An ideal gas at atmospheric pressure is adiabatically compressed so that its density becomes $32$ times its initial value. If the final pressure of the gas is $128$ atmospheres,the value of $\gamma$ for the gas is:

$A$ diatomic gas $\left(\gamma = \frac{7}{5}\right)$ is compressed adiabatically to volume $\frac{V_0}{32}$,where $V_0$ is its initial volume. The initial temperature of the gas is $T_i$ in Kelvin and the final temperature is $xT_i$ in Kelvin. The value of $x$ is:

$Assertion:$ In adiabatic compression,the internal energy and temperature of the system decrease.
$Reason:$ Adiabatic compression is a slow process.

Consider a spherical shell of radius $R$ at temperature $T$. The black body radiation inside it can be considered as an ideal gas of photons with internal energy per unit volume $E = \frac{U}{V} \propto T^4$ and pressure $P = \frac{1}{3} \left( \frac{U}{V} \right)$. If the shell now undergoes an adiabatic expansion,the relation between $T$ and $R$ is:

$A$ gas is suddenly compressed to $1/4$ th of its original volume at normal temperature. The increase in its temperature is ....... $K$ $(\gamma = 1.5)$

Five moles of hydrogen initially at $STP$ is compressed adiabatically so that its temperature becomes $673 \, K$. The increase in internal energy of the gas, in kilo joule is $(R=8.3 \, J/mol-K; \gamma=1.4$ for diatomic gas$)$