Calculate the root mean square speed of hydrogen gas molecules at 300 K.

• Temperature, $T = 300 \text{ K}$
• Molar mass of hydrogen (H₂), $M = 2.016 \text{ g/mol}$
• Ideal gas constant, $R = 8.314 \text{ J mol}^{-1} \text{ K}^{-1}$

Convert the molar mass from g/mol to kg/mol:

$$M = 2.016 \text{ g/mol} \times \frac{1 \text{ kg}}{1000 \text{ g}} = 0.002016 \text{ kg/mol}$$

The formula is:

$$v_{rms} = \sqrt{\frac{3RT}{M}}$$

$$v_{rms} = \sqrt{\frac{3 \times 8.314 \text{ J mol}^{-1} \text{ K}^{-1} \times 300 \text{ K}}{0.002016 \text{ kg mol}^{-1}}}$$ $$v_{rms} = \sqrt{\frac{7482.6 \text{ J mol}^{-1}}{0.002016 \text{ kg mol}^{-1}}}$$

Since $1 \text{ J} = 1 \text{ kg m}^2 \text{ s}^{-2}$, the units simplify to $\text{m}^2 \text{ s}^{-2}$:

$$v_{rms} = \sqrt{\frac{7482.6 \text{ kg m}^2 \text{ s}^{-2}}{0.002016 \text{ kg}}}$$ $$v_{rms} = \sqrt{3711607.142857 \text{ m}^2 \text{ s}^{-2}}$$ $$v_{rms} \approx 1926.558 \text{ m/s}$$

$$v_{rms} \approx 1927 \text{ m/s}$$

The root mean square speed of hydrogen molecules at 300 K is:

$$1927 \text{ m/s}$$