Root Mean Square (RMS) Speed

The square root of the mean of the squares of the speeds of different molecules of a gas is called Root Mean Square Speed (v_rms).

The temperature at which the rms speed of molecules of a gas becomes zero (i.e., T = 0 K, v_rms = 0) is called absolute zero.

\[v_{rms}=\sqrt{\frac{v_1^2+v_2^2+v_3^2+v_4^2+\ldots}{N}}\]

where v₁, v₂, v₃… are speeds of individual molecules and N = total number of molecules.

RMS speed varies directly with the square root of temperature:

With rise in temperature, v_rms of gas molecules increases.

RMS speed decreases with increase in molecular weight:

• Independence from Pressure: v_rms​ does not depend on the pressure of the gas at constant temperature. If pressure is increased n times, density also increases n times, but v_rms remains constant (Boyle's Law: p ∝ ρ).
• Atmosphere of Planets: A planet or satellite will have an atmosphere only if v_rms < v_e​ (escape velocity). Moon has no atmosphere because v_rms of its gas molecules is greater than escape velocity (v_e​).