Explanation
z =4 \text { (for } f C C)
M =197 \text { (for gold) }
a =4.07 \text { A( given) }
=407 \mathrm{pm}
P =\frac{2 \times M}{N a \times a^{3} \times 10^{-30}} g \mathrm{~cm}^{-3}
=\frac{4 \times 191}{6.022 \times 10^{23} \times(401)^{3} \times 10^{-30}}
=\frac{7899}{401}=19.4 \mathrm{~g} \mathrm{~cm}^{-3}
Given that, Cu atoms forms CCP arrangement which is analogous to FCC type arrangement.So,
Number of atoms of Cu present in unit cell =4
So,
Number of octahedral voids = Number of atoms in FCC arrangement =4
Number of tetrahedral voids = 2 \times Number of octahedral voids
= 2 \times 4 = 8
The given hypothetical formula is Cu_4Ag_3Au
This is possible if \left (\cfrac 14 \right)^{th} of the octahedral voids and \left (\cfrac 32 \right)^{th} of the tetrahedral voids are occupied.
Face centred cubic (fcc) lattice has a packing efficiency of 74%.
Body centred cubic (bcc) lattice has a packing efficiency of 68%.
Simple/ Primitive cubic (pc) lattice has a packing efficiency of 52.4%.
Hence, the correct order is \text {fcc > bcc > pc}.
CsBr has body centered cubic (BCC) structure. The edge length of unit cell is given by 4.4 \ \mathring A
Now, for a cube of edge a units, the length of the diagonal is \sqrt 3 a units.
The shortest ionic distance between Cs^+ and Br^-= \cfrac {\sqrt 3a}{2}
= \cfrac {\sqrt 3 \times 4.4}{2}= 3.81 \ \mathring A
In NaCl lattice, Cl^- ions forms FCC and Na^+ ions are present at edge centers and body center.
Number of Na^+ ions present = \cfrac 14 \times 12 = 3
Now, charge on Al^{3+} = +3 and charge on Na^+ =1.
So, each Al^{3+} ion will replace 3 Na^+ ions and two vacant sites will be created.
Now, 3/4 moles of Na^+ ions will be replaced by 1/4 moles of Al^{3+} ions to maintain electrical neutrality. Now, remaining 1/4 moles of Na^+ ions and 1/4 moles of Al^{3+} ion will occupy 1/2 moles of lattice sites.
So, Number of vacancies = \cfrac 12 \ moles = \cfrac 12 \times 6.022 \times 10^{23}= 3.0125 \times 10^{23}
We know that,
density \ = \cfrac {Mass}{Volume} \Rightarrow Volume = \cfrac {Mass}{Density}
\Rightarrow Mass = density \times volume
Now, given densities of gold and quartz are respectively d_1 and d_2
Masses of gold and quartz are x \ g and y \ g respectively.
Volume \ of \ gold = \cfrac {x}{d_1}.....(i)
Volume \ of \ quartz = \cfrac {y}{d_2}.....(ii)
Volume \ of \ Nugget = \cfrac {Mass \ of \ nugget}{density \ of \ Nugget}
=\cfrac {x+y}{d}.....(iii)
Now, volume \ of \ nugget = volume \ of \ gold + volume \ of \ quartz
\Rightarrow \cfrac {x+y}{d}= \cfrac {x}{d_1} + \cfrac {y}{d_2}
If x is the length of body diagonal, then the distance between two nearest cations in rock salt structure is:
In NaCl structure, Cl^- ions form FCC and Na^+ are present at body center and edge center.
Now, if a is the edge length of the cube, then body diagonal is given by
d= \sqrt3 a
Given, d=x
\Rightarrow x = \sqrt 3 a.......(i)
Since, atoms at face centere and corners are touching each other, so distance between two nearest cation = \cfrac {\sqrt 2 a}2........(ii)
(i) and (ii)
\Rightarrow \cfrac {x}{\sqrt 3}= \sqrt 2 d’
\Rightarrow d’ = \cfrac {x}{\sqrt 6}
Area od square of side length l=l^2
Number of atoms in unit cell = 1+ 4 \times \cfrac 14 = 2
Let, radius of each circle be r
Area occupied = \pi r^2 \times 2
Now, diagonal of square = l \sqrt 2
\Rightarrow 4r = l \sqrt 2
r= \cfrac l{2 \sqrt 2}
Now, Packing efficiency = \cfrac {Area \ occupied}{Total area} \times 100%
= \cfrac {2 \pi r^2}{l^2} \times 100%
=\cfrac {2 \pi l^2}{(2 \sqrt 2)^2 \times l^2} \times 100%
= \cfrac {100 \times 2 \pi}{8}%
\Rightarrow Packing Efficiency = 78.54%
Different types of crystal system arised due to difference in the arrangement of atoms in the space.
Theoretical density of crystal, \rho = \cfrac {nM}{N_o a^3} \ g/cm^3
In NaCl crystal, there are 4 formula units per unit cell. So, n=4
\Rightarrow a^3 = \cfrac {nM}{N_o \rho}= \cfrac {4 \times 58.5}{6.022 \times 10^{23} \times 3.165}
\Rightarrow a= (123)^{(1/3)} \times (10^{-24})^{1/3} \ cm
\Rightarrow a = 4.97 \times 10^{-8} \ cm = 497 \ pm
Now, in NaCl, Cl^- ions are present at corners and at face centers and Na^+ ions are present at edge centers and body centers.
So, distance between Na^+ and Cl^-= \cfrac a2= \cfrac {497}{2}=248.5 \ pm
Since, cube close packing is analogous to face-centered cubic (fcc) structure and number of atoms in a fcc crystal is 4 \ per \ unit \ cell
So, number of tetrahedral voids = 2 \times Number \ of \ atoms \ in \ unit \ cell
Hence, Option "D" is the correct answer.
Square close packing in 2D: In this kind of arrangement the AAA type of pattern of stacking is followed. The coordination number of each sphere in this arrangement will be 4.
Hexagonal close packing in 2D has triangular voids as shown in the figure.
Na_2O structure is antifluorite type structure in which O^{2-} ions forms ccp (cubic close packing) and Na^{+} occupy all tetrahedral voids.
Please disable the adBlock and continue. Thank you.