Respuesta :
Answer:
(a). The distance from the center of one silicon atom to the center of its nearest neighbor is [tex]2.35\ \AA[/tex]
(b). The number density of silicon atoms is [tex]4.99\times10^{22}\ cm^{3}[/tex].
(c). The mass density is [tex]2.33\ g/cm^{3}[/tex]
Explanation:
Given that,
Lattice constant of silicon [tex]a= 5.43\ \AA[/tex]
We know that,
The nearest atom distance is
[tex]d=\dfrac{\sqrt{3}a}{4}[/tex]
The radius of each atom is
[tex]r=\dfrac{d}{2}[/tex]
Put the value into the formula
[tex]r=\dfrac{\dfrac{\sqrt{3}a}{4}}{2}[/tex]
[tex]r=\dfrac{\sqrt{3}\times5.43\times10^{-10}}{8}[/tex]
[tex]r=1.175\ \AA[/tex]
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
[tex]\text{nearest neighbor distance}=2\times1.175[/tex]
[tex]\text{nearest neighbor distance}=2.35\ \AA[/tex]
(b). We need to calculate the number density of silicon atoms
Using formula for the number density of silicon atoms
[tex]\text{Number density}=\dfrac{\text{number of atoms per unit cell}}{\text{Volume of unit cell}}[/tex]
[tex]\text{Number density}=\dfrac{8}{(0.543\times10^{-7})^3}[/tex]
[tex]\text{Number density}=4.99\times10^{22}\ cm^{3}[/tex]
(c). We need to calculate the number of atoms
Using formula for number of atoms
[tex]\text{number of atoms}=\dfrac{\text{Avogadro number}}{\text{moleculer weight}}[/tex]
Put the value into the formula
[tex]\text{number of atoms}=\dfrac{6.02\times10^{23}}{28.09}[/tex]
[tex]\text{number of atoms}=2.143\times10^{22}[/tex]
We need to calculate the mass density
Using formula of mass density
[tex]\text{mass density}=\dfrac{\text{number density}}{\text{number of atoms}}[/tex]
Put the value into the formula
[tex]\text{mass density}=\dfrac{4.99\times10^{22}}{2.143\times10^{22}}[/tex]
[tex]\text{mass density}=2.33\ g/cm^{3}[/tex]
Hence, (a). The distance from the center of one silicon atom to the center of its nearest neighbor is [tex]2.35\ \AA[/tex]
(b). The number density of silicon atoms is [tex]4.99\times10^{22}\ cm^{3}[/tex].
(c). The mass density is [tex]2.33\ g/cm^{3}[/tex]
The distance between the center of the silicon atom is 2.35 Å, the number density of silicon atoms is 4.99 × 10²² cm³, the mass density is 2.33 g/cm³.
What is a crystalline lattice?
The symmetrical 3-D structural arrangement of the ions, atoms, or molecules inside a crystalline lattice solid as a point.
The lattice constant of silicon is 5.43 Å.
Then the nearest distance of the atom will be
[tex]\rm d = \dfrac{\sqrt{3}a}{4}[/tex]
The radius of each atom will be
[tex]\rm r = \dfrac{d}{2}\\\\d = 2r[/tex]
Then we have
[tex]\rm r = \dfrac{ \dfrac{\sqrt{3}a}{4}}{2}\\\\\\r = \dfrac{\sqrt{3}*5.43*10^{-10}}{8} \\\\\\r = 1.175 \ \AA[/tex]
A. The center of one silicon atom to the center of the nearest neighbor atom is twice the radius of the silicon atom.
[tex]\rm d = 2\ r\\\\d = 2 * 1.175\\\\d = 2.35 \AA[/tex]
B. The number density of the silicon atoms will be
[tex]\rm Number \ density = \dfrac{number \ of \ atoms \ per\ unit \ cell}{Volume of unit cell}\\\\\\Number \ density = \dfrac{8}{0.543*10^{-7}}^3\\\\\\Number \ density = 4.99*10^{22}\ \ cm^3[/tex]
C. The number of atoms will be
[tex]\rm Number\ of\ atoms = \dfrac{Avogadro \ number }{molecular \ weight}\\\\Number\ of\ atoms =\dfrac{6.023*10^{23}}{28.09}\\\\Number\ of\ atoms = 2.143*10^{22}[/tex]
Then the mass density will be
[tex]\rm Mass \ density = \dfrac{number\ density}{number\ of\ atoms}\\\\Mass \ density = \dfrac{4.99*10^{22}}{2.143*10^{22}}\\\\Mass \ density = 2.33\ \ g/cm^3[/tex]
More about the crystalline lattice link is given below.
https://brainly.com/question/10951564
