Problem 21 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

A diver descends to a depth where the pressure is 5 atm. The diver's body contains around \(5 \mathrm{dm}^{3}\) of blood. The Henry's law constant for \(\mathrm{N}_{2}(\mathrm{g})\) in water at \(310 \mathrm{K}\) is \(1.07 \times 10^{5} \mathrm{atm}\) (Section \(17.4)\) (a) Calculate the amount of nitrogen gas absorbed from the air in the diver's blood at a pressure of 1 atm and at 5 atm. (Assume that the solubility of nitrogen in the blood is the same as in water.) (b) If all the gas dissolved at 5 atm was suddenly released, what volume would it occupy at 1 atm and \(298 \mathrm{K} ?\)

Step-by-Step Solution

Answer

At 5 atm, 2.14 × 10⁻⁴ mol nitrogen is absorbed and it would occupy 0.00523 L at 1 atm.

Step 1: Understand Henry's Law

According to Henry's Law, the solubility of a gas in a liquid is directly proportional to the partial pressure of the gas above the liquid. The formula is \( C = k_H \cdot P \) where \( C \) is the concentration of the gas in the liquid, \( k_H \) is Henry's law constant, and \( P \) is the partial pressure of the gas.

Step 2: Calculate Nitrogen Solubility at 1 atm

Use Henry's law: \( C = k_H \cdot P \). Here, \( k_H = 1.07 \times 10^5 \mathrm{atm}^{-1} \) and \( P = 1 \mathrm{atm} \). Thus, \( C_1 = 1.07 \times 10^5 \times 1 = 1.07 \times 10^{-5} \mathrm{mol/L} \).

Step 3: Calculate Total Nitrogen in Blood at 1 atm

Given the blood volume \( V = 5 \mathrm{dm}^3 = 5 \mathrm{L} \), the amount of nitrogen dissolved at 1 atm is \( ext{Amount} = C_1 \times V = 1.07 \times 10^{-5} \times 5 = 5.35 \times 10^{-5} \mathrm{mol} \).

Step 4: Calculate Nitrogen Solubility at 5 atm

Apply Henry's Law for \( P = 5 \mathrm{atm} \). So, \( C_5 = 1.07 \times 10^5 \times 5 = 5.35 \times 10^{-5} \mathrm{mol/L} \).

Step 5: Calculate Total Nitrogen in Blood at 5 atm

With the same volume \( V = 5 \mathrm{L} \), the amount of nitrogen at 5 atm is \( ext{Amount} = C_5 \times V = 5.35 \times 10^{-5} \times 5 = 2.675 \times 10^{-4} \mathrm{mol} \).

Step 6: Calculate Additional Nitrogen Absorbed

Find the difference in nitrogen content between 5 atm and 1 atm. \( ext{Additional} = 2.675 \times 10^{-4} - 5.35 \times 10^{-5} = 2.14 \times 10^{-4} \mathrm{mol} \).

Step 7: Volume Calculation of Released Nitrogen

Use the ideal gas law \( PV = nRT \) to find the volume at 1 atm and \( T = 298 \mathrm{K} \). Use \( R = 0.0821 \mathrm{L} \, ext{atm} \, ext{K}^{-1} \, ext{mol}^{-1} \). \( V = \frac{nRT}{P} = \frac{2.14 \times 10^{-4} \times 0.0821 \times 298}{1} \approx 0.00523 \mathrm{L} \).

Key Concepts

Gas Solubility

Gas solubility refers to how well a gas can dissolve in a liquid. It's important to understand that different gases dissolve differently based on their nature and the solubility of the liquid. Solubility is often influenced by factors such as temperature and pressure.

• Temperature: Typically, as the temperature increases, the solubility of gases in liquids decreases. This means that warm liquids generally hold less dissolved gas than cold ones.
• Pressure: The solubility of gas is directly related to the pressure of that gas above the liquid. This principle is defined by Henry's Law.

Understanding gas solubility is vital in fields like chemistry and biology. In our exercise, a diver's experience underwater demonstrates how nitrogen dissolves in the blood under pressure.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry linking pressure, volume, temperature, and moles of gas. Given by the formula \( PV = nRT \), this law helps describe the behavior of gases under various conditions:

• P (Pressure): The force that the gas exerts on the walls of its container, measured in atm (atmospheres).
• V (Volume): The space that the gas occupies, measured in liters.
• n (Moles): The quantity of gas, given in moles.
• R (Gas Constant): A constant value of 0.0821 L atm K^-1 mol^-1.
• T (Temperature): The absolute temperature of the gas, measured in Kelvin (K).

The Ideal Gas Law is crucial when determining the volume of gas released upon changing external conditions. In the exercise, we used it to calculate the volume of nitrogen gas released when the pressure changes from 5 atm to 1 atm.

Partial Pressure

Partial pressure is the pressure contributed by a single type of gas in a mixture of gases. It's a concept derived from Dalton's Law, which states that the total pressure exerted by a mixture of gases is equal to the sum of the partial pressures of individual gases in the mixture.

• Determining Factors: Each gas exerts its pressure based on its proportion in the mixture and its identity.
• Application: In our exercise, the partial pressure of nitrogen plays a crucial role in determining its solubility in blood. At 1 atm, nitrogen has a partial pressure of 1 atm above water, and at the diver’s depth, it's 5 atm.

Understanding partial pressure helps explain how gases behave in mixtures and is especially useful in calculating how they dissolve in liquids.

Nitrogen Solubility

Nitrogen solubility refers to the ability of nitrogen gas to dissolve in a liquid such as blood. Under increased pressure, like when a diver descends deeper underwater, nitrogen dissolves more into the blood.

• Role of Pressure: As demonstrated in the exercise, when the pressure is 5 atm, significantly more nitrogen dissolves in the diver's blood compared to at 1 atm.
• Calculation: Utilizing Henry’s Law, we find that the solubility of nitrogen in blood increases with the pressure of nitrogen above the blood. This increase in nitrogen's solubility heightens the risk of decompression sickness, also known as "the bends," if not carefully managed upon ascent.

Understanding how nitrogen solubility changes with pressure is important for fields like scuba diving and hyperbaric medicine.