$A$ gas having a volume of $1 \ mm^3$ at $1 \ atm$ pressure is compressed from a temperature of $27^{\circ}C$ to $627^{\circ}C$. What will be the final pressure if the process is adiabatic? (For the gas,$\gamma = 1.5$)

$A$ thermally insulating cylinder has a thermally insulating and frictionless movable partition in the middle,as shown in the figure below. On each side of the partition,there is one mole of an ideal gas,with specific heat at constant volume,$C_v = 2R$. Here,$R$ is the gas constant. Initially,each side has a volume $V_0$ and temperature $T_0$. The left side has an electric heater,which is turned on at very low power to transfer heat $Q$ to the gas on the left side. As a result,the partition moves slowly towards the right,reducing the right side volume to $V_0 / 2$. Consequently,the gas temperatures on the left and the right sides become $T_L$ and $T_R$,respectively. Ignore the changes in the temperatures of the cylinder,heater,and the partition.
$(1)$ The value of $\frac{T_R}{T_0}$ is
$(A)$ $\sqrt{2}$ $(B)$ $\sqrt{3}$ $(C)$ $2$ $(D)$ $3$
$(2)$ The value of $\frac{Q}{RT_0}$ is
$(A)$ $4(2\sqrt{2}+1)$ $(B)$ $4(2\sqrt{2}-1)$ $(C)$ $(5\sqrt{2}+1)$ $(D)$ $(5\sqrt{2}-1)$

In adiabatic compression, the decrease in volume is associated with

The volume of an ideal gas increases $8$ times and the temperature becomes $1/4$ of the initial temperature during a reversible adiabatic change. If there is no exchange of heat in this process $( \Delta Q=0 )$, identify the gas from the following options:

$A$ monatomic gas at $630 \,K$ expands adiabatically to $27$ times its initial volume. The final temperature of the gas is (in $\,K$)

$A$ monoatomic ideal gas, initially at temperature $T_1$, is enclosed in a cylinder fitted with a massless, frictionless piston. By releasing the piston suddenly, the gas is allowed to expand adiabatically to a temperature $T_2$. If $L_1$ and $L_2$ are the lengths of the gas columns before and after expansion respectively, then $(T_2 / T_1)$ is given by