Three moles of an ideal gas undergo a cyclic process $ABCA$ as shown in the figure. The pressure, volume, and absolute temperature at points $A, B,$ and $C$ are respectively $(P_1, V_1, T_1)$, $(P_2, 3V_1, T_1)$, and $(P_2, V_1, T_2)$. Then the total work done in the cycle $ABCA$ is (where $R$ is the universal gas constant).

The internal energy of the air in a room of volume $V$ at temperature $T$, with outside pressure $P$ increasing linearly with time, varies as

Two cylinders $A$ and $B$ fitted with pistons contain an equal amount of an ideal diatomic gas at temperature $T$ $K$. The piston of cylinder $A$ is free to move,while that of $B$ is held fixed. The same amount of heat is given to the gas in each cylinder. If the rise in temperature of the gas in $A$ is $dT_{A}$,then the rise in temperature of the gas in cylinder $B$ is (where $\gamma = \frac{C_{P}}{C_{V}}$):

$n$ moles of a perfect gas undergo a cyclic process $ABCA$ (see figure) consisting of the following processes:
$A \rightarrow B :$ Isothermal expansion at temperature $T$ so that the volume is doubled from $V_{1}$ to $V_{2}=2V_{1}$ and pressure changes from $P_{1}$ to $P_{2}$.
$B \rightarrow C :$ Isobaric compression at pressure $P_{2}$ to initial volume $V_{1}$.
$C \rightarrow A :$ Isochoric change leading to a change of pressure from $P_{2}$ to $P_{1}$.
Total work done in the complete cycle $ABCA$ is

An ideal gas undergoes a cyclic process as shown in the density-pressure graph. During the process $AB$, the work done $|W_{AB}| = 70\,J$. During the process $BC$, the gas absorbs $150\,J$ of heat. During the process $CA$, the gas undergoes expansion and does $210\,J$ of work. Which of the following statements is correct?

There are two samples $A$ and $B$ of a certain gas,which are initially at the same temperature and pressure. Both are compressed from volume $V$ to $\frac{V}{2}$. Sample $A$ is compressed isothermally while sample $B$ is compressed adiabatically. The final pressure of $A$ is