At 300K , the rms speed of oxygen molecules is (α + 5)/(α) times to...

RMS speed is given by, ${v}_{rms} = \sqrt{\dfrac{3RT}{M}}$ and average speed is given by, ${v}_{avg} = \sqrt{\dfrac{8}{\pi}\dfrac{RT}{M}}$.

As per question,

${v}_{rms} = \sqrt{\dfrac{\alpha + 5}{\alpha}}{v}_{avg}$

$⇒ \sqrt{\dfrac{3RT}{M}} = \sqrt{\dfrac{\alpha + 5}{\alpha}}\sqrt{\dfrac{8}{\pi}\dfrac{RT}{M}}$

$⇒ 3 = \dfrac{\alpha + 5}{\alpha} \times \dfrac{8}{\dfrac{22}{7}}$

$⇒ \alpha = 28$