Problem 146 - A Complete Resource Book in Chemistry for JEE Main

Question

A follows parallel path Ist order reactions giving \(\mathrm{B}\) and C as shown: If initial concentration of \(\mathrm{A}\) is \(0.25 \mathrm{M}\), calculate the concentration of \(\mathrm{C}\) after 5 hour of reaction. Given, \(\lambda_{1}=1.5 \times 10^{-5} \mathrm{~s}^{-1}, \lambda_{2}=5 \times 10^{-6} \mathrm{~s}^{-1}\) (a) \(7.55 \times 10^{-3} \mathrm{M}\) (b) \(1.89 \times 10^{-2} \mathrm{M}\) (c) \(5.53 \times 10^{-3} \mathrm{M}\) (d) \(3.51 \times 10^{-3} \mathrm{M}\)

Step-by-Step Solution

Answer

The concentration of C is \(7.55 \times 10^{-3} \text{ M}\), option (a).

Step 1: Understanding the Problem

We need to find the concentration of product C after 5 hours when starting with a concentration of 0.25 M of A. The reaction is a parallel first-order reaction, where A breaks down into B and C simultaneously with different rate constants, \(\lambda_1\) and \(\lambda_2\).

Step 2: Finding the Total Rate Constant

In a parallel reaction, the total rate constant \(k\) is the sum of the individual rate constants of the pathways to products B and C. Therefore, \(k = \lambda_1 + \lambda_2\). Substitute the given values:\[\lambda_1 = 1.5 \times 10^{-5} \, \text{s}^{-1}\text{ and }\lambda_2 = 5 \times 10^{-6} \, \text{s}^{-1}\] Hence, \[k = 2 \times 10^{-5} \, \text{s}^{-1}\].

Step 3: Using First-Order Kinetics to Find C's Concentration

The concentration of C is given by the expression for first-order parallel reactions:\[ [\text{C}] = \frac{\lambda_2}{\lambda_1 + \lambda_2} [\text{A}_0] (1 - e^{-kt}) \] Substituting the known values:\[ [\text{C}] = \frac{5 \times 10^{-6}}{2 \times 10^{-5}} \times 0.25 \, \text{M}\times (1 - e^{-2 \times 10^{-5} \times 5 \times 60 \times 60}) \].

Step 4: Calculating the Exponential Term

First, calculate the exponential decay term: \[ e^{-kt} = e^{-2 \times 10^{-5} \times 18000} \Rightarrow e^{-0.36}. \]

Step 5: Evaluating Exponential Expression

Using a calculator, \(e^{-0.36} \approx 0.6977\). Hence, \[1 - e^{-0.36} \approx 0.3023\].

Step 6: Final Calculation

Now substitute back into the formula to find \([\text{C}]\):\[ [\text{C}] = \frac{5 \times 10^{-6}}{2 \times 10^{-5}} \times 0.25 \times 0.3023 = 0.25 \times 0.25 \times 0.3023 = 0.0756 \times 0.3023\approx 0.00755.\]

Step 7: Conclusion

Thus, the concentration of C after 5 hours is \(7.55 \times 10^{-3} \text{ M}\). This corresponds to option (a).

Key Concepts

Parallel Reaction Kinetics

In chemical reactions, parallel reaction kinetics refers to processes where a single reactant species can transform into multiple product species simultaneously through different pathways. This happens when the reactant can decompose or react in different directions, each governed by its own rate constant. Typically, these reactions are distinguished by their order, and in our case, it involves a first-order reaction. This means each transformation pathway depends linearly on the concentration of the reactant.

In a first-order parallel reaction, two or more competing reactions occur at the same time. Each pathway is characterized by its own first-order rate constant. For example, if compound A decomposes into products B and C simultaneously, the path to B might have a different rate constant than the path to C. The overall result is influenced by both pathways acting at once, and the observed rates reflect the sum of these contributions. Understanding parallel reactions is crucial to determining how fast a chemical species will be consumed or produced over time.

Rate Constants

In the context of chemical kinetics, a rate constant is a numerical representation of the speed of a reaction under specific conditions. It is a fundamental parameter in the rate equation of a reaction and helps describe how fast reactants are converted to products. It depends largely on factors such as temperature, pressure, and the nature of the reactants.

For first-order reactions, which are common in parallel reaction kinetics, the rate constant is particularly significant because it precisely determines the reaction speed. In our example, two different rate constants are associated with the conversion of A to products B and C. These constants, denoted as \( \lambda_1 \) and \( \lambda_2 \) , represent the specific rates of each parallel pathway. To determine the overall effect or the combined kinetics of parallel reactions, these rate constants can be added together. This summation reflects the cumulative probability of the reactant A transforming into either product B or C in a given period.

Exponential Decay

Exponential decay is a mathematical concept often used to describe the decrease in the concentration of a reactant over time in first-order reactions. It follows a specific form where the concentration of the reactant diminishes exponentially as the reaction proceeds. This can be modeled by the function \( e^{-kt} \) , where **k** is the overall reaction rate constant and **t** is the time.

In the scenario of parallel kinetics with a first-order mechanism, each decomposition or transformation path contributes to the exponential decay observed. The decay function describes how fast reactant molecules disappear, which directly impacts the formation of products like B and C. These processes embody ideas from exponential decay because the speed at which these reactions occur is always proportional to the current concentration of reactants. Calculations involving exponential decay allow us to predict the amount of reactant remaining or the amount of product formed at any given time during the reaction.

Chemical Concentration Calculation

Calculating chemical concentrations, especially in reactions involving parallel kinetics, involves analyzing how reactant molecules transform into product molecules over time. This requires understanding and applying rate equations tailored for first-order reactions. In parallel reactions, the respective contributions to products depend on the initial reactant concentration and individual rate constants.

To find the concentration of a product like C from our example, one needs to use the appropriate concentration formulas involving given rate constants and initial concentrations. The formula for the concentration of C, \([C]\) , is structured to account for the specific path leading to C involving the rate constant \( \lambda_2 \) . Calculations entail plugging these constants into the formula \([C] = \frac{\lambda_2}{\lambda_1 + \lambda_2} [A_0] (1 - e^{-kt})\) , where \([A_0]\) is the initial concentration of A, and \(1 - e^{-kt}\) reflects the fraction of A that has transformed into products over time. Successfully applying these calculations enables the determination of how much of each product, such as C, is formed after a specific time.