An engine operates by taking $n$ moles of an ideal gas through the cycle $ABCDA$ shown in the figure. The thermal efficiency of the engine is: (Take $C_v = 1.5 R$,where $R$ is the gas constant)

Column $I$ contains a list of processes involving the expansion of an ideal gas. Match this with Column $II$ describing the thermodynamic change during this process.

Column $I$	Column $II$
$(A)$ An insulated container has two chambers separated by a valve. Chamber $I$ contains an ideal gas and Chamber $II$ has a vacuum. The valve is opened.	$(p)$ The temperature of the gas decreases
$(B)$ An ideal monoatomic gas expands to twice its original volume such that its pressure $P \propto V^{-2}$	$(q)$ The temperature of the gas increases or remains constant
$(C)$ An ideal monoatomic gas expands to twice its original volume such that its pressure $P \propto V^{-4/3}$	$(r)$ The gas loses heat
$(D)$ An ideal monoatomic gas expands such that its pressure $P$ and volume $V$ follow the behavior shown in the graph	$(s)$ The gas gains heat

One mole of an ideal monoatomic gas undergoes the process $A \rightarrow B \rightarrow C \rightarrow D \rightarrow A$ as shown in the graph. The work done during the process is

The internal energy of an ideal diatomic gas corresponding to volume $V$ and pressure $P$ is $2.5 PV$. The gas expands from $1 \text{ litre}$ to $2 \text{ litre}$ at a constant pressure of $10^5 \text{ N/m}^2$. The heat supplied to the gas is: (in $\text{ J}$)

Obtain the relation between the coefficient of performance and the efficiency of a heat engine.

Two identical vessels $A$ and $B$ contain equal amounts of an ideal monoatomic gas. The piston of $A$ is fixed,while the piston of $B$ is free to move. The same amount of heat $\Delta Q$ is absorbed by both $A$ and $B$. If the internal energy of $B$ increases by $100 \, J$,what is the change in the internal energy of $A$ (in $, J$)?