Homework Help is Here – Start Your Trial Now!
Engineering
Chemical Engineering
7.21 In a crystal with a face-centred cubic structure, the basic cell can be taken as a cube of edge a with its centre at the origin of coordinates and its edges parallel to the Cartesian coordinate axes; atoms are sited at the eight corners and at the centre of each face. However, other basic cells are possible. One is the rhomboid shown in figure 7.17, which has the three vectors b, c and d as edges. (a) Show that the volume of the rhomboid is one-quarter that of the cube. (b) Show that the angles between pairs of edges of the rhomboid are 60° and that the corresponding angles between pairs of edges of the rhomboid defined by the reciprocal vectors to b, c, d are each 109.5°. (This rhomboid can be used as the basic cell of a body-centred cubic structure, more easily visualised as a cube with an atom at each corner and one at its centre.)

7.21 In a crystal with a face-centred cubic structure, the basic cell can be taken as a cube of edge a with its centre at the origin of coordinates and its edges parallel to the Cartesian coordinate axes; atoms are sited at the eight corners and at the centre of each face. However, other basic cells are possible. One is the rhomboid shown in figure 7.17, which has the three vectors b, c and d as edges. (a) Show that the volume of the rhomboid is one-quarter that of the cube. (b) Show that the angles between pairs of edges of the rhomboid are 60° and that the corresponding angles between pairs of edges of the rhomboid defined by the reciprocal vectors to b, c, d are each 109.5°. (This rhomboid can be used as the basic cell of a body-centred cubic structure, more easily visualised as a cube with an atom at each corner and one at its centre.)

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
See similar textbooks
icon
Related questions
Question
100%
**Figure 7.17: A Face-Centred Cubic Crystal** This diagram illustrates a face-centred cubic (FCC) crystal structure. In an FCC lattice, atoms are located at each of the corners and the centers of all the cube faces in the unit cell. **Diagram Explanation:** - The cube is represented with dotted lines indicating its edges. - The letter "a" denotes the side length of the unit cell. - The vectors labeled "b," "c," and "d" indicate directions within the crystal. These vectors are typically used to represent atomic positions or directions of interest, though their specific orientation here isn't detailed. - The spheres at the corners and on the faces represent the atom positions in the crystal lattice. - Solid lines connect these atomic positions to illustrate the repeating pattern within the crystal lattice. This type of structure is important in understanding the properties of various metals and alloys, as the arrangement affects how atoms interact with one another.
Transcribed Image Text:**Figure 7.17: A Face-Centred Cubic Crystal** This diagram illustrates a face-centred cubic (FCC) crystal structure. In an FCC lattice, atoms are located at each of the corners and the centers of all the cube faces in the unit cell. **Diagram Explanation:** - The cube is represented with dotted lines indicating its edges. - The letter "a" denotes the side length of the unit cell. - The vectors labeled "b," "c," and "d" indicate directions within the crystal. These vectors are typically used to represent atomic positions or directions of interest, though their specific orientation here isn't detailed. - The spheres at the corners and on the faces represent the atom positions in the crystal lattice. - Solid lines connect these atomic positions to illustrate the repeating pattern within the crystal lattice. This type of structure is important in understanding the properties of various metals and alloys, as the arrangement affects how atoms interact with one another.
### 7.21 In a crystal with a face-centred cubic structure, the basic cell can be taken as a cube of edge \(a\) with its centre at the origin of coordinates and its edges parallel to the Cartesian coordinate axes; atoms are sited at the eight corners and at the centre of each face. However, other basic cells are possible. One is the rhomboid shown in figure 7.17, which has the three vectors \( \mathbf{b}, \mathbf{c}, \) and \( \mathbf{d} \) as edges. **(a)** Show that the volume of the rhomboid is one-quarter that of the cube. **(b)** Show that the angles between pairs of edges of the rhomboid are \(60^\circ\) and that the corresponding angles between pairs of edges of the rhomboid defined by the reciprocal vectors to \( \mathbf{b}, \mathbf{c}, \mathbf{d} \) are each \(109.5^\circ\). (This rhomboid can be used as the basic cell of a body-centred cubic structure, more easily visualised as a cube with an atom at each corner and one at its centre.)
Transcribed Image Text:### 7.21 In a crystal with a face-centred cubic structure, the basic cell can be taken as a cube of edge \(a\) with its centre at the origin of coordinates and its edges parallel to the Cartesian coordinate axes; atoms are sited at the eight corners and at the centre of each face. However, other basic cells are possible. One is the rhomboid shown in figure 7.17, which has the three vectors \( \mathbf{b}, \mathbf{c}, \) and \( \mathbf{d} \) as edges. **(a)** Show that the volume of the rhomboid is one-quarter that of the cube. **(b)** Show that the angles between pairs of edges of the rhomboid are \(60^\circ\) and that the corresponding angles between pairs of edges of the rhomboid defined by the reciprocal vectors to \( \mathbf{b}, \mathbf{c}, \mathbf{d} \) are each \(109.5^\circ\). (This rhomboid can be used as the basic cell of a body-centred cubic structure, more easily visualised as a cube with an atom at each corner and one at its centre.)
Expert Solution
steps

Step by step

Solved in 3 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Introduction to Chemical Engineering Thermodynami…
Introduction to Chemical Engineering Thermodynami…
Chemical Engineering
ISBN:
9781259696527
Author:
J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Publisher:
McGraw-Hill Education
Elementary Principles of Chemical Processes, Bind…
Elementary Principles of Chemical Processes, Bind…
Chemical Engineering
ISBN:
9781118431221
Author:
Richard M. Felder, Ronald W. Rousseau, Lisa G. Bullard
Publisher:
WILEY
Elements of Chemical Reaction Engineering (5th Ed…
Elements of Chemical Reaction Engineering (5th Ed…
Chemical Engineering
ISBN:
9780133887518
Author:
H. Scott Fogler
Publisher:
Prentice Hall
Process Dynamics and Control, 4e
Process Dynamics and Control, 4e
Chemical Engineering
ISBN:
9781119285915
Author:
Seborg
Publisher:
WILEY
Industrial Plastics: Theory and Applications
Industrial Plastics: Theory and Applications
Chemical Engineering
ISBN:
9781285061238
Author:
Lokensgard, Erik
Publisher:
Delmar Cengage Learning
Unit Operations of Chemical Engineering
Unit Operations of Chemical Engineering
Chemical Engineering
ISBN:
9780072848236
Author:
Warren McCabe, Julian C. Smith, Peter Harriott
Publisher:
McGraw-Hill Companies, The
SEE MORE TEXTBOOKS