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Packing Efficiency Since the layering pattern in all of the lattices leaves empty space between the particles, the unit cell is not completely occupied by atoms (here we are treating atoms like hard spheres). The packing efficiency, which is the percentage of occupied space in the cube, is not 100%. The packing efficiency is not the same for all 3 cubic lattices. A more densely packed unit cell will have a higher packing efficiency than a less densely packed one. The packing efficiency of a lattice structure measures how well the space inside of a unit cell is utilized. It is the percent ratio of volume occupied by the particles in a unit cell to its total volume. Packing efficiency - (VoccupiedVista) X 100% The occupied volume (Vecoupied) is related to the number of particles occupying the cell and their location within the cell. The edge length of each unit cell is derived using trigonometric relationships (see Figure directly below) where / is the edge length and ris the radius of the sphere. Figure 2. Geometric relotionships showing how the edge length is related to the atomic radilus for simple cubic, body-centered cubic, ond foce-centered cubic unit cells Unit Cell Edge length in terms of radius Simple cubic I- 2r Body-centered cubic Face-centered cubic Voccupied- ( particles) xar Viotat = In a simple cubic unit cell, all the atoms are comer atoms. There are 8 comer atoms in a simple cubic unit cell. Remember from the lecture that for a corner atom, only 1/8th of the sphere is inside the unit cell. Hence, the total number of equivalent particle inside a simple cubic unit cell is 1 (because 1/8 times 8 atoms = 1). How many equivalent particles are inside a face-centered cubic unit cell? --> particles How many equivalent particles are inside a body-centered cubic unit cell? --> particles

Packing Efficiency Since the layering pattern in all of the lattices leaves empty space between the particles, the unit cell is not completely occupied by atoms (here we are treating atoms like hard spheres). The packing efficiency, which is the percentage of occupied space in the cube, is not 100%. The packing efficiency is not the same for all 3 cubic lattices. A more densely packed unit cell will have a higher packing efficiency than a less densely packed one. The packing efficiency of a lattice structure measures how well the space inside of a unit cell is utilized. It is the percent ratio of volume occupied by the particles in a unit cell to its total volume. Packing efficiency - (VoccupiedVista) X 100% The occupied volume (Vecoupied) is related to the number of particles occupying the cell and their location within the cell. The edge length of each unit cell is derived using trigonometric relationships (see Figure directly below) where / is the edge length and ris the radius of the sphere. Figure 2. Geometric relotionships showing how the edge length is related to the atomic radilus for simple cubic, body-centered cubic, ond foce-centered cubic unit cells Unit Cell Edge length in terms of radius Simple cubic I- 2r Body-centered cubic Face-centered cubic Voccupied- ( particles) xar Viotat = In a simple cubic unit cell, all the atoms are comer atoms. There are 8 comer atoms in a simple cubic unit cell. Remember from the lecture that for a corner atom, only 1/8th of the sphere is inside the unit cell. Hence, the total number of equivalent particle inside a simple cubic unit cell is 1 (because 1/8 times 8 atoms = 1). How many equivalent particles are inside a face-centered cubic unit cell? --> particles How many equivalent particles are inside a body-centered cubic unit cell? --> particles

Chemistry & Chemical Reactivity
Chemistry & Chemical Reactivity
10th Edition
ISBN:9781337399074
Author:John C. Kotz, Paul M. Treichel, John Townsend, David Treichel
Publisher:John C. Kotz, Paul M. Treichel, John Townsend, David Treichel
Chapter12: The Solid State
Section: Chapter Questions
Problem 50GQ: Consider the three types of cubic units cells. (a) Assuming that the spherical atoms or ions in a...
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QUESTION 5 Packing Efficiency Since the layering pattern in all of the lattices leaves empty space between the particles, the unit cell is not completely occupied by atoms (here we are treating atoms like hard spheres). The packing efficiency, which is the percentage of occupied space in the cube, is not 100%. The packing efficiency is not the same for all 3 cubic lattices. A more densely packed unit cell will have a higher packing efficiency than a less densely packed one. The packing efficiency of a lattice structure measures how well the space inside of a unit cell is utilized. It is the percent ratio of volume occupied by the particles in a unit cell to its total volume. Packing efficiency = (Vaccupied/Viptal) x 100% The occupied volume (Veccupied) is related to the number of particles occupying the cell and their location within the cell. The edge length of each unit cell is derived using trigonometric relationships (see Figure directly below) where / is the edge length and ris the radius of the sphere. Figure 2. Geometric relotionships showing how the edge length is related to the atomic radius for simple cubic, body-centered cubic, and foce-centered cubic unit cells. Unit Cell Edge length in terms of radius Simple cubic l = 2r 4r Body-centered cubic Face-centered cubic 1 = 2VZr Voccupied = (# particles) Vrotat = 1 In a simple cubic unit cell, all the atoms are corner atoms. There are 8 corner atoms in a simple cubic unit cell. Remember from the lecture that for a corner atom, only 1/8th of the sphere is inside the unit cell. Hence, the total number of equivalent particle inside a simple cubic unit cell is 1 (because 1/8 times 8 atoms = 1). How many equivalent particles are inside a face-centered cubic unit cell? --> particles How many equivalent particles are inside a body-centered cubic unit cell? --> particles
Transcribed Image Text:QUESTION 5 Packing Efficiency Since the layering pattern in all of the lattices leaves empty space between the particles, the unit cell is not completely occupied by atoms (here we are treating atoms like hard spheres). The packing efficiency, which is the percentage of occupied space in the cube, is not 100%. The packing efficiency is not the same for all 3 cubic lattices. A more densely packed unit cell will have a higher packing efficiency than a less densely packed one. The packing efficiency of a lattice structure measures how well the space inside of a unit cell is utilized. It is the percent ratio of volume occupied by the particles in a unit cell to its total volume. Packing efficiency = (Vaccupied/Viptal) x 100% The occupied volume (Veccupied) is related to the number of particles occupying the cell and their location within the cell. The edge length of each unit cell is derived using trigonometric relationships (see Figure directly below) where / is the edge length and ris the radius of the sphere. Figure 2. Geometric relotionships showing how the edge length is related to the atomic radius for simple cubic, body-centered cubic, and foce-centered cubic unit cells. Unit Cell Edge length in terms of radius Simple cubic l = 2r 4r Body-centered cubic Face-centered cubic 1 = 2VZr Voccupied = (# particles) Vrotat = 1 In a simple cubic unit cell, all the atoms are corner atoms. There are 8 corner atoms in a simple cubic unit cell. Remember from the lecture that for a corner atom, only 1/8th of the sphere is inside the unit cell. Hence, the total number of equivalent particle inside a simple cubic unit cell is 1 (because 1/8 times 8 atoms = 1). How many equivalent particles are inside a face-centered cubic unit cell? --> particles How many equivalent particles are inside a body-centered cubic unit cell? --> particles
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