Root mean square velocity of a gas molecule is proportional to — States of Matter and Gaseous State Chemistry Question

💡 Solution & Explanation

# RMS Velocity of Gas Molecules **Formula for RMS velocity:** $$v_{rms} = \sqrt{\frac{3kT}{m}} = \sqrt{\frac{3RT}{M}}$$ where $k$ = Boltzmann constant, $T$ = absolute temperature, $m$ = mass of one molecule, $R$ = gas constant, $M$ = molar mass **Step-by-step analysis:** 1. **Temperature dependence:** $v_{rms} \propto \sqrt{T}$ — RMS velocity increases with the square root of absolute temperature. 2. **Molar mass dependence:** $v_{rms} \propto \frac{1}{\sqrt{M}}$ — RMS velocity is inversely proportional to the square root of molar mass (heavier molecules move slower). 3. **Combined proportionality:** $$v_{rms} \propto \sqrt{\frac{T}{M}}$$ **Therefore, the correct answer is C: $\sqrt{\frac{T}{M}}$** (or equivalently $\sqrt{T}$ and inversely with $\sqrt{M}$) *Note: Pressure does NOT appear in this expression because at constant temperature and volume, pressure depends on the number of molecules, not on individual molecular velocities.*