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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 gas undergoes a four step cycle as shown in the \( P{-}V \) diagram below. During this cycle, heat is absorbed by the gas in

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 ________.

In the given P–V diagram, a monoatomic gas \(\left( \gamma = \dfrac{5}{3}\right) \) is first compressed adiabatically from state \( A \) to state \( B \). Then it expands isothermally from state \( B \) to state \( C \). [Given: \( \left(\dfrac{1}{3}\right)^{0.6} \simeq 0.5 \), \( \ln 2 \simeq 0.7 \)]

Which of the following statement(s) is(are) correct?

List I describes thermodynamic processes in four different systems. List II gives the magnitudes (either exactly or as a close approximation) of possible changes in the internal energy of the system due to the process.

Which one of the following options is correct?

An incompressible liquid is kept in a container having a weightless piston with a hole. A capillary tube of inner radius \( 0.1 \, \text{mm} \) is dipped vertically into the liquid through the airtight piston hole, as shown in the figure. The air in the container is isothermally compressed from its original volume \( V_0 \) to \( \frac{100}{101} V_0 \) with the movable piston. Considering air as an ideal gas, the height (\( h \)) of the liquid column in the capillary above the liquid level in \( \text{cm} \) is ____________.
[Given : Surface tension of the liquid is \( 0.075 \, \text{N m}^{-1} \), atmospheric pressure is \( 10^{5} \, \text{N m}^{-2} \) acceleration due to gravity (\( g \)) is \( 10 \, \text{m s}^{-2} \), density of the liquid is \( 10^{3} \, \text{kg m}^{-3} \) and contact angle of capillary surface with the liquid is zero]

One mole of an ideal gas expands adiabatically from an initial state \( (T_A, V_0) \) to final state \( (T_f, 5V_0) \). Another mole of the same gas expands isothermally from a different initial state \( (T_B, V_0) \) to the same final state \( (T_f, 5V_0) \). The ratio of the specific heats at constant pressure and constant volume of this ideal gas is \( \gamma \). What is the ratio \( T_A / T_B \)?

An ideal gas of density \( \rho = 0.2\,\text{kg}\,\text{m}^{-3} \) enters a chimney of height \( h \) at the rate of \( \alpha = 0.8\,\text{kg}\,\text{s}^{-1} \) from its lower end, and escapes through the upper end as shown in the figure. The cross-sectional area of the lower end is \( A_1 = 0.1\,\text{m}^2 \) and the upper end is \( A_2 = 0.4\,\text{m}^2 \). The pressure and the temperature of the gas at the lower end are \( 600\,\text{Pa} \) and \( 300\,\text{K} \), respectively, while its temperature at the upper end is \( 150\,\text{K} \). The chimney is heat insulated so that the gas undergoes adiabatic expansion.
Take \( g = 10\,\text{m}\,\text{s}^{-2} \) and the ratio of specific heats of the gas \( \gamma = 2 \). Ignore atmospheric pressure.

Which of the following statement(s) is(are) correct?

A thermally insulating cylinder has a thermally insulating and frictionless movable partition in the middle, as shown in the figure below. On each side of the partition, there is one mole of an ideal gas, with specific heat at constant volume, \( C_v = 2R \). Here, \( R \) is the gas constant. Initially, each side has a volume \( V_0 \) and temperature \( T_0 \). The left side has an electric heater, which is turned on at very low power to transfer heat \( Q \) to the gas on the left side. As a result the partition moves slowly towards the right reducing the right side volume to \( V_0/2 \). Consequently, the gas temperatures on the left and the right sides become \( T_L \) and \( T_R \), respectively. Ignore the changes in the temperatures of the cylinder, heater and the partition.

The value of \( \frac{T_R}{T_0} \) is:

A thermally insulating cylinder has a thermally insulating and frictionless movable partition in the middle, as shown in the figure below. On each side of the partition, there is one mole of an ideal gas, with specific heat at constant volume, \( C_v = 2R \). Here, \( R \) is the gas constant. Initially, each side has a volume \( V_0 \) and temperature \( T_0 \). The left side has an electric heater, which is turned on at very low power to transfer heat \( Q \) to the gas on the left side. As a result the partition moves slowly towards the right reducing the right side volume to \( V_0/2 \). Consequently, the gas temperatures on the left and the right sides become \( T_L \) and \( T_R \), respectively. Ignore the changes in the temperatures of the cylinder, heater and the partition.

The value of \( \frac{Q}{R T_0} \) 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:

As shown in the figure, an insulated container is fitted with a thermally conducting but immovable partition \( (P_1) \) and a freely movable but thermally insulated piston \( (P_2) \). The partition \( P_1 \) with thermal conductivity \( K \), cross sectional area \( A \) and width \( x \) divides the container into two sections, \( S_1 \) and \( S_2 \), each containing one mole of a monoatomic gas. The piston \( P_2 \) moves freely such that the gas in \( S_2 \) is always at the atmospheric pressure. Initially, the difference between the temperatures of \( S_1 \) and \( S_2 \) is \( \Delta T_0 \). The time it takes for the temperature difference to become \( \dfrac{\Delta T_0}{2} \) is \( \dfrac{n x R}{K A} \), where \( R \) is the universal gas constant. The value of \( n \) is: [Given: \( \ln 2 \approx 0.7 \)]

A bubble has surface tension \( S \). The ideal gas inside the bubble has ratio of specific heats \( \gamma = \dfrac{5}{3} \). The bubble is exposed to the atmosphere and it always retains its spherical shape. When the atmospheric pressure is \( P_{a1} \), the radius of the bubble is found to be \( r_1 \) and the temperature of the enclosed gas is \( T_1 \). When the atmospheric pressure is \( P_{a2} \), the radius of the bubble and the temperature of the enclosed gas are \( r_2 \) and \( T_2 \), respectively.
Which of the following statement(s) is(are) correct?

As shown in the figure, five Carnot engines, each with efficiency \( \eta \) and same number of cycles per unit time, are operating between six heat reservoirs. The amount of heat released per cycle by one engine is completely absorbed by the next engine. Consider \( Q_0 \) to be the amount of heat absorbed per cycle by the first engine and \( W \) as the amount of total work done by all the engines per cycle, then the net efficiency of the system is found to be \( \eta_{\text{net}} = \dfrac{W}{Q_0} = \dfrac{211}{243} \). The value of \( \eta \) is:

Ten moles of an ideal monoatomic gas, initially in state \( a \) at atmospheric pressure and temperature \( T_a = 27^\circ C \), is enclosed in a metal cylinder of volume \( V_0 \) fitted with a frictionless piston. The gas is suddenly compressed to state \( b \) with volume \( \frac{V_0}{3} \). Now, keeping the piston stationary, the cylinder is submerged in a water bath of temperature \( 11^\circ C \) until the gas reaches the temperature of the water bath, which is denoted as state \( c \). Finally, while still in the water bath, the piston is brought slowly to its initial position, which is denoted as state \( f \). If \( R \) is universal gas constant, then the correct option(s) is/are: [Given: \( 9^{1/3} = 2.08 \)]

A quasi-static cycle of a monoatomic ideal gas contains an isothermal process \( (ab) \), followed by an isochoric process \( (bc) \) and an adiabatic process \( (ca) \) as shown in the figure. The volumes of the gas are \( V_{1} \) and \( V_{2} \) at \( a \) and \( b \), respectively. If the cycle has heat input \( Q_{\text{in}} \) and output \( Q_{\text{out}} \), then the efficiency of the cycle is defined as \( \eta = \frac{Q_{\text{in}} - Q_{\text{out}}}{Q_{\text{in}}} \). The correct statement(s) is/are: [Given: \( \ln 2 \approx 0.7 \) ]

One mole of a monatomic ideal gas undergoes the cyclic process \( J \rightarrow K \rightarrow L \rightarrow M \rightarrow J \), as shown in the \( P \)-\( T \) diagram.

Match the quantities mentioned in List-I with their values in List-II and choose the correct option. [\( R \) is the gas constant.]