(a) Place the following gases in order of increasing average molecular speed at $300 \mathrm{~K}: \mathrm{CO}, \mathrm{SF}_{6}, \mathrm{H}_{2} \mathrm{~S}, \mathrm{Cl}_{2}, \mathrm{HBr}$. (b) Calculate the rms speeds of \(\mathrm{CO}\) and \(\mathrm{Cl}_{2}\) molecules at \(300 \mathrm{~K}\). (c) Calculate the most probable speeds of \(\mathrm{CO}\) and \(\mathrm{Cl}_{2}\) molecules at \(300 \mathrm{~K}\).

According to the Kinetic Gas theory, the average molecular speed, u, of an ideal gas can be determined using the formula: \[u\propto\sqrt{\frac{k_{B}T}{m_{g}}}\] where \(k_{B}\) is Boltzmann's constant, \(T\) is the temperature, and \(m_{g}\) is the molar mass of the gas. To order the gases, we can ignore the constants and compare the gases based on their molar masses. So, we have to find the molar mass of each gas: - CO (Carbon Monoxide): \(12.01+16.00 = 28.01 \mathrm{~g/mol}\) - SF6 (Sulfur Hexafluoride): \(32.07+6\times 19.00 = 146.07 \mathrm{~g/mol}\) - H2S (Hydrogen Sulfide): \(1.01\times 2+32.07 = 34.08 \mathrm{~g/mol}\) - Cl2 (Chlorine Gas): \(2\times 35.45 = 70.90 \mathrm{~g/mol}\) - HBr (Hydrogen Bromide): \(1.008+79.90 = 80.91 \mathrm{~g/mol}\) Since the speed is inversely proportional to the molar mass, the order in increasing average molecular speed is : SF6, HBr, Cl2, H2S, CO. Step 2: Calculate rms speeds for CO and Cl2 molecules

The root-mean-square (rms) speed, \(v_{rms}\), can be calculated using the formula: \[v_{rms}=\sqrt{\frac{3k_{B}T}{m_{g}}}\] where \(k_{B}\) is Boltzmann's constant in J/K/mol, \(T\) is the temperature in Kelvin, and \(m_{g}\) is the molar mass of the gas in kg/mol. For CO, we have: \[v_{rms}(\mathrm{CO})=\sqrt{\frac{3(8.314)(300)}{0.02801}}=336.47 \mathrm{~m/s}\] For Cl2, we have: \[v_{rms}(\mathrm{Cl}_{2})=\sqrt{\frac{3(8.314)(300)}{0.07090}}=206.44 \mathrm{~m/s}\] Step 3: Calculate most probable speeds for CO and Cl2 molecules

The most probable speed, \(v_{p}\), can be calculated using the formula: \[v_{p}=\sqrt{\frac{2k_{B}T}{m_{g}}}\] For CO, we have: \[v_{p}(\mathrm{CO})=\sqrt{\frac{2(8.314)(300)}{0.02801}}=272.62 \mathrm{~m/s}\] For Cl2, we have: \[v_{p}(\mathrm{Cl}_{2})=\sqrt{\frac{2(8.314)(300)}{0.07090}}=168.29 \mathrm{~m/s}\] To summarize the results: 1. The order of the gases in increasing average molecular speed is: SF6, HBr, Cl2, H2S, CO. 2. The rms speeds for CO and Cl2 molecules at 300K are 336.47 m/s and 206.44 m/s, respectively. 3. The most probable speeds for CO and Cl2 molecules at 300K are 272.62 m/s and 168.29 m/s, respectively.