Question

A diprotic acid, \(\mathrm{H}_{2} \mathrm{~A}\), has the following ionization constants: \(K_{\mathrm{a}_{1}}=1.1 \times 10^{-3}\) and \(K_{\mathrm{a}_{2}}=2.5 \times 10^{-6}\) To make up a buffer solution of \(\mathrm{pH} 5.80,\) which combination would you choose: \(\mathrm{NaHA} / \mathrm{H}_{2} \mathrm{~A}\) or Na A/NaHA?

Short answer

Use Na A and NaHA combination for a pH 5.80 buffer.

Step-by-step solution

Analyze the Ionization Constants

First, recognize that the diprotic acid can release two protons, thus having two ionization constants: \(K_{\mathrm{a}_1}\) and \(K_{\mathrm{a}_2}\). Given values are \(K_{\mathrm{a}_1}=1.1 \times 10^{-3}\) representing the ionization of \(H_2A\) to \(HA^-\), and \(K_{\mathrm{a}_2}=2.5 \times 10^{-6}\) representing the ionization of \(HA^-\) to \(A^{2-}\).

Recall the Henderson-Hasselbalch Equation

The pH of a buffer solution can be calculated using the Henderson-Hasselbalch equation: \( \text{pH} = \text{pKa} + \log \left( \frac{[\text{base}]}{[\text{acid}]} \right) \). For diprotic acids, choose the step and corresponding \( \text{pK}_a \) that makes the buffer pH closest to your target pH.

Determine pKa Values

Calculate \(\text{pK}_{\mathrm{a}_1}\) and \(\text{pK}_{\mathrm{a}_2}\): \(\text{pK}_{\mathrm{a}_1} = -\log(1.1 \times 10^{-3}) \approx 2.96 \) and \(\text{pK}_{\mathrm{a}_2} = -\log(2.5 \times 10^{-6}) \approx 5.60\).

Compare Desired pH with pKa Values

The desired pH of 5.80 is closer to \(\text{pK}_{\mathrm{a}_2} = 5.60\) compared to \(\text{pK}_{\mathrm{a}_1} = 2.96\). Therefore, the buffer system involving \(HA^-\) and \(A^{2-}\) is appropriate.

Select the Correct Buffer Components

To form a buffer with pH 5.80, use the second ionization equilibrium \(\text{pK}_{\mathrm{a}_2}\). Select Na A (providing \(A^{2-}\)) and NaHA (providing \(HA^-\)).

Key concepts

Henderson-Hasselbalch equation

The Henderson-Hasselbalch equation is a crucial formula used to calculate the pH of buffer solutions. It is derived from the expression for the equilibrium constant of weak acids and bases. The formula is expressed as:
\[ \text{pH} = \text{pKa} + \log \left( \frac{[\text{base}]}{[\text{acid}]} \right) \]
This equation relates the pH of a solution to the pKa of a weak acid and the ratio of the concentrations of its conjugate base and the acid. In simpler terms, it helps us understand how much the pH will change when we adjust the amounts of acid and base in a solution.

• pH: The measure of acidity or basicity of a solution.
• pKa: The negative logarithm of the acid dissociation constant, indicating the strength of the acid.
• [base] and [acid]: Concentrations of the base and the acid in the solution, respectively.

For a diprotic acid, which can release two protons, there are two possible pH values depending on whether the first or second ionization is more relevant to the pH being targeted. Using this equation helps you efficiently create buffer solutions with a desired pH.

Diprotic acid

A diprotic acid is a type of acid that can donate two hydrogen ions (protons) per molecule during dissociation. Each stage of dissociation has a distinct acid dissociation constant, known as \( K_{\text{a}_1} \) and \( K_{\text{a}_2} \).

• First Ionization: The first dissociation leads to the formation of a hydrogen ion and a singly charged anion (e.g., from \( \text{H}_2\text{A} \) to \( \text{HA}^- \)). This process is characterized by the constant \( K_{\text{a}_1} \).
• Second Ionization: The second dissociation further splits the singly charged anion to release another hydrogen ion, forming a doubly charged anion (e.g., from \( \text{HA}^- \) to \( \text{A}^{2-} \)). This stage has the constant \( K_{\text{a}_2} \).

The behavior of a diprotic acid in solution is important for understanding how to create buffer solutions. When choosing a diprotic acid for a specific pH, you need to evaluate which of the two ionization steps matches the pH most closely.

pKa values

The \( \text{pKa} \) value is an important parameter that helps determine an acid's strength. It is the negative logarithm of the acid dissociation constant \( K_{\text{a}} \), calculated as \( \text{pKa} = -\log(K_{\text{a}}) \). A lower \( \text{pKa} \) indicates a stronger acid, which more readily donates protons.
For diprotic acids, there are two \( \text{pKa} \) values corresponding to each ionization step:

• \( \text{pKa}_1 \): Calculated from \( K_{\text{a}_1} \), this value corresponds to the acid's strength in its first ionization process, where the acid loses its first proton.
• \( \text{pKa}_2 \): Derived from \( K_{\text{a}_2} \), it describes the acid's strength in the second ionization, shedding the second proton.

When preparing buffer solutions, the \( \text{pKa} \) value closest to the desired pH suggests which stage of ionization and which components should be used. For example, in the given problem, the desired pH of 5.80 is closer to \( \text{pKa}_2 = 5.60 \), implying the need to use components from the second ionization step.