Root-mean-square speed

Also called RMS speed, v_rms

The root-mean-square speed is the speed an atom would need in order to have exactly the average kinetic energy of the sample. It is the square root of the mean of the squared speeds, which is not the average speed.

The AP Physics 2 sheet prints it inside a single line with two equals signs:

K_{\text{avg}} = \frac{3}{2} k_B T = \frac{1}{2} m v_{\text{rms}}^2

Setting the outer expressions equal and solving gives

v_{\text{rms}} = \sqrt{\frac{3 k_B T}{m}}

That rearranged form is not printed. The sheet hands you the equality and expects the algebra, so rehearse the rearrangement rather than the result.

Two symbols are traps. $m$ is the mass of one atom in kilograms, not the mass of the sample and not the molar mass; given a molar mass $M$ in grams per mole, convert to kilograms per mole and divide by $N_0 = 6.02 \times 10^{23}\ \mathrm{mol^{-1}}$ . And $T$ is in kelvin, since a squared speed cannot be negative.

It is not the average speed. Squaring before averaging gives extra weight to the fast atoms, so the root-mean-square speed sits above the ordinary mean speed, which in turn sits above the most probable speed at the peak of the Maxwell-Boltzmann distribution. Three different speeds read off one curve, and the AP equation uses only this one.

The dependencies are worth having ready. Quadruple the absolute temperature and $v_{\text{rms}}$ doubles. Quadruple the atomic mass at fixed temperature and it halves. Change the volume at constant temperature and it does not move at all.

Essential knowledge 9.1.B.1.ii is where the CED states the relationship.

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