# UY1: Root-mean-square speed of the gas particles

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 (Vrms) 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 H2, O2, N2)

– Hydrogen in stratosphere: H2 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 H2) 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.

Next: Mean Free Path

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