Hydrogen, M = 2.016 g/mol, is at 300 K. (a) Find the average speed of its molecules…

Step 1. The average speed of hydrogen. $\bar{u} = \sqrt{\dfrac{8 \times 8.314 \times 300}{\pi \times 2.016 \times 10^{-3}}} = 1775\ \text{m/s}$.

Step 2. The temperature of methane. We want $T$, so from $u_{\text{rms}}^2 = \dfrac{3RT}{M}$, $T = \dfrac{Mu_{\text{rms}}^2}{3R} = \dfrac{16.04 \times 10^{-3} \times 1775^2}{3 \times 8.314} = 2026\ \text{K}$.

Step 3. The ratio. The ratio of $u_{\text{rms}}$ to $u_{\text{mp}}$ is $\sqrt{3/2} = 1.225$ for every gas and temperature. So at $2026\ \text{K}$ the most probable speed of methane is $\dfrac{1775}{1.225} = 1449\ \text{m/s}$.