Two compartments of a thermally isolated container of length L are separated by a thermally conducti

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Question

Two compartments of a thermally isolated container of length L are separated by a thermally conducting movable piston of area A. Left compartment has 3/2 moles, right has 1 mole of ideal gas. In the left compartment, the piston is attached by a spring of constant k and natural length 2L/5. In equilibrium, piston is at L/2 from each edge. If pressure in right compartment is P = βkL/A, find β.

Accepted Answer

Correct Answer: β = 0.20Step-by-step Solution:At equilibrium, the piston is at L/2 from the left edge and L/2 from the right edge.The spring has natural length L₀ = 2L/5 and is attached to the left wall and the piston.Current length of spring = distance from left wall to piston = L/2.Extension of spring: x = L/2 − 2L/5 = 5L/10 − 4L/10 = L/10.Spring force on piston: F_spring = kx = kL/10 (pushing piston rightward, since spring is compressed if L/2 > L₀... wait: L/2 = 5L/10 > 2L/5 = 4L/10, so the spring is stretched, pulling piston leftward).Wait: natural length = 2L/5 and current length = L/2. Since L/2 > 2L/5, the spring is EXTENDED. A spring extended pulls both ends toward each other, so it pulls piston toward left wall (leftward on piston).Force balance on piston (taking rightward as positive):P₂·A − P₁·A − k·x = 0 (spring pulls piston left, so its contribution is −kx on the rightward direction... actually if spring is extended and pulls piston leftward: P₂·A = P₁·A + kx)Wait, the spring is in the left compartment connecting left wall and piston. Extended spring: force on piston = kx toward left wall (leftward). Pressure P₁ acts on piston from left (rightward) and P₂ from right (leftward).Force balance: P₁·A − P₂·A − k·x = 0... No: P₁ pushes piston right, P₂ pushes piston left, spring pulls piston left (extended spring).P₁·A = P₂·A + kxP₁ = P₂ + kx/A = P₂ + k(L/10)/A = P₂ + kL/(10A).Since the partition is thermally conducting, temperatures equilibrate: T₁ = T₂ = T (common temperature).Ideal gas for right compartment: P₂·(L/2)·A = 1·RT → P₂ = 2RT/(LA).For left compartment: P₁·(L/2)·A = (3/2)RT → P₁ = 3RT/(LA).Check: P₁ = P₂ + kL/(10A) → 3RT/(LA) = 2RT/(LA) + kL/(10A)RT/(LA) = kL/(10A) → RT/L = kL/10 → RT = kL²/10.So: P₂ = 2RT/(LA) = 2kL/(10LA·L) · L² ... P₂ = 2RT/(LA) = 2(kL²/10)/(LA) = 2kL/(10A) = kL/(5A) = 0.2·kL/A.Therefore: P₂ = β·kL/A → β = 0.20.Key Concepts:Mechanical equilibrium of piston: The net force on a piston in equilibrium is zero: ΣF = P₁A − P₂A ± F_spring = 0. The spring force depends on whether it's compressed or extended relative to its natural length.Thermal equilibrium through conducting partition: A thermally conducting wall allows heat transfer until temperatures equalize. At equilibrium, both compartments are at the same temperature T.Ideal gas law for both compartments: P₁V₁ = n₁RT and P₂V₂ = n₂RT (same T). The ratio P₁/P₂ = (n₁V₂)/(n₂V₁) = (3/2 × L/2)/(1 × L/2) = 3/2. This gives P₁ = (3/2)P₂, consistent with our derivation.Spring extension/compression: Always check whether the current length > or < natural length to determine the sign of the spring force. Extended spring pulls (tension); compressed spring pushes (compression).

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