Chapter 12, Problem 51GQ

The solid-state structure of silicon is shown below.

Unit cell for silicon

1. (a) Describe this crystal as pc, bcc, or fcc.
2. (b) What type of holes are occupied in the lattice?
3. (c) How many Si atoms are there per unit cell?
4. (d) Calculate the density of silicon in g/cm³ (given that the cube edge has a length of 543.1 pm).
5. (e) Estimate the radius of the silicon atom. (Note: The Si atoms on the edges do not touch one another.)

Interpretation Introduction

Interpretation:

Given silicon crystal has to be described for PC, BCC or FCC.

Concept introduction:

• An ionic radii are the radius of an atom's ion in ionic crystals structure.
• An ionic solid is made up cations and anions held together by electrostatic forces in a rigid array or lattice.
• Positive charge ions are cations and negative charge ions are anions.
• Lattice Energy is mainly depends on the charge on the ion and radius or size of the ion.
• Ionic radius increases from top to bottom on the periodic table.

  Ionic radius decreases from left to right the periodic table.

• The density of the unit cell:

  $\frac{\text{Mass}}{\text{volume}}\text{ = density}$

• The face-centered cubic system:

  It has lattice points on the faces of the cube, that each gives exactly one half contributions, in addition to the corner lattice points, giving a total of 4 lattice points per unit cell

Answer to Problem 51GQ

The types of the lattice is face center cubic crystal

Explanation of Solution

The types of the lattice is face center cubic crystal, because It has lattice points on the faces of the cube, that each gives exactly one half contribution, in addition to the corner lattice points, giving a total of 4 lattice points per unit cell

Interpretation Introduction

Interpretation:

The types of holes that are occupied in the lattice has to be determined.

Concept introduction:

• An ionic radii are the radius of an atom's ion in ionic crystals structure.
• An ionic solid is made up cations and anions held together by electrostatic forces in a rigid array or lattice.
• Positive charge ions are cations and negative charge ions are anions.
• Lattice Energy is mainly depends on the charge on the ion and radius or size of the ion.
• Ionic radius increases from top to bottom on the periodic table.

  Ionic radius decreases from left to right the periodic table.

• The density of the unit cell:

  $\frac{\text{Mass}}{\text{volume}}\text{ = density}$

• The face-centered cubic system:

  It has lattice points on the faces of the cube, that each gives exactly one half contribution, in addition to the corner lattice points, giving a total of 4 lattice points per unit cell

Answer to Problem 51GQ

Silicon atoms are located in one half of the tetrahedral holes

Explanation of Solution

Silicon atoms are located in one half of the tetrahedral holes.

Interpretation Introduction

Interpretation:

Number of silicon atoms per unit cell has to be determined.

Concept introduction:

• An ionic radii are the radius of an atom's ion in ionic crystals structure.
• An ionic solid is made up cations and anions held together by electrostatic forces in a rigid array or lattice.
• Positive charge ions are cations and negative charge ions are anions.
• Lattice Energy is mainly depends on the charge on the ion and radius or size of the ion.
• Ionic radius increases from top to bottom on the periodic table.

  Ionic radius decreases from left to right the periodic table.

• The density of the unit cell:

  $\frac{\text{Mass}}{\text{volume}}\text{ = density}$

• The face-centered cubic system:

  It has lattice points on the faces of the cube, that each gives exactly one half contribution, in addition to the corner lattice points, giving a total of 4 lattice points per unit cell

Answer to Problem 51GQ

Totally eight atoms in the unit cell

Explanation of Solution

Silicon atoms are located in one half of the tetrahedral holes.

Interpretation Introduction

Interpretation:

The density of the silicon atom has to be identified.

Concept introduction:

• An ionic radii are the radius of an atom's ion in ionic crystals structure.
• An ionic solid is made up cations and anions held together by electrostatic forces in a rigid array or lattice.
• Positive charge ions are cations and negative charge ions are anions.
• Lattice Energy is mainly depends on the charge on the ion and radius or size of the ion.
• Ionic radius increases from top to bottom on the periodic table.

  Ionic radius decreases from left to right the periodic table.

• The density of the unit cell:

  $\frac{\text{Mass}}{\text{volume}}\text{ = density}$

• The face-centered cubic system:

  It has lattice points on the faces of the cube, that each gives exactly one half contribution, in addition to the corner lattice points, giving a total of 4 lattice points per unit cell

Answer to Problem 51GQ

Density is $\text{2}{\text{.329 g/cm}}^{\text{3}}$

Explanation of Solution

The density of the silicon atom is given below,

$\begin{array}{l}\frac{\text{8 Si atoms}}{\text{unit cell}}\text{ × }\frac{\text{28}\text{.086 g}}{\text{1 mol Si}}\text{ ×}\frac{\text{1 mol Ir}}{\text{6}{\text{.0221 × 10}}^{\text{23}}\text{ atoms}}\\ \text{= 3}{\text{.731×10}}^{\text{–22}}\text{g/unit cell}\\ \text{volume}\text{of}\text{the}\text{unit}\text{cell}\text{=}{\text{a}}^{\text{3}}\\ \text{volume}\text{of}\text{the}\text{unit}\text{cell}\text{=}{(5.431\times {10}^{-8}\text{cm})}^3\text{= 1}{\text{.602×10}}^{\text{–22}}{\text{cm}}^{\text{3}}\\ \text{Determining the density of the unit cell:}\\ \frac{\text{Mass}}{\text{volume}}\text{ = density}\\ \frac{\text{3}{\text{.731×10}}^{\text{–22}}\text{g}}{\text{1}{\text{.602×10}}^{\text{–22}}{\text{cm}}^{\text{3}}}\text{ = density}\\ \text{density = 2}{\text{.329 g/cm}}^{\text{3}}\end{array}$

Interpretation Introduction

Interpretation:

The radius of the silicon atom has to be identified.

Concept introduction:

• An ionic radii are the radius of an atom's ion in ionic crystals structure.
• An ionic solid is made up cations and anions held together by electrostatic forces in a rigid array or lattice.
• Positive charge ions are cations and negative charge ions are anions.
• Lattice Energy is mainly depends on the charge on the ion and radius or size of the ion.
• Ionic radius increases from top to bottom on the periodic table.

  Ionic radius decreases from left to right the periodic table.

• The density of the unit cell:

  $\frac{\text{Mass}}{\text{volume}}\text{ = density}$

• The face-centered cubic system:

  It has lattice points on the faces of the cube, that each gives exactly one half contribution, in addition to the corner lattice points, giving a total of 4 lattice points per unit cell

Answer to Problem 51GQ

Radius is $\text{117}\text{.6}\text{pm}$.

Explanation of Solution

 The radius of silicon atom can be calculated as,

$\begin{array}{l}\text{ face diagonal = }\sqrt{\text{2}}\text{×edge length =}\sqrt{\text{2}}\text{× 543}\text{.1 pm }\\ \text{ace diagonal = 768}\text{.1 pm }\\ \text{ ½ face diagonal = 384}\text{.1 pm }\\ \text{ sin}\left(\text{109}\text{.5º/2}\right)\text{ = }\frac{{}^{\text{1}}{\text{/}}_{\text{2}}\text{(384}\text{.1 pm)}}{\text{Si-Si distance}}\\ \text{Si-Si distance = 235}\text{.1 pm }\\ \text{ Si radius = 117}\text{.6 pm}\end{array}$