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Answer the following questions and show your complete solution. This means the studentt must include drawings or schematics if necessary. 1. Calculate the volume of a BCC unit cell in terms of the atomic radius R. 2. The atomic packing factor (APF) for a BCC unit cell is 0.68. Show how this value was obtained/calculated.

Answer the following questions and show your complete solution. This means the studentt must include drawings or schematics if necessary. 1. Calculate the volume of a BCC unit cell in terms of the atomic radius R. 2. The atomic packing factor (APF) for a BCC unit cell is 0.68. Show how this value was obtained/calculated.

Introduction to Chemical Engineering Thermodynamics
Introduction to Chemical Engineering Thermodynamics
8th Edition
ISBN:9781259696527
Author:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Publisher:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Chapter1: Introduction
Section: Chapter Questions
Problem 1.1P
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**Title: Calculating the Volume and Atomic Packing Factor of a BCC Unit Cell** **Instructions:** Answer the following questions and show your complete solution. This means the student must include drawings or schematics if necessary. **Problems:** 1. Calculate the volume of a BCC unit cell in terms of the atomic radius \( R \). 2. The atomic packing factor (APF) for a BCC unit cell is 0.68. Show how this value was obtained/calculated. **Detailed Explanation:** To solve these problems, you need to understand the structure of a Body-Centered Cubic (BCC) unit cell: - **BCC Unit Cell Structure:** - A BCC unit cell has atoms at each of its eight corners and one atom at the center of the cell. - **Volume Calculation:** - To find the volume of the unit cell in terms of the atomic radius \( R \), you first need to determine the relationship between the atomic radius and the edge length \( a \) of the unit cell. - In a BCC structure, the body diagonal of the cube is equal to \( 4R \) (the atomic radius), as it passes through the center atom from one corner of the cube to the opposite corner. - The body diagonal can also be expressed in terms of the edge length \( a \) using the Pythagorean theorem in three dimensions: \( \sqrt{3}a \). - Therefore, \( \sqrt{3}a = 4R \). - Solving for \( a \), we get \( a = \frac{4R}{\sqrt{3}} \). - The volume \( V \) of the cube is \( a^3 \), so \( V = \left(\frac{4R}{\sqrt{3}}\right)^3 \). - **Atomic Packing Factor (APF) Calculation:** - APF is the ratio of the volume occupied by atoms in the unit cell to the total volume of the unit cell. - In a BCC structure, there are 2 atoms per unit cell (one from the center atom and a total equivalent of one atom from the eight corner atoms, each contributing \( \frac{1}{8} \) of an atom). - The volume of one atom, considered as a sphere, is \( \frac{4}{3} \pi R^3 \).
Transcribed Image Text:**Title: Calculating the Volume and Atomic Packing Factor of a BCC Unit Cell** **Instructions:** Answer the following questions and show your complete solution. This means the student must include drawings or schematics if necessary. **Problems:** 1. Calculate the volume of a BCC unit cell in terms of the atomic radius \( R \). 2. The atomic packing factor (APF) for a BCC unit cell is 0.68. Show how this value was obtained/calculated. **Detailed Explanation:** To solve these problems, you need to understand the structure of a Body-Centered Cubic (BCC) unit cell: - **BCC Unit Cell Structure:** - A BCC unit cell has atoms at each of its eight corners and one atom at the center of the cell. - **Volume Calculation:** - To find the volume of the unit cell in terms of the atomic radius \( R \), you first need to determine the relationship between the atomic radius and the edge length \( a \) of the unit cell. - In a BCC structure, the body diagonal of the cube is equal to \( 4R \) (the atomic radius), as it passes through the center atom from one corner of the cube to the opposite corner. - The body diagonal can also be expressed in terms of the edge length \( a \) using the Pythagorean theorem in three dimensions: \( \sqrt{3}a \). - Therefore, \( \sqrt{3}a = 4R \). - Solving for \( a \), we get \( a = \frac{4R}{\sqrt{3}} \). - The volume \( V \) of the cube is \( a^3 \), so \( V = \left(\frac{4R}{\sqrt{3}}\right)^3 \). - **Atomic Packing Factor (APF) Calculation:** - APF is the ratio of the volume occupied by atoms in the unit cell to the total volume of the unit cell. - In a BCC structure, there are 2 atoms per unit cell (one from the center atom and a total equivalent of one atom from the eight corner atoms, each contributing \( \frac{1}{8} \) of an atom). - The volume of one atom, considered as a sphere, is \( \frac{4}{3} \pi R^3 \).
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