In an adiabatic process,the state of a gas is changed from $(P_1, V_1, T_1)$ to $(P_2, V_2, T_2)$. Which of the following relations is correct?

$A$ cylindrical vessel of uniform cross-section containing a gas with $\gamma = 1.5$ is divided into two parts $A$ and $B$ using a piston. Initially, the piston is kept fixed such that part $A$ has pressure $p$ and volume $5V$, and part $B$ has pressure $8p$ and volume $V$. If the piston is released and the gas is allowed to undergo an adiabatic process, then the final volume of the gas in part $A$ is:

An ideal gas having initial pressure $P$, volume $V$ and temperature $T$ is allowed to expand adiabatically until its volume becomes $4V$, while its temperature falls to $T/2$. If the work done by the gas during the expansion is $\alpha PV$, the value of $\alpha$ is

Assertion $(A)$: When an ideal gas is compressed adiabatically,its temperature and the average kinetic energy of the gas molecules increase.
Reason $(R)$: The kinetic energy increases because of collisions of molecules with the moving parts of the wall.

In an adiabatic expansion of a gas, the initial and final temperatures are $T_1$ and $T_2$ respectively. Then, the change in internal energy of the gas is:
$[R = \text{gas constant}, \gamma = \text{adiabatic ratio}]$

$V$ cc volume of a gas having $\gamma = \frac{5}{2}$ is suddenly compressed to $\frac{V}{4}$ cc. The initial pressure of the gas is $P$. The final pressure of the gas will be