Problem 47 - University Physics with Modern Physics

Question

For diatomic carbon dioxide gas \(\left(\mathrm{CO}_{2},\) molar mass \right. 44.0 \(\mathrm{g} / \mathrm{mol} )\) at \(T=300 \mathrm{K}\) , calculate (a) the most probable speed \(v_{\mathrm{mp}} ;\) (b) the average speed \(v_{\mathrm{av}} ;(\mathrm{c})\) the root-mean-square speed \(v_{\mathrm{rms}}\)

Step-by-Step Solution

Answer

\(v_{mp} = 337.94 \, \mathrm{m/s}\), \(v_{av} = 369.62 \, \mathrm{m/s}\), \(v_{rms} = 394.45 \, \mathrm{m/s}\).

Step 1: Understand the formulas

To solve the problem, we need to calculate three different speeds for the CO2 gas using these formulas: most probable speed \(v_{mp} = \sqrt{\frac{2RT}{M}}\), average speed \(v_{av} = \sqrt{\frac{8RT}{\pi M}}\), and root-mean-square speed \(v_{rms} = \sqrt{\frac{3RT}{M}}\), where \(R\) is the universal gas constant \(8.314 \frac{J}{mol \cdot K}\), \(T\) is the temperature in Kelvin, and \(M\) is the molar mass in kg/mol.

Step 2: Convert molar mass to kg/mol

The molar mass given is \(44.0 \, \mathrm{g/mol}\). We convert it to kilograms by dividing by 1000, resulting in \(M = 0.044 \, \mathrm{kg/mol}\).

Step 3: Calculate the most probable speed

Substitute the known values into the formula for the most probable speed: \[v_{mp} = \sqrt{\frac{2 \times 8.314 \times 300}{0.044}} \]This simplifies to:\[v_{mp} = \sqrt{\frac{4988.4}{0.044}}\approx 337.94 \, \mathrm{m/s}\]

Step 4: Calculate the average speed

Substitute the known values into the formula for the average speed: \[v_{av} = \sqrt{\frac{8 \times 8.314 \times 300}{\pi \times 0.044}} \]This simplifies to:\[v_{av} = \sqrt{\frac{19953.6}{0.1382}} \approx 369.62 \, \mathrm{m/s}\]

Step 5: Calculate the root-mean-square speed

Substitute the known values into the formula for the root-mean-square speed: \[v_{rms} = \sqrt{\frac{3 \times 8.314 \times 300}{0.044}} \]This simplifies to:\[v_{rms} = \sqrt{\frac{7482.6}{0.044}} \approx 394.45 \, \mathrm{m/s}\]

Step 6: Final Answer

The most probable speed \(v_{mp}\) is \(337.94 \, \mathrm{m/s}\), the average speed \(v_{av}\) is \(369.62 \, \mathrm{m/s}\), and the root-mean-square speed \(v_{rms}\) is \(394.45 \, \mathrm{m/s}\).

Key Concepts

Most Probable Speed

When we talk about gases, the kinetic molecular theory provides valuable insights into the behavior of gas molecules. One key concept is the 'most probable speed'. This term refers to the speed at which the largest number of gas molecules are moving. It's computed using the formula:\[ v_{mp} = \sqrt{\frac{2RT}{M}} \]where:

• \( R \) is the universal gas constant, approximately 8.314 \( \text{J/mol K} \).
• \( T \) is the temperature in Kelvin.
• \( M \) is the molar mass of the gas in \( \text{kg/mol} \).

For the gas carbon dioxide \( \text{CO}_2 \) at 300 K, with a molar mass of 44.0 g/mol (converted to 0.044 kg/mol), the most probable speed is calculated to be approximately 337.94 m/s. This speed is significant because it represents the peak of the distribution curve of molecular speeds in a gas, meaning it's most common for the molecules of this gas at this temperature.

Average Speed

The second speed of interest is the 'average speed'. While the most probable speed is about where most molecules are, the average speed considers a balance of all molecular speeds. It is slightly higher than the most probable speed because it takes into account that some molecules move very fast.The formula for average speed is:\[ v_{av} = \sqrt{\frac{8RT}{\pi M}} \]In this formula:

• \( \pi \) is approximately 3.14159.
• Other variables, \( R \), \( T \), and \( M \), remain the same as defined for the most probable speed.

Using this formula for \( \text{CO}_2 \) at 300 K, we find the average speed is around 369.62 m/s. This indicates that when considering all speeds, the collective average is slightly faster than the peak speed. This is due to the presence of a few molecules with significantly higher speeds skewing the average.

Root-Mean-Square Speed

The 'root-mean-square speed' (often abbreviated as rms speed) is another important measure. It is useful because it's directly related to the kinetic energy of the gas molecules, which depends on the square of the speed. This makes the root-mean-square speed particularly significant when considering energy and temperature relations.The formula to calculate the root-mean-square speed is:\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]Here's a breakdown of the root-mean-square speed:

• The constant 3 in the formula brings balance between the energy framework (proportional to \( v^2 \)) and the actual speeds.
• As before, \( R \), \( T \), and \( M \) are the same.

For \( \text{CO}_2 \), when plugged into the equation, the root-mean-square speed turns out to be about 394.45 m/s. This value is typically always the highest among the three types of speeds because it's influenced heavily by the few fastest moving molecules, highlighting their impact on the overall temperature and energy of the gas.