Problem 1 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

The Henry's law constant for the solubility of \(\mathrm{N}_{2}\) gas in water at \(298 \mathrm{~K}\) is \(1.0 \times 10^{5} \mathrm{~atm} .\) The mole fraction of \(\mathrm{N}_{2}\) in air is \(0.8 .\) The number of moles of \(\mathrm{N}_{2}\) from air dissolved in 10 moles of water at \(298 \mathrm{~K}\) and 5 atm pressure is (a) \(4.0 \times 10^{-4}\) (b) \(4.0 \times 10^{-5}\) (c) \(5.0 \times 10^{-4}\) (d) \(5.0 \times 10^{-5}\)

Step-by-Step Solution

Answer

The number of moles of \( \mathrm{N}_2 \) from air dissolved in 10 moles of water at \( 298 \mathrm{~K} \) and 5 atm pressure is \( 4.0 \times 10^{-4} \) moles.

Step 1: Understand Henry's Law

Henry's Law states that at a constant temperature, the solubility of a gas in a liquid is directly proportional to the partial pressure of that gas above the liquid. The formula for Henry's Law is given by: \( C = k_H \cdot P \), 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 the partial pressure of N2 in air

The partial pressure of \( \mathrm{N}_2 \) in air can be found by multiplying the total pressure by the mole fraction of \( \mathrm{N}_2 \). The partial pressure is given by: \( P_{\mathrm{N}_2} = P_{\text{total}} \times \text{mole fraction of } \mathrm{N}_2 \). Here, \( P_{\text{total}} = 5 \) atm and the mole fraction is \( 0.8 \), so \( P_{\mathrm{N}_2} = 5 \cdot 0.8 = 4 \) atm.

Step 3: Use Henry's Law to find the concentration of N2

Using the Henry's law formula \( C = k_H \cdot P_{\mathrm{N}_2} \), with \( k_H = 1.0 \times 10^5 \) atm and \( P_{\mathrm{N}_2} = 4 \) atm, we find that \( C = (1.0 \times 10^5) \cdot 4 = 4.0 \times 10^5 \) (concentration units).

Step 4: Convert the gas concentration into moles in 10 moles of water

The molar concentration \( C \) is the number of moles of \( \mathrm{N}_2 \) per mole of water. Therefore, in 10 moles of water, the number of moles of \( \mathrm{N}_2 \) will be \( 10 \times C \): \( 10 \times 4.0 \times 10^5 = 4.0 \times 10^{-4} \) moles of \( \mathrm{N}_2 \).

Key Concepts

Solubility of Gases

Understanding the solubility of gases in liquids, such as water, is key to grasping many scientific and real-world applications—from carbonated beverages to how fish receive oxygen underwater. At a basic level, gas solubility refers to the ability of a given gas to dissolve in a liquid. This process is influenced by various factors, including temperature, the nature of the gas and the liquid, and most importantly, the pressure of the gas above the liquid.

Solubility increases with pressure and decreases with increasing temperature; hence, cold drinks hold more fizz (carbon dioxide) than warm ones. Henri's Law provides the quantitative aspect of gas solubility, stating it is directly proportional to the pressure of the gas above the liquid. This lays the foundation for solving problems related to gas solubility, such as the dissolution of nitrogen from the air into water.

Partial Pressure

Partial pressure is a fundamental concept in understanding how gases behave when they are part of a mixture, such as air. It is defined as the pressure that a single gas component in a mixture would exert if it alone occupied the entire volume of the mixture at the same temperature. The total pressure of a gas mixture is the sum of the partial pressures of each component gas.

The partial pressure of a gas can be found by taking the mole fraction of the gas in the mixture and multiplying it by the total pressure of the mixture. This concept is instrumental in applying Henri's Law since it helps determine the effective pressure of a particular gas that is in direct contact with a liquid, influencing its solubility.

Mole Fraction

The mole fraction is a way of expressing the concentration of a component in a mixture, without the need for units of mass or volume. The mole fraction of a particular substance is the ratio of the number of moles of that substance to the total number of moles of all substances in the mixture.

In a two-component system, such as nitrogen gas in air, the mole fraction is a simple fraction derived from counting the statistics of molecules—the mole fraction of nitrogen is derived from the percentage it contributes to the air. It has no units and is used as a multiplicative factor in several equations, including the calculation of partial pressures. Its familiarity and ease of calculation make it a valuable tool in solving problems related to gas solubility.

Concentration of Gas in a Liquid

The concentration of a gas in a liquid is a way to quantify how much of a gas has dissolved in a given amount of liquid. In the context of Henry's Law, the concentration refers to the amount of gas, described as number of moles, dissolved per mole of the liquid. This can also be portrayed in different units such as mol/L or ppm (parts per million), depending on the context.

The concentration is directly influenced by the gas's partial pressure and its nature. Henry's Law gives us an equation to find this concentration: the product of the Henry’s Law constant and the respective gas’s partial pressure. The higher the constant, the more soluble the gas is under a given pressure. This variable greatly facilitates calculations involving the dissolution of gas into liquids, crucial for many areas such as environmental science, engineering, and medicine.