Answer
The ratio of the rms speed of the argon atoms to that of the neon atoms is $0.707$
Work Step by Step
Let $m_n$ be the mass of a neon atom. We can find the rms speed of the neon atoms:
$\overline{KE} = \frac{3}{2}~k~T$
$\frac{1}{2}m_n~v_{rms,n}^2 = \frac{3}{2}~k~T$
$v_{rms,n} = \sqrt{\frac{3~k~T}{m_n}}$
Let $m_a$ be the mass of an argon atom. Note that $m_a = 2~m_n$. We can find the rms speed of the argon atoms:
$\overline{KE} = \frac{3}{2}~k~T$
$\frac{1}{2}m_a~v_{rms,a}^2 = \frac{3}{2}~k~T$
$v_{rms,a} = \sqrt{\frac{3~k~T}{m_a}}$
$v_{rms,a} = \sqrt{\frac{3~k~T}{2~m_n}}$
$v_{rms,a} = \frac{\sqrt{2}}{2}\times \sqrt{\frac{3~k~T}{m_n}}$
$v_{rms,a} = \frac{\sqrt{2}}{2}\times v_{rms,n}$
$\frac{v_{rms,a}}{v_{rms,n}}~ = \frac{\sqrt{2}}{2}$
$\frac{v_{rms,a}}{v_{rms,n}}~ = 0.707$
The ratio of the rms speed of the argon atoms to that of the neon atoms is $0.707$
