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 l1, and that below the piston is l2, such that l1l2 . 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)

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 piston is l1, and that below the piston is l2, such that l1>l2 . 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) (R T/n g)[(l1-3 l2/l1 l2)] (B) (n R T/g)[(l1-l2/l1 l2)] (C) (n R T/g)[(1/l2)+(1/l1)] (D) (R T/g l)[(2 l1+l2/l1 l2)]

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="Solution" src="https://cdn.tardigrade.in/q/nta/p-rkqwwfqkyxt0.png" /> P1+(m g/A)=P2 ⇒ (n R T/l1 A)+(m g/A)=(n R T/l2 A) ⇒ m=(n R T/g)((1/l2)-(1/l1)) =(n R T/g)((l1-l2/l1 l2))

$\frac{RT}{n g} [\frac{l _{1} - 3 l _{2}}{l _{1} l _{2}}]$

$\frac{n RT}{g} [\frac{l _{1} - l _{2}}{l _{1} l _{2}}]$

$\frac{n RT}{g} [\frac{1}{l _{2}} + \frac{1}{l _{1}}]$

$\frac{RT}{g l} [\frac{2 l _{1} + l _{2}}{l _{1} l _{2}}]$

$P_{1} + \frac{m g}{A} = P_{2}$
$\Rightarrow \frac{n RT}{l _{1} A} + \frac{m g}{A} = \frac{n RT}{l _{2} A}$
$\Rightarrow m = \frac{n RT}{g} (\frac{1}{l _{2}} - \frac{1}{l _{1}})$
$= \frac{n RT}{g} (\frac{l _{1} - l _{2}}{l _{1} l _{2}})$

ngRT​[l1​l2​l1​−3l2​​]

gnRT​[l1​l2​l1​−l2​​]

gnRT​[l2​1​+l1​1​]

glRT​[l1​l2​2l1​+l2​​]

P1​+Amg​=P2​ ⇒l1​AnRT​+Amg​=l2​AnRT​ ⇒m=gnRT​(l2​1​−l1​1​) =gnRT​(l1​l2​l1​−l2​​)

$\frac{RT}{n g} [\frac{l _{1} - 3 l _{2}}{l _{1} l _{2}}]$

$\frac{n RT}{g} [\frac{l _{1} - l _{2}}{l _{1} l _{2}}]$

$\frac{n RT}{g} [\frac{1}{l _{2}} + \frac{1}{l _{1}}]$

$\frac{RT}{g l} [\frac{2 l _{1} + l _{2}}{l _{1} l _{2}}]$

$P_{1} + \frac{m g}{A} = P_{2}$
$\Rightarrow \frac{n RT}{l _{1} A} + \frac{m g}{A} = \frac{n RT}{l _{2} A}$
$\Rightarrow m = \frac{n RT}{g} (\frac{1}{l _{2}} - \frac{1}{l _{1}})$
$= \frac{n RT}{g} (\frac{l _{1} - l _{2}}{l _{1} l _{2}})$