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 and 1 moles of an ideal gas, respectively. [2025]

Question

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 $\frac{3}{2}$ 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 $\frac{2 L}{5}$ . In thermodynamic equilibrium, the piston is at a distance $\frac{L}{2}$ from the left and right edges of the container as shown in the figure. Under the above conditions, if the pressure in the right compartment is $P = \frac{k L}{A} α,$ then the value of $α$ is ________. [2025]

Smart Achievers · Accepted Answer

(0.20)

$Extension in spring$

$x = 0.5 L - 0.4 L = 0.1 L$

FBD of piston

$k x + P_{2} A = P_{1} A$

$P_{2} A = P_{1} A - k x$

$P_{2} = P_{1} - \frac{k L}{A (10)} \dots (i)$

$P_{1} V = n_{1} R T and P_{2} V = n_{2} R T$

$\frac{P_{1}}{P_{2}} = \frac{n_{1}}{n_{2}} = \frac{3}{2}$

$P_{1} = \frac{3}{2} P_{2} \dots (i i)$

$P_{2} = \frac{3}{2} P_{2} - \frac{k L}{10 A}$

$\frac{P_{2}}{2} = \frac{k L}{10 A}$

$P_{2} = \frac{k L}{5 A} = \frac{k L}{A} α$

$∴$ $α = \frac{1}{5} = 0.2$

Solution

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Solution

Ans. (0.20) Extension in spring x=0.5L-0.4L=0.1L FBD of piston kx+P2A=P1A P2A=P1A-kx P2=P1-kLA(10) ⋯(i) P1V=n1RT and P2V=n2RT P1P2=n1n2=32 P1=32P2 ⋯(ii) P2=32P2-kL10A P22=kL10A P2=kL5A=kLAα ∴ α=15=0.2

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