${\text{V}}_{\text{rms}}=\sqrt{\frac{\text{3RT}}{\text{M}}}$

                    ${\text{V}}_{\text{rms}}\propto \sqrt{\frac{\text{T}}{\text{M}}}$                              ...(i)

When oxygen gas dissociates into atomic oxygen, its
atomic mass M will become half.
Temperature is doubled.
So, from Eq. (i) V_rms will become two times.

$\frac{{\text{V}}_{\text{rms}}^{\text{'}}}{{\text{V}}_{\text{rms}}}=\frac{\sqrt{\frac{\text{2T}}{\frac{\text{M}}{\text{2}}}}}{\sqrt{\frac{\text{T}}{\text{M}}}}$

$=\sqrt{4}=2$

$\Rightarrow {\text{V}}_{\text{rms}}^{\text{'}}=2{\text{V}}_{\text{rms}}$