Question

A thermally isolated cylindrical closed vessel of height $8m$ is kept vertically. It is divided into two equal parts by a diathermic (perfect thermal conductor) frictionless partition of mass $8.3kg$. Thus the partition is held initially at a distance of $4m$ 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 $300K$. 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 $m$ ) will be ______.
(take the acceleration due to gravity $=10m{s}^{-2}$ and the universal gas constant $=8.3Jmo{l}^{-1}{K}^{-1}$ ).

Answer 1

Answer: 6

Assuming the gas to be an ideal mono-atomic gas
$\therefore C_v=\frac{3}{2}R$
and area of cross section to be A.
Initially
$P_{0}4A=0.1R\times 300$ (For both)
Let piston shifts by $\times$ meter and final temperature is $T$.
Then
$P_{2}A(4-x)=0.1RT$
$\therefore P_{2}A=\frac{0.1RT}{4-x}=\frac{RT}{10(4-x)}$
And $P_{1}A=\frac{RT}{10(4+x)}$
But finally $P_{2}A-P_{1}A=83$
$=\frac{RT}{10}\left(\frac{1}{4-x}-\frac{1}{4+x}\right)$
$\Rightarrow \frac{RT}{10}\left(\frac{2x}{16-x^2}\right)=83$
$0.2C_v\times 300+mg\cdot x=0.2C_{v}T$
$\Rightarrow \frac{3}{2}R\times \frac{2}{10}\times 300+83x=\frac{2}{10}\times \frac{3}{2}RT$
$\Rightarrow \left(\frac{900R}{10}+83x\right)=3\left(\frac{RT}{10}\right)$
$\left(\frac{300R}{10}+\frac{83}{3}x\right)=\frac{RT}{10}$
$\left(30R+\frac{83x}{3}\right)\left(\frac{2x}{16-x^2}\right)=83$
$\Rightarrow 83\left(3+\frac{x}{3}\right)\left(\frac{2x}{16-x^2}\right)=83$
$\Rightarrow \frac{(9+x)}{3}\frac{(2x)}{\left(16-x^2\right)}=1$
$\Rightarrow 18x+2x^2=48-3x^2$
$\Rightarrow 5x^2+18x-48=0$
$\therefore x=1.78\approx 2$
So, the distance from top is $6m$.