Problem 67 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

The initial rate of hydrolysis of methyl acetate (1M) by a weak acid \((\mathrm{H} A, 1 \mathrm{M})\) is \(1 / 100^{\text {th }}\) of that of a strong acid \((\mathrm{H} X, 1 \mathrm{M})\), at \(25^{\circ} \mathrm{C}\). The \(K_{a}\) of \(\mathrm{H} A\) is (a) \(1 \times 10^{-4}\) (b) \(1 \times 10^{-5}\) (c) \(1 \times 10^{-6}\) (d) \(1 \times 10^{-3}\)

Step-by-Step Solution

Answer

The correct answer is (a) \(1 \times 10^{-4}\).

Step 1: Understand the Problem

We are given the initial rate of hydrolysis of methyl acetate by a weak acid (HA, 1M) is \( \frac{1}{100} \) of that by a strong acid (HX, 1M) at \( 25^{\circ} C \). We need to find the dissociation constant \( K_a \) of the weak acid HA.

Step 2: Write the Rate Expressions

The rate of reaction is dependent on the concentration of hydrogen ions (\( H^+ \)). For a strong acid like HX, which fully dissociates, the concentration of \( H^+ \) is 1M. For the weak acid HA, which partially dissociates, the concentration of \( H^+ \) is \( \sqrt{K_a} \).

Step 3: Use the Given Rate Comparison

The rate of reaction with the weak acid is \( \frac{1}{100} \) of that with the strong acid. Thus, the concentration of \( H^+ \) for the weak acid is \( \frac{1}{100} \) of 1M (the strength of the strong acid), so we have \( \sqrt{K_a} = \frac{1}{100} \).

Step 4: Solve for Ka

Square both sides of the equation to solve for \( K_a \):\[K_a = \left(\frac{1}{100}\right)^2 = \frac{1}{10000} = 10^{-4}\]

Step 5: Make a Choice

According to the options provided, \( K_a = 1 \times 10^{-4} \), which corresponds to option (a).

Key Concepts

Hydrolysis of Esters

Hydrolysis of esters is an important chemical reaction where esters react with water to form alcohols and acids. Think of esters as being like seeds that require water to "grow" into different molecules. In acid hydrolysis, the presence of an acid (like hydrochloric acid or sulfuric acid) accelerates the reaction.

Here’s how it generally looks:

• The ester (\[\text{RCOOR'}\]) reacts with water to form an acid (\[\text{RCOOH}\]) and an alcohol (\[\text{R'OH}\]).
• This reaction is reversible, meaning it can go forward and backward.
• Ester hydrolysis can be catalyzed by either acids or bases, but in our scenario, we’re focusing on acid-catalyzed hydrolysis.

Understanding how different acids influence the rate of ester hydrolysis is crucial. Strong acids completely dissociate in water, providing many hydrogen ions (\[H^+\]), which speed up the reaction. Weak acids, however, only partially dissociate, resulting in fewer \[H^+\], and consequently, a slower rate of hydrolysis.

Weak Acids

Weak acids only partially dissociate in solution, as opposed to strong acids that dissociate completely. This means that when a weak acid like \[ HA \] is added to water, it doesn’t release all its hydrogen ions (\[ H^+\]) into the solution.

Imagine trying to fill a cup rapidly with water from a faucet. A strong faucet (acid) would have a high flow rate, filling the cup quickly. Conversely, a weak faucet would have a gentle trickle, filling the cup much slower. In chemistry:

• Weak acids have a characteristic constant represented by \[K_a\].
• \[K_a\] helps quantify an acid's strength — the lower the \[K_a\], the weaker the acid.
• The partial dissociation is what makes weak acids crucial in buffering solutions, maintaining stability in pH changes.

Studying weak acids gives insight into the subtleties of chemical reactions and equilibrium.

Dissociation Constant

The dissociation constant (\[K_a\]) is a measure of the strength of an acid in solution. It represents how well an acid can donate its hydrogen ions (\[H^+\]) when dissolved.

Think of \[K_a\] as the scorecard for acids:

• A high \[K_a\] means a strong acid with complete dissociation.
• A low \[K_a\] indicates a weak acid with partial dissociation.

For the exercise, determining the \[K_a\] of weak acid \[HA\] was crucial. We used the rate of reaction to relate the \[K_a\] of \[HA\] with the known rate of a strong acid. This allowed us to predict and calculate \[K_a\] through simple mathematical relationships derived from observed reaction rates.

The dissociation constant gives an understanding of many qualities of the acid:

• It helps compare the strengths of different acids.
• It also plays a crucial role in understanding chemical equilibria in solutions.

Reaction Rates

The rate of a chemical reaction refers to how quickly the reactants are transformed into products. Think of it like how fast you can bake a cake after mixing the ingredients.

Chemical reaction rates are influenced by several factors:

• Concentration of reactants: More concentrated solutions generally react faster.
• Temperature: Higher temperatures increase particle movement and therefore reaction rates.
• Catalysts: Substances that speed up reactions without being consumed.
• Nature of the reactants: Different substances react at different speeds.

In our specific scenario, the reaction rate of ester hydrolysis depends on the concentration of hydrogen ions \[H^+\]. Since strong acids provide more \[H^+\], they naturally have a faster reaction rate compared to weak acids like \[HA\].

Understanding reaction rates is vital because it allows chemists to predict how changes in conditions can affect the speed of reactions. This knowledge is applied in everything from designing industrial chemical processes to creating the products you use every day.