Kinetic Theory of Gases Root Mean Square Speed

General Chemistry • Gases

Written by STEM Calculators Team Published October 21, 2025 Updated February 24, 2026

Kinetic–Molecular Theory — Root-Mean-Square Speed \(u_\mathrm{rms}\)

For an ideal gas, the root-mean-square (rms) molecular speed is \(u_\mathrm{rms}=\sqrt{ \dfrac{3RT}{M} }\), where \(R\) is the gas constant (\(8.314462618\ \mathrm{J\,mol^{-1}\,K^{-1}} = \mathrm{kg\,m^{2}\,s^{-2}\,mol^{-1}\,K^{-1}}\)), \(T\) is absolute temperature (K), and \(M\) is the molar mass in \(\mathrm{kg\,mol^{-1}}\).

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Root-Mean-Square Speed \(u_\mathrm{rms}\)

In the kinetic–molecular theory, gas molecules move randomly and collide elastically. The spread of their speeds in a sample at temperature \(T\) is described by the Maxwell–Boltzmann distribution. Three characteristic speeds are commonly used:

\[ \boxed{ u_m=\sqrt{\dfrac{2RT}{M}},\quad u_{\text{av}}=\sqrt{\dfrac{8RT}{\pi M}},\quad u_{\mathrm{rms}}=\sqrt{\dfrac{3RT}{M}} } \]

Here \(R=8.314462618\ \mathrm{J\,mol^{-1}\,K^{-1}}\), \(M\) is the molar mass in \(\mathrm{kg\,mol^{-1}}\), and speeds are in \(\mathrm{m\,s^{-1}}\). At a fixed temperature, lighter gases (smaller \(M\)) have higher characteristic speeds. Raising \(T\) increases all three speeds. The typical ordering is \(u_m < u_{\text{av}} < u_{\mathrm{rms}}\).

Unit consistency

The formula for \(u_{\mathrm{rms}}\) produces speed units:

\[ \begin{aligned} \left[u_{\mathrm{rms}}\right] &= \sqrt{ \dfrac{ \mathrm{J\,mol^{-1}\,K^{-1}} \cdot \mathrm{K} }{ \mathrm{kg\,mol^{-1}} } } \\ &= \sqrt{ \dfrac{ \mathrm{kg\,m^{2}\,s^{-2}} }{ \mathrm{kg} } } \\ &= \mathrm{m\,s^{-1}} \end{aligned} \]

Interpretation

The rms speed equals the square root of the mean of \(u^2\) over all molecules, so it slightly exceeds the average speed. Distributions become broader and shift to higher values as \(T\) increases or \(M\) decreases.

Frequently Asked Questions

What is the root-mean-square speed of a gas and how is it calculated?

The rms speed is a characteristic molecular speed from the Maxwell–Boltzmann distribution. For an ideal gas, u_rms = sqrt(3RT/M), where T is in kelvin and M is molar mass in kg/mol.

What is the difference between u_m, u_av, and u_rms?

u_m is the most probable speed, u_av is the average speed, and u_rms is the root-mean-square speed. For an ideal gas: u_m = sqrt(2RT/M), u_av = sqrt(8RT/(pi M)), and u_rms = sqrt(3RT/M), and typically u_m < u_av < u_rms.

Why does molar mass need to be in kg/mol for the rms speed formula?

The gas constant R is in J/(mol K), which is equivalent to kg m^2 s^-2 (mol^-1 K^-1). Using M in kg/mol makes the units reduce to m/s for speed; the calculator converts g/mol to kg/mol automatically.

How do temperature and molar mass affect u_rms?

u_rms increases with temperature because it scales with sqrt(T). It decreases with molar mass because it scales with 1/sqrt(M), so lighter gases move faster at the same temperature.

Kinetic–Molecular Theory — Root-Mean-Square Speed \(u_\mathrm{rms}\)

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