At an underwater depth of 250 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?

Verified step by step guidance

1. Identify the given values: total pressure at depth is 8.38 atm, desired partial pressure of oxygen is 0.21 atm.
2. Use the formula for partial pressure: \( P_{\text{O}_2} = X_{\text{O}_2} \times P_{\text{total}} \), where \( P_{\text{O}_2} \) is the partial pressure of oxygen, \( X_{\text{O}_2} \) is the mole fraction of oxygen, and \( P_{\text{total}} \) is the total pressure.
3. Rearrange the formula to solve for the mole fraction of oxygen: \( X_{\text{O}_2} = \frac{P_{\text{O}_2}}{P_{\text{total}}} \).
4. Substitute the known values into the equation: \( X_{\text{O}_2} = \frac{0.21\, \text{atm}}{8.38\, \text{atm}} \).
5. Convert the mole fraction to mole percent by multiplying by 100: \( \text{Mole percent of oxygen} = X_{\text{O}_2} \times 100 \).

Key Concepts

Partial Pressure

Partial pressure refers to the pressure exerted by a single component of a gas mixture. According to Dalton's Law of Partial Pressures, the total pressure of a gas mixture is the sum of the partial pressures of each individual gas. In this context, understanding how to calculate the partial pressure of oxygen in a diving gas mixture is essential for determining the appropriate mole percent of oxygen.

Mole Fraction and Mole Percent

Mole fraction is the ratio of the number of moles of a specific component to the total number of moles in a mixture. Mole percent is simply the mole fraction expressed as a percentage. To find the mole percent of oxygen needed in the diving gas, one must relate the desired partial pressure of oxygen to the total pressure at the given depth, using the formula: mole fraction = partial pressure of oxygen / total pressure.

Gas Laws

Gas laws describe the behavior of gases under various conditions of temperature and pressure. The ideal gas law (PV=nRT) is particularly relevant here, as it relates pressure, volume, and temperature to the number of moles of gas. Understanding these laws helps in calculating how gases behave in different environments, such as underwater, where pressure significantly affects gas mixtures.