Problem 158 - A Complete Resource Book in Chemistry for JEE Main

Question

In a certain reaction \(8 \%\) of the reactant decomposes in 30 minutes, \(24 \%\) in 90 minutes and \(48 \%\) in 180 minutes. What are the dimensions of the rate constant? (a) Mol lit \(^{-1} \sec ^{-1}\) (b) \(\mathrm{Mol} \mathrm{sec}^{-1}\) (c) Lit \(\mathrm{mol}^{-1} \mathrm{sec}^{-1}\) (d) Hour \(^{-1}\)

Step-by-Step Solution

Answer

The dimensions of the rate constant are Hour \(^{-1}\), option (d).

Step 1: Assume Reaction Order

For this type of problem, we must assume a reaction order. Since the percentage decomposition over different times suggests a changing rate not proportional to concentration, assume it is a first-order reaction. The rate of a first-order reaction can be represented as \( \text{Rate} = k[A] \) where \( k \) is the rate constant and \( [A] \) is the concentration of the reactant.

Step 2: First-Order Reaction Rate Equation

For a first-order reaction, the integrated rate equation is given as \( \ln \left(\frac{[A_0]}{[A]}\right) = kt \), where \([A_0]\) is the initial concentration, \([A]\) is the concentration at time \(t\), and \(k\) is the rate constant.

Step 3: Determine Rate Constant Units

In the equation \( \ln \left(\frac{[A_0]}{[A]}\right) = kt \), \( \ln \left(\frac{[A_0]}{[A]}\right) \) is dimensionless. Thus, the dimensions of \(kt\) must also be dimensionless. Since \(t\) is in time units like seconds, the dimension of the rate constant \(k\) must be the reciprocal of time, typically \( ext{time}^{-1} \).

Step 4: Identify Time Units in Problem

The times given in the problem are in minutes and hours (90 and 180 minutes correspond to 1.5 and 3 hours respectively). Therefore, it is reasonable to expect the units of time for the rate constant to be in hours.

Step 5: Conclude Rate Constant Dimension

As the rate constant is the reciprocal of time, the dimension for \(k\) is \( ext{Hour}^{-1} \). This matches option (d) in the provided list of choices.

Key Concepts

First-Order Reaction

In reaction kinetics, a first-order reaction is characterized by the rate of reaction being directly proportional to the concentration of a single reactant. This means the reaction rate will change as the concentration of the reactant fluctuates. Generally, this is represented mathematically as:\[ \text{Rate} = k[A] \]where:

• \( k \) is the rate constant.
• \([A]\) is the concentration of the reactant.

The concept of first-order reactions is a cornerstone in understanding how chemical kinetics work because it often applies to many natural and industrial processes, like radioactive decay or simple chemical reactions in solution.
This kind of reaction is particularly simple because it involves just one reactant whose concentration directly affects the rate of reaction. This simplicity makes first-order reactions a great starting point for students learning about reaction kinetics.

Rate Constant Units

The rate constant \( k \) is crucial in the study of reaction kinetics, acting as a factor that scales the rate of reaction based on the concentration of reactants. For first-order reactions, finding the units for the rate constant is relatively straightforward. The equation linking the rate constant to reaction rate and concentration is:\[\ln \left(\frac{[A_0]}{[A]}\right) = kt\]Here, \( \ln \left(\frac{[A_0]}{[A]}\right) \) is a dimensionless term, which means that \( kt \) must also be dimensionless.

This implies that the units of \( k \) automatically become the reciprocal of the time units used in the problem because time (\( t \)) is the only dimensional term remaining. For instance, if time is measured in hours, then the rate constant \( k \) will have the units \( \text{Hour}^{-1} \).

Understanding the units of the rate constant allows students to evaluate reaction rates and predict how the concentration of reactants will change over time in a given reaction.

Integrated Rate Equation

The Integrated Rate Equation for a first-order reaction provides the necessary framework to understand how concentrations change over time starting from initial conditions. This equation is usually expressed as:\[\ln \left(\frac{[A_0]}{[A]}\right) = kt \]where:

• \([A_0]\) is the initial concentration.
• \([A]\) is the concentration at time \( t \).
• \(k\) is the rate constant.

This equation helps calculate the concentration of reactants at any given point in time, provided you know the initial concentration and the rate constant. It is especially useful because it simplifies to a linear equation, making it easy to plot and interpret the data. By calculating \(k\) using the known amounts of reactant decomposed at different times, you can predict how the system will behave in the future. For students, mastering the integrated rate equation is fundamental to gaining a deeper understanding of reaction kinetics and reaction dynamics.