$A$ monoatomic ideal gas is compressed adiabatically to $\left(\frac{1}{27}\right)$ of its initial volume. If the initial temperature of the gas is $T \ K$ and the final temperature is $xT \ K$, the value of $x$ is:

$A$ diatomic gas undergoes an adiabatic change. Its pressure $P$ and temperature $T$ are related as $P \propto T^{x}$, where $x$ is:

$A$ monoatomic gas having $\gamma = 5/3$ is stored in a thermally insulated container and the gas is suddenly compressed to $(1/8)^{th}$ of its initial volume. The ratio of final pressure to initial pressure is: ($\gamma$ is the ratio of specific heats of the gas at constant pressure and at constant volume).

For an adiabatic process,if $\gamma = 2.5$ and the volume becomes $1/8$ of its initial volume,then the new pressure $P'$ is (initial pressure $= P$):

Two different adiabatic paths for the same gas intersect two isothermal curves as shown in the $P-V$ diagram. The relation between the ratio $\frac{V_a}{V_d}$ and the ratio $\frac{V_b}{V_c}$ is:

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