An ideal gas expands adiabatically, $(\gamma=1 \cdot 5)$. To reduce the r.m.s. velocity of the molecules 4 times, the gas has to be expanded

The root mean square velocity is given by $v_{\text{rms}} = \sqrt{\frac{3RT}{M}}$, hence $v_{\text{rms}} \propto \sqrt{T}$. A fourfold reduction in velocity requires $\frac{v'_{\text{rms}}}{v_{\text{rms}}} = \frac{1}{4} = \sqrt{\frac{T'}{T}}$, yielding $T' = \frac{T}{16}$ after squaring both sides. $\frac{1}{4} = \sqrt{\frac{T'}{T}}$ $T' = \frac{T}{16}$ For adiabatic expansion with $\gamma = 1.5$, $TV^{\gamma-1}$ remains constant. Using $T_2 = \frac{T_1}{16}$: $T_1 V_1^{0.5} = \frac{T_1}{16} V_2^{0.5}$ $\frac{V_2^{0.5}}{V_1^{0.5}} = 16$ $\frac{V_2}{V_1} = 256$

The gas must be expanded 256 times, corresponding to option A.