Problem 82 - Chemistry The Central Science In Si Units

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 \(\mathrm{CO}\) and \(\mathrm{Cl}_{2}\) molecules at \(300 \mathrm{~K}\). (c) Calculate the most probable speeds of \(\mathrm{CO}\) and \(\mathrm{Cl}_{2}\) molecules at \(300 \mathrm{~K}\).

Step-by-Step Solution

Answer

(a) Order: SF\(_6\), HBr, Cl\(_2\), H\(_2\)S, CO. (b) RMS Speeds: CO = 517 m/s, Cl\(_2\) = 292 m/s. (c) Most Probable Speeds: CO = 422 m/s, Cl\(_2\) = 238 m/s.

Step 1: Understand Molecular Speed Relationship

The average molecular speed of gases is related to the temperature and molar mass. According to the kinetic molecular theory, lighter molecules move faster, so we use the molecular mass for comparison.

Step 2: List Molar Masses of Gases

Find the molar masses of the given gases by considering the atomic masses: \( \mathrm{CO (28 \ g/mol)}, \ \mathrm{SF_6 (146 \ g/mol)}, \ \mathrm{H_2S (34 \ g/mol)}, \ \mathrm{Cl_2 (71 \ g/mol)}, \ \mathrm{HBr (81 \ g/mol)} \).

Step 3: Order Gases by Molar Mass

The gas with the smallest molar mass will have the highest speed. Based on molar masses, order the gases from lightest to heaviest: \( \mathrm{CO}, \mathrm{H_2S}, \mathrm{Cl_2}, \mathrm{HBr}, \mathrm{SF_6} \).

Step 4: Rearrange Gases by Increasing Speed

Since lighter gases move faster, arrange the gases in order of increasing average molecular speed: \( \mathrm{SF_6}, \mathrm{HBr}, \mathrm{Cl_2}, \mathrm{H_2S}, \mathrm{CO} \).

Step 5: Use the RMS Speed Formula

To calculate the root mean square (rms) speed, use the formula: \( v_{rms} = \sqrt{\frac{3RT}{M}} \), where \( R = 8.314 \ J/mol \cdot K \) and \( M \) is the molar mass in kg/mol.

Step 6: Convert Molar Mass to Kilograms

Convert the molar masses of \( \mathrm{CO} \) and \( \mathrm{Cl_2} \) from g/mol to kg/mol: \( \mathrm{CO: 0.028 \ kg/mol} \), \( \mathrm{Cl_2: 0.071 \ kg/mol} \).

Step 7: Calculate RMS Speed for CO

Plug the values for \( \mathrm{CO} \) into the rms speed formula: \[ v_{rms, CO} = \sqrt{\frac{3 \times 8.314 \times 300}{0.028}} \approx 517 \ m/s. \]

Step 8: Calculate RMS Speed for Cl2

Use the rms speed formula for \( \mathrm{Cl_2} \): \[ v_{rms, Cl_2} = \sqrt{\frac{3 \times 8.314 \times 300}{0.071}} \approx 292 \ m/s. \]

Step 9: Use Most Probable Speed Formula

The most probable speed is calculated using: \( v_{mp} = \sqrt{\frac{2RT}{M}} \).

Step 10: Calculate Most Probable Speed for CO

Apply the most probable speed formula to \( \mathrm{CO} \): \[ v_{mp, CO} = \sqrt{\frac{2 \times 8.314 \times 300}{0.028}} \approx 422 \ m/s. \]

Step 11: Calculate Most Probable Speed for Cl2

Calculate for \( \mathrm{Cl_2} \) using the most probable speed formula: \[ v_{mp, Cl_2} = \sqrt{\frac{2 \times 8.314 \times 300}{0.071}} \approx 238 \ m/s. \]

Key Concepts

Kinetic Molecular Theory

The Kinetic Molecular Theory provides a framework for understanding the behavior of gases. It's built on the idea that gases are composed of randomly moving particles, which collide with each other and the walls of their container. A key point is that these particles move faster when the temperature increases, as heat adds energy to the system.

• Particles are in constant motion and collisions are perfectly elastic.
• Temperature is directly proportional to the average kinetic energy of gas molecules.
• Lighter gas molecules move faster on average than heavier ones.

Using this theory, the speed of gas molecules can be linked directly to their molar mass and temperature, explaining why at the same temperature, lighter gases like \( ext{H}_2S\) travel faster than heavier ones like \( ext{SF}_6\). The kinetic molecular theory is foundational in deriving formulas for molecular speeds, including the root mean square (rms) speed.

RMS Speed

Root mean square (rms) speed is a measure of the average speed of particles in a gas. It's a statistical measure that provides insights into how fast molecules move in a gas. Mathematically, rms speed is given by the formula: \[ v_{rms} = \sqrt{\frac{3RT}{M}} \] where \(R\) is the gas constant \(8.314 \, J/mol \cdot K\), \(T\) is temperature in Kelvin, and \(M\) is the molar mass in kilograms per mole.

• The formula shows that rms speed increases with temperature.
• Lighter molecules (lower \(M\)) have higher rms speeds.

For instance, \(CO\), with a molar mass of \(0.028 \, kg/mol\), moves faster than \(Cl_2\) due to its smaller molar mass. At \(300 \, K\), calculated rms speeds are approximately \(517 \, m/s\) for \(CO\) and \(292 \, m/s\) for \(Cl_2\), highlighting how molecular mass influences speed.

Molar Mass

Molar mass is critical in determining a molecule's speed. It represents the mass of one mole of a substance and is expressed in grams per mole (g/mol). Here's how molar mass affects molecular speed:

• Lighter gases (lower molar mass) have higher average speeds.
• Heavier gases (higher molar mass) move more slowly on average.

In our example, gases are ordered by increasing molar masses as follows: \(\text{CO (28 \, g/mol)}, \, \text{H}_2\text{S (34 \, g/mol)}, \, \text{Cl}_2 (71 \, g/mol), \, \text{HBr (81 \, g/mol)}, \, \text{SF}_6 (146 \, g/mol)\). This order directly influences their molecular speeds, as shown in the exercise, where the lightest gas \(CO\) moves fastest.

Most Probable Speed

The most probable speed is the speed at which the highest number of molecules are moving. Calculated using the formula: \[ v_{mp} = \sqrt{\frac{2RT}{M}} \] this speed also depends on the temperature and molar mass, similar to rms speed. However, the factor of 2 instead of 3 in the formula implies it's slightly less than rms speed.

• Most probable speed is typically lower than rms speed.
• It also increases with temperature and decreases with higher molar mass.

For \(CO\) and \(Cl_2\) at \(300 \, K\), the most probable speeds are calculated to be \(422 \, m/s\) and \(238 \, m/s\) respectively, illustrating how even among the same group of molecules, not all have precisely the same speed.