Question 44 of 210

A vertical closed cylinder is separated into two parts by a frictionless piston of mass $m$ and of negligible thickness. The piston is free to move along the length of the cylinder. The length of the cylinder above piston is ${l}_{1},$ and that below the piston is ${l}_{2},$ such that ${l}_{1}\gt {l}_{2}.$ Each part of the cylinder contains $n$ moles of an ideal gas at equal temperature $T.$ If the piston is stationary, its mass $m$ will be given by:

($R$ is universal gas constant and $g$ is the acceleration due to gravity)

A$\dfrac{RT}{ng}\left[\dfrac{{l}_{1}-3{l}_{2}}{{l}_{1}{l}_{2}}\right]$B$\dfrac{nRT}{g}\left[\dfrac{{l}_{1}-{l}_{2}}{{l}_{1}{l}_{2}}\right]$C$\dfrac{nRT}{g}\left[\dfrac{{1}_{}}{{l}_{2}}+\dfrac{1}{{l}_{1}}\right]$D$\dfrac{RT}{gl}\left[\dfrac{2{l}_{1}+{l}_{2}}{{l}_{1}{l}_{2}}\right]$

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Answer: (B) $\dfrac{nRT}{g}\left[\dfrac{{l}_{1}-{l}_{2}}{{l}_{1}{l}_{2}}\right]$

${P}_{1}+\dfrac{mg}{A}={P}_{2}$

$\Rightarrow \dfrac{nRT}{{l}_{1}A}+\dfrac{mg}{A}=\dfrac{nRT}{{l}_{2}A}$

$\Rightarrow m=\dfrac{nRT}{g}\left(\dfrac{1}{{l}_{2}}-\dfrac{1}{{l}_{1}}\right)$

$=\dfrac{nRT}{g}\left(\dfrac{{l}_{1}-{l}_{2}}{{l}_{1}{l}_{2}}\right)$