Question

If the lattice constant of silicon is 5.43 Å, calculate?

Answer 1

Answer:

(a). The distance from the center of one silicon atom to the center of its nearest neighbor is $2.35\ \AA$

(b). The number density of silicon atoms is $4.99\times10^{22}\ cm^{3}$.

(c). The mass density is $2.33\ g/cm^{3}$

Explanation:

Given that,

Lattice constant of silicon $a= 5.43\ \AA$

We know that,

The nearest atom distance is

$d=\dfrac{\sqrt{3}a}{4}$

The radius of each atom is

$r=\dfrac{d}{2}$

Put the value into the formula

$r=\dfrac{\dfrac{\sqrt{3}a}{4}}{2}$

$r=\dfrac{\sqrt{3}\times5.43\times10^{-10}}{8}$

$r=1.175\ \AA$

The center of the one silicon atom to the center of its nearest neighbor is equal to the twice the radius of the atom

(a). We need to calculate the nearest neighbor distance

Using formula of distance

$\text{nearest neighbor distance}=2\times1.175$

$\text{nearest neighbor distance}=2.35\ \AA$

(b). We need to calculate the number density of silicon atoms

Using formula for the number density of silicon atoms

$\text{Number density}=\dfrac{\text{number of atoms per unit cell}}{\text{Volume of unit cell}}$

$\text{Number density}=\dfrac{8}{(0.543\times10^{-7})^3}$

$\text{Number density}=4.99\times10^{22}\ cm^{3}$

(c). We need to calculate the number of atoms

Using formula for number of atoms

$\text{number of atoms}=\dfrac{\text{Avogadro number}}{\text{moleculer weight}}$

Put the value into the formula

$\text{number of atoms}=\dfrac{6.02\times10^{23}}{28.09}$

$\text{number of atoms}=2.143\times10^{22}$

We need to calculate the mass density

Using formula of mass density

$\text{mass density}=\dfrac{\text{number density}}{\text{number of atoms}}$

Put the value into the formula

$\text{mass density}=\dfrac{4.99\times10^{22}}{2.143\times10^{22}}$

$\text{mass density}=2.33\ g/cm^{3}$

Hence, (a). The distance from the center of one silicon atom to the center of its nearest neighbor is $2.35\ \AA$

(b). The number density of silicon atoms is $4.99\times10^{22}\ cm^{3}$.

(c). The mass density is $2.33\ g/cm^{3}$