Problem 37 - Chemistry: The Central Science

Question

The Henry's law constant for helium gas in water at \(30^{\circ} \mathrm{C}\) is \(3.7 \times 10^{-4} \mathrm{M} / \mathrm{atm}\) and the constant for \(\mathrm{N}_{2}\) at \(30^{\circ} \mathrm{C}\) is \(6.0 \times 10^{-4} \mathrm{M} / \mathrm{atm}\). If the two gases are each present at \(1.5\) atm pressure, calculate the solubility of each gas.

Step-by-Step Solution

Answer

At a pressure of 1.5 atm and 30°C, the solubility of helium in water is \(5.55 \times 10^{-4} \mathrm{M}\), and the solubility of nitrogen is \(9.0 \times 10^{-4} \mathrm{M}\).

Step 1: First, let's find the solubility of helium

To find the solubility of helium, use the given values for the Henry's law constant for helium and the partial pressure of helium: \[C_\mathrm{He} = k_{H,\mathrm{He}}P_\mathrm{He}\] Plug the values (3.7 × 10^{-4} M / atm and 1.5 atm) into the equation: \[C_\mathrm{He} = (3.7 \times 10^{-4} \mathrm{M} / \mathrm{atm})(1.5 \mathrm{atm})\]

Step 2: Calculate the solubility of helium

Multiply the value of the Henry's law constant for helium and the partial pressure of helium: \[C_\mathrm{He} = 5.55 \times 10^{-4} \mathrm{M}\] So the solubility of helium in water at 30°C and 1.5 atm pressure is 5.55 × 10^{-4} M.

Step 3: Now let's find the solubility of nitrogen

To find the solubility of nitrogen, use the given values for the Henry's law constant for nitrogen and the partial pressure of nitrogen: \[C_\mathrm{N_2} = k_{H,\mathrm{N_2}}P_\mathrm{N_2}\] Plug the values (6.0 × 10^{-4} M / atm and 1.5 atm) into the equation: \[C_\mathrm{N_2} = (6.0 \times 10^{-4} \mathrm{M} / \mathrm{atm})(1.5 \mathrm{atm})\]

Step 4: Calculate the solubility of nitrogen

Multiply the value of the Henry's law constant for nitrogen and the partial pressure of nitrogen: \[C_\mathrm{N_2} = 9.0 \times 10^{-4} \mathrm{M}\] So the solubility of nitrogen in water at 30°C and 1.5 atm pressure is 9.0 × 10^{-4} M. In conclusion, at a pressure of 1.5 atm and 30°C, the solubility of helium in water is 5.55 × 10^{-4} M, and the solubility of nitrogen is 9.0 × 10^{-4} M.

Key Concepts

Gas Solubility

Gas solubility refers to the ability of a gas to dissolve in a liquid, such as water. It is an important concept in chemistry and environmental science. Solubility depends on several factors:

• Type of gas: Different gases exhibit different solubilities. For example, carbon dioxide is more soluble in water than oxygen.
• Temperature: Typically, as the temperature of the liquid increases, the solubility of a gas decreases.
• Pressure: According to Henry's Law, solubility is directly proportional to the pressure of the gas above the liquid. The higher the pressure, the more gas dissolves.

Gas solubility is important in various applications. In carbonated drinks, carbon dioxide is dissolved under high pressure to provide effervescence. Our atmosphere's composition, climate modeling, and even fish survival in water are influenced by gas solubility. Remember, the solubility of a gas can change significantly with different conditions.

Partial Pressure

Partial pressure is the individual pressure exerted by a single gas in a mixture of gases. Each gas in a mixture contributes to the overall pressure with its partial pressure. It's a critical concept in understanding gas behavior and calculating solubility. The sum of all partial pressures in a gas mixture equals the total pressure. This idea comes from Dalton's Law of Partial Pressures. The partial pressure of a gas in a mixture can be found by multiplying the mole fraction of the gas by the total pressure of the mixture. For example, in a mixture of oxygen and nitrogen, if oxygen accounts for 20% of the mixture and the total pressure is 1 atm, then the partial pressure of oxygen is 0.2 atm. Partial pressure is crucial for processes like respiration, where gases move based on their partial pressures. Understanding partial pressures helps us predict how gases dissolve, react, and behave under different conditions, as demonstrated in the Henry's Law calculations.

Henry's Law Constant

Henry's Law is a fundamental principle that states the amount of gas that dissolves in a liquid is proportional to the pressure of the gas above the liquid. This relationship is expressed using Henry's Law Constant \(k_H\), unique to each gas at a given temperature.The formula is:\[C = k_H \times P\]Where:

• \(C\) is the solubility of the gas in the liquid (molarity).
• \(k_H\) is Henry's Law Constant (M/atm).
• \(P\) is the partial pressure of the gas (atm).

For example, with a known \(k_H\) for helium, we can calculate its solubility in water at a specific pressure. We see Henry's Law in practice with everyday phenomena like scuba diving, where gas solubility changes with water depth and pressure, affecting how divers manage their breathing gases.Understanding Henry's Law Constant provides insight into the behavior of gases when they come into contact with liquids, helping us calculate and predict gas solubility in various scientific and industrial applications.