Question

What is the \(\mathrm{pH}\) at \(25^{\circ} \mathrm{C}\) of water saturated with \(\mathrm{CO}_{2}\) at a partial pressure of \(1.10 \mathrm{~atm}\) ? The Henry's law constant for \(\mathrm{CO}_{2}\) at \(25^{\circ} \mathrm{C}\) is \(3.1 \times 10^{-2} \mathrm{~mol} / \mathrm{L}-\mathrm{atm}\).

Short answer

The pH of the water saturated with CO₂ at 25°C and a partial pressure of 1.10 atm is approximately 1.77.

Step-by-step solution

Determine the concentration of dissolved CO₂ using Henry's Law

Henry's Law states that the concentration of a gas in a liquid is proportional to the partial pressure of the gas above the liquid. The Henry's Law constant (KH) relates the concentration (C) of the dissolved gas to its partial pressure (P) in the following manner: \[C = K_H \cdot P\] The partial pressure of CO₂ (P) is given as 1.10 atm and the Henry's Law constant (KH) for CO₂ at 25°C is \(3.1 \times 10^{-2} \frac{mol}{L \cdot atm}\). We can now use this information to determine the concentration of dissolved CO₂.

Calculate the dissolved CO₂ concentration

Using Henry's Law equation from Step 1, we can calculate the dissolved CO₂ concentration (C) in the water: \[C = K_H \cdot P = (3.1 \times 10^{-2} \frac{mol}{L \cdot atm}) \cdot (1.10 \, atm) = 3.41 \times 10^{-2} \frac{mol}{L}\] The concentration of dissolved CO₂ in the water is \(3.41 \times 10^{-2} mol/L\).

Calculate the concentration of H⁺ ions

When CO₂ dissolves in water, it reacts with water to form carbonic acid (H₂CO₃), which then dissociates predominantly into bicarbonate ions (HCO₃⁻) and H⁺ ions: \[CO₂(aq) + H₂O(l) \rightleftharpoons H₂CO₃(aq) \rightleftharpoons HCO₃⁻(aq) + H⁺(aq)\] The concentration of H⁺ ions is equal to the concentration of HCO₃⁻ ions: \[[H⁺] = [HCO₃⁻] = \frac{1}{2}[CO₂(aq)]\] Let's calculate the concentration of H⁺ ions in the water: \[[H⁺] = \frac{1}{2}(3.41 \times 10^{-2} \frac{mol}{L}) = 1.705 \times 10^{-2} \frac{mol}{L}\] The concentration of H⁺ ions in the water is \(1.705 \times 10^{-2} mol/L\).

Calculate the pH of the solution

Finally, we can calculate the pH using the definition of pH, which is the negative base-10 logarithm of the concentration of hydrogen ions: \[pH = -\log_{10} [H⁺]\] \[pH = -\log_{10} (1.705 \times 10^{-2}) = 1.77\] The pH of the water saturated with CO₂ at 25°C and a partial pressure of 1.10 atm is approximately 1.77.

Key concepts

Henry's Law

Understanding Henry's Law is crucial for grasping how gases dissolve in liquids, particularly in our discussion about pH calculation involving carbon dioxide in water. The law states that at a constant temperature, the concentration of a dissolved gas is directly proportional to its partial pressure in the gas phase.

In mathematical terms, the law is given as: \[C = K_H \times P\]where C is the concentration of the gas in the liquid, K_H is Henry's Law constant for the particular gas at a given temperature, and P is the partial pressure of the gas.

This relationship allows us to calculate how much CO₂ is dissolved in water at a specific partial pressure, which is the starting point for determining the pH level of carbonated water. The constant (K_H) varies for different gases and depends on the temperature, enabling us to use it as a unique factor for a given gas, such as CO₂, at a given condition.

Dissociation of Carbonic Acid

When carbon dioxide dissolves in water, it forms carbonic acid (\(H_2CO_3\)), a reaction key to understanding the acidity of solutions such as carbonated water. The dissociation of carbonic acid into bicarbonate (\(HCO_3^-\)) and hydrogen (\(H^+\)) ions can be represented by the equilibrium:\[CO_2(aq) + H_2O(l) \rightleftharpoons H_2CO_3(aq) \rightleftharpoons HCO_3^-(aq) + H^+(aq)\]
The degree of this dissociation affects the concentration of hydrogen ions, hence affecting the pH. While carbonic acid is considered a weak acid because it dissociates partially in water, the process significantly contributes to the acidity of solutions like soda water. Understanding the balance between carbonic acid and its dissociated ions is pivotal for accurately predicting the pH of such solutions.

Concentration of Hydrogen Ions

The concentration of hydrogen ions (\([H^+]\)) is directly tied to the pH level of a solution. As per the calculation from the provided exercise, the dissociation of carbonic acid results in the production of hydrogen ions, which determines the acidity.

The reaction between dissolved CO₂ and water produces carbonic acid, which then dissociates into hydrogen and bicarbonate ions. The concentration of hydrogen ions is derived from the half reaction of carbonic acid dissociation:\[[H^+] = \frac{1}{2}[CO_2(aq)]\]Since the concentration of hydrogen ions is available, you can calculate the pH level of the solution, illustrating the importance of this concept in understanding the chemistry of acids and bases.

Acid-base Equilibrium

The acid-base equilibrium in a solution refers to the reversible chemical reactions involving the transfer of hydrogen ions between reactants. In the context of the dissolution of CO₂ in water, carbonic acid (\(H_2CO_3\)) forms and slightly dissociates to give bicarbonate (\(HCO_3^-\)) and hydrogen (\(H^+\)) ions.

This equilibrium can be expressed as:\[H_2CO_3(aq) \rightleftharpoons HCO_3^-(aq) + H^+(aq)\]The position of this equilibrium is influenced by various factors, including the concentration of dissolved CO₂, the temperature, and the presence of other ions in the solution. Understanding this equilibrium is essential for predicting the pH and the behavior of carbonated drinks, ocean acidification, and physiological processes in biology.