The Root mean square is that the reasonably average that you simply want to use when it’s the square of the values that you’re fascinated by. This pops up surprisingly often.

For example, if you've got a mass oscillating back and forth on a spring, it's kinetic energy(K.E.) depends on the square of velocity, while its (P.E.) 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 wish to speak a few representative displacements or a representative speed, RMS is that the thanks to going: they end in the proper average energies.

Root Mean Square Velocity

Since we now know the way to relate temperature and kinetic energy we can relate temperature to the speed of gas molecules. Note since these are distributions the values ( E_kEk​ 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 up to the molar mass of the molecule in kg/mol. the foundation means square velocity is that the root of the average of the square of the speed. As such, it's units of velocity. the explanation we use the RMS velocity rather than the average is that for a typical gas sample the web velocity is zero since the particles are acquiring all directions. this is often a key formula because the velocity of the particles is what determines both the diffusion and effusion rates.

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