
Concept explainers
Interpretation:
The packing efficiencies of a simple cubic, body-centered, and face-centered cell are to be calculated
Concept introduction:
A crystal lattice is made of small repeating unit cells.
A unit cell is of primitive or centered type. The spheres in a unit cell can represent an atom, a molecule, or an ion.
Each sphere is shared by adjacent cells. A sphere at corners is shared by eight unit cells. A sphere in the center of each face is shared by two unit cells.
The volume of the unit cell is calculated as:
Here,
is the edge length.
The formula to calculate the volume of a sphere is as:
Here,
is the radius of the sphere.
Packing efficiency is the ratio of the volume of atoms in a unit cell to the volume of the unit cell. Its expression is:

Answer to Problem 136AP
Solution:
Explanation of Solution
Given information: For SCC,
In a simple cubic cell, spheres are present only at the eight corners of the unit cell.
The contribution of 8 spheres at the corners to one unit cell is:
Thus, one sphere is present in the unit cell.
The formula to calculate the volume of a unit cell is as follows:
Substitute
for
in the above expression as:
The volume of a sphere is
The formula to calculate packing efficiency is as:
Substitute
for the volume of atoms in the unit cell and
for the volume of the unit cell in the above expression as:
In BCC, there are spheres at the eight corners of the unit cell and one sphere is present in the center of the unit cell.
The contribution of 8 spheres at the corners to one unit cell is:
One sphere is in the center of the unit cell.
Thus, two spheres are present in a body centered unit cell.
The volume of a sphere is
Thus, the volume of two spheres will be as:
Substitute
in equation (1) as:
Substitute
for the volume of atoms in the unit cell and
for the volume of the unit cell in equation (2) as follows:
In FCC cubic cell, there are spheres at the eight corners of the unit cell and one sphere in the center of six faces.
The contribution of
The contribution of
spheres present in the center of each face to one-unit cell is:
Thus, four spheres are present in a face-centered unit cell.
The volume of a sphere is
Thus, the volume of four spheres is:
Substitute
for
in equation (1):
Substitute
for the volume of atoms in the unit cell and
for the volume of the unit cell in equation (2) as follows:
The packing efficiencies of simple cubic, body-centered, and face-centered cell are
and
Want to see more full solutions like this?
Chapter 11 Solutions
EBK CHEMISTRY
- 1. Identify the following alkenes as E or Z NH₂ Br 2. Draw the structures based on the IUPAC names (3R,4R)-3-bromo-4-fluoro- 1-hexene (Z)-4-bromo-2-iodo-3-ethyl- 3-heptene تر 3. For the following, predict all possible elimination product(s) and circle the major product. HO H₂SO4 Heat 80 F4 OH H2SO4 Heat 어요 F5 F6 1 A DII 4 F7 F8 F9 % & 5 6 7 * ∞ 8 BAB 3 E R T Y U 9 F D G H J K O A F11 F10arrow_forwardDraw the major product of this reaction. Ignore inorganic byproducts. ○ O 1. H₂O, pyridine 2. neutralizing work-up a N W X 人 Parrow_forward✓ Check the box under each molecule that has a total of five ẞ hydrogens. If none of the molecules fit this description, check the box underneath the table. tab OH CI 0 Br xx Br None of these molecules have a total of five ẞ hydrogens. esc Explanation Check caps lock shift 1 fn control 02 F2 W Q A N #3 S 80 F3 E $ t 01 205 % 5 F5 & 7 © 2025 McGraw Hill LLC. All Rights Reserved. Terms of Use | Privacy Center | Accessibility FT * 8 R T Y U כ F6 9 FIG F11 F D G H J K L C X V B < N M H option command P H + F12 commandarrow_forward
- Draw the major product of this reaction. Ignore inorganic byproducts and the carboxylic acid side product. O 1. CHзMgBr (excess) 2. H₂O ✓ W X 人arrow_forwardIf cyclopentyl acetaldehyde reacts with NaOH, state the product (formula).arrow_forwardDraw the major product of this reaction. Ignore inorganic byproducts. N S S HgCl2, H2SO4 く 8 W X Parrow_forward
- tab esc く Drawing the After running various experiments, you determine that the mechanism for the following reaction occurs in a step-wise fashion. Br + OH + Using this information, draw the correct mechanism in the space below. 1 Explanation Check F2 F1 @2 Q W A os lock control option T S # 3 80 F3 Br $ 4 0105 % OH2 + Br Add/Remove step X C F5 F6 6 R E T Y 29 & 7 F D G H Click and drag to start drawing a structure. © 2025 McGraw Hill LLC. All Rights Reserved. Terms of Use | Privacy Ce A F7 DII F8 C Ո 8 * 9 4 F10 F C J K L C V Z X B N M H command P ge Coarrow_forwardIndicate compound A that must react with ethylbenzene to obtain 4-ethylbenzene-1-sulfonic acid. 3-bromo-4-ethylbenzene-1-sulfonic acid.arrow_forwardPart 1 of 2 Draw the structure of A, the minor E1 product of the reaction. esc I Skip Part Check H₂O, D 2 A + Click and drag to start drawing a structure. -0- F1 F2 1 2 # 3 Q A 80 F3 W E S D F4 $ 4 % 5 F5 ㅇ F6 R T Y F G X 5 & 7 + Save 2025 McGraw Hill LLC. All Rights Reserved. DII F7 F8 H * C 80 J Z X C V B N 4 F9 6arrow_forward
- File Preview The following is a total synthesis of the pheromone of the western pine beetle. Such syntheses are interesting both because of the organic chemistry, and because of the possibility of using species specific insecticides, rather than broad band insecticides. Provide the reagents for each step. There is some chemistry from our most recent chapter in this synthesis, but other steps are review from earlier chapters. (8 points) COOEt COOEt A C COOEt COOEt COOH B OH OTS CN D E See the last homework set F for assistance on this one. H+, H₂O G OH OH The last step is just nucleophilic addition reactions, taking the ketone to an acetal, intramolecularly. But it is hard to visualize the three dimensional shape as it occurs. Frontalin, pheromone of the western pine beetlearrow_forwardFor the reaction below: 1. Draw all reasonable elimination products to the right of the arrow. 2. In the box below the reaction, redraw any product you expect to be a major product. C Major Product: Check + ◎ + X ง © Cl I F2 80 F3 I σ F4 I F5 NaOH Click and drawing F6 A 2025 McGraw Hill LLC. All Rights E F7 F8 $ # % & 2 3 4 5 6 7 8 Q W E R T Y U A S D F G H Jarrow_forwardCan I please get help with this graph. If you can show exactly where it needs to pass through.arrow_forward
- Chemistry & Chemical ReactivityChemistryISBN:9781337399074Author:John C. Kotz, Paul M. Treichel, John Townsend, David TreichelPublisher:Cengage LearningChemistryChemistryISBN:9781305957404Author:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCostePublisher:Cengage LearningChemistry: An Atoms First ApproachChemistryISBN:9781305079243Author:Steven S. Zumdahl, Susan A. ZumdahlPublisher:Cengage Learning
- Chemistry: Principles and PracticeChemistryISBN:9780534420123Author:Daniel L. Reger, Scott R. Goode, David W. Ball, Edward MercerPublisher:Cengage LearningPhysical ChemistryChemistryISBN:9781133958437Author:Ball, David W. (david Warren), BAER, TomasPublisher:Wadsworth Cengage Learning,





