Problem 120 - Chemistry The Molecular Science

Question

A student calculated a pH of 5.89 for a \(1.0 \times 10^{-7} \mathrm{~mol} / \mathrm{L}\) acetic acid solution. This answer is incorrect because the student made two invalid assumptions in the calculation. Identify the two incorrect assumptions.

Step-by-Step Solution

Answer

The assumptions were: acetic acid fully dissociates and water doesn't contribute to \(\text{H}^+\).

Step 1: Understanding Acetic Acid Properties

Acetic acid is a weak acid and does not dissociate completely in solution. Therefore, the student's assumption that it fully dissociates is incorrect as it affects the pH calculation.

Step 2: Considering Water Ionization

The student likely overlooked the contribution of hydrogen ions from the water itself. In very dilute solutions, like this one, the ionization of water can significantly affect the pH.

Step 3: Calculating the Expected Water Contribution

In pure water at 25°C, the concentration of hydrogen ions is about \(1.0 \times 10^{-7} \mathrm{~mol/L}\). Thus, the total concentration of hydrogen ions would approximately be twice this value, since both acetic acid ionizes and water contributes equally.

Step 4: Correcting the pH Calculation

Considering contributions from both the acetic acid and water, we calculate the total hydrogen ion concentration as \(2.0 \times 10^{-7} \mathrm{~mol/L}\). This leads to a pH of \(6.67\).

Key Concepts

Understanding Weak Acid Dissociation

Weak acids like acetic acid are characterized by their incomplete dissociation in water. This means that when added to water, only a small fraction of the acetic acid molecules actually release hydrogen ions. Most of the molecules remain intact. For example, if you dissolve acetic acid in water, only a few of the molecules will split into hydrogen ions (H⁺) and acetate ions (CH₃COO⁻). The strength of an acid is typically indicated by its dissociation constant, denoted as Ka. Acetic acid has a low Ka value, which means it is a weak acid and does not dissociate significantly.

• The assumption of complete dissociation is a common error when dealing with weak acids.
• This can significantly alter pH calculations as it overestimates the concentration of hydrogen ions in the solution.

When calculating the pH of a solution containing a weak acid, it is crucial to consider its partial dissociation. Ignoring this can lead to incorrect pH values.

Role of Water Ionization

Water undergoes a self-ionization process even in its pure form. This means that water molecules tend to slightly dissociate into hydrogen ions (H⁺) and hydroxide ions (OH⁻). The equilibrium constant for this process is the ion product of water, Kw, which is equal to \[1.0 \times 10^{-14} \mathrm{~mol^2/L^2} \text{ at 25°C}. \]In very dilute solutions, like with the acetic acid in the problem, the concentration of hydrogen ions from water becomes significant compared to that from the acetic acid. The self-ionization of water means it is always contributing a baseline level of hydrogen ions to the pH of the solution.

• This can lead to errors in pH calculations if not taken into account.
• Especially in solutions with very low concentrations of a weak acid, the water's ionization effect cannot be ignored.

Therefore, when calculating pH in such scenarios, one must include this contribution to achieve accurate results.

Importance of Hydrogen Ion Concentration

Hydrogen ion concentration is central to determining the pH of a solution. pH is defined mathematically by the expression: \[pH = -\log[H⁺]\]where \([H⁺]\) is the molarity of hydrogen ions in the solution. For accurate pH calculation, we must consider all possible sources of hydrogen ions. In the given scenario, both the acetic acid's dissociation and water's ionization contribute to the hydrogen ion pool.

• At low concentrations, each source of hydrogen ions potentially plays a more pronounced role.
• A naive approach that overlooks any of these sources can predict incorrect pH values.

The assumption that a highly dilute acetic acid solution solely dictates \([H⁺]\) overlooks water's contribution. In this particular exercise, recognizing that water provides approximately the same concentration of hydrogen ions as the acetic acid itself was key to arriving at the correct pH. Therefore, always double-check which contributions to \([H⁺]\) have been considered to ensure pH calculations are reliable.