Problem 107 - Chemistry The Central Science

Question

Which of the following solutions has the higher pH? (a) a \(0.1 M\) solution of a strong acid or a \(0.1 M\) solution of a weak acid, (b) a \(0.1 \mathrm{M}\) solution of an acid with \(K_{a}=2 \times 10^{-3}\) or one with \(K_{a}=8 \times 10^{-6},(\mathrm{c}) \mathrm{a} 0.1 \mathrm{M}\) solution of a base with \(\mathrm{p} K_{b}=4.5\) or one with \(\mathrm{p} K_{b}=6.5\).

Step-by-Step Solution

Answer

(a) The 0.1 M solution of a weak acid has a higher pH than the 0.1 M solution of a strong acid. (b) The 0.1 M solution of an acid with \(K_{a}=8 \times 10^{-6}\) has a higher pH. (c) The 0.1 M solution of a base with \(pK_{b}=4.5\) has a higher pH.

Step 1: (a) Comparing a 0.1 M solution of a strong acid and a 0.1 M solution of a weak acid

Recall that the pH of a solution is defined as the negative base 10 logarithm of the hydrogen ion concentration, given by the formula: pH = -log10([H+]). For strong acids, they ionize completely in water and, for weak acids, only partially ionize. For the 0.1 M solution of a strong acid, since it ionizes completely, the concentration of H+ ions would be 0.1 M and the pH would be -log10(0.1) = 1. For the 0.1 M solution of a weak acid, the concentration of H+ ions would be less than 0.1 M. Therefore, the pH would be greater than 1 since the pH is inversely proportional to the concentration of H+ ions. Conclusion: The 0.1 M solution of a weak acid has a higher pH than the 0.1 M solution of a strong acid.

Step 2: (b) Comparing 0.1 M solution of an acid with \(K_{a}=2 \times 10^{-3}\) and one with \(K_{a}=8 \times 10^{-6}\)

For weak acids, the presence of the dissociation constant (\(K_a\)) is crucial. A higher \(K_a\) value indicates the acid is stronger (dissociates to a greater extent) and will have a higher concentration of H+ ions, which means a lower pH. Comparing the given acids with \(K_{a}\) values of \(2 \times 10^{-3}\) and \(8 \times 10^{-6}\), we can see that the first one has a higher \(K_a\) value and thus will be stronger than the second one. Hence, the second acid will have higher pH than the first one. Conclusion: The 0.1 M solution of an acid with \(K_{a}=8 \times 10^{-6}\) has a higher pH.

Step 3: (c) Comparing 0.1 M solution of a base with \(pK_{b}=4.5\) and \(pK_{b}=6.5\)

For bases, we look at their dissociation constant (\(K_b\)) or their \(pK_b\) value. A lower \(pK_b\) means a stronger base (dissociates more in water). The pH of a solution of a base will be more than 7. We are given the bases with \(pK_b\) values of 4.5 and 6.5. The base with a \(pK_b\) of 4.5 is stronger than the base with a \(pK_b\) of 6.5. As a stronger base dissociates more in water and produces a larger amount of hydroxide ions, this will result in a higher pH. Conclusion: The 0.1 M solution of a base with \(pK_{b}=4.5\) has a higher pH.

Key Concepts

Strong and Weak Acids

When discussing acids, the terms "strong" and "weak" refer to their ability to dissociate in water.

**Strong acids** fully dissociate in water, meaning they release all their hydrogen ions (H⁺) into the solution. Common examples include hydrochloric acid (HCl) and sulfuric acid (H₂SO₄). Because they release more H⁺ ions, they tend to have a lower pH

**Weak acids**, on the other hand, only partly dissociate in water, meaning only some of their hydrogen ions are released. Acetic acid (CH₃COOH), found in vinegar, is an example of a weak acid. Since they release fewer H⁺ ions, weak acids result in a higher pH compared to strong acids at the same concentration.

• Complete vs Partial Dissociation
• Examples: HCl (strong) vs CH₃COOH (weak)
• Impact on pH: More H⁺ ions lower pH

Acid Dissociation Constant

The acid dissociation constant, often referred to as the **Ka value**, measures the strength of an acid by indicating how well it dissociates in water.

A larger Ka value means stronger acid, as it points to more dissociation and more H⁺ ions in the solution. Conversely, a smaller Ka value indicates a weaker acid.

Understanding Ka helps in predicting and comparing the acidity of different substances.

• Ka Related to Dissociation: High Ka = Strong Acid
• Impact on pH: Higher Ka means lower pH
• Examples of Ka in use: Comparing acids like H₂SO₄ (strong, high Ka) vs. HF (weaker, lower Ka)

When comparing two acids with different Ka values, the one with the higher Ka value is the stronger acid, resulting in a lower pH.

Bases and pKb

The pKb value is a key concept when discussing bases. It is similar to the Ka for acids but applies to bases and their dissociation in water.

pKb stands for the negative logarithm of the base dissociation constant, Kb. A lower pKb value signifies a stronger base, meaning it dissociates more in water.

The more dissociation, the higher the concentration of hydroxide ions (OH⁻), which raises the pH of the solution.

• pKb vs Kb: Lower pKb, stronger base
• Impact on pH: More OH⁻ ions mean higher pH
• Examples: Comparing common bases like ammonia (NH₃, weaker) vs sodium hydroxide (NaOH, stronger)

So, when you need to predict the pH of a basic solution, always look at the pKb. A lower pKb will result in a higher pH, indicating a more strongly basic solution.