Question

EX. 3 Calculate the percentage of the total volume is occupied by spheres in (a) a simple cube, (b) a body-centered cube, and (c) a face-centered cube in which all atoms are identical. (c) A face-centered cube (fcc)

Answer 1

Ans 3a: Packing efficiency in simple cube

• In simple cubic lattice (SCC), we have atoms in 8 corners of a unit cell
• Atoms present at the corner will contribute 1/8 to the unit cell
• Number of atoms per unit cell in SCC, Z = (8 x 1/8) = 1

1 A 1 B Edge length = a = AB = 2r

• We can see from the diagram that edge length AB = a = 2r

     

• Packing efficiency =
  (Volume of 1 atom x Z) Volume of unit cell
  x 100

      Packing efficiency =
п. r3. 1) аз
x 100 =
. 1. r3.1) (2r)3
x 100

      Packing efficiency = 52.8%

Ans 3b: Packing efficiency in body-cantered cube (bcc)

• In BCC, along with the atoms in 8 corners of a unit cell, we will have 1 atom sitting at the body centre
• Atoms present at the corner will contribute 1/8 to the unit cell
• Atoms present at the centre of unit cell contribute 1 to the unit cell
• A cube will have 8 corners and 1 body centre
• Number of atoms per unit cell in BCC, Z = (8 x 1/8) + (1 x 1) = 2

— = body diagonal=f= AB = face diagonal = b = BC Edge length = a = CD=BD= AC

• In triangle BCD, using Pythagorean theorem we can write

            BC² = BD² + CD²

            b² = a² + a² = 2a²

• Similarly, in triangle ABC we can write

            AB² = AC² + BC²

            f² = a² + b² = a² + 2a² = 3a²

• We can also see from the diagram that, f = 4r. So,

            (4r)² = 3a²

            4r = a

            r

• Packing efficiency =
  (Volume of 1 atom x Z) Volume of unit cell
  x 100

            Packing efficiency =
+ п. r3. 2) аз
x 100 =
4. 1. 73.2) 13
x 100

            Packing efficiency = 68%

Ans 3c: Packing efficiency in face-centred cube (fcc)

• In FCC, along with the atoms in the 8 corner of a unit cell, we will have 1 atom sitting at the centre of each face.
• Atoms present at the corner will contribute 1/8 to unit cell
• Atoms present at the centre of face contribute 1/2 to unit cell
• A cube will have 8 corners and 6 faces.
• Number of atoms per unit cell of FCC, Z = (8 x 1/8) + (6 x 1/2) = 4

1 = radius of the atom F = Length AC = face diagonal a = Length of AB/BC = Edge lenth

• In triangle ABC, using Pythagorean theorem we can write

            AC² = AB² + BC²

            F² = a² + a² = 2a²

• We can also see from the diagram that, F = 4r. So,

            (4r)² = 2a²

            16r² = 2a²

            a =

• Packing efficiency =
  (Volume of 1 atom x Z) Volume of unit cell
  x 100

            Packing efficiency =
+ п. r3. 4) аз
x 100 =
. 1. r3.4) (78. r)3
x 100

            Packing efficiency = 74%

EMPIRICAL FORMULA:

Face corner Internal edge internal Vanadium Sulfide

Each corner atom gets shared between 6 hexagons

Each edge atom gets shared between 3 hexagons

Each face atom gets shared between 2 hexagons

Internal atoms entirely belong to the cell

	@corner	@edge	@face	@internal	Net
V2+	12 x (1/6) =2	6 x (1/3) = 2	2 x (1/2) = 1	1	6
S2-	-	-	-	6	6

Formula = V₆S₆

Empirical formula = V₁S₁