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| {{Refimprove|date=December 2009}}
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| '''Root-mean-square speed''' is the measure of the [[speed]] of particles in a gas that is most convenient for problem solving within the [[kinetic theory]] of gases. It is defined as the square root of the average velocity-squared of the molecules in a gas. It is given by the formula<ref>
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| {{cite book
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| | title = College Physics, Volume 1
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| | edition = 9th
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| | author = Raymond A. Serway, Jerry S. Faughn, and Chris Vuille
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| | publisher = 9780840068484
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| | year = 2011
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| | isbn =
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| | page = 352
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| | url = http://books.google.com/books?id=HLxV-IKYO5IC&pg=PA352
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| }}</ref>
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| :<math>v_\mathrm{rms} = \sqrt {{3RT}\over{M_m}}</math>
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| where ''v''<sub>rms</sub> is the [[root mean square]] of the speed in meters per second, ''M<sub>m</sub>'' is the [[molar mass]] of the gas in kilograms per mole, ''R'' is the [[molar gas constant]], and ''T'' is the [[temperature]] in [[kelvin]]. Although the molecules in a sample of gas have an average kinetic energy (and therefore an average speed) the individual molecules move at various speeds and they stop and change direction according to the law of density measurements and isolation, i.e. they exhibit a distribution of speeds. Some move fast, others relatively slowly. Collisions change individual molecular speeds but the distribution of speeds remains the same. This equation is derived from kinetic theory of gases using [[Maxwell–Boltzmann distribution]] function. The higher the temperature, the greater the mean velocity will be. This works well for both nearly ideal, atomic gases like [[helium]] and for [[Molecule|molecular gas]]es like diatomic [[oxygen]]. This is because despite the larger internal energy in many molecules (compared to that for an atom), 3''RT''/2 is still the mean [[Translation (physics)|translational]] [[kinetic energy|kinetic]] energy.
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| This can also be written in terms of the [[Boltzmann constant]] (''k'') as
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| :<math>v_\mathrm{rms} = \sqrt {{3kT}\over{m}}</math>
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| where ''m'' is the mass of one molecule of the gas.
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| This can be derived with energy methods:
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| :<math>E_\mathrm{k} = {{3}\over{2}}nRT = \frac{3}{2}NkT</math>
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| where ''E''<sub>k</sub> is the [[kinetic energy]] and ''N'' is the number of gas molecules.
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| :<math>E_\mathrm{k,molecule} = {{1}\over{2}}mv^2</math>
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| Given that ''v''<sup>2</sup> ignores direction, it is logical to assume that the formula can be extended to the entire sample, replacing ''m'' with the entire sample's mass, equal to the molar mass times the number of moles ''n'' yielding
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| :<math>{{1}\over{2}}nMv^2 = E_\mathrm{k}</math>
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| Therefore
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| :<math>v_\mathrm{rms} = \sqrt {{2E_\mathrm{k}}\over{m}}</math> | |
| which is equivalent.
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| The same result is obtained by solving the [[Gaussian integral]] containing the [[Maxwell speed distribution]], ''p''(''v''):
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| <!-- The \,\! is to keep the formula rendered as PNG instead of HTML. Please don't remove it.-->
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| :<math>v_\mathrm{rms} = \sqrt{\int_{0}^{\infty} v^2\ p(v)dv}\,\! </math>
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| ::<math> = \sqrt{\int_{0}^{\infty} 4 \pi \left (\frac{m}{2 \pi k T} \right )^\frac{3}{2} v^4\ e^{-\frac{v^{2}m}{2kT}}dv}\,\!</math> | |
| ::<math> = \sqrt{4 \pi \left (\frac{m}{2 \pi k T} \right )^\frac{3}{2} \frac{3}{8} \pi^{\frac{1}{2}} \left( \frac{2kT}{m}\right)^{\frac{5}{2}}}\,\!</math>
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| ::<math> = \sqrt{\frac{3kT}{m}}</math>
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| ==References==
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| {{reflist}}
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| {{DEFAULTSORT:Root Mean Square Speed}}
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| [[Category:Gases]]
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