Question

Fish breathe the dissolved air in water through their gills. Assuming the partial pressures of oxygen and nitrogen in air to be 0.20 and 0.80 atm, respectively, calculate the mole fractions of oxygen and nitrogen in the air dissolved in water at \(298 \mathrm{~K}\). The solubilities of \(\mathrm{O}_{2}\) and \(\mathrm{N}_{2}\) in water at \(298 \mathrm{~K}\) are \(1.3 \times 10^{-3} \mathrm{~mol} / \mathrm{L} \cdot \mathrm{atm}\) and \(6.8 \times 10^{-4} \mathrm{~mol} / \mathrm{L} \cdot\) atm, respectively. Comment on your results.

Short answer

Oxygen mole fraction: 0.323; Nitrogen mole fraction: 0.677. Oxygen is more soluble than nitrogen in water.

Step-by-step solution

Determine the Solubility

Use the given solubility coefficients to determine the solubility of each gas in water. For oxygen, solubility is given by \( S_{O_2} = 1.3 \times 10^{-3} \text{ mol/L atm} \). For nitrogen, solubility is \( S_{N_2} = 6.8 \times 10^{-4} \text{ mol/L atm} \).

Calculate the Concentrations in Water

Multiply the solubility coefficients by their respective partial pressures to find the concentration of each gas in water. For oxygen, the concentration \( C_{O_2} \) is calculated as \( 1.3 \times 10^{-3} \times 0.20 = 2.6 \times 10^{-4} \text{ mol/L} \). For nitrogen, \( C_{N_2} \) is \( 6.8 \times 10^{-4} \times 0.80 = 5.44 \times 10^{-4} \text{ mol/L} \).

Determine the Total Concentration

Add the concentrations of both gases to find the total concentration of dissolved gases in water: \( C_{\text{total}} = 2.6 \times 10^{-4} + 5.44 \times 10^{-4} = 8.04 \times 10^{-4} \text{ mol/L} \).

Calculate the Mole Fractions

Find the mole fractions of oxygen and nitrogen by dividing their respective concentrations by the total concentration. For oxygen, \( X_{O_2} = \frac{2.6 \times 10^{-4}}{8.04 \times 10^{-4}} \approx 0.323 \). For nitrogen, \( X_{N_2} = \frac{5.44 \times 10^{-4}}{8.04 \times 10^{-4}} \approx 0.677 \).

Comment on the Results

Although oxygen has a much smaller partial pressure in air, its higher solubility leads to a significant mole fraction in water compared to nitrogen. This indicates that oxygen is more readily available for aquatic life, such as fish, which breathe using dissolved oxygen.

Key concepts

Partial Pressures

Partial pressure refers to the pressure exerted by a single gas in a mixture of gases, like air. It is important because it determines how much of that gas will dissolve in water. For example, air is composed mostly of nitrogen and oxygen, with partial pressures of 0.80 atm and 0.20 atm, respectively. These values tell us how much pressure each gas contributes to the total air pressure. The higher the partial pressure of a gas, the more it will tend to dissolve in water. This concept is crucial in understanding how gases mix and dissolve, especially in scenarios like aquatic environments where fish rely on dissolved oxygen.

Gaseous Solubility

Gaseous solubility is a measure of how well a gas can dissolve in a liquid, like water. Different gases have different solubility levels based on their interaction with the liquid and their surrounding environment.In our exercise, the solubility of oxygen in water is given as \(1.3 \times 10^{-3} \text{ mol/L atm}\), and for nitrogen, it's \(6.8 \times 10^{-4} \text{ mol/L atm}\). The higher the solubility, the more gas can be dissolved in water at a given pressure. Thus, even though the partial pressure of nitrogen is higher than that of oxygen in air, the latter still dissolves in water in notable amounts due to its higher solubility. This is why fish can rely on dissolved oxygen for breathing.

Mole Fraction Calculation

Mole fraction is a way to express the concentration of a component in a mixture. It is calculated as the ratio of the concentration of that component to the total concentration of all components. To find the mole fraction, you first need to determine how much of each gas dissolves in water. For example, by using the solubility and partial pressure, we calculated that oxygen has a concentration of \(2.6 \times 10^{-4} \text{ mol/L}\) and nitrogen \(5.44 \times 10^{-4} \text{ mol/L}\) in water. The total concentration is the sum of these, or \(8.04 \times 10^{-4} \text{ mol/L}\).To find the mole fraction of oxygen: \[ X_{O_2} = \frac{2.6 \times 10^{-4}}{8.04 \times 10^{-4}} \approx 0.323 \]And for nitrogen: \[ X_{N_2} = \frac{5.44 \times 10^{-4}}{8.04 \times 10^{-4}} \approx 0.677 \]These mole fractions tell us the proportion of each gas in water, which is essential for understanding how aquatic organisms access these gases.

Henry's Law

Henry's Law describes the relationship between the solubility of a gas in a liquid and the partial pressure of that gas above the liquid. It tells us that the solubility of a gas is directly proportional to its partial pressure.The law is mathematically expressed as:\[ C = kH \cdot P \]where \(C\) is the concentration of the gas in the liquid, \(kH\) is the Henry's Law constant (solubility), and \(P\) is the partial pressure of the gas.In this exercise, the solubility constants for oxygen and nitrogen were provided. By using these constants and the respective partial pressures, we calculated how much of each gas dissolves in water. For instance:- Oxygen: \(C_{O_2} = 1.3 \times 10^{-3} \cdot 0.20 = 2.6 \times 10^{-4} \text{ mol/L} \)- Nitrogen: \(C_{N_2} = 6.8 \times 10^{-4} \cdot 0.80 = 5.44 \times 10^{-4} \text{ mol/L} \)Henry's Law helps predict how gases behave under different pressures and is fundamental in fields such as environmental science and chemical engineering.