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 \(\mathrm{pK}_{b}=4.5\) or one with \(\mathrm{pK}_{b}=6.5\).

Short answer

In summary, (a) the weak acid solution has a higher pH, (b) the solution with Ka = 8×10^(-6) has a higher pH, and (c) the solution with pKb = 4.5 has a higher pH.

Step-by-step solution

Part (a) Strong Acid vs. Weak Acid

First, let's evaluate part (a) which asks us to compare a 0.1 M solution of a strong acid and a 0.1 M solution of a weak acid. Since strong acids ionize completely in water, the concentration of H+ ions equals the concentration of the strong acid. For a 0.1 M strong acid solution, the concentration of H+ is 0.1 M. The pH of a solution can be calculated using: \(pH = -\log[H^{+}]\) For a 0.1 M strong acid: \(pH = -\log(0.1) = 1\) On the other hand, weak acids don't release all their H+ ions into the solution and have an associated acidity constant (Ka). We are not provided with a particular Ka value for the weak acid in question, but since Ka values for weak acids are always less than 1, the concentration of H+ ions in the weak acid solution must be lower than 0.1 M. Therefore, the pH of the weak acid solution will be higher than 1 (resulting from the strong acid). The weak acid solution will have a higher pH.

Part (b) Acid with Different Ka values

Now, let's evaluate part (b). We have two acidic solutions with different Ka values. For an acidic solution with Ka = 2×10^(-3), we need to compare its pH with another acidic solution with Ka = 8×10^(-6). Higher Ka values indicate stronger acids, as they dissociate more in solution, releasing more H+ ions. Since the acidic solution with Ka = 2×10^(-3) is stronger, it will have a higher concentration of H+ ions in the solution, resulting in a lower pH value. Therefore, the solution with Ka = 8×10^(-6) will have a higher pH.

Part (c) Bases with Different pKb values

Lastly, let's evaluate part (c). We need to compare the pH levels of two basic solutions with pKb values of 4.5 and 6.5. Recall that for bases, a lower pKb value corresponds to a stronger base. To find the pH of a basic solution, we can use the following relationship: \(pH = 14 - pOH\) where pOH can be obtained from the pKb value: \(pOH = pK_b - \log[B^{-}]\) Here, B- represents the concentration of the base. Since both basic solutions have the same concentration (0.1 M), the difference in their pH values will be determined by their pKb values. The base with a lower pKb value (4.5) is stronger and will have a higher concentration of OH- ions, resulting in a lower pOH. Thus, it will have a higher pH value. Therefore, the solution with pKb = 4.5 will have a higher pH.

Key concepts

Strong Acid vs Weak Acid

In the world of chemistry, when we talk about acids, the distinction between strong acids and weak acids is crucial. A strong acid, like hydrochloric acid (HCl), ionizes completely in water. This means it releases all its hydrogen ions (H+) into the solution, providing a high concentration of H+ ions. For example, a 0.1 M solution of a strong acid will produce 0.1 M H+ ions. When you calculate pH using the formula \( pH = -\log[H^+] \), you find it has a low pH value of 1.

In contrast, a weak acid, such as acetic acid (CH3COOH), only partially ionizes in water. This means it doesn't release all available H+ ions, leading to a lower concentration of hydrogen ions in the solution. This partial ionization results in a higher pH value compared to a strong acid at the same concentration. Thus, for the same molarity, a weak acid's solution will have a higher pH than that of a strong acid. This is because of the difference in ionization abilities between strong and weak acids.

Acid Dissociation Constant (Ka)

The acid dissociation constant, abbreviated as \( K_a \), is a numerical value that tells us how strongly an acid can dissociate into its ions in water. When an acid dissolves in water, it releases H+ ions, and \( K_a \) helps us quantify this ability. A larger \( K_a \) value means the acid is stronger because it releases more H+ ions.

For instance, if you have two acids with different \( K_a \) values, say \( K_a = 2 \times 10^{-3} \) and \( K_a = 8 \times 10^{-6} \), the acid with \( K_a = 2 \times 10^{-3} \) is stronger. Consequently, it produces a higher concentration of H+ ions, which results in a lower pH value for the solution. The relationship between \( K_a \) values and pH is important for predicting how acidic a solution will be.

Base Dissociation Constant (pKb)

When discussing bases, the base dissociation constant, denoted as \( pK_b \), provides insight into how effectively a base can produce hydroxide ions (OH-) in a solution. The relationship between \( pK_b \) and base strength is inverse; a lower \( pK_b \) value means a stronger base. This is because a stronger base releases more OH- ions into the solution, decreasing the pOH and, since \( pH = 14 - pOH \), increasing the pH.

For example, consider two bases with \( pK_b \) values of 4.5 and 6.5. The base with a \( pK_b \) of 4.5 is stronger, releasing more OH- ions. It will have a lower pOH and, thus, a higher pH in comparison to the base with a \( pK_b \) of 6.5. Understanding \( pK_b \) is crucial when comparing the basicity of different solutions.