Problem 67
Question
At an underwater depth of \(250 \mathrm{ft}\), the pressure is \(8.38 \mathrm{~atm}\). What should the mole percent of oxygen be in the diving gas for the partial pressure of oxygen in the mixture to be \(0.21 \mathrm{~atm}\), the same as in air at \(1 \mathrm{~atm}\) ?
Step-by-Step Solution
Verified Answer
In order for the partial pressure of oxygen in the diving gas to be 0.21 atm at an underwater depth of 250 ft, the mole percent of oxygen in the mixture should be approximately 2.51%.
1Step 1: Understand the relationship between partial pressure and mole fraction
In a gas mixture, according to Dalton's Law, the partial pressure of a component is directly proportional to its mole fraction in the mixture.
\( P_{component} = X_{component} * P_{total} \)
Where \(P_{component}\) is the partial pressure of the component, \(X_{component}\) is its mole fraction, and \(P_{total}\) is the total pressure of the system.
2Step 2: Write down given information
We are given the following information:
\(P_{total} = 8.38 \mathrm{~atm}\) and
\(P_{O_2(in air)} = 0.21 \mathrm{~atm}\)
Our task is to find the mole fraction of oxygen (\(X_{O_2}\)) in the diving gas mixture, such that the partial pressure of oxygen in the mixture (\(P_{O_2}\)) is equal to the partial pressure of oxygen in the air.
3Step 3: Find the mole fraction of oxygen
Using the relationship between partial pressure and mole fraction, we can find the mole fraction of oxygen based on the desired partial pressure of oxygen and the total pressure of the system:
\(P_{O_2} = X_{O_2} * P_{total} \)
\(0.21 \mathrm{~atm} = X_{O_2} * 8.38 \mathrm{~atm}\)
Now, we can solve for the mole fraction of oxygen (\(X_{O_2}\)):
\(X_{O_2} = \frac{0.21 \mathrm{~atm}}{8.38 \mathrm{~atm}}\)
\(X_{O_2} \approx 0.0251\)
4Step 4: Calculate the mole percent of oxygen
Now that we have the mole fraction of oxygen in the diving gas, we can convert it to mole percent by multiplying by 100:
Mole percent of oxygen = (\(X_{O_2}\)) * 100
Mole percent of oxygen = \(0.0251 * 100\)
Mole percent of oxygen ≈ \(2.51\%\)
In order for the partial pressure of oxygen in the diving gas to be 0.21 atm at an underwater depth of 250 ft, the mole percent of oxygen in the mixture should be approximately 2.51%.
Key Concepts
Dalton's LawMole FractionUnderwater Diving Gas MixturesPressure Calculations
Dalton's Law
When studying gas mixtures, Dalton's Law is a fundamental principle to understand. It states that in a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of individual gases. For clarity, consider the partial pressure as the pressure that each gas would exert if it were alone occupying the entire volume of the mixture.
Understanding this helps us in environments with varying pressure, such as underwater diving or high-altitude flying, because it allows for calculations on how gas concentrations must be adjusted to maintain safe and breathable air. The relevance of Dalton's Law is evident in applying the appropriate concentrations of oxygen in underwater diving gas mixtures to prevent hypoxia or oxygen toxicity.
Understanding this helps us in environments with varying pressure, such as underwater diving or high-altitude flying, because it allows for calculations on how gas concentrations must be adjusted to maintain safe and breathable air. The relevance of Dalton's Law is evident in applying the appropriate concentrations of oxygen in underwater diving gas mixtures to prevent hypoxia or oxygen toxicity.
Mole Fraction
The mole fraction is a way of expressing the concentration of a component in a mixture. It is the ratio of the number of moles of that component to the total number of moles of all substances present. Mathematically, it is denoted as:\begin{center} \( X_i = \frac{n_i}{n_{total}} \)
Underwater Diving Gas Mixtures
In underwater diving, the right gas mixture is critical for ensuring the diver's safety. The deeper a diver goes, the more the pressure increases, and the composition of the breathing gas must be adjusted accordingly to prevent problems such as nitrogen narcosis or decompression sickness. Custom mixtures like nitrox, trimix, and heliox are used to tailor the amounts of nitrogen, oxygen, and sometimes helium needed at specific depths.
Using Dalton's Law and the concept of mole fraction, divers and technicians can create a mixture that offers enough oxygen for breathing without reaching toxic levels, while also reducing the potential for nitrogen narcosis by adjusting nitrogen levels or substituting it with helium in deeper dives.
Using Dalton's Law and the concept of mole fraction, divers and technicians can create a mixture that offers enough oxygen for breathing without reaching toxic levels, while also reducing the potential for nitrogen narcosis by adjusting nitrogen levels or substituting it with helium in deeper dives.
Pressure Calculations
Pressure calculations are key components in many scientific and practical applications, especially when dealing with gases. Whether you're trying to find the force exerted by a gas in a container, or figuring out the proper breathing mix for an underwater dive, it boils down to using formulas that relate pressure, volume, temperature, and the amount of gas.
The example provided shows how to solve for the mole fraction and thus the percentage of a specific gas (oxygen) in a mixture, taking into account the ambient pressure underwater. Given a total pressure and the desired partial pressure of oxygen, you can rearrange the formulas to calculate the required concentration. Such calculations are not only academically important but also crucial in real-world applications like diving, where precise gas mixtures can mean the difference between a safe dive and a life-threatening situation.
The example provided shows how to solve for the mole fraction and thus the percentage of a specific gas (oxygen) in a mixture, taking into account the ambient pressure underwater. Given a total pressure and the desired partial pressure of oxygen, you can rearrange the formulas to calculate the required concentration. Such calculations are not only academically important but also crucial in real-world applications like diving, where precise gas mixtures can mean the difference between a safe dive and a life-threatening situation.
Other exercises in this chapter
Problem 64
A sample of \(4.00 \mathrm{~mL}\) of diethylether \(\left(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OC}_{2} \mathrm{H}_{5},\right.\), density \(=0.7134 \mathrm{~g}
View solution Problem 65
A mixture of gases contains \(0.75 \mathrm{~mol} \mathrm{~N}_{2}, 0.30 \mathrm{~mol} \mathrm{O}_{2}\) and \(0.15 \mathrm{~mol} \mathrm{CO}_{2}\). If the total p
View solution Problem 68
(a) What are the mole fractions of each component in a mixture of \(5.08 \mathrm{~g}\) of \(\mathrm{O}_{2}, 7.17 \mathrm{~g}\) of \(\mathrm{N}_{2}\), and \(1.32
View solution Problem 69
A quantity of \(\mathrm{N}_{2}\) gas originally held at \(4.75\) atm pressure in a 1.00-L container at \(26^{\circ} \mathrm{C}\) is transferred to a 10.0-L cont
View solution