Calculate the \(\mathrm{pH}\) of \(1.00 \mathrm{~L}\) of the buffer \(1.00 \mathrm{M}\) \(\mathrm{CH}_{3} \mathrm{COONa} / 1.00 \mathrm{M} \mathrm{CH}_{3} \mathrm{COOH}\) before and after the addition of (a) \(0.080 \mathrm{~mol} \mathrm{NaOH},\) (b) \(0.12 \mathrm{~mol} \mathrm{HCl}\). (Assume that there is no change in volume.)

The Henderson-Hasselbalch equation provides a straightforward method for calculating the pH of buffer solutions. It's a transformed version of the acid dissociation constant (Ka) and connects the pH directly to the concentrations of an acid and its conjugate base in solution. The formula is given by:
\[ pH = pKa + \log \left( \frac{[\mathrm{A}^-]}{[\mathrm{HA}]} \right)\]
where \(pKa\) is the acid's dissociation constant, \([A^-]\) is the concentration of the conjugate base, and \([HA]\) is the concentration of the acid.
It's most effective with buffer systems, where the acid and base are present in similar concentrations, and the solution can resist changes in pH. This formula was utilized to determine the starting pH of the buffer solution in the given exercise, revealing the buffer's initial acidity or basicity before any outside substance was added.

Acid-base buffer systems are pivotal for maintaining a nearly constant pH in a variety of chemical and biological contexts. A buffer is composed of a weak acid and its conjugate base or a weak base and its conjugate acid. This dual substance setup allows buffers to neutralize added acids or bases.
Buffers function according to Le Chatelier's principle—when a system in equilibrium is disturbed, it adjusts to minimize the disturbance. When an acid (like HCl) is added to a buffer, it reacts with the conjugate base in the buffer solution, thus minimizing any increase in hydrogen ion concentration and preventing a significant change in pH. Conversely, when a base (like NaOH) is added to a buffer, it's neutralized by the weak acid in the buffer, again preventing a significant change in pH.
In the exercise, the buffer was able to maintain its pH within a narrow range even after the addition of NaOH or HCl, exemplifying the functionality of acid-base buffer systems.

Determining the pH of a solution is crucial in identifying its acidity or basicity, which is essential for chemical reactions, biological processes, and environmental science. The pH scale ranges from 0 to 14, where 7 is neutral, values less than 7 are acidic, and values greater than 7 are basic.
The pH is most commonly determined using the concentration of hydrogen ions (H+) or hydronium ions (H3O+) in a solution, with the formula:\[ pH = -\log[H^+] \]
The exercise demonstrates pH determination first by using the Henderson-Hasselbalch equation—a practical tool for buffers—and then by observing how the pH changes when the buffer reacts with additional acids or bases, calculated by updating the ratio of conjugate base to acid.

Buffer capacity is a measure of a buffer solution's ability to resist changes in pH upon the addition of an acid or base. It depends on both the concentrations of the acid and conjugate base in the buffer and the absolute concentration of the buffer components. In essence, the higher the concentration of the acid and conjugate base, the greater the buffer capacity, and the more resistant the solution is to changes in pH.
The reactions that occur in a buffer upon the addition of acids or bases are key to its buffering action. For example, the addition of a strong acid like HCl to a buffer results in the formation of a weak acid from its conjugate base, with a negligible change in pH. Likewise, adding a strong base like NaOH leads to the consumption of the weak acid in the buffer and the formation of its conjugate base with again a minor impact on the pH. The exercise provides a clear example where NaOH and HCl are added to a buffer system and the pH remains relatively stable thanks to these buffering reactions.