Problem 64 - 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 \(2,000 \mathrm{m}\) 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) Approximately 2.61 × 10^21 oxygen molecules per breath at sea level. (b) Approximately 2.06 × 10^21 oxygen molecules per breath at 2,000 m elevation. (c) Increased altitude reduces oxygen intake per breath due to decreased pressure.

Step 1: Calculate the Ideal Gas Law parameters

To find the number of oxygen molecules inhaled at sea level, we first apply the ideal gas law. The formula is given by \[ PV = nRT \] where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the ideal gas constant \((0.0821 \ \text{L} \cdot \text{atm} / \text{mol} \cdot \text{K})\), and \( T \) is the temperature in Kelvin. Convert \( 20.0^{\circ}\text{C} \) to Kelvin using \( T = 20 + 273.15 = 293.15\text{ K} \).

Step 2: Calculate moles of air inhaled

Use the values in the ideal gas law:\[ n = \frac{PV}{RT} = \frac{(1.00 \ \text{atm})(0.50 \ \text{L})}{(0.0821 \ \text{L} \cdot \text{atm}/\text{mol} \cdot \text{K})(293.15\text{ K})} \approx 0.0207 \ \text{mol of air} \]

Step 3: Determine oxygen moles inhaled

Since the air is 21\% oxygen, calculate the moles of oxygen: \[ \text{Moles of oxygen} = 0.0207 \times 0.21 \approx 0.00434 \ \text{mol} \]

Step 4: Convert moles of oxygen to molecules

Use Avogadro's number to determine the number of oxygen molecules:\[ \text{Number of molecules} = 0.00434 \ \text{mol} \times 6.022 \times 10^{23} \ \text{molecules/mol} \approx 2.61 \times 10^{21} \text{ molecules} \]

Step 5: Calculate oxygen molecules at high elevation

At an elevation of 2,000 m, atmospheric pressure decreases to approximately 0.79 atm. Recalculate the moles of air:\[ n = \frac{(0.79 \ \text{atm})(0.50 \ \text{L})}{(0.0821 \ \text{L} \cdot \text{atm}/\text{mol} \cdot \text{K})(293.15\text{ K})} \approx 0.0163 \ \text{mol of air} \]\[ \text{Moles of oxygen} = 0.0163 \times 0.21 \approx 0.00342 \ \text{mol} \]\[ \text{Number of molecules} = 0.00342 \ \text{mol} \times 6.022 \times 10^{23} \ \text{molecules/mol} \approx 2.06 \times 10^{21} \text{ molecules} \]

Step 6: Explain shortness of breath at high elevation

The number of oxygen molecules inhaled per breath decreases at higher elevations due to lower atmospheric pressure, yet the body requires the same amount of oxygen to function. This reduction in oxygen intake per breath can lead to shortness of breath as the body struggles to obtain sufficient oxygen to meet its needs.

Key Concepts

Oxygen Molecules

Oxygen molecules are vital for our survival. They are diatomic, which means each oxygen molecule is composed of two oxygen atoms. These molecules are necessary for the physiological processes that release energy in our body. At sea level, when a person inhales air, about 21% of that air consists of oxygen. A typical breath at normal atmospheric conditions involves inhaling approximately 2.61 × 10^{21} oxygen molecules, playing a crucial role in cellular respiration.

• Essential for energy: Oxygen molecules are used to convert glucose into energy through the process of aerobic respiration.
• Role in cellular functions: Oxygen is critical for cell metabolism, enabling cells to perform various functions necessary for life.

Understanding how oxygen molecules contribute to energy production helps us appreciate their importance in maintaining health and well-being.

High Elevation

High elevation refers to locations situated at significant altitudes above sea level, like mountains. At higher elevations, such as 2,000 meters, the atmospheric conditions vary significantly compared to sea level. One major change is the decrease in atmospheric pressure, which affects the amount of oxygen available in the air.

• Lower pressure: Atmospheric pressure decreases as elevation increases, meaning fewer air molecules are present, including oxygen molecules.
• Environmental conditions: Even with the same temperature, the air is less dense, causing individuals to inhale fewer molecules of oxygen with each breath.

This change in conditions requires the body to adapt, as the decreased oxygen availability can have various physiological effects.

Atmospheric Pressure

Atmospheric pressure is the force exerted by the weight of the air in the Earth's atmosphere. It is greatest at sea level and decreases with altitude because the air becomes less dense. This reduction in pressure means that at higher elevations, such as 2,000 meters, there's a noticeable decrease in the number of oxygen molecules per breath compared to sea level.

• Impact on breathing: At lower atmospheric pressure, the same volume of inhaled air contains fewer oxygen molecules.
• Pressure differences: Whereas sea level pressure is typically around 1.00 atm, at 2,000 meters, it drops to about 0.79 atm.

This reduction is crucial for understanding why the quantity of oxygen molecules inhaled decreases at higher altitudes, necessitating adjustments in respiratory function.

Shortness of Breath

Shortness of breath, medically known as "dyspnea," occurs when the body perceives an inadequate supply of oxygen. At high elevations, the number of oxygen molecules per breath decreases due to lower atmospheric pressure. As a result, although the volume of inhaled air may remain the same, the amount of actual oxygen available is less.

• Physiological challenge: The body must work harder to obtain enough oxygen to meet its needs, increasing breathing rate and depth.
• Oxygen demand: The body's requirement for oxygen remains the same, despite the reduction in available oxygen per breath.

This discrepancy leads to the sensation of shortness of breath, as the body adapts to cope with the lower oxygen availability, often experienced by individuals who are not acclimated to high altitudes.