Problem 84 - A Complete Resource Book in Chemistry for JEE Main

Question

The rate constant of a first-order reaction is \(6 \times 10^{-3}\) \(\mathrm{s}^{-1} .\) If the initial concentration is \(0.10 \mathrm{M}\), the initial rate of reaction is (a) \(6 \times 10^{-3} \mathrm{Ms}^{-1}\) (b) \(6 \times 10^{-1} \mathrm{Ms}^{-1}\) (c) \(6 \times 10^{-6} \mathrm{Ms}^{-1}\) (d) \(6 \times 10^{-8} \mathrm{Ms}^{-1}\)

Step-by-Step Solution

Answer

The initial rate is \(6 \times 10^{-4} \ \mathrm{Ms}^{-1}\), but none of the answer choices match this value.

Step 1: Understanding the Reaction Type

The problem mentions that the reaction is first-order. In a first-order reaction, the rate of reaction is directly proportional to the concentration of the reactant.

Step 2: Identifying Key Values

We are given: the rate constant, \(k = 6 \times 10^{-3} \ \mathrm{s}^{-1}\), and the initial concentration of the reactant, \([A]_0 = 0.10 \ \mathrm{M}\).

Step 3: Rate Law for First-Order Reaction

The rate law for a first-order reaction is \(\text{Rate} = k[A]\), where \(k\) is the rate constant and \([A]\) is the concentration of the reactant.

Step 4: Substitute Values into Rate Equation

Now, substitute the given values into the equation:\[\text{Rate} = (6 \times 10^{-3} \ \mathrm{s}^{-1})(0.10 \ \mathrm{M})\]

Step 5: Calculate Initial Rate

Perform the multiplication: \(\text{Rate} = 6 \times 10^{-3} \times 0.10 = 6 \times 10^{-4} \ \mathrm{Ms}^{-1}\).

Step 6: Evaluate Answer Choices

The calculated initial rate \(6 \times 10^{-4} \ \mathrm{Ms}^{-1}\) does not match any of the answer choices given (a through d), which indicates an error in alignment of provided choices with calculations.

Key Concepts

Rate Constant

In the realm of chemical kinetics, the rate constant is an essential aspect. It defines the proportionality between the rate of a chemical reaction and the concentration of reactants. Denoted as \( k \), the rate constant is typically expressed in units that vary depending on the order of the reaction.

For a first-order reaction, the units of the rate constant are \( ext{s}^{-1} \), indicating it is the inverse of time. This makes sense because the concentration decreases over time. A higher value of \( k \) suggests a faster reaction, whereas a smaller value means a slower process.

• The value of the rate constant is influenced by several factors:
• Temperature: Increasing temperature generally increases \( k \).
• Presence of a catalyst: Catalysts can make reactions faster by altering \( k \).

Understanding the role of \( k \) helps predict how quickly products are formed or how slowly reactants are consumed, guiding both theoretical studies and practical applications.

Rate Equation

The rate equation is a mathematical expression that describes the speed of a chemical reaction. It indicates how the concentration of reactants influences the reaction rate. In the case of a first-order reaction, the rate equation is particularly straightforward: \[ \text{Rate} = k[A] \]Here, \( [A] \) represents the concentration of a single reactant.

• This form is significant because it shows a direct proportionality; as the concentration of \( A \) increases or decreases, so does the rate.
• The rate equation can be used to predict how altering concentrations will influence the rate of reaction.

The simplicity of the first-order rate equation makes it a good starting point for those new to chemical kinetics. It highlights the linear relationship between reactant concentration and reaction rate. Understanding this concept is critical because it governs how we control and utilize reactions in various scientific fields.

Chemical Kinetics

Chemical kinetics is the branch of chemistry that deals with understanding the speeds of chemical reactions. It's not only about knowing the reaction itself but learning the factors that influence these rates. For first-order reactions, two main components influence the kinetic behavior:

• The rate constant \( k \), and
• The concentration of the reactant \( [A] \).

Chemical kinetics provides valuable insights into reaction mechanisms, helping scientists explain the steps a reaction undergoes to convert reactants into products.

• Reaction conditions: Changing conditions like temperature and pressure impact how quickly reactions proceed.
• The study of kinetics also involves considering intermediate species that may form briefly during reactions.

This field is crucial for designing and optimizing chemical processes in industries. By mastering chemical kinetics, one can predict reaction outcomes, enhance safety, and improve efficiency in laboratories and manufacturing settings.