The left and right compartments of a thermally isolated container...

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{\mathrm{k} L}{\mathrm{~A}} \alpha

, then the value of

\alpha

\qquad

Correct answer

Extension in spring

\begin{aligned} & x=0.5 \mathrm{~L}-0.4 \mathrm{~L} \\ & =0.1 \mathrm{~L} \end{aligned}

FBD of piston The left and right compartments of a thermally isolated container of length L are separated by a thermally conducting,

\begin{aligned} & \mathrm{kx}+\mathrm{P}_2 \mathrm{~A}=\mathrm{P}_1 \mathrm{~A} \\ & \mathrm{P}_2 \mathrm{~A}=\mathrm{P}_1 \mathrm{~A}-\mathrm{kx}\end{aligned}

P_2=P_1-\frac{k L}{A(10)} \qquad ....(i)

\begin{aligned} & \mathrm{P}_1 \mathrm{~V}=\mathrm{n}_1 \mathrm{RT} \\ & \mathrm{P}_2 \mathrm{~V}=\mathrm{n}_2 \mathrm{RT} \\ & \frac{\mathrm{P}_1}{\mathrm{P}_2}=\frac{\mathrm{n}_1}{\mathrm{n}_2}=\frac{3}{2}\end{aligned}

\mathrm{P}_1=\frac{3}{2} \mathrm{P}_2 \qquad ....(ii)

\begin{aligned} & 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} \alpha \\ & \alpha=\frac{1}{5}=0.2\end{aligned}