For the titration of \(50.00 \mathrm{~mL}\) of \(0.1000 \mathrm{M}\) ammonia with \(0.1000 \mathrm{M} \mathrm{HCl}\), calculate the \(\mathrm{pH}\) (a) before the addition of any HCl solution, (b) after \(20.00 \mathrm{~mL}\) of the acid has been added, (c) after half of the \(\mathrm{NH}_{3}\) has been neutralized, and (d) at the equivalence point.

Understanding pH is crucial in chemistry, as it quantifies the acidity or basicity of a solution. It's defined as the negative logarithm of the hydrogen ion concentration, or in mathematical terms, \(\text{pH} = -\log[H^+]\). In a titration involving a weak base like ammonia (NH3), pH calculation starts by determining the concentration of hydroxide ions (OH-) generated by NH3 in water. Because NH3 is a weak base, it doesn't dissociate completely, and using the base dissociation constant (Kb), we can find the equilibrium concentration of OH-. With the pOH then known as \(\text{pOH} = -\log[OH^-]\), we can find the pH using the relation \(\text{pH} + \text{pOH} = 14\). Throughout the titration process, as acid is added, the pH continuously changes, reflecting the dynamic balance between the acidic and basic species in solution.

Titration serves as a quantitative analytical method to determine concentration. When a weak base like ammonia is titrated with a strong acid such as hydrochloric acid (HCl), the reaction proceeds until the weak base is neutralized. During a weak base titration, the pH shifts from basic toward neutral as the strong acid reacts with and neutralizes the base. Analyzing such a titration involves stepwise calculations to ascertain pH at various stages by accounting for the stoichiometry of the neutralization reaction and the concentrations of species in the resultant solution. This methodical approach allows chemists to identify significant points such as half-neutralization and the equivalence point, which have distinct pH characteristics.

A buffer solution is a mixture that resists changes in pH when acids or bases are added. It typically consists of a weak acid and its conjugate base or a weak base and its conjugate acid. During the titration of NH3 with HCl, a buffer solution forms when approximately half of the NH3 has been neutralized, resulting in comparable amounts of NH3 and its conjugate acid, NH4+. This buffer system can absorb moderate amounts of additional acid or base without significantly altering its pH. The buffering capacity is at its highest when the concentration of the weak base and its conjugate acid are equal because the buffer system can neutralize either added acid or base effectively.

The equivalence point in a titration is the moment when the moles of titrant added are stoichiometrically equivalent to the moles of substance present in the sample. For the titration of NH3 with HCl, it occurs when all the NH3 present in the solution has been converted to its conjugate acid, NH4+, signifying the end of the neutralization reaction. Determining the equivalence point is crucial as it is indicative of the completion of the reaction. At this juncture, the pH will not be neutral (pH 7) if the titrant and analyte are a weak base and strong acid, respectively; instead, it will lean towards the acidic range due to the formation of the weak acid (NH4+) upon complete neutralization.

The Henderson-Hasselbalch equation is a mathematical representation of the relationship between pH, pKa, and the ratio of the concentrations of the conjugate base to the weak acid in a buffer solution. Expressed as \(\text{pH} = \text{pKa} + \log(\frac{[A^-]}{[HA]})\), it simplifies the process of estimating the pH of buffer solutions. The equation hinges on several assumptions, such as the dissociation of the weak acid is negligible and that the concentration of the acid and its conjugate base at the start of the reaction is equivalent to their concentration in equilibrium. This equation is particularly handy during the halfway point to the equivalence point in a weak base titration, where the concentrations of the weak base (NH3) and its conjugate acid (NH4+) are equal, simplifying the log term to zero and the pH can be directly determined as equal to the pKa of the conjugate acid.