By what factor should the volume of an ideal gas $(\gamma = 1.5)$ be increased through an adiabatic expansion so that its $rms$ speed becomes half?

Under an adiabatic process,the volume of an ideal gas gets doubled. Consequently,the mean collision time between the gas molecules changes from $\tau_{1}$ to $\tau_{2}$. If $\frac{C_{p}}{C_{v}}=\gamma$ for this gas,then a good estimate for $\frac{\tau_{2}}{\tau_{1}}$ is given by:

$Assertion :$ Air quickly leaking out of a balloon becomes cooler.
$Reason :$ The leaking air undergoes adiabatic expansion.

$A$ polyatomic gas is compressed to $\left(\frac{1}{8}\right)^{\text{th}}$ of its original volume adiabatically. If its initial pressure is $P_0$,what will be its new pressure (in $P_0$)? (Given: $\gamma = \frac{C_p}{C_v} = \frac{4}{3}$)

The heat capacity of one mole of an ideal gas is found to be $C_V = \frac{3R(1 + aRT)}{2}$,where $a$ is a constant. The equation obeyed by this gas during a reversible adiabatic expansion is

The temperature of a hypothetical gas increases to $\sqrt{2}$ times when compressed adiabatically to half the volume. Its equation can be written as