Problem 21 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

\(2.5 \mathrm{~mL}\) of \((2 / 5) \mathrm{M}\) weak monoacidic base \(\left(K_{b}=1 \times 10^{-12}\right.\) at \(\left.25^{\circ}\right)\) is titrated \((2 / 15) \mathrm{M} \mathrm{HCl}\) in water at \(25^{\circ} \mathrm{C}\). The concentration of \(\mathrm{H}^{+}\)at equivalence point is \(\left(K_{w}=1 \times 10^{-14}\right.\) at \(\left.25^{\circ} \mathrm{C}\right)\) (a) \(3.7 \times 10^{-14} \mathrm{M}\) (b) \(3.2 \times 10^{-7} \mathrm{M}\) (c) \(\quad 3.2 \times 10^{-2} \mathrm{M}\) (d) \(2.7 \times 10^{-2} \mathrm{M}\)

Step-by-Step Solution

Answer

The concentration of \(\text{H}^+\) at the equivalence point is \(3.2 \times 10^{-2} \text{ M}\), corresponding to option (c).

Step 1: Determine the number of moles of base

First, calculate the number of moles of the base solution used in the titration. The formula for the number of moles is: \[\text{Moles of base} = \text{Volume} \times \text{Molarity} = 2.5\, \text{mL} \times \frac{2}{5}\, \text{M}\]Convert the volume from mL to L: \[2.5\, \text{mL} = 2.5 \times 10^{-3}\, \text{L}\]Therefore:\[\text{Moles of base} = 2.5 \times 10^{-3}\, \text{L} \times \frac{2}{5}\, \text{M} = 1 \times 10^{-3}\, \text{moles}\]

Step 2: Calculate the volume of HCl required to reach equivalence

Calculate the volume of \(2/15\, \text{M HCl}\) needed to reach the equivalence point, where moles of base equal moles of HCl. The relationship is given by:\[\text{Moles of HCl} = \text{Volume} \times \text{Molarity} = 1 \times 10^{-3} \text{moles}\]So, \[\text{Volume of HCl} = \frac{1 \times 10^{-3}}{\frac{2}{15}} = 7.5 \times 10^{-3}\, \text{L or } 7.5\, \text{mL}\]

Step 3: Determine the concentration of \([\text{H}^+]\) at equivalence point

At the equivalence point, the solution contains the salt of the weak base and acts as a weak acid.Using the hydrolysis equation of the weak base: \(K_w = [\text{OH}^-][\text{H}^+]\) Given that \(K_b = 1 \times 10^{-12}\) and \(K_w = 1 \times 10^{-14}\), we find \(K_a\) using the equation \(K_a = \frac{K_w}{K_b}\). Thus, \[K_a = \frac{1 \times 10^{-14}}{1 \times 10^{-12}} = 1 \times 10^{-2}\]

Step 4: Solve for H+ concentration

In the hydrolyzed solution at equivalence, \[\text{H}^+ \approx \sqrt{C K_a}\], where \(C\) is the concentration of the conjugate acid:\[C = \frac{\text{moles of base}}{\text{total volume}} = \frac{1 \times 10^{-3}\, \text{moles}}{10\, \text{mL}} = 0.1\, \text{M} \]\[[\text{H}^+] \approx \sqrt{0.1 \times 1 \times 10^{-2}} = \sqrt{1 \times 10^{-3}} = 3.2 \times 10^{-2}\, \text{M}\]

Step 5: Compare with given options

The concentration of \(\text{H}^+\) at the equivalence point is calculated to be \(3.2 \times 10^{-2} \text{ M}\), which corresponds to option (c).

Key Concepts

Acid-Base Titration

An acid-base titration is a laboratory technique used to determine the concentration of a certain acid or base in a solution. It involves gradually adding a solution of known concentration, called the titrant, to a solution of unknown concentration until a chemical reaction reaches its completion point. The point at which the amount of titrant added is chemically equivalent to the quantity of the substance determined in the reaction is known as the equivalence point.

• The titrant is usually placed in a burette and added to the unknown solution in a flask.
• An indicator, often a pH probe or a specific dye, is used to signal when the equivalence point has been reached.
• This process helps to identify the concentration of the unknown solution by using stoichiometry.

This method is quite useful for solutions involving both strong and weak acids or bases. Titrations involving weak acids or bases require special consideration to understand how their dissociation and buffering capacity impact the result, especially when determining the pH at different points in the titration.

Weak Base

A weak base is a chemical base that does not ionize fully in an aqueous solution. Unlike strong bases, which dissociate completely, weak bases only partially dissociate to form hydroxide ions and the conjugate acid.

• The extent of dissociation is described by the base dissociation constant, denoted by \(K_b\).
• Weak bases often have a \(K_b\) value much less than 1, indicating that at equilibrium, most of the base remains un-dissociated.
• This partial dissociation influences the pH of the solution in which the weak base is dissolved.

When involved in titration, understanding the behavior of a weak base is crucial for accurate pH calculation at various titration stages. The hydrolysis of the salt formed at the equivalence point can also impact the resulting pH.

Equivalence Point

The equivalence point in a titration is reached when the number of moles of titrant added equals the number of moles of substance originally present in the sample. At this point, the reaction is considered complete. However, it's essential to note that the pH at equivalence differs depending on the nature of the acid or base involved.

• In strong acid and strong base titrations, the pH at the equivalence point is typically neutral (pH 7).
• For a weak acid and strong base, the equivalence point is above pH 7 due to the formation of a conjugate base.
• Conversely, for a weak base and strong acid, the pH will be below 7 due to the formation of a conjugate acid.

The equivalence point is vital in titration analysis, as it allows for precise calculations of unknown concentrations. It also provides insight into the chemical nature of the solution post-reaction.

pH Calculation

Calculating the pH in titration, particularly at the equivalence point, requires an understanding of the chemistry of the substances involved—whether strong or weak acids and bases. pH is a scale used to specify the acidity or basicity of an aqueous solution.

• Before the equivalence point, the solution's pH is mainly determined by the excess reactant.
• At equivalence, the solution is solely composed of a salt, whose properties must be considered.
• The \(K_a\) of the conjugate acid or \(K_b\) of the conjugate base (derived from the original weak base or acid) is used to determine the resulting pH.

For weak acids or bases, their conjugate’s ability to hydrolyze will adjust the pH. Utilizing the formula \[[\text{H}^+] = \sqrt{C \times K_a}\], where \(C\) is the concentration of the resulting equivalence solution, allows the calculation of pH based on the concentration of hydrogen ions.