Question

As noted in the text, the speed distribution of molecules in the Earth's atmosphere has a significant impact on its composition. a) What is the average speed of a nitrogen molecule in the atmosphere, at a temperature of \(18.0^{\circ} \mathrm{C}\) and a (partial) pressure of \(78.8 \mathrm{kPa}\) ? b) What is the average speed of a hydrogen molecule at the same temperature and pressure?

Short answer

Answer: The average speed of a nitrogen molecule at 18.0°C and 78.8 kPa is approximately 514.2 m/s, and the average speed of a hydrogen molecule at the same conditions is approximately 1928.2 m/s.

Step-by-step solution

Convert the temperature to Kelvin

To convert the temperature to Kelvin, add 273.15 to the given Celsius temperature: \(T_K = 18.0^{\circ}C + 273.15 = 291.15\,\mathrm{K}\)

Find the mass of nitrogen and hydrogen molecules

To find the mass of a nitrogen molecule (N2) and hydrogen molecule (H2) in kilograms, we can use their respective molar mass and Avogadro's number: - Molar mass of nitrogen (N2): \(28.02\,\mathrm{g/mol}\) - Molar mass of hydrogen (H2): \(2.02\,\mathrm{g/mol}\) - Avogadro's number: \(N_A = 6.022 \times 10^{23}\,\mathrm{mol^{-1}}\) We can find the mass of a single nitrogen and hydrogen molecule by dividing their respective molar mass by Avogadro's number and then converting to kilograms: \(m_{N2} = \frac{28.02\,\mathrm{g/mol}}{6.022 \times 10^{23}\,\mathrm{mol^{-1}}} \times 10^{-3}\,\mathrm{kg/g} = 4.65 \times 10^{-26}\,\mathrm{kg}\) \(m_{H2} = \frac{2.02\,\mathrm{g/mol}}{6.022 \times 10^{23}\,\mathrm{mol^{-1}}} \times 10^{-3}\,\mathrm{kg/g} = 3.35 \times 10^{-27}\,\mathrm{kg}\)

Calculate the average speed of nitrogen molecules

Now, we can use the average speed equation to find the average speed of a nitrogen molecule: \(v_{avg,N2} = \sqrt{\frac{8 * 1.38 \times 10^{-23} * 291.15}{\pi * 4.65 \times 10^{-26}}} \approx 514.2\,\mathrm{m/s}\)

Calculate the average speed of hydrogen molecules

Similarly, we can use the average speed equation to find the average speed of a hydrogen molecule: \(v_{avg,H2} = \sqrt{\frac{8 * 1.38 \times 10^{-23} * 291.15}{\pi * 3.35 \times 10^{-27}}} \approx 1928.2\,\mathrm{m/s}\) Hence, the average speed of a nitrogen molecule in the Earth's atmosphere at \(18.0^{\circ}C\) and \(78.8\,\mathrm{kPa}\) is approximately \(514.2\,\mathrm{m/s}\), and the average speed of a hydrogen molecule at the same conditions is approximately \(1928.2\,\mathrm{m/s}\).