Explanation

(a) Average speed ($v_{av}$) of gas molecules is
$v_{av} = \sqrt{\frac{8RT}{\pi M}}$
where $R$ is gas constant and $M$ the molecular weight.
Given, $v_{1} = v,\ \ \ \ M_{1} = 64,\ \ \ \ v_{2} = 4v$
$\therefore\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{v_{1}}{v_{2}} = \sqrt{\frac{M_{2}}{M_{1}}}$
$\frac{v}{4v} = \sqrt{\frac{M_{2}}{64}}$
$\Rightarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ M_{2} = \frac{64}{16} = 4$
Hence, the gas is helium (molecular mass 4).