Cálculo do Fator de Empacotamento CFC e CS - Exercícios Resolvidos Callister (07)

Explica Professor!

2 Nov 202009:08

Summary

TLDRThis educational video script focuses on exercises related to the calculation of the unit cell volume and packing factor of simple cubic and face-centered cubic crystal structures. The instructor guides through determining the unit cell volume in relation to atomic radius and explains the concept of packing factor by comparing the volume occupied by atoms to the cell volume. The script includes visual aids and 3D models to clarify the number of atoms within the unit cells and their arrangement, ultimately revealing the packing efficiency of these crystalline structures.

Takeaways

🔬 The lecture focuses on exercises related to the calculation of the unit cell volume and packing factor of simple cubic and face-centered cubic structures.
📐 In a simple cubic structure, the unit cell volume is calculated to be 8R^3, where R is the atomic radius.
📊 The packing factor for a simple cubic structure is approximately 0.52, indicating a lower density of atomic packing.
🔍 For a face-centered cubic structure, the unit cell volume is determined to be 16R^3, with R being the atomic radius.
🔢 The packing factor in a face-centered cubic structure is calculated to be around 0.74, which is higher than that of a simple cubic structure, indicating a denser atomic arrangement.
🧠 The number of atoms in a simple cubic unit cell is one, while in a face-centered cubic unit cell, it's calculated to be four.
📚 The lecture emphasizes the importance of understanding the theoretical part of the material before attempting exercises for better comprehension.
📖 The script mentions that the theoretical part of the lecture is available in a separate video, which is recommended for viewing to aid in understanding the exercises.
💡 The exercises are designed to help students understand the relationship between atomic radius, unit cell volume, and packing factor in different crystal structures.
👨‍🏫 The instructor uses visual aids and 3D models to explain the atomic arrangement within the unit cells and how it affects the packing factor.
🤔 The lecture concludes with an invitation for students to ask questions if they have any doubts, encouraging interaction and clarification.

Q & A

What is the main topic of the video?
-The main topic of the video is the calculation of the unit cell volume and the packing factor of simple cubic and face-centered cubic structures in materials science.
What is the relationship between the edge of a simple cubic cell and the atomic radius?
-The edge of a simple cubic cell is twice the atomic radius, as each edge of the cube is defined by the distance between two atomic centers.
How is the unit cell volume of a simple cubic structure calculated?
-The unit cell volume of a simple cubic structure is calculated by cubing the edge length, which is 2R where R is the atomic radius, resulting in 8R^3.
What is the packing factor of a simple cubic structure?
-The packing factor of a simple cubic structure is calculated by dividing the volume occupied by the atoms (using the rigid sphere model) by the unit cell volume. For a simple cubic structure, it is (4/3)πR^3/8R^3 = π/6 ≈ 0.52.
What is the difference between a simple cubic and a face-centered cubic structure?
-In a simple cubic structure, each corner of the cube contains one atom, while in a face-centered cubic structure, atoms are also located at the center of each face of the cube.
How is the unit cell volume of a face-centered cubic structure related to the atomic radius?
-The unit cell volume of a face-centered cubic structure is related to the atomic radius by the formula V = (16/3)a^3, where a is the edge length of the cube, and a = 2R for the face-centered cubic structure.
What is the number of atoms per unit cell in a face-centered cubic structure?
-In a face-centered cubic structure, there are 4 atoms per unit cell, with one atom at each corner and one atom at the center of each face of the cube.
How is the packing factor calculated for a face-centered cubic structure?
-The packing factor for a face-centered cubic structure is calculated by dividing the total volume of the atoms (considering each as a sphere of radius R) by the unit cell volume, which results in a packing factor of approximately 0.74.
Why is the packing factor lower in a simple cubic structure compared to a face-centered cubic structure?
-The packing factor is lower in a simple cubic structure because the atoms are less efficiently packed, with more empty space between them compared to the face-centered cubic structure where atoms are located at the centers of the cube faces, leading to a denser packing.
What does the video suggest about the compactness of metals with a simple cubic structure?
-The video suggests that metals with a simple cubic structure are not very compact, as indicated by the low packing factor, which is not typical for metals where a more efficient packing is usually observed.

Outlines

00:00

🔬 Calculation of Unit Cell Volume and Packing Factor in Simple Cubic Structures

This paragraph introduces a science materials exercise focused on calculating the unit cell volume and packing factor for simple cubic structures without face-centered cubic (FCC) characteristics. The speaker emphasizes the importance of understanding the theoretical lesson available in a companion card and provides a step-by-step guide to determine the unit cell volume in relation to the atomic radius (r), using the formula V = 8 * (4/3) * π * r^3. The explanation includes the concept of the packing factor, which is the volume occupied by the atoms divided by the unit cell volume, and is calculated for a simple cubic structure with one atom per cell, resulting in a low packing factor of approximately 0.52, indicating a non-compact arrangement typical for metals.

05:01

📐 Determining Unit Cell Volume and Packing Factor for Body-Centered Cubic Structures

The second paragraph delves into the calculations for body-centered cubic (BCC) structures, starting with the determination of the unit cell volume in relation to the atomic radius (r), using the formula V = a^3, where a is the length of the cube's edge and is related to r by the equation a = 4r. The speaker then explains the process of calculating the packing factor for BCC structures, which involves counting the number of atoms within the cell and their respective volumes. The summary of atoms includes one-eighth of an atom at each of the eight corners and one-half of an atom at the center of the cube, leading to a total of 2 atoms per unit cell. The packing factor is then calculated by dividing the total volume of these atoms by the unit cell volume, resulting in a value of approximately 0.68, which is higher than that of a simple cubic structure, indicating a more compact arrangement.

Mindmap

Keywords

💡Material Science

Material Science is an interdisciplinary field involving the properties of matter and its applications to various areas of science and engineering. It is the main theme of the video, as the script discusses the calculation of volume and packing factors in crystalline structures, which are essential concepts in understanding material properties.

💡Unit Cell

A unit cell is the smallest repeating unit in a crystal lattice that can be used to describe the atomic arrangement in a crystalline material. The script focuses on calculating the volume of a unit cell for different types of crystal structures, which is crucial for understanding the material's density and atomic packing.

💡Atomic Radius (R)

The atomic radius (R) is a measure of the size of an atom, typically defined as the distance from the nucleus to the outermost shell of electrons. In the script, the atomic radius is used to calculate the volume of the unit cell in simple cubic and face-centered cubic structures.

💡Simple Cubic Structure

A simple cubic structure is a type of crystal structure where each lattice point is occupied by one atom, and atoms are dispersed at the corners of a cube. The script explains how to calculate the volume of the unit cell and the packing factor for this structure.

💡Packing Factor

The packing factor, also known as the packing efficiency, is a measure of how efficiently space is filled by atoms in a crystal structure. The script discusses calculating the packing factor for simple cubic and face-centered cubic structures to understand the compactness of the atomic arrangement.

💡Face-Centered Cubic (FCC) Structure

A face-centered cubic structure is a crystal lattice where each face of the cube is centered with an atom, in addition to the atoms at the corners. The script provides an example of calculating the volume of the unit cell and the packing factor for an FCC structure.

💡Lattice Parameter

A lattice parameter is a fundamental dimension of a unit cell in a crystal lattice, typically the length of the edges or the angles between them. In the script, the lattice parameter is used to relate the unit cell's volume to the atomic radius.

💡Rigid Sphere Model

The rigid sphere model is a simplistic representation of atoms as incompressible spheres that pack together to form a crystal lattice. The script uses this model to calculate the volume occupied by atoms and, subsequently, the packing factor.

💡Volume Calculation

Volume calculation is a mathematical process used to determine the space occupied by atoms within a unit cell. The script demonstrates how to calculate the volume of the unit cell and the volume occupied by atoms in different crystal structures.

💡Crystalline Structures

Crystalline structures refer to the ordered arrangement of atoms, ions, or molecules in a crystalline material. The video's script explores the calculation of unit cell volume and packing factors for different crystalline structures, which are key to understanding material properties.

💡Metallic Packing

Metallic packing refers to the way atoms are arranged in a metal's crystal lattice. The script mentions that metals typically do not exhibit a high packing factor, indicating that their atomic arrangement is not as compact as in some other materials.

Highlights

Introduction to the exercise on calculating unit cell volume and packing factor for crystal structures.

Explanation of the theoretical lesson on the topic, available in a previous video.

The task is to determine the volume of a simple cubic unit cell as a function of atomic radius.

Detailed steps to find the volume of a simple cubic unit cell by relating the lattice parameter to the atomic radius.

Introduction of the concept of the packing factor, defined as the volume occupied by atoms in a unit cell divided by the volume of the unit cell.

The realization that the simple cubic structure is not common in nature, especially in metals, due to its low packing efficiency.

Calculation of the packing factor for the simple cubic structure, resulting in a low value of 0.52.

Transition to the second exercise, focusing on a face-centered cubic (FCC) structure.

Calculation of the unit cell volume for the FCC structure by relating the atomic radius to the lattice parameter.

Introduction to the more complex process of determining the packing factor for the FCC structure, which involves counting the number of atoms per unit cell.

Explanation of how to visualize the distribution of atoms in the FCC unit cell, including a 3D representation.

The packing factor for the FCC structure is calculated to be 0.74, higher than that of the simple cubic structure.

The FCC structure's higher packing factor is explained as being more representative of metals due to better atomic packing.

Final summary of the differences between simple cubic and FCC structures in terms of packing efficiency.

Encouragement to review the video again if there are any doubts and a request to like and share the content.