Problem 46 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

When glycinium hydrochloride \(\left(\mathrm{NH}_{2} \mathrm{CH}_{2}\right.\) COOH.HCl) is titrated against \(\mathrm{NaOH}\), \(\mathrm{pH}\) at the first half equivalence point is \(2.40\) and the \(\mathrm{pH}\) at second half equivalence point is \(9.60\). The \(\mathrm{pH}\) at first equivalence point is (a) \(2.40\) (b) \(9.60\) (c) \(6.00\) (d) \(7.20\)

Step-by-Step Solution

Answer

The pH at the first equivalence point is (c) 6.00, calculated by averaging the pH values at the two half-equivalence points.

Step 1: Identify the Type of Titration

Glycinium hydrochloride is a diprotic acid (has two ionizable protons) being titrated with NaOH, a strong base. The pH changes recorded correspond to the two half-equivalence points of the titration.

Step 2: Use the Half-Equivalence Point pH

At half-equivalence point, the concentration of the weak acid (HA) equals the concentration of its conjugate base (A-), meaning the solution consists of a buffer. For such a solution, Henderson-Hasselbalch equation applies: \(pH = pKa + \log\frac{[A^-]}{[HA]}\). At the half-equivalence point, \(\frac{[A^-]}{[HA]} = 1\), so \(pH = pKa\).

Step 3: Determine the pH at First Equivalence Point

At the first equivalence point, all the first ionizable protons have reacted with the base. The pH at this point is determined by the dissociation of the conjugate base formed from the first ionizable proton. Since we have a diprotic acid, the first equivalence point is somewhere between the two pKa values. To find it, we take the average of the two pKa values which are equal to the pH at the half-equivalence points given.

Step 4: Calculate the Average pH

The average pH at first equivalence point is the average of the pH values at the two half-equivalence points: \(\text{Average pH} = \frac{2.40 + 9.60}{2} = 6.00\).

Key Concepts

Diprotic Acid Titration

Understanding diprotic acid titration is crucial for chemistry students as it deals with acids that have two ionizable hydrogen atoms, like glycinium hydrochloride. In a titration process involving a diprotic acid and a strong base such as NaOH, there are two points of interest: the first and second equivalence points. The first equivalence point occurs when one mole of base has reacted with one mole of the first ionizable hydrogen of the acid. The second equivalence point is reached when the base has neutralized both of the acid's ionizable hydrogens.

During titration, there are dramatic pH changes near these equivalence points due to the acid losing its hydrogen ions. Recognizing these points is essential when determining the concentrations and properties of the solutions involved in the titration. It’s also important to note the difference between half-equivalence points, where half of a specific hydrogen ion has reacted, and full equivalence points where all such ions have reacted.

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is a key tool in understanding buffer solutions and pH calculation in diprotic acid titrations. It is given as:
\[ pH = pKa + \text{log}(\frac{[A^-]}{[HA]}) \]
This equation relates the pH of a solution to the pKa (acid dissociation constant) and the ratio of the concentration of the deprotonated form ([A-]) to the protonated form ([HA]) of the acid.

At the half-equivalence points of a diprotic acid titration, the concentrations of [HA] and [A-] are equal, making the log term equal to zero, thus simplifying the equation to \( pH = pKa \). This is a critical concept as it explains why the pH at half-equivalence points is directly equal to the pKa values of the acid, making these points ideal for determining the pKa values in a titration experiment.

Equivalence Point pH Calculation

Calculating the pH at the equivalence point of a titration involves understanding the chemistry occurring in the solution at that specific moment. For a diprotic acid, the first equivalence point occurs when all of the first set of ionizable protons have been neutralized by the base. At this stage, the pH is not simply the average of the initial pH and the pKa of the second ionizable proton, but rather depends on the properties of the conjugate base formed from the first proton's dissociation.

In the provided exercise, the pH at the first equivalence point is calculated by averaging the pH at the two half-equivalence points, as these correspond to the pKa values of the acid. By taking the mean of these pH values, which are 2.40 and 9.60 respectively, the pH at the first equivalence point is determined to be 6.00. This is a pivotal concept because it dictates the exact point at which the solution transitions from being acidic to neutral in a titration involving a diprotic acid.

Buffer Solutions

Buffer solutions play a significant role in titration, especially in the context of half-equivalence points. A buffer solution is one that resists changes in pH upon the addition of small amounts of either an acid or a base. They are typically composed of a weak acid and its conjugate base or a weak base and its conjugate acid.

In the context of diprotic acid titrations, the half-equivalence point creates an ideal buffer situation: the amounts of weak acid ([HA]) and its conjugate base ([A-]) are equal, leading to maximum buffering capacity. Understanding buffer solutions is vital as they maintain pH at a nearly constant level even after the addition of more titrant, until the solution reaches the equivalence point where the buffer's capacity is overrun by the addition of the strong base.