The root-mean-square speed of the particles is given by:

$$\sqrt{ \left< v^{2} \right>} = \sqrt{\frac{3 k_{B} T}{m}} = \sqrt{\frac{3RT}{M}}$$

,where

M is the molar mass

m is the molecular mass

The root-mean-square speed (V_{rms}) is the square root of the mean of the square of the individual speeds.

This at any given temperature, lighter molecules move faster on average than heavier molecules.

**Implications:**

– Note: The speed of sound can only propagate as fast as the speed of the gas molecules. Hence, the speed of sound is higher in the lighter gases.

$$v_{sound} = \sqrt{\frac{\gamma}{3}} v_{rms}$$

, where

γ is the heat capacity ratio (=1.4 for H_{2}, O_{2}, N_{2})

– Hydrogen in stratosphere: H_{2} has a high speed which enables its escape from Earth’s gravity (escape velocity at Earth’s surface = 11,200 m/s)

– Vacuum pumping: High-speed molecules (such as H_{2}) are very difficult to pump out in ultra-high vacuum chambers by turbo-molecular pumps (which works as giant fans).

– Diffusion: High speed molecules diffuse faster.