Indium has a tetragonal unit cell for which the a and c lattice parameters are 0.459 and 0.495 nm, respectively. (a) if the atomic packing factor and atomic radius are 0.693 and 0.1625 nm, respectively, determine the number of atoms in each unit cell. (b) the atomic weight of indium is 114.82 g/mol; compute its theoretical density.

a) = 4.02 = 4 atoms / unit cell

b) Density = 7.31 g/(cm)³

Explanation:

a)

Tetragonal structure, one of the structural groups to which crystalline solids may be classified. In this process, the crystals are referred to as three mutually perpendicular poles, i.e 90° from each other, two of which are of equal length. It has a square base with area a×a and height c.  

Given data:

a =  0.459 nm

c = 0.495 nm

Atomic packing factor is the measurement of the actual volume of the imaginary atomic scale "shell" filled by the atoms. In simple words, it is the ratio of  volume of atoms within the unit cell to the total volume of the unit cell.

atomic packing factor = ( volume of atoms within the unit cell) / (total volume of the unit cell)

APF =

where is the number of particles in the unit cell, is the volume of each particle, and is the volume occupied by the unit cell.

Given data:

APF = 0.693

Atomic radius = r = 0.1625 nm

Volume of unit cell = a × a × c

= 0.459 × 0.459 × 0.495

= 0.1043 (nm)³ = 1.043 × 10⁻²⁸  m³

Volume of each particle = 4/3(π)r³

4/3(3.14)(0.1625)³ = 0.0179 (nm)³ = 1.79 × 10⁻²⁹ m

APF =

0.693 = [ × 0.0179 (nm)³] ÷ 0.1043 (nm)³

= 4.02 = 4 atoms / unit cell

b)

Density is the mass of an object per unit volume. It tell us that how much mass of an object in one unit volume.Metals has higher density than gases because atoms of metals are closely packed than gases. General formula for density (d) is

d = mass/Volume

To calculate density here, we use density formula related to molecules.

No. of atoms = (Avogadro's number×density×Volume)/(Molecular weight)

The atomic weight of indium = 114.82 g/mol

Avogadro's number = 6.022 × 10²³

Theoretical density = D = ?

density = (4 × 114.82 ) / (6.022 × 10²³ × 1.043 × 10⁻²⁸)

Density = 7.31 g/(cm)³

3. dark matter

empty space could also be a simple vacuum