An ideal gas is expanded adiabatically. How many times has the gas to be expanded to reduce the r.m.s. speed of molecules $3$ times (in $\times$)? $(\gamma = 1.5)$

In a thermodynamic system,$\Delta U$ represents the increase in internal energy and $W$ represents the work done by the system. Which of the following statements is true?

$A$ spherical bubble inside water has radius $R$. Take the pressure inside the bubble and the water pressure to be $p_0$. The bubble now gets compressed radially in an adiabatic manner so that its radius becomes $(R-a)$. For $a \ll R$,the magnitude of the work done in the process is given by $(4 \pi p_0 R a^2) X$,where $X$ is a constant and $\gamma = C_p / C_V = 41 / 30$. The value of $X$ is:

$A$ cylindrical vessel of uniform cross-section containing a gas with $\gamma = 1.5$ is divided into two parts $A$ and $B$ using a piston. Initially, the piston is kept fixed such that part $A$ has pressure $p$ and volume $5V$, and part $B$ has pressure $8p$ and volume $V$. If the piston is released and the gas is allowed to undergo an adiabatic process, then the final volume of the gas in part $A$ 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).

An ideal gas $(\gamma = 1.5)$ is expanded adiabatically. To reduce the root mean square velocity of molecules two times, the gas should be expanded (in $\times$)