What are the \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) and the \(\mathrm{pH}\) of a buffer that consists of \(0.55 M \mathrm{HNO}_{2}\) and \(0.75 \mathrm{M} \mathrm{KNO}_{2}\left(\mathrm{~K}_{\mathrm{a}}\right.\) of \(\left.\mathrm{HNO}_{2}=7.1 \times 10^{-4}\right) ?\)

The Henderson-Hasselbalch equation is essential for understanding buffer solutions. This equation links together the pH, pKa, and the concentrations of an acid and its conjugate base. The formula looks like this: \[\text{pH} = \text{p}K_a + \text{log}\frac{[\text{A}^-]}{[\text{HA}]}\] Here, \( [\text{A}^-] \) is the concentration of the conjugate base (e.g., \(\text{KNO}_2\)) and \( [\text{HA}] \) is the concentration of the acid (e.g., \( \text{HNO}_2 \)). This equation allows us to predict the pH of a solution when we know the pKa and the ratio of the concentrations of the base and the acid.

• The term \( \text{p}K_a \) is derived from the acid dissociation constant \( K_a \).

Understanding how to use this equation is crucial for accurately calculating the pH in acid-base chemistry.

Calculating pKa from Ka is a fundamental step in using the Henderson-Hasselbalch equation. The pKa is found using the following equation: \[\text{p}K_a = -\text{log}(\text{K}_a)\] Given an acid dissociation constant \(\text{K}_a\), you can find its pKa. For our exercise, the \( \text{K}_a \) for \( \text{HNO}_2 \) is \( 7.1 \times 10^{-4} \). So, \[\text{p}K_a = -\text{log}(7.1 \times 10^{-4}) \approx 3.15\]

• This means \( \text{HNO}_2 \) has a pKa of approximately 3.15.

Knowing the pKa value is crucial because it simplifies the calculation of pH when using the Henderson-Hasselbalch equation.

Acid-base equilibrium is at the heart of understanding how buffers work. A buffer solution consists of a weak acid and its conjugate base or a weak base and its conjugate acid. These components work together to resist changes in pH when small amounts of acid or base are added. Here’s what you need to know:

• The weak acid (e.g., \( \text{HNO}_2 \)) partially dissociates in water, releasing \( \text{H}^+ \) ions.
• The conjugate base (e.g., \( \text{KNO}_2 \)) can accept \( \text{H}^+ \) ions, neutralizing added acids.

The equilibrium between these forms is what allows the buffer to maintain a stable pH within a certain range. This balance is directly described by the Henderson-Hasselbalch equation.

The pH of a solution indicates its acidity or alkalinity. Calculating the pH of a buffer involves a few straightforward steps:

• First, use the Henderson-Hasselbalch equation: \[\text{pH} = 3.15 + \text{log}\frac{0.75 \text{ M}}{0.55 \text{ M}} \approx 3.15 + 0.13 \approx 3.28\]
• This gives the pH of the buffer solution as approximately 3.28.
• Next, calculate the hydronium ion concentration \( \text{[H}_3\text{O}^+] \) from the pH: \[\text{[H}_3\text{O}^+] = 10^{-\text{pH}} \approx 10^{-3.28} \approx 5.2 \times 10^{-4} \text{ M}\]

Understanding these calculations is key to working with buffers. It ensures you can predict how changes in the system will affect the overall pH.