Problem 60 - A Complete Resource Book in Chemistry for JEE Main

Question

For a first-order reaction, \(\mathrm{t}_{0.75}\) is \(138.6\) seconds. Its specific reaction rate constant (in sec \(^{-1}\) ) is (a) \(10^{-2}\) (b) \(10^{-4}\) (c) \(10^{-5}\) (d) \(10^{-6}\)

Step-by-Step Solution

Answer

The specific reaction rate constant is \(10^{-2} \text{ s}^{-1}\) (option a).

Step 1: Understanding the Problem

We are given the time it takes for a first-order reaction to reach three-quarters completion, denoted as \( t_{0.75} = 138.6 \) seconds. We need to find the rate constant \( k \) for this reaction from the given choices.

Step 2: Formula for a First-Order Reaction

For a first-order reaction, the relationship between the concentration and time is expressed by the formula:\[ t_{0.75} = \frac{0.75}{k} \ln\left( \frac{1}{0.25} \right) \]Since \( t_{0.75} = 138.6 \) seconds and \( \ln(4) \approx 1.386 \), we will use these values in the next step.

Step 3: Solving for the Rate Constant \( k \)

Rearrange the formula to solve for \( k \):\[ k = \frac{1.386}{t_{0.75}}\]Substitute \( t_{0.75} = 138.6 \) into the equation:\[ k = \frac{1.386}{138.6} = 0.01 \, \text{s}^{-1}\]

Step 4: Comparison with Given Options

From the previous step, the calculated reaction rate constant \( k \) is \( 0.01 \, \text{s}^{-1} \), which is equivalent to \( 10^{-2} \, \text{s}^{-1} \). Thus, option (a) is correct.

Key Concepts

Reaction Rate Constant

In chemical kinetics, the reaction rate constant, often symbolized by \( k \), is a crucial parameter that quantifies the rate at which a reaction occurs. For first-order reactions, the rate constant can be directly related to the time it takes for the reaction to reach a certain level of completion.
This parameter helps scientists predict how quickly a reaction will proceed under certain conditions.

• For first-order reactions, the units of \( k \) are typically expressed in \( ext{s}^{-1} \).
• The reaction rate constant depends on multiple factors such as temperature and the presence of catalysts.

Understanding the rate constant allows chemists to manipulate reaction conditions to optimize the speed and efficiency of chemical processes.

Three-Quarters Completion

The term "three-quarters completion" refers to the point in a reaction where 75% of the initial reactants have been converted to products. This is a valuable milestone for predicting how long a reaction will take to substantially progress.
In the context of first-order reactions, this specific point gives us a basis to calculate the reaction rate constant using the time it takes to reach this level of conversion. The time required for three-quarters completion is denoted as \( t_{0.75} \).

• This time can be experimentally determined, providing a practical means to calculate \( k \).
• It is one of the characteristic times that reflects the kinetics of the reaction.

Knowing \( t_{0.75} \) allows us to use specific formulas to derive more detailed kinetic information about the reaction.

First-Order Reaction Formula

For first-order reactions, the relationship between the reactant concentration and time follows an exponential decay model. The primary formula describing this relationship is:
\[ t_{0.75} = \frac{1}{k} \ln\left(\frac{1}{0.25}\right) \]
Here, \( t_{0.75} \) is the time it takes for the reaction to reach 75% completion. The natural logarithm term accounts for the decay process. The formula can be rearranged to solve for the rate constant \( k \).

• This formulation allows the calculation of \( k \) when time measurements are known.
• It highlights the power of logarithmic functions in predicting reaction behavior.

Using this formula effectively bridges experimental observations with theoretical kinetics.

Chemical Kinetics

Chemical kinetics is the study of reaction rates and the factors that influence them. It provides insight into the speed of reactions and is fundamental in designing chemical processes.
At its core, chemical kinetics helps explain the steps through which a reaction proceeds and enables chemists to verify reaction mechanisms.

• Kinetics consider variables such as concentration, temperature, and catalysts.
• It allows the determination of rate laws and reaction order, which describe how the concentration of reactants affects the rate.

The field of chemical kinetics plays a critical role in industries where controlling reaction speed is necessary to optimize production and ensure safety.