The Solid State Problems - Numericals

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01.

Chromium has mono atomic body-centred cubic structure. Its cell edge is 300 pm, what is its density? (Molar mass of Cr = 52 g mol^-1, Avogadro number N = 6.023 x 10²³ mol).

Suggested solution:

Given:

z = 2 since structure is body-centered cubic

M = 52 g mol^-1

a = 300 pm = 300 x 10^-10 cm

N = 6.023 x 10²³ mol^-1

02.

A solid AB has NaCl structure. If the radius of cation A⁺ is 170 pm, Calculate the maximum possible value of radius of the anion B^-.

Suggested solution:

For NaCl structure (coordination number 6), the range of radius ratio

For maximum possible radius or anion B^- the radius ratio will be the minimum, i.e., 0.414.

03.

An element of atomic mass 98.0 g mol^-1 occurs in FCC structure. If the unit cell edge length is 500 pm and its density is 5.22 g cm^-3. What is the value of the Avogadro's constant?

Suggested solution:

Given z = 4

Since the structure is FCC:

M = 98.0 g mol^-1

A = 500 pm = 500 x 10^-10 cm = 5 x 10^-8 cm

P = 5.22 g cm^-3

= 6.0 x 10²³ mol^-1

04.

The length of the unit cell edge of a body centered cubic metal crystal is 352 pm. Calculate the radius of an atom of the metal.

Suggested solution:

In body-centered cubic unit cell, the radius r of the atom is related to the edge of the unit cell by the relation.

05.

Sodium chloride crystal has FCC structure. Its density is 2.163 x10³ kg m^-3. Calculate the edge of the unit cell cube.

Suggested solution:

(M_NaCl = 58.45 x 10^-3 kg mol^-1 ; NA = 6.023 x 10²³)

Z = 4 for a FCC structures

a³ = 179.5 x 10^-30m³

or a = 5.64 x 10^-10 m = 564 pm.

06.

Ionic compound AB has NaCl type of structure. If the radius of A⁺ ion is 41.4 cm.

(i) What is the ideal radius of B^- ion?

(ii) If the radius of A⁺ ion decreases and becomes equal to 30pm, then in what type of structure would AB crystallize and what would be coordination number of A⁺ ion?

Suggested solution:

Radius ratio is in 0.225 - 0.414 range which is for tetrahedral.

It crystallizes in ZnS type of structure

Coordination number of A+ ion changes from 6 (in NaCl) to 4 (in ZnS type of structure).

07.

In mineral A₂B, atoms B form CCP structure while A atoms occupy the voids within this structure. What are the coordination numbers of atoms A and B ? Name the voids in which atoms A are present and what fraction of these voids they occupy?

Suggested solution:

B atoms form CCP structure

Number of octahedral voids = Number of B atoms

Number of tetrahedral voids = 2 x Number of B atoms.

Since the number of A atoms = 2 x Number of B atoms therefore they occupy all of tetrahedral voids.

Coordination number of A atoms = Four as they occupy tetrahedral voids

Coordination number of B atoms = Twice the coordination number of A

Since B atoms are half of A atoms therefore their coordination number is twice the coordination number of A.

Coordination number of B = 8.

08.

Calculate the density of Mo which forms body centered cubic crystal in which the distance between the centers of closest atoms is 274 pm. Atomic mass of Mo is 95.94.

Suggested solution:

In a body centered cubic crystal, the closest distance between touching atom is

and here z = 2

or P = 10.06 g cm^-3.

09.

NH₄Cl crystallises in a body centered cubic lattice with a, edge length of 400 pm. Calculate the

a) Distance between oppositely charged ions in the lattice.

b) Radius of Cl^- ions if radius of that of NH₄⁺ ion is 160 pm.

Suggested solution:

10.

Assuming aluminium (At mass = 27) has a density of 2.7 g cm^-3 and its unit cell has an edge of 400 pm, what is the nature of the crystal lattice for the metal?

Suggested solution:

z = 3.85

z = 4

Hence Al crystallises in face-centered cubic lattice.