Problem 82 - Chemistry and Chemical Reactivity

Question

A A study of climbers who reached the summit of Mount Everest without supplemental oxygen showed that the partial pressures of \(\mathrm{O}_{2}\) and \(\mathrm{CO}_{2}\) in their lungs were \(35 \mathrm{mm}\) Hg and \(7.5 \mathrm{mm}\) Hg, respectively. The barometric pressure at the summit was \(253 \mathrm{mm}\) Hg. Assume the lung gases are saturated with moisture at a body temperature of \(37^{\circ} \mathrm{C}\) [which means the partial pressure of water vapor in the lungs is \(P\left(\mathrm{H}_{2} \mathrm{O}\right)=47.1 \mathrm{mm} \mathrm{Hg}\) ]. If you assume the lung gases consists of only \(\mathrm{O}_{2}, \mathrm{N}_{2}, \mathrm{CO}_{2},\) and \(\mathrm{H}_{2} \mathrm{O},\) what is the partial pressure of \(\mathrm{N}_{2} ?\)

Step-by-Step Solution

Answer

The partial pressure of \(\text{N}_2\) is 163.4 mm Hg.

Step 1: Understanding Partial Pressure

According to Dalton's law of partial pressures, the total pressure of a gas mixture is equal to the sum of the partial pressures of individual gases. Here, the total barometric pressure at the summit is given as 253 mm Hg.

Step 2: List Given Partial Pressures

We are given the partial pressures of oxygen (\(\text{O}_2\)), carbon dioxide (\(\text{CO}_2\)), and water vapor (\(\text{H}_2\text{O}\)): \(\text{P}({\text{O}_2}) = 35 \text{ mm Hg}\), \(\text{P}({\text{CO}_2}) = 7.5 \text{ mm Hg}\), and \(\text{P}({\text{H}_2\text{O}}) = 47.1 \text{ mm Hg}\).

Step 3: Apply Dalton's Law

To find the partial pressure of nitrogen (\(\text{N}_2\)), we subtract the sum of the known partial pressures from the total pressure. Use the equation: \(\text{Total Pressure} = \text{P}({\text{O}_2}) + \text{P}({\text{CO}_2}) + \text{P}({\text{H}_2\text{O}}) + \text{P}({\text{N}_2})\).

Step 4: Calculate Sum of Known Partial Pressures

Add the partial pressures of \(\text{O}_2\), \(\text{CO}_2\), and \(\text{H}_2\text{O}\): \(35 \text{ mm Hg} + 7.5 \text{ mm Hg} + 47.1 \text{ mm Hg} = 89.6 \text{ mm Hg}\).

Step 5: Solve for \(\text{P}({\text{N}_2})\)

Subtract the sum of the known partial pressures from the total barometric pressure to find the pressure of \(\text{N}_2\): \(\text{P}({\text{N}_2}) = 253 \text{ mm Hg} - 89.6 \text{ mm Hg} = 163.4 \text{ mm Hg}\).

Key Concepts

Partial Pressures

In the realm of gases, understanding how individual components behave within a mixture is crucial. This brings us to the concept of partial pressures, a key principle when working with gases. According to Dalton's Law of Partial Pressures, the total pressure exerted by a mixture of gases is equal to the sum of the pressures that each gas would exert if it were alone in the container. Each gas in this mixture has its own pressure, known as its partial pressure.
These partial pressures are proportional to the gas's mole fraction within the mixture. For example, in a gaseous environment like the lungs of a mountain climber, several gases are present, such as oxygen, carbon dioxide, and nitrogen. Each of these gases has its own partial pressure and contributes to the overall well-being and function of the biological system.

• The formula for calculating total pressure is: \[ P_{ ext{total}} = P_{ ext{O}_2} + P_{ ext{CO}_2} + P_{ ext{H}_2 ext{O}} + P_{ ext{N}_2} \]

This formula helps determine the pressure of any single gas component, like nitrogen, if others are already known.

Gas Mixtures

Gas mixtures are combinations of different gases that exist together within the same space. In the context of human lungs, particularly at high altitudes, proper gas mixtures are vital for survival. A mix typically consists of oxygen (\( ext{O}_2\)), nitrogen (\( ext{N}_2\)), carbon dioxide (\( ext{CO}_2\)), and water vapor (\( ext{H}_2 ext{O}\)).
These gases work simultaneously to ensure that essential physiological processes occur efficiently. Oxygen is crucial for cellular respiration, while nitrogen helps keep the lungs inflated, and carbon dioxide is a byproduct to be expelled after oxygen utilization.

• Understanding the proportion of gases allows for calculations of their partial pressures.
• The existence of water vapor showcases the humid environment within the human body even at high elevation.

These mixtures change with environmental conditions, such as altitude, impacting how gases are balanced, and breathing becomes more challenging during a high-altitude expedition.

Mount Everest

Mount Everest presents one of the most extreme environments on Earth due to its high altitude. At its summit, atmospheric conditions change drastically compared to sea level, affecting climbers physiologically. This environment forces a closer look into how gas mixtures within the body are adjusted.
The major factor is decreased oxygen availability—a phenomenon known as hypoxia, which can considerably impact physical performance. Falling barometric pressures as one ascends lead to a significant drop in partial pressures of vital gases too, necessitating adaptations or supplemental oxygen for safety.

• Climbers experience lower oxygen partial pressure, contributing to difficulty in breathing.
• Water vapor still exists in lung gases, needing consideration in calculations.

Evolving alongside this unique environment, bodies may undergo acclimatization, where physiological changes help improve oxygen uptake.

Barometric Pressure

Barometric pressure, also known as atmospheric pressure, is the force exerted by the weight of air in the atmosphere. On Mount Everest, this pressure is significantly lower than at sea level. Climbers face around 253 mm Hg at the summit, compared to the typical 760 mm Hg at sea level.
Changes in barometric pressure directly affect the partial pressures of gases inhaled. With a reduced barometric pressure, individual gas partial pressures decrease, crucially the partial pressure of oxygen, which is vital for biological functions. Understanding how barometric pressure impacts gas behavior is vital for safety in high-altitude adventures.

• It plays a significant role in determining how efficiently gases are exchanged in the lungs.
• Affects how climbers plan their ascents to allow safe acclimatization.

Monitoring barometric pressure helps climbers and researchers assess the readiness and risks during high-altitude treks.