Explanation

(a) Root mean square velocity ($v_{rms}$), given by
$v_{rms} = \sqrt{\frac{3RT}{M}}$
where $R$ is gas constant, $T$ the temperature and $M$ molecular weight.
Given, $T_{1} = 27℃ = 273 + 27 = 300\ K,\ \ \ $
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ T_{2} = 327℃ = 327 + 273 = 600$ K
$\therefore\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{\left( v_{rms} \right)_{1}}{\left( v_{rms} \right)_{2}} = \sqrt{\frac{300}{600}} = \ \sqrt{\frac{1}{2}}$
$\Rightarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( v_{rms} \right)_{2} = \sqrt{2}\ \left( v_{rms} \right)_{1}$
Hence, rms speed increases $\sqrt{2}$ times.