# Chapter 13 - Problems - Page 499: 70

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 $0.707$

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