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The Henderson-Hasselbalch equation is an important tool in chemistry that helps us understand the relationship between the pH of a buffer solution and its components. This formula is given by:\[ \text{pH} = \text{pKa} + \log \frac{[\text{conjugate base}]}{[\text{acid}]} \]This equation is particularly useful for calculating the pH of buffer solutions, which are solutions that resist changes in pH upon the addition of small amounts of acid or base. By knowing the ratio of the concentrations of the conjugate base and the acid in a buffer solution, you can predict how the pH will change.
In our exercise, the conjugate base is \(\text{HPO}_4^{2-}\) and the acid is \(\text{H}_2\text{PO}_4^{-}\). By plugging the concentrations of these into the equation, we can determine the pH of the buffer at equilibrium.

The \(\text{pKa}\) value is a measure of the strength of an acid. It represents the negative logarithm of the acid dissociation constant \(K_a\), which tells us how well an acid can donate a proton (H\(^+\)) in a solution. The lower the \(\text{pKa}\), the stronger the acid, because it dissociates more to form \(H\(^+\)\) ions.
For phosphoric acid, the \(\text{pKa}\) value is a key component because each dissociation step has its own \(\text{pKa}\). When setting up a buffer involving \(\text{H}_2\text{PO}_4^{-}\), the relevant \(\text{pKa}\) is for its second dissociation, which is 7.21. Knowing this helps us in plugging the correct value into the Henderson-Hasselbalch equation to calculate the pH of the buffer.
Calculating the \(\text{pKa}\) accurately is crucial for designing buffer solutions with precise pH levels.

A conjugate acid-base pair consists of two species that transform into each other by the gain or loss of a proton. In any given buffer solution, you will typically find a weak acid paired with its conjugate base, or a weak base with its conjugate acid.
In this particular problem, \(\text{H}_2\text{PO}_4^{-}\) is the acid, and \(\text{HPO}_4^{2-}\) is its conjugate base. When the acid donates a proton, it becomes its conjugate base. Conversely, this conjugate base can accept a proton to become the conjugate acid again. This equilibrium between donating and accepting protons is what gives buffer solutions their ability to withstand pH changes.
Understanding the concept of conjugate acid-base pairs is essential for seeing how buffers can maintain a relatively constant pH even when external conditions attempt to disrupt them.