# 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.