The Root mean square is the kind of average that you want to use when it’s the square of the values that you’re interested in. This pops up surprisingly often.

For example, if you have a mass oscillating back and forth on a spring, its kinetic energy depends on the square of velocity, while its potential energy depends on the square of position. (Note that, in such a case, the arithmetic means of both position and velocity are useless; for oscillatory motion, both are zero, which doesn’t tell you anything interesting.) So, if you want to talk about a representative displacement, or a representative speed, RMS is the way to go: they result in the correct average energies.

Root Mean Square Velocity

Since we now know how to relate temperature and kinetic energy, we can relate temperature to the velocity of gas molecules. Note since these are distributions the values (E_k or velocity) that we are talking about are always averages.

                E_{k\ }\ =\ \frac{3}{2}RT\

                E_k\ =\ \frac{1}{2}mv^2

setting these two equal and solving for the average square velocity we get

                v^2=\ \frac{3RT}{m}

The root mean square velocity or v_{rms} is the square root of the average square velocity and is

            v_{rms\ }=\ \sqrt{\frac{3RT}{M}}

Where M is equal to the molar mass of the molecule in kg/mol. The root mean square velocity is the square root of the average of the square of the velocity. As such, it has units of velocity. The reason we use the rms velocity instead of the average is that for a typical gas sample the net velocity is zero since the particles are moving in all directions. This is a key formula as the velocity of the particles is what determines both the diffusion and effusion rates.

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Mean vs. Average :

Most people understand the word average as describing a representative value within a group. For example, the average age of a group of three people aged 10, 16 and 40 is (10 + 16 + 40) / 3, or 22. When speaking statistically, this average age of 22 is referred to as the mean age. Notice that the average age is not very close in value to any of the individual ages. This is because there is a wide range between the lowest value, 10, and the highest, 40.