πr}}^{\text{3}}}{{\text{s}}^{\text{3}}}\text{×100}$

And the relation between r and s:

  $s=\frac{4r}{\sqrt{2}}$

There are four atoms present in the face-centered cubic cell.

Therefore,

  $\text{Packing efficiency=}\frac{\text{4×}\frac{\text{4}}{\text{3}}{\text{πr}}^{\text{3}}}{{\left(\frac{\text{4r}}{\sqrt{\text{2}}}\right)}^{\text{3}}}\text{×100}$

Packing efficiency = 74 %

(c)

Interpretation Introduction

Interpretation:

The packing efficiency of a body-centered cubic cell (2 atoms/cell) is to be determined.

Concept Introduction:

Packing efficiency of the cubic cell is calculated by the following formula:

  $\text{Packing efficiency=}\frac{\text{volume of atom in cell}}{\text{volume of cell}}\text{×100}$

Answer to Problem 62QAP

The packing efficiency of a body-centered cubic cell (2 atom/cell) is 68 %.

Explanation of Solution

Given:

The volume of one atom is $\frac{4}{3}\pi r^3$ and the volume of the cell is s³.

The percentage of the total of solid occupied by the spherical atoms in the cell is said to be packing efficiency. The mathematical formula for the packing efficiency is:

  $\text{Packing efficiency=}\frac{\text{volume of atom in cell}}{\text{volume of cell}}\text{×100}$ .......(1)

In cubic cell, edge length is denoted as ‘s’ and the radius as ‘r’.

The relation between edge length and radius of the body-centered cubic cell is:

  $4r=\sqrt{3}s$

  $s=\frac{4r}{\sqrt{3}}$

Put the above values in (1):

  $\text{Packing efficiency=}\frac{\frac{\text{4}}{\text{3}}{\text{πr}}^{\text{3}}}{{\text{s}}^{\text{3}}}\text{×100}$

The relation between r and s:

  $s=\frac{4r}{\sqrt{3}}$

There are two atoms present in the body-centered cubic cell.

Therefore,

  $\text{Packing efficiency=}\frac{\text{2×}\frac{\text{4}}{\text{3}}{\text{πr}}^{\text{3}}}{{\left(\frac{\text{4r}}{\sqrt{\text{3}}}\right)}^{\text{3}}}\text{×100}$

Packing efficiency = 68 %