Question

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 the piston is $\ell_1$, and that below the piston is $\ell_2$ , such that $\ell_1$>$\ell_2$. 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. $\frac{nRT}{g} \bigg[\frac{1}{\ell_2} + \frac{1}{\ell_1}\bigg]$
• B. $\frac{nRT}{g} \bigg[\frac{\ell_1 - \ell_2}{\ell_1\ell_2} \bigg]$
• C. $\frac{RT}{g} \bigg[\frac{2\ell_1 + \ell_2}{\ell_1 \ell_2} \bigg]$
• D. $\frac{RT}{g} \bigg[\frac{\ell_1 - 3\ell_2}{\ell_1\ell_2} \bigg]$

Answer 1

Answer: (B)

$\frac{nRT}{g} \bigg[\frac{\ell_1 - \ell_2}{\ell_1\ell_2} \bigg]$

Answer 2

$P_2 A \, = \, P_1A \, + \, mg$
$\frac{nRT.A}{A\ell_2} \, = \frac{nRT.A}{A\ell_2} +mg$
$nRT \bigg(\frac{1}{\ell_2} \, - \, \frac{1}{\ell_1}\bigg) = mg$
$m \, = \, \frac{nRT}{g} \, \bigg(\frac{\ell_1\ell_2}{\ell_1 .\ell_2}\bigg)$