Problem 57 - College Physics

Question

In human lungs, oxygen and carbon dioxide are exchanged in the blood within many small sacs called alveoli. Alveoli provide a large surface area for gas exchange. Recent careful measurements show that the total number of alveoli in a typical human lung pair is about \(480 \times 10^{6}\) and that the average volume of a single alveolus is \(4.2 \times 10^{6} \mu \mathrm{m}^{3}\). (Recall the equation for the volume of a sphere, \(V=\frac{4}{3} \pi r^{3},\) and the equation for the area of a sphere, \(\left.A=4 \pi r^{2} .\right)\) What is the total volume of the gas-exchanging region of the lungs? A. \(2000 \mu \mathrm{m}^{3}\) B. \(2 \mathrm{~m}^{3}\) C. \(2.0 \mathrm{~L}\) D. \(120 \mathrm{~L}\)

Step-by-Step Solution

Answer

The total volume of the gas-exchanging region of the lungs is 2.0 L (Option C).

Step 1: Understand the Problem

We are given the total number of alveoli and the average volume of a single alveolus in a human lung pair. We need to find the total volume of the gas-exchanging region of the lungs, which involves multiplying the number of alveoli by the volume of one alveolus.

Step 2: Define the Total Volume Formula

The total volume of the alveoli can be calculated by multiplying the number of alveoli by the volume of a single alveolus. The formula is:\[\text{Total Volume} = \text{Number of Alveoli} \times \text{Volume of One Alveolus}\]

Step 3: Plug Values into the Formula

Given:- Number of Alveoli = \( 480 \times 10^6 \)- Volume of One Alveolus = \( 4.2 \times 10^6 \mu \mathrm{m}^3 \)Substitute these values into the formula:\[\text{Total Volume} = (480 \times 10^6) \times (4.2 \times 10^6)\]

Step 4: Calculate the Total Volume

To calculate the total volume, we perform the multiplication:\[\text{Total Volume} = 480 \times 4.2 \times 10^{6+6} = 2016 \times 10^{12} \mu \mathrm{m}^3\]After adjustment, this becomes:\[2.016 \times 10^{15} \mu \mathrm{m}^3\]

Step 5: Convert the Total Volume to Liters

Convert \( \mu \mathrm{m}^3 \) to \( \, \mathrm{L} \):- 1 Liter = \( 10^{15} \mu \mathrm{m}^3 \)Thus, total volume in liters is:\[2.016 \times 10^{15} \mu \mathrm{m}^3 /10^{15} = 2.016 \, \mathrm{L}\]

Step 6: Find the Correct Answer Option

From the given options, 2.0 L is option (C), which matches our calculated value of 2.016 L with consideration of significant figures.

Key Concepts

Alveoli Structure

Alveoli are tiny, balloon-like sacs within the lungs where the exchange of gases takes place. These small structures play a critical role in our breathing process, allowing oxygen to enter the bloodstream and removing carbon dioxide from the body. The alveoli are so numerous in the lungs that they provide a massive surface area for gas exchange. This is crucial because it maximizes the amount of oxygen and carbon dioxide that can be exchanged during each breath.

The structure of an alveolus is designed for efficiency. Each alveolus is surrounded by a network of capillaries, which are small blood vessels. This proximity enables a rapid diffusion of gases due to the short distance between blood and air space. The wall of each alveolus is extremely thin, consisting of a single layer of cells, further facilitating the efficient exchange of gases. This thin barrier, coupled with the large surface area, makes the alveoli exceptionally well-adapted for their function of gas exchange.

Lung Volume Calculation

Calculating the total volume of the lungs' gas-exchanging region involves considering the total number of alveoli and the volume of a single alveolus. Here, we're given that a typical human lung has about 480 million (480 \times 10^{6}) alveoli, and each alveolus has an average volume of 4.2 \times 10^{6} cubic micrometers. To find the total volume, we simply multiply these numbers together.

• Number of Alveoli: 480 \times 10^{6}
• Volume of One Alveolus: 4.2 \times 10^{6} \( \mu \mathrm{m}^{3} \)
• Total Volume Formula:\(\text{Total Volume} = \text{Number of Alveoli} \times \text{Volume of One Alveolus}\)

Perform the multiplication: 480 \times 4.2 \times 10^{6+6} = 2016 \times 10^{12} \mu \mathrm{m}^{3}, which can be adjusted to 2.016 \times 10^{15} \mu \mathrm{m}^{3}.

This calculation provides an estimate of the total volume available for gas exchange in the lungs, emphasizing the massive capacity that our alveoli network offers.

Conversion of Units

Conversion of units is essential in making sense of the calculations and conveying the data in understandable and usable terms. In this instance, we convert cubic micrometers (\(\mu \mathrm{m}^{3}\)) to liters (\(\mathrm{L}\)), a more practical unit for volume that is easier to comprehend in everyday contexts.

The conversion factor we use is that 1 liter is equivalent to \(10^{15} \mu \mathrm{m}^{3}\). With this conversion factor, you can convert any volume from microcubic meters to liters:

• Total Volume: 2.016 \times 10^{15} \mu \mathrm{m}^{3}
• Convert to liters: \frac{2.016 \times 10^{15} \mu \mathrm{m}^{3}}{10^{15}} = 2.016 \, \mathrm{L}

When performing this conversion, the volume calculated is 2.016 L. This is rounded based on significant figures, leading us to the answer: 2.0 L as a sensible representation of the lungs' gas-exchange volume.