# How to understand molecular speed derived from average KE?

Mean kinetic energy is related to temperature: $$\langle K_E\rangle=\frac{3}{2}kT=\frac{1}{2}mv^2$$

where $$k$$ is the Boltzmann constant and $$T$$ the temperature in Kelvin.

For example, the average molar mass of air is 29 g/mol, the average $$K_E$$ of air molecules at 20°C (293K) is approximately $$6\times10^{-21}$$ J calculated from the equation above. (I assume air is an ideal gas which it isn’t, but it gives an approximate answer)

The average mass of one molecule of air is $$4.8\times10^{-26}$$ kg and the average speed of one molecule of air at 20°C (293K) is approximately 500 m/s based on my calculation.

I couldn't understand why the theoretical average speed of air molecules is so high. Perhaps my understanding is completely wrong, but does that mean an air molecule is travelling above the speed of sound in air at STP which is only 343 m/s? Also, does the average speed of air molecules impact wind speed? It sounds quite frantic, but the highest wind speed ever recorded on Earth is only 113 m/s compared to an average speed of 500 m/s of an air molecule.

• The average speed of air molecules you calculate is not only theoretical :) Jul 5, 2022 at 9:25

Indeed the root mean square speed of molecules is that high but what you must also realise is the the motion is random in direction so the average velocity of the gas molecules is zero.
This means that if you have a gas in a container at rest relative to the laboratory then the centre of mass of the gas is not moving.
If you how transport the gas, which is equivalent to there being a wind, superimposed on the random motion of the gas molecules is the velocity at which the gas is being transported.

. . . but does that mean an air molecule is travelling above the speed of sound in air . . . .

Indeed yes but some molecules are travelling slower.

The surprise that you show as to the speeds of gas molecules being similar to that to the speed of sound in a gas is perhaps reduced when the mechanism of sound propagation through a gas in noted.

The interactions between (ideal) gas molecules are only via collisions.
So if one gas molecule has to interact with another gas molecule the speed at which it can do that is governed by the speed of the molecules.
Thus the speed at which information about the local pressure in one region of a gas to another region of a gas (the speed of a sound wave) is limited by the "thermal" speed of the molecules.
Amongst other factors the speed of sound in a gas is proportional to the square root of the absolute temperature, but the square root of the absolute temperature is proportional to the square root of the mean square speed of the gas molecules, which in turn is proportion to the mean speed of the molecules.

Higher temperature $$\Rightarrow$$ molecules move faster $$\Rightarrow$$ molecules take less time between collisions $$\Rightarrow$$ information about local conditions travels faster $$\Rightarrow$$ speed of sound higher.

Yes, molecules travel faster than the speed of sound. There are some caveats here, since air is not an ideal gas, and sound could not propagate in an ideal gas due to the absence of collisions. Sound is pressure waves, where different layers of gas interact, as if they were closed reservoirs, each with its equilibrium pressure (the technical term is local equilibrium). The walls between the reservoirs are replaced by the fact that a molecule is likely to be scattered back within $$5\times 10^{-6}$$cm.

A related question: How does hot air rise?