Problem 82 - Chemistry The Central Science

Question

(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 \(\mathrm{K}\) .

Step-by-Step Solution

Answer

(a) The order of increasing average molecular speed at 300 K is: \(SF_{6} < HBr < Cl_{2} < H_{2}S < CO\). (b) The root-mean-square (rms) speeds of CO and Cl2 at 300 K are 338.59 m/s and 203.00 m/s, respectively. (c) The most probable speeds of CO and Cl2 at 300 K are 296.52 m/s and 177.11 m/s, respectively.

Step 1: Part (a): Ordering the gases based on their average molecular speed

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: \[SF_{6} < HBr < Cl_{2} < H_{2}S < CO\]

Step 2: Part (b): Calculate the rms speeds of CO and Cl2 molecules at 300 K

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

Step 3: Part (c): Calculate the most probable speeds of CO and Cl2 molecules at 300 K

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

Key Concepts

rms speed

The Root Mean Square (rms) speed is a measure of the average speed of particles in a gas. It's important for understanding how gas molecules move and how frequently they collide. The formula for rms speed is given by \(v_{rms} = \sqrt{\frac{3RT}{M}}\), where:

• \(R\) is the universal gas constant, which is 8.314 J/mol K.
• \(T\) is the temperature in Kelvin. In this context, it's 300 K.
• \(M\) is the molar mass of the gas in kg/mol.

This speed is called "root mean square" because it's the square root of the average of the squares of the velocities. It provides a useful means of relating energy to temperature. Lighter gases move faster at a given temperature because the kinetic energy depends on both mass and speed. So, the rms speed decreases with increasing molar mass for gases at the same temperature.

most probable speed

The most probable speed, denoted as \(v_p\), refers to the speed at which the greatest number of molecules in a gas sample are moving. It is different from average and rms speed but is often used to analyze gases' behaviors at the molecular level. The equation for calculating the most probable speed is \(v_p = \sqrt{\frac{2RT}{M}}\), where:

• \(R\) is still our universal gas constant (8.314 J/mol K).
• \(T\) is the given temperature in Kelvin, such as 300 K for this problem.
• \(M\) is the molar mass of the gas in kg/mol.

The most probable speed is generally less than the rms speed because it considers the point at which the maximum number of particles are likely to be found. For lighter molecules, the most probable speed is higher because they're less massive, leading to faster velocities at the same temperature.

molar mass

Molar mass is the mass of one mole of a substance, usually expressed in grams per mole (g/mol). For gases, it is crucial for calculations involving speed and kinetic energy, as seen in exercises determining rms and most probable speeds. To convert molar mass to kg/mol (as needed for our equations), you multiply by \(1 \text{ kg} / 1000 \text{ g}\). It's pivotal to know how molecular mass affects gas speed:

• Lighter gases (lower molar mass) move faster at the same temperature.
• Heavier gases (higher molar mass) move slower.

In this example, understanding how to order gases by molar mass helps in predicting molecular speed. Thus, a gas like CO, with a lower molar mass of 28.01 g/mol, will move faster than Cl\(_2\), which has a molar mass of 70.90 g/mol at the same temperature.