Concept explainers
Interpretation:
The volume of the atoms in bcc and fcc unit cell in comparison to the volume of the unit cell itself has to be calculated. From these data, the fraction of space occupied by atoms in bcc and fcc unit cell has to be calculated.
Concept Introduction:
Bcc unit cell:
Eight atoms occupy the corner position of a cube each contributing
Relationship between unit cell edge length and radius of a unit cell of a bcc arrangement:
Figure 1
Applying Pythagoras theorem, from the diagram we can clearly concluded that
Body diagonal
Fcc unit cell:
Eight atoms occupy the corner position of a cube each contributing
Relationship between unit cell edge length and radius of a unit cell of a fcc arrangement:
Figure 2
In fcc, the corner spheres are in touch with the face centered sphere as shown in the above figure. Hence, the face diagonal
Consider right angled triangle ACD.
Volume of a cubic unit cell:
Volume of cubic unit cell
Volume of a sphere:
Volume of a sphere
Packing fraction:
Packing fraction is the fraction of space occupied by total number of atoms per unit cell.
Mathematically, it can be represented as given below.
Fraction of space occupied by atoms in a unit cell
Where,
Answer to Problem 115QRT
The fraction of space occupied by atoms in bcc and fcc unit cell is
Explanation of Solution
The atoms can be assumed to be spherical in shape with radius
There are two atoms present per one bcc unit cell. The relationship between unit cell edge length and radius of a unit cell of a bcc arrangement is given below.
Now, the total volume of two atoms present in a bcc unit cell can be calculated as given below.
There are four atoms present per one fcc unit cell. The relationship between unit cell edge length and radius of a unit cell of a fcc arrangement is given below.
Now, the total volume of two atoms present in a bcc unit cell can be calculated as given below.
Volume of cubic unit cell ,
Now, the fraction of space occupied by atoms can be calculated as given below.
For bcc unit cell:
Fraction of space occupied by atoms
For fcc unit cell:
Fraction of space occupied by atoms
Therefore, the fraction of space occupied by atoms in bcc and fcc unit cell is
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Chapter 9 Solutions
Chemistry: The Molecular Science
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