Question

A steel cylinder with sulfur dioxide, sulfur trioxide, and oxygen gases is at \(825^{\circ} \mathrm{C}\) and \(1.00 \mathrm{~atm} .\) If the partial pressure of sulfur dioxide is \(150 \mathrm{~mm} \mathrm{Hg}\) and sulfur trioxide is \(475 \mathrm{~mm}\) \(\mathrm{Hg}\), what is the partial pressure of oxvgen in \(\mathrm{mm}\) Hg?

Short answer

The partial pressure of oxygen is 135 mm Hg.

Step-by-step solution

Identify the total pressure in consistent units

Given that the total pressure of the gas mixture is 1.00 atm, we need to convert this into mm Hg for consistency with the given partial pressures. Since 1 atm equals 760 mm Hg, the total pressure in the cylinder is 760 mm Hg.

Apply Dalton's Law of Partial Pressures

According to Dalton’s Law, the total pressure of a gas mixture is the sum of the partial pressures of each individual gas in the mixture. Thus, we have:\[ P_{\text{Total}} = P_{\text{SO}_2} + P_{\text{SO}_3} + P_{\text{O}_2} \]where \(P_{\text{Total}} = 760 \ \text{mm Hg}\), \(P_{\text{SO}_2} = 150 \ \text{mm Hg}\), and \(P_{\text{SO}_3} = 475 \ \text{mm Hg}\). We need to find \(P_{\text{O}_2}\).

Calculate the partial pressure of oxygen

Rearrange the equation from Step 2 to solve for the partial pressure of oxygen:\[ P_{\text{O}_2} = P_{\text{Total}} - P_{\text{SO}_2} - P_{\text{SO}_3} \]Substitute the known values:\[ P_{\text{O}_2} = 760 \ \text{mm Hg} - 150 \ \text{mm Hg} - 475 \ \text{mm Hg} \]\[ P_{\text{O}_2} = 135 \ \text{mm Hg} \]

Verify the solution

Add up the partial pressures: \(150 \ \text{mm Hg}\) for sulfur dioxide, \(475 \ \text{mm Hg}\) for sulfur trioxide, and \(135 \ \text{mm Hg}\) for oxygen to check if it equals the total pressure:\[150 + 475 + 135 = 760 \ \text{mm Hg} \]This sum equals the total pressure, confirming that the partial pressure of oxygen is correctly calculated.

Key concepts

Partial Pressure

The concept of partial pressure is central to understanding how a mixture of gases behaves. Imagine a single gas molecule existing in an isolated container. This solitary scenario allows for a specific pressure to be exerted by that gas, known as its partial pressure. When we place multiple gases in the same container, each gas continues to exert the same pressure it would if it were alone. Consequently, the total pressure inside the container is simply the sum of the individual partial pressures. This is Dalton's Law of Partial Pressures. To formulate this mathematically, we write:

• \(P_{\text{Total}} = P_1 + P_2 + P_3 + \ldots \)

where \(P_1, P_2, P_3, \ldots\) are the partial pressures of the individual gases. In our case, the sum of the pressures from sulfur dioxide, sulfur trioxide, and oxygen gives us the total pressure in the vessel.

Sulfur Dioxide

Sulfur dioxide is a significant component in the gas mixture from the exercise. It’s a colorless gas and has a pungent aroma often associated with burning matches. Commonly used in industries for making sulfuric acid, SO i 2 plays a crucial role in environmental chemistry due to its involvement in acid rain formation. In the context of gas mixtures, its behavior under conditions of constant temperature and pressure aligns with Dalton's Law, where the contribution of sulfur dioxide to the total pressure is known as its partial pressure. Thus, when considering a cylinder containing sulfur dioxide along with other gases, the amount of pressure SO 2 contributes depends on its proportion in the mixture.

Oxygen Gas

In the context of gas mixtures, oxygen plays an equally important role. Although commonly known for its vital necessity for respiration, here it acts as one of the gases according to Dalton's Law. The partial pressure of oxygen is effectively the share of pressure it adds to the overall pressure in the container. Calculating this requires subtracting the known partial pressures of other gases, like sulfur dioxide and sulfur trioxide, from the total gas pressure. The equation used in the exercise was:

• \(P_{\text{O}_2} = P_{\text{Total}} - P_{\text{SO}_2} - P_{\text{SO}_3}\)

This formula highlights how the pressure exerted by each component of the gas mixture adds up to the overall pressure, emphasizing the individual contributions each gas makes in a shared environment.