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. $\frac{nRT}{g}[\frac{1}{{\ell }_2}+\frac{1}{{\ell }_1}]$
• B. $\frac{nRT}{g}\left[\frac{{\ell }_1-{\ell }_2}{{\ell }_1{\ell }_2}\right]$
• C. $\frac{RT}{g}\left[\frac{2{\ell }_1+{\ell }_2}{{\ell }_1{\ell }_2}\right]$
• D. $\frac{RT}{g}\left[\frac{{\ell }_1-3{\ell }_2}{{\ell }_1{\ell }_2}\right]$

Answer 1

The correct option is B $\frac{nRT}{g}\left[\frac{{\ell }_1-{\ell }_2}{{\ell }_1{\ell }_2}\right]$

$P_{2}A=P_{1}A+mg$
$\frac{nRT.A}{A{\ell }_2}=\frac{nRT.A}{A{\ell }_1}+mg$
$nRT(\frac{1}{{\ell }_2}-\frac{1}{{\ell }_1})=mg$
$m=\frac{nRT}{g}\left(\frac{{\ell }_1-{\ell }_2}{{\ell }_1{\ell }_2}\right)$