Problem 66 - University Physics with Modern Physics

Question

A person at rest inhales 0.50 \(\mathrm{L}\) of air with each breath at a pressure of 1.00 atm and a temperature of \(20.0^{\circ} \mathrm{C}\) . The inhaled air is 21.0\(\%\) oxygen. (a) How many oxygen molecules does this person inhale with each breath? (b) Suppose this person is now resting at an elevation of 2000 \(\mathrm{m}\) but the temperature is still \(20.0^{\circ} \mathrm{C}\) . Assuming that the oxygen percentage and volume per inhalation are the same as stated above, how many oxygen molecules does this person now inhale with each breath? (c) Given that the body still requires the same number of oxygen molecules per second as at sea level to maintain its functions, explain why some people report "shortness of breath" at high elevations.

Step-by-Step Solution

Answer

(a) ~\(2.62 \times 10^{21}\) molecules/breath; (b) ~\(2.10 \times 10^{21}\) molecules/breath; (c) Fewer molecules per breath at high elevations.

Step 1: Understand the Given Quantities

For part (a), we have the initial conditions: Volume \( V = 0.50 \, \mathrm{L} = 0.50 \, \mathrm{m}^3 \times 10^{-3} \), Pressure \( P = 1.00 \, \mathrm{atm} \), which converts to \( P = 1.013 \times 10^5 \, \mathrm{Pa} \), and Temperature \( T = 20.0^{\circ} \mathrm{C} = 293 \mathrm{K} \). The air contains 21% oxygen. The molar volume of gas at standard temperature and pressure (STP) is \( 22.4 \, \mathrm{L/mol} \approx 22.4 \times 10^{-3} \, \mathrm{m}^3/mol \), and \( R = 8.314 \, \mathrm{J/(mol\cdot K)} \) is the ideal gas constant.

Step 2: Calculate Moles of Air

Using the ideal gas law equation \( PV = nRT \), we can solve for \( n \), the moles of air. \[ n = \frac{PV}{RT} = \frac{1.013 \times 10^5 \times 0.50 \times 10^{-3}}{8.314 \times 293} \approx 2.07 \times 10^{-2} \, \mathrm{mol} \]

Step 3: Calculate Moles of Oxygen Inhaled

Since air is 21% oxygen, the moles of oxygen \( n_{O_2} \) inhaled are \( 0.21 \times n \). \[ n_{O_2} = 0.21 \times 2.07 \times 10^{-2} \approx 4.35 \times 10^{-3} \, \mathrm{mol} \]

Step 4: Calculate Oxygen Molecules Inhaled at Sea Level

The number of molecules is calculated using Avogadro's number \( N_A = 6.022 \times 10^{23} \, \mathrm{molecules/mol} \). The number of oxygen molecules \( N \) inhaled at sea level is: \[ N = n_{O_2} \times N_A = 4.35 \times 10^{-3} \times 6.022 \times 10^{23} \approx 2.62 \times 10^{21} \]

Step 5: Recall Change in Atmospheric Pressure at Elevation

At an elevation of 2000 m, atmospheric pressure decreases to roughly 0.8 atm. Convert this to Pascals: \( 0.8 \times 1.013 \times 10^5 \approx 8.10 \times 10^4 \, \mathrm{Pa} \).

Step 6: Calculate Moles of Air at Higher Elevation

Solve for moles \( n \) using the pressure at 2000 m: \[ n = \frac{(8.10 \times 10^4) \times (0.50 \times 10^{-3})}{8.314 \times 293} \approx 1.66 \times 10^{-2} \, \mathrm{mol} \]

Step 7: Calculate Oxygen Molecules Inhaled at Higher Elevation

Moles of oxygen inhaled \( n_{O_2} \) at higher elevation is \( 0.21 \times 1.66 \times 10^{-2} \approx 3.49 \times 10^{-3} \, \mathrm{mol} \). The number of oxygen molecules \( N \) is: \[ N = n_{O_2} \times N_A = 3.49 \times 10^{-3} \times 6.022 \times 10^{23} \approx 2.10 \times 10^{21} \]

Step 8: Explain Shortness of Breath at High Elevation

Although the percentage of oxygen remains constant, the lower pressure at higher altitudes leads to fewer oxygen molecules being inhaled per breath. The body requires the same amount of oxygen molecules to function as at sea level, so fewer molecules being inhaled can cause shortness of breath due to not meeting body demands.

Key Concepts

Understanding Oxygen Molecules

Oxygen molecules are essential for human survival as they are crucial for cellular respiration. In the context of breathing, an average person's breath contains a mixture of gases from the air, primarily nitrogen and oxygen. Oxygen makes up about 21% of the air we breathe. When this oxygen is inhaled, it's transported via the bloodstream to various parts of the body, where it fuels vital processes.
A single oxygen molecule is represented by two oxygen atoms bonded together, symbolized as \( \mathrm{O}_2 \). These molecules are tiny, but there are a tremendous number of them due to their small size. For instance, in standard conditions, one mole of gas (about 22.4 liters) contains Avogadro's number of molecules, which is approximately \( 6.022 \times 10^{23} \) molecules.
By applying the ideal gas law, \( PV = nRT \), where \( P \) represents pressure, \( V \) volume, \( n \) number of moles, \( R \) the gas constant, and \( T \) temperature, you can calculate the number of moles of gas, and consequently the number of molecules present in a given volume of air.

High Elevation Breathing

Breathing at high elevations presents unique challenges because of shifts in atmospheric pressure. As elevation increases, the air pressure decreases. This decrease affects the density of oxygen molecules in the air, meaning there are fewer oxygen molecules in each liter of air at high altitudes compared to sea level.

• At sea level, with a pressure usually around 1 atm, the air is denser and contains more oxygen molecules.
• At higher altitudes, like 2000 meters, the atmospheric pressure drops to roughly 0.8 atm, making it harder to intake the same amount of oxygen molecules with each breath.

This reduced oxygen availability means your body has to work harder to get the oxygen it needs. Even if the volume of each breath remains the same, the total number of oxygen molecules inhaled decreases. This is why individuals might experience symptoms like shortness of breath and fatigue when they first arrive at high elevations, as their bodies adjust to the change.

Impact of Atmospheric Pressure on Breathing

Atmospheric pressure is the force exerted by the weight of air in the atmosphere on a given surface. At sea level, this pressure is typically 1 atm or 1013 hPa (hectopascals). As you ascend to higher altitudes, atmospheric pressure decreases because there is less air above you.
This reduction in pressure affects respiration because it directly impacts the partial pressure of oxygen, a measurement of how much oxygen is present in the air. Lower atmospheric pressure results in a lower partial pressure of oxygen, which means fewer oxygen molecules are available for the lungs to absorb.
Although the percentage of oxygen in the air remains constant at about 21%, the actual number of oxygen molecules you inhale per breath decreases. Hence, climbers or people living at high altitudes often need to acclimatize or perform conditioning to adjust to these lower oxygen conditions. Supplemental oxygen or pressure chambers are sometimes employed to alleviate the physiological strain of high-elevation exposure.