A \(0.054 M \mathrm{HNO}_{2}\) solution is titrated with a \(\mathrm{KOH}\) solution. What is \(\left[\mathrm{H}^{+}\right]\) at half way to the equivalence point?

Understanding the Henderson-Hasselbalch equation is essential when studying acid-base chemistry, especially during titrations. This equation is a simplified version of the acid dissociation constant expression and offers a direct relationship between the pH of a solution and the pKa (acid dissociation constant) of the acid in question. It is written as:
\[ pH = pKa + \log \left(\frac{[A^-]}{[HA]}\right) \]
The equation indicates that if we know the ratio of the concentration of the conjugate base, \([A^-]\), to that of the acid, \([HA]\), we can determine the pH of the solution. This becomes particularly handy during titrations. It's important to note that at the half-equivalence point of a titration, the concentrations of the acid and its conjugate base are equal, simplifying the equation to \(pH = pKa\). This is because the log of 1 (the ratio at half-equivalence point) is zero.

The half-equivalence point in a titration is a crucial milestone where exactly half of the acid has been neutralized by the base. At this juncture, the amount of acid, \([HA]\), present in the solution is equal to the amount of its conjugate base, \([A^-]\). This equality provides a unique opportunity to determine the inherent strength of the weak acid being titrated without extra calculations.
For a weak acid titration, the pH at the half-equivalence point is equal to the pKa of the weak acid. Therefore, not only does this point give insight into the acid's dissociation but also serves as a key reference for buffer capacity, which is at its maximum at the half-equivalence point because the solution contains equal amounts of the acid and its conjugate base, able to neutralize either added acids or bases.

The pKa is a numerical scale used to express the strength of an acid. To calculate pKa, one must first understand Ka, the acid dissociation constant, which measures the extent of dissociation of an acid in solution. The pKa is simply the negative logarithm of the Ka value and is calculated as follows:
\[ pKa = -\log(Ka) \]
In the context of titration, once the Ka of the weak acid is known, calculating its pKa is straightforward. Using the values, as in our example with nitrous acid (HNO2), where \(Ka = 4.5 \times 10^{-4}\), you would compute the pKa like this:
\[ pKa = -\log(4.5 \times 10^{-4}) \]
Which gives you a pKa of 3.35, indicating the acid strength of HNO2 at 25°C. Knowing the pKa facilitates various calculations in acid-base equilibrium, including those at the half-equivalence point of a titration.

The concentration of hydrogen ions, denoted as \([H^+]\), is a direct measure of a solution's acidity. To calculate the \([H^+]\) concentration from pH, we use the following logarithmic relationship:
\[ [H^+] = 10^{-pH} \]
During a titration, once you have determined the pH—as it's done at the half-equivalence point where pH equals pKa—we can easily find the concentration of hydrogen ions. This step is critical in understanding the acid's behavior and the titration progress. In our weak acid titration example, when the pH is 3.35, we calculate the hydrogen ion concentration by inverting the pH value:
\[ [H^+] = 10^{-3.35} \]
Thus, the concentration of hydrogen ions in the solution would be \(4.47 \times 10^{-4} M\). This value is indispensable for chemists to predict how the acid and base will interact as the titration approaches the equivalence point.