Problem 96 - Chemistry The Molecular Nature of Matter

Question

For the gases \(\mathrm{CO}, \mathrm{CO}_{2}, \mathrm{NO},\) and \(\mathrm{NO}_{2},\) which gas will effuse the fastest? Which gas the slowest? Which gases will be the most difficult to separate by effusion?

Step-by-Step Solution

Answer

\( \mathrm{CO} \) will effuse the fastest and \( \mathrm{NO}_{2} \) will effuse the slowest. \( \mathrm{CO} \) and \( \mathrm{NO} \) will be the most difficult to separate by effusion due to their similar molar masses.

Step 1: Understand Graham's Law of Effusion

To solve this problem, we need to apply Graham's Law of Effusion, which states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass. Mathematically, it can be expressed as: Rate of effusion \( \alpha \frac{1}{\sqrt{M}} \) where \(M\) is the molar mass of the gas.

Step 2: Determine the Molar Mass of Each Gas

Calculate the molar mass of each gas by adding the atomic masses of its constituent elements. The molar masses (g/mol) are approximately: \( \mathrm{CO} \): 28.01, \( \mathrm{CO}_{2} \): 44.01, \( \mathrm{NO} \): 30.01, \( \mathrm{NO}_{2} \): 46.01.

Step 3: Rank the Gases by Rate of Effusion

Using Graham's Law, we know that lighter gases effuse faster. Thus, we can order the gases from fastest to slowest effusion based on their molar masses: \( \mathrm{CO} \)(28.01) > \( \mathrm{NO} \)(30.01) > \( \mathrm{CO}_{2} \)(44.01) > \( \mathrm{NO}_{2} \)(46.01).

Step 4: Identify the Most Difficult Gases to Separate

Gases with similar molar masses will have similar rates of effusion and thus will be harder to separate. In this case, \( \mathrm{CO} \) (28.01 g/mol) and \( \mathrm{NO} \) (30.01 g/mol) have the closest molar masses and hence will be the most difficult to separate by effusion.

Key Concepts

Rate of Effusion

Understanding the rate of effusion is crucial when predicting how different gases will behave under the same conditions. Effusion is the process by which gas molecules escape through a tiny hole into a vacuum. According to Graham's Law of Effusion, the rate of effusion of a gas is inversely proportional to the square root of its molar mass, meaning lighter gases effuse more quickly.

For instance, if we consider a balloon filled with helium and another with carbon dioxide, the helium-filled balloon will deflate faster due to helium's lower molar mass, thus a higher rate of effusion—this concept explains the transient nature of helium balloons compared to those filled with air.

Molar Mass

Molar mass, typically expressed in grams per mole (g/mol), is the mass of one mole of a substance. It plays a pivotal role in various chemical calculations and has a direct impact on the rate of effusion as outlined by Graham's Law. When it comes to gas effusion, the key takeaway is that the lighter the molar mass, the faster the gas will effuse.

In our exercise, we calculated the molar mass of each gas to determine their respective effusion rates. These calculations not only aid in theoretical predictions but also in practical applications such as the production of chemical compounds where controlling gas effusion rates is essential.

Gas Separation by Effusion

Gas separation by effusion is a practical application of Graham's Law used in industrial processes. It leverages the different rates of effusion of gases to separate them from a mixture. For example, in the enrichment of uranium, the slight difference in molar mass between the isotopes of uranium allows for their separation using effusion.

In our textbook problem, identifying which gases would be most difficult to separate is based on their similar molar masses—closely matched molar masses result in similar rates of effusion, making separation more challenging. Advanced gas separation techniques often require precise control over the conditions to enhance the separation of gases with close molar masses.

Comparing Effusion Rates

Comparing effusion rates among different gases allows us to anticipate which gas will effuse the fastest or slowest under identical conditions. Using Graham's Law, one can calculate or estimate these rates based on known molar masses. For instructional and practical scenarios, understanding this comparison helps in predicting and controlling processes like gas leaks, respiratory therapy, and chemical synthesis.

In our exercise, by calculating and comparing the molar masses, we could predict the order of effusion rates for CO, CO2, NO, and NO2, proceeding from the fastest to the slowest effuser. This comprehension is essential for students and professionals who work with gas mixtures in scientific and industrial environments.