Chapter 12, Problem 46GQ

Consider the three types of cubic units cells.

1. (a) Assuming that the spherical atoms or ions in a primitive cubic unit cell just touch along the cube’s edges, calculate the percentage of occupied space within the unit cell. (Recall that the volume of a sphere is (4/3)πr³, where r is the radius of the sphere.)
2. (b) Compare the percentage of occupied space in the primitive cell (pc) with the bcc and fcc unit cells. Based on this, will a metal in these three forms have the same or different densities? If different, in which is it most dense? In which is it least dense?

Interpretation Introduction

Interpretation:

Percentage of occupied space within the unit cell has to be calculated.

Concept introduction:

$\begin{array}{l}\text{The packing efficiency can be calculated }\\ \text{by the percent of space occupied by spheres present in a unit cell}\text{.}\\ \text{packing efficiency}\text{=}\frac{\text{volumeoccupied}\text{by}\text{the}\text{spheres}\text{present}\text{in}\text{unit}\text{cell}}{\text{Total}\text{volume}\text{of}\text{the}\text{unit}\text{cell}}\text{×}\text{100}\end{array}$

${\text{Volume of the cube = a}}^{\text{3}}$

Answer to Problem 46GQ

Percentage of occupied space within the unit cell is given below

$\begin{array}{l}\text{Packing efficiency of simple primitive cell =}52.\text{4}\text{\%}\hfill \\ \text{Packing efficiency of BCC unit cell =}68\text{\%}\hfill \\ \text{Packing efficiency of FCC unit cell =}\text{74}\text{\%}\hfill \end{array}$

More dense is FCC and least dense is simple primitive

Explanation of Solution

Calculate the packing efficiency of primitive cell

$\begin{array}{l}\text{volume}\text{of}\text{the}\text{sphere}\text{=}\frac{\text{4}}{\text{3}}{\text{Πr}}^{\text{3}}\\ \text{Volume of the cube =}{\left(\text{2r}\right)}^{\text{3}}\\ \text{One unit cell of the simple cubic contains only one atom therefore,}\\ \text{volume}\text{of}\text{four}\text{atoms}\text{=}1\text{×}\frac{\text{4}}{\text{3}}{\text{Πr}}^{\text{3}}\\ \text{The packing efficiency can be calculated }\\ \text{by the percent of space occupied by spheres present in a unit cell}\text{.}\\ \text{Packing efficiency}\text{=}\frac{\text{volumeoccupied}\text{by}\text{the}\text{spheres}\text{present}\text{in}\text{unit}\text{cell}}{\text{Total}\text{volume}\text{of}\text{the}\text{unit}\text{cell}}\\ \text{Packing efficiency}\text{=}\frac{\text{1×}\frac{\text{4}}{\text{3}}{\text{Πr}}^{\text{3}}}{{\left(\text{2r}\right)}^{\text{3}}}\text{×}\text{100}\\ \text{Packing efficiency}\text{of}\text{FCC=}52.4\text{\%}\end{array}$

Calculate the packing efficiency of FCC

One unit cell of the fcc contains four atoms therefore,

$\begin{array}{l}\text{volume}\text{of}\text{the}\text{sphere}\text{=}\frac{\text{4}}{\text{3}}{\text{Πr}}^{\text{3}}\\ \text{Volume of the cube =}{\left(\text{2r}\sqrt{\text{2}}\right)}^{\text{3}}\\ \text{One unit cell of the fcc contains four atoms therefore,}\\ \text{volume}\text{of}\text{four}\text{atoms}\text{=}\text{4×}\frac{\text{4}}{\text{3}}{\text{Πr}}^{\text{3}}\\ \text{The packing efficiency can be calculated }\\ \text{by the percent of space occupied by spheres present in a unit cell}\text{.}\\ \text{Packing efficiency}\text{=}\frac{\text{volumeoccupied}\text{by}\text{the}\text{spheres}\text{present}\text{in}\text{unit}\text{cell}}{\text{Total}\text{volume}\text{of}\text{the}\text{unit}\text{cell}}\\ \text{Packing efficiency}\text{=}\frac{\text{4×}\frac{\text{4}}{\text{3}}{\text{Πr}}^{\text{3}}}{{\left(\text{2r}\sqrt{\text{2}}\right)}^{\text{3}}}\text{×}\text{100}\\ \text{Packing efficiency}\text{of}\text{FCC=}\text{74}\text{\%}\end{array}$

Calculate the packing efficiency of BCC

$\begin{array}{l}\text{volume}\text{of}\text{the}\text{sphere}\text{=}\frac{\text{4}}{\text{3}}{\text{Πr}}^{\text{3}}\\ \text{Volume of the cube =}{\left(\frac{\text{4r}}{\sqrt{\text{3}}}\right)}^{\text{3}}\\ \text{One unit cell of the fcc contains two atoms therefore,}\\ \text{volume}\text{of}\text{four}\text{atoms}\text{=}\text{2×}\frac{\text{4}}{\text{3}}{\text{Πr}}^{\text{3}}\\ \text{The packing efficiency can be calculated }\\ \text{by the percent of space occupied by spheres present in a unit cell}\text{.}\\ \text{Packing efficiency}\text{=}\frac{\text{volumeoccupied}\text{by}\text{the}\text{spheres}\text{present}\text{in}\text{unit}\text{cell}}{\text{Total}\text{volume}\text{of}\text{the}\text{unit}\text{cell}}\\ \text{Packing efficiency}\text{=}\frac{\text{2×}\frac{\text{4}}{\text{3}}{\text{Πr}}^{\text{3}}}{{\left(\frac{\text{4r}}{\sqrt{\text{3}}}\right)}^{\text{3}}}\text{×}\text{100}\\ \text{Packing efficiency}\text{of}\text{BCC=}\text{68}\text{\%}\end{array}$

$\begin{array}{l}\text{Packing efficiency of simple primitive cell =}52.\text{4}\text{\%}\hfill \\ \text{Packing efficiency of BCC unit cell =}68\text{\%}\hfill \\ \text{Packing efficiency of FCC unit cell =}\text{74}\text{\%}\hfill \end{array}$

By comparing packing efficiency percentage of the above three unit cell more the percentage, more the dense, therefore,

More dense is FCC and least dense is simple primitive

Interpretation Introduction

Interpretation:

On comparing the percentage of occupied space in PC with BCC and FCC, most dense and least dense has to be identified.

Concept introduction:

$\begin{array}{l}\text{The packing efficiency can be calculated }\\ \text{by the percent of space occupied by spheres present in a unit cell}\text{.}\\ \text{packing efficiency}\text{=}\frac{\text{volumeoccupied}\text{by}\text{the}\text{spheres}\text{present}\text{in}\text{unit}\text{cell}}{\text{Total}\text{volume}\text{of}\text{the}\text{unit}\text{cell}}\text{×}\text{100}\end{array}$

${\text{Volume of the cube = a}}^{\text{3}}$

Answer to Problem 46GQ

More dense is FCC and least dense is simple primitive

Explanation of Solution

Percentage of occupied space within the unit cell is given below

$\begin{array}{l}\text{Packing efficiency of simple primitive cell =}52.\text{4}\text{\%}\hfill \\ \text{Packing efficiency of BCC unit cell =}68\text{\%}\hfill \\ \text{Packing efficiency of FCC unit cell =}\text{74}\text{\%}\hfill \end{array}$

By comparing packing efficiency percentage of the above three unit cell more the percentage, more the dense, therefore,

More dense is FCC and least dense is simple primitive