Problem 57 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

The rate constant for an isomerisation reaction, \(A \rightarrow B\) is \(4.5 \times 10^{3}\) \(\min ^{1}\). If the initial concentration of \(A\) is \(1 \mathrm{M}\), calculate the rate of the reaction after \(1 \mathrm{~h}\). [1999-4 Marks]

Step-by-Step Solution

Answer

The reaction rate after 1 hour is essentially zero due to the large rate constant.

Step 1: Understanding the Formula for Reaction Rate

For a rate constant \( k \) given by \( k = 4.5 \times 10^3 \text{ min}^{-1} \) and an initial concentration of \([A]_0 = 1 \text{ M}\), the rate of a first-order reaction is calculated using the formula \( r = k[A] \). In this step, we gather all necessary information.

Step 2: Determine the Remaining Concentration at 1 Hour

We need to calculate the concentration of \( [A] \) remaining after \( 1 \) hour. Since 1 hour is \( 60 \) minutes, we can apply the first-order reaction concentration formula: \([A] = [A]_0 e^{-kt}\), where \( t = 60 \text{min} \). Substituting the known values: \([A] = 1 \times e^{-4.5 \times 10^3 \times 60}\). The exponent is quite large, indicating \([A]\) will effectively approach zero.

Step 3: Calculate the Reaction Rate

As the concentration of \( A \) becomes negligible in 1 hour due to the large rate constant, the rate \( r = k[A] \) becomes approximately zero. The reaction reaches completion very rapidly.

Step 4: Conclusion

Calculating with the given large rate constant, the concentration of \( A \) after 1 hour becomes negligible almost instantaneously. Thus, the rate soon after starting is nearly zero.

Key Concepts

Rate Constant

The rate constant is a significant parameter in chemical kinetics, particularly when discussing the speed of a chemical reaction. It is denoted by the symbol \( k \). For the given exercise, \( k \) was given as \( 4.5 \times 10^3 \text{ min}^{-1} \). This rate constant is an indicator of how quickly an isomerization reaction \( A \rightarrow B \) progresses.

The value of the rate constant depends on several factors:

• Temperature: Higher temperatures generally increase the rate constant, speeding up the reaction.
• Nature of Reactants: Different substances react at different speeds.
• Catalysts: These substances can alter the rate constant by offering an alternative reaction pathway with a lower activation energy.

The rate constant in a first-order reaction is unique because it is independent of the concentration of the reactants. It gives us a direct measure of reaction dynamics and allows for predicting how fast a reaction can proceed under specific conditions.

First-Order Reaction

A first-order reaction is characterized by its dependence on the concentration of a single reactant. The rate equation is given by

\[ r = k[A] \]

where \( r \) is the reaction rate, \( k \) is the rate constant, and \( [A] \) is the concentration of reactant A. In a first-order reaction, the rate at which the reaction occurs is directly proportional to the concentration of the reactant.

We can calculate the concentration of a reactant at any time \( t \) using the formula:

\[ [A] = [A]_0 e^{-kt} \]

Here,

• \( [A]_0 \) is the initial concentration of the reactant.
• \( e \) is the base of the natural logarithm.
• \( t \) is the time elapsed.

This equation shows that the concentration decreases exponentially over time. For an extremely large rate constant as in the given exercise, the concentration of \( A \) diminishes very quickly, reaching negligible levels almost instantly. This leads to a rapid conclusion of the reaction.

Reaction Rate Calculation

The reaction rate is a measure of how quickly a reactant turns into products. For the isomerization reaction in the exercise, calculating the rate involves understanding the role of concentration and the rate constant. Initially, when the concentration \( [A]_0 = 1 \text{ M} \), the reaction starts quickly because of the reactive nature of the initial concentration.

Using the formula:

\[ r = k[A] \]

Substitute \( k = 4.5 \times 10^3 \text{ min}^{-1} \) and \( [A] = [A]_0 e^{-kt} \).

Upon substituting values and recognizing the exponential term's effect after one hour, the concentration of \( A \) drops to nearly zero. This means that after one hour, further progression of the reaction is negligible, and the rate of reaction appears almost zero. This illustrates the difference in reaction speeds based on initial concentration and the large magnitude of a rate constant in a first-order reaction. It emphasizes that as reactant concentrations decrease, so does the reaction rate.