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JEE Advanced 2020 Paper 2

Section: Physics

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Question : 6 of 54

Marks: +1, -0

A thermally isolated cylindrical closed vessel of height

8\ \text{m}

is kept vertically. It is divided into two equal parts by a diathermic (perfect thermal conductor) frictionless partition of mass

8.3\ \text{kg}.

Thus the partition is held initially at a distance of

4\ \text{m}

from the top, as shown in the schematic figure below. Each of the two parts of the vessel contains 0.1 mole of an ideal gas at temperature

300\ \text{K}

. The partition is now released and moves without any gas leaking from one part of the vessel to the other. When equilibrium is reached, the distance of the partition from the top (in

\text{m}

) will be ______.

(take the acceleration due to gravity

=10\ \text{ms}^{-2}

and the universal gas constant

=8.3\ \text{J}\ \text{mol}^{-1}\ \text{K}^{-1})

[JEE Adv 2020 P2]

Your Answer:

Solution:

Assuming the gas to be an ideal mono-atomic gas

∴

C_{v}=\frac{3}{2}R

and area of cross section to be A.

Initially

P_{0} 4 A=0.1 R \times 300

(For both)

Let piston shifts by × meter and final temperature is T.

Then

P_{2} A(4-x)=0.1 R T

\therefore\ P_{2} A= \frac{0.1 R T}{4-x}= \frac{R T}{10(4-x)}

And

P_{1} A= \frac{R T}{10(4+x)}

But finally

P_{2} A-P_{1} A=83

= \frac{R T}{10}\left(\frac{1}{4-x} - \frac{1}{4+x}\right)

\Rightarrow\ \frac{R T}{10}\left(\frac{2x}{16-x^{2}}\right)=83

0.2 C_{v} \times 300 + mg \cdot x

=0.2 C_{v} T

\Rightarrow\ \frac{3}{2} R \times \frac{2}{10} \times 300 + 83 x

= \frac{2}{10} \times \frac{3}{2} R T

\Rightarrow \left(\frac{900 R}{10}+83 x\right)=3\left(\frac{R T}{10}\right)

\left(\frac{300 R}{10}+ \frac{83}{3} x\right)= \frac{R T}{10}

\left(30 R+ \frac{83x}{3}\right)\left(\frac{2x}{16-x^{2}}\right)=83

\Rightarrow\ 83 \left(3+ \frac{x}{3}\right)\left(\frac{2x}{16-x^{2}}\right)=83

\Rightarrow\ \frac{9+x}{3}\frac{2x}{16-x^{2}}=1

\Rightarrow 18x+2x^{2}=48-3x^{2}

\Rightarrow 5x^{2}+18x-48=0

\therefore\ x=1.78 \approx 2

So, the distance from top is

6\ \text{m}

.

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