Question

An airplane is pressurized with air to \(650 \mathrm{mmHg}\). a. If air is \(21 \%\) oxygen, what is the partial pressure of oxygen on the plane? b. If the partial pressure of oxygen drops below \(100 \mathrm{mmHg}\), passengers become drowsy. If this happens, oxygen masks are released. What is the total cabin pressure at which oxygen masks are dropped?

Short answer

a. 136.5 mmHg b. 476.2 mmHg

Step-by-step solution

Understanding Partial Pressure

Partial pressure can be found by multiplying the total pressure by the fraction of the gas in question within the mixture. Here, air is composed of 21% oxygen.

Calculating Partial Pressure of Oxygen

The total pressure in the airplane is 650 mmHg and 21% of this is oxygen. Thus, the partial pressure of oxygen: \( P_{\text{O}_2} = 0.21 \times 650 \text{ mmHg} = 136.5 \text{ mmHg} \)

Determining Critical Oxygen Partial Pressure

If the partial pressure of oxygen drops below 100 mmHg, passengers become drowsy and oxygen masks must be released.

Calculating Total Cabin Pressure for Oxygen Mask Deployment

To find the total cabin pressure at which oxygen masks are deployed, set the partial pressure of oxygen at 100 mmHg and solve for the total pressure using the formula: \( P_{\text{total}} \times 0.21 = 100 \text{ mmHg} \). Thus, \( P_{\text{total}} = \frac{100}{0.21} \approx 476.2 \text{ mmHg} \)

Key concepts

Dalton's Law of Partial Pressures

Dalton's Law of Partial Pressures is really important when working with gases in a mixture. It says that the total pressure of a gas mixture is equal to the sum of the pressures of each individual gas. Each gas in the mixture behaves independently and contributes to the total pressure by an amount known as its partial pressure.
To put it simply, if you have a mixture of gases, you can calculate the total pressure by adding up the contributions from each gas. This is very useful for figuring out things like how much oxygen you have in the air you are breathing.
In our exercise, we used Dalton's Law to find the partial pressure of oxygen in the airplane cabin. Since the total pressure is 650 mmHg and oxygen makes up 21% of the air, we can use the law to determine that the partial pressure of oxygen is 136.5 mmHg.

Partial Pressure

Partial pressure is the pressure that one particular gas in a mixture would exert if it occupied the entire volume by itself. In simpler terms, it’s the contribution of one type of gas to the total pressure.
To calculate the partial pressure of a gas, you multiply the total pressure by the fraction of the mixture that the gas makes up. In our case, the airplane cabin has a total pressure of 650 mmHg and oxygen makes up 21% of the air. This means we multiply 650 mmHg by 0.21 to get the partial pressure of oxygen, which is 136.5 mmHg.
Understanding partial pressure is crucial in fields like medicine, aviation, and environmental science. For instance, knowing the partial pressure of oxygen is critical for ensuring that the passengers on an airplane have enough oxygen to breathe.

Oxygen Concentration

Oxygen concentration refers to the amount of oxygen present in a mixture of gases. It is often expressed as a percentage but can also be looked at in terms of partial pressure.
In everyday air, oxygen makes up about 21%. This means that if you have a total pressure of 650 mmHg, as in our airplane example, the partial pressure of oxygen is 21% of that, or 136.5 mmHg.
If the partial pressure of oxygen drops too low, people can become drowsy because their bodies aren't getting enough oxygen. This is why oxygen masks are deployed in airplanes if the partial pressure drops below 100 mmHg, ensuring that passengers can breathe properly.
So, oxygen concentration isn't just a number—it's a vital measurement for safety, especially in controlled environments like airplane cabins.