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 part (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 buffer solution pH remains 4.76 after dilution, while the acetic acid pH changes from 2.52 to 3.00.

Step-by-step solution

Calculate initial pH of Buffer Solution

A buffer solution can be described by the Henderson-Hasselbalch equation: \( \mathrm{pH} = \mathrm{pKa} + \log \left( \frac{[\mathrm{A}^-]}{[\mathrm{HA}]} \right) \).Here, acetic acid (\( \mathrm{CH}_3\mathrm{COOH} \)) is \( \mathrm{HA} \) and its conjugate base, acetate (\( \mathrm{CH}_3\mathrm{COO}^- \)), is \( \mathrm{A}^- \).Given that \( \mathrm{pKa} \) for acetic acid is 4.76, both concentrations of \( \mathrm{CH}_3\mathrm{COOH} \) and \( \mathrm{CH}_3\mathrm{COONa} \) are \( 0.500 \mathrm{M} \),\[\mathrm{pH} = 4.76 + \log \left( \frac{0.500}{0.500} \right) = 4.76 + 0 = 4.76.\]

Calculate pH after 10-fold dilution

When the buffer solution is diluted ten times, both concentrations of \( \mathrm{CH}_3\mathrm{COOH} \) and \( \mathrm{CH}_3\mathrm{COONa} \) become \( 0.050 \mathrm{M} \).Applying the Henderson-Hasselbalch equation again:\[\mathrm{pH}_{\text{diluted}} = 4.76 + \log \left( \frac{0.050}{0.050} \right) = 4.76.\]The pH remains unchanged at 4.76.

Calculate pH of 0.500 M Acetic Acid Solution

For a simple acidic solution, use the expression for weak acid dissociation:\[\mathrm{pH} = -\log[H^+]\]Considering the dissociation: \( \mathrm{CH}_3\mathrm{COOH} \Leftrightarrow \mathrm{H}^+ + \mathrm{CH}_3\mathrm{COO}^- \), where \( K_a = 1.8 \times 10^{-5} \).\[K_a = \frac{[\mathrm{H}^+][\mathrm{CH}_3\mathrm{COO}^-]}{[\mathrm{CH}_3\mathrm{COOH}]}\approx \frac{x^2}{0.500-x} \approx \frac{x^2}{0.500} = 1.8 \times 10^{-5}\]Solving for \( x \), \( x = [\mathrm{H}^+] = \sqrt{0.500 \times 1.8 \times 10^{-5}} \approx 0.003 \), thus\[\mathrm{pH} = -\log(0.003) \approx 2.52.\]

Calculate pH after 10-fold dilution of Acetic Acid

When the acetic acid is diluted 10-fold, its concentration becomes \( 0.050 \mathrm{M} \).Repeat the pH calculation for this new concentration:\[K_a = \frac{x^2}{0.050} = 1.8 \times 10^{-5}\]Solving for \( x \), \([ ext{H}^+]= \sqrt{0.050 \times 1.8 \times 10^{-5}} \approx 0.001 \), thus\[\mathrm{pH} = -\log(0.001) = 3.00.\]

Compare Buffer and Acid Solution pH Changes

The buffer solution shows a pH change from 4.76 to 4.76 with dilution (no change), maintaining its pH due to resistance provided by the buffer system. The acetic acid solution shows a pH change from 2.52 to 3.00, indicating the sensitivity of simple acid solutions to dilution. This demonstrates that buffer solutions resist pH change better than simple acid solutions when diluted.

Key concepts

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is central to understanding buffer solutions, which are mixtures that resist changes in pH. It is given by:\[pH = ext{pKa} + \log \left( \frac{[A^-]}{[HA]} \right)\]Here,

• \([HA]\) represents the concentration of the weak acid, while
• \([A^-]\) stands for the concentration of the conjugate base.

This equation is particularly useful for calculating the pH of a buffer solution. It takes into account the acid dissociation constant (\( ext{pKa}\)) and the ratio of the concentrations of the conjugate base and acid. Thus, it highlights why buffer systems maintain their pH even when diluted. This resilient property is demonstrated when diluting acetic acid and sodium acetate solutions, where the pH largely remains unchanged.

pH Calculation

Calculating pH is an essential practice in chemistry, especially for understanding acidity and basicity. In general terms, pH is defined by the negative logarithm of the hydrogen ion concentration:\[pH = -\log [H^+]\]For buffer solutions, we often rely on the Henderson-Hasselbalch equation, as these solutions involve weak acids.

• For the acetic acid buffer solution, \( ext{pH} = 4.76\) was calculated using concentrations of \(0.500\,\text{M}\) for both acetic acid and sodium acetate.
• Upon dilution, the new concentrations were \(0.050\,\text{M}\), yet the pH remained constant at \( ext{pH} = 4.76\).

This calculation highlights the buffer's ability to resist pH changes, a vital property that differentiates it from unbuffered solutions.

Weak Acid Dissociation

Weak acid dissociation is the process wherein a weak acid partially ionizes in solution. This process is represented by an equilibrium:\[\mathrm{HA} \rightleftharpoons \mathrm{H}^+ + \mathrm{A}^- \]Here,

• \(HA\) is the weak acid, and
• \(A^-\) is the conjugate base formed.

The dissociation is characterized by the acid dissociation constant \(K_a\), which quantifies the strength of the acid. Using the equilibrium expression, the pH can be calculated for a simple acidic solution:\[K_a = \frac{[\mathrm{H}^+][\mathrm{A}^-]}{[\mathrm{HA}]}\]For acetic acid, the pH was calculated to be \(2.52\) at an initial concentration of \(0.500\,\text{M}\). This highlights how weak acids, unlike buffer solutions, change more with dilution, as shown by the shift to \(3.00\) upon tenfold dilution.

Acetic Acid Buffer

An acetic acid buffer solution is composed of acetic acid (\(\text{CH}_3\text{COOH}\)) and its conjugate base, acetate (\(\text{CH}_3\text{COO}^-\)). This buffer system provides a stable pH environment, crucial for various chemical and biological applications. The buffer works by maintaining a constant equilibrium between the acid and its conjugate base:

• The buffer can neutralize added acids or bases, keeping the pH stable.
• This stability results from the presence of both the weak acid and its conjugate base.

In the exercise, the acetic acid buffer showed remarkable resistance to pH changes, maintaining \(\text{pH} = 4.76\) even when diluted tenfold. This contrasts sharply with a simple acetic acid solution, where dilution significantly altered the pH. Thus, acetic acid buffers are essential in stabilizing pH in a variety of settings.