Question

1. Vanadium crystallizes in a body-centered cubic lattice, and the length of the edge of a unit cell is 305 pm. what is the density of V?

2. Vanadium crystallizes in a face-centered cubic lattice, and the length of the edge of a unit cell is 305 pm. what is the density of V?

3. Vanadium crystallizes in a single-centered cubic lattice, and the length of the edge of a unit cell is 305 pm. what is the density of V?

Please explain the works.

Answer 1

Sol 1.

In body - centered cubic lattice ,  

Number of atoms per unit cell = Z

= Number of atoms at corner × Contribution by each corner + Number of atoms at centre × Contribution by centre

= 8 × 1/8 + 1 × 1 = 1 + 1 = 2  

Molar Mass of Vanadium , V = M = 50.9415 g/mol

Edge length = a = 305 pm = 305 × 10-10 cm

Avogadro's Number = NA = 6.023 × 1023  

So , Density of V  

= ZM / a3NA  

= 2 × 50.9415 g/mol / ( ( 305 × 10-10 cm )3 × 6.023 × 1023 )

=    5.962    g/cm3  

Sol 2 .

In face - centered cubic lattice ,  

Z = Number of atoms at corner × Contribution by each corner + Number of atoms at faces × Contribution by each face

= 8 × 1/8 + 6 × 1/2 = 1 + 3 = 4  

Density of V

= ZM / a3NA

= 4 × 50.9415 g/mol / ( ( 305 × 10-10 cm )3 × 6.023 × 1023 )

=    11.924    g/cm3

Sol 3.

In single - centered cubic lattice , Z = 1

Density of V  

= ZM / a3NA

= 1 × 50.9415 g/mol / ( ( 305 × 10-10 cm )3 × 6.023 × 1023 )

=    2.981    g/cm3