JEE Advanced 2023 — Paper 2 — CHEM-II solution

JEE Advanced2023Paper 2CHEM-II

Atoms of metals $x$ , $y$ , and $z$ form face-centred cubic (fcc) unit cell of edge length $L_{x}$ , body-centred cubic (bcc) unit cell of edge length $L_{y}$ , and simple cubic unit cell of edge length $L_{z}$ , respectively. If

$r_{z} = \frac{3}{2} r_{y}$ ; $r_{y} = \frac{8}{3} r_{x}$ ; $M_{z} = \frac{3}{2} M_{y}$ and $M_{z} = 3 M_{x}$ ,

then the correct statement(s) is (are)

[Given: $M_{x}$ , $M_{y}$ , and $M_{z}$ are molar masses of metals $x$ , $y$ , and $z$ , respectively. $r_{x}$ , $r_{y}$ , and $r_{z}$ are atomic radii of metals $x$ , $y$ , and $z$ , respectively.]

A
Packing efficiency of unit cell of $x$ $>$ Packing efficiency of unit cell of $y$ $>$ Packing efficiency of unit cell of $z$
B
$L_{y} > L_{z}$
C
$L_{x} > L_{y}$
D
Density of $x$ $>$ Density of $y$

Solution

Edge lengths. For fcc, $L_{x} = 22 r_{x}$ . For bcc, $L_{y} = \frac{4}{3} r_{y}$ . For simple cubic, $L_{z} = 2 r_{z}$ .

Express everything in terms of $r_{x}$ using the given radius ratios.

$r_{y} = \frac{8}{3} r_{x}$ , hence $r_{z} = \frac{3}{2} \cdot \frac{8}{3} r_{x} = 4 r_{x}$ .

Therefore $L_{x} = 22 r_{x} \approx 2.83 r_{x}$ , $L_{y} = \frac{4}{3} \cdot \frac{8}{3} r_{x} = \frac{32}{3} r_{x} \approx 10.67 r_{x}$ , and $L_{z} = 2 \cdot 4 r_{x} = 8 r_{x}$ .

So $L_{y} > L_{z} > L_{x}$ . Option (B) is correct; (C) is incorrect.

Packing efficiency. fcc: $π / (32) \approx 74%$ . bcc: $3 π /8 \approx 68%$ . Simple cubic: $π /6 \approx 52%$ . Thus PE( $x$ ) $>$ PE( $y$ ) $>$ PE( $z$ ). Option (A) is correct.

Densities. Density $= \frac{Z M}{N _{A} L ^{3}}$ . For fcc $Z = 4$ and for bcc $Z = 2$ . From $M_{z} = 3 M_{x}$ and $M_{z} = \frac{3}{2} M_{y}$ , we get $M_{y} = 2 M_{x}$ .

$\frac{d _{x}}{d _{y}} = \frac{4 M _{x}}{2 M _{y}} \cdot \frac{L _{y}^{3}}{L _{x}^{3}} = \frac{4 M _{x}}{2 ( 2 M _{x} )} \cdot (\frac{32/3}{2 2})^{3} = (\frac{32/3}{2 2})^{3} ≫ 1$ .

So density of $x$ is much greater than density of $y$ . Option (D) is correct.

Correct options: (A), (B), (D).

← Back to full paper

Practice more CHEM-II

Concept-wise practice with instant solutions on Fundamenthol.

Start practicing →