Problem 82 - Chemistry: The Central Science in SI Units, Global Edition

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

The order of the gases in increasing average molecular speed is: SF6, HBr, Cl2, H2S, CO. The rms speeds for CO and Cl2 molecules at 300K are 336.47 m/s and 206.44 m/s, respectively. The most probable speeds for CO and Cl2 molecules at 300K are 272.62 m/s and 168.29 m/s, respectively.

Step 1: Determine the average molecular speed

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

Step 2: Determine the rms speeds

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

Step 3: Determine the most probable speeds

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.

Key Concepts

Average Molecular Speed

The average molecular speed of a gas provides an insight into the kinetic energy and behavior of gas molecules at a given temperature. This concept is rooted in the Kinetic Molecular Theory, which helps us understand gases as collections of particles that are in constant, random motion. The average molecular speed depends on several factors:

1. **Temperature:** The speed is directly proportional to the square root of the temperature. As temperature increases, molecules move faster.
2. **Molar Mass:** The speed is inversely proportional to the square root of the molar mass. Lighter molecules move more quickly than heavier ones.

To give a practical example, when we compare different gases like CO, SF₆, H₂S, Cl₂, and HBr at a constant temperature of 300 K, we find that the gas with the lowest molar mass, CO, moves the fastest. Calculating the molar masses and arranging them accordingly, we have the order from slowest to fastest as SF₆, HBr, Cl₂, H₂S, and CO.

Root Mean Square Speed (rms)

Root mean square speed (rms) is an important concept in gas dynamics, combining both the average speed and the distribution of speeds in a gas. It is calculated using the formula:
\[v_{rms} = \sqrt{\frac{3k_{B}T}{m_{g}}}\]
where:

• \(k_{B}\) is the Boltzmann constant.
• \(T\) is the temperature in Kelvin.
• \(m_{g}\) is the molar mass of the gas in kg/mol.

This formula demonstrates that rms speed increases with temperature and decreases with a larger molar mass. This gives us a better "mean" speed that accounts for all particles in the system. In practice, when we calculate rms speeds for CO and Cl₂ at 300 K, we find they are 336.47 m/s and 206.44 m/s, respectively. This illustrates how CO, being a lighter molecule than Cl₂, has a higher rms speed.

Most Probable Speed

The most probable speed is the speed at which the largest number of molecules in a gas sample move. It is a statistical measurement derived from the Maxwell-Boltzmann distribution and is calculated with the formula:

\[v_{p} = \sqrt{\frac{2k_{B}T}{m_{g}}}\]

This speed is most informative in understanding how likely a particular speed is within a gas mixture. For gases like CO and Cl₂ at 300 K, the most probable speeds are 272.62 m/s and 168.29 m/s, respectively. The smaller this speed relative to rms speed, the more uniform the speed distribution is. It's important to note that the most probable speed usually differs slightly from the average and rms speeds because it takes into account the non-uniform distribution of molecular speeds. Hence, knowing this speed allows us to better predict the common behavior of gas molecules at given conditions.