Problem 73 - Chemistry The Central Science

Question

At an underwater depth of \(250 \mathrm{ft}\), the pressure is 8.38 atm. What should the mole percent of oxygen be in the diving gas for the partial pressure of oxygen in the mixture to be 0.21 atm, the same as in air at 1 atm?

Step-by-Step Solution

Answer

The mole percent of oxygen in the diving gas should be approximately 2.51% for the partial pressure of oxygen in the mixture to be 0.21 atm, the same as in air at 1 atm.

Step 1: Identify the given information

We are given: - The total pressure at the underwater depth, \(P_t = 8.38 \, \mathrm{atm}\) - The partial pressure of oxygen in air at 1 atm, \(P_{O_2}^{air} = 0.21 \, \mathrm{atm}\)

Step 2: Define formula for partial pressure

Recall that the partial pressure of a gas in a mixture can be calculated using the formula: \[P_i = X_i \cdot P_t\] Where \(P_i\) is the partial pressure of the gas, \(X_i\) is the mole fraction of the gas, and \(P_t\) is the total pressure of the mixture. In this problem, we need to find the mole fraction of oxygen (\(X_{O_2}\)) in the diving gas, such that the partial pressure of oxygen (\(P_{O_2}\)) is the same as in air at 1 atm.

Step 3: Set up the equation for mole fraction of oxygen

We can write the equation for the partial pressure of oxygen in the diving gas mixture as: \[P_{O_2} = X_{O_2} \cdot P_t\] We know that \(P_{O_2} = P_{O_2}^{air} = 0.21 \, \mathrm{atm}\) and \(P_t = 8.38 \, \mathrm{atm}\). So, we can rewrite the equation as: \[0.21 = X_{O_2} \cdot 8.38\]

Step 4: Solve for the mole fraction of oxygen

Now, we can solve for \(X_{O_2}\) by dividing both sides by 8.38: \[X_{O_2} = \frac{0.21}{8.38} \approx 0.0251\]

Step 5: Calculate the mole percent of oxygen

The mole percent of oxygen is found by multiplying the mole fraction by 100: \[Mole \, Percent \, of \, O_2 = X_{O_2} \cdot 100\] \[Mole \, Percent \, of \, O_2 \approx 0.0251 \cdot 100 = 2.51\%\] Therefore, for the partial pressure of oxygen in the diving gas to be the same as in air at 1 atm, the mole percent of oxygen in the diving gas should be approximately 2.51%.

Key Concepts

Diving Gas Mixture

When diving deep underwater, divers breathe a specially prepared mixture of gases. This is necessary because water pressure increases with depth. At 250 ft, the pressure is around 8.38 atm, which is much higher than the atmospheric pressure at sea level, which is 1 atm. Using a regular mix of atmospheric oxygen would be harmful due to the increased pressure. Thus, divers use mixtures like Nitrox, Heliox, or Trimix. These are carefully calculated to provide the necessary oxygen for breathing without causing toxic effects. The correct mixture ensures divers can safely stay underwater for extended periods.

Mole Fraction

In chemistry, the mole fraction is a way of expressing the concentration of a component in a mixture. It is defined as the ratio of the moles of one component to the total moles of all components in the mixture. The mole fraction is a unitless number, typically defined as:- For component A in a mixture: \[ X_A = \frac{n_A}{n_{\text{total}}} \] where \( n_A \) is the moles of component A, and \( n_{\text{total}} \) is the total moles in the mixture.This concept is crucial for calculating partial pressures in gas mixtures, such as in diving gases, ensuring each component contributes appropriately to the overall pressure without exceeding safe limits.

Oxygen

Oxygen is crucial for human survival, but its concentration must be managed carefully, especially under high pressures found underwater. At sea level, the partial pressure of oxygen in air is 0.21 atm. Divers need to maintain this pressure in the breathing gas mixture to avoid both hypoxia (insufficient oxygen) and oxygen toxicity (too much oxygen), which can occur at higher pressures. In the diving example, the challenge is adjusting the oxygen's mole fraction in the mix so that its partial pressure remains at 0.21 atm. This involves modifying the gas proportions to match the increased pressure environment.

Total Pressure

Total pressure in a gas mixture is the sum of each gas's partial pressures. This follows from Dalton's Law, which states that the total pressure exerted by a gaseous mix is equal to the sum of the pressures of each gas in the mix. At 250 ft underwater, the total pressure encountered by a diver is 8.38 atm. To maintain a partial pressure of oxygen at 0.21 atm (the same as air at 1 atm), divers must adjust the gas mixture. By understanding the total pressure and the role of each component, divers and scientists can prepare safe breathing mixtures that maintain necessary life support conditions even under immense underwater pressure.