In the given $P-V$ diagram,a monoatomic gas $\left(\gamma = \frac{5}{3}\right)$ is first compressed adiabatically from state $A$ to state $B$. Then it expands isothermally from state $B$ to state $C$. [Given: $\left(\frac{1}{3}\right)^{0.6} \simeq 0.5, \ln 2 \simeq 0.7$].
Which of the following statement$(s)$ is(are) correct?
$(A)$ The magnitude of the total work done in the process $A \rightarrow B \rightarrow C$ is $144 \text{ kJ}$.
$(B)$ The magnitude of the work done in the process $B \rightarrow C$ is $84 \text{ kJ}$.
$(C)$ The magnitude of the work done in the process $A \rightarrow B$ is $60 \text{ kJ}$.
$(D)$ The magnitude of the work done in the process $C \rightarrow A$ is zero.

An ideal monoatomic gas is confined in a horizontal cylinder by a spring-loaded piston (as shown in the figure). Initially,the gas is at temperature $T_1$,pressure $P_1$,and volume $V_1$,and the spring is in its relaxed state. The gas is then heated very slowly to temperature $T_2$,pressure $P_2$,and volume $V_2$. During this process,the piston moves out by a distance $x$. Ignoring the friction between the piston and the cylinder,the correct statement$(s)$ is(are):
$(A)$ If $V_2=2V_1$ and $T_2=3T_1$,then the energy stored in the spring is $\frac{1}{4}P_1V_1$
$(B)$ If $V_2=2V_1$ and $T_2=3T_1$,then the change in internal energy is $3P_1V_1$
$(C)$ If $V_2=3V_1$ and $T_2=4T_1$,then the work done by the gas is $\frac{7}{3}P_1V_1$
$(D)$ If $V_2=3V_1$ and $T_2=4T_1$,then the heat supplied to the gas is $\frac{41}{6}P_1V_1$

$A$ gas is enclosed in a cylinder with a movable frictionless piston. Its initial thermodynamic state at pressure $P_i = 10^5 \text{ Pa}$ and volume $V_i = 10^{-3} \text{ m}^3$ changes to a final state at $P_f = (1/32) \times 10^5 \text{ Pa}$ and $V_f = 8 \times 10^{-3} \text{ m}^3$ in an adiabatic quasi-static process, such that $P^3 V^5 = \text{constant}$. Consider another thermodynamic process that brings the system from the same initial state to the same final state in two steps: an isobaric expansion at $P_i$, followed by an isochoric (isovolumetric) process at volume $V_f$. The amount of heat supplied to the system in the two-step process is approximately: (in $\text{ J}$)

The graph for $y = 1 - e^{-x}$ is represented by (for $x > 0$):

$A$ Carnot cycle for a gas is represented in the pressure-volume $(P-V)$ diagram shown below. Consider the following statements:
$I.$ The area of $ABCD$ is the work done by the gas.
$II.$ The area of $ABCD$ is the total heat absorbed.
$III.$ The change in internal energy of the cycle is zero.
Which of the following is correct?

$A$ soap bubble of radius $r$ contains a monoatomic ideal gas. The gas is heated in such a manner that the bubble remains in mechanical equilibrium. Assuming that the soap material of the bubble has no heat capacity,the molar heat capacity of the gas in the process will be (Neglect atmospheric pressure). (in $R$)