Problem 22 - Chemistry and Chemical Reactivity

Question

The Henry's law constant for \(\mathrm{O}_{2}\) in water at \(25^{\circ} \mathrm{C}\) is 1.66 \(\times 10^{-6} \mathrm{M} / \mathrm{mm}\) Hg. Which of the following is a reasonable constant when the temperature is \(50^{\circ} \mathrm{C}\) ? Explain the reason for your choice. (a) \(8.80 \times 10^{-7} \mathrm{M} / \mathrm{mm} \mathrm{Hg}\) (b) \(3.40 \times 10^{-6} \mathrm{M} / \mathrm{mm} \mathrm{Hg}\) (c) \(1.66 \times 10^{-6} \mathrm{M} / \mathrm{mm} \mathrm{Hg}\) (d) \(8.40 \times 10^{-5} \mathrm{M} / \mathrm{mm} \mathrm{Hg}\)

Step-by-Step Solution

Answer

Option (a) \( 8.80 \times 10^{-7} \mathrm{M} / \mathrm{mm} \mathrm{Hg} \) is reasonable because Henry's Law constant decreases as temperature increases.

Step 1: Understand Henry's Law

Henry's Law states that the solubility of a gas in a liquid is directly proportional to the partial pressure of the gas above the liquid, represented by the equation \( C = k_H P \), where \( C \) is the concentration of the gas, \( k_H \) is the Henry's Law constant, and \( P \) is the partial pressure.

Step 2: Influence of Temperature on Henry's Law Constant

The Henry's Law constant \( k_H \) generally decreases with an increase in temperature for gases dissolving in liquids. This is because increasing temperature tends to decrease gas solubility in liquids.

Step 3: Analyze Given Options

Since the temperature is increasing from \( 25^{ ext{o}} ext{C} \) to \( 50^{ ext{o}} ext{C} \), we expect \( k_H \) to decrease. Therefore, option (a) \( 8.80 \times 10^{-7} \mathrm{M} / \mathrm{mm} \mathrm{Hg} \) is a reasonable constant because it is lower than the original value \( 1.66 \times 10^{-6} \mathrm{M} / \mathrm{mm} \mathrm{Hg} \).

Step 4: Examine Other Options

Option (b) \( 3.40 \times 10^{-6} \mathrm{M} / \mathrm{mm} \mathrm{Hg} \) and option (d) \( 8.40 \times 10^{-5} \mathrm{M} / \mathrm{mm} \mathrm{Hg} \) have increased substantially compared to the original \( k_H \) and are not consistent with decreasing solubility. Option (c) remains unchanged, which does not reflect the expected decrease in solubility at higher temperatures.

Step 5: Determine Reasonable Constant

The expected outcome is that \( k_H \) decreases with increasing temperature. Hence, the option that fits this expectation is (a) \( 8.80 \times 10^{-7} \mathrm{M} / \mathrm{mm} \mathrm{Hg} \). This proposed constant reflects the inverse relationship between solubility and temperature.

Key Concepts

Gas Solubility

Gas solubility is a term used to describe how much of a gas can be dissolved in a liquid. This concept is crucial in understanding how gases behave in different environments, such as natural bodies of water or even in beverages like soda. Solubility indicates the maximum concentration of a gas that can dissolve in a liquid under specific conditions:

• Temperature: Typically, gases are less soluble in liquids at higher temperatures.
• Pressure: Increasing the pressure of the gas above the liquid generally increases solubility, according to Henry's Law.
• Nature of Gas and Liquid: Different gases and liquids interact differently; some gases dissolve better in certain liquids.

The balance of these factors can significantly affect how much of a gas can be dissolved in water or any other liquid.

Henry's Law Constant

The Henry's Law constant, represented as \( k_H \), is a crucial part of Henry's Law, which expresses the relationship between the solubility of a gas and the pressure of that gas above a liquid. It is defined by the equation \( C = k_H P \), where \( C \) denotes the concentration of the gas in the liquid and \( P \) is the partial pressure of the gas:

• Specific to Gas and Liquid: The \( k_H \) value is unique for every specific pair of gas and liquid.
• Units: Typically expressed in \( \, \text{M} / \text{mm Hg} \), indicating molarity per unit of pressure.
• Temperature Dependent: The Henry's Law constant varies with temperature, often decreasing as temperature increases, leading to reduced solubility of most gases in liquids.

Understanding \( k_H \) is essential for predicting how gases like oxygen or carbon dioxide will dissolve under different environmental and experimental conditions.

Temperature Effects on Solubility

The effect of temperature on solubility is particularly important in understanding Henry's Law and is a common observation in everyday life. As a general rule, as temperature increases, the solubility of gases in liquids tends to decrease. This phenomenon can be influenced by:

• Kinetic Energy: Higher temperatures increase the kinetic energy of molecules, making it harder for gas molecules to stay dissolved, as they become more likely to escape back into the gas phase.
• Inverse Relationship: Due to this temperature-solubility relationship, Henry's Law constants typically decrease with rising temperature for most gas-liquid combinations.
• Applications: This rule is applied in various fields, such as environmental science where it helps to understand oxygen levels in warm vs. cold water, affecting aquatic life.

This temperature impact is crucial in explaining why, in the given exercise, a lower Henry's Law constant at a higher temperature aligns with expected scientific principles.