$A$ rigid diatomic gas undergoes adiabatic change. Its pressure $P$ and temperature $T$ are related as $P \propto T^x$ where $x$ is (in $.5$)

An ideal gas is expanded adiabatically at an initial temperature of $300 \ K$ so that its volume is doubled. The final temperature of the hydrogen gas is $(\gamma = 1.40)$.

If $\Delta E_{int}$ represents the increase in internal energy and $W$ represents the work done by the system,which of the following statements is correct for a thermodynamic system?

An ideal gas at $27 \, ^\circ C$ is compressed adiabatically such that its volume becomes $8/27$ of its original volume. If $\gamma = 5/3$ for the gas,the increase in temperature of the gas is ..... $K$.

The pressure and density of a diatomic gas $\left(\gamma=\frac{7}{5}\right)$ changes adiabatically from $(P, d)$ to $(P^{\prime}, d^{\prime})$. If $\frac{d^{\prime}}{d}=32$, then $\frac{P^{\prime}}{P}$ is:

$A$ gas is suddenly compressed such that its absolute temperature is doubled. If the ratio of the specific heat capacities of the gas is $1.5$, then the percentage decrease in the volume of the gas is