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

💡 Solution & Explanation

I need to derive the relationship for root mean square (rms) velocity. **Starting from kinetic theory of gases:** The average kinetic energy of a gas molecule is: $$\frac{1}{2}m\langle v^2 \rangle = \frac{3}{2}k_BT$$ where $m$ is molecular mass, $k_B$ is Boltzmann constant, and $T$ is absolute temperature. **Solving for rms velocity:** $$v_{rms} = \sqrt{\langle v^2 \rangle} = \sqrt{\frac{3k_BT}{m}}$$ **Therefore:** $$v_{rms} \propto \sqrt{\frac{T}{M}}$$ where $M$ is molar mass (or $m$ for individual molecule). **Proportionality relationships:** - $v_{rms} \propto \sqrt{T}$ (increases with temperature) - $v_{rms} \propto \frac{1}{\sqrt{M}}$ (decreases with molar mass) **The correct answer is C: $\sqrt{\frac{T}{M}}$ or $\sqrt{T}$ and inversely proportional to $\sqrt{M}$** This shows rms velocity increases with temperature and decreases with molecular weight—lighter molecules at higher temperatures move faster.