(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 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 \(\mathrm{K}\) .

First, find the molar mass of each gas: 1. CO: C = 12.01 g/mol, O = 16.00 g/mol, total = 28.01 g/mol 2. SF6: S = 32.07 g/mol, F = 19.00 g/mol (6 atoms), total = 32.07 + 6 × 19.00 = 146.07 g/mol 3. H2S: H = 1.01 g/mol (2 atoms), S = 32.07 g/mol, total = 2 × 1.01 + 32.07 = 34.09 g/mol 4. Cl2: Cl = 35.45 g/mol (2 atoms), total = 2 × 35.45 = 70.90 g/mol 5. HBr: H = 1.01 g/mol, Br = 79.90 g/mol, total = 1.01 + 79.90 = 80.91 g/mol Now, order the gases from the lowest to highest molar mass: \(CO < H_{2}S < Cl_{2} < HBr < SF_{6}\) At the same temperature, lighter molecules have higher average speeds. So, the order of increasing average molecular speed at 300 K is: \[SF_{6} < HBr < Cl_{2} < H_{2}S < CO\]

The equation for rms speed (\(v_{rms}\)) is: \[v_{rms} = \sqrt{\frac{3RT}{M}}\] Where: - R is the universal gas constant (8.314 J/mol K) - T is the temperature in Kelvin (300 K) - M is the molar mass of the gas in kg/mol Now, convert the molar mass of CO and Cl2 into kg/mol: CO = 28.01 g/mol × (1 kg/1000 g) = 0.02801 kg/mol Cl2 = 70.90 g/mol × (1 kg/1000 g) = 0.07090 kg/mol Next, calculate the rms speeds for CO and Cl2: For CO: \(v_{rms} = \sqrt{\frac{3(8.314 \,\text{J/mol K})(300 \,\text{K})}{0.02801\, \text{kg/mol}}}= 338.59 \, \text{m/s}\) For Cl2: \(v_{rms} = \sqrt{\frac{3(8.314 \,\text{J/mol K})(300\, \text{K})}{0.07090\, \text{kg/mol}}}= 203.00 \, \text{m/s}\) Thus, the rms speeds of CO and Cl2 at 300 K are: CO: 338.59 m/s Cl2: 203.00 m/s

The equation for the most probable speed (\(v_p\)) is: \[v_{p} = \sqrt{\frac{2RT}{M}}\] To calculate the most probable speeds for CO and Cl2, we will use the same values for R, T, and molar mass as in part (b). For CO: \(v_{p} = \sqrt{\frac{2(8.314\, \text{J/mol K})(300\, \text{K})}{0.02801\, \text{kg/mol}}} = 296.52\, \text{m/s}\) For Cl2: \(v_{p} = \sqrt{\frac{2(8.314\, \text{J/mol K})(300\, \text{K})}{0.07090\, \text{kg/mol}}} = 177.11\, \text{m/s}\) Thus, the most probable speeds of CO and Cl2 at 300 K are: CO: 296.52 m/s Cl2: 177.11 m/s