Question

Which of the following gases has the highest root-mean-square speed? a) nitrogen at \(1 \mathrm{~atm}\) and \(30^{\circ} \mathrm{C}\) b) argon at \(1 \mathrm{~atm}\) and \(30^{\circ} \mathrm{C}\) c) argon at 2 atm and \(30^{\circ} \mathrm{C}\) d) oxygen at 2 atm and \(30^{\circ} \mathrm{C}\) e) nitrogen at 2 atm and \(15^{\circ} \mathrm{C}\)

Short answer

Answer: Nitrogen at 1 atm and 30°C has the highest root-mean-square speed, with a value of 516.98 m/s.

Step-by-step solution

Convert temperatures to Kelvin

We'll convert the given temperatures in Celsius to Kelvin by adding 273.15 to each temperature. For the gases at \(30^{\circ} \mathrm{C}\): \(T_1 = 30 + 273.15 = 303.15 \mathrm{K}\) For the gases at \(15^{\circ} \mathrm{C}\): \(T_2 = 15 + 273.15 = 288.15 \mathrm{K}\)

Determine molar masses

We need to find the molar mass of each gas in kg/mol. a) Nitrogen: (N2) \(M_a = 28 \times 10^{-3} \mathrm{kg/mol}\) b) Argon: (Ar) \(M_b = 40 \times 10^{-3} \mathrm{kg/mol}\) d) Oxygen: (O2) \(M_d = 32 \times 10^{-3} \mathrm{kg/mol}\) (Nitrogen and Oxygen are diatomic molecules.)

Calculate RMS speed

Now, we'll use the equation \(V_{rms} = \sqrt{\frac{3RT}{M}}\) to calculate RMS speed for each gas. Note that for this problem, the pressure doesn't affect RMS speed, which can be neglected. - The gas constant \(R = 8.314 \mathrm{J/(mol \cdot K)}\) \(V_{rms-a} = \sqrt{\frac{3 \times 8.314 \times 303.15}{28\times10^{-3}}}\) \(V_{rms-b} = \sqrt{\frac{3 \times 8.314 \times 303.15}{40\times10^{-3}}}\) \(V_{rms-c} = \sqrt{\frac{3 \times 8.314 \times 303.15}{40\times10^{-3}}}\) \(V_{rms-d} = \sqrt{\frac{3 \times 8.314 \times 303.15}{32\times10^{-3}}}\) \(V_{rms-e} = \sqrt{\frac{3 \times 8.314 \times 288.15}{28\times10^{-3}}}\)

Compare RMS speeds

We can now compare the calculated RMS speeds to find the highest value: \(V_{rms-a} = 516.98 \mathrm{m/s}\) \(V_{rms-b} = 431.76 \mathrm{m/s}\) \(V_{rms-c} = 431.76 \mathrm{m/s}\) \(V_{rms-d} = 483.58 \mathrm{m/s}\) \(V_{rms-e} = 499.69 \mathrm{m/s}\) Clearly, nitrogen at \(1\mathrm{~atm}\) and \(30^{\circ} \mathrm{C}\) (option a) has the highest RMS speed of 516.98 m/s.