$A$ monoatomic gas of pressure $P$ having volume $V$ expands isothermally to a volume $2V$ and then adiabatically to a volume $16V$. The final pressure of the gas is (ratio of specific heats $\gamma = \frac{5}{3}$)

$1\,g$ of a liquid is converted to vapour at $3 \times 10^5\,Pa$ pressure. If $10\%$ of the heat supplied is used for increasing the volume by $1600\,cm^3$ during this phase change,then the increase in internal energy in the process will be $............\,J$.

In the figure,a container is shown to have a movable (frictionless) piston on top. The container and the piston are made of perfectly insulating material,allowing no heat transfer between the outside and inside. The container is divided into two compartments by a rigid partition made of a thermally conducting material that allows slow heat transfer. The lower compartment is filled with $2$ moles of an ideal monatomic gas at $700 \ K$,and the upper compartment is filled with $2$ moles of an ideal diatomic gas at $400 \ K$. The heat capacities per mole are: for monatomic gas,$C_v = \frac{3}{2} R, C_p = \frac{5}{2} R$; for diatomic gas,$C_v = \frac{5}{2} R, C_p = \frac{7}{2} R$.
$1.$ Consider the partition to be rigidly fixed so that it does not move. When equilibrium is achieved,the final temperature of the gases will be:
$(A) 550 \ K$ $(B) 525 \ K$ $(C) 513 \ K$ $(D) 490 \ K$
$2.$ Now consider the partition to be free to move without friction so that the pressure of gases in both compartments is the same. Then the total work done by the gases until they achieve equilibrium will be:
$(A) 250 \ R$ $(B) 200 \ R$ $(C) 100 \ R$ $(D) -100 \ R$
Give the answer for questions $1$ and $2$.

An ideal monoatomic gas is confined in a horizontal cylinder by a spring-loaded piston (as shown in the figure). Initially,the gas is at temperature $T_1$,pressure $P_1$,and volume $V_1$,and the spring is in its relaxed state. The gas is then heated very slowly to temperature $T_2$,pressure $P_2$,and volume $V_2$. During this process,the piston moves out by a distance $x$. Ignoring the friction between the piston and the cylinder,the correct statement$(s)$ is(are):
$(A)$ If $V_2=2V_1$ and $T_2=3T_1$,then the energy stored in the spring is $\frac{1}{4}P_1V_1$
$(B)$ If $V_2=2V_1$ and $T_2=3T_1$,then the change in internal energy is $3P_1V_1$
$(C)$ If $V_2=3V_1$ and $T_2=4T_1$,then the work done by the gas is $\frac{7}{3}P_1V_1$
$(D)$ If $V_2=3V_1$ and $T_2=4T_1$,then the heat supplied to the gas is $\frac{41}{6}P_1V_1$

Two moles of helium gas are taken over the cycle $ABCDA$, as shown in the $P-T$ diagram. The net work done on the gas in the cycle $ABCDA$ is ...... $R$.

$A$ cyclic process is shown in the $p-v$ diagram. Which of the following statement$(s)$ is/are true?