3 questions from 2025
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 \( \dfrac{3}{2} \) 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 \( \dfrac{2L}{5} \). In thermodynamic equilibrium, the piston is at a distance \( \dfrac{L}{2} \) 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 \( P = \dfrac{kL}{A}\alpha \), then the value of \( \alpha \) is ____.
An ideal monatomic gas of \( n \) moles is taken through a cycle \( WXYZW \) consisting of consecutive adiabatic and isobaric quasi-static processes, as shown in the schematic \( V \)-\( T \) diagram. The volume of the gas at \( W \), \( X \) and \( Y \) points are, \( 64 \, \text{cm}^3 \), \( 125 \, \text{cm}^3 \) and \( 250 \, \text{cm}^3 \), respectively. If the absolute temperature of the gas \( T_w \) at the point \( W \) is such that \( nRT_w = 1 \, \text{J} \) (\( R \) is the universal gas constant), then the amount of heat absorbed (in \( \text{J} \)) by the gas along the path \( XY \) is ________. 
The efficiency of a Carnot engine operating with a hot reservoir kept at a temperature of \( 1000 \, \text{K} \) is \( 0.4 \). It extracts \( 150 \, \text{J} \) of heat per cycle from the hot reservoir. The work extracted from this engine is being fully used to run a heat pump which has a coefficient of performance \( 10 \). The hot reservoir of the heat pump is at a temperature of \( 300 \, \text{K} \). Which of the following statements is/are correct:
