Problem 111 - Chemistry: The Central Science

Question

Which of the following solutions has the higher pH? (a) a \(0.1 \mathrm{M}\) solution of a strong acid or a \(0.1 \mathrm{M}\) 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 \(K_{a}=8 \times 10^{-6}\), (c) a 0.1 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

In summary, (a) the 0.1 M solution of a weak acid will have a higher pH, (b) the 0.1 M solution of the acid with \(K_a = 8 \times 10^{-6}\) will have a higher pH, and (c) the 0.1 M solution of the base with \(pK_b = 4.5\) will have a higher pH.

Step 1: Understanding strong and weak acids

A strong acid is one that completely dissociates in water, producing a higher concentration of hydronium ions (\(\mathrm{H_3O^+}\)). In contrast, a weak acid only partially dissociates, so the concentration of hydronium ions is lower in comparison to a strong acid of the same concentration. Therefore, solutions of strong acids generally have lower pH values than solutions of weak acids (with the same concentration). Now let's analyze the comparison (a).

Step 2: Comparison (a): Strong acid vs. Weak acid

In this comparison, we have a 0.1 M solution of a strong acid and a 0.1 M solution of a weak acid. Since strong acids produce more hydronium ions than weak acids at the same concentration, the solution of the strong acid will have a lower pH. Therefore, the 0.1 M solution of the weak acid will have a higher pH.

Step 3: Understanding the relationship between Ka and acid strength

The acid dissociation constant (\(K_a\)) gives us an idea of the strength of an acid. The higher the \(K_a\) value, the more the acid dissociates, and the stronger the acid is. In comparison (b), we have two different \(K_a\) given. Let's analyze them.

Step 4: Comparison (b): Different Ka values

In this comparison, we have a 0.1 M solution of an acid with \(K_a = 2 \times 10^{-3}\) and another 0.1 M solution of an acid with \(K_a = 8 \times 10^{-6}\). Since the first acid has a higher \(K_a\), it is a stronger acid. This means that it will produce more hydronium ions and thus have a lower pH compared to a solution with the weaker acid (lower \(K_a\)). Therefore, the 0.1 M solution of the acid with \(K_a = 8 \times 10^{-6}\) will have a higher pH.

Step 5: Understanding the relationship between pKb and base strength

The base dissociation constant (\(K_b\)) is used to quantify the strength of a base in solution. The \(pK_b\) value is the negative logarithm of the \(K_b\) and hence, a lower \(pK_b\) value indicates a stronger base. In comparison (c), we have two different \(pK_b\) values. Let's analyze them.

Step 6: Comparison (c): Different pKb values

In this comparison we have a 0.1 M solution of a base with \(pK_b = 4.5\) and another 0.1 M solution of a base with \(pK_b = 6.5\). The base with the lower \(pK_b\) value (4.5) is stronger, therefore implying that it will produce more hydroxide ions (\(\mathrm{OH^-}\)) and have a higher pH as compared to the base with a higher \(pK_b\) value (6.5). Therefore, the 0.1 M solution of the base with \(pK_b = 4.5\) will have a higher pH.

Key Concepts

Strong and Weak Acids

When we talk about the acidity of solutions, we need to understand the distinction between strong and weak acids. Strong acids, such as hydrochloric acid (HCl) and sulfuric acid (H₂SO₄), fully dissociate in water, releasing a greater number of hydronium ions (\(\mathrm{H_3O^+}\)). This results in a lower pH value for the solution, indicating higher acidity.

Weak acids like acetic acid (CH₃COOH) do not completely dissociate in water; only a fraction of their molecules release protons to form hydronium ions. As a result, the pH of a weak acid solution is higher compared to a strong acid of the same molarity. Understanding this fundamental difference is crucial for predicting the behavior of acids in chemical reactions and in various processes like buffering and acid-base titrations.

Acid Dissociation Constant (Ka)

The strength of an acid in aqueous solution can be quantified by its acid dissociation constant, denoted as \(K_a\). It is a measure of how readily the acid gives up its protons to the surrounding water molecules. The general equation for an acid's dissociation in water is:
\[\mathrm{HA} \rightleftharpoons \mathrm{H^+} + \mathrm{A^-}\]
The \(K_a\) is calculated using the concentrations of the products and reactants at equilibrium:
\[K_a = \frac{[\mathrm{H^+}][\mathrm{A^-}]}{[\mathrm{HA}]}\]
A higher \(K_a\) indicates a greater concentration of hydronium ions, meaning the acid is stronger and thus has a lower pH. Learning how to calculate and interpret the \(K_a\) is vital for students as it allows for the prediction of the extent of dissociation of acids in solution.

Base Dissociation Constant (Kb)

Similar to acids, bases have a corresponding dissociation constant, referred to as the base dissociation constant or \(K_b\). It measures a base's propensity to accept protons and form hydroxide ions (\(\mathrm{OH^-}\)) in solution. The higher the \(K_b\), the stronger the base is, meaning it can more effectively pull protons from water molecules which leads to an increase in the solution's pH. Familiarity with \(K_b\) values is essential for predicting the behavior of bases in neutralization reactions and other chemical processes. It is important to note that the relationship between \(K_b\) and basic strength is inverse to that of \(K_a\) and acidic strength because bases and acids are on opposite ends of the pH scale.

pH and pKb Relationship

Understanding the relationship between pH and \(pK_b\) is crucial in acid-base chemistry. The \(pK_b\) is derived by taking the negative logarithm of \(K_b\), which provides a more manageable number to work with. A lower \(pK_b\) indicates a stronger base, as it corresponds to a higher \(K_b\). Since stronger bases generate a greater concentration of hydroxide ions, they also lead to a higher pH. This inverse logarithmic relationship helps students comprehend the effect of different bases on the pH of a solution and is essential when comparing the strengths of bases through their \(pK_b\) values.