A vertical closed cylinder is separated into two parts by a...

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} > {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 vertical closed cylinder is separated into two parts by a frictionless piston of mass m and of negligible thickness.

A vertical closed cylinder is separated into two parts by a frictionless piston of mass m and of negligible thickness.

Options

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 $\dfrac{RT}{gl}\left[\dfrac{2{l}_{1} + {l}_{2}}{{l}_{1}{l}_{2}}\right]$

Correct answer

B. $\dfrac{nRT}{g}\left[\dfrac{{l}_{1} - {l}_{2}}{{l}_{1}{l}_{2}}\right]$

Step-by-step solution

A vertical closed cylinder is separated into two parts by a frictionless piston of mass m and of negligible thickness.

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

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

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

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

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