Starting with the same initial conditions,an ideal gas expands from volume $V_{i}$ to $V_{f}$ in three different ways. The work done by the gas is $W_{1}$ if the process is purely isothermal,$W_{2}$ if the process is purely adiabatic,and $W_{3}$ if the process is purely isobaric. Then,choose the correct option.

An engine runs between a reservoir at temperature $200 \,K$ and a hot body which is initially at temperature of $600 \,K$. If the hot body cools down to a temperature of $400 \,K$ in the process,then the maximum amount of work that the engine can do (while working in a cycle) is (the heat capacity of the hot body is $1 \,J/K$).

Two moles of monoatomic gas is expanded from $(P_0, V_0)$ to $(P_0, 2V_0)$ under isobaric condition. Let $\Delta Q_1$,$\Delta W_1$,and $\Delta U_1$ be the heat given to the gas,the work done by the gas,and the change in internal energy,respectively. Now,the monoatomic gas is replaced by a diatomic gas,with other conditions remaining the same. The corresponding values in this case are $\Delta Q_2$,$\Delta W_2$,and $\Delta U_2$. Then:

$A$ 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 $R$)

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

$A$ heat engine is involved with an exchange of heat of $1915\, J,$ $-40\, J,$ $+125\, J,$ and $-Q\, J$ during one cycle,achieving an efficiency of $50.0 \%$. The value of $Q$ is ....... $J$.