If during an adiabatic process the pressure of a mixture of gases is found to be proportional to the square of its absolute temperature,the ratio of $C_p / C_v$ for the mixture of gases is .........

The mean kinetic energy of monoatomic gas molecules under standard conditions is $\langle E_1 \rangle$. If the gas is compressed adiabatically $8$ times to its initial volume, the mean kinetic energy of gas molecules changes to $\langle E_2 \rangle$. The ratio $\frac{\langle E_2 \rangle}{\langle E_1 \rangle}$ is

The process in which no heat enters or leaves the system is termed as

Given below are two statements:
Statement-$I$: When $\mu$ amount of an ideal gas undergoes adiabatic change from state $(P_1, V_1, T_1)$ to state $(P_2, V_2, T_2)$,the work done is $W = \frac{\mu R(T_2 - T_1)}{1 - \gamma}$,where $\gamma = \frac{C_P}{C_V}$ and $R$ is the universal gas constant.
Statement-$II$: In the above case,when work is done on the gas,the temperature of the gas would rise.
Choose the correct answer from the options given below:

For an adiabatic expansion of an ideal gas,the fractional change in its pressure is equal to (where $\gamma$ is the ratio of specific heats):

$A$ rigid diatomic gas $(\gamma = 7/5)$ is compressed adiabatically to volume $(V_i/32)$, where $V_i$ is the initial volume. The initial temperature of the gas is $T_i \text{ K}$ and the final temperature is $x T_i \text{ K}$. The value of $x$ is