What are the rms speeds of helium atoms, and nitrogen, hydrogen, and oxygen molecules at \(25^{\circ} \mathrm{C} ?\)

To convert from Celsius to Kelvin, add 273.15: \(T=25^{\circ} \mathrm{C} + 273.15 = 298.15\ \mathrm{K}\) Now, we will find the molar mass of each gas particle and convert it to the mass of a single particle in kilograms: For helium atom: \(M_{He} = 4.0026\ \mathrm{g/mol}\) or \(4.0026 \times 10^{-3}\ \mathrm{kg/mol}\) For nitrogen molecule: \(M_{N_2} = 28.0134\ \mathrm{g/mol}\) or \(28.0134 \times 10^{-3}\ \mathrm{kg/mol}\) For hydrogen molecule: \(M_{H_2} = 2.01588\ \mathrm{g/mol}\) or \(2.01588 \times 10^{-3}\ \mathrm{kg/mol}\) For oxygen molecule: \(M_{O_2} = 31.9988\ \mathrm{g/mol}\) or \(31.9988 \times 10^{-3}\ \mathrm{kg/mol}\) The Avogadro constant denoted as \(N_A\) is \(6.022 \times 10^{23}\ \mathrm{mol^{-1}}\). Now, we will calculate the mass of a single particle for each type of gas using the formula: \(m = \frac{M}{N_A}\) For helium atom: \(m_{He} = \frac{4.0026 \times 10^{-3}}{6.022 \times 10^{23}}\ \mathrm{kg}\) For nitrogen molecule: \(m_{N_2} = \frac{28.0134 \times 10^{-3}}{6.022 \times 10^{23}}\ \mathrm{kg}\) For hydrogen molecule: \(m_{H_2} = \frac{2.01588 \times 10^{-3}}{6.022 \times 10^{23}}\ \mathrm{kg}\) For oxygen molecule: \(m_{O_2} = \frac{31.9988 \times 10^{-3}}{6.022 \times 10^{23}}\ \mathrm{kg}\) Now, we will calculate the rms speed for each particle using the equation: \(v_{rms} = \sqrt{\frac{3kT}{m}}\) For helium atom:

\(v_{rms_{He}} = \sqrt{\frac{3(1.38 \times 10^{-23})(298.15)}{4.0026 \times 10^{-3}/(6.022 \times 10^{23})}} \approx 1367\ \mathrm{m/s}\) For nitrogen molecule:

\(v_{rms_{N_2}} = \sqrt{\frac{3(1.38 \times 10^{-23})(298.15)}{28.0134 \times 10^{-3}/(6.022 \times 10^{23})}} \approx 515\ \mathrm{m/s}\) For hydrogen molecule:

\(v_{rms_{H_2}} = \sqrt{\frac{3(1.38 \times 10^{-23})(298.15)}{2.01588 \times 10^{-3}/(6.022 \times 10^{23})}} \approx 1929\ \mathrm{m/s}\) For oxygen molecule:

\(v_{rms_{O_2}} = \sqrt{\frac{3(1.38 \times 10^{-23})(298.15)}{31.9988 \times 10^{-3}/(6.022 \times 10^{23})}} \approx 482\ \mathrm{m/s}\) The rms speeds at \(25^{\circ} \mathrm{C}\) are approximately: Helium atoms: 1367 m/s Nitrogen molecules: 515 m/s Hydrogen molecules: 1929 m/s Oxygen molecules: 482 m/s