Answer
Verified
351k+ views
Hint: For a better solution it is very important to understand what is packing fraction So Packing fraction is defined as a way of expressing the variation of isotopic mass from the whole mass number (atomic mass). This fraction can have positive or can have negative sign. This difference is due to the transformation of mass into energy in the formation of nucleus.
Complete step by step solution:
Let us know what is face centered cubic unit cell and then solve for its packing fraction:
Packing faction or Packing efficiency is the percentage of total space filled by the particles.
The face centered unit cell (FCC) contains atoms at all the corners of the crystal lattice and at the center of all the faces of the cube. The atom present at the face centered is shared between 2 adjacent unit cells and only 1/2 of each atom belongs to an individual cell. The packing efficiency of FCC lattice is 74%.
Let r be the radius of the sphere and $ a $ be the edge length of the cube and the number of atoms or spheres is n that is equal to 4.
As there are 4 sphere in FCC unit cell
Volume of four spheres is given by
$ V = 4 \times \dfrac{4}{3}\pi {r^3} $
In FCC, the corner spheres are in touch with the face centered sphere so the relation between the edge length and radius of the sphere is given by
$ r = \dfrac{a}{{4\sqrt 2 }} $
Substituting the above values in the below equation and solving:
$ packing{\text{ }}efficiency = \dfrac{{volume{\text{ }}occupied{\text{ }}by{\text{ }}all{\text{ }}the{\text{ }}spheres{\text{ in }}unit{\text{ }}cell}}{{Total{\text{ }}volume{\text{ }}of{\text{ }}the{\text{ }}unit{\text{ }}cell}} \times 100 $
$ Packing{\text{ }}fraction = \dfrac{{n \times \dfrac{4}{3}\pi {r^3}}}{{{a^3}}} $
$ packing{\text{ }}fraction = \dfrac{{4 \times \dfrac{4}{3} \times {{\left( {\dfrac{a}{{4\sqrt 2 }}} \right)}^3}}}{{{a^3}}} = \dfrac{\pi }{{3\sqrt 2 }} = 0.74 $
Hence the correct option is (A).
Note:
The relationship between the edge length of the cube unit and the radius of the sphere can be properly derived by a proper diagram and mathematical theorems. The packing efficiency of different cubic unit cells vary according to their structure and shape.
Complete step by step solution:
Let us know what is face centered cubic unit cell and then solve for its packing fraction:
Packing faction or Packing efficiency is the percentage of total space filled by the particles.
The face centered unit cell (FCC) contains atoms at all the corners of the crystal lattice and at the center of all the faces of the cube. The atom present at the face centered is shared between 2 adjacent unit cells and only 1/2 of each atom belongs to an individual cell. The packing efficiency of FCC lattice is 74%.
Let r be the radius of the sphere and $ a $ be the edge length of the cube and the number of atoms or spheres is n that is equal to 4.
As there are 4 sphere in FCC unit cell
Volume of four spheres is given by
$ V = 4 \times \dfrac{4}{3}\pi {r^3} $
In FCC, the corner spheres are in touch with the face centered sphere so the relation between the edge length and radius of the sphere is given by
$ r = \dfrac{a}{{4\sqrt 2 }} $
Substituting the above values in the below equation and solving:
$ packing{\text{ }}efficiency = \dfrac{{volume{\text{ }}occupied{\text{ }}by{\text{ }}all{\text{ }}the{\text{ }}spheres{\text{ in }}unit{\text{ }}cell}}{{Total{\text{ }}volume{\text{ }}of{\text{ }}the{\text{ }}unit{\text{ }}cell}} \times 100 $
$ Packing{\text{ }}fraction = \dfrac{{n \times \dfrac{4}{3}\pi {r^3}}}{{{a^3}}} $
$ packing{\text{ }}fraction = \dfrac{{4 \times \dfrac{4}{3} \times {{\left( {\dfrac{a}{{4\sqrt 2 }}} \right)}^3}}}{{{a^3}}} = \dfrac{\pi }{{3\sqrt 2 }} = 0.74 $
Hence the correct option is (A).
Note:
The relationship between the edge length of the cube unit and the radius of the sphere can be properly derived by a proper diagram and mathematical theorems. The packing efficiency of different cubic unit cells vary according to their structure and shape.
Recently Updated Pages
Basicity of sulphurous acid and sulphuric acid are
Three beakers labelled as A B and C each containing 25 mL of water were taken A small amount of NaOH anhydrous CuSO4 and NaCl were added to the beakers A B and C respectively It was observed that there was an increase in the temperature of the solutions contained in beakers A and B whereas in case of beaker C the temperature of the solution falls Which one of the following statements isarecorrect i In beakers A and B exothermic process has occurred ii In beakers A and B endothermic process has occurred iii In beaker C exothermic process has occurred iv In beaker C endothermic process has occurred
What is the stopping potential when the metal with class 12 physics JEE_Main
The momentum of a photon is 2 times 10 16gm cmsec Its class 12 physics JEE_Main
How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE
Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE
Trending doubts
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Fill the blanks with proper collective nouns 1 A of class 10 english CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Select the word that is correctly spelled a Twelveth class 10 english CBSE
What organs are located on the left side of your body class 11 biology CBSE
What is the z value for a 90 95 and 99 percent confidence class 11 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers