The root mean square speed of an ideal gas at $300$ K is $500$ m/s. What is the root mean square speed at $1200$ K?

Solution

1. The formula for root mean square speed is $u_{rms}=\sqrt{\frac{3RT}{M}}$.
2. This implies $u_{rms}\propto \sqrt{T}$.
3. Ratio of speeds: $$\begin{array}{rl}\frac{u_2}{u_1}& =\sqrt{\frac{T_2}{T_1}}\\ & =\sqrt{\frac{1200}{300}}\\ & =\sqrt{4}\\ & =2\end{array}$$.
4. $$\begin{array}{rl}{u}_2& =2\cdot 500\\ & =1000\end{array}$$ m/s.
5. Hence the answer is (A).