Problem 80 - 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 mm Hg and 7.5 mm Hg, respectively. The barometric pressure at the summit was 253 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 \(\left.P\left(\mathrm{H}_{2} \mathrm{O}\right)=47.1 \mathrm{mm} \mathrm{Hg}\right]\). If you assume the lung gases consist of only \(\mathbf{O}_{2}, \mathbf{N}_{2}, \mathbf{C O}_{2},\) and \(\mathrm{H}_{2} \mathrm{O},\) what is the partial pressure of \(\mathrm{N}_{2} ?\)

Step-by-Step Solution

Answer

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

Step 1: Identify Known Pressures

We know the partial pressures for each component, except for nitrogen. The partial pressures provided are: \( P(O_2) = 35 \, \text{mm Hg} \), \( P(CO_2) = 7.5 \, \text{mm Hg} \), and \( P(H_2O) = 47.1 \, \text{mm Hg} \).

Step 2: Apply Dalton's Law of Partial Pressures

Dalton's Law states that the total pressure is the sum of the partial pressures of all gases present. Therefore, \( P_{\text{total}} = P(O_2) + P(CO_2) + P(N_2) + P(H_2O) \).

Step 3: Substitute Known Values into Dalton's Equation

Substituting the known values into the equation from Step 2 gives: \( 253 = 35 + 7.5 + P(N_2) + 47.1 \).

Step 4: Solve for the Partial Pressure of Nitrogen

To find \( P(N_2) \), rearrange the equation from Step 3: \( P(N_2) = 253 - (35 + 7.5 + 47.1) \).

Step 5: Calculate the Value of \( P(N_2) \)

Substitute and compute: \( P(N_2) = 253 - 89.6 = 163.4 \, \text{mm Hg} \).

Key Concepts

Partial Pressure

Partial pressure is a crucial concept in chemistry and physics, especially when it comes to understanding gas behavior. It refers to the pressure that a specific gas in a mixture of gases would exert if it were the only gas present in a given volume. This concept is useful because it allows us to analyze the contributions of each individual gas in a mixture to the total pressure.

Dalton's Law of Partial Pressures is the principle that describes how to calculate the total pressure of a gas mixture. According to Dalton's Law, the total pressure (\( P_{\text{total}} \) ) is the sum of the partial pressures of all gases involved.

• \( P_{\text{total}} = P_{O_2} + P_{CO_2} + P_{N_2} + P_{H_2O} \)

By breaking down the total pressure into partial pressures, we can isolate and analyze the properties of individual gases. This concept is essential for students to grasp when studying gas mixtures and understanding how different gases interact within a system.

Barometric Pressure

Barometric pressure, also known as atmospheric pressure, is the pressure exerted by the weight of air in the atmosphere above us. It plays a vital role in weather patterns, altitude, and even the physiological behavior of gases at various altitudes.

At higher altitudes, such as the summit of Mount Everest, barometric pressure decreases significantly due to the thinning atmosphere. This decrease in pressure affects the partial pressures of gases and, consequently, human physiology. It is because of this lower pressure that climbers often require supplemental oxygen to maintain adequate \( O_2 \) levels for survival. Understanding barometric pressure is essential for navigating situations where atmospheric conditions vary, such as in aviation, meteorology, and mountaineering.

Nitrogen Partial Pressure

Nitrogen partial pressure is the specific pressure exerted by nitrogen in a mixture of gases. Nitrogen (\( N_2 \)) is the most abundant gas in Earth’s atmosphere, making its partial pressure a significant component of total atmospheric pressure.

When considering a scenario such as the summit of Mount Everest, where there is only a limited amount of air available, nitrogen still constitutes a major portion of the gas mixture in the lungs. To determine the partial pressure of nitrogen in such a mixture, we use Dalton’s Law.
The calculation involves knowing the total pressure and subtracting the known partial pressures of other gases like \( O_2 \), \( CO_2 \), and \( H_2O \).
Through this process, we found the nitrogen partial pressure to be 163.4 mm Hg. This calculation helps illustrate how each gas contributes to the overall pressure and can influence breathing—and ultimately survival—for climbers at high altitudes.