Two samples of gas $A$ and $B$ are initially at the same pressure and temperature. They are compressed from volume $V$ to $V/2$. If $A$ is compressed isothermally and $B$ is compressed adiabatically,then the final pressure of $A$ is:

One mole of an ideal gas is taken through a cyclic process with alternating isothermal and adiabatic curves. In the $P-V$ diagram, $AB, CD, EF$ are isothermal curves at absolute temperatures $T_1, T_2,$ and $T_3$ respectively, and $BC, DE,$ and $FA$ are adiabatic curves. If $\frac{V_B}{V_A} = 2$ and $\frac{V_D}{V_C} = 2$, then for the cycle shown in the figure, four statements are made below. (Figure is not drawn to scale)
Statement $1$: Ratio of volumes $\frac{V_E}{V_F} = 4$
Statement $2$: Magnitude of work done in isothermal compression $EF$ is $2RT_3 \ln(2)$
Statement $3$: Ratio of heat supplied to the gas in process $AB$ to heat rejected by the gas in process $EF$ is $\frac{T_1}{T_3}$
Statement $4$: Net work done by the gas in the cycle $ABCDEFA$ is $(T_1 + T_2 - 2T_3) R \ln(2)$
Find the number of correct statements given for the cyclic process followed by the gas.

$A$ heating element of resistance $r$ is fitted inside an adiabatic cylinder which carries a frictionless piston of mass $m$ and cross-sectional area $A$. The cylinder contains one mole of a diatomic gas. The temperature of the gas varies with time $t$ as $T = \alpha t + \frac{1}{2} \beta t^2$ (where $\alpha$ and $\beta$ are constants),while the pressure remains constant. The atmospheric pressure above the piston is $P_0$. Then:

In the thermodynamic processes, which of the following statements is $NOT$ true?

For the $P-V$ diagram of a thermodynamic cycle as shown in the figure,processes $BC$ and $DA$ are isothermal. Which of the corresponding graphs is correct?

Heat energy of $735\,J$ is given to a diatomic gas allowing the gas to expand at constant pressure. Each gas molecule rotates around an internal axis but does not oscillate. The increase in the internal energy of the gas will be $..........\,J$