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 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)

Engineering

Physics

Pressure Derivation Ideal Gas Equation and Gas Laws

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 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)

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

$\frac{RT}{ng} [\frac{ℓ_{1} - 3 ℓ_{2}}{ℓ_{1} ℓ_{2}}]$

$\frac{nRT}{g} [\frac{1}{ℓ_{2}} + \frac{1}{ℓ_{1}}]$

$\frac{nRT}{g} [\frac{ℓ_{1} - ℓ_{2}}{ℓ_{1} ℓ_{2}}]$

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P₂A – P₁A = mg

$m = \frac{1}{g} (\frac{P_{1} {Aℓ}_{1}}{ℓ_{1}} - \frac{P_{2} {Aℓ}_{2}}{ℓ_{2}})$

$m = \frac{1}{g} (\frac{nRT}{ℓ_{1}} - \frac{nRT}{ℓ_{2}})$

$m = \frac{nRT}{g} (\frac{1}{ℓ_{1}} - \frac{1}{ℓ_{2}})$