Question

Which of the following graph correctly represents the plots of \(K_H\) at 1 bar gases in water versus temperature?

• A.
• B.
• C.
• D.

Hint:

For gases in water, the temperature dependence of Henry’s Law constant shows an initial decrease followed by an increase in solubility at higher temperatures. This behavior varies across different gases.

Answer 1

The Correct Option is D

As temperature increases, solubility first decreases then increases, hence \( K_H \) first increases, then decreases. At moderate temperature, the value of \( K_H \) follows the order: \[ \text{He}>\text{N}_2>\text{CH}_4 \]

Answer 2

The solution requires an understanding of Henry's Law and the factors that influence the Henry's Law constant (\( K_H \)).

1. Henry's Law: This law relates the partial pressure (\( p \)) of a gas above a liquid to its mole fraction (\( x \)) in the liquid phase: \[ p = K_H \cdot x \] From this equation, the solubility of a gas (represented by \( x \)) is inversely proportional to \( K_H \) for a given partial pressure. A higher \( K_H \) value means lower solubility. \[ \text{Solubility} \propto \frac{1}{K_H} \]
2. Effect of Temperature on \( K_H \): The dissolution of gases in liquids is typically an exothermic process. A simplified model suggests that increasing the temperature decreases solubility, which would mean \( K_H \) monotonically increases with temperature. However, the actual behavior for many nonpolar gases in water is more complex. The value of \( K_H \) first increases with temperature, reaches a maximum (corresponding to a minimum in solubility), and then begins to decrease at higher temperatures. This non-monotonic behavior is a more accurate representation.
3. Dependence of \( K_H \) on the Nature of the Gas: The value of \( K_H \) is specific to the gas-solvent pair. For nonpolar gases in a polar solvent like water, the solubility is governed by London dispersion forces. Gases that are smaller and less polarizable interact more weakly with water, leading to lower solubility and a higher \( K_H \) value.

Step-by-Step Solution:

Step 1: Analyze the Temperature Dependence shown in the graphs.

The graphs show different trends for \( K_H \) versus temperature (\( t/^\circ\text{C} \)).

• Graphs (2) and (3) show a simple, monotonic increase of \( K_H \) with temperature. This represents the introductory model where gas solubility always decreases with increasing temperature.
• Graphs (1) and (4) show a more complex, non-monotonic relationship: \( K_H \) first increases, reaches a maximum value, and then decreases. This is a more accurate depiction of the behavior of many real gases (like He, N\(_2\), O\(_2\), CH\(_4\)) in water.

Given the options, we should consider the more accurate, non-monotonic trend as the correct representation of the physical phenomenon.

Step 2: Analyze the relative order of \( K_H \) values for the gases in option (4).

Option (4) plots \( K_H \) for Helium (He), Nitrogen (N\(_2\)), and Methane (CH\(_4\)). We need to determine the correct order of their \( K_H \) values.

• Helium (He): A very small noble gas with extremely low polarizability. It has very weak intermolecular interactions with water.
• Nitrogen (N\(_2\)): A nonpolar diatomic molecule, larger and more polarizable than He.
• Methane (CH\(_4\)): A nonpolar molecule, larger and more polarizable than N\(_2\).

Greater polarizability leads to stronger London dispersion forces with water molecules, resulting in higher solubility.

Therefore, the order of solubility is: \( \text{Solubility(CH}_4\text{)} > \text{Solubility(N}_2\text{)} > \text{Solubility(He)} \).

Since \( K_H \) is inversely proportional to solubility, the order of the Henry's Law constants must be the reverse:

\[ K_H(\text{He}) > K_H(\text{N}_2) > K_H(\text{CH}_4) \]

Step 3: Evaluate Graph (4) based on the analysis.

Let's check if Graph (4) is consistent with our findings.

1. Shape of the Curves: The graph correctly shows the non-monotonic behavior where \( K_H \) first increases and then decreases with temperature.
2. Relative Positions: At any given temperature on the graph, the curve for He is the highest, followed by N\(_2\), and then CH\(_4\) is the lowest. This corresponds to the order \( K_H(\text{He}) > K_H(\text{N}_2) > K_H(\text{CH}_4) \), which we determined to be correct.

Conclusion:

Graph (4) correctly represents both the sophisticated temperature dependence and the relative magnitudes of the Henry's Law constants for He, N\(_2\), and CH\(_4\) in water. The other graphs are incorrect because they either show a simplified temperature dependence (3) or an incorrect relative ordering of the \( K_H \) values (1, 2).

Therefore, the correct representation is given by option (4).