The following figure shows the $P-T$ graph for four processes $A, B, C,$ and $D$. Select the correct alternative.

Consider one mole of helium gas enclosed in a container at initial pressure $P_1$ and volume $V_1$. It expands isothermally to volume $4 V_1$. After this,the gas expands adiabatically and its volume becomes $32 V_1$. The work done by the gas during isothermal and adiabatic expansion processes are $W_{\text{iso}}$ and $W_{\text{adia}}$,respectively. If the ratio $\frac{W_{\text{iso}}}{W_{\text{adia}}} = f \ln 2$,then $f$ is:

$A$ 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?

The ratio of the slopes of isothermal and adiabatic curves is

The ratio of the specific heat capacities of a gas is $1.5$. When the gas undergoes an adiabatic process, its volume is doubled and its pressure becomes $P_1$. When the gas undergoes an isothermal process, its volume is doubled and its pressure becomes $P_2$. If $P_1 = P_2$, what is the ratio of the initial pressures of the gas when it undergoes adiabatic and isothermal processes?

Two identical adiabatic vessels are filled with oxygen at pressure $P_1$ and $P_2$ $(P_1 > P_2)$. The vessels are interconnected with each other by a non-conducting pipe. If $U_{01}$ and $U_{02}$ denote the initial internal energy of oxygen in the first and second vessel respectively,and $U_{f1}$ and $U_{f2}$ denote the final internal energy values,then: