Solid State Test

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Solid State Test - 47

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Question 1
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How many unit cells are there in $$1.00$$ gm cube shaped ideal crystal $$AB (MF = 60)$$ having $$NaCl$$ type lattice?

A
$$6.02 \times 10^{23}$$

B
$$2.50 \times 10^{21}$$

C
$$1.00 \times 10^{22}$$

D
$$6.02 \times 10^{24}$$

Solution

In $$NaCl$$ type lattice,
$$Na^{+} \longrightarrow$$present octahedral voids
Thus, no. of $$Na^{+}$$ ions=4
$$Cl^{-}$$ is present in fcc structure.
Thus, no. of $$Cl^{-}$$ ions=4
1 unit cell contains four $$NaCl$$ molecules.
Thus, 1 unit cell contains 4$$Na^{+}$$ ion and 4$$Cl^{-}$$ ions.
No. of moles in 1 g of crystal of AB=$$\cfrac{1}{60}$$moles
1 mole contains=$$6.023 \times 10^{23}$$ ions
$$\cfrac{1}{60}$$ moles contains=$$\cfrac{6.023 \times 10^{23}}{60}$$ ions or molecules.
1 unit cell has=8 ions or 4 molecules
or 1 ion=$$\cfrac{1}{8}$$ unit cell
$$\cfrac{6.023 \times 10^{23}}{60}$$ ions=$$\cfrac{1}{8} \times \cfrac{6.023 \times 10^{23}}{60}$$ unit cell
$$\cfrac{6.023 \times 10^{23}}{60}$$ molecules=$$\cfrac{1}{4} \times \cfrac{6.023 \times 10^{23}}{60}$$ unit cell
Thus, 1 g of crystal contains $$2.5 \times 10^{21}$$ unit cell.
Question 2
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In a cubic close packing of spheres in three dimensions, the co-ordination number of each sphere?

A
6

B
9

C
3

D
12

Solution

In cubic close packing the atoms are present at corners and face center of the unit cell.
As the atom present at face center touches $$12$$ atoms of face centers . $$4$$ in its plane , $$8$$ outside the plane..
hence the co- ordination number is $$12$$.
Question 3
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In a hypothetical solid, C atoms are found to form cubical close-packed lattice . A atoms occupy att tetrahedral voids and B atoms occupy all octahedral voids.
A and B atoms are of appropriate size, so that there is no distortion in the ccp lattice of C atoms .Now if a plane as shown in the following figure is cut, then the cross section of this plane will look like?

A

B

C

D

Solution

From the sizes of octahedral and tetrahedral voids, it is clear that the atoms occupying these voids will not touch each other as we have along body diagonal FCC.
Question 4
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The unit cell cube length for $$LiCl$$ (just like $$NaCl$$ structure) is $$5.14 \ \mathring { A } $$. Assuming anion-anion contact, the ionic radius for chloride ion is:

A
$$1.815 \ \mathring { A } $$

B
$$2.8 \ \mathring { A } $$

C
$$3.8 \ \mathring { A } $$

D
$$4.815 \ \mathring { A } $$

Solution

In NaCl structure, anions have face centered cubic arrangement.
In such a unit cell face diagonal is four times the radius of anions.
Face diagonal $$=\sqrt{2}a=\sqrt{2}\times5.14\mathring {A}$$
Face diagonal $$=4r$$
$$4r=\sqrt{2}\times5.14$$
$$r=\dfrac{\sqrt{2}\times5.14}{4}=1.817\mathring A$$
Question 5
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A compound $$CuCl$$ has face centered cubic structure. Its density is $$3.4 \ g \ cm^{-3}$$. The length of unit cell is :

A
$$5.783 \ \mathring { A } $$

B
$$6.783 \ \mathring { A } $$

C
$$7.783 \ \mathring { A } $$

D
$$8.783 \ \mathring { A } $$

Solution

$$Z_{eff}=4; \ \rho=3.4 \ g/cm^3; \ $$ Molecular weight $$=99; \ N_A = $$ Avogadro Number $$= 6.02 \times 10^{23}$$
$$\rho = \cfrac {M_A \times Z_A}{N_A \times a^3}$$
$$a^3= \cfrac {99 \times 4}{6.02 \times 10^23 \times 3.4}=19.34 \times 10^{-23}$$
$$a^3=193.4 \times 10^{-24} \ cm$$
$$a=5.783 \ \mathring A$$
Question 6
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The Aluminium chloride crystallizes in BCC lattice with an edge length of unit cell equal to $$387\ pm $$. If the size of $$Cl^- $$ ion is $$181\ pm $$, the size of $$NH_4^{+} $$ ion would be:

A
$$ 116\ pm$$

B
$$ 154\ pm$$

C
$$174\ pm$$

D
$$206\ pm$$

Solution

In BCC lattice,

$$\sqrt{3}\times edge\ length = 2(r_{+} + r_{-})$$

$$\Rightarrow \dfrac{\sqrt{3}\times 387}{2 } = {^{r}NH_{4}^{+}} +$$ $$ ^{r}Cl^{-}$$

$$\Rightarrow {^{r}NH_{4}^{+}} = \dfrac{387\times \sqrt{3}}{2} - 181\ pm$$

$$\Rightarrow{^{r}NH_{4}^{+}} = 154\ pm$$

Hence, option $$(2)$$ is correct.
Question 7
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Number of space lattices present in triclinic system is:

A
four

B
three

C
two

D
one

Solution

A triclinic system is a crystal described by a vector of unequal length, as in orthorhombic system. Also, the angles between these vectors must all be different and may include $$90^{\circ}$$. It has the minimum symmetry of all lattices.
It has four space lattices:
In it, $$a\ne b\ne c$$ and $$\alpha \ne \beta\ne \gamma=90^{\circ}$$
It is seen in $$K_2Cr_2O_7,CuSO_4.5H_2O$$ and $$H_3BO_3$$
Question 8
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Which is correct statement?

A
Schottky defect occurs when radius of cation is smaller

B
When temperature increases then number of defects decreases

C
Frenkel defect occurs when radius of cation is smaller

D
None of these

Solution

As temperature and number of defects are inversely propotional, with increase in temperature number of defects decreases.
In schottky effect, size of ions are almost same while in frenkel defect, size of cation is small .
Question 9
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For bcc structure, the coordination number and packing are ______ respectively.

A
$$8, \cfrac {\pi \sqrt 3}{8}$$

B
$$6, \cfrac {\pi \sqrt 2}{6}$$

C
$$8, \cfrac {\pi \sqrt 2}{6}$$

D
none of the above

Solution

For bcc structure, $$4r= a
\sqrt 3 \Rightarrow a = \cfrac {4r}{\sqrt 3}$$ (along body diagonal)
$$\text{Packing Fraction}= \cfrac {2 \times \cfrac
43 \pi r^3}{a^3}= \cfrac {2 \times \cfrac 43 \pi r^3}{\cfrac {64 r^3}{3 \sqrt 3}}=
\cfrac {\pi \sqrt 3}{8}$$
Since, the body center atom is surrounded by $$8$$ other atoms. Therefore coordination number is $$8$$.
Question 10
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An element X (At. wt. = 80 g / mol ) having fcc structure, calculate no. of unit cells in 8 gm of X:

A
$$0.4 \times N_A$$

B
$$0.1 \times N_A$$

C
$$4 \times N_A$$

D
None of these

Solution

Effective no. of atoms in a unit cell $$= 4$$N
No. of atoms $$= \dfrac{8}{80} \times N_{A}$$
$$\therefore$$ No. of unit cell $$= \dfrac{N_{A}}{10} \times \dfrac{1}{4}$$