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# Root mean square speed of an ideal gas

The root mean square speed ($v_{\text{rms}}$) of the particles of a gas is the root of the mean of the squares of the individual speeds of the particles. It is a type of "average" speed of the gas particles.

The mean velocity of gas particles within a stationary container would be zero as there is no effective movement of the gas as a whole.

The root mean square speed gives a meaningful summary of the speed of the particles of a gas and is easily used in calculations.

The root mean square speed is given by:$$v_{\text{rms}}=\sqrt{\langle v^{2}\rangle}=\sqrt\frac{3kT}{m}=\sqrt\frac{3RT}{mN_{\text{A}}}=\sqrt\frac{3RT}{m_{\text{r}}}$$

$k=$Boltzmann constant; $m=$mass of a particle; $m_{r}=$molar mass; $R=$molar gas constant; $T=$thermodynamic temperature of the gas.