JEE Advanced 2023 · Paper 2 · Chemistry Q6

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ChemistryMultiple correct+4 / -2Question 40 of 51

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 $r_z = \dfrac{\sqrt3}{2}r_y$ ; $r_y = \dfrac{8}{\sqrt3}r_x$ ; $M_z = \dfrac32 M_y$ and $M_z = 3M_x$ , 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.]

Packing efficiency of unit cell of x > Packing efficiency of unit cell of y > Packing efficiency of unit cell of z

L_y > L_z

L_x > L_y

Density of x > Density of y

Answer and worked solution

Official answer: A, B, D

Edge lengths in terms of $r_x$ .

fcc: $L_x=2\sqrt2\,r_x$ .
bcc: $L_y=\frac{4r_y}{\sqrt3}$ . With $r_y=\frac{8}{\sqrt3}r_x$ , $L_y=\frac{32}{3}r_x\approx10.7\,r_x$ .
simple cubic: $L_z=2r_z$ . With $r_z=\frac{\sqrt3}2r_y=4r_x$ , $L_z=8r_x$ .

(A) True. Packing efficiencies are 74% (fcc) > 68% (bcc) > 52% (sc).

(B) True. $L_y\approx10.7\,r_x>L_z=8\,r_x$ .

(C) False. $L_x\approx2.83\,r_x<L_y$ .

(D) True. $\rho=\dfrac{ZM}{N_AL^3}$ , and $M_y=\frac23M_z=2M_x$ :

$\rho_x\propto\dfrac{4M_x}{(2\sqrt2)^3}=0.177\,M_x$ ;
$\rho_y\propto\dfrac{2\cdot2M_x}{(32/3)^3}=0.0033\,M_x$ .

So x is much denser.

Topics:Packing efficiency Crystal lattices and unit cells Density of crystals

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