JEE Advanced · Chemistry · 15. Solid State

Paragraph:
In hexagonal system of crystals, a frequently encountered arrangement of atoms is described as a hexagonal prism. Here, the top and bottom of the cell are regular hexagons and three atoms are sandwiched in between them. A space-filling model of this structure, called hexagonal close-packed (HCP), is constituted of a sphere on a flat surface surrounded in the same plane by six identical spheres as closely as possible. Three spheres are then placed over the first layer so that they touch each other and represent the second layer. Each one of the three spheres touches three spheres of the bottom layer. Finally, the second layer is covered with a third layer that is identical to the bottom layer in relative position. Assume radius of every sphere to be ' $r$ '.
Question:
The volume of this $\mathrm{HCP}$ unit cell is

A $24 \sqrt{2} r^3$
B $16 \sqrt{2} r^3$
C $12 \sqrt{2} r^3$
D $\frac{64}{3 \sqrt{3}} r^3$

Verified Solution

Answer & Solution

Correct Answer

(A) $24 \sqrt{2} r^3$

Step-by-step Solution

Detailed explanation

Height of unit cell $=\sqrt{\frac{2}{4}} 4 r$
$\text {Base area }=6 \times \frac{\sqrt{3}}{4}(2 r)^2 $
$ \text {Volume }=\frac{6 \sqrt{3}}{4}(2 r)^2 \cdot \sqrt{\frac{2}{3}} 4 r=24 \sqrt{2} r^3$

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

The atomic masses of $He$ and $Ne$ are $4$ and $20 amu$ , respectively. The value of the de Broglie wavelength of $He$ gas at $- 73 ° C$ is "M" times that of the de Broglie wavelength of $Ne$ at $727 ° C$ . M is:

JEE Advanced 2013 Medium

The standard reduction potential data at

25 ° C

is given below.
$E ^o\left( Fe ^{3+}, Fe ^{2+}\right)=+0.77 V ; $
$ E ^o\left( Fe ^{2+}, Fe \right)=-0 . 44 V $
$ E ^o\left( Cu ^{2+}, Cu \right)=+0.34 V ; $
$ E ^o\left( Cu ^{+}, Cu \right)=+0.52 V $
$ E ^o\left[ O _2(g)+4 H ^{+}+4 e ^{-} \rightarrow 2 H _2 O \right]=$ $+1.23~V $
$ E ^o\left[ O _2(g)+2 H _2 O +4 e^{-} \longrightarrow \text { 4OH}^{-}\right]=$ $+0.40~V $
$ E ^o\left( Cr ^{3+}, Cr \right)=-0.74 V ; $
$ E ^o\left( Cr ^{2+}, Cr \right)=-0.91 V$
Match E^o of the redox pair in List I with the values given in List II and select the correct answer using the code given below the lists :

	List I		List II
A.	$E^{ο} ({Fe}^{3 +}, Fe)$	P.	$- 0.18 V$
B.	$E^{ο} (4 H_{2} O ⇌ 4 H^{+} + 4 {OH}^{-})$	Q.	$- 0.4 V$
C	$E^{ο} ({Cu}^{2 +} + Cu \to 2 {Cu}^{+})$	R.	$- 0.04 V$
D.	$E^{ο} ({Cr}^{3 +}, {Cr}^{2 +})$	S.	$- 0.83 V$

JEE Advanced 2013 Hard

The unbalanced chemical reactions given in List I show missing reagent or condition (?) which are provided in list II. match List I with List II and select the correct answer using the code given below the lists :

List - I	List - II
A. $PbO _2+ H _2 SO _4 \xrightarrow{?}$ $PbSO _4 + O _2+$ other product	P. $\text{NO}$
B. $Na _2 S_2 O _3+ H _2 O \xrightarrow{?}$ $NaHSO _4~+$ other product	Q. $\text{I}_2$
C. $N _2 H _4 \xrightarrow{?} N_2 +$ other product	R. Warm
D. $XeF _2 \xrightarrow{?} Xe ~+$ other product	S. $\text{Cl}_2$

JEE Advanced 2013 Easy

When $O_{2}$ is adsorbed on a metallic surface, electron transfer occurs from the metal to $O_{2}$ . The TRUE statement(s) regarding this adsorption is(are)

JEE Advanced 2015 Hard

Explore more questions on app

From JEE Advanced

Explore more questions on app

List - I	List - II
A. \(PbO _2+ H _2 SO _4 \xrightarrow{?}\) \(PbSO _4 + O _2+\) other product	P. \(\text{NO}\)
B. \(Na _2 S_2 O _3+ H _2 O \xrightarrow{?}\) \(NaHSO _4~+\) other product	Q. \(\text{I}_2\)
C. \(N _2 H _4 \xrightarrow{?} N_2 +\) other product	R. Warm
D. \(XeF _2 \xrightarrow{?} Xe ~+\) other product	S. \(\text{Cl}_2\)