A vertical closed cylinder is separated into two parts by a frictionless piston of mass m and negligible thickness. The piston is free to move along the length of the cylinder. The length above the piston is l₁ and that below the piston is l₂, such that l₁ l₂. Each part contains n moles of an ide…

Question

A vertical closed cylinder is separated into two parts by a frictionless piston of mass m and negligible thickness. The piston is free to move along the length of the cylinder. The length above the piston is l₁ and that below the piston is l₂, such that l₁ > l₂. Each part 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, g is acceleration due to gravity)

Accepted Answer

(4) (nRT)/g [(l₁ - l₂)/(l₁l₂)]. Let the cylinder's cross-section be A. The upper gas occupies Al₁, the lower Al₂. Each part holds n moles at the same T, so P₁=(nRT)/(Al₁) (above) and P₂=(nRT)/(Al₂) (below). The piston is in equilibrium: the gas below must support the gas above plus the piston's weight. P₂A=P₁A+mg⇒ P₂-P₁=(mg)/A. (nRT)/A(1/(l₂)-1/(l₁))=(mg)/A, and the A cancels. m=(nRT)/g((l₁-l₂)/(l₁l₂)).

Step-by-step solution

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