Advertisements
Advertisements
Question
Calculate the packing efficiency for bcc lattice.
Advertisements
Solution
Packing efficiency of metal crystal in body-centred cubic (bcc) lattice:
- Step 1: Radius of sphere (particle):
In bcc unit cell, particles occupy the corners and in addition one particle is at the centre of the cube. The particle at the centre of the cube touches two corner particles along the diagonal of the cube. To obtain radius of the particle (sphere) Pythagoras theorem is applied. For triangle FED, ∠FED = 90°.
∴ FD2 = FE2 + ED2 = a2 +a2 = 2a2 (because FE = ED = a) …(1)
For triangle AFD, ∠ADF = 90°
∴ AF2 = AD2 + FD2 …(2)
Substitution of equation (1) into (2) yields
AF2 = a2 + 2a2 = 3a2 (because AD = a)
or AF = `sqrt3` a …(3)
From the figure, AF = 4r.
Substitution for AF from equation (3) gives
`sqrt3`a = 4r and hence, r = `sqrt3/4`a ....(4) - Step 2: Volume of sphere:
Volume of sphere particle =`4/3 π r^3`
Substitution for r from equation (4) gives
Volume of one particle = `4/3 pi xx (sqrt3/4 "a")^3 = 4/3 pi xx (sqrt3)^3/64 "a"^3 = (sqrt3 pi "a"^3)/16` - Step 3: Total volume of particles:
Unit cell bcc contains 2 particles.
Hence, volume occupied by particles in bcc unit cell
`= (2sqrt3pi"a"^3)/16 = (sqrt3 pi"a"^3)/8` ......(5) - Step 4: Packing efficiency:
Packing efficiency = `"Volume occupied by particles in unit cell"/"total volume of unit cell" xx 100`
`= (sqrt3 pi"a"^3)/(8"a"^3) xx 100 = 68%`
Thus, 68% of the total volume in bcc unit lattice is occupied by atoms and 32 % is empty space or void volume.
RELATED QUESTIONS
Answer the following in one or two sentences.
Which of the three types of packing used by metals makes the most efficient use of space and which makes the least efficient use?
Answer the following in one or two sentences.
Mention two properties that are common to both hcp and ccp lattices.
Answer the following in brief.
Calculate the packing efficiency of metal crystal that has simple cubic structure.
Answer the following in brief.
Cesium chloride crystallizes in a cubic unit cell with Cl– ions at the corners and a Cs+ ion in the center of the cube. How many CsCl molecules are there in the unit cell?
The density of iridium is 22.4 g/cm3. The unit cell of iridium is fcc. Calculate the radius of iridium atom. Molar mass of iridium is 192.2 g/mol.
In ionic crystalline solid atoms of element Y form hcp lattice. The atoms of element X occupy one-third of tetrahedral voids. What is the formula of the compound?
An element has a bcc structure with a unit cell edge length of 288 pm. How many unit cells and a number of atoms are present in 200 g of the element? (1.16 × 1024, 2.32 × 1024)
A compound is formed by elements A and B. This crystallizes in the cubic structure when atoms A are at the comers of the cube and atoms B are at the centre of the body. The simplest formula of the compounds is ____________.
A compound of X and Y crystallizes in ccp structure in which the X atoms occupy the lattice points at the corners of cube and Y atoms occupy the centres of each of the cube faces. The formula of this compound is ____________.
The number of particles in 1 g of a metallic crystal is equal to ____________.
The coordination number of each sphere in simple cubic lattice is ____________.
Which among the following crystal structures the edge length of unit cell is equal to twice the radius of one atom?
Which of the following is INCORRECT?
Atoms of elements A and B crystallize in hep lattice to form a molecule. Element A occupies 2/3 of tetrahedral voids, the formula of molecule is ______.
Copper crystallizes as face centered cubic lattice, with edge length of unit cell 361 pm. Calculate the radius of copper atom.
Copper crystallises with fee unit cell. If the radius of copper atom is 127.8 pm, calculate the density of copper? (At. mass: Cu = 63.55 g mol-1)
What is the percentage of void space in bcc type of in unit cell?
Silver crystallizes in face centred cubic structure, if radius of silver atom is 144.5 pm. What is the edge length of unit cell?
AB is an ionic solid. If the ratio of ionic radius of A+ and B- is 0.52, what is the coordination number of B- ?
The relation between the radius of the sphere and the edge length in the body-centred cubic lattice is given by the formula ______.
A compound has a hep structure. Calculate the number of voids in 0.4 mol of it.
Find the edge length of bcc unit cell if radius of metal atom is 126 pm.
