Problem 80 - A Complete Resource Book in Chemistry for JEE Main

Question

What will be the initial rate of a reaction if its con stant is \(10^{-3} \mathrm{~min}^{-1}\) and the concentration of reactant is \(0.2 \mathrm{~mol} \mathrm{dm}^{3} ?\) (a) \(0.02 \mathrm{~mol} \mathrm{dm}^{-3} \mathrm{~min}^{-1}\) (b) \(0.002 \mathrm{~mol} \mathrm{dm}^{-3} \mathrm{~min}^{-1}\) (c) \(0.0002 \mathrm{~mol} \mathrm{dm}^{-3} \mathrm{~min}^{-1}\) (d) \(2 \mathrm{~mol} \mathrm{dm}^{-3} \mathrm{~min}^{-1}\)

Step-by-Step Solution

Answer

The initial rate is \( 0.0002 \mathrm{~mol} \mathrm{dm}^{-3} \mathrm{~min}^{-1} \), option (c).

Step 1: Understand the Rate Law Formula

The rate of a reaction can generally be given by the formula: \( \text{Rate} = k [A]^n \), where \( k \) is the rate constant, \( [A] \) is the concentration of the reactant, and \( n \) is the order of the reaction.

Step 2: Identify Given Values

We are given that the rate constant \( k = 10^{-3} \mathrm{~min}^{-1} \) and the concentration of the reactant \( [A] = 0.2 \mathrm{~mol~dm}^{-3} \).

Step 3: Assume the Reaction Order

Assume this is a first-order reaction, meaning \( n = 1 \), as no other information about the reaction order is given. This means the rate law can be simplified to \( \text{Rate} = k [A] \).

Step 4: Calculate the Rate

Substitute the values into the formula: \( \text{Rate} = 10^{-3} \times 0.2 = 0.0002 \mathrm{~mol~dm}^{-3} \mathrm{~min}^{-1} \).

Step 5: Verify and Match with Given Answers

The calculated initial rate is \( 0.0002 \mathrm{~mol~dm}^{-3} \mathrm{~min}^{-1} \), which matches option (c).

Key Concepts

Rate Constant

In chemical kinetics, the rate constant plays a crucial role in determining the speed of a chemical reaction. It's a proportionality factor in the rate law equation, and its value gives us insight into how fast a reaction proceeds. The rate constant, often denoted by the letter "k," is unique for every reaction and depends on factors like temperature and the presence of catalysts.

The units of the rate constant vary depending on the order of the reaction. For a first-order reaction, like the one described in the exercise, the units of the rate constant are typically \( ext{min}^{-1} \) or \( ext{s}^{-1} \). This unit indicates how the concentration of the reactant changes over time.

A small rate constant, such as \( 10^{-3} \text{min}^{-1} \), suggests that the reaction proceeds relatively slowly under the conditions given. Understanding and calculating the rate constant is essential for chemists to design and control chemical processes effectively.

Reaction Rate

The reaction rate is a measure of how quickly reactants are converted to products in a chemical reaction. It gives us the speed of the reaction as a change in concentration of a reactant or product per unit time.

In the rate law expression \( \text{Rate} = k[A]^n \), the reaction rate is influenced by the rate constant \( k \) and the concentration of the reactants \( [A] \). For a first-order reaction, the dependence on the concentration is linear, meaning the rate directly decreases as the concentration decreases.

To find the initial reaction rate, you substitute the given values for the rate constant and concentration into the rate law equation. This gives you the speed at which the reaction starts. Accurate calculation of the reaction rate is critical for predicting how fast the reaction will proceed initially and helps in optimizing various industrial and lab processes.

First-Order Reaction

A first-order reaction is one where the rate is directly proportional to the concentration of one reactant. This implies that the power of the concentration term in the rate law is one \( n = 1 \). As such, the equation simplifies to \( \text{Rate} = k[A] \).

The hallmark of first-order reactions is their exponential decay in the concentration of reactants over time. This type of reaction follows a characteristic mathematical form that can be described using logarithmic functions, which allows chemists to predict concentration changes over time.

In the exercise provided, assuming the reaction is first-order is necessary when no other information about the reaction's order is available. This assumption allows you to use a straightforward calculation to determine the initial reaction rate, simplifying both the understanding and application for students and professionals alike.