Problem 69 - A Complete Resource Book in Chemistry for JEE Main

Question

The rate constant of first-order reaction is \(3 \times 10^{-6}\) per second. The initial concentration is \(0.10 \mathrm{M}\). The initial rate is (a) \(3 \times 10^{-7} \mathrm{Ms}^{-1}\) (b) \(3 \times 10^{-4} \mathrm{Ms}^{-1}\) (c) \(3 \times 10^{-5} \mathrm{Ms}^{-1}\) (d) \(3 \times 10^{-6} \mathrm{Ms}^{-1}\)

Step-by-Step Solution

Answer

The initial rate is \(3 \times 10^{-7} \mathrm{Ms}^{-1}\), which is option (a).

Step 1: Understanding the formula for rate of reaction

The rate of a first-order reaction can be calculated using the formula: \( \text{rate} = k [A]_0 \), where \( k \) is the rate constant and \( [A]_0 \) is the initial concentration of reactant.

Step 2: Substituting known values into the formula

Substitute \( k = 3 \times 10^{-6} \mathrm{s^{-1}} \) and \( [A]_0 = 0.10 \mathrm{M} \) into the rate formula: \[ \text{rate} = (3 \times 10^{-6} \mathrm{s^{-1}})(0.10 \mathrm{M}) \]

Step 3: Calculating the rate

Calculate the rate: \[ \text{rate} = 3 \times 10^{-6} \times 0.10 = 3 \times 10^{-7} \mathrm{Ms}^{-1} \]

Step 4: Selecting the correct answer

Comparing the calculated rate \( 3 \times 10^{-7} \mathrm{Ms}^{-1} \) to the given choices, we find that it matches option (a).

Key Concepts

Understanding Rate Constant in First-Order Reactions

In a first-order reaction, the rate constant, often denoted by the symbol \( k \), plays a crucial role in determining how fast a reaction proceeds.
The rate constant is unique for every reaction and is affected by factors such as temperature and the presence of a catalyst.
In our original exercise, we were given a rate constant of \( 3 \times 10^{-6} \) per second, which directly relates to how quickly the reactant is being converted into product over time.

• The units of the rate constant for a first-order reaction are in \( ext{s}^{-1} \).
• A larger rate constant indicates a faster reaction.
• This value is constant throughout the reaction at a given temperature when no catalysts are involved.

Understanding the rate constant allows chemists to predict the time frame in which a certain amount of reactant will have reacted.

The Role of Initial Concentration

In any chemical reaction, the initial concentration of the reactant, denoted as \( [A]_0 \), is vital when calculating the reaction rate.
This measure refers to the concentration of reactants present at the start, before the reaction begins to proceed.
In our example, it is \( 0.10 \mathrm{M} \).

• The initial concentration influences the initial reaction rate directly in first-order reactions, as seen in the formula \( ext{rate} = k [A]_0 \).
• Higher initial concentrations result in a higher initial rate of reaction, provided the rate constant remains the same.
• Although the initial concentration decreases as the reaction proceeds, it is used to calculate the starting rate of reaction.

Knowing the initial concentration helps in modeling how the reaction will decrease in concentration over time and aids in kinetics studies.

Exploring Reaction Rate

The reaction rate is a measure of how fast reactants are converted into products in a chemical reaction.
It provides crucial insight into the speed of the reaction and is essential for understanding reaction kinetics.
In our problem, the calculated initial reaction rate is \( 3 \times 10^{-7} \mathrm{Ms}^{-1} \), indicating the speed at which the concentration changes initially.

• For first-order reactions, the rate is calculated as \( ext{rate} = k [A]_0 \).
• Reaction rates are typically reported in terms of concentration changes per unit time, such as \( ext{Ms}^{-1} \).
• A faster rate means reactants are converted into products quicker.

Understanding reaction rates is key for processes where time is an important factor, such as in industrial chemistry and biochemistry.

Diving into Chemical Kinetics

Chemical kinetics is the branch of chemistry that explores the rates of chemical reactions and the factors affecting them.
It helps scientists understand how different conditions like concentration, temperature, and catalysts influence the speed of reactions.
This field provides the mathematical framework necessary for calculating rates, as demonstrated in our exercise.

• Studying kinetics involves evaluating rate laws and understanding mechanisms of reaction steps.
• It assists in the design and optimization of chemical processes in laboratories and industries.
• Kinetics not only help predict reaction behavior but also aid in developing new chemical reactions.

By understanding chemical kinetics, chemists can manipulate reactions to be faster, more efficient, and suitable for specific applications.