Number of atoms ∈ a unit cell (n) = (1/8) × number of corner atoms + (1/2) × number of face-centered atoms + 1 × number of body-centered atoms
Density (ρ) = (n × M)/(N × a³)
a = (n × M)/(N × ρ)¹/³
Interplanar distance (d) = a/(h² + k² + l²)¹/²
Radius ratio (r⁺/r⁻) = (r⁺ + r⁻)/(2 × r⁻) for tetrahedral void
r⁺/r⁻ = (r⁺ + r⁻)/(4 × r⁻) for octahedral void
Packing efficiency = (π × n × r³)/(3 × √2 × a³) for fcc
Packing efficiency = (π × n × r³)/(3 × √3 × a³) for hcp
Length of the unit cell (a) = 2 × (r⁺ + r⁻) for ionic compounds

🪤 The 5 Mistakes That Cost Marks

Not understanding the concept of packing efficiency and its relation to the crystal structure
Incorrectly calculating the number of atoms ∈ a unit cell
Not knowing the difference between fcc and hcp structures
Forgetting to consider the voids ∈ the crystal lattice
Incorrectly applying the formula for interplanar distance

✏️ 3 Solved PYQs

Calculate the number of atoms ∈ a unit cell of a metal with a face-centered cubic structure. Step 1: Identify the number of corner atoms, face-centered atoms, and body-centered atoms. Step 2: Apply the formula n = (1/8) × number of corner atoms + (1/2) × number of face-centered atoms + 1 × number of body-centered atoms. Step 3: Calculate the value of n.
The ionic radii of Na⁺ and Cl⁻ are 95 ± and 181 ± respectively. Calculate the ratio of the radius of the cation to that of the anion. Step 1: Identify the given values of the ionic radii. Step 2: Calculate the ratio of the radius of the cation to that of the anion using the formula r⁺/r⁻.
A solid has a bcc structure with a cell edge length of 200 ±. If the density of the solid is 5 g/cm³, calculate the number of atoms ∈ the unit cell. Step 1: Identify the given values of the cell edge length and density. Step 2: Apply the formula ρ = (n × M)/(N × a³) to calculate the value of n.

🧠 The One Thing Most Students Get Wrong

Most students struggle with understanding the concept of crystal structures, specifically the difference between fcc and hcp structures, and how to calculate the packing efficiency.
The key to concept is to practice calculating the packing efficiency for different crystal structures and to understand the relationship between the crystal structure and the physical properties of the solid.

👁️ Ayush's Note

To solve problems related to solid state, it is essential to have a strong understanding of the concepts of crystal structures, packing efficiency, and the relationship between the crystal structure and the physical properties of the solid.
Practice is key to mastering these concepts, so make sure to practice a variety of problems, including those that involve calculating the number of atoms ∈ a unit cell, the packing efficiency, and the interplanar distance.
For JEE Advanced and NEET level shortcuts, focus on understanding the underlying concepts and principles, rather than just memorizing formulas and equations.

🔁 Last 5 Minutes Box

Quickly review the formulas for calculating the number of atoms ∈ a unit cell, packing efficiency, and interplanar distance.
Make sure to understand the difference between fcc and hcp structures and how to calculate the packing efficiency for each.
Practice calculating the radius ratio and the length of the unit cell for ionic compounds.
Review the concepts of crystal structures and their relationship to the physical properties of the solid.

📝 Practice MCQs

1. What is the number of atoms ∈ a unit cell of a metal with a face-centered cubic structure?

A) 2

B) 4

C) 6

D) 8

Answer: B) 4. Explanation: In a face-centered cubic structure, there are 8 corner atoms, each shared by 8 unit cells, and 6 face-centered atoms, each shared by 2 unit cells. Therefore, the total number of atoms ∈ the unit cell is (1/8) × 8 + (1/2) × 6 = 4.

2. What is the packing efficiency of a crystal with a simple cubic structure?

A) 34%

B) 52%

C) 68%

D) 74%

Answer: B) 52%. Explanation: The packing efficiency of a crystal with a simple cubic structure is given by the formula (π × n × r³)/(3 × a³), where n is the number of atoms ∈ the unit cell, r is the radius of the atom, and an is the length of the unit cell. For a simple cubic structure, n = 1 and a = 2r, so the packing efficiency is (π × 1 × r³)/(3 × (2r)³) = π/6 = 0.52 or 52%.

3. What is the ratio of the radius of the cation to that of the anion ∈ an ionic compound with a radius ratio of 0.5?

A) 0.5

B) 1.0

C) 1.5

D) 2.0

Answer: A) 0.5. Explanation: The radius ratio is given by the formula r⁺/r⁻, where r⁺ is the radius of the cation and r⁻ is the radius of the anion. Therefore, if the radius ratio is 0.5, the ratio of the radius of the cation to that of the anion is 0.5.

4. What is the length of the unit cell of a crystal with a face-centered cubic structure and a packing efficiency of 74%?

A) 200 ±

B) 400 ±

C) 600 ±

D) 800 ±

Answer: B) 400 ±. Explanation: The packing efficiency of a crystal with a face-centered cubic structure is given by the formula (π × n × r³)/(3 × √2 × a³), where n is the number of atoms ∈ the unit cell, r is the radius of the atom, and an is the length of the unit cell. For a face-centered cubic structure, n = 4 and the packing efficiency is 74%, so we can rearrange the formula to solve for a.

5. What is the number of atoms ∈ a unit cell of a crystal with a hexagonal close-packed structure?

A) 2

B) 4

C) 6

D) 8

Answer: B) 4. Explanation: In a hexagonal close-packed structure, there are 6 corner atoms, each shared by 6 unit cells, and 2 face-centered atoms, each shared by 2 unit cells, and 3 atoms ∈ the center of the unit cell. Therefore, the total number of atoms ∈ the unit cell is (1/6) × 6 + (1/2) × 2 + 3 = 4.

🚀 Ready to Ace Your Exam?

Put your knowledge to the test! Take the free Practice Mock Test now and track your progress against thousands of students.

📚 Academic References

Content verified against peer-reviewed research:

Selected Performance Indicators of University-Model Schools — Aquila Digital Community (University of Southern Mississippi) (2019) 🔓 — DOI ↗
All Of Chinese Literature Condensed: A Sourcebook From The Playwr... — Journal of International Crisis and Risk Communication Research (2013) 🔓 — DOI ↗

🔓 = Open Access article

🎬 Watch video explanations on YouTube →

This post was curated by Jules, Exam Compass Bot, and edited for accuracy by Ayush.

📚 Related Topics

Continue your revision with these related guides:

Number of atoms ∈ a unit cell (n) = (1/8) × number of corner atoms + (1/2) × number of face-centered atoms + 1 × number of body-centered atoms
Density (ρ) = (n × M)/(N × a³)
a = (n × M)/(N × ρ)¹/³
Interplanar distance (d) = a/(h² + k² + l²)¹/²
Radius ratio (r⁺/r⁻) = (r⁺ + r⁻)/(2 × r⁻) for tetrahedral void
r⁺/r⁻ = (r⁺ + r⁻)/(4 × r⁻) for octahedral void
Packing efficiency = (π × n × r³)/(3 × √2 × a³) for fcc
Packing efficiency = (π × n × r³)/(3 × √3 × a³) for hcp
Length of the unit cell (a) = 2 × (r⁺ + r⁻) for ionic compounds

🪤 The 5 Mistakes That Cost Marks

Not understanding the concept of packing efficiency and its relation to the crystal structure
Incorrectly calculating the number of atoms ∈ a unit cell
Not knowing the difference between fcc and hcp structures
Forgetting to consider the voids ∈ the crystal lattice
Incorrectly applying the formula for interplanar distance

✏️ 3 Solved PYQs

Calculate the number of atoms ∈ a unit cell of a metal with a face-centered cubic structure. Step 1: Identify the number of corner atoms, face-centered atoms, and body-centered atoms. Step 2: Apply the formula n = (1/8) × number of corner atoms + (1/2) × number of face-centered atoms + 1 × number of body-centered atoms. Step 3: Calculate the value of n.
The ionic radii of Na⁺ and Cl⁻ are 95 ± and 181 ± respectively. Calculate the ratio of the radius of the cation to that of the anion. Step 1: Identify the given values of the ionic radii. Step 2: Calculate the ratio of the radius of the cation to that of the anion using the formula r⁺/r⁻.
A solid has a bcc structure with a cell edge length of 200 ±. If the density of the solid is 5 g/cm³, calculate the number of atoms ∈ the unit cell. Step 1: Identify the given values of the cell edge length and density. Step 2: Apply the formula ρ = (n × M)/(N × a³) to calculate the value of n.

🧠 The One Thing Most Students Get Wrong

Most students struggle with understanding the concept of crystal structures, specifically the difference between fcc and hcp structures, and how to calculate the packing efficiency.
The key to concept is to practice calculating the packing efficiency for different crystal structures and to understand the relationship between the crystal structure and the physical properties of the solid.

👁️ Ayush's Note

To solve problems related to solid state, it is essential to have a strong understanding of the concepts of crystal structures, packing efficiency, and the relationship between the crystal structure and the physical properties of the solid.
Practice is key to mastering these concepts, so make sure to practice a variety of problems, including those that involve calculating the number of atoms ∈ a unit cell, the packing efficiency, and the interplanar distance.
For JEE Advanced and NEET level shortcuts, focus on understanding the underlying concepts and principles, rather than just memorizing formulas and equations.

🔁 Last 5 Minutes Box

Quickly review the formulas for calculating the number of atoms ∈ a unit cell, packing efficiency, and interplanar distance.
Make sure to understand the difference between fcc and hcp structures and how to calculate the packing efficiency for each.
Practice calculating the radius ratio and the length of the unit cell for ionic compounds.
Review the concepts of crystal structures and their relationship to the physical properties of the solid.

📝 Practice MCQs

1. What is the number of atoms ∈ a unit cell of a metal with a face-centered cubic structure?

A) 2

B) 4

C) 6

D) 8

2. What is the packing efficiency of a crystal with a simple cubic structure?

A) 34%

B) 52%

C) 68%

D) 74%

3. What is the ratio of the radius of the cation to that of the anion ∈ an ionic compound with a radius ratio of 0.5?

A) 0.5

B) 1.0

C) 1.5

D) 2.0

4. What is the length of the unit cell of a crystal with a face-centered cubic structure and a packing efficiency of 74%?

A) 200 ±

B) 400 ±

C) 600 ±

D) 800 ±

5. What is the number of atoms ∈ a unit cell of a crystal with a hexagonal close-packed structure?

A) 2

B) 4

C) 6

D) 8

🚀 Ready to Ace Your Exam?

Put your knowledge to the test! Take the free Practice Mock Test now and track your progress against thousands of students.

📚 Academic References

Content verified against peer-reviewed research:

Selected Performance Indicators of University-Model Schools — Aquila Digital Community (University of Southern Mississippi) (2019) 🔓 — DOI ↗
All Of Chinese Literature Condensed: A Sourcebook From The Playwr... — Journal of International Crisis and Risk Communication Research (2013) 🔓 — DOI ↗

🔓 = Open Access article

🎬 Watch video explanations on YouTube →

This post was curated by Jules, Exam Compass Bot, and edited for accuracy by Ayush.

📚 Related Topics

Continue your revision with these related guides: