Question

In a series of experiments, the U.S. Navy developed an undersea habitat. In one experiment, the mole percent composition of the atmosphere in the undersea habitat was \(79.0 \%\) He, \(17.0 \% \mathrm{~N}_{2}\), and \(4.0 \% \mathrm{O}_{2}\). What will the partial pressure of each gas be when the habitat is \(58.8 \mathrm{~m}\) below sea level, where the pressure is 6.91 atm?

Short answer

The partial pressures are: Helium = 5.46 atm, Nitrogen = 1.17 atm, Oxygen = 0.28 atm.

Step-by-step solution

Calculate Partial Pressure of Helium

The mole fraction of helium is given as 79.0%. Convert this percentage to a decimal by dividing by 100, so the mole fraction of helium, \(X_{\text{He}}\), is 0.79. The partial pressure of helium can be found using the formula for partial pressure: \[ P_{\text{He}} = X_{\text{He}} \times P_{\text{total}} \]Substitute the known values: \[ P_{\text{He}} = 0.79 \times 6.91 = 5.46 \text{ atm} \]

Calculate Partial Pressure of Nitrogen

The mole fraction of nitrogen is given as 17.0%. Convert this percentage to a decimal by dividing by 100, so \(X_{\text{N}_2} = 0.17\). Use the partial pressure formula:\[ P_{\text{N}_2} = X_{\text{N}_2} \times P_{\text{total}} \]Substitute the known values:\[ P_{\text{N}_2} = 0.17 \times 6.91 = 1.17 \text{ atm} \]

Calculate Partial Pressure of Oxygen

The mole fraction of oxygen is given as 4.0%. Convert this percentage to a decimal by dividing by 100, giving \(X_{\text{O}_2} = 0.04\). Use the partial pressure formula:\[ P_{\text{O}_2} = X_{\text{O}_2} \times P_{\text{total}} \]Substitute the known values:\[ P_{\text{O}_2} = 0.04 \times 6.91 = 0.28 \text{ atm} \]

Key concepts

Mole Fraction

The mole fraction is a way to express the composition of a mixture of gases. It represents the proportion of a gas in a mixture, based on the number of moles it contributes. To calculate the mole fraction, you divide the number of moles of the specific gas by the total number of moles of all gases present in the mixture.
For example, if you know the percent composition of a gas, like helium being 79%, you can convert it to a mole fraction by dividing by 100, resulting in 0.79.
Remember, the sum of the mole fractions in a mixture is always equal to one. This makes it a useful tool to understand the composition of gas mixtures in various scenarios, such as an undersea habitat, where maintaining proper balance of gases is crucial.

Gas Laws

Gas laws describe how gases behave under different conditions of pressure, volume, and temperature. They are essential for predicting how gases will react to changes in these conditions. There are several fundamental gas laws, including Boyle's Law, Charles's Law, and Avogadro's Law.
Together with the Ideal Gas Law, which is expressed as \( PV = nRT \), these laws help us understand the relationship between the different properties of gases. In the context of undersea habitats, understanding these laws is vital to ensure the safety and effectiveness of any gas system used.

• Boyle’s Law: Pressure and volume are inversely proportional when temperature is constant. If the pressure increases, the volume decreases.
  
• Charles's Law: Volume and temperature are directly proportional when pressure is constant. A rise in temperature increases the volume.
  

Knowing these interactions can help anticipate the behavior of gases at various depths and pressures underwater.

Dalton's Law

Dalton's Law of Partial Pressures is a key principle in understanding gas mixtures. It states that the total pressure exerted by a mixture of non-reacting gases is equal to the sum of the partial pressures of each individual gas.
To calculate the partial pressure of a particular gas, you multiply the mole fraction of that gas by the total pressure of the mixture.
This concept helps in environments like undersea habitats, where different gases need to be mixed to provide a breathable atmosphere. Each gas, such as helium, nitrogen, and oxygen, contributes to the overall pressure, and ensuring the correct partial pressures helps maintain human safety and equipment performance in the depths of the ocean. By understanding Dalton's Law, we can effectively manage and adjust gas mixtures to adapt to specific conditions.

Undersea Habitat

An undersea habitat is a human living environment located below the ocean surface. It requires a controlled atmosphere to support life, where factors such as gas pressure and composition must be carefully regulated.
At a certain depth, like 58.8 meters below sea level, atmospheric pressure increases significantly. This can affect the behavior of gases and, consequently, the health of those inhabiting the habitat.

• Pressure Considerations: Increased pressure means gases are more compressed. Understanding how to adjust gas mixtures correctly is important for preventing adverse effects like decompression sickness.
• Gas Composition: The safe mix of gases is critical. Usually, more helium is used than nitrogen to reduce the narcotic effects nitrogen can have at high pressures.

Designing an effective undersea habitat requires a keen understanding of how gases behave under pressure, and tailoring the gas mixtures accordingly. Such knowledge ensures the safety and comfort of the inhabitants by providing a balanced and breathable environment.