JEE Advanced 2025 Paper 2 · Q13 · Kinetic Theory of Gases
The left and right compartments of a thermally isolated container of length $L$ are separated by a thermally conducting, movable piston of area $A$. The left and right compartments are filled with $3/2$ and $1$ moles of an ideal gas, respectively. In the left compartment the piston is attached by a spring with spring constant $k$ and natural length $2L/5$. In thermodynamic equilibrium, the piston is at a distance $L/2$ from each edge of the container. Under the above conditions, if the pressure in the right compartment is $P_2 = \alpha\,(k L)/A$, then the value of $\alpha$ is __________.
Reveal answer + step-by-step solution
Correct answer:0.2
Solution
Spring extension $x = L/2 - 2L/5 = L/10$ (spring is stretched). The stretched spring pulls the piston to the LEFT (toward the natural-length position), so it acts on the piston in the same direction as the right gas pressure. Force balance on piston: $P_1 A = P_2 A + k x$. Volumes are equal ($V_1 = V_2 = A L/2$), and the piston is thermally conducting so $T_1 = T_2 = T$. Ideal gas: $P_1/P_2 = n_1/n_2 = (3/2)/1 = 3/2 \to P_1 = 3 P_2/2$. Sub: $3 P_2/2 - P_2 = k L/(10 A) \to P_2/2 = k L/(10 A) \to P_2 = k L/(5 A) = 0.2\,(k L/A)$. Hence $\alpha = 0.2$.
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