• id: JTHERM-02-N03
• title: RMS Speed and Its Temperature Dependence
• tags: rms-speed, temperature, molar-mass

The root mean square speed $v_{\text{rms}} = \sqrt{\overline{v^2}} = \sqrt{3k_BT/m} = \sqrt{3RT/M}$. Key dependencies: $v_{\text{rms}} \propto \sqrt{T}$ (at constant $M$) and $v_{\text{rms}} \propto 1/\sqrt{M}$ (at constant $T$). Lighter molecules move faster — at 300 K, hydrogen ($M = 2 \times 10^{-3}$ kg/mol) has $v_{\text{rms}} \approx 1920$ m/s while oxygen ($M = 32 \times 10^{-3}$) has $v_{\text{rms}} \approx 480$ m/s. Doubling the absolute temperature increases $v_{\text{rms}}$ by a factor of $\sqrt{2}$, not 2. This $\sqrt{T}$ dependence is one of the most common JEE traps.