The root mean square or rms speed of a particle is an important concept because for an ideal at a temperature
the average kinetic energy of a particle of mass
is 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 variance
of the speeds of the gas particles,
Where
is the expectation value of the square of the speed of the particles and
is the mean speed of the particles.
is the square of the rms speed
so the variance can be written as
Since the variance is always greater than 0,