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 \(\mathrm{Na}_{2} \mathrm{~A} / \mathrm{NaHA} ?\)

Short answer

To make a buffer solution of pH 5.80, the combination to choose would be \(\mathrm{Na}_{2}\mathrm{A} / \mathrm{NaHA}\).

Step-by-step solution

Identify the acid and base pairs

For a diprotic acid, the first ionization step is \(\mathrm{H}_{2}\mathrm{A} \rightarrow \mathrm{HA}^- + \mathrm{H}^+\), forming the pair \(\mathrm{H}_{2}\mathrm{A} / \mathrm{HA}^- \). And the second step is \(\mathrm{HA}^- \rightarrow \mathrm{A}^{2-} + \mathrm{H}^+\), forming the pair \(\mathrm{HA}^- / \mathrm{A}^{2-}\). Therefore, \(\mathrm{NaHA} / \mathrm{H}_{2} \mathrm{A}\) gives us the pair for the first dissociation, and \(\mathrm{Na}_{2} \mathrm{A} / \mathrm{NaHA}\) for the second.

Identify appropriate buffer range

Each ${K}_a$ relates to a particular ionization step. At ${K}_{a1}$, the buffer pair would create a pH range within one pH unit of ${pK}_{a1}$. Same for ${K}_{a2}$, within one pH unit of ${pK}_{a2}$. ${pK}_{a} = -log{K}_{a}$, so we calculate ${pK}_{a1}$ and ${pK}_{a2}$ to see in which buffer range the pH 5.80 falls.

Calculate $pK_{a1}$ and $pK_{a2}$

${pK}_{a1} = -log(1.1 \times 10^{-3}) = 3.00$ and ${pK}_{a2} = -log(2.5 \times 10^{-6}) = 5.60$

Compare $pK_{a1}$ and $pK_{a2}$ to given pH

Given pH is 5.80. Comparing this with ${pK}_{a1} = 3.00$ and ${pK}_{a2} = 5.60$, it is clear that the pH 5.80 is within one pH unit of ${pK}_{a2}$ (between 4.60 and 6.60), which corresponds to the second ionization, so the buffer pair is \(\mathrm{Na}_{2}\mathrm{A} / \mathrm{NaHA}\)

Key concepts

Understanding Diprotic Acid

Diprotic acids are fascinating compounds that have the ability to donate two protons (hydrogen ions, H⁺) per molecule during the ionization process. They are also referred to as polyprotic acids due to their multiple protons.
For instance, a common example of a diprotic acid is sulfuric acid ( H₂SO₄ ), which ionizes in two steps. The first step involves the loss of one proton, and the second step involves the loss of another proton.
The diprotic acid considered in this exercise, H₂A , follows this pattern:

• First ionization: H₂A → HA⁻ + H⁺
• Second ionization: HA⁻ → A²⁻ + H⁺

Each step has its own ionization constant and impacts the acid's behavior in solutions, particularly when forming buffer solutions. Understanding diprotic acids requires grasping how these ionization steps work independently and together under various pH conditions.

Diving into Ionization Constants

Ionization constants are a crucial aspect of understanding how acids and bases react in solution. These constants, often represented as K_a , are a quantitative measure of the strength of an acid in solution.
For diprotic acids, we have two constants:

• The first ionization constant, K_{a1} , indicates the strength of the first proton donation in the reaction H₂A → HA⁻ + H⁺ .
• The second ionization constant, K_{a2} , relates to the second proton donation in the reaction HA⁻ → A²⁻ + H⁺ .

These constants are crucial because they help predict the pH of the solution and guide decisions regarding buffer solutions.
A higher K_a value means a stronger acid, capable of donating its protons more readily. By calculating these constants, students can understand the predicted behavior of an acid under different conditions, which is fundamental when preparing buffer solutions.

Deciphering pKa Values

The term pK_a is derived from the ionization constant K_a , and it provides a more intuitive way to express acid strength using a logarithmic scale. The formula used is ext{p}K_a = - ext{log}(K_a) .
When dealing with a diprotic acid, each ionization constant ( K_{a1} and K_{a2} ) has a corresponding pK_a a dneesa lue ( pK_{a1} and pK_{a2} ).
This exercise highlights why it's useful: comparing a given pH to pK_a values helps determine which buffer pair is appropriate for maintaining or achieving a specific pH.
For instance, if the desired pH is closer to pK_{a2} , then the second ionization step will dominate, guiding the selection of components for a buffer. This conceptual understanding is essential as it informs how students can manipulate pH in practical applications, like creating stable buffers for chemical reactions or biological systems.