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In the physics of gas molecules, the root-mean-square speed is defined as the square root of the average squared-speed. The RMS speed of an ideal gas is calculated using the following equation: v RMS = 3 R T M {\displaystyle v_{\text{RMS}}={\sqrt {3RT \over M}}}
Mathematically, the Maxwell–Boltzmann distribution is the chi distribution with three degrees of freedom (the components of the velocity vector in Euclidean space), with a scale parameter measuring speeds in units proportional to the square root of / (the ratio of temperature and particle mass). [2]
The most probable (or mode) speed is 81.6% of the root-mean-square speed , and the mean (arithmetic mean, or average) speed ¯ is 92.1% of the rms speed (isotropic distribution of speeds). See: Average, Root-mean-square speed; Arithmetic mean; Mean; Mode (statistics)
Effusion occurs through an orifice smaller than the mean free path of the particles in motion, whereas diffusion occurs through an opening in which multiple particles can flow through simultaneously. In physics and chemistry, effusion is the process in which a gas escapes from a container through a hole of diameter considerably smaller than the ...
The thermal energy can be used to calculate the root-mean-square speed of the atoms, which turns out to be inversely proportional to the square root of the atomic mass. The root mean square speeds found at room temperature accurately reflect this, ranging from 1370 m/s for helium, down to 240 m/s for xenon.
The TKE can be defined to be half the sum of the variances σ² (square of standard deviations σ) of the fluctuating velocity components: = (+ +) = ((′) ¯ + (′) ¯ + (′) ¯), where each turbulent velocity component is the difference between the instantaneous and the average velocity: ′ = ¯ (Reynolds decomposition).
In the gas phase, is often defined as the diffusional mean free path, by assuming that a simple approximate relation is exact: =, where is the root mean square speed of the gas molecules: =, where is the mass of the diffusing species. This approximate equation becomes exact when used to define the diffusional mean free path.
The standard deviation of the particle displacement is given in terms of the Lagrangian velocity autocorrelation following by () = ′ () where ′ is the root mean square velocity. This result corresponds with the result originally obtained by Taylor.