Question

Calculate the \(\mathrm{pH}\) at \(25^{\circ} \mathrm{C}\) of a solution containing \(0.10 \mathrm{M}\) sodium acetate and \(0.03 \mathrm{M}\) acetic acid. The apparent \(\mathrm{pK}\) for acetic acid at this ionic strength is \(4.57\).

Short answer

Using the Henderson-Hasselbalch equation, the pH of the buffer solution containing \(0.10\,\mathrm{M}\) sodium acetate and \(0.03\,\mathrm{M}\) acetic acid at \(25^{\circ} \mathrm{C}\) can be calculated as follows: \(\mathrm{pH} = 4.57 + \log_{10}\left(\frac{0.10\,\mathrm{M}}{0.03\,\mathrm{M}}\right) \approx 4.08\).

Step-by-step solution

Recall the Henderson-Hasselbalch equation

The Henderson-Hasselbalch equation helps us find the pH of a buffer solution. It is given by: \[ \mathrm{pH} = \mathrm{pK_a} + \log_{10}\left(\frac{[\mathrm{A}^-]}{[\mathrm{HA}]}\right) \] where \(\mathrm{pH}\) is the pH of the solution, \(\mathrm{pK_a}\) is the apparent pKa of the weak acid, \([\mathrm{A}^-]\) is the concentration of the weak base (sodium acetate), and \([\mathrm{HA}]\) is the concentration of the weak acid (acetic acid). 2. Plug the given values into the equation

Plug the given values into the equation

We are given the concentrations of sodium acetate and acetic acid, as well as the apparent pKa. We will plug these values into the Henderson-Hasselbalch equation: \[ \mathrm{pH} = 4.57 + \log_{10}\left(\frac{0.10\,\mathrm{M}}{0.03\,\mathrm{M}}\right) \] 3. Calculate the pH

Calculate the pH

Now, we can calculate the pH by performing the arithmetic operation: \[ \mathrm{pH} = 4.57 + \log_{10}\left(\frac{1}{3}\right) \approx 4.08 \] Thus, the pH of the solution at \(25^{\circ} \mathrm{C}\) containing \(0.10\,\mathrm{M}\) sodium acetate and \(0.03\,\mathrm{M}\) acetic acid is approximately \(4.08\).

Key concepts

Henderson-Hasselbalch equation

The Henderson-Hasselbalch equation is a valuable tool in chemistry for calculating the pH of a buffer solution. It is derived from the expression for the equilibrium constant of a weak acid dissociation. The equation is expressed as:\[\mathrm{pH} = \mathrm{pK_a} + \log_{10}\left(\frac{[\mathrm{A}^-]}{[\mathrm{HA}]}\right)\]Here,

• \( \mathrm{pH} \) is the measure of acidity in the solution.
• \( \mathrm{pK_a} \) represents the acid dissociation constant, showing the strength of the weak acid.
• \([\mathrm{A}^-]\) indicates the concentration of the conjugate base, such as sodium acetate in our exercise.
• \([\mathrm{HA}]\) denotes the concentration of the weak acid, like acetic acid.

By using this equation, you can easily determine how the pH changes with varying concentrations of the weak acid and its conjugate base. This equation is particularly useful because it relates pH directly to easily measurable quantities.

buffer solution

A buffer solution is one that maintains a nearly constant pH when small amounts of acid or base are added. This is important in many chemical and biological systems where pH stability is crucial. Buffer solutions typically consist of:

• A weak acid and its conjugate base (like acetic acid and sodium acetate).
• A weak base and its conjugate acid.

These components work together to resist changes in pH. When an acid is added, the weak base in the buffer neutralizes it, diminishing the pH change. Similarly, when a base is added, the weak acid neutralizes it.
The key to a buffer's functionality is the equilibrium between the weak acid and its conjugate base. This balance allows the buffer to adjust to the addition of acids or bases, thereby maintaining the pH within a desirable range.

weak acid and weak base equilibrium

The equilibrium between a weak acid and a weak base is fundamental to the function of buffer solutions. A weak acid only partially dissociates into its ions in solution, establishing an equilibrium:\[\mathrm{HA} \rightleftharpoons \mathrm{H}^+ + \mathrm{A}^-\]Here,

• \(\mathrm{HA}\) is the weak acid.
• \(\mathrm{H}^+\) is the hydrogen ion.
• \(\mathrm{A}^-\) is the conjugate base.

For weak bases, the equilibrium can be similarly described. In a buffer solution, the weak acid and its conjugate base are present in considerable amounts, allowing the solution to absorb added H\(^+\) or OH\(^-\) ions without a significant change in pH. This equilibrium ensures the solution is capable of minimizing pH fluctuations, thus acting effectively as a buffer.

pKa and pH relationship

The relationship between pKa and pH is central to understanding buffer systems. The pKa value indicates how easily an acid donates its proton, thus showing its acid strength. It is defined as:\[\mathrm{pKa} = -\log_{10}(K_a)\]Where \(K_a\) is the acid dissociation constant. A smaller pKa means a stronger acid.
In buffer solutions, the pKa value helps determine the buffer's effective pH range. Ideally, a buffer works best when its pH is close to the pKa of the weak acid it contains. This ensures the concentration ratio of the conjugate base to the acid remains close to 1, leading to an effective buffering capacity.
Thus, understanding how pKa influences pH enables better design and use of buffer systems in various scientific applications, from laboratory experiments to natural biological processes.