Problem 29 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

In a first order reaction the concentration of reactant decreases from \(800 \mathrm{~mol} / \mathrm{dm}^{3}\) to \(50 \mathrm{~mol} / \mathrm{dm}^{3}\) in \(2 \times 10^{4} \mathrm{sec}\). The rate constant of reaction in \(\mathrm{sec}^{-1}\) is: [2003S](a) \(2 \times 10^{4}\) (b) \(3.45 \times 10^{-5}\) (c) \(1.386 \times 10^{-4}\) (d) \(2 \times 10^{-4}\)

Step-by-Step Solution

Answer

(c) \(1.386 \times 10^{-4}\) \(\mathrm{sec}^{-1}\).

Step 1: Understand the First Order Reaction Formula

For a first order reaction, the rate law is expressed as:\[\ln \left( \frac{[A]_0}{[A]} \right) = kt\]where \([A]_0\) is the initial concentration, \([A]\) is the final concentration, \(k\) is the rate constant, and \(t\) is the time taken for the reaction.

Step 2: Plug in the Given Values

We know from the problem statement:\([A]_0 = 800 \, \text{mol/dm}^3\), \([A]= 50 \, \text{mol/dm}^3\), and \(t = 2 \times 10^4\, \text{sec}\).Substituting these values into the first order reaction formula, we get:\[\ln \left( \frac{800}{50} \right) = k \cdot 20000\]

Step 3: Simplify the Equation

Calculate \(\frac{800}{50}\):\[\frac{800}{50} = 16\]Now the equation becomes:\[\ln (16) = k \cdot 20000\]

Step 4: Calculate Logarithm and Solve for k

Find the natural logarithm of 16:\[\ln (16) = 2.7726\]Substituting this back into the equation:\[2.7726 = k \cdot 20000\]Solve for \(k\) by dividing both sides by 20000:\[k = \frac{2.7726}{20000} = 1.3863 \times 10^{-4} \, \text{sec}^{-1}\]

Step 5: Choose the Correct Answer

Based on the calculation, the correct rate constant is \(k = 1.386 \times 10^{-4} \, \text{sec}^{-1}\).Therefore, the correct answer is (c) \(1.386 \times 10^{-4}\) \(\mathrm{sec}^{-1}\).

Key Concepts

Rate Constant

The rate constant is a crucial factor in the study of reaction kinetics, especially for first order reactions. It is a proportionality constant that links the rate of a reaction to the concentration of the reactants. The higher the rate constant, the faster the reaction occurs. In the context of a first order reaction, such as the one where the concentration decreases over time, the rate constant is specifically denoted by the symbol \(k\).

For this type of reaction, the rate equation is given by the formula \( k = \frac{1}{t} \ln \left( \frac{[A]_0}{[A]} \right) \), where \( [A]_0 \) is the initial concentration, \([A]\) is the final concentration, and \(t\) is the time taken for the reaction to occur.

• It is important to note that the unit of the rate constant, in this case, is \( \text{sec}^{-1} \), which indicates that the reaction depends solely on time, not on the concentration of other reactants. • Understanding the rate constant helps predict how fast a reaction progresses and how long it will take to reach a certain concentration.

Natural Logarithm

A natural logarithm (denoted as \( \ln \)) plays an essential role in calculating reaction rates, particularly in the context of first order reactions. It is the logarithm to the base of the mathematical constant \(e\), roughly equal to 2.71828.

In the first order reaction formula \( \ln \left( \frac{[A]_0}{[A]} \right) = kt \), the natural logarithm is utilized to linearize the relationship between time and concentration, allowing for easier calculations of the rate constant. Breaking down the reactions into these units can greatly simplify problems that involve exponential decay, as is common in first order kinetics.

• Calculating \( \ln \) involves determining how many times you need to multiply \(e\) to achieve the specified number. • In this particular exercise, the natural logarithm of 16 is calculated as approximately 2.7726. This conversion allows for the direct computation of the rate constant \(k\) by rearranging the equation as illustrated in the solution.

Reaction Kinetics

Reaction kinetics is the study of the speed or rate at which chemical reactions occur. It provides insights into the mechanisms of reactions, helping us understand which steps of a reaction are fast or slow, and how conditions like temperature affect the rate. Reaction kinetics not only supports the calculation of the rate constant but also aids in designing chemical processes and optimizing industrial reactions.

For first order reactions, kinetics is simplistically depicted, as the rate of reaction is directly proportional to the concentration of a single reactant. The kinetics of such reactions are described using equations that incorporate the rate constant \(k\) and natural logarithms to solve for concentrations over time.

• First order reactions provide a clearer and more straightforward understanding of kinetics because they involve only one concentration term in the rate equation.• Analyzing reaction kinetics helps predict outcomes of reactions under various conditions, tailoring processes to achieve desired results efficiently.