Primitive Unit Cell (Simple Cubic) and Centred Unit Cells Class 12 Notes | EduRev

Chemistry Class 12

Created by: Mohit Rajpoot

Class 12 : Primitive Unit Cell (Simple Cubic) and Centred Unit Cells Class 12 Notes | EduRev

The document Primitive Unit Cell (Simple Cubic) and Centred Unit Cells Class 12 Notes | EduRev is a part of the Class 12 Course Chemistry Class 12.
All you need of Class 12 at this link: Class 12

Simple cubic (SC) unit cell
A simple cubic unit cell consists of eight corner atoms. In a simple cubic unit cell, a corner atom touches with another corner atom. The simple cubic unit cell is shown Primitive Unit Cell (Simple Cubic) and Centred Unit Cells Class 12 Notes | EduRev

Fig: Simple cubic unit cell of a crystal lattice


Number of atoms in a simple cubic unit cell
A simple cubic unit cell consists of 8 corner atoms. Each and every corner atom is shared by eight adjacent unit cells. Therefore, one corner atom contributes 1/8th of its parts to one unit cell. Since, there are eight corner atoms in a unit cell, the total number of atoms is (1/8)* 8=1. Therefore, the number of atoms in a simple cubic unit cell is one.

Atomic radius
In a simple cubic lattice, a corner atom touches with another corner atom.
Therefore, 2r=a. So, the atomic radius of an atom in a simple cubic unit cell is a/2.

Coordination number
Consider a corner atom in a simple cubic unit cell. It has four nearest neighbours in its own plane. In a lower plane, it has one more nearest neighbour and in an upper plane, it has one more nearest neighbour. Therefore, the total number of nearest neighbour is six.

Packing density
The packing density of a simple cubic unit cell is calculated as follows:
theunit cell Number of atoms p  
Primitive Unit Cell (Simple Cubic) and Centred Unit Cells Class 12 Notes | EduRev
Substituting, r=a/2, we get,
Primitive Unit Cell (Simple Cubic) and Centred Unit Cells Class 12 Notes | EduRev
The packing density of a simple cubic unit cell is 0.52. It means, 52% of the volume of the unit cell is occupied by atoms and the remaining 48% volume is vacant.

Primitive unit cell:
In a Primitive unit cell, atoms are present at the corner only. An atom is shared by 8 unit cell. Hence the only 1/8th of the atom belonging to the particular unit cell. There are total 8 corners in each unit cell, therefore,1/8 part of 8 atoms would give the value of the number of atoms per unit cell.
In brief 8 corners, 1 atom at each corner, 1/8 of each atom in unit cell

Primitive Unit Cell (Simple Cubic) and Centred Unit Cells Class 12 Notes | EduRev

Fig: A primitive cubic unit cell 
(a) open structure
(b) space-filling structure
(c) actual portions of atoms belonging to one unit cell


Centred unit cell (body- centred):

Body- centred unit cell:
The number of atoms present at the different position is as follows:
(a) 8 corners, 1 atom at each corner (1/8 of each atom in the unit cell)
(b) 1 atom at the centre of the body (1 atom completely present in the unit cell)

Primitive Unit Cell (Simple Cubic) and Centred Unit Cells Class 12 Notes | EduRev

Fig: Body centered cubic cell


The total number of atom present in each unit cell
= Contribution of atoms at corner + contribution of atom at the centre of the body

Packing Efficiency

Primitive Unit Cell (Simple Cubic) and Centred Unit Cells Class 12 Notes | EduRevFace-centred unit cell:
The number of atoms present at the different position is as follows:
(a) 8 corners, 1 atom at each corner (1/8 of each atom in unit cell)
(b) 6 faces, 1 atom at each face of the unit cell (1/6; of each atom in the unit cell)Primitive Unit Cell (Simple Cubic) and Centred Unit Cells Class 12 Notes | EduRev

Fig: Face Centered cubic unit cell

Primitive Unit Cell (Simple Cubic) and Centred Unit Cells Class 12 Notes | EduRev

Fig: A face-centred cubic unit cell 
(a) open structure 
(b) space filling structure  
(c) actual portions of atoms belonging to one unit cell


The total number of atom present in each unit cell = Contribution of atoms at corner + contribution of atom at the face of the body


Packing Efficiency

Primitive Unit Cell (Simple Cubic) and Centred Unit Cells Class 12 Notes | EduRev


Q.1. An element crystallizes into a structure which may be described by a cubic type of unit cell having one atom in each corner of the cube and two atoms on one of its face diagonals. If the volume of this unit cell is 24 x 10-24 cm3 and density of the element is 7.20 gm/cm3, calculate no. of atoms present in 200 gm of the element.
Ans: Number of atoms contributed in one unit cell = one atom from the eight corners + one atom from the two face diagonals = 1+1 = 2 atoms
Mass of one unit cell = volume × its density
= 24 × 10–24 cm3 × 7.2 gm cm3 
= 172.8 × 10–24 gm
∴ 172.8 10–24 gm is the mass of one – unit cell i.e., 2 atoms
∴ 200 gm is the mass = 2 × 200 / 172.8 × 10–24 atoms = 2.3148 × 1024 atoms


Q.2. Calculate the void fraction for the structure formed by A and B atoms such that A form hexagonal closed packed structure and B occupies 2/3 of octahedral voids. Assuming that B atoms exactly fitting into octahedral voids in the HCP formed by A.
Ans: Total volume of A atom = 6 × 4 / 3 πrA3 
Total volume of B atoms = 4 × 4/3 πrA3 
=4 × 4/3 π(0.414rA)3 
Since rB/rA as B is in octahedral void of A
Volume of HCP = 24√2rA3 
Packing fraction = 6 × 4/3 πrA3 + 4 × 4/3 π (0.414rA)3 / 24√2rA3 = 0.7756
Void fraction = 1-0.7756 = 0.2244


Q.3: An element with molar mass 2.7×10-2kg mol-1 forms a cubic unit cell with edge length 405 pm. If its density is 2.7×103kg-3, what is the nature of the cubic unit cell? 
Ans: d = 2.7×103kg-3
M = 2.7×10 -kg mol-1
a= 405 pm = 405 X 10-12
NA= 6.023 X 1023
Using the formula

Primitive Unit Cell (Simple Cubic) and Centred Unit Cells Class 12 Notes | EduRev

Z=4
Unit cell is fcc unit cell.
Volume of 54 g of the element = 0.054/(2.7 x 103) = 2 X 10-6
Number of unit cell in this volume = volume of 554 g of element/volume of each unit cell = 2 x 10-6/(405 x 10-12)3 = 3.012 x 1022

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