Why is RMS Speed of Gas Particles Greater Than Mean Speed

The root mean square or rms speed of a particle is an important concept because for an ideal at a temperaturethe average kinetic energy of a particle of massis related to the temperature via the equation

The mean speed of the particles is also important because it is the mean speed of the gas particles as they bounce back from a surface that appears in the expression for the pressure and force exerted on that surface.

The rms speed is always greater than the mean speed. We can see this by considering the expression for the varianceof the speeds of the gas particles,

Whereis the expectation value of the square of the speed of the particles andis the mean speed of the particles.

is the square of the rms speedso the variance can be written as

Since the variance is always greater than 0,

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