Question

One way to distinguish a buffer solution with an acid solution is by dilution. (a) Consider a buffer solution made of \(0.500 \mathrm{M} \mathrm{CH}_{3} \mathrm{COOH}\) and \(0.500 \mathrm{M}\) \(\mathrm{CH}_{3} \mathrm{COONa}\). Calculate its \(\mathrm{pH}\) and the \(\mathrm{pH}\) after it has been diluted 10 -fold. (b) Compare the result in (a) with the pHs of a \(0.500 \mathrm{M} \mathrm{CH}_{3} \mathrm{COOH}\) solution before and after it has been diluted 10 -fold

Short answer

The pH of the buffer solution is 4.76 and remains the same (4.76) after it has been diluted 10-fold. For the \(0.500 \mathrm{M} \mathrm{CH}_{3} \mathrm{COOH}\) solution, the original pH is 0.301 and after 10-fold dilution, the pH increases to 1.301. So, when diluted, the pH of the buffer solution remains nearly constant while that of the acid solution increases.

Step-by-step solution

Analyze the Buffer Solution

A buffer solution is prepared by mixing \(0.500 \mathrm{M} \mathrm{CH}_{3}\mathrm{COOH}\) (Weak Acid) and \(0.500 \mathrm{M} \mathrm{CH}_{3} \mathrm{COONa}\) (Weak Base, Sodium Acetate). Using the Henderson-Hasselbalch equation (which is \(\mathrm{pH} = \mathrm{pKa} + \log(\frac{{[\mathrm{A}^-]}}{{[\mathrm{HA}]}})\), where \(\mathrm{A}^-\) is the concentration of the base (anion), \(\mathrm{HA}\) is the concentration of the acid, and \(\mathrm{pKa}\) is the acid dissociation constant), we can find the pH of the buffer solution by substituting the values. Here, \(\mathrm{pKa}\) for \(\mathrm{CH}_{3}\mathrm{COOH}\) is 4.76 (from the acid table). Both the acid and its conjugate base concentrations are equal, so the logarithmic term will be 0. Hence, \( \mathrm{pH} = 4.76 + \log(1) = 4.76 \).

Calculate the pH of the Buffer Solution after Dilution

When buffer solution is diluted, the concentrations of the acid and its conjugate base are reduced by the same factor. Thus, the ratio \(\frac{{[\mathrm{A}^-]}}{{[\mathrm{HA}]}}\) remains unchanged. Hence, according to the Henderson-Hasselbalch equation, the pH of the buffer, even after 10-fold dilution, will remain 4.76.

Analyze the Acid Solution

The \(\mathrm{CH}_{3}\mathrm{COOH}\) solution is a pure weak acid solution. To calculate the original pH, we will use the definition of \( \mathrm{pH} = - \log [\mathrm{H^{+}}]\). Since acetic acid barely ionizes, we can assume that \([\mathrm{H^{+}}] = [\mathrm{CH}_{3}\mathrm{COOH}]\). Hence, using this into the above equation we get \(pH = -\log (0.500) = 0.301\).

Calculate the pH of the Acid Solution after Dilution

When the \(\mathrm{CH}_{3}\mathrm{COOH}\) solution is diluted 10-fold, its concentration becomes \(0.500 \mathrm{M}/10 = 0.050 \mathrm{M}\). Substituting this into the pH equation, we get \( pH = - \log (0.05) = 1.301\). Hence, the pH increases when the acid solution is diluted.

Key concepts

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is a significant relationship used in chemistry to estimate the pH of a buffer solution. It is mathematically expressed as: \[\begin{equation}\text{pH} = \text{pKa} + \log\left(\frac{[\text{A}^-]}{[\text{HA}]}\right)\end{equation}\]where \text{pKa} is the acid dissociation constant, [\text{A}^-] is the concentration of the conjugate base, and [\text{HA}] is the concentration of the acid.
For a buffer solution where the acid and its conjugate base are present in equal concentrations, the log term becomes log(1), thus simplifying the equation to: \[\text{pH} = \text{pKa}\]In the context of the exercise provided, this equation allows us to determine that the pH of a buffer solution remains unchanged even after dilution, since the ratio of concentrations of its components remains constant.

Acid Dissociation Constant (pKa)

The acid dissociation constant, represented by \text{pKa}, is a measure of the strength of an acid in solution. It is defined as the negative base-10 logarithm of the equilibrium constant (\[\text{Ka}\]) for a chemical's dissociation reaction in water. The lower the \text{pKa} value, the stronger the acid as it dissociates more completely in solution. Conversely, a high \text{pKa} value indicates a weaker acid.
In our exercise, the \text{pKa} of acetic acid (\(\text{CH}_3\text{COOH}\)) is given as 4.76, which helps us establish the pH of the buffer solution created using acetic acid and its conjugate base, sodium acetate (\(\text{CH}_3\text{COONa}\)).

Weak Acid and Base Equilibria

Weak acids and bases do not completely ionize in solution, which leads to an equilibrium between the non-ionized acid or base and the ions produced. The position of this equilibrium is crucial for understanding the buffer capacity and pH of solutions.Weak acids, for instance, partially dissociate into hydrogen ions (\(\text{H}^+\)) and their conjugate bases, whereas weak bases partially bind hydrogen ions to form their conjugate acids. This dynamic is captured in the acid dissociation constant (\text{Ka}) and its logarithmic expression \text{pKa}. When applying these concepts to buffer solutions, as in our exercise, we can deduce that the buffer’s ability to maintain pH upon dilution arises from the weak acid-base equilibria established in the solution.

pH Calculation upon Dilution

The pH of a solution is fundamentally related to the concentration of hydrogen ions (\(\text{H}^+\)) present. For a strong acid or base, the pH changes significantly upon dilution because it affects the concentration of these ions directly. However, with buffer solutions, the presence of both an acid and its conjugate base allows them to resist changes in pH when diluted. This is because the ratio of their concentrations remains the same, keeping the pH stable, as demonstrated by the Henderson-Hasselbalch equation. Conversely, in the case of weak acids without a conjugate base, as shown in our exercise, dilution increases the pH value because the concentration of hydrogen ions decreases correspondingly.